Equation form expr-00103be7101dbb1d

$g(x,y) = f_x(y)$

Read as: g open parenthesis x comma y close parenthesis equals f sub x open parenthesis y close parenthesis

Means: A recursive-function relation or equation stating: g open parenthesis x comma y close parenthesis equals f sub x open parenthesis y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015144

Equation form expr-00364439c5956aa3

$h(1) & = 2\cdot h(0) = 2,\\ h(2) & = 2\cdot h(1) = 2\cdot 2,\\ h(3) & = 2 \cdot h(2) = 2\cdot 2 \cdot 2,\\ & \vdots$

Read as: h open parenthesis one close parenthesis equals two times h open parenthesis zero close parenthesis equals two comma next row h open parenthesis two close parenthesis equals two times h open parenthesis one close parenthesis equals two times two comma next row h open parenthesis three close parenthesis equals two times h open parenthesis two close parenthesis equals two times two times two comma next row and so on vertically

Means: A source-ordered system, calculation, or case table stating: h open parenthesis one close parenthesis equals two times h open parenthesis zero close parenthesis equals two comma next row h open parenthesis two close parenthesis equals two times h open parenthesis one close parenthesis equals two times two comma next row h open parenthesis three close parenthesis equals two times h open parenthesis two close parenthesis equals two times two times two comma next row and so on vertically. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014387

Equation form expr-00ad65da3dd95c7b

$\fn{sequenceBound}(x,k)$

Read as: the function sequence bound open parenthesis x comma k close parenthesis

Means: Recursive-function notation denoting: the function sequence bound open parenthesis x comma k close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015064

Equation form expr-00b249a8d9a46a59

$g_3(x)$

Read as: g sub three open parenthesis x close parenthesis

Means: Recursive-function notation denoting: g sub three open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015163

Equation form expr-00f405007f60a533

$h(x) = 2^x$

Read as: h open parenthesis x close parenthesis equals two superscript x

Means: A recursive-function relation or equation stating: h open parenthesis x close parenthesis equals two superscript x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014389

Equation form expr-038bc2816a9da1e4

$h_0(\vec x, 0) & = f_0(\vec x) \\ h_1(\vec x, 0) & = f_1(\vec x) \\ h_0(\vec x, y+1) & = g_0(\vec x, y, h_0(\vec x, y), h_1(\vec x, y)) \\ h_1(\vec x, y+1) & = g_1(\vec x, y, h_0(\vec x, y), h_1(\vec x, y))$

Read as: h sub zero open parenthesis vector x comma zero close parenthesis equals f sub zero open parenthesis vector x close parenthesis next row h sub one open parenthesis vector x comma zero close parenthesis equals f sub one open parenthesis vector x close parenthesis next row h sub zero open parenthesis vector x comma y plus one close parenthesis equals g sub zero open parenthesis vector x comma y comma h sub zero open parenthesis vector x comma y close parenthesis comma h sub one open parenthesis vector x comma y close parenthesis close parenthesis next row h sub one open parenthesis vector x comma y plus one close parenthesis equals g sub one open parenthesis vector x comma y comma h sub zero open parenthesis vector x comma y close parenthesis comma h sub one open parenthesis vector x comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: h sub zero open parenthesis vector x comma zero close parenthesis equals f sub zero open parenthesis vector x close parenthesis next row h sub one open parenthesis vector x comma zero close parenthesis equals f sub one open parenthesis vector x close parenthesis next row h sub zero open parenthesis vector x comma y plus one close parenthesis equals g sub zero open parenthesis vector x comma y comma h sub zero open parenthesis vector x comma y close parenthesis comma h sub one open parenthesis vector x comma y close parenthesis close parenthesis next row h sub one open parenthesis vector x comma y plus one close parenthesis equals g sub one open parenthesis vector x comma y comma h sub zero open parenthesis vector x comma y close parenthesis comma h sub one open parenthesis vector x comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015112

Equation form expr-0419464ce26ee408

$\fn{element}(s,i)$

Read as: the function element open parenthesis s comma i close parenthesis

Means: A sequence- or tree-coding expression denoting: the function element open parenthesis s comma i close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015019
• Occurrence projected-formula-0015036

Equation form expr-043a718774c572bd

$s$

Read as: s

Means: Recursive-function notation denoting: s. This meaning is freshly authored from the complete recursive-functions source packet.

27 occurrences in this chapter

• Occurrence projected-formula-0014994
• Occurrence projected-formula-0014998
• Occurrence projected-formula-0015004
• Occurrence projected-formula-0015007
• Occurrence projected-formula-0015011
• Occurrence projected-formula-0015013
• Occurrence projected-formula-0015016
• Occurrence projected-formula-0015021
• Occurrence projected-formula-0015025
• Occurrence projected-formula-0015028
• Occurrence projected-formula-0015031
• Occurrence projected-formula-0015039
• Occurrence projected-formula-0015041
• Occurrence projected-formula-0015050
• Occurrence projected-formula-0015054
• Occurrence projected-formula-0015057
• Occurrence projected-formula-0015061
• Occurrence projected-formula-0015063
• Occurrence projected-formula-0015072
• Occurrence projected-formula-0015102
• Occurrence projected-formula-0015104
• Occurrence projected-formula-0015240
• Occurrence projected-formula-0015241
• Occurrence projected-formula-0015249
• Occurrence projected-formula-0015250
• Occurrence projected-formula-0015251
• Occurrence projected-formula-0015253

Equation form expr-0525641564d7ee83

$\fn{IsZero}(\left|x - y\right|)$

Read as: the zero-test predicate open parenthesis the absolute value of x minus y close parenthesis

Means: A recursive-function relation or equation stating: the zero-test predicate open parenthesis the absolute value of x minus y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014783

Equation form expr-05ca6c059dde249a

$\Char{\lnot P}(\vec x)$

Read as: the characteristic function of not P open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of not P open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014798

Equation form expr-05d2116af8b1c471

$\Proj{3}{2}$

Read as: the three place projection with index two

Means: A primitive-recursive construction or operator denoting: the three place projection with index two. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014498
• Occurrence projected-formula-0014622
• Occurrence projected-formula-0014683

Equation form expr-067cc339ff1fde60

$h(x_0, \dots, x_{k-1}, y+1)$

Read as: h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014439

Equation form expr-06a4f0f2e779f114

$\cfind{e_d}(e_d) \fundefined$

Read as: partial recursive function phi sub e sub d open parenthesis e sub d close parenthesis is undefined

Means: A partial-computation statement expressing: partial recursive function phi sub e sub d open parenthesis e sub d close parenthesis is undefined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015307

Equation form expr-076252f478748bc5

$p \le \fact{x}$

Read as: p is less than or equal to x factorial

Means: A recursive-function relation or equation stating: p is less than or equal to x factorial. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014976

Equation form expr-08622f37fb7d4a85

$g(x, y, z) = \Succ(\Proj{3}{2}(x, y, z)).$

Read as: g open parenthesis x comma y comma z close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x comma y comma z close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x comma y comma z close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x comma y comma z close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014492

Equation form expr-08c6aa8d5824f2de

$\Succ$

Read as: the successor function

Means: A primitive-recursive construction or operator denoting: the successor function. This meaning is freshly authored from the complete recursive-functions source packet.

13 occurrences in this chapter

• Occurrence projected-formula-0014495
• Occurrence projected-formula-0014565
• Occurrence projected-formula-0014573
• Occurrence projected-formula-0014590
• Occurrence projected-formula-0014594
• Occurrence projected-formula-0014616
• Occurrence projected-formula-0014617
• Occurrence projected-formula-0014621
• Occurrence projected-formula-0014635
• Occurrence projected-formula-0014648
• Occurrence projected-formula-0014681
• Occurrence projected-formula-0014708
• Occurrence projected-formula-0014716

Equation form expr-0904146399813bd8

$U(s)$

Read as: U open parenthesis s close parenthesis

Means: Recursive-function notation denoting: U open parenthesis s close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015233

Equation form expr-095b358cbc4ad8dd

$f_i$

Read as: f sub i

Means: Recursive-function notation denoting: f sub i. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0015146
• Occurrence projected-formula-0015148
• Occurrence projected-formula-0015170

Equation form expr-0a4af79d897c3431

$\fn{max}(x,y) \defis x + (y \tsub x).$

Read as: the maximum function open parenthesis x comma y close parenthesis is defined as x plus open parenthesis y truncated minus x close parenthesis

Means: Recursive-function notation denoting: the maximum function open parenthesis x comma y close parenthesis is defined as x plus open parenthesis y truncated minus x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014752

Equation form expr-0a566b1db5acde7f

$g_0$

Read as: g sub zero

Means: Recursive-function notation denoting: g sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

10 occurrences in this chapter

• Occurrence projected-formula-0014469
• Occurrence projected-formula-0014499
• Occurrence projected-formula-0014515
• Occurrence projected-formula-0014519
• Occurrence projected-formula-0014555
• Occurrence projected-formula-0014560
• Occurrence projected-formula-0014577
• Occurrence projected-formula-0014581
• Occurrence projected-formula-0014654
• Occurrence projected-formula-0015201

Equation form expr-0b52c0cb12adb97f

$R(\vec x, z)$

Read as: R open parenthesis vector x comma z close parenthesis

Means: Recursive-function notation denoting: R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

6 occurrences in this chapter

• Occurrence projected-formula-0014805
• Occurrence projected-formula-0014810
• Occurrence projected-formula-0014853
• Occurrence projected-formula-0014857
• Occurrence projected-formula-0014878
• Occurrence projected-formula-0014880

Equation form expr-0b85d3b66169d1fe

$\fn{IsZero}(x)$

Read as: the zero-test predicate open parenthesis x close parenthesis

Means: Recursive-function notation denoting: the zero-test predicate open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014778

Equation form expr-0bab920452cd75b4

$h(x_0, \dots, x_{k-1}, 0)$

Read as: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014437

Equation form expr-0d33326592643dc4

$n = 2$

Read as: n equals two

Means: A recursive-function relation or equation stating: n equals two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014514

Equation form expr-0d8178df168b7b20

$\le \fact{x}+1$

Read as: less than or equal to x factorial plus one

Means: A recursive-function relation or equation stating: less than or equal to x factorial plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014978

Equation form expr-0e4763978622e187

$f_1$

Read as: f sub one

Means: Recursive-function notation denoting: f sub one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015139

Equation form expr-0f1b7c672fb64d9e

$\fn{append}(\fn{append}(\dots \fn{append}(\emptyseq,s_0) \dots),s_k).$

Read as: the function append open parenthesis the function append open parenthesis and so on the function append open parenthesis the empty sequence code comma s sub zero close parenthesis and so on close parenthesis comma s sub k close parenthesis

Means: A sequence- or tree-coding expression denoting: the function append open parenthesis the function append open parenthesis and so on the function append open parenthesis the empty sequence code comma s sub zero close parenthesis and so on close parenthesis comma s sub k close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015038

Equation form expr-10e6003fed243490

$\bmin{z < y}{R(\vec{x}, z)},$

Read as: the least z less than y such that R open parenthesis vector x comma z close parenthesis

Means: A recursive-function relation or equation stating: the least z less than y such that R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014860

Equation form expr-10fc318a34834bdf

$s \concat t$

Read as: s concatenated with t

Means: A sequence- or tree-coding expression denoting: s concatenated with t. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015052

Equation form expr-1255294d6e0899de

$h(\vec x, 0)$

Read as: h open parenthesis vector x comma zero close parenthesis

Means: Recursive-function notation denoting: h open parenthesis vector x comma zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014693
• Occurrence projected-formula-0014700
• Occurrence projected-formula-0015114

Equation form expr-12622fa5b3240e76

$\fn{concat}(s,t)$

Read as: the concatenate function open parenthesis s comma t close parenthesis

Means: Recursive-function notation denoting: the concatenate function open parenthesis s comma t close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015044
• Occurrence projected-formula-0015053

Equation form expr-12c24165ec0a21e2

$m_R(\vec{x}, 0) & = 0\\ m_R(\vec{x}, y+1) & = \begin{cases} m_R(\vec{x}, y) & \text{if $m_R(\vec x, y) \neq y$}\\ y & \text{if $m_R(\vec x, y) = y$ and $R(\vec{x}, y)$}\\ y+1 & \text{otherwise.} \end{cases}$

Read as: m sub R open parenthesis vector x comma zero close parenthesis equals zero next row m sub R open parenthesis vector x comma y plus one close parenthesis equals cases begin; row one: m sub R open parenthesis vector x comma y close parenthesis, if m sub R open parenthesis vector x comma y close parenthesis is not equal to y; row two: y, if m sub R open parenthesis vector x comma y close parenthesis equals y and R open parenthesis vector x comma y close parenthesis; row three: y plus one, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: m sub R open parenthesis vector x comma zero close parenthesis equals zero next row m sub R open parenthesis vector x comma y plus one close parenthesis equals cases begin; row one: m sub R open parenthesis vector x comma y close parenthesis, if m sub R open parenthesis vector x comma y close parenthesis is not equal to y; row two: y, if m sub R open parenthesis vector x comma y close parenthesis equals y and R open parenthesis vector x comma y close parenthesis; row three: y plus one, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014876

Equation form expr-12cb5b344dd4a38e

$\Add(x,y)$

Read as: the addition function open parenthesis x comma y close parenthesis

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014418

Equation form expr-130a1d632ea45a2c

$\left|x-y\right|$

Read as: the absolute value of x minus y

Means: A recursive-function relation or equation stating: the absolute value of x minus y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014742

Equation form expr-143122bc4b4cc29e

$y > 0$

Read as: y is greater than zero

Means: A recursive-function relation or equation stating: y is greater than zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015130

Equation form expr-1431b986377b5430

$x = 2$

Read as: x equals two

Means: A recursive-function relation or equation stating: x equals two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014409

Equation form expr-148de9c5a7a44d19

$p$

Read as: p

Means: Recursive-function notation denoting: p. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014942
• Occurrence projected-formula-0014965
• Occurrence projected-formula-0014967

Equation form expr-151169f9933258d1

$m = 1$

Read as: m equals one

Means: A recursive-function relation or equation stating: m equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014835

Equation form expr-156f5402c1d0baa8

$x+1$

Read as: x plus one

Means: Recursive-function notation denoting: x plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015060

Equation form expr-158d6ecc1df30470

$g_k(\vec x)$

Read as: g sub k open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: g sub k open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015208

Equation form expr-15e892a32e1d61ee

$g_n$

Read as: g sub n

Means: Recursive-function notation denoting: g sub n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015158

Equation form expr-1634b4749c11037a

$\fn{element}(s,i) = \begin{cases} 0 & \mbox{if $i \geq \len{s}$} \\ \bmin{a < s}{(p_i^{a+2} \nmid s)} & \text{otherwise.} \end{cases}$

Read as: the function element open parenthesis s comma i close parenthesis equals cases begin; row one: zero, if i is greater than or equal to the length of s; row two: the least a less than s such that open parenthesis p sub i superscript a plus two does not divide s close parenthesis, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the function element open parenthesis s comma i close parenthesis equals cases begin; row one: zero, if i is greater than or equal to the length of s; row two: the least a less than s such that open parenthesis p sub i superscript a plus two does not divide s close parenthesis, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015034

Equation form expr-17353f1e3ded0f24

$f_x(y)$

Read as: f sub x open parenthesis y close parenthesis

Means: Recursive-function notation denoting: f sub x open parenthesis y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015175
• Occurrence projected-formula-0015186

Equation form expr-1741a20fd8620fe3

$f(x) \simeq g(x)$

Read as: f open parenthesis x close parenthesis has the same definedness and value as g open parenthesis x close parenthesis

Means: A partial-computation statement expressing: f open parenthesis x close parenthesis has the same definedness and value as g open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015197

Equation form expr-175651834a5356f2

$l_1$

Read as: l sub one

Means: Recursive-function notation denoting: l sub one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015081

Equation form expr-17acf2e05eea73b7

$\bexists{z < y}{R(\vec x, z)}$

Read as: there exists z less than y such that R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: there exists z less than y such that R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014813

Equation form expr-17b16968e31c63ba

$p(0) & = 2\\ p(x+1) & = \fn{nextPrime}(p(x))$

Read as: p open parenthesis zero close parenthesis equals two next row p open parenthesis x plus one close parenthesis equals the function next prime open parenthesis p open parenthesis x close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: p open parenthesis zero close parenthesis equals two next row p open parenthesis x plus one close parenthesis equals the function next prime open parenthesis p open parenthesis x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014943

Equation form expr-1841916b64ed0232

$p(x)$

Read as: p open parenthesis x close parenthesis

Means: Recursive-function notation denoting: p open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014931
• Occurrence projected-formula-0014936
• Occurrence projected-formula-0014952

Equation form expr-1893fffb4ea13da8

$\Add(2,0) = 2$

Read as: the addition function open parenthesis two comma zero close parenthesis equals two

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis two comma zero close parenthesis equals two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014410

Equation form expr-18ac3e7343f01689

$d$

Read as: d

Means: Recursive-function notation denoting: d. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014767
• Occurrence projected-formula-0015294
• Occurrence projected-formula-0015313

Equation form expr-19175254495df302

$(s)_{k-1}$

Read as: open parenthesis s close parenthesis sub k minus one

Means: Recursive-function notation denoting: open parenthesis s close parenthesis sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015043

Equation form expr-196db4d36f4e7dd2

$R(i, s)$

Read as: R open parenthesis i comma s close parenthesis

Means: Recursive-function notation denoting: R open parenthesis i comma s close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014995

Equation form expr-1a22a13fe4649f4d

$\lnot P(\vec x)$

Read as: not P open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: not P open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014788
• Occurrence projected-formula-0014796

Equation form expr-1b14da63db175569

$h\colon \Nat \to \Nat$

Read as: h from the natural numbers to the natural numbers

Means: Recursive-function notation denoting: h from the natural numbers to the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014385

Equation form expr-1b16b1df538ba12d

$n$

Read as: n

Means: Recursive-function notation denoting: n. This meaning is freshly authored from the complete recursive-functions source packet.

21 occurrences in this chapter

• Occurrence projected-formula-0014360
• Occurrence projected-formula-0014472
• Occurrence projected-formula-0014473
• Occurrence projected-formula-0014490
• Occurrence projected-formula-0014521
• Occurrence projected-formula-0014522
• Occurrence projected-formula-0014523
• Occurrence projected-formula-0014558
• Occurrence projected-formula-0014562
• Occurrence projected-formula-0014567
• Occurrence projected-formula-0014579
• Occurrence projected-formula-0014653
• Occurrence projected-formula-0014826
• Occurrence projected-formula-0015048
• Occurrence projected-formula-0015073
• Occurrence projected-formula-0015090
• Occurrence projected-formula-0015097
• Occurrence projected-formula-0015098
• Occurrence projected-formula-0015154
• Occurrence projected-formula-0015260
• Occurrence projected-formula-0015261

Equation form expr-1c019cc38ea951c7

$h(\vec x, 0) & = & f(\vec x) \\ h(\vec x, y + 1) & = & g(\vec x, y, h(\vec x, y))$

Read as: h open parenthesis vector x comma zero close parenthesis equals f open parenthesis vector x close parenthesis next row h open parenthesis vector x comma y plus one close parenthesis equals g open parenthesis vector x comma y comma h open parenthesis vector x comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: h open parenthesis vector x comma zero close parenthesis equals f open parenthesis vector x close parenthesis next row h open parenthesis vector x comma y plus one close parenthesis equals g open parenthesis vector x comma y comma h open parenthesis vector x comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014687

Equation form expr-1c87ccc1cf53a9f9

$P(\vec x) \land Q(\vec x)$

Read as: P open parenthesis vector x close parenthesis and Q open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: P open parenthesis vector x close parenthesis and Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014789

Equation form expr-1c9879b76c4da2fb

$\Nat^n$

Read as: the n-fold Cartesian power of the natural numbers

Means: Recursive-function notation denoting: the n-fold Cartesian power of the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014570

Equation form expr-1ca70a4e084a739a

$f\colon \Nat^n \to \Nat$

Read as: f maps the n-fold Cartesian power of the natural numbers to the natural numbers

Means: Recursive-function notation denoting: f maps the n-fold Cartesian power of the natural numbers to the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014358

Equation form expr-1d1d862ac905fb9d

$g_f(s,k)=\begin{cases}\emptyseq & \text{if }k=0,\\ f((s)_0)\concat\dots\concat f((s)_{k-1}) & \text{if }k>0.\end{cases}$

Read as: g sub f returns the code of the first k sequence values transformed by f, with zero values yielding the empty-sequence code

Means: A sequence- or tree-coding expression denoting: g sub f returns the code of the first k sequence values transformed by f, with zero values yielding the empty-sequence code. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015100

Equation form expr-1dbeca42e2c54a76

$T(e, x, s)$

Read as: T open parenthesis e comma x comma s close parenthesis

Means: Recursive-function notation denoting: T open parenthesis e comma x comma s close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015232

Equation form expr-1de68b290bd4632a

$\left| x-y \right| = (x \tsub y) + (y \tsub x)$

Read as: the absolute value of x minus y equals open parenthesis x truncated minus y close parenthesis plus open parenthesis y truncated minus x close parenthesis

Means: A recursive-function relation or equation stating: the absolute value of x minus y equals open parenthesis x truncated minus y close parenthesis plus open parenthesis y truncated minus x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014743

Equation form expr-1e9de57bf29ef2a6

$h'$

Read as: h prime

Means: Recursive-function notation denoting: h prime. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014632
• Occurrence projected-formula-0014643

Equation form expr-1f206b11c23e28cc

$x=1$

Read as: x equals one

Means: A recursive-function relation or equation stating: x equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014377

Equation form expr-1f97d653b7ed2d2b

$n+1$

Read as: n plus one

Means: Recursive-function notation denoting: n plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015095

Equation form expr-20213f439aa72395

$\len{s} = \begin{cases} 0 & \text{if $s = 0$ or $s = 1$} \\ 1 + \bmin{i < s}{R(i, s)} & \text{otherwise} \end{cases}$

Read as: the length of s equals cases begin; row one: zero, if s equals zero or s equals one; row two: one plus the least i less than s such that R open parenthesis i comma s close parenthesis, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the length of s equals cases begin; row one: zero, if s equals zero or s equals one; row two: one plus the least i less than s such that R open parenthesis i comma s close parenthesis, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015008

Equation form expr-20792d2b94351456

$h(x_0, x_1) = f(\Proj{2}{1}(x_0, x_1),\Proj{2}{0}(x_0, x_1)).$

Read as: h open parenthesis x sub zero comma x sub one close parenthesis equals f open parenthesis the two place projection with index one open parenthesis x sub zero comma x sub one close parenthesis comma the two place projection with index zero open parenthesis x sub zero comma x sub one close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: h open parenthesis x sub zero comma x sub one close parenthesis equals f open parenthesis the two place projection with index one open parenthesis x sub zero comma x sub one close parenthesis comma the two place projection with index zero open parenthesis x sub zero comma x sub one close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014512

Equation form expr-20d3379dffa08348

$x \equiv y \mod n$

Read as: x is congruent to y modulo n

Means: Recursive-function notation denoting: x is congruent to y modulo n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014825

Equation form expr-215710a0ef11a250

$\cfind{y}(y) \fdefined$

Read as: partial recursive function phi sub y open parenthesis y close parenthesis is defined

Means: A partial-computation statement expressing: partial recursive function phi sub y open parenthesis y close parenthesis is defined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015292

Equation form expr-217331b710731876

$f(x_0) = 1$

Read as: f open parenthesis x sub zero close parenthesis equals one

Means: A recursive-function relation or equation stating: f open parenthesis x sub zero close parenthesis equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014634

Equation form expr-223a8b94b371dfab

$p_{k-1}$

Read as: p sub k minus one

Means: Recursive-function notation denoting: p sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015059

Equation form expr-2247a4a374fd1ebe

$h(x) & = & g(x,x) + 1 \\ & = & f_x(x) +1.$

Read as: h open parenthesis x close parenthesis equals g open parenthesis x comma x close parenthesis plus one next row equals f sub x open parenthesis x close parenthesis plus one

Means: A source-ordered system, calculation, or case table stating: h open parenthesis x close parenthesis equals g open parenthesis x comma x close parenthesis plus one next row equals f sub x open parenthesis x close parenthesis plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015145

Equation form expr-22a8f59b46b3dd3d

$x \ge y$

Read as: x is greater than or equal to y

Means: A recursive-function relation or equation stating: x is greater than or equal to y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014754

Equation form expr-238c8e56b56be6ef

$\bforall{z < y}{R(\vec x, z)}$

Read as: for every z less than y, R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: for every z less than y, R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014807

Equation form expr-252f10c83610ebca

$f$

Read as: f

Means: Recursive-function notation denoting: f. This meaning is freshly authored from the complete recursive-functions source packet.

41 occurrences in this chapter

• Occurrence projected-formula-0014445
• Occurrence projected-formula-0014461
• Occurrence projected-formula-0014465
• Occurrence projected-formula-0014467
• Occurrence projected-formula-0014476
• Occurrence projected-formula-0014483
• Occurrence projected-formula-0014493
• Occurrence projected-formula-0014524
• Occurrence projected-formula-0014533
• Occurrence projected-formula-0014543
• Occurrence projected-formula-0014548
• Occurrence projected-formula-0014553
• Occurrence projected-formula-0014559
• Occurrence projected-formula-0014575
• Occurrence projected-formula-0014580
• Occurrence projected-formula-0014583
• Occurrence projected-formula-0014587
• Occurrence projected-formula-0014603
• Occurrence projected-formula-0014607
• Occurrence projected-formula-0014610
• Occurrence projected-formula-0014626
• Occurrence projected-formula-0014641
• Occurrence projected-formula-0014652
• Occurrence projected-formula-0014665
• Occurrence projected-formula-0014668
• Occurrence projected-formula-0014675
• Occurrence projected-formula-0014688
• Occurrence projected-formula-0014703
• Occurrence projected-formula-0014705
• Occurrence projected-formula-0014723
• Occurrence projected-formula-0014833
• Occurrence projected-formula-0015187
• Occurrence projected-formula-0015190
• Occurrence projected-formula-0015193
• Occurrence projected-formula-0015195
• Occurrence projected-formula-0015234
• Occurrence projected-formula-0015242
• Occurrence projected-formula-0015258
• Occurrence projected-formula-0015264
• Occurrence projected-formula-0015266
• Occurrence projected-formula-0015273

Equation form expr-259072e6c653cc56

$f(x)$

Read as: f open parenthesis x close parenthesis

Means: Recursive-function notation denoting: f open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0015099
• Occurrence projected-formula-0015198
• Occurrence projected-formula-0015252

Equation form expr-25976bbb4ee521e3

$h(s) = g_{\fn{ISubtrees}}(s, \len{s})$

Read as: h open parenthesis s close parenthesis equals g sub the function immediate subtrees open parenthesis s comma the length of s close parenthesis

Means: A sequence- or tree-coding expression denoting: h open parenthesis s close parenthesis equals g sub the function immediate subtrees open parenthesis s comma the length of s close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015103

Equation form expr-26b1545c8762f573

$\Char{\fn{IsZero}}(0) & = 1,\\ \Char{\fn{IsZero}}(x+1) & = 0.$

Read as: the characteristic function of the zero-test predicate open parenthesis zero close parenthesis equals one comma next row the characteristic function of the zero-test predicate open parenthesis x plus one close parenthesis equals zero

Means: A source-ordered system, calculation, or case table stating: the characteristic function of the zero-test predicate open parenthesis zero close parenthesis equals one comma next row the characteristic function of the zero-test predicate open parenthesis x plus one close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014781

Equation form expr-27ac1d6f9a948981

$p_x$

Read as: p sub x

Means: Recursive-function notation denoting: p sub x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014937

Equation form expr-2874121c6701e126

$d(e_d)$

Read as: d open parenthesis e sub d close parenthesis

Means: Recursive-function notation denoting: d open parenthesis e sub d close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015302
• Occurrence projected-formula-0015309

Equation form expr-28a8dfe72ff8c45f

$\Proj{2}{0}$

Read as: the two place projection with index zero

Means: A primitive-recursive construction or operator denoting: the two place projection with index zero. This meaning is freshly authored from the complete recursive-functions source packet.

4 occurrences in this chapter

• Occurrence projected-formula-0014518
• Occurrence projected-formula-0014534
• Occurrence projected-formula-0014540
• Occurrence projected-formula-0014541

Equation form expr-2ac06f7d9f5bec32

$g(x_0, y, z)$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis

Means: Recursive-function notation denoting: g open parenthesis x sub zero comma y comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014612

Equation form expr-2be7aa126eb30a02

$\fn{subseq}(s, i, n)$

Read as: subseq of s, i, and n

Means: Recursive-function notation denoting: subseq of s, i, and n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015071

Equation form expr-2cf31e1fd0dda933

$\fn{Prime}(x) \defiff x \geq 2 \land \bforall{y \leq x}{(y \mid x \lif y = 1 \lor y = x)}$

Read as: the primality predicate open parenthesis x close parenthesis is defined exactly when x is greater than or equal to two and for every y less than or equal to x, open parenthesis y divides x implies y equals one or y equals x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the primality predicate open parenthesis x close parenthesis is defined exactly when x is greater than or equal to two and for every y less than or equal to x, open parenthesis y divides x implies y equals one or y equals x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014925

Equation form expr-2d711642b726b044

$x$

Read as: x

Means: Recursive-function notation denoting: x. This meaning is freshly authored from the complete recursive-functions source packet.

42 occurrences in this chapter

• Occurrence projected-formula-0014380
• Occurrence projected-formula-0014399
• Occurrence projected-formula-0014403
• Occurrence projected-formula-0014406
• Occurrence projected-formula-0014420
• Occurrence projected-formula-0014426
• Occurrence projected-formula-0014428
• Occurrence projected-formula-0014728
• Occurrence projected-formula-0014740
• Occurrence projected-formula-0014746
• Occurrence projected-formula-0014753
• Occurrence projected-formula-0014760
• Occurrence projected-formula-0014846
• Occurrence projected-formula-0014887
• Occurrence projected-formula-0014892
• Occurrence projected-formula-0014894
• Occurrence projected-formula-0014895
• Occurrence projected-formula-0014911
• Occurrence projected-formula-0014918
• Occurrence projected-formula-0014924
• Occurrence projected-formula-0014932
• Occurrence projected-formula-0014941
• Occurrence projected-formula-0014948
• Occurrence projected-formula-0014958
• Occurrence projected-formula-0015056
• Occurrence projected-formula-0015128
• Occurrence projected-formula-0015164
• Occurrence projected-formula-0015179
• Occurrence projected-formula-0015184
• Occurrence projected-formula-0015191
• Occurrence projected-formula-0015192
• Occurrence projected-formula-0015196
• Occurrence projected-formula-0015213
• Occurrence projected-formula-0015216
• Occurrence projected-formula-0015226
• Occurrence projected-formula-0015237
• Occurrence projected-formula-0015243
• Occurrence projected-formula-0015244
• Occurrence projected-formula-0015248
• Occurrence projected-formula-0015255
• Occurrence projected-formula-0015283
• Occurrence projected-formula-0015318

Equation form expr-2e4ffc307aff38dc

$g_0(x_0)$

Read as: g sub zero open parenthesis x sub zero close parenthesis

Means: Recursive-function notation denoting: g sub zero open parenthesis x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014507

Equation form expr-2e604bf4f935f434

$G(x) = g_x(x)$

Read as: G open parenthesis x close parenthesis equals g sub x open parenthesis x close parenthesis

Means: A recursive-function relation or equation stating: G open parenthesis x close parenthesis equals g sub x open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015166

Equation form expr-2ed52d51edc5e0ab

$h(x)$

Read as: h open parenthesis x close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

5 occurrences in this chapter

• Occurrence projected-formula-0014369
• Occurrence projected-formula-0014381
• Occurrence projected-formula-0014383
• Occurrence projected-formula-0014395
• Occurrence projected-formula-0014464

Equation form expr-304c6693e2e2e163

$f(\vec x)$

Read as: f open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: f open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014694

Equation form expr-30502f423dbe9c3e

$y \tsub x = 0$

Read as: y truncated minus x equals zero

Means: A recursive-function relation or equation stating: y truncated minus x equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014755

Equation form expr-3078a09641b0267f

$x_0$

Read as: x sub zero

Means: Recursive-function notation denoting: x sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014441
• Occurrence projected-formula-0014691

Equation form expr-308fda5bda779517

$f_x$

Read as: f sub x

Means: Recursive-function notation denoting: f sub x. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015141
• Occurrence projected-formula-0015176

Equation form expr-31116f778b485d26

$m_R(\vec x, 0)$

Read as: m sub R open parenthesis vector x comma zero close parenthesis

Means: Recursive-function notation denoting: m sub R open parenthesis vector x comma zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014875

Equation form expr-314b62cde7182298

$\cfind{e_d}(e_d) \fdefined$

Read as: partial recursive function phi sub e sub d open parenthesis e sub d close parenthesis is defined

Means: A partial-computation statement expressing: partial recursive function phi sub e sub d open parenthesis e sub d close parenthesis is defined. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015300
• Occurrence projected-formula-0015310

Equation form expr-31995b8f7c5225c9

$\Add(x_0, 0) & = f(x_0) = x_0\\ \Add(x_0, y+1) & = g(x_0, y, \Add(x_0, y)) = \Succ(\Add(x_0, y))$

Read as: the addition function open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals x sub zero next row the addition function open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis equals the successor function open parenthesis the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: the addition function open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals x sub zero next row the addition function open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis equals the successor function open parenthesis the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014454
• Occurrence projected-formula-0014605

Equation form expr-3206e909c0a3e4ba

$\tuple{0,l}$

Read as: the code of the two-entry sequence zero, l

Means: A sequence- or tree-coding expression denoting: the code of the two-entry sequence zero, l. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015080

Equation form expr-327e88b1ea78f2fd

$\Succ(z) = z+1$

Read as: applied to z equals z plus one

Means: A primitive-recursive construction or operator denoting: applied to z equals z plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014417

Equation form expr-333e0a1e27815d0c

$G$

Read as: G

Means: Recursive-function notation denoting: G. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014671

Equation form expr-3364140985aee02f

$h(x+1)$

Read as: h open parenthesis x plus one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014370
• Occurrence projected-formula-0014382
• Occurrence projected-formula-0014394

Equation form expr-33ea9e6b1b108a22

$\Char{P}(\vec x) \cdot \Char{Q}(\vec x)$

Read as: the characteristic function of P open parenthesis vector x close parenthesis times the characteristic function of Q open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of P open parenthesis vector x close parenthesis times the characteristic function of Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014802

Equation form expr-346cf456a77b1ce5

$p_n$

Read as: p sub n

Means: Recursive-function notation denoting: p sub n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014953

Equation form expr-34c14001ab1c2005

$k=0$

Read as: k equals zero

Means: A recursive-function relation or equation stating: k equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014629

Equation form expr-3543fe426014be71

$\Mult(x,y)$

Read as: the multiplication function open parenthesis x comma y close parenthesis

Means: A primitive-recursive construction or operator denoting: the multiplication function open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014425
• Occurrence projected-formula-0014427

Equation form expr-359492d1aa9966c8

$R(i, s) \text{ iff } p_i \mid s \land p_{i+1} \nmid s.$

Read as: R open parenthesis i comma s close parenthesis if and only if p sub i divides s and p sub i plus one does not divide s

Means: A primitive-recursive relation or bounded-quantifier statement expressing: R open parenthesis i comma s close parenthesis if and only if p sub i divides s and p sub i plus one does not divide s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014996

Equation form expr-3650daa721307966

$g_f(s,0)=\emptyseq \\ g_f(s,k+1)=g_f(s,k)\concat f((s)_k)$

Read as: g sub f of s and zero equals the empty-sequence code; and g sub f of s and k plus one equals g sub f of s and k concatenated with f of the k-th entry of s

Means: A source-ordered system, calculation, or case table stating: g sub f of s and zero equals the empty-sequence code; and g sub f of s and k plus one equals g sub f of s and k concatenated with f of the k-th entry of s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015101

Equation form expr-36613f93d5f91b7d

$h(x) = f(g(x))$

Read as: h open parenthesis x close parenthesis equals f open parenthesis g open parenthesis x close parenthesis close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis x close parenthesis equals f open parenthesis g open parenthesis x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014463

Equation form expr-367264cfdb704d7a

$\fn{pred}'(x, 0) & = \Zero(x) = 0,\\ \fn{pred}'(x, y+1) & = \Proj{3}{1}(x, y, \fn{pred'}(x, y)) = y.$

Read as: the predecessor function prime open parenthesis x comma zero close parenthesis equals the constant zero function open parenthesis x close parenthesis equals zero comma next row the predecessor function prime open parenthesis x comma y plus one close parenthesis equals the three place projection with index one open parenthesis x comma y comma the function pred prime open parenthesis x comma y close parenthesis close parenthesis equals y

Means: A source-ordered system, calculation, or case table stating: the predecessor function prime open parenthesis x comma zero close parenthesis equals the constant zero function open parenthesis x close parenthesis equals zero comma next row the predecessor function prime open parenthesis x comma y plus one close parenthesis equals the three place projection with index one open parenthesis x comma y comma the function pred prime open parenthesis x comma y close parenthesis close parenthesis equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014732

Equation form expr-36840676dffa30a4

$R(x,\vec z)$

Read as: R open parenthesis x comma vector z close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma vector z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015222
• Occurrence projected-formula-0015227

Equation form expr-36c99c0bb63416fc

$g_i$

Read as: g sub i

Means: Recursive-function notation denoting: g sub i. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014477
• Occurrence projected-formula-0014706
• Occurrence projected-formula-0015204

Equation form expr-380918b946a52664

$=$

Read as: equals

Means: A recursive-function relation or equation stating: equals. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014908
• Occurrence projected-formula-0015209

Equation form expr-38bcea832ba5496b

$h(\vec x, y) \defis \prod_{z = 0}^y f(\vec x, z).$

Read as: h of vector x and y is defined as the product, from z equals zero through y, of f of vector x and z

Means: A recursive-function relation or equation stating: h of vector x and y is defined as the product, from z equals zero through y, of f of vector x and z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014771

Equation form expr-3978289f4037c489

$d(y) \fdefined$

Read as: d open parenthesis y close parenthesis is defined

Means: A partial-computation statement expressing: d open parenthesis y close parenthesis is defined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015288

Equation form expr-3a1ee4a84807a810

$\fn{append}(s,a)$

Read as: append of s and a

Means: A sequence- or tree-coding expression denoting: append of s and a. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015014

Equation form expr-3a1fe9e699b7b5f4

$\fn{nextPrime}(x)$

Read as: the function next prime open parenthesis x close parenthesis

Means: Recursive-function notation denoting: the function next prime open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014940

Equation form expr-3a374a50eda6d8c8

$\fn{hSubtreeSeq}(t, 0) & = \tuple{t} \\ \fn{hSubtreeSeq}(t, n+1) & = \fn{hSubtreeSeq}(t, n) \concat h(\fn{hSubtreeSeq}(t, n)).$

Read as: the height-subtree-sequence function of t and zero equals the code of the one-entry sequence t; at n plus one it equals the preceding coded sequence concatenated with the coded sequence returned by h

Means: A source-ordered system, calculation, or case table stating: the height-subtree-sequence function of t and zero equals the code of the one-entry sequence t; at n plus one it equals the preceding coded sequence concatenated with the coded sequence returned by h. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015106

Equation form expr-3a375b2c74e350fb

$d(y) \fundefined$

Read as: d open parenthesis y close parenthesis is undefined

Means: A partial-computation statement expressing: d open parenthesis y close parenthesis is undefined. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015286
• Occurrence projected-formula-0015291

Equation form expr-3a488f515ac10699

$h(\vec x, 0) & = f(\vec x) \\ h(\vec x, y+1) & = g(\vec x, y, h(k(\vec x), y))$

Read as: h open parenthesis vector x comma zero close parenthesis equals f open parenthesis vector x close parenthesis next row h open parenthesis vector x comma y plus one close parenthesis equals g open parenthesis vector x comma y comma h open parenthesis k open parenthesis vector x close parenthesis comma y close parenthesis close parenthesis

Means: This is the changing-parameters recursion scheme. The recursive call passes k of vector x as the next side-parameter input. The source does not specify whether that value is scalar or tuple-coded, so the reader preserves the displayed notation without adding a shape claim.

1 occurrence in this chapter

• Occurrence projected-formula-0015137

Equation form expr-3b4a67bf78b2061e

$\fn{const}_2(x)$

Read as: the constant-two function applied to x

Means: Recursive-function notation denoting: the constant-two function applied to x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014718

Equation form expr-3bdc55e2c2a897b2

$f(x) \simeq U(\umin{s}{T(e, x, s)})$

Read as: f open parenthesis x close parenthesis has the same definedness and value as U open parenthesis the least s such that T open parenthesis e comma x comma s close parenthesis close parenthesis

Means: A partial-computation statement expressing: f open parenthesis x close parenthesis has the same definedness and value as U open parenthesis the least s such that T open parenthesis e comma x comma s close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015236

Equation form expr-3bfa341bc0271c4a

$\cfind{y}(y) \fundefined$

Read as: partial recursive function phi sub y open parenthesis y close parenthesis is undefined

Means: A partial-computation statement expressing: partial recursive function phi sub y open parenthesis y close parenthesis is undefined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015289

Equation form expr-3d2b86c8d04ffb33

$\fn{append}(s,a) = \begin{cases} 2^{a+1} & \text{if $s = 0$ or $s = 1$} \\ s \cdot p_{\len{s}}^{a+1} & \text{otherwise.} \end{cases}$

Read as: the function append open parenthesis s comma a close parenthesis equals cases begin; row one: two superscript a plus one, if s equals zero or s equals one; row two: s times p sub the length of s superscript a plus one, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the function append open parenthesis s comma a close parenthesis equals cases begin; row one: two superscript a plus one, if s equals zero or s equals one; row two: s times p sub the length of s superscript a plus one, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015018

Equation form expr-3dac54dbafb34c1d

$\tuple{k, d_1, \dots, d_k}$

Read as: the code of the sequence whose first entry is k and whose remaining entries are the immediate-subtree codes d sub one through d sub k

Means: A sequence- or tree-coding expression denoting: the code of the sequence whose first entry is k and whose remaining entries are the immediate-subtree codes d sub one through d sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015075

Equation form expr-3e62f055a251be10

$\fn{exp}$

Read as: the exponentiation function

Means: Recursive-function notation denoting: the exponentiation function. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014721

Equation form expr-3e897855985e8b9b

$\fn{Comp}_{k,n}[F,G_0,\dots,G_{k-1}]$

Read as: the composition operator with k inner functions and n inputs, applied to F, G sub zero, through G sub k minus one

Means: A primitive-recursive construction or operator denoting: the composition operator with k inner functions and n inputs, applied to F, G sub zero, through G sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014661

Equation form expr-3f79bb7b435b0532

$e$

Read as: e

Means: Recursive-function notation denoting: e. This meaning is freshly authored from the complete recursive-functions source packet.

12 occurrences in this chapter

• Occurrence projected-formula-0015235
• Occurrence projected-formula-0015238
• Occurrence projected-formula-0015245
• Occurrence projected-formula-0015247
• Occurrence projected-formula-0015254
• Occurrence projected-formula-0015256
• Occurrence projected-formula-0015259
• Occurrence projected-formula-0015263
• Occurrence projected-formula-0015265
• Occurrence projected-formula-0015267
• Occurrence projected-formula-0015274
• Occurrence projected-formula-0015279

Equation form expr-3fabf135074b7b16

$g(x_0, y, z) = z + 1$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis equals z plus one

Means: A recursive-function relation or equation stating: g open parenthesis x sub zero comma y comma z close parenthesis equals z plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014452

Equation form expr-40fc8d77c687fe2d

$h(g_0(\vec x),\dots,g_k(\vec x))$

Read as: h open parenthesis g sub zero open parenthesis vector x close parenthesis comma and so on comma g sub k open parenthesis vector x close parenthesis close parenthesis

Means: Recursive-function notation denoting: h open parenthesis g sub zero open parenthesis vector x close parenthesis comma and so on comma g sub k open parenthesis vector x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015203

Equation form expr-4124c64d4022edfc

$\fn{sconcat}(\tuple{s_0, \dots, s_k}) = s_0 \concat \dots \concat s_k.$

Read as: the sequence-concatenate function applied to the code of the sequence s sub zero through s sub k equals s sub zero concatenated through s sub k

Means: A sequence- or tree-coding expression denoting: the sequence-concatenate function applied to the code of the sequence s sub zero through s sub k equals s sub zero concatenated through s sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015068

Equation form expr-412e442e7b659467

$\Add(2,3)$

Read as: the addition function open parenthesis two comma three close parenthesis

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis two comma three close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014408

Equation form expr-419c54f0f0586afd

$y \nmid x$

Read as: y does not divide x

Means: A recursive-function relation or equation stating: y does not divide x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014922

Equation form expr-4281249074e07c99

$Q(\vec x)$

Read as: Q open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014787
• Occurrence projected-formula-0014793

Equation form expr-42b0225751bf4a04

$x \tsub y$

Read as: x truncated minus y

Means: Recursive-function notation denoting: x truncated minus y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014737

Equation form expr-42cb82a7a1a7890a

$g(x_0, y, z) = \Succ(z).$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis equals the successor function open parenthesis z close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x sub zero comma y comma z close parenthesis equals the successor function open parenthesis z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014613

Equation form expr-42d29d9d2a195573

$l(x,y) = g(\Proj{2}{0}(x,y),\Proj{2}{0}(x,y),\Proj{2}{1}(x,y)),$

Read as: l open parenthesis x comma y close parenthesis equals g open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: l open parenthesis x comma y close parenthesis equals g open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014537

Equation form expr-4443576302eb4cf1

$\fn{exp}(x, y) = x^y$

Read as: applied to x and y equals x to the power y

Means: A recursive-function relation or equation stating: applied to x and y equals x to the power y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014720

Equation form expr-4454461bf5348bb9

$\bexists{z < 0}{R(\vec x, z)}$

Read as: there exists z less than zero such that R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: there exists z less than zero such that R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014817

Equation form expr-4569dc409f0ddf21

$\Mult(2,3)$

Read as: the multiplication function open parenthesis two comma three close parenthesis

Means: A primitive-recursive construction or operator denoting: the multiplication function open parenthesis two comma three close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014430
• Occurrence projected-formula-0014434

Equation form expr-45c75507f7a8bad6

$\fn{min}(x,y)$

Read as: the minimum function open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: the minimum function open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014762

Equation form expr-4733f05694df9788

$z \le y$

Read as: z is less than or equal to y

Means: A recursive-function relation or equation stating: z is less than or equal to y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014905

Equation form expr-480dfc2600de3c0a

$\Mult(2,2)$

Read as: the multiplication function open parenthesis two comma two close parenthesis

Means: A primitive-recursive construction or operator denoting: the multiplication function open parenthesis two comma two close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014433

Equation form expr-48fe546d733e9534

$m_R(\vec x, 0) = 0$

Read as: m sub R open parenthesis vector x comma zero close parenthesis equals zero

Means: A recursive-function relation or equation stating: m sub R open parenthesis vector x comma zero close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014864

Equation form expr-49db45fb1a3f8106

$\cfind{e}(x) \simeq f(x)$

Read as: partial recursive function phi sub e open parenthesis x close parenthesis has the same definedness and value as f open parenthesis x close parenthesis

Means: A partial-computation statement expressing: partial recursive function phi sub e open parenthesis x close parenthesis has the same definedness and value as f open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015271

Equation form expr-4a27b0062516f89c

$R(x, y-1)$

Read as: R open parenthesis x comma y minus one close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma y minus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014852

Equation form expr-4a422e3fcbd0db21

$g^n(x)$

Read as: the n-fold iterate of g, evaluated at x

Means: Recursive-function notation denoting: the n-fold iterate of g, evaluated at x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015152

Equation form expr-4a73096daa3399ce

$i+1$

Read as: i plus one

Means: Recursive-function notation denoting: i plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015005

Equation form expr-4aaf63602f7fc3f2

$\Add(x,y) = x+y$

Read as: applied to x and y equals x plus y

Means: A primitive-recursive construction or operator denoting: applied to x and y equals x plus y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014601

Equation form expr-4b183d86304ee69b

$\langle 2, 7, 3, 0, 0 \rangle$

Read as: the five-entry tuple two, seven, three, zero, zero

Means: A sequence- or tree-coding expression denoting: the five-entry tuple two, seven, three, zero, zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014983

Equation form expr-4ba54a6a818f6522

$y+1$

Read as: y plus one

Means: Recursive-function notation denoting: y plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014401

Equation form expr-4bd8ce42d5f1e9a3

$m_R(\vec{x}, y+1) = y$

Read as: m sub R open parenthesis vector x comma y plus one close parenthesis equals y

Means: A recursive-function relation or equation stating: m sub R open parenthesis vector x comma y plus one close parenthesis equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014871

Equation form expr-4c26f56e5ef1fafa

$\fn{Rec}_k[F,G]$

Read as: the k-place recursion operator applied to F and G

Means: A primitive-recursive construction or operator denoting: the k-place recursion operator applied to F and G. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014673

Equation form expr-4cdf4e1946b887c2

$h(x_0, \dots, x_{n-1}) = f(g_0(x_0, \dots, x_{n-1}), \dots, g_{k-1}(x_0, \dots, x_{n-1}))$

Read as: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals f open parenthesis g sub zero open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis comma and so on comma g sub k minus one open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals f open parenthesis g sub zero open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis comma and so on comma g sub k minus one open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014475

Equation form expr-4dc4e01def696af8

$\Proj{n}{i}$

Read as: the n place projection with index i

Means: A primitive-recursive construction or operator denoting: the n place projection with index i. This meaning is freshly authored from the complete recursive-functions source packet.

4 occurrences in this chapter

• Occurrence projected-formula-0014489
• Occurrence projected-formula-0014574
• Occurrence projected-formula-0014649
• Occurrence projected-formula-0014709

Equation form expr-4ddd6f419b334e54

$f(x) \fdefined$

Read as: f open parenthesis x close parenthesis is defined

Means: A partial-computation statement expressing: f open parenthesis x close parenthesis is defined. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015189
• Occurrence projected-formula-0015239

Equation form expr-4e07408562bedb8b

$3$

Read as: three

Means: Recursive-function notation denoting: three. This meaning is freshly authored from the complete recursive-functions source packet.

7 occurrences in this chapter

• Occurrence projected-formula-0014453
• Occurrence projected-formula-0014459
• Occurrence projected-formula-0014497
• Occurrence projected-formula-0014500
• Occurrence projected-formula-0014526
• Occurrence projected-formula-0014682
• Occurrence projected-formula-0014927

Equation form expr-4e5347e07e38c76e

$k=2$

Read as: k equals two

Means: A recursive-function relation or equation stating: k equals two. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014503
• Occurrence projected-formula-0014513

Equation form expr-4fb6819c3daab3e6

$2^x \cdot x$

Read as: two superscript x times x

Means: Recursive-function notation denoting: two superscript x times x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015162

Equation form expr-4fc82b26aecb47d2

$11$

Read as: eleven

Means: Recursive-function notation denoting: eleven. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014930

Equation form expr-4fd0ce6d4813340f

$\fn{SubtreeSeq}(t)$

Read as: subtree sequence of t

Means: Recursive-function notation denoting: subtree sequence of t. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015085

Equation form expr-500259889cffd163

$\Char{P}(\vec x, y)$

Read as: the characteristic function of P open parenthesis vector x comma y close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of P open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014819

Equation form expr-50dfb0911bb1dfff

$x \mid y \defiff \bexists{z \leq y}{(x \cdot z) = y}.$

Read as: x divides y is defined exactly when there exists z less than or equal to y such that open parenthesis x times z close parenthesis equals y

Means: A primitive-recursive relation or bounded-quantifier statement expressing: x divides y is defined exactly when there exists z less than or equal to y such that open parenthesis x times z close parenthesis equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014909

Equation form expr-50e62753e571d66c

$\fn{max}$

Read as: the maximum function

Means: Recursive-function notation denoting: the maximum function. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014821

Equation form expr-5120a35e3ac4dee8

$\fn{concat}(\tuple{a_0, \dots, a_k}, \tuple{b_0, \dots, b_l}) = \tuple{a_0, \dots, a_k, b_0, \dots, b_l}.$

Read as: concatenating the code of sequence a sub zero through a sub k with the code of sequence b sub zero through b sub l yields the code of the combined sequence

Means: A sequence- or tree-coding expression denoting: concatenating the code of sequence a sub zero through a sub k with the code of sequence b sub zero through b sub l yields the code of the combined sequence. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015046

Equation form expr-51af7ceb518b1b82

$R(x, z)$

Read as: R open parenthesis x comma z close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014849

Equation form expr-51afe980e791613d

$\bexists{x < y+1}{\dots x \dots}$

Read as: there exists x less than y plus one such that the displayed surrounding condition holds at x

Means: A primitive-recursive relation or bounded-quantifier statement expressing: there exists x less than y plus one such that the displayed surrounding condition holds at x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014824

Equation form expr-53b42c0095c19854

$\Add(x,y+1)$

Read as: the addition function open parenthesis x comma y plus one close parenthesis

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis x comma y plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014419

Equation form expr-549f5465eaa5cf11

$\Add(2, 2) = 3 + 1 = 4$

Read as: the addition function open parenthesis two comma two close parenthesis equals three plus one equals four

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis two comma two close parenthesis equals three plus one equals four. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014412

Equation form expr-55b4ecee7498e559

$h(e, n) = \begin{cases} 1 & \text{if computation $e$ halts on input $n$}\\ 0 & \text{otherwise,} \end{cases}$

Read as: h open parenthesis e comma n close parenthesis equals cases begin; row one: one, if computation e halts on input n; row two: zero, otherwise comma; cases end

Means: A source-ordered system, calculation, or case table stating: h open parenthesis e comma n close parenthesis equals cases begin; row one: one, if computation e halts on input n; row two: zero, otherwise comma; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015262

Equation form expr-56c840afa529f071

$h(e_d, e_d) = 0$

Read as: h open parenthesis e sub d comma e sub d close parenthesis equals zero

Means: A recursive-function relation or equation stating: h open parenthesis e sub d comma e sub d close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015303
• Occurrence projected-formula-0015305

Equation form expr-570e2f6841c723e6

$f(y_0,y_1)$

Read as: f open parenthesis y sub zero comma y sub one close parenthesis

Means: Recursive-function notation denoting: f open parenthesis y sub zero comma y sub one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014505

Equation form expr-575d86b85365c7b4

$G_0$

Read as: G sub zero

Means: Recursive-function notation denoting: G sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014657

Equation form expr-594e519ae499312b

$z$

Read as: z

Means: Recursive-function notation denoting: z. This meaning is freshly authored from the complete recursive-functions source packet.

7 occurrences in this chapter

• Occurrence projected-formula-0014811
• Occurrence projected-formula-0014815
• Occurrence projected-formula-0014855
• Occurrence projected-formula-0014882
• Occurrence projected-formula-0014900
• Occurrence projected-formula-0014902
• Occurrence projected-formula-0015134

Equation form expr-5959d6d76c6e4a85

$g_1(x_0)$

Read as: g sub one open parenthesis x sub zero close parenthesis

Means: Recursive-function notation denoting: g sub one open parenthesis x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014508

Equation form expr-59664c8716127f12

$k+2$

Read as: k plus two

Means: Recursive-function notation denoting: k plus two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014586

Equation form expr-596cfa458d470596

$\fn{exp}(x, 0) & = 1\\ \fn{exp}(x, y+1) & = \Mult(x, \fn{exp}(x,y)). \intertext{Strictly speaking, this is not a recursive definition from primitive recursive functions. Officially, though, we have:} \fn{exp}(x, 0) & = f(x)\\ \fn{exp}(x, y+1) & = g(x, y, \fn{exp}(x,y)). \intertext{where} f(x) & = \Succ(\Zero(x)) = 1\\ g(x, y, z) & = \Mult(\Proj{3}{0}(x, y, z), \Proj{3}{2}(x, y, z)) = x \cdot z$

Read as: the exponentiation function open parenthesis x comma zero close parenthesis equals one next row the exponentiation function open parenthesis x comma y plus one close parenthesis equals the multiplication function open parenthesis x comma the exponentiation function open parenthesis x comma y close parenthesis close parenthesis Strictly speaking comma this is not a recursive definition from primitive recursive functions Officially comma though comma we have the exponentiation function open parenthesis x comma zero close parenthesis equals f open parenthesis x close parenthesis next row the exponentiation function open parenthesis x comma y plus one close parenthesis equals g open parenthesis x comma y comma the exponentiation function open parenthesis x comma y close parenthesis close parenthesis where f open parenthesis x close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis x close parenthesis close parenthesis equals one next row g open parenthesis x comma y comma z close parenthesis equals the multiplication function open parenthesis the three place projection with index zero open parenthesis x comma y comma z close parenthesis comma the three place projection with index two open parenthesis x comma y comma z close parenthesis close parenthesis equals x times z

Means: A source-ordered system, calculation, or case table stating: the exponentiation function open parenthesis x comma zero close parenthesis equals one next row the exponentiation function open parenthesis x comma y plus one close parenthesis equals the multiplication function open parenthesis x comma the exponentiation function open parenthesis x comma y close parenthesis close parenthesis Strictly speaking comma this is not a recursive definition from primitive recursive functions Officially comma though comma we have the exponentiation function open parenthesis x comma zero close parenthesis equals f open parenthesis x close parenthesis next row the exponentiation function open parenthesis x comma y plus one close parenthesis equals g open parenthesis x comma y comma the exponentiation function open parenthesis x comma y close parenthesis close parenthesis where f open parenthesis x close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis x close parenthesis close parenthesis equals one next row g open parenthesis x comma y comma z close parenthesis equals the multiplication function open parenthesis the three place projection with index zero open parenthesis x comma y comma z close parenthesis comma the three place projection with index two open parenthesis x comma y comma z close parenthesis close parenthesis equals x times z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014722

Equation form expr-5abe9172ea8287b1

$p(1) = 3$

Read as: p open parenthesis one close parenthesis equals three

Means: A recursive-function relation or equation stating: p open parenthesis one close parenthesis equals three. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014934

Equation form expr-5b23e599d3ec3b36

$\fn{hSubtreeSeq}$

Read as: the height-subtree-sequence function

Means: Recursive-function notation denoting: the height-subtree-sequence function. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015105
• Occurrence projected-formula-0015111

Equation form expr-5c2bbfb80bdbbc6e

$r(x,y)$

Read as: r open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: r open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015127
• Occurrence projected-formula-0015131

Equation form expr-5c62e091b8c0565f

$P$

Read as: P

Means: Recursive-function notation denoting: P. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014839
• Occurrence projected-formula-0014840
• Occurrence projected-formula-0014844

Equation form expr-5cb73b049d09fef0

$h(x_0, \dots, x_{k-1}, y)$

Read as: h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014436
• Occurrence projected-formula-0014440

Equation form expr-5ccb84c570fcf376

$\vec x$

Read as: vector x

Means: Recursive-function notation denoting: vector x. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014690
• Occurrence projected-formula-0014808
• Occurrence projected-formula-0015205

Equation form expr-5d42a3e20328f891

$f(x_0) = x_0$

Read as: f open parenthesis x sub zero close parenthesis equals x sub zero

Means: A recursive-function relation or equation stating: f open parenthesis x sub zero close parenthesis equals x sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014451

Equation form expr-5d88fe8893a198e6

$> 0$

Read as: greater than zero

Means: A recursive-function relation or equation stating: greater than zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014897

Equation form expr-5ead97aab6f88888

$\fn{Prime}(x)$

Read as: the primality predicate open parenthesis x close parenthesis

Means: Recursive-function notation denoting: the primality predicate open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014923

Equation form expr-5fe6dc6b2281b3d2

$\Zero$

Read as: the constant zero function

Means: A primitive-recursive construction or operator denoting: the constant zero function. This meaning is freshly authored from the complete recursive-functions source packet.

7 occurrences in this chapter

• Occurrence projected-formula-0014572
• Occurrence projected-formula-0014589
• Occurrence projected-formula-0014593
• Occurrence projected-formula-0014636
• Occurrence projected-formula-0014647
• Occurrence projected-formula-0014707
• Occurrence projected-formula-0014715

Equation form expr-5feceb66ffc86f38

$0$

Read as: zero

Means: Recursive-function notation denoting: zero. This meaning is freshly authored from the complete recursive-functions source packet.

20 occurrences in this chapter

• Occurrence projected-formula-0014362
• Occurrence projected-formula-0014364
• Occurrence projected-formula-0014365
• Occurrence projected-formula-0014368
• Occurrence projected-formula-0014373
• Occurrence projected-formula-0014457
• Occurrence projected-formula-0014595
• Occurrence projected-formula-0014816
• Occurrence projected-formula-0014841
• Occurrence projected-formula-0014885
• Occurrence projected-formula-0014912
• Occurrence projected-formula-0014968
• Occurrence projected-formula-0014984
• Occurrence projected-formula-0014987
• Occurrence projected-formula-0015022
• Occurrence projected-formula-0015023
• Occurrence projected-formula-0015092
• Occurrence projected-formula-0015120
• Occurrence projected-formula-0015173
• Occurrence projected-formula-0015297

Equation form expr-6061b6960ac8a2eb

$\fn{hconcat}(s,t,0) & = s\\ \fn{hconcat}(s,t,n+1) & = \fn{append}(\fn{hconcat}(s,t,n),(t)_n) \intertext{Then we can define $\fn{concat}$ by} \fn{concat}(s,t) & = \fn{hconcat}(s,t,\len{t}).$

Read as: the function helper concatenate open parenthesis s comma t comma zero close parenthesis equals s next row the function helper concatenate open parenthesis s comma t comma n plus one close parenthesis equals the function append open parenthesis the function helper concatenate open parenthesis s comma t comma n close parenthesis comma open parenthesis t close parenthesis sub n close parenthesis Then we can define the concatenate function by the concatenate function open parenthesis s comma t close parenthesis equals the function helper concatenate open parenthesis s comma t comma the length of t close parenthesis

Means: A source-ordered system, calculation, or case table stating: the function helper concatenate open parenthesis s comma t comma zero close parenthesis equals s next row the function helper concatenate open parenthesis s comma t comma n plus one close parenthesis equals the function append open parenthesis the function helper concatenate open parenthesis s comma t comma n close parenthesis comma open parenthesis t close parenthesis sub n close parenthesis Then we can define the concatenate function by the concatenate function open parenthesis s comma t close parenthesis equals the function helper concatenate open parenthesis s comma t comma the length of t close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015051

Equation form expr-612990c5c626fdc9

$\#(0) & = & \langle 0 \rangle \\ \#(S) & = & \langle 1 \rangle \\ \#(\Proj{n}{i}) & = & \langle 2, n, i \rangle \\ \#(\fn{Comp}_{k,l}[H,G_0,\dots,G_{k-1}]) & = & \langle 3,k,l,\#(H),\#(G_0),\dots,\#(G_{k-1}) \rangle \\ \#(\fn{Rec}_l[G,H]) & = & \langle 4, l, \#(G), \#(H) \rangle$

Read as: coding equations, in source order: the code of zero is the one-entry tuple zero; the code of successor is the one-entry tuple one; the code of the n-place projection with index i is the tuple two, n, i; the code of composition with k inner functions and l inputs is the tuple beginning three, k, l, followed by the codes of H and of G sub zero through G sub k minus one; and the code of l-place recursion from G and H is the tuple four, l, the code of G, the code of H

Means: A source-ordered system, calculation, or case table stating: coding equations, in source order: the code of zero is the one-entry tuple zero; the code of successor is the one-entry tuple one; the code of the n-place projection with index i is the tuple two, n, i; the code of composition with k inner functions and l inputs is the tuple beginning three, k, l, followed by the codes of H and of G sub zero through G sub k minus one; and the code of l-place recursion from G and H is the tuple four, l, the code of G, the code of H. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015169

Equation form expr-61524c3ccaa0dea9

$\Char{\lnot P}(\vec x) = \begin{cases} 0 & \text{if $\Char{P}(\vec x) = 1$}\\ 1 & \text{otherwise} \end{cases}$

Read as: the characteristic function of not P open parenthesis vector x close parenthesis equals cases begin; row one: zero, if the characteristic function of P open parenthesis vector x close parenthesis equals one; row two: one, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the characteristic function of not P open parenthesis vector x close parenthesis equals cases begin; row one: zero, if the characteristic function of P open parenthesis vector x close parenthesis equals one; row two: one, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014797

Equation form expr-6158f77c3833268f

$p_i$

Read as: p sub i

Means: Recursive-function notation denoting: p sub i. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0015001
• Occurrence projected-formula-0015006
• Occurrence projected-formula-0015030

Equation form expr-61d2d6985fbd6781

$\Proj{n}{i}(x_0, \dots, x_{n-1}) = x_i$

Read as: the n place projection with index i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals x sub i

Means: A primitive-recursive construction or operator denoting: the n place projection with index i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals x sub i. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014487

Equation form expr-61f0cb69c9e8e1d3

$d_k$

Read as: d sub k

Means: Recursive-function notation denoting: d sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015078

Equation form expr-6286bb29e3d8c824

$p_i \le x$

Read as: p sub i is less than or equal to x

Means: A recursive-function relation or equation stating: p sub i is less than or equal to x. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014966
• Occurrence projected-formula-0014969

Equation form expr-62c66a7a5dd70c31

$m$

Read as: m

Means: Recursive-function notation denoting: m. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014361
• Occurrence projected-formula-0014837

Equation form expr-63e6d97172cc88a5

$\mu x \; R(x,\vec z)$

Read as: the least x such that R open parenthesis x comma vector z close parenthesis

Means: A partial-computation statement expressing: the least x such that R open parenthesis x comma vector z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015223
• Occurrence projected-formula-0015225

Equation form expr-6411abc35515ce88

$\cfind{e}(x) \simeq U(\mu s \; T(e, x, s))$

Read as: partial recursive function phi sub e open parenthesis x close parenthesis has the same definedness and value as U open parenthesis the least s such that T open parenthesis e comma x comma s close parenthesis close parenthesis

Means: A partial-computation statement expressing: partial recursive function phi sub e open parenthesis x close parenthesis has the same definedness and value as U open parenthesis the least s such that T open parenthesis e comma x comma s close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015268

Equation form expr-644f0340595a771d

$h(x_0, \dots, x_{n-1}) = f(g_0(x_0, \dots, x_{n-1}), \dots, g_{k-1}(x_0, \dots, x_{n-1})).$

Read as: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals f open parenthesis g sub zero open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis comma and so on comma g sub k minus one open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals f open parenthesis g sub zero open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis comma and so on comma g sub k minus one open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014564

Equation form expr-6471c9ed0d4ec520

$f(0,\vec z), f(1,\vec z), \dots, f(x,\vec z)$

Read as: f open parenthesis zero comma vector z close parenthesis comma f open parenthesis one comma vector z close parenthesis comma and so on comma f open parenthesis x comma vector z close parenthesis

Means: Recursive-function notation denoting: f open parenthesis zero comma vector z close parenthesis comma f open parenthesis one comma vector z close parenthesis comma and so on comma f open parenthesis x comma vector z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015214

Equation form expr-648c9482f974bd13

$h(x) = \Add(x, x)$

Read as: h open parenthesis x close parenthesis equals the addition function open parenthesis x comma x close parenthesis

Means: A primitive-recursive construction or operator denoting: h open parenthesis x close parenthesis equals the addition function open parenthesis x comma x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014501

Equation form expr-6495ba8709ca4c05

$f(z) = \Succ(\Zero(z))$

Read as: f open parenthesis z close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis z close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: f open parenthesis z close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis z close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014642

Equation form expr-65fdd75ff6b59995

$p_i \nmid p+1$

Read as: p sub i does not divide p plus one

Means: A recursive-function relation or equation stating: p sub i does not divide p plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014972

Equation form expr-6669250149368b78

$m_R(\vec{x}, y)$

Read as: m sub R open parenthesis vector x comma y close parenthesis

Means: Recursive-function notation denoting: m sub R open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014854

Equation form expr-66c5e34270e1daf7

$g(x_0, y, z) = \Succ(\Proj{3}{2}(x_0, y, z))$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x sub zero comma y comma z close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014620

Equation form expr-6784f03e13056c8a

$y + 1$

Read as: y plus one

Means: Recursive-function notation denoting: y plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014865

Equation form expr-67aa902e375d4f8b

$P(\vec x, y) \defiff \bforall{z < y}{R(\vec x, z)}$

Read as: P open parenthesis vector x comma y close parenthesis is defined exactly when for every z less than y, R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: P open parenthesis vector x comma y close parenthesis is defined exactly when for every z less than y, R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014818

Equation form expr-67b4a7bdc829adf1

$P(\vec x) \lor Q(\vec x)$

Read as: P open parenthesis vector x close parenthesis or Q open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: P open parenthesis vector x close parenthesis or Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014790

Equation form expr-692dc9a1552174fa

$x \in \Nat$

Read as: x is in the natural numbers

Means: A recursive-function relation or equation stating: x is in the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014384
• Occurrence projected-formula-0015272

Equation form expr-6962c50750001b7f

$p_1=3$

Read as: p sub one equals three

Means: A recursive-function relation or equation stating: p sub one equals three. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014939

Equation form expr-696bb1f61825d9d0

$R_{m-1}(\vec x)$

Read as: R sub m minus one open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: R sub m minus one open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014832

Equation form expr-698cd4222a287d75

$S_i$

Read as: S sub i

Means: Recursive-function notation denoting: S sub i. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014596
• Occurrence projected-formula-0014598

Equation form expr-69f39223632549c8

$h(x_0,\dots,x_{k-1},0) & = f(x_0,\dots,x_{k-1}) \\ h(x_0,\dots,x_{k-1},y+1) & = g(x_0,\dots,x_{k-1}, y, h(x_0,\dots,x_{k-1}, y))$

Read as: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis equals f open parenthesis x sub zero comma and so on comma x sub k minus one close parenthesis next row h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis equals g open parenthesis x sub zero comma and so on comma x sub k minus one comma y comma h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis equals f open parenthesis x sub zero comma and so on comma x sub k minus one close parenthesis next row h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis equals g open parenthesis x sub zero comma and so on comma x sub k minus one comma y comma h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014552

Equation form expr-6a48418a64fd666c

$h(e_d, e_d) = 1$

Read as: h open parenthesis e sub d comma e sub d close parenthesis equals one

Means: A recursive-function relation or equation stating: h open parenthesis e sub d comma e sub d close parenthesis equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015299

Equation form expr-6b7fc046b34c5afb

$x \tsub y = \begin{cases} 0 & \text{if $x < y$}\\ x-y & \text{otherwise} \end{cases}$

Read as: x truncated minus y equals cases begin; row one: zero, if x is less than y; row two: x minus y, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: x truncated minus y equals cases begin; row one: zero, if x is less than y; row two: x minus y, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014738

Equation form expr-6b86b273ff34fce1

$1$

Read as: one

Means: Recursive-function notation denoting: one. This meaning is freshly authored from the complete recursive-functions source packet.

11 occurrences in this chapter

• Occurrence projected-formula-0014416
• Occurrence projected-formula-0014494
• Occurrence projected-formula-0014630
• Occurrence projected-formula-0014680
• Occurrence projected-formula-0014838
• Occurrence projected-formula-0014842
• Occurrence projected-formula-0014913
• Occurrence projected-formula-0014914
• Occurrence projected-formula-0014971
• Occurrence projected-formula-0014985
• Occurrence projected-formula-0015298

Equation form expr-6b881f36ea9d7e75

$\fn{nextPrime}(x) = \bmin{y \leq \fact{x}+1}{(y > x \land \fn{Prime}(y))}.$

Read as: the function next prime open parenthesis x close parenthesis equals the least y less than or equal to x factorial plus one such that open parenthesis y is greater than x and the primality predicate open parenthesis y close parenthesis close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the function next prime open parenthesis x close parenthesis equals the least y less than or equal to x factorial plus one such that open parenthesis y is greater than x and the primality predicate open parenthesis y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014950

Equation form expr-6bab81b22118df78

$\Char{R}(\vec x) = 1$

Read as: the characteristic function of R open parenthesis vector x close parenthesis equals one

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of R open parenthesis vector x close parenthesis equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014776

Equation form expr-6bc75720af9b6f3b

$h(e, x) = 0$

Read as: h open parenthesis e comma x close parenthesis equals zero

Means: A recursive-function relation or equation stating: h open parenthesis e comma x close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015277

Equation form expr-6d178a3ce74e984f

$R(\vec{x}, y)$

Read as: R open parenthesis vector x comma y close parenthesis

Means: Recursive-function notation denoting: R open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014870

Equation form expr-6db4fd387e47e209

$\fn{pred}(0) & = 0 \text{ and}\\ \fn{pred}(y+1) & = y.$

Read as: the predecessor function open parenthesis zero close parenthesis equals zero and next row the predecessor function open parenthesis y plus one close parenthesis equals y

Means: A source-ordered system, calculation, or case table stating: the predecessor function open parenthesis zero close parenthesis equals zero and next row the predecessor function open parenthesis y plus one close parenthesis equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014727

Equation form expr-6eeeedda664b846e

$(k+1)$

Read as: k plus one

Means: Recursive-function notation denoting: k plus one. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014550
• Occurrence projected-formula-0014662

Equation form expr-6fdf6ffa6e221abc

$i = k$

Read as: i equals k

Means: A recursive-function relation or equation stating: i equals k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015003

Equation form expr-70ae369529f710ea

$r(x,0) = 0$

Read as: r open parenthesis x comma zero close parenthesis equals zero

Means: A recursive-function relation or equation stating: r open parenthesis x comma zero close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015136

Equation form expr-70c99f9414ddaf50

$\fn{const}_n(x) = n$

Read as: const sub n of x equals n

Means: A recursive-function relation or equation stating: const sub n of x equals n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014714

Equation form expr-72d1fe772549df11

$\Proj{n}{i}$

Read as: P superscript n sub i

Means: A primitive-recursive construction or operator denoting: P superscript n sub i. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014488

Equation form expr-7348e247f28a28c7

$f_0,f_1,\dots$

Read as: f sub zero comma f sub one comma and so on

Means: Recursive-function notation denoting: f sub zero comma f sub one comma and so on. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015183

Equation form expr-73e8b28c2b36adf8

$p(0) = 2$

Read as: p open parenthesis zero close parenthesis equals two

Means: A recursive-function relation or equation stating: p open parenthesis zero close parenthesis equals two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014933

Equation form expr-7471edf760a11d0b

$\Proj{1}{0}$

Read as: the one place projection with index zero

Means: A primitive-recursive construction or operator denoting: the one place projection with index zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014676
• Occurrence projected-formula-0014713

Equation form expr-75c88e4446b4c231

$(s)_i$

Read as: open parenthesis s close parenthesis sub i

Means: Recursive-function notation denoting: open parenthesis s close parenthesis sub i. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015035

Equation form expr-76a11c131097cc8f

$g_0(\vec x)$

Read as: g sub zero open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: g sub zero open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014829
• Occurrence projected-formula-0015207

Equation form expr-7731c7052b278ab5

$h(\vec x, 1)$

Read as: h open parenthesis vector x comma one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis vector x comma one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014695
• Occurrence projected-formula-0014697
• Occurrence projected-formula-0014701

Equation form expr-773ba08a905e1f03

$x \cdot z = y$

Read as: x times z equals y

Means: A recursive-function relation or equation stating: x times z equals y. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014901
• Occurrence projected-formula-0014906

Equation form expr-7809ef9964aafac8

$x \leq y$

Read as: x is less than or equal to y

Means: A recursive-function relation or equation stating: x is less than or equal to y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014784

Equation form expr-785bbea9a2e3c724

$e_d$

Read as: e sub d

Means: Recursive-function notation denoting: e sub d. This meaning is freshly authored from the complete recursive-functions source packet.

4 occurrences in this chapter

• Occurrence projected-formula-0015295
• Occurrence projected-formula-0015306
• Occurrence projected-formula-0015311
• Occurrence projected-formula-0015314

Equation form expr-7879e6ffd79c577c

$g_0(x) & = & x+1 \\ g_{n + 1}(x) & = & g_n^x(x)$

Read as: g sub zero open parenthesis x close parenthesis equals x plus one next row g sub n plus one open parenthesis x close parenthesis equals g sub n superscript x open parenthesis x close parenthesis

Means: The two-row recursive definition sets g sub zero at x equal to x plus one, and sets g sub n plus one at x equal to the x-fold iterate of g sub n evaluated at x.

1 occurrence in this chapter

• Occurrence projected-formula-0015157

Equation form expr-7902699be42c8a8e

$7$

Read as: seven

Means: Recursive-function notation denoting: seven. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014929

Equation form expr-794d66ff4ebd363e

$x=0$

Read as: x equals zero

Means: A recursive-function relation or equation stating: x equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014374

Equation form expr-79d4c7f9c8579543

$\Nat$

Read as: the natural numbers

Means: Recursive-function notation denoting: the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014396
• Occurrence projected-formula-0014571

Equation form expr-7a3369f7710c1b8d

$z < 0$

Read as: z is less than zero

Means: A recursive-function relation or equation stating: z is less than zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014861
• Occurrence projected-formula-0014863

Equation form expr-7af3433babb00679

$R(\vec x)$

Read as: R open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: R open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014773
• Occurrence projected-formula-0014775

Equation form expr-7b8ba390a71f512b

$\Zero(x) = 0$

Read as: Zero of x equals zero

Means: A primitive-recursive construction or operator denoting: Zero of x equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014569

Equation form expr-7c35c5a1785d2070

$k-1$

Read as: k minus one

Means: Recursive-function notation denoting: k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014482

Equation form expr-7c54527fdada89e4

$\fn{pred}(y) = \begin{cases} 0 & \text{if $y=0$}\\ y-1 & \text{otherwise} \end{cases}$

Read as: the predecessor function open parenthesis y close parenthesis equals cases begin; row one: zero, if y equals zero; row two: y minus one, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the predecessor function open parenthesis y close parenthesis equals cases begin; row one: zero, if y equals zero; row two: y minus one, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014726

Equation form expr-7df5d15034517447

$\fn{pred}(y+1)$

Read as: the predecessor function open parenthesis y plus one close parenthesis

Means: Recursive-function notation denoting: the predecessor function open parenthesis y plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014730

Equation form expr-7f43b8fe74a47b59

$\emptyseq$

Read as: the empty sequence code

Means: A sequence- or tree-coding expression denoting: the empty sequence code. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014986

Equation form expr-7f47a818a507b08d

$e \in \Nat$

Read as: e is in the natural numbers

Means: A recursive-function relation or equation stating: e is in the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015270

Equation form expr-80ea33767c851351

$g(\vec x, y) \defis \sum_{z = 0}^y f(\vec x, z).$

Read as: g of vector x and y is defined as the sum, from z equals zero through y, of f of vector x and z

Means: A recursive-function relation or equation stating: g of vector x and y is defined as the sum, from z equals zero through y, of f of vector x and z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014769

Equation form expr-81a5c46a5de68012

$m_R(\vec{x}, y+1) = y+1$

Read as: m sub R of vector x and y plus one equals y plus one

Means: A recursive-function relation or equation stating: m sub R of vector x and y plus one equals y plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014874

Equation form expr-81c3b50d0de42ffc

$P(\vec x) \lif Q(\vec x)$

Read as: P open parenthesis vector x close parenthesis implies Q open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: P open parenthesis vector x close parenthesis implies Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014791

Equation form expr-8254c329a92850f6

$k$

Read as: k

Means: Recursive-function notation denoting: k. This meaning is freshly authored from the complete recursive-functions source packet.

14 occurrences in this chapter

• Occurrence projected-formula-0014468
• Occurrence projected-formula-0014471
• Occurrence projected-formula-0014544
• Occurrence projected-formula-0014554
• Occurrence projected-formula-0014557
• Occurrence projected-formula-0014576
• Occurrence projected-formula-0014584
• Occurrence projected-formula-0014651
• Occurrence projected-formula-0014664
• Occurrence projected-formula-0015040
• Occurrence projected-formula-0015055
• Occurrence projected-formula-0015058
• Occurrence projected-formula-0015076
• Occurrence projected-formula-0015126

Equation form expr-82d2d4a0ab550b46

$R(x,z)$

Read as: R open parenthesis x comma z close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014845

Equation form expr-8385116703b3fad7

$h(0) & = 1\\ h(x+1) & = 2 \cdot h(x)$

Read as: h open parenthesis zero close parenthesis equals one next row h open parenthesis x plus one close parenthesis equals two times h open parenthesis x close parenthesis

Means: A source-ordered system, calculation, or case table stating: h open parenthesis zero close parenthesis equals one next row h open parenthesis x plus one close parenthesis equals two times h open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014386

Equation form expr-838c9d5992ee25fd

$y \tsub x = y - x$

Read as: y truncated minus x equals y minus x

Means: A recursive-function relation or equation stating: y truncated minus x equals y minus x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014758

Equation form expr-839af1f5a31a76e3

$h(x_0, \dots, x_{k-1}, 0) & = f(x_0, \dots, x_{k-1})\\ h(x_0, \dots, x_{k-1}, y+1) & = g(x_0, \dots, x_{k-1}, y, h(x_0, \dots, x_{k-1}, y))$

Read as: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis equals f open parenthesis x sub zero comma and so on comma x sub k minus one close parenthesis next row h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis equals g open parenthesis x sub zero comma and so on comma x sub k minus one comma y comma h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: h open parenthesis x sub zero comma and so on comma x sub k minus one comma zero close parenthesis equals f open parenthesis x sub zero comma and so on comma x sub k minus one close parenthesis next row h open parenthesis x sub zero comma and so on comma x sub k minus one comma y plus one close parenthesis equals g open parenthesis x sub zero comma and so on comma x sub k minus one comma y comma h open parenthesis x sub zero comma and so on comma x sub k minus one comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014448

Equation form expr-8429f6938e150109

$\fn{id}(x) = x$

Read as: applied to x equals x

Means: A recursive-function relation or equation stating: applied to x equals x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014712

Equation form expr-84424f7a24d2a950

$x \tsub 0 & = x\\ x \tsub (y+1) & = \fn{pred}(x \tsub y)$

Read as: x truncated minus zero equals x next row x truncated minus open parenthesis y plus one close parenthesis equals the predecessor function open parenthesis x truncated minus y close parenthesis

Means: A source-ordered system, calculation, or case table stating: x truncated minus zero equals x next row x truncated minus open parenthesis y plus one close parenthesis equals the predecessor function open parenthesis x truncated minus y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014739

Equation form expr-866007381d5c48a8

$d_1$

Read as: d sub one

Means: Recursive-function notation denoting: d sub one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015077

Equation form expr-86aad91554a7b587

$G_{k-1}$

Read as: G sub k minus one

Means: Recursive-function notation denoting: G sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014658

Equation form expr-86b127802106e784

$g(g(\dots g(x)))$

Read as: g open parenthesis g open parenthesis and so on g open parenthesis x close parenthesis close parenthesis close parenthesis

Means: Recursive-function notation denoting: g open parenthesis g open parenthesis and so on g open parenthesis x close parenthesis close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015153

Equation form expr-86ed9d8929b7e282

$\fn{cond}(x,y,z)$

Read as: cond of x, y, and z

Means: Recursive-function notation denoting: cond of x, y, and z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014827

Equation form expr-86eee16f1f1eaa29

$\umin{x}{x \neq x}$

Read as: the least x such that x is not equal to x

Means: A partial-computation statement expressing: the least x such that x is not equal to x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015285

Equation form expr-8815c24f3342dceb

$h(\vec x, y) & = \begin{cases} g(\vec x, y, h(\vec x, k(\vec x, y))) & \text{if $k(\vec x, y) < y$} \\ f(\vec x) & \text{otherwise} \end{cases}$

Read as: h open parenthesis vector x comma y close parenthesis equals cases begin; row one: g open parenthesis vector x comma y comma h open parenthesis vector x comma k open parenthesis vector x comma y close parenthesis close parenthesis close parenthesis, if k open parenthesis vector x comma y close parenthesis is less than y; row two: f open parenthesis vector x close parenthesis, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: h open parenthesis vector x comma y close parenthesis equals cases begin; row one: g open parenthesis vector x comma y comma h open parenthesis vector x comma k open parenthesis vector x comma y close parenthesis close parenthesis close parenthesis, if k open parenthesis vector x comma y close parenthesis is less than y; row two: f open parenthesis vector x close parenthesis, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015122

Equation form expr-883b324e432edae4

$\Char{P}$

Read as: chi sub P

Means: A primitive-recursive relation or bounded-quantifier statement expressing: chi sub P. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014794

Equation form expr-88a51162b7219690

$(s)_0$

Read as: open parenthesis s close parenthesis sub zero

Means: Recursive-function notation denoting: open parenthesis s close parenthesis sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015042

Equation form expr-88b04d3080ebcab7

$\tuple{}(a_0, \dots, a_k) = \tuple{a_0, \dots, a_k}$

Read as: from a sub zero through a sub k to the natural-number code of that sequence

Means: The empty-tuple glyph is used here as the source's name for the tuple-coding map; applying it to the listed entries yields the natural-number code of that tuple.

1 occurrence in this chapter

• Occurrence projected-formula-0014991

Equation form expr-8951af8642d04083

$p_i \mid s$

Read as: p sub i divides s

Means: A recursive-function relation or equation stating: p sub i divides s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015002

Equation form expr-8982bca6d544bc0c

$d(x, y)$

Read as: d open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: d open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014979

Equation form expr-89859566ae631bdd

$\Mult(2,0) & = 0\\ \Mult(2,1) & = \Mult(2,0+1) = \Add(\Mult(2,0), 2) = \Add(0, 2) = 2\\ \Mult(2,2) & = \Mult(2,1+1) = \Add(\Mult(2,1), 2) = \Add(2, 2) = 4\\ \Mult(2,3) & = \Mult(2,2+1) = \Add(\Mult(2,2), 2) = \Add(4, 2) = 6$

Read as: the multiplication function open parenthesis two comma zero close parenthesis equals zero next row the multiplication function open parenthesis two comma one close parenthesis equals the multiplication function open parenthesis two comma zero plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma zero close parenthesis comma two close parenthesis equals the addition function open parenthesis zero comma two close parenthesis equals two next row the multiplication function open parenthesis two comma two close parenthesis equals the multiplication function open parenthesis two comma one plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma one close parenthesis comma two close parenthesis equals the addition function open parenthesis two comma two close parenthesis equals four next row the multiplication function open parenthesis two comma three close parenthesis equals the multiplication function open parenthesis two comma two plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma two close parenthesis comma two close parenthesis equals the addition function open parenthesis four comma two close parenthesis equals six

Means: A source-ordered system, calculation, or case table stating: the multiplication function open parenthesis two comma zero close parenthesis equals zero next row the multiplication function open parenthesis two comma one close parenthesis equals the multiplication function open parenthesis two comma zero plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma zero close parenthesis comma two close parenthesis equals the addition function open parenthesis zero comma two close parenthesis equals two next row the multiplication function open parenthesis two comma two close parenthesis equals the multiplication function open parenthesis two comma one plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma one close parenthesis comma two close parenthesis equals the addition function open parenthesis two comma two close parenthesis equals four next row the multiplication function open parenthesis two comma three close parenthesis equals the multiplication function open parenthesis two comma two plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis two comma two close parenthesis comma two close parenthesis equals the addition function open parenthesis four comma two close parenthesis equals six. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014435

Equation form expr-8ac3b91bd5510f4f

$\Char{P}(\vec x, 0) & = 1\\ \Char{P}(\vec x, y+1) & = \fn{min}(\Char{P}(\vec x, y), \Char{R}(\vec x, y)).$

Read as: the characteristic function of P open parenthesis vector x comma zero close parenthesis equals one next row the characteristic function of P open parenthesis vector x comma y plus one close parenthesis equals the minimum function open parenthesis the characteristic function of P open parenthesis vector x comma y close parenthesis comma the characteristic function of R open parenthesis vector x comma y close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: the characteristic function of P open parenthesis vector x comma zero close parenthesis equals one next row the characteristic function of P open parenthesis vector x comma y plus one close parenthesis equals the minimum function open parenthesis the characteristic function of P open parenthesis vector x comma y close parenthesis comma the characteristic function of R open parenthesis vector x comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014820

Equation form expr-8c2574892063f995

$R$

Read as: R

Means: Recursive-function notation denoting: R. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014997

Equation form expr-8ce37d0b21a90209

$R(\vec{x}, z)$

Read as: R open parenthesis vector x comma z close parenthesis

Means: Recursive-function notation denoting: R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

4 occurrences in this chapter

• Occurrence projected-formula-0014862
• Occurrence projected-formula-0014867
• Occurrence projected-formula-0014873
• Occurrence projected-formula-0014884

Equation form expr-8cf9f0ef3fca196b

$S_{i+1}$

Read as: S sub i plus one

Means: Recursive-function notation denoting: S sub i plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014597

Equation form expr-8dc6f75010fead63

$\fn{min}(\Char{P}(\vec x), \Char{Q}(\vec x))$

Read as: the minimum function open parenthesis the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the minimum function open parenthesis the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014803

Equation form expr-8e35c2cd3bf6641b

$q$

Read as: q

Means: Recursive-function notation denoting: q. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014973

Equation form expr-8eaed08a534580c6

$q \mid p+1$

Read as: q divides p plus one

Means: A recursive-function relation or equation stating: q divides p plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014961

Equation form expr-8efdfa96eef0dc09

$\Add(x, y)$

Read as: the addition function open parenthesis x comma y close parenthesis

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014398

Equation form expr-8f70a77bd6291ef5

$m_R$

Read as: m sub R

Means: Recursive-function notation denoting: m sub R. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014859

Equation form expr-90e35b98b5809a14

$\fn{cond}(x,y,z) & = \begin{cases} y & \text{if $x = 0$} \\ z & \text{otherwise}. \end{cases} \intertext{This is defined recursively by} \fn{cond}(0,y,z) & = y,\\ \fn{cond}(x+1,y,z) & = z.$

Read as: the conditional function open parenthesis x comma y comma z close parenthesis equals cases begin; row one: y, if x equals zero; row two: z, otherwise; cases end This is defined recursively by the conditional function open parenthesis zero comma y comma z close parenthesis equals y comma next row the conditional function open parenthesis x plus one comma y comma z close parenthesis equals z

Means: A source-ordered system, calculation, or case table stating: the conditional function open parenthesis x comma y comma z close parenthesis equals cases begin; row one: y, if x equals zero; row two: z, otherwise; cases end This is defined recursively by the conditional function open parenthesis zero comma y comma z close parenthesis equals y comma next row the conditional function open parenthesis x plus one comma y comma z close parenthesis equals z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014828

Equation form expr-919dbf7fb26c9783

$f(x_0) = \Succ(\Zero(x_0))$

Read as: f open parenthesis x sub zero close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis x sub zero close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: f open parenthesis x sub zero close parenthesis equals the successor function open parenthesis the constant zero function open parenthesis x sub zero close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014637

Equation form expr-91a8946193fdf882

$\langle 2, 7, 3\rangle$

Read as: the three-entry tuple two, seven, three

Means: A sequence- or tree-coding expression denoting: the three-entry tuple two, seven, three. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014982

Equation form expr-91d4137d9a3d2145

$q < p+1$

Read as: q is less than p plus one

Means: A recursive-function relation or equation stating: q is less than p plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014962

Equation form expr-92cea14920147b76

$\fn{Comp}_{1,3}[\Succ,\Proj{3}{2}]$

Read as: the composition operator with one inner function and three inputs, applied to the successor function and the three-place projection with index two

Means: A primitive-recursive construction or operator denoting: the composition operator with one inner function and three inputs, applied to the successor function and the three-place projection with index two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014679

Equation form expr-932414051cf3c7a0

$h(1) = h(0+1)$

Read as: h open parenthesis one close parenthesis equals h open parenthesis zero plus one close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis one close parenthesis equals h open parenthesis zero plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014375

Equation form expr-93777d281a5d3240

$R(i,s)$

Read as: R open parenthesis i comma s close parenthesis

Means: Recursive-function notation denoting: R open parenthesis i comma s close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015000
• Occurrence projected-formula-0015010

Equation form expr-93deebf22eb3ed81

$h(x,y) = f(\Proj{2}{0}(x,y), g(\Proj{2}{0}(x,y), \Proj{2}{0}(x,y), \Proj{2}{1}(x,y)), \Proj{2}{1}(x,y)).$

Read as: h open parenthesis x comma y close parenthesis equals f open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma g open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: h open parenthesis x comma y close parenthesis equals f open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma g open parenthesis the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index zero open parenthesis x comma y close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis comma the two place projection with index one open parenthesis x comma y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014531

Equation form expr-942b50f4e0bf3cad

$\fn{concat}$

Read as: the concatenate function

Means: Recursive-function notation denoting: the concatenate function. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015045
• Occurrence projected-formula-0015065

Equation form expr-94953145943a3013

$f(\vec x, z)$

Read as: f open parenthesis vector x comma z close parenthesis

Means: Recursive-function notation denoting: f open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014768
• Occurrence projected-formula-0014770

Equation form expr-94aea7d857cac4e6

$f(x_0) = x_0 = \Proj{1}{0}(x_0)$

Read as: f open parenthesis x sub zero close parenthesis equals x sub zero equals the one place projection with index zero open parenthesis x sub zero close parenthesis

Means: A primitive-recursive construction or operator denoting: f open parenthesis x sub zero close parenthesis equals x sub zero equals the one place projection with index zero open parenthesis x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014609

Equation form expr-94fa7dc15ec49291

$f(0,\vec z), f(1,\vec z), f(2,\vec z)$

Read as: f open parenthesis zero comma vector z close parenthesis comma f open parenthesis one comma vector z close parenthesis comma f open parenthesis two comma vector z close parenthesis

Means: Recursive-function notation denoting: f open parenthesis zero comma vector z close parenthesis comma f open parenthesis one comma vector z close parenthesis comma f open parenthesis two comma vector z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015220

Equation form expr-968a6035541b05d2

$x \nmid y$

Read as: x does not divide y

Means: A recursive-function relation or equation stating: x does not divide y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014898

Equation form expr-971387061bac6803

$m_R(\vec{x}, y+1) = m_R(\vec{x}, y)$

Read as: m sub R open parenthesis vector x comma y plus one close parenthesis equals m sub R open parenthesis vector x comma y close parenthesis

Means: A recursive-function relation or equation stating: m sub R open parenthesis vector x comma y plus one close parenthesis equals m sub R open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014868

Equation form expr-973975dfb53609aa

$g_m(\vec x)$

Read as: g sub m open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: g sub m open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014830

Equation form expr-97952e1958b54679

$R(x, 1)$

Read as: R open parenthesis x comma one close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014851

Equation form expr-979a2739166a2174

$\Char{Q}$

Read as: chi sub Q

Means: A primitive-recursive relation or bounded-quantifier statement expressing: chi sub Q. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014795

Equation form expr-97b37ee44274a337

$y=x$

Read as: y equals x

Means: A recursive-function relation or equation stating: y equals x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014917

Equation form expr-98032d3a5157a88e

$\fn{nextPrime}(x)$

Read as: the function next prime open parenthesis x close parenthesis

Means: Recursive-function notation denoting: the function next prime open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014944
• Occurrence projected-formula-0014951

Equation form expr-98174b73dbda257c

$d(x, y) = \lfloor x/y \rfloor$

Read as: d of x and y equals the floor of x divided by y

Means: A recursive-function relation or equation stating: d of x and y equals the floor of x divided by y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014764

Equation form expr-982d20d1e491c9e1

$f\colon \Nat \to \Nat$

Read as: f from the natural numbers to the natural numbers

Means: Recursive-function notation denoting: f from the natural numbers to the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015269

Equation form expr-98550483d35bb33d

$f_2$

Read as: f sub two

Means: Recursive-function notation denoting: f sub two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015140

Equation form expr-98c0252df9654e57

$\tsub$

Read as: truncated minus

Means: Recursive-function notation denoting: truncated minus. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014745
• Occurrence projected-formula-0014751

Equation form expr-98f0455530b5b3d8

$g_{k-1}$

Read as: g sub k minus one

Means: Recursive-function notation denoting: g sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

7 occurrences in this chapter

• Occurrence projected-formula-0014470
• Occurrence projected-formula-0014520
• Occurrence projected-formula-0014556
• Occurrence projected-formula-0014561
• Occurrence projected-formula-0014578
• Occurrence projected-formula-0014582
• Occurrence projected-formula-0014655

Equation form expr-993dfacb9c1d5689

$g(x,y)$

Read as: g open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: g open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

6 occurrences in this chapter

• Occurrence projected-formula-0015143
• Occurrence projected-formula-0015174
• Occurrence projected-formula-0015177
• Occurrence projected-formula-0015178
• Occurrence projected-formula-0015181
• Occurrence projected-formula-0015182

Equation form expr-9ae4d3d3a50c4b39

$\fn{fac}(0) &= 1\\ \fn{fac}(y+1) & = \fn{fac}(y) \cdot (y+1). \intertext{Officially, we have to first define a two-place function $h$} h(x, 0) & = \fn{const}_1(x)\\ h(x, y+1) & = g(x, y, h(x, y)) \intertext{where $g(x, y, z) = \Mult(\Proj{3}{2}(x, y, z), \Succ(\Proj{3}{1}(x, y, z)))$ and then let} \fn{fac}(y) & = h(\Proj{1}{0}(y), \Proj{1}{0}(y)) = h(y,y).$

Read as: the factorial function open parenthesis zero close parenthesis equals one next row the factorial function open parenthesis y plus one close parenthesis equals the factorial function open parenthesis y close parenthesis times open parenthesis y plus one close parenthesis Officially comma we have to first define a two-place function h. Next equation. h open parenthesis x comma zero close parenthesis equals the constant function sub one open parenthesis x close parenthesis next row h open parenthesis x comma y plus one close parenthesis equals g open parenthesis x comma y comma h open parenthesis x comma y close parenthesis close parenthesis where g open parenthesis x comma y comma z close parenthesis equals the multiplication function open parenthesis the three place projection with index two open parenthesis x comma y comma z close parenthesis comma the successor function open parenthesis the three place projection with index one open parenthesis x comma y comma z close parenthesis close parenthesis close parenthesis and then let the factorial function open parenthesis y close parenthesis equals h open parenthesis the one place projection with index zero open parenthesis y close parenthesis comma the one place projection with index zero open parenthesis y close parenthesis close parenthesis equals h open parenthesis y comma y close parenthesis

Means: A source-ordered system, calculation, or case table stating: the factorial function open parenthesis zero close parenthesis equals one next row the factorial function open parenthesis y plus one close parenthesis equals the factorial function open parenthesis y close parenthesis times open parenthesis y plus one close parenthesis Officially comma we have to first define a two-place function h. Next equation. h open parenthesis x comma zero close parenthesis equals the constant function sub one open parenthesis x close parenthesis next row h open parenthesis x comma y plus one close parenthesis equals g open parenthesis x comma y comma h open parenthesis x comma y close parenthesis close parenthesis where g open parenthesis x comma y comma z close parenthesis equals the multiplication function open parenthesis the three place projection with index two open parenthesis x comma y comma z close parenthesis comma the successor function open parenthesis the three place projection with index one open parenthesis x comma y comma z close parenthesis close parenthesis close parenthesis and then let the factorial function open parenthesis y close parenthesis equals h open parenthesis the one place projection with index zero open parenthesis y close parenthesis comma the one place projection with index zero open parenthesis y close parenthesis close parenthesis equals h open parenthesis y comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014736

Equation form expr-9b19467654aed1a2

$k=1$

Read as: k equals one

Means: A recursive-function relation or equation stating: k equals one. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014450
• Occurrence projected-formula-0014496

Equation form expr-9b96421cae19455c

$h(e_d, e_d)$

Read as: h open parenthesis e sub d comma e sub d close parenthesis

Means: Recursive-function notation denoting: h open parenthesis e sub d comma e sub d close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015296

Equation form expr-9c12f517727ede14

$\fn{append}$

Read as: the function append

Means: A sequence- or tree-coding expression denoting: the function append. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015017

Equation form expr-9e9e388296cda0af

$\cfind{e}(x) \fundefined$

Read as: partial recursive function phi sub e open parenthesis x close parenthesis is undefined

Means: A partial-computation statement expressing: partial recursive function phi sub e open parenthesis x close parenthesis is undefined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015278

Equation form expr-9ee230f22d8cf57f

$x \mid y$

Read as: x divides y

Means: A recursive-function relation or equation stating: x divides y. This meaning is freshly authored from the complete recursive-functions source packet.

6 occurrences in this chapter

• Occurrence projected-formula-0014889
• Occurrence projected-formula-0014890
• Occurrence projected-formula-0014899
• Occurrence projected-formula-0014904
• Occurrence projected-formula-0014907
• Occurrence projected-formula-0014910

Equation form expr-9f694f9407126df0

$g_2(x)$

Read as: g sub two open parenthesis x close parenthesis

Means: Recursive-function notation denoting: g sub two open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015161

Equation form expr-a079646eb13e84be

$f(x) = 2 \cdot x$

Read as: f open parenthesis x close parenthesis equals two times x

Means: A recursive-function relation or equation stating: f open parenthesis x close parenthesis equals two times x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014717

Equation form expr-a07a48d217aa4849

$\fn{hSubtreeSeq}(t, t)$

Read as: the height-subtree-sequence function open parenthesis t comma t close parenthesis

Means: Recursive-function notation denoting: the height-subtree-sequence function open parenthesis t comma t close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015110

Equation form expr-a09012b9ba076c9d

$h(\vec x, y) = g(\vec x, y, \tuple{h(\vec x, 0), \dots, h(\vec x, y-1)})$

Read as: h of vector x and y equals g applied to vector x, y, and the code of the sequence of earlier values h of vector x and zero through h of vector x and y minus one

Means: A sequence- or tree-coding expression denoting: h of vector x and y equals g applied to vector x, y, and the code of the sequence of earlier values h of vector x and zero through h of vector x and y minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015117

Equation form expr-a0bcbfe80a5958dc

$\fn{fac}(x) = \fact{x} = 1 \cdot 2 \cdot 3 \cdot \dots \cdot x$

Read as: the function factorial open parenthesis x close parenthesis equals x factorial equals one times two times three times and so on times x

Means: The equality chain states that the factorial function at x equals x factorial and also equals the product one times two times three and so on through x.

1 occurrence in this chapter

• Occurrence projected-formula-0014735

Equation form expr-a13c77e678b707fe

$x + (y \tsub x) = x + (y - x) = y$

Read as: x plus open parenthesis y truncated minus x close parenthesis equals x plus open parenthesis y minus x close parenthesis equals y

Means: A recursive-function relation or equation stating: x plus open parenthesis y truncated minus x close parenthesis equals x plus open parenthesis y minus x close parenthesis equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014759

Equation form expr-a1fce4363854ff88

$y$

Read as: y

Means: Recursive-function notation denoting: y. This meaning is freshly authored from the complete recursive-functions source packet.

29 occurrences in this chapter

• Occurrence projected-formula-0014402
• Occurrence projected-formula-0014407
• Occurrence projected-formula-0014421
• Occurrence projected-formula-0014429
• Occurrence projected-formula-0014443
• Occurrence projected-formula-0014741
• Occurrence projected-formula-0014747
• Occurrence projected-formula-0014757
• Occurrence projected-formula-0014761
• Occurrence projected-formula-0014809
• Occurrence projected-formula-0014812
• Occurrence projected-formula-0014847
• Occurrence projected-formula-0014856
• Occurrence projected-formula-0014858
• Occurrence projected-formula-0014883
• Occurrence projected-formula-0014888
• Occurrence projected-formula-0014891
• Occurrence projected-formula-0014893
• Occurrence projected-formula-0014896
• Occurrence projected-formula-0014945
• Occurrence projected-formula-0014947
• Occurrence projected-formula-0015119
• Occurrence projected-formula-0015124
• Occurrence projected-formula-0015129
• Occurrence projected-formula-0015132
• Occurrence projected-formula-0015142
• Occurrence projected-formula-0015180
• Occurrence projected-formula-0015185
• Occurrence projected-formula-0015290

Equation form expr-a20f0ea99bc9b1dc

$\Add(x,0) & = x\\ \Add(x,y+1) & = \Add(x,y)+1$

Read as: the addition function open parenthesis x comma zero close parenthesis equals x next row the addition function open parenthesis x comma y plus one close parenthesis equals the addition function open parenthesis x comma y close parenthesis plus one

Means: A source-ordered system, calculation, or case table stating: the addition function open parenthesis x comma zero close parenthesis equals x next row the addition function open parenthesis x comma y plus one close parenthesis equals the addition function open parenthesis x comma y close parenthesis plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014404

Equation form expr-a2277e0b98ac28a5

$g(x)$

Read as: g open parenthesis x close parenthesis

Means: Recursive-function notation denoting: g open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015199

Equation form expr-a269f7e0d4a25436

$n=1$

Read as: n equals one

Means: A recursive-function relation or equation stating: n equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014504

Equation form expr-a318c24216defe20

$+$

Read as: plus

Means: Recursive-function notation denoting: plus. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014415
• Occurrence projected-formula-0014744
• Occurrence projected-formula-0014750

Equation form expr-a33a76c1c0ac39d4

$\mu x \; (1 \tsub \Char{R}(x,\vec z))$

Read as: the least x such that relation R holds of x and vector z

Means: A partial-computation statement expressing: the least x such that relation R holds of x and vector z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015224

Equation form expr-a35eda3a0c6a247d

$h'(x_0, 0) & = f(x_0) = 1 \\ h'(x_0, y+1) & = g(x_0, y, h'(x_0, y)) = 2 \cdot h'(x_0, y).$

Read as: h prime open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals one next row h prime open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma h prime open parenthesis x sub zero comma y close parenthesis close parenthesis equals two times h prime open parenthesis x sub zero comma y close parenthesis

Means: A source-ordered system, calculation, or case table stating: h prime open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals one next row h prime open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma h prime open parenthesis x sub zero comma y close parenthesis close parenthesis equals two times h prime open parenthesis x sub zero comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014633

Equation form expr-a59c27c9ee664da4

$h(e_d, e_d) \neq 1$

Read as: h open parenthesis e sub d comma e sub d close parenthesis is not equal to one

Means: Recursive-function notation denoting: h open parenthesis e sub d comma e sub d close parenthesis is not equal to one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015304

Equation form expr-a7501dff9eccca9f

$a_k \ge 1$

Read as: a sub k is greater than or equal to one

Means: A recursive-function relation or equation stating: a sub k is greater than or equal to one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014990

Equation form expr-a75dbfeaae9c1c65

$f_0$

Read as: f sub zero

Means: Recursive-function notation denoting: f sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015138

Equation form expr-a7711d2e52daeb33

$\tuple{2, \tuple{0, l_2}, \tuple{0, l_3}, l_1}$

Read as: the code of the tree sequence two, the code of the leaf labelled l sub two, the code of the leaf labelled l sub three, and the root label l sub one

Means: A sequence- or tree-coding expression denoting: the code of the tree sequence two, the code of the leaf labelled l sub two, the code of the leaf labelled l sub three, and the root label l sub one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015084

Equation form expr-a7993d06483a098c

$i < n$

Read as: i is less than n

Means: A recursive-function relation or equation stating: i is less than n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014568

Equation form expr-a7a88239cad25c82

$\bexists{x \leq y}{\dots x\dots}$

Read as: there exists x less than or equal to y such that the displayed surrounding condition holds at x

Means: A primitive-recursive relation or bounded-quantifier statement expressing: there exists x less than or equal to y such that the displayed surrounding condition holds at x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014823

Equation form expr-a80d58bc5c86f001

$\fn{hconcat}(s,t,n)$

Read as: the function helper concatenate open parenthesis s comma t comma n close parenthesis

Means: Recursive-function notation denoting: the function helper concatenate open parenthesis s comma t comma n close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015047

Equation form expr-a94ee8c72b6b483b

$\fn{tail}(\emptyseq) & = 0 \text{ and}\\ \fn{tail}(\tuple{s_0, \dots, s_{k}}) & = \tuple{s_1, \dots, s_{k}}.$

Read as: tail of the empty-sequence code equals zero; and tail of the code for sequence s sub zero through s sub k equals the code for sequence s sub one through s sub k

Means: A source-ordered system, calculation, or case table stating: tail of the empty-sequence code equals zero; and tail of the code for sequence s sub zero through s sub k equals the code for sequence s sub one through s sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015070

Equation form expr-aa0944382fc03cf0

$\langle a_0, a_1, a_2, \dots, a_k \rangle$

Read as: the tuple a sub zero, a sub one, a sub two, and so on through a sub k

Means: A sequence- or tree-coding expression denoting: the tuple a sub zero, a sub one, a sub two, and so on through a sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014980

Equation form expr-aaa9402664f1a41f

$h$

Read as: h

Means: Recursive-function notation denoting: h. This meaning is freshly authored from the complete recursive-functions source packet.

37 occurrences in this chapter

• Occurrence projected-formula-0014367
• Occurrence projected-formula-0014372
• Occurrence projected-formula-0014388
• Occurrence projected-formula-0014390
• Occurrence projected-formula-0014391
• Occurrence projected-formula-0014392
• Occurrence projected-formula-0014393
• Occurrence projected-formula-0014438
• Occurrence projected-formula-0014444
• Occurrence projected-formula-0014447
• Occurrence projected-formula-0014474
• Occurrence projected-formula-0014478
• Occurrence projected-formula-0014527
• Occurrence projected-formula-0014530
• Occurrence projected-formula-0014532
• Occurrence projected-formula-0014551
• Occurrence projected-formula-0014563
• Occurrence projected-formula-0014645
• Occurrence projected-formula-0014646
• Occurrence projected-formula-0014650
• Occurrence projected-formula-0014660
• Occurrence projected-formula-0014663
• Occurrence projected-formula-0014672
• Occurrence projected-formula-0014686
• Occurrence projected-formula-0015121
• Occurrence projected-formula-0015123
• Occurrence projected-formula-0015125
• Occurrence projected-formula-0015147
• Occurrence projected-formula-0015150
• Occurrence projected-formula-0015200
• Occurrence projected-formula-0015206
• Occurrence projected-formula-0015275
• Occurrence projected-formula-0015280
• Occurrence projected-formula-0015281
• Occurrence projected-formula-0015293
• Occurrence projected-formula-0015312
• Occurrence projected-formula-0015315

Equation form expr-ab8a75907c76983d

$p = p_0\cdot p_1 \cdot \dots \cdot p_n$

Read as: p equals p sub zero times p sub one times and so on times p sub n

Means: A recursive-function relation or equation stating: p equals p sub zero times p sub one times and so on times p sub n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014955

Equation form expr-abaff34ef46989e0

$x = z\times y + r(x,y)$

Read as: x equals z times y plus r open parenthesis x comma y close parenthesis

Means: A recursive-function relation or equation stating: x equals z times y plus r open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015133

Equation form expr-acac86c0e609ca90

$l$

Read as: l

Means: Recursive-function notation denoting: l. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014535
• Occurrence projected-formula-0014538
• Occurrence projected-formula-0015079

Equation form expr-ad294477c6ade43d

$h(x_0, \dots, x_{n-1}) = f(y_0, \dots, y_{k-1})$

Read as: h of x sub zero through x sub n minus one equals f of y sub zero through y sub k minus one

Means: A recursive-function relation or equation stating: h of x sub zero through x sub n minus one equals f of y sub zero through y sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014484

Equation form expr-ae109cb4887930a7

$\fn{ISubtrees}(t) = \fn{subseq}(t, 1, (t)_0)$

Read as: the function immediate subtrees open parenthesis t close parenthesis equals the subsequence function open parenthesis t comma one comma open parenthesis t close parenthesis sub zero close parenthesis

Means: A recursive-function relation or equation stating: the function immediate subtrees open parenthesis t close parenthesis equals the subsequence function open parenthesis t comma one comma open parenthesis t close parenthesis sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015087

Equation form expr-ae3fa5fe5bb8b35e

$\tuple{n_1, \dots, n_k}$

Read as: n sub one through n sub k

Means: A sequence- or tree-coding expression denoting: n sub one through n sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014359

Equation form expr-ae8614f81f3eab91

$\fn{Rec}_1[\Proj{1}{0},\fn{Comp}_{1,3}[\Succ,\Proj{3}{2}]].$

Read as: the one-place recursion operator applied to the one-place projection with index zero and to the composition of successor with the three-place projection with index two

Means: A primitive-recursive construction or operator denoting: the one-place recursion operator applied to the one-place projection with index zero and to the composition of successor with the three-place projection with index two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014684

Equation form expr-ae98c7b6a7e4af07

$> x$

Read as: greater than x

Means: A recursive-function relation or equation stating: greater than x. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014974
• Occurrence projected-formula-0014977

Equation form expr-aed54b42651f1880

$\Char{\fn{IsZero}}$

Read as: chi for the zero-test relation

Means: A primitive-recursive relation or bounded-quantifier statement expressing: chi for the zero-test relation. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014780

Equation form expr-aedc7f9854e15b6d

$y \le x$

Read as: y is less than or equal to x

Means: A recursive-function relation or equation stating: y is less than or equal to x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014920

Equation form expr-af6ff7a7bda75f24

$h(1)$

Read as: h open parenthesis one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014379

Equation form expr-afaab97eb7e01d41

$f(x_0) = 0$

Read as: f open parenthesis x sub zero close parenthesis equals zero

Means: A recursive-function relation or equation stating: f open parenthesis x sub zero close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014456

Equation form expr-afd975476ac29b85

$x = y$

Read as: x equals y

Means: A recursive-function relation or equation stating: x equals y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014782

Equation form expr-b07b76bcffde3315

$h(\vec x, y)$

Read as: h open parenthesis vector x comma y close parenthesis

Means: Recursive-function notation denoting: h open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014699
• Occurrence projected-formula-0014702
• Occurrence projected-formula-0015115

Equation form expr-b0c752611f8bcf99

$p_i^{a+1} \mid s$

Read as: p sub i superscript a plus one divides s

Means: A recursive-function relation or equation stating: p sub i superscript a plus one divides s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015032

Equation form expr-b0d426d00081371f

$\vec z$

Read as: vector z

Means: Recursive-function notation denoting: vector z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015317

Equation form expr-b18954ba2cf5c193

$\Char{P \land Q}(\vec x) = \begin{cases} 1 & \text{if $\Char{P}(\vec x) = \Char{Q}(\vec x) = 1$}\\ 0 & \text{otherwise} \end{cases}$

Read as: the characteristic function of P and Q open parenthesis vector x close parenthesis equals cases begin; row one: one, if the characteristic function of P open parenthesis vector x close parenthesis equals the characteristic function of Q open parenthesis vector x close parenthesis equals one; row two: zero, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: the characteristic function of P and Q open parenthesis vector x close parenthesis equals cases begin; row one: one, if the characteristic function of P open parenthesis vector x close parenthesis equals the characteristic function of Q open parenthesis vector x close parenthesis equals one; row two: zero, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014800

Equation form expr-b1ebfd534777e14f

$R(x, 0)$

Read as: R open parenthesis x comma zero close parenthesis

Means: Recursive-function notation denoting: R open parenthesis x comma zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014850

Equation form expr-b2a62ddd97c4a290

$\Proj{n}{i}(x_0,\dots,x_{n-1}) = x_i,$

Read as: the n place projection with index i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals x sub i

Means: A primitive-recursive construction or operator denoting: the n place projection with index i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis equals x sub i. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014566

Equation form expr-b2dbb62440facda5

$\bforall{z < 0}{R(\vec x, z)}$

Read as: for every z less than zero, R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: for every z less than zero, R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014814

Equation form expr-b595b214b5d4865e

$2x$

Read as: two times x

Means: Recursive-function notation denoting: two times x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015160

Equation form expr-b60126f0f6cfe15f

$g(x, y, z) = \Succ(z)$

Read as: g open parenthesis x comma y comma z close parenthesis equals the successor function open parenthesis z close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x comma y comma z close parenthesis equals the successor function open parenthesis z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014485

Equation form expr-b63914d2aa2ce5fd

$g_0,g_1,\dots$

Read as: g sub zero comma g sub one comma and so on

Means: Recursive-function notation denoting: g sub zero comma g sub one comma and so on. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015156

Equation form expr-b7358701e22a7a6c

$y = 0$

Read as: y equals zero

Means: A recursive-function relation or equation stating: y equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014765

Equation form expr-b77483d29eb72dc8

$f(x) = \Mult(\fn{const}_2(x), \Proj{1}{0}(x))$

Read as: f open parenthesis x close parenthesis equals the multiplication function open parenthesis the constant function sub two open parenthesis x close parenthesis comma the one place projection with index zero open parenthesis x close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: f open parenthesis x close parenthesis equals the multiplication function open parenthesis the constant function sub two open parenthesis x close parenthesis comma the one place projection with index zero open parenthesis x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014719

Equation form expr-b7cdd52ed0c34518

$\Mult(x,0) & = 0 \\ \Mult(x,y+1) & = \Add(\Mult(x,y),x)$

Read as: the multiplication function open parenthesis x comma zero close parenthesis equals zero next row the multiplication function open parenthesis x comma y plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis x comma y close parenthesis comma x close parenthesis

Means: A source-ordered system, calculation, or case table stating: the multiplication function open parenthesis x comma zero close parenthesis equals zero next row the multiplication function open parenthesis x comma y plus one close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis x comma y close parenthesis comma x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014422

Equation form expr-b9711ee2915de9c0

$\Add$

Read as: the addition function

Means: A primitive-recursive construction or operator denoting: the addition function. This meaning is freshly authored from the complete recursive-functions source packet.

10 occurrences in this chapter

• Occurrence projected-formula-0014405
• Occurrence projected-formula-0014414
• Occurrence projected-formula-0014424
• Occurrence projected-formula-0014449
• Occurrence projected-formula-0014486
• Occurrence projected-formula-0014506
• Occurrence projected-formula-0014600
• Occurrence projected-formula-0014602
• Occurrence projected-formula-0014606
• Occurrence projected-formula-0014710

Equation form expr-bab3121cf95a2287

$n = p_0^{a_0} \cdot p_1^{a_1} \cdot \dots \cdot p_k^{a_k}$

Read as: n equals p sub zero superscript a sub zero times p sub one superscript a sub one times and so on times p sub k superscript a sub k

Means: A recursive-function relation or equation stating: n equals p sub zero superscript a sub zero times p sub one superscript a sub one times and so on times p sub k superscript a sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014989

Equation form expr-bac52f5ad56b34af

$f(x, y) = 2^{(2^{\iddots^{2^{x}}})}\raisebox{1ex}{\bigg\rbrace} \raisebox{1ex}{\text {$y$ $2$'s}}$

Read as: f of x and y equals an exponent tower containing y copies of two, with x as the top exponent

Means: A primitive-recursive construction or operator denoting: f of x and y equals an exponent tower containing y copies of two, with x as the top exponent. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014763

Equation form expr-bc3b5864a25454f5

$x = 0$

Read as: x equals zero

Means: A recursive-function relation or equation stating: x equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014779

Equation form expr-bc5f18eacb554029

$p_0=2$

Read as: p sub zero equals two

Means: A recursive-function relation or equation stating: p sub zero equals two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014938

Equation form expr-bc6e68768f4df913

$\fn{pred}'(x, y)$

Read as: the predecessor function prime open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: the predecessor function prime open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014731

Equation form expr-bd11fad62d2c9dfc

$(k+2)$

Read as: k plus two

Means: Recursive-function notation denoting: k plus two. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014547
• Occurrence projected-formula-0014666

Equation form expr-bd522885def92f18

$\mu x \; (f(x,\vec z) = 0)$

Read as: the least x such that open parenthesis f open parenthesis x comma vector z close parenthesis equals zero close parenthesis

Means: A partial-computation statement expressing: the least x such that open parenthesis f open parenthesis x comma vector z close parenthesis equals zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015230

Equation form expr-be6153780ebdd5b1

$\Add(2, 3) = 4 + 1 = 5$

Read as: the addition function open parenthesis two comma three close parenthesis equals four plus one equals five

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis two comma three close parenthesis equals four plus one equals five. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014413

Equation form expr-bf173953a9e841d7

$g_1$

Read as: g sub one

Means: Recursive-function notation denoting: g sub one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014516

Equation form expr-bf57d113efc4ce1a

$p(2) = 5$

Read as: p open parenthesis two close parenthesis equals five

Means: A recursive-function relation or equation stating: p open parenthesis two close parenthesis equals five. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014935

Equation form expr-bff58268aa6eeedc

$\mu x \; f(x,\vec z)$

Read as: the least x such that f of x and vector z equals zero and f is defined at every earlier input

Means: A partial-computation statement expressing: the least x such that f of x and vector z equals zero and f is defined at every earlier input. This meaning is freshly authored from the complete recursive-functions source packet.

6 occurrences in this chapter

• Occurrence projected-formula-0015212
• Occurrence projected-formula-0015217
• Occurrence projected-formula-0015218
• Occurrence projected-formula-0015219
• Occurrence projected-formula-0015221
• Occurrence projected-formula-0015229

Equation form expr-c1533de8cf0e95f7

$h(0)$

Read as: h open parenthesis zero close parenthesis

Means: Recursive-function notation denoting: h open parenthesis zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014371
• Occurrence projected-formula-0014376

Equation form expr-c165821f92f9541b

$g(\vec x, 0) & = f(\vec x, 0)\\ g(\vec x, y+1) & = g(\vec x, y) + f(\vec x, y+1).$

Read as: g open parenthesis vector x comma zero close parenthesis equals f open parenthesis vector x comma zero close parenthesis next row g open parenthesis vector x comma y plus one close parenthesis equals g open parenthesis vector x comma y close parenthesis plus f open parenthesis vector x comma y plus one close parenthesis

Means: A source-ordered system, calculation, or case table stating: g open parenthesis vector x comma zero close parenthesis equals f open parenthesis vector x comma zero close parenthesis next row g open parenthesis vector x comma y plus one close parenthesis equals g open parenthesis vector x comma y close parenthesis plus f open parenthesis vector x comma y plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014772

Equation form expr-c1c366f26da43334

$g_k$

Read as: g sub k

Means: Recursive-function notation denoting: g sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015202

Equation form expr-c2a54fdf1c4380b9

$\Proj{2}{1}$

Read as: the two place projection with index one

Means: A primitive-recursive construction or operator denoting: the two place projection with index one. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014517
• Occurrence projected-formula-0014536
• Occurrence projected-formula-0014542

Equation form expr-c489f23430817d0e

$g(x_0, y, z) & = g'(\Proj{3}{2}(x_0, y, z)) \intertext{and $g'$ in turn can be defined by composition as} g'(z) & = \Mult(g''(z), \Proj{1}{0}(z)) \intertext{and} g''(z) & = \Succ(f(z)),$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis equals g prime open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis and g prime in turn can be defined by composition as g prime open parenthesis z close parenthesis equals the multiplication function open parenthesis g prime prime open parenthesis z close parenthesis comma the one place projection with index zero open parenthesis z close parenthesis close parenthesis and g prime prime open parenthesis z close parenthesis equals the successor function open parenthesis f open parenthesis z close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x sub zero comma y comma z close parenthesis equals g prime open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis and g prime in turn can be defined by composition as g prime open parenthesis z close parenthesis equals the multiplication function open parenthesis g prime prime open parenthesis z close parenthesis comma the one place projection with index zero open parenthesis z close parenthesis close parenthesis and g prime prime open parenthesis z close parenthesis equals the successor function open parenthesis f open parenthesis z close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014640

Equation form expr-c49641d768f5b8cb

$y=0$

Read as: y equals zero

Means: A recursive-function relation or equation stating: y equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014400
• Occurrence projected-formula-0015135

Equation form expr-c5824a5708aa1bd9

$h(2) = h(1+1)$

Read as: h open parenthesis two close parenthesis equals h open parenthesis one plus one close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis two close parenthesis equals h open parenthesis one plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014378

Equation form expr-c59dc4e44ff99288

$+1$

Read as: plus one

Means: Recursive-function notation denoting: plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014363

Equation form expr-c65e342838690a21

$y \mid x$

Read as: y divides x

Means: A recursive-function relation or equation stating: y divides x. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014915
• Occurrence projected-formula-0014919

Equation form expr-c80904897ed65f23

$\tuple{s_0, \dots, s_k}$

Read as: the code of the sequence s sub zero through s sub k

Means: A sequence- or tree-coding expression denoting: the code of the sequence s sub zero through s sub k. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015037

Equation form expr-c8882bc19dd1a481

$\fn{max}(x,y)$

Read as: the maximum function open parenthesis x comma y close parenthesis

Means: Recursive-function notation denoting: the maximum function open parenthesis x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014748
• Occurrence projected-formula-0014749

Equation form expr-c8f5191d2c850bdc

$m_R(\vec x, y) \neq y$

Read as: m sub R open parenthesis vector x comma y close parenthesis is not equal to y

Means: Recursive-function notation denoting: m sub R open parenthesis vector x comma y close parenthesis is not equal to y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014879

Equation form expr-c8fe935309f027bc

$f(x,\vec z)$

Read as: f open parenthesis x comma vector z close parenthesis

Means: Recursive-function notation denoting: f open parenthesis x comma vector z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

4 occurrences in this chapter

• Occurrence projected-formula-0015211
• Occurrence projected-formula-0015228
• Occurrence projected-formula-0015231
• Occurrence projected-formula-0015316

Equation form expr-c97dd470afa92249

$h(x,y) = f(x,g(x,x,y),y).$

Read as: h open parenthesis x comma y close parenthesis equals f open parenthesis x comma g open parenthesis x comma x comma y close parenthesis comma y close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis x comma y close parenthesis equals f open parenthesis x comma g open parenthesis x comma x comma y close parenthesis comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014529

Equation form expr-ca3700e68890701c

$p_i^{a+1}$

Read as: p sub i superscript a plus one

Means: Recursive-function notation denoting: p sub i superscript a plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015029

Equation form expr-ca84fb5032e1e543

$f(\vec x) = \begin{cases} g_0(\vec x) & \text{if $R_0(\vec{x})$} \\ g_1(\vec x) & \text{if $R_1(\vec{x})$ and not $R_0(\vec{x})$} \\ \vdots & \\ g_{m-1}(\vec x) & \text{if $R_{m-1}(\vec{x})$ and none of the previous hold} \\ g_m(\vec x) & \mbox{otherwise} \end{cases}$

Read as: f open parenthesis vector x close parenthesis equals cases begin; row one: g sub zero open parenthesis vector x close parenthesis, if R sub zero open parenthesis vector x close parenthesis; row two: g sub one open parenthesis vector x close parenthesis, if R sub one open parenthesis vector x close parenthesis and not R sub zero open parenthesis vector x close parenthesis; row three: and so on vertically; row four: g sub m minus one open parenthesis vector x close parenthesis, if R sub m minus one open parenthesis vector x close parenthesis and none of the previous hold; row five: g sub m open parenthesis vector x close parenthesis, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: f open parenthesis vector x close parenthesis equals cases begin; row one: g sub zero open parenthesis vector x close parenthesis, if R sub zero open parenthesis vector x close parenthesis; row two: g sub one open parenthesis vector x close parenthesis, if R sub one open parenthesis vector x close parenthesis and not R sub zero open parenthesis vector x close parenthesis; row three: and so on vertically; row four: g sub m minus one open parenthesis vector x close parenthesis, if R sub m minus one open parenthesis vector x close parenthesis and none of the previous hold; row five: g sub m open parenthesis vector x close parenthesis, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014834

Equation form expr-ca978112ca1bbdca

$a$

Read as: a

Means: Recursive-function notation denoting: a. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015015
• Occurrence projected-formula-0015026

Equation form expr-cb5a7299b00d81e2

$h(x_0, x_1) = f(x_1, x_0)$

Read as: h open parenthesis x sub zero comma x sub one close parenthesis equals f open parenthesis x sub one comma x sub zero close parenthesis

Means: A recursive-function relation or equation stating: h open parenthesis x sub zero comma x sub one close parenthesis equals f open parenthesis x sub one comma x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014511

Equation form expr-cb8e10e73bf6a18c

$p+1$

Read as: p plus one

Means: Recursive-function notation denoting: p plus one. This meaning is freshly authored from the complete recursive-functions source packet.

5 occurrences in this chapter

• Occurrence projected-formula-0014957
• Occurrence projected-formula-0014959
• Occurrence projected-formula-0014960
• Occurrence projected-formula-0014964
• Occurrence projected-formula-0014970

Equation form expr-ccc7c1fc434e2e9d

$y_i = g_i(x_0, \dots, x_{n-1})$

Read as: y sub i equals g sub i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis

Means: A recursive-function relation or equation stating: y sub i equals g sub i open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014480

Equation form expr-ccf15e4f87ae4337

$i = 0$

Read as: i equals zero

Means: A recursive-function relation or equation stating: i equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014481

Equation form expr-cd0aa9856147b6c5

$g$

Read as: g

Means: Recursive-function notation denoting: g. This meaning is freshly authored from the complete recursive-functions source packet.

30 occurrences in this chapter

• Occurrence projected-formula-0014446
• Occurrence projected-formula-0014462
• Occurrence projected-formula-0014466
• Occurrence projected-formula-0014491
• Occurrence projected-formula-0014525
• Occurrence projected-formula-0014539
• Occurrence projected-formula-0014546
• Occurrence projected-formula-0014549
• Occurrence projected-formula-0014585
• Occurrence projected-formula-0014588
• Occurrence projected-formula-0014604
• Occurrence projected-formula-0014608
• Occurrence projected-formula-0014611
• Occurrence projected-formula-0014614
• Occurrence projected-formula-0014615
• Occurrence projected-formula-0014618
• Occurrence projected-formula-0014619
• Occurrence projected-formula-0014623
• Occurrence projected-formula-0014627
• Occurrence projected-formula-0014638
• Occurrence projected-formula-0014667
• Occurrence projected-formula-0014669
• Occurrence projected-formula-0014677
• Occurrence projected-formula-0014689
• Occurrence projected-formula-0014704
• Occurrence projected-formula-0014724
• Occurrence projected-formula-0015118
• Occurrence projected-formula-0015151
• Occurrence projected-formula-0015155
• Occurrence projected-formula-0015188

Equation form expr-cd2150e8aed591a7

$h\colon\Nat \to \Nat$

Read as: h from the natural numbers to the natural numbers

Means: Recursive-function notation denoting: h from the natural numbers to the natural numbers. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014366

Equation form expr-cdfd500e7880a526

$\fn{tail}(s)$

Read as: tail of s

Means: Recursive-function notation denoting: tail of s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015069

Equation form expr-ce0a7ad973bc52f0

$f(x) \fundefined$

Read as: f open parenthesis x close parenthesis is undefined

Means: A partial-computation statement expressing: f open parenthesis x close parenthesis is undefined. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015194

Equation form expr-ce7e0b0e2a8c45f4

$< p+1$

Read as: less than p plus one

Means: A recursive-function relation or equation stating: less than p plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014975

Equation form expr-cf917227556fc1f4

$l_2$

Read as: l sub two

Means: Recursive-function notation denoting: l sub two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015082

Equation form expr-d086805bfd665865

$h(0) & = 1 \\ h(y+1) & = 2 \cdot h(y).$

Read as: h open parenthesis zero close parenthesis equals one next row h open parenthesis y plus one close parenthesis equals two times h open parenthesis y close parenthesis

Means: A source-ordered system, calculation, or case table stating: h open parenthesis zero close parenthesis equals one next row h open parenthesis y plus one close parenthesis equals two times h open parenthesis y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014628

Equation form expr-d2416246840fc803

$\fn{pred}(x) = \fn{pred}'(\Zero(x), \Proj{1}{0}(x))$

Read as: the predecessor function open parenthesis x close parenthesis equals the predecessor function prime open parenthesis the constant zero function open parenthesis x close parenthesis comma the one place projection with index zero open parenthesis x close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: the predecessor function open parenthesis x close parenthesis equals the predecessor function prime open parenthesis the constant zero function open parenthesis x close parenthesis comma the one place projection with index zero open parenthesis x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014734

Equation form expr-d3ef735b35769c75

$\cfind{e_d} \simeq d$

Read as: partial recursive function phi sub e sub d has the same definedness and value as d

Means: A partial-computation statement expressing: partial recursive function phi sub e sub d has the same definedness and value as d. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015301
• Occurrence projected-formula-0015308

Equation form expr-d4098e4d6311558c

$n \ge 2$

Read as: n is greater than or equal to two

Means: A recursive-function relation or equation stating: n is greater than or equal to two. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014988

Equation form expr-d43e11fe7ee7f6da

$\Mult(x,y) = x \cdot y$

Read as: applied to x and y equals x times y

Means: A primitive-recursive construction or operator denoting: applied to x and y equals x times y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014624

Equation form expr-d4735e3a265e16ee

$2$

Read as: two

Means: Recursive-function notation denoting: two. This meaning is freshly authored from the complete recursive-functions source packet.

6 occurrences in this chapter

• Occurrence projected-formula-0014397
• Occurrence projected-formula-0014528
• Occurrence projected-formula-0014631
• Occurrence projected-formula-0014843
• Occurrence projected-formula-0014926
• Occurrence projected-formula-0015165

Equation form expr-d4a66fb1aa33ca18

$P(\vec x)$

Read as: P open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: P open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014786
• Occurrence projected-formula-0014792

Equation form expr-d58c1d0fd20a74b3

$h(\vec x, 0) & = f(\vec x) \\ h(\vec x, y+1) & = g(\vec x, y, \tuple{h(\vec x, 0), \dots, h(\vec x, y)}).$

Read as: h of vector x and zero equals f of vector x; and h of vector x and y plus one equals g applied to vector x, y, and the code of the sequence of values h of vector x and zero through h of vector x and y

Means: A source-ordered system, calculation, or case table stating: h of vector x and zero equals f of vector x; and h of vector x and y plus one equals g applied to vector x, y, and the code of the sequence of values h of vector x and zero through h of vector x and y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015116

Equation form expr-d600a7b2756ca6bb

$h(x_0) = \Add(\Proj{1}{0}(x_0),\Proj{1}{0}(x_0)).$

Read as: h open parenthesis x sub zero close parenthesis equals the addition function open parenthesis the one place projection with index zero open parenthesis x sub zero close parenthesis comma the one place projection with index zero open parenthesis x sub zero close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: h open parenthesis x sub zero close parenthesis equals the addition function open parenthesis the one place projection with index zero open parenthesis x sub zero close parenthesis comma the one place projection with index zero open parenthesis x sub zero close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014502

Equation form expr-d65c49298cf86509

$\Add(x_0, 0) & = \Proj{1}{0}(x_0) = x_0\\ \Add(x_0, y+1) & = \Succ(\Proj{3}{2}(x_0, y, \Add(x_0, y))) = \Add(x_0, y) +1$

Read as: the addition function open parenthesis x sub zero comma zero close parenthesis equals the one place projection with index zero open parenthesis x sub zero close parenthesis equals x sub zero next row the addition function open parenthesis x sub zero comma y plus one close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis close parenthesis equals the addition function open parenthesis x sub zero comma y close parenthesis plus one

Means: A source-ordered system, calculation, or case table stating: the addition function open parenthesis x sub zero comma zero close parenthesis equals the one place projection with index zero open parenthesis x sub zero close parenthesis equals x sub zero next row the addition function open parenthesis x sub zero comma y plus one close parenthesis equals the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma the addition function open parenthesis x sub zero comma y close parenthesis close parenthesis close parenthesis equals the addition function open parenthesis x sub zero comma y close parenthesis plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014674

Equation form expr-d76c929dee4a1980

$\cfind{e}$

Read as: partial recursive function phi sub e

Means: A partial-computation statement expressing: partial recursive function phi sub e. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015257

Equation form expr-d8b12b6b58c473d3

$\Add(2,1) = 2 + 1 = 3$

Read as: the addition function open parenthesis two comma one close parenthesis equals two plus one equals three

Means: A primitive-recursive construction or operator denoting: the addition function open parenthesis two comma one close parenthesis equals two plus one equals three. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014411

Equation form expr-d966a207d30630ec

$\Char{P \lor Q}(\vec x) & = \fn{max}(\Char{P}(\vec x), \Char{Q}(\vec x)) \text{ and}\\ \Char{P \lif Q}(\vec x) & = \fn{max}(1 \tsub \Char{P}(\vec x), \Char{Q}(\vec x)).$

Read as: the characteristic function of P or Q open parenthesis vector x close parenthesis equals the maximum function open parenthesis the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis and next row the characteristic function of P implies Q open parenthesis vector x close parenthesis equals the maximum function open parenthesis one truncated minus the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis

Means: A source-ordered system, calculation, or case table stating: the characteristic function of P or Q open parenthesis vector x close parenthesis equals the maximum function open parenthesis the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis and next row the characteristic function of P implies Q open parenthesis vector x close parenthesis equals the maximum function open parenthesis one truncated minus the characteristic function of P open parenthesis vector x close parenthesis comma the characteristic function of Q open parenthesis vector x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014804

Equation form expr-d9791973c1dded6c

$\Proj{1}{0}(x_0)$

Read as: the one place projection with index zero open parenthesis x sub zero close parenthesis

Means: A primitive-recursive construction or operator denoting: the one place projection with index zero open parenthesis x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014509

Equation form expr-d9ca66e7c23867a7

$l_3$

Read as: l sub three

Means: Recursive-function notation denoting: l sub three. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015083

Equation form expr-d9eb0e49bd1e9839

$h(\vec x, y+1)$

Read as: h open parenthesis vector x comma y plus one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis vector x comma y plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015113

Equation form expr-da12b6fe87dcf5f3

$\fn{sconcat}(s)$

Read as: s-concat of s

Means: Recursive-function notation denoting: s-concat of s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015067

Equation form expr-db5d799ef1a0f9c3

$f(\vec x) = \fn{cond}(\Char{\lnot R_0}(\vec x),g_0(\vec x),g_1(\vec x)).$

Read as: f open parenthesis vector x close parenthesis equals the conditional function open parenthesis the characteristic function of not R sub zero open parenthesis vector x close parenthesis comma g sub zero open parenthesis vector x close parenthesis comma g sub one open parenthesis vector x close parenthesis close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: f open parenthesis vector x close parenthesis equals the conditional function open parenthesis the characteristic function of not R sub zero open parenthesis vector x close parenthesis comma g sub zero open parenthesis vector x close parenthesis comma g sub one open parenthesis vector x close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014836

Equation form expr-dc4e94c8eee03c1f

$d(y) = \begin{cases} 1 & \text{if $h(y, y) = 0$}\\ \umin{x}{x \neq x} & \text{otherwise.} \end{cases}$

Read as: d open parenthesis y close parenthesis equals cases begin; row one: one, if h open parenthesis y comma y close parenthesis equals zero; row two: the least x such that x is not equal to x, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: d open parenthesis y close parenthesis equals cases begin; row one: one, if h open parenthesis y comma y close parenthesis equals zero; row two: the least x such that x is not equal to x, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015282

Equation form expr-dddec4cec1599e3c

$\Mult$

Read as: the multiplication function

Means: A primitive-recursive construction or operator denoting: the multiplication function. This meaning is freshly authored from the complete recursive-functions source packet.

5 occurrences in this chapter

• Occurrence projected-formula-0014423
• Occurrence projected-formula-0014455
• Occurrence projected-formula-0014625
• Occurrence projected-formula-0014685
• Occurrence projected-formula-0014711

Equation form expr-de7d1b721a1e0632

$i$

Read as: i

Means: Recursive-function notation denoting: i. This meaning is freshly authored from the complete recursive-functions source packet.

10 occurrences in this chapter

• Occurrence projected-formula-0014992
• Occurrence projected-formula-0015009
• Occurrence projected-formula-0015012
• Occurrence projected-formula-0015020
• Occurrence projected-formula-0015024
• Occurrence projected-formula-0015027
• Occurrence projected-formula-0015074
• Occurrence projected-formula-0015149
• Occurrence projected-formula-0015171
• Occurrence projected-formula-0015172

Equation form expr-df0dad555a9c7289

$1 \tsub \Char{P}(\vec x)$

Read as: one truncated minus the characteristic function of P open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: one truncated minus the characteristic function of P open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014799

Equation form expr-df9955829d714544

$\Succ(\Proj{3}{2}(x_0, y, z))$

Read as: the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: the successor function open parenthesis the three place projection with index two open parenthesis x sub zero comma y comma z close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014678

Equation form expr-dfa441daf9f4df8a

$h(e, x) = \begin{cases} 1 & \text{if $\cfind{e}(x) \fdefined$}\\ 0 & \text{otherwise.} \end{cases}$

Read as: h open parenthesis e comma x close parenthesis equals cases begin; row one: one, if partial recursive function phi sub e open parenthesis x close parenthesis is defined; row two: zero, otherwise; cases end

Means: A source-ordered system, calculation, or case table stating: h open parenthesis e comma x close parenthesis equals cases begin; row one: one, if partial recursive function phi sub e open parenthesis x close parenthesis is defined; row two: zero, otherwise; cases end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015276

Equation form expr-dfb4f746dd2f3b1b

$\simeq$

Read as: has the same definedness and value as

Means: A partial-computation statement expressing: has the same definedness and value as. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015210

Equation form expr-e1e6765855c0925c

$p_i^{a+2} \nmid s$

Read as: p sub i superscript a plus two does not divide s

Means: A recursive-function relation or equation stating: p sub i superscript a plus two does not divide s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015033

Equation form expr-e302e6f07b3d70a5

$h(y) = h'(\Proj{1}{0}(y),\Proj{1}{0}(y))$

Read as: h open parenthesis y close parenthesis equals h prime open parenthesis the one place projection with index zero open parenthesis y close parenthesis comma the one place projection with index zero open parenthesis y close parenthesis close parenthesis

Means: A primitive-recursive construction or operator denoting: h open parenthesis y close parenthesis equals h prime open parenthesis the one place projection with index zero open parenthesis y close parenthesis comma the one place projection with index zero open parenthesis y close parenthesis close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014644

Equation form expr-e3077567b15fa4af

$z<y$

Read as: z is less than y

Means: A recursive-function relation or equation stating: z is less than y. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014869
• Occurrence projected-formula-0014877

Equation form expr-e38b7b58eb4f7b3a

$R_0(\vec x)$

Read as: R sub zero open parenthesis vector x close parenthesis

Means: Recursive-function notation denoting: R sub zero open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014831

Equation form expr-e39db7114af81e97

$g'(z) = 2 \cdot z$

Read as: g prime open parenthesis z close parenthesis equals two times z

Means: A recursive-function relation or equation stating: g prime open parenthesis z close parenthesis equals two times z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014639

Equation form expr-e3a38f7294a58e31

$s = p_0^{a_0+1} \cdot \dots \cdot p_{k}^{a_{k}+1},$

Read as: s equals p sub zero superscript a sub zero plus one times and so on times p sub k superscript a sub k plus one

Means: A recursive-function relation or equation stating: s equals p sub zero superscript a sub zero plus one times and so on times p sub k superscript a sub k plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014999

Equation form expr-e3b98a4da31a127d

$t$

Read as: t

Means: Recursive-function notation denoting: t. This meaning is freshly authored from the complete recursive-functions source packet.

10 occurrences in this chapter

• Occurrence projected-formula-0015049
• Occurrence projected-formula-0015086
• Occurrence projected-formula-0015088
• Occurrence projected-formula-0015091
• Occurrence projected-formula-0015093
• Occurrence projected-formula-0015094
• Occurrence projected-formula-0015096
• Occurrence projected-formula-0015107
• Occurrence projected-formula-0015108
• Occurrence projected-formula-0015109

Equation form expr-e3cb3f4dce5e370a

$d(x, y) = 0$

Read as: d open parenthesis x comma y close parenthesis equals zero

Means: A recursive-function relation or equation stating: d open parenthesis x comma y close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014766

Equation form expr-e60d53cec3f4568f

$x_{k-1}$

Read as: x sub k minus one

Means: Recursive-function notation denoting: x sub k minus one. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014442
• Occurrence projected-formula-0014692

Equation form expr-e6304ffd5391c53c

$m'_R(\vec{x}, y)$

Read as: m prime sub R open parenthesis vector x comma y close parenthesis

Means: Recursive-function notation denoting: m prime sub R open parenthesis vector x comma y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014881

Equation form expr-e632b7095b0bf32c

$T$

Read as: T

Means: Recursive-function notation denoting: T. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015246

Equation form expr-e6605a6f18171dc9

$h(y,y) \neq 0$

Read as: h open parenthesis y comma y close parenthesis is not equal to zero

Means: Recursive-function notation denoting: h open parenthesis y comma y close parenthesis is not equal to zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015287

Equation form expr-e6799051fc36316c

$\Char{P \land Q}(\vec x)$

Read as: the characteristic function of P and Q open parenthesis vector x close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of P and Q open parenthesis vector x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014801

Equation form expr-e6915bf887e556eb

$z < y$

Read as: z is less than y

Means: A recursive-function relation or equation stating: z is less than y. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014848
• Occurrence projected-formula-0014866

Equation form expr-e76d8eb331140f83

$\Mult(2,0)$

Read as: the multiplication function open parenthesis two comma zero close parenthesis

Means: A primitive-recursive construction or operator denoting: the multiplication function open parenthesis two comma zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014431

Equation form expr-e77de3ec8e962e37

$\len{s}$

Read as: length, applied to s

Means: A sequence- or tree-coding expression denoting: length, applied to s. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014993

Equation form expr-e7d1c9237b449a64

$g_1(x)$

Read as: g sub one open parenthesis x close parenthesis

Means: Recursive-function notation denoting: g sub one open parenthesis x close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015159

Equation form expr-e8748a5040d97e23

$\fn{IsZero}(x \tsub y)$

Read as: the zero-test predicate open parenthesis x truncated minus y close parenthesis

Means: Recursive-function notation denoting: the zero-test predicate open parenthesis x truncated minus y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014785

Equation form expr-e8c17252cf9632d5

$\Char{R}$

Read as: the characteristic function of R

Means: A primitive-recursive relation or bounded-quantifier statement expressing: the characteristic function of R. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014777
• Occurrence projected-formula-0014886

Equation form expr-e95355777b59cb5e

$& \bforall{z < y}{R(\vec x, z)} \text{ and}\\ & \bexists{z < y}{R(\vec x, z)}.$

Read as: for every z less than y, R of vector x and z; and there exists z less than y such that R of vector x and z

Means: A source-ordered system, calculation, or case table stating: for every z less than y, R of vector x and z; and there exists z less than y such that R of vector x and z. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014806

Equation form expr-eae5274810fb2e34

$z < y+1$

Read as: z is less than y plus one

Means: A recursive-function relation or equation stating: z is less than y plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014872

Equation form expr-ec4651e8e7f851e0

$y = 1$

Read as: y equals one

Means: A recursive-function relation or equation stating: y equals one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014916

Equation form expr-ec53b9efae68ebaf

$y > x$

Read as: y is greater than x

Means: A recursive-function relation or equation stating: y is greater than x. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014921
• Occurrence projected-formula-0014946

Equation form expr-eea62bad037aa99b

$\leq y$

Read as: less than or equal to y

Means: A recursive-function relation or equation stating: less than or equal to y. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014903

Equation form expr-ef2d127de37b942b

$5$

Read as: five

Means: Recursive-function notation denoting: five. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014928

Equation form expr-ef59500f593f24e8

$x + (y \tsub x) = x + 0 = x$

Read as: x plus open parenthesis y truncated minus x close parenthesis equals x plus zero equals x

Means: A recursive-function relation or equation stating: x plus open parenthesis y truncated minus x close parenthesis equals x plus zero equals x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014756

Equation form expr-efb95e73a322391a

$\fn{Comp}_{k,n}$

Read as: the function composition sub k comma n

Means: A primitive-recursive construction or operator denoting: the function composition sub k comma n. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014659

Equation form expr-f10f7239cc7ed303

$\le x$

Read as: less than or equal to x

Means: A recursive-function relation or equation stating: less than or equal to x. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014954
• Occurrence projected-formula-0014956
• Occurrence projected-formula-0014963

Equation form expr-f17d8b48f6a69641

$\Mult(2,1)$

Read as: the multiplication function open parenthesis two comma one close parenthesis

Means: A primitive-recursive construction or operator denoting: the multiplication function open parenthesis two comma one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014432

Equation form expr-f19333325be3b4d7

$h(\vec x, 1) & = g(\vec x, 0, h(\vec x, 0)) = g(\vec x, 0, f(\vec x)). \intertext{We can go on in this way and compute} h(\vec x, 2) & = g(\vec x, 1, h(\vec x, 1)) = g(\vec x, 1, g(\vec x, 0, f(\vec x)))\\ h(\vec x, 3) & = g(\vec x, 2, h(\vec x, 2)) = g(\vec x, 2, g(\vec x, 1, g(\vec x, 0, f(\vec x))))\\ h(\vec x, 4) & = g(\vec x, 3, h(\vec x, 3)) = g(\vec x, 3, g(\vec x, 2, g(\vec x, 1, g(\vec x, 0, f(\vec x)))))\\ & \vdots$

Read as: h open parenthesis vector x comma one close parenthesis equals g open parenthesis vector x comma zero comma h open parenthesis vector x comma zero close parenthesis close parenthesis equals g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis We can go on in this way and compute h open parenthesis vector x comma two close parenthesis equals g open parenthesis vector x comma one comma h open parenthesis vector x comma one close parenthesis close parenthesis equals g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis next row h open parenthesis vector x comma three close parenthesis equals g open parenthesis vector x comma two comma h open parenthesis vector x comma two close parenthesis close parenthesis equals g open parenthesis vector x comma two comma g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis close parenthesis next row h open parenthesis vector x comma four close parenthesis equals g open parenthesis vector x comma three comma h open parenthesis vector x comma three close parenthesis close parenthesis equals g open parenthesis vector x comma three comma g open parenthesis vector x comma two comma g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis close parenthesis close parenthesis next row and so on vertically

Means: A source-ordered system, calculation, or case table stating: h open parenthesis vector x comma one close parenthesis equals g open parenthesis vector x comma zero comma h open parenthesis vector x comma zero close parenthesis close parenthesis equals g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis We can go on in this way and compute h open parenthesis vector x comma two close parenthesis equals g open parenthesis vector x comma one comma h open parenthesis vector x comma one close parenthesis close parenthesis equals g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis next row h open parenthesis vector x comma three close parenthesis equals g open parenthesis vector x comma two comma h open parenthesis vector x comma two close parenthesis close parenthesis equals g open parenthesis vector x comma two comma g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis close parenthesis next row h open parenthesis vector x comma four close parenthesis equals g open parenthesis vector x comma three comma h open parenthesis vector x comma three close parenthesis close parenthesis equals g open parenthesis vector x comma three comma g open parenthesis vector x comma two comma g open parenthesis vector x comma one comma g open parenthesis vector x comma zero comma f open parenthesis vector x close parenthesis close parenthesis close parenthesis close parenthesis close parenthesis next row and so on vertically. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014698

Equation form expr-f1e5183bd7411038

$S_0$

Read as: S sub zero

Means: Recursive-function notation denoting: S sub zero. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014592

Equation form expr-f212c1cf438fc09c

$\bexists{z < y}{R(\vec x, z)} \liff \lnot \bforall{z < y}{\lnot R(\vec x, z)}$

Read as: there exists z less than y such that R open parenthesis vector x comma z close parenthesis if and only if not for every z less than y, not R open parenthesis vector x comma z close parenthesis

Means: A primitive-recursive relation or bounded-quantifier statement expressing: there exists z less than y such that R open parenthesis vector x comma z close parenthesis if and only if not for every z less than y, not R open parenthesis vector x comma z close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014822

Equation form expr-f25413b063beb2ef

$p_0^{a_0+1} \cdot p_1^{a_1+1} \cdot p_2^{a_2+1} \cdot \dots \cdot p_k^{a_k+1}.$

Read as: p sub zero superscript a sub zero plus one times p sub one superscript a sub one plus one times p sub two superscript a sub two plus one times and so on times p sub k superscript a sub k plus one

Means: Recursive-function notation denoting: p sub zero superscript a sub zero plus one times p sub one superscript a sub one plus one times p sub two superscript a sub two plus one times and so on times p sub k superscript a sub k plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014981

Equation form expr-f2dc52e623459ded

$\Proj{n}{j}$

Read as: P superscript n sub j

Means: A primitive-recursive construction or operator denoting: P superscript n sub j. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014591

Equation form expr-f303ce6dda74413f

$\fn{sequenceBound}(x,k) = p_{k-1}^{k \cdot (x+1)},$

Read as: the function sequence bound open parenthesis x comma k close parenthesis equals p sub k minus one superscript k times open parenthesis x plus one close parenthesis

Means: A recursive-function relation or equation stating: the function sequence bound open parenthesis x comma k close parenthesis equals p sub k minus one superscript k times open parenthesis x plus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015062

Equation form expr-f394eff20da25b50

$\fn{hSubtreeSeq}(t,n)$

Read as: the height-subtree-sequence function open parenthesis t comma n close parenthesis

Means: Recursive-function notation denoting: the height-subtree-sequence function open parenthesis t comma n close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015089

Equation form expr-f3b9fd9c94e230f5

$x \neq x$

Read as: x is not equal to x

Means: Recursive-function notation denoting: x is not equal to x. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015284

Equation form expr-f4c0417aafe3ccec

$\Mult(x_0,0) & = f(x_0) = 0 \\ \Mult(x_0,y+1) & = g(x_0, y, \Mult(x_0,y)) = \Add(\Mult(x_0,y), x_0)$

Read as: the multiplication function open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals zero next row the multiplication function open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma the multiplication function open parenthesis x sub zero comma y close parenthesis close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis x sub zero comma y close parenthesis comma x sub zero close parenthesis

Means: A source-ordered system, calculation, or case table stating: the multiplication function open parenthesis x sub zero comma zero close parenthesis equals f open parenthesis x sub zero close parenthesis equals zero next row the multiplication function open parenthesis x sub zero comma y plus one close parenthesis equals g open parenthesis x sub zero comma y comma the multiplication function open parenthesis x sub zero comma y close parenthesis close parenthesis equals the addition function open parenthesis the multiplication function open parenthesis x sub zero comma y close parenthesis comma x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014460

Equation form expr-f67ab10ad4e4c531

$F$

Read as: F

Means: Recursive-function notation denoting: F. This meaning is freshly authored from the complete recursive-functions source packet.

3 occurrences in this chapter

• Occurrence projected-formula-0014656
• Occurrence projected-formula-0014670
• Occurrence projected-formula-0015168

Equation form expr-f6c7bb8c5df25008

$\fn{pred}$

Read as: the predecessor function

Means: Recursive-function notation denoting: the predecessor function. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014733

Equation form expr-f6edbd6f847b69e7

$S = \bigcup_{i \in \Nat} S_i$

Read as: S equals the union, over all natural numbers i, of S sub i

Means: S is the union of all finite construction stages. The reading is cumulative: an identity composition promotes each stage function to the next stage even though the frozen prose does not state that promotion separately.

1 occurrence in this chapter

• Occurrence projected-formula-0014599

Equation form expr-f782dd90b07e2129

$\fn{pred}(y)$

Read as: the predecessor function open parenthesis y close parenthesis

Means: Recursive-function notation denoting: the predecessor function open parenthesis y close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0014725
• Occurrence projected-formula-0014729

Equation form expr-f9fae44fbe241f45

$1 = 0 + 1$

Read as: one equals zero plus one

Means: A recursive-function relation or equation stating: one equals zero plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014696

Equation form expr-fa7d6b524c7003f2

$f(y_0, y_1)$

Read as: f open parenthesis y sub zero comma y sub one close parenthesis

Means: Recursive-function notation denoting: f open parenthesis y sub zero comma y sub one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014510

Equation form expr-fb4d66121f534e20

$h(x_0, \dots, x_{n-1})$

Read as: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis

Means: Recursive-function notation denoting: h open parenthesis x sub zero comma and so on comma x sub n minus one close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014479

Equation form expr-fb7e3d8aa70fbb2e

$k\ge 1$

Read as: k is greater than or equal to one

Means: A recursive-function relation or equation stating: k is greater than or equal to one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014545

Equation form expr-fcbfca902c664ca4

$f(x,\vec z) = 0$

Read as: f open parenthesis x comma vector z close parenthesis equals zero

Means: A recursive-function relation or equation stating: f open parenthesis x comma vector z close parenthesis equals zero. This meaning is freshly authored from the complete recursive-functions source packet.

2 occurrences in this chapter

• Occurrence projected-formula-0015215
• Occurrence projected-formula-0015319

Equation form expr-fd3e84dc55a773ef

$\#(F)$

Read as: the code of open parenthesis F close parenthesis

Means: A notation-coding expression denoting: the code of open parenthesis F close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015167

Equation form expr-fdcf076edc5108f3

$\fact{x}+1$

Read as: x factorial plus one

Means: Recursive-function notation denoting: x factorial plus one. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014949

Equation form expr-fe96ac5aba439557

$g(x_0, y, z) = \Add(z,x_0)$

Read as: g open parenthesis x sub zero comma y comma z close parenthesis equals the addition function open parenthesis z comma x sub zero close parenthesis

Means: A primitive-recursive construction or operator denoting: g open parenthesis x sub zero comma y comma z close parenthesis equals the addition function open parenthesis z comma x sub zero close parenthesis. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014458

Equation form expr-ff057e3908f4aca2

$\fn{concat}(s,t)=\begin{cases}\emptyseq & \text{if }\len{s}+\len{t}=0,\\ \bmin{v < \fn{sequenceBound}(s+t,\len{s}+\len{t})}{\len{v}=\len{s}+\len{t} \land \bforall{i<\len{s}}{(v)_i=(s)_i} \land \bforall{j<\len{t}}{(v)_{\len{s}+j}=(t)_j} }& \text{otherwise.}\end{cases}$

Read as: concatenate of s and t is the empty-sequence code when their total length is zero; otherwise it is the least bounded sequence code v having the combined length, the entries of s as its prefix, and the entries of t as its suffix

Means: A source-ordered system, calculation, or case table stating: concatenate of s and t is the empty-sequence code when their total length is zero; otherwise it is the least bounded sequence code v having the combined length, the entries of s as its prefix, and the entries of t as its suffix. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0015066

Equation form expr-ff6e5d65faad1ad4

$\Char{R}(\vec x) = \left\{ \begin{array}{ll} 1 & \mbox{if $R(\vec x)$} \\ 0 & \mbox{otherwise} \end{array} \right.$

Read as: the characteristic function of R open parenthesis vector x close parenthesis equals table begin; row one: one, if R open parenthesis vector x close parenthesis; row two: zero, otherwise; table end

Means: A source-ordered system, calculation, or case table stating: the characteristic function of R open parenthesis vector x close parenthesis equals table begin; row one: one, if R open parenthesis vector x close parenthesis; row two: zero, otherwise; table end. This meaning is freshly authored from the complete recursive-functions source packet.

1 occurrence in this chapter

• Occurrence projected-formula-0014774

Power of two primitive recursion display

The first row sets h at zero equal to one. The second row sets h at the successor of x equal to two times h at x.

Source

Initial power of two computation display

The rows compute h at one as two times h at zero and hence two, h at two as two times h at one and hence two times two, and h at three as two times h at two and hence three factors of two, followed by an ellipsis.

Source

Addition recursion display

The base row sets addition of x and zero equal to x. The successor row sets addition of x and the successor of y equal to the successor of addition of x and y.

Source

Multiplication recursion display

The base row sets multiplication of x and zero equal to zero. The successor row sets multiplication of x and the successor of y equal to addition of the preceding product and x.

Source

Multiplication by two computation display

The rows compute multiplication of two by zero, one, two, and three in source order, expanding each successor step through addition and obtaining zero, two, four, and six.

Source

Parameterized primitive recursion display

The base row gives h on parameters x sub zero through x sub k minus one and zero as f of those parameters. The successor row gives h on the same parameters and the successor of y as g of the parameters, y, and the preceding value of h.

Source

Addition as primitive recursion display

The base row identifies addition of x sub zero and zero with f of x sub zero and with x sub zero. The successor row identifies the next addition value with g applied to x sub zero, y, and the preceding addition value, then with successor of that value.

Source

Multiplication as primitive recursion display

The base row identifies multiplication of x sub zero and zero with f of x sub zero and with zero. The successor row identifies the next product with g applied to x sub zero, y, and the preceding product, then with addition of that product and x sub zero.

Source

Definition of primitive recursion

The definition assumes a function f of k arguments, with k at least one, and a function g of k plus two arguments. It defines h by primitive recursion as a function of k plus one arguments. Its nested equation display first gives h at the parameter vector and zero as f of that vector, then gives h at the successor of y as g of the same parameters, y, and the preceding value of h.

Source

Nested primitive recursion equation display

This display inside the definition gives the two defining equations for h. The first applies f at recursion value zero, and the second applies g to the fixed parameters, the current recursion value, and the already computed value of h.

Source

Definition of composition

The definition assumes an outer function f of k arguments and k inner functions from g sub zero through g sub k minus one, each of n arguments. It defines the n place function h by applying every inner function to the same n inputs and feeding the resulting k values, in order, to f.

Source

Definition of primitive recursive functions

The definition concerns natural number valued functions of any finite arity and gives five inductive clauses. Zero and successor are primitive recursive, every projection is primitive recursive, composition of primitive recursive functions is primitive recursive, and primitive recursion from primitive recursive base and step functions is primitive recursive.

Source

Addition is primitive recursive proposition

The proposition states that the function sending x and y to their sum is primitive recursive.

Source

Addition proof equation display

The base row presents addition of x sub zero and zero as the projection f and hence x sub zero. The successor row presents addition at the successor of y as g applied to the parameters and preceding value, then as successor of the preceding sum.

Source

Multiplication is primitive recursive proposition

The proposition states that the function sending x and y to their product is primitive recursive.

Source

Multiplication proof exercise

This exercise is intentionally left unsolved. It asks the reader to prove that multiplication is primitive recursive by placing its recursive definition in the required primitive recursion form and showing that the corresponding base function f and step function g are primitive recursive.

Source

First primitive recursion example

The example begins with h at zero equal to one and each successor value equal to twice the preceding value. It explains the zero parameter obstacle and the constants one and two, introduces a dummy argument and a function h prime, builds its base function from successor and zero, builds its step through multiplication, projections, and two auxiliary functions, then recovers h from h prime by composition and concludes that h is primitive recursive.

Source

Nested initial recursion example display

This first display inside the example sets h at zero equal to one and h at the successor of y equal to two times h at y.

Source

Nested dummy argument recursion display

This second display inside the example introduces h prime. Its base value is f of the dummy argument and equals one, while its successor value is g of the dummy argument, y, and the preceding h prime value, and equals twice that preceding value.

Source

Nested auxiliary composition display

This third display inside the example defines g by applying g prime to the third projection, defines g prime by multiplying g double prime with the first projection, and defines g double prime as successor applied to f.

Source

Addition operator notation display

The base row writes addition of x sub zero and zero as the one place projection and hence x sub zero. The successor row writes the next sum as successor applied to the third projection of the parameter, recursion value, and preceding sum, and hence as the preceding sum plus one.

Source

Multiplication notation exercise

This exercise is intentionally left unsolved. It asks the reader to give the complete primitive recursive operator notation for multiplication.

Source

Vector primitive recursion display

The first row gives h of vector x and zero as f of vector x. The second row gives h of vector x and the successor of y as g of vector x, y, and h of vector x and y.

Source

Successive primitive recursion computation display

The rows expand h of vector x at one, two, three, and four by repeatedly applying g to the current recursion index and the preceding value, beginning with f of vector x, and then indicate continuation by an ellipsis.

Source

Exponentiation is primitive recursive proposition

The proposition states that exponentiation, sending x and y to x raised to the power y, is primitive recursive.

Source

Exponentiation construction display

The display first gives exponentiation at zero as one and at a successor as x times the preceding power. It then gives the official primitive recursion form using f and g, defines f as successor of zero and hence one, and defines g through multiplication of the appropriate first and third projections and hence x times z.

Source

Predecessor is primitive recursive proposition

The proposition defines predecessor by cases, returning zero at zero and y minus one otherwise, and states that this function is primitive recursive.

Source

Predecessor recursion display

The first row sets predecessor at zero equal to zero. The second row sets predecessor at the successor of y equal to y.

Source

Dummy argument predecessor display

The first row sets predecessor prime at x and zero equal to the zero function at x and hence zero. The second row sets its successor value equal to the second projection of x, y, and the preceding value, and hence y.

Source

Factorial is primitive recursive proposition

The proposition identifies factorial of x with the product of the positive natural numbers through x and states that factorial is primitive recursive.

Source

Factorial construction display

The display first gives factorial at zero as one and factorial at a successor as the preceding factorial times that successor. It then introduces a two place function h with a constant one base and a step function g, defines g through multiplication of the third projection with successor of the second projection, and finally recovers factorial of y from h at y and y.

Source

Truncated subtraction is primitive recursive proposition

The proposition defines truncated subtraction of y from x by cases, returning zero when x is less than y and ordinary subtraction otherwise, and states that it is primitive recursive.

Source

Truncated subtraction recursion display

The base row gives x truncated by zero as x. The successor row gives x truncated by the successor of y as predecessor of x truncated by y.

Source

Distance is primitive recursive proposition

The proposition states that the absolute value of x minus y, viewed as the distance between x and y, is primitive recursive.

Source

Maximum is primitive recursive proposition

The proposition states that the maximum of x and y is primitive recursive.

Source

Minimum is primitive recursive proposition

The proposition states that the minimum of x and y is primitive recursive.

Source

Minimum proof exercise

This exercise is intentionally left unsolved. It asks the reader to prove the preceding proposition that minimum is primitive recursive.

Source

Exponent tower exercise

This exercise is intentionally left unsolved. It asks the reader to show that the function whose value is a tower containing y copies of two with x as the top exponent is primitive recursive.

Source

Integer division construction exercise

This exercise is intentionally left unsolved. It asks the reader to show that integer division, which discards the fractional part and returns zero when the divisor is zero, is primitive recursive, and to give an explicit construction using primitive recursion and composition.

Source

Finite sums and products closure proposition

The proposition states two closure results in source order. A finite sum through y of values of a primitive recursive function is primitive recursive, and a finite product through y of values of a primitive recursive function is primitive recursive.

Source

Finite sum recursion display

The base row sets g of vector x and zero equal to f of vector x and zero. The successor row sets the next finite sum equal to the preceding sum plus f of vector x at the new index.

Source

Definition of primitive recursive relations

The definition says that a relation on vector x is primitive recursive when its characteristic function is primitive recursive. Its displayed characteristic function returns one when the relation holds and zero otherwise.

Source

Zero test characteristic function display

The first row gives the characteristic function of the zero test at zero as one. The second row gives its value at every successor as zero.

Source

Boolean closure proposition for relations

The proposition states that if relations P and Q are primitive recursive, then their negation, conjunction, disjunction, and implication are also primitive recursive, in that source order.

Source

Disjunction and implication characteristic display

The first row defines the characteristic function of the disjunction of P and Q as the maximum of their characteristic values. The second row defines the characteristic function of their implication as the maximum of one truncated by the characteristic value of P and the characteristic value of Q.

Source

Bounded quantification closure proposition

The proposition states that if R of vector x and z is primitive recursive, then both the bounded universal relation and the bounded existential relation with z less than y are primitive recursive. It explains that the universal relation holds exactly when R holds for every z below y and says the existential case is analogous.

Source

Nested bounded quantifier display

This display inside the proposition presents, in order, the bounded universal statement that R holds for every z below y and the bounded existential statement that R holds for some z below y.

Source

Bounded universal characteristic recursion display

The base row sets the characteristic function of P at vector x and zero equal to one. The successor row sets its next value equal to the minimum of its preceding value and the characteristic value of R at vector x and y.

Source

Congruence relation exercise

This exercise is intentionally left unsolved. It asks the reader to show that the three place relation saying x is congruent to y modulo n is primitive recursive.

Source

Conditional function display

The display first defines the conditional function by cases, returning y when x is zero and z otherwise. It then gives its primitive recursion equations, with value y at zero and value z at every successor.

Source

Piecewise primitive recursion proposition

The proposition assumes primitive recursive functions from g sub zero through g sub m and primitive recursive relations from R sub zero through R sub m minus one. It defines f by an ordered case list, choosing each g sub i when its relation holds and no earlier relation holds, and choosing g sub m otherwise, then states that f is primitive recursive.

Source

Bounded minimization proposition

The proposition states that when R of vector x and z is primitive recursive, the function m sub R returns the least z below y for which R holds, if one exists, and returns y otherwise. It also introduces the bounded minimization notation for that function.

Source

Bounded minimization recursion display

The base row sets m sub R at vector x and zero equal to zero. At the successor of y, the case display keeps an earlier witness when one was found, returns y when y is the first witness, and returns the successor of y when no witness has appeared.

Source

Alternative bounded search exercise

This exercise is intentionally left unsolved. It asks the reader to define, by primitive recursion from the characteristic function of R, a bounded search function that returns the least z below y satisfying R and returns zero when no such z exists.

Source

Prime enumeration recursion display

The base row sets p at zero equal to two. The successor row sets the next value of p equal to the first prime larger than the preceding value.

Source

Bounded minimization division exercise

This exercise is intentionally left unsolved. It asks the reader to define integer division of x by y using bounded minimization.

Source

Sequence length proposition

The proposition states that the function returning the length of the coded sequence s is primitive recursive.

Source

Sequence append proposition

The proposition states that the function returning the result of appending the entry a to the coded sequence s is primitive recursive.

Source

Sequence element proposition

The proposition states that the element function is primitive recursive. It returns the entry at index i of sequence s, counting the initial entry as index zero, and returns zero when i is at least the length of s.

Source

Sequence concatenation proposition

The proposition states that the function concatenating coded sequences s and t is primitive recursive.

Source

Iterated concatenation recursion display

The base row sets the helper concatenation function at step zero equal to s. The successor row appends entry n of t to the preceding helper value. The final row defines concatenation of s and t by running the helper for the length of t.

Source

Corrected bounded search concatenation display

The source display attempts to obtain the concatenation of s and t by bounded search for a sequence v of the combined length whose initial entries match s and whose remaining entries match t. The reader correction supplies the missing predicate scope and an explicit empty sequence case, while the frozen malformed source display remains available separately.

Source

Sequence list concatenation exercise

This exercise is intentionally left unsolved. It asks the reader to show that a primitive recursive function can concatenate every coded sequence from s sub zero through s sub k into one sequence.

Source

Sequence tail exercise

This exercise is intentionally left unsolved. It asks the reader to construct a primitive recursive tail function that returns zero on the empty sequence and otherwise removes the initial entry, as stated in its nested display.

Source

Nested sequence tail equation display

This display inside the unsolved exercise gives the required behavior of tail. It returns zero on the empty sequence and maps the sequence from s sub zero through s sub k to the sequence from s sub one through s sub k.

Source

Subsequence is primitive recursive proposition

The proposition states that the function returning the length n subsequence of s beginning at entry i is primitive recursive.

Source

Subsequence proof exercise

This exercise is intentionally left unsolved. It asks the reader to prove the preceding proposition that the subsequence function is primitive recursive.

Source

Subtree sequence is primitive recursive proposition

The proposition states that the function returning a coded sequence of the codes of all subtrees of the tree coded by t is primitive recursive.

Source

Corrected subtree accumulator display

The source display starts an accumulator with f applied to the first entry and then appends f applied to each next entry. The reader correction starts with the empty sequence and at each successor step appends f applied to the entry at the preceding index, removing the empty input and off by one defect while preserving the frozen source display separately.

Source

Subtree iteration display

The base row sets the helper subtree sequence at t and zero equal to the one entry sequence containing t. The successor row concatenates the preceding helper sequence with h applied to that preceding sequence.

Source

Duplicate free subtree exercise

This exercise is intentionally left unsolved. It notes that the helper subtree sequence from the preceding proof can repeat subtree codes and asks for an alternative definition in which every subtree code occurs only once.

Source

Simultaneous recursion display

The first two rows give the base values of h sub zero and h sub one from f sub zero and f sub one. The next two rows give their successor values through g sub zero and g sub one, with each step function receiving both preceding h values.

Source

Course of values recursion display

The base row gives h of vector x and zero as f of vector x. The successor row gives h at the successor of y by applying g to vector x, y, and the coded sequence of all earlier h values from zero through y.

Source

Earlier value selection display

The case display gives h of vector x and y through g using the earlier value selected by k when that selected index is below y, and gives f of vector x otherwise.

Source

Remainder by course of values exercise

This exercise is intentionally left unsolved. It asks the reader to define remainder by course of values recursion, requiring a remainder below a positive divisor y whose addition to a multiple of y gives x, and stipulating zero when y is zero.

Source

Changing parameters recursion display

The base row gives h of vector x and zero as f of vector x. The successor row applies g to vector x, y, and a preceding h computation whose parameter vector has been changed by k.

Source

Diagonal successor function display

The first row defines h of x as g of x and x plus one. The second row identifies that value with f sub x at x plus one, producing the diagonal disagreement.

Source

Fast growing function hierarchy display

The base row defines g sub zero at x as x plus one. The successor row defines g sub n plus one at x by iterating g sub n exactly x times starting at x.

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Primitive recursive notation coding display

The rows assign codes in source order. Zero receives the one entry tuple zero, successor receives the one entry tuple one, and a projection receives the tuple containing two, its arity, and its index. A composition receives a tuple beginning with three and the two relevant arities, followed by the codes of its outer and inner notations. A recursion receives the tuple containing four, its arity, and the codes of its base and step notations.

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Definition of partial recursive functions

The definition gives the partial recursive functions as the smallest class of partial natural number functions of varying arities that contains zero, successor, and projections and is closed under composition, primitive recursion, and unbounded search.

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Definition of recursive functions

The definition says that recursive functions are exactly the total functions among the partial recursive functions.

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Kleene normal form theorem

The theorem states that there are a primitive recursive relation T and a primitive recursive function U such that every partial recursive function f has some index e for which f at x has the same definedness and value as U applied to the least s satisfying T of e, x, and s, for every x.

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Halting function nonrecursiveness theorem

The theorem states that the halting function h is not partial recursive.

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Definition of general recursive functions

The definition gives the general recursive functions as the smallest class of total natural number functions of varying arities that contains zero, successor, and projections and is closed under composition, primitive recursion, and unbounded search applied only to regular functions.

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Cross-reference reference-000554

Addition construction reference to primitive recursion

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Cross-reference reference-000555

Multiplication exercise reference to multiplication proposition

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Cross-reference reference-000556

Multiplication exercise reference to primitive recursion

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Cross-reference reference-000557

First example obstacle reference to primitive recursion

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Cross-reference reference-000558

Minimum exercise reference to minimum proposition

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Cross-reference reference-000559

Subsequence exercise reference to subsequence proposition

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Cross-reference reference-000560

Duplicate free subtree exercise reference

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Cross-reference reference-000561

Initial comparison reference to general recursion definition

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Cross-reference reference-000562

Initial comparison reference to recursive function definition

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Cross-reference reference-000563

Misnomer comparison reference to general recursion definition

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Cross-reference reference-000564

Equivalence comparison reference to recursive function definition

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Source disclosures

• TR030-SOURCE-001: The frozen source reverses the dependency in its concluding sentence. The reader restores the direction used in the preceding examples; the source remains available unchanged. source
• TR030-SOURCE-002: The frozen source changes h's n input variables to k input variables in the final computation sentence. The reader restores n on h's argument list. source
• TR030-SOURCE-003: The frozen source switches from n to k when naming the just-defined projection family. The reader consistently uses n. source
• TR030-SOURCE-009: The reader corrects the adjective in the section title; the frozen title is retained in source replay. source
• TR030-SOURCE-010: The reader supplies the missing subject and names the constant-two function used by the displayed composition. source
• TR030-SOURCE-017: The reader names the displayed non-strict relation accurately; its formula and characteristic function are unchanged. source
• TR030-SOURCE-011: The reader changes 'than' to 'that'; the mathematics and source formulas are unchanged. source
• TR030-SOURCE-004: The frozen source changes the fixed parameter vector from x to z in the third case. The reader restores vector x. source
• TR030-SOURCE-005: The frozen parenthetical reverses divisor and dividend. The reader follows the surrounding definition: divide y by x. source
• TR030-SOURCE-012: The reader restores the missing closing parenthesis in this prose sentence. source
• TR030-SOURCE-013: The reader separates the function name from its argument so MathML and speech expose application structure. source
• TR030-SOURCE-014: The reader corrects the indefinite article. source
• TR030-SOURCE-006: The frozen concatenation display has a misplaced empty minimization argument and does not handle two empty inputs. The reader supplies an explicit empty case and gives bounded minimization the intended three-part predicate; the original display remains available. source
• TR030-SOURCE-015: The reader supplies the missing verb in the definition of immediate subtrees. source
• TR030-SOURCE-007: The frozen helper counts through index k but is called with the sequence length. The reader uses a length-counting recursion that visits exactly the valid entries. source
• TR030-SOURCE-016: The reader restores the direction described by the preceding coding equations; some numbers still do not code notations. source
• TR030-SOURCE-008: The frozen proof mixes two indexing conventions. The reader follows the convention stated immediately above the halting function, under which every natural e determines a partial recursive function. The diagonal contradiction for the known index e sub d is unchanged. source