Methods

Induction

Equation form expr-002bf3eb1573e012

k+1k+1

Read as: k plus one

Means: k plus one

7 occurrences in this chapter

Equation form expr-0277ebaf6ee04abd

P(k)P(k)

Read as: capital P of k

Means: capital P of k

12 occurrences in this chapter

Equation form expr-043a718774c572bd

ss

Read as: s

Means: s

5 occurrences in this chapter

Equation form expr-046b4724fc2febf0

ana_n

Read as: a sub n

Means: a sub n

1 occurrence in this chapter

Equation form expr-063bda123453a086

<l2/2< l_2/2

Read as: is less than l sub two divided by two

Means: is less than l sub two divided by two

1 occurrence in this chapter

Equation form expr-066590f91f51f4ba

t⊑tt \sqsubseteq t

Read as: t is a subterm of t

Means: t is a subterm of t

1 occurrence in this chapter

Equation form expr-067d6191247c957f

l2<kl_2 < k

Read as: l sub two is less than k

Means: l sub two is less than k

1 occurrence in this chapter

Equation form expr-07b87a6c7402098f

[s1∘[s_1 \circ {}

Read as: the prefix consisting of an opening square bracket, s sub one, then a circle

Means: the prefix consisting of an opening square bracket, s sub one, then a circle

2 occurrences in this chapter

Equation form expr-0800fc467307cd20

[b∘[b \circ {}

Read as: the prefix consisting of an opening square bracket, b, then a circle

Means: the prefix consisting of an opening square bracket, b, then a circle

1 occurrence in this chapter

Equation form expr-0be586ab08aff4f0

l(t)l(t)

Read as: l of t

Means: l of t

2 occurrences in this chapter

Equation form expr-0fdb2ff26a22c17a

sk+1=(k+1)(k+2)/2s_{k+1} = (k+1)(k+2)/2

Read as: s sub k plus one equals the product of k plus one and k plus two, divided by two

Means: s sub k plus one equals the product of k plus one and k plus two, divided by two

2 occurrences in this chapter

Equation form expr-16265f79a0459c8c

m1<l1/2m_1 < l_1/2

Read as: m sub one is less than l sub one divided by two

Means: m sub one is less than l sub one divided by two

1 occurrence in this chapter

Equation form expr-16abc7757d726a17

a⊑a\mathrm{a} \sqsubseteq \mathrm{a}

Read as: the letter a is a subterm of the letter a

Means: the letter a is a subterm of the letter a

2 occurrences in this chapter

Equation form expr-175651834a5356f2

l1l_1

Read as: l sub one

Means: l sub one

2 occurrences in this chapter

Equation form expr-1a57de5c03f06b4d

<n/2< n/2

Read as: is less than n divided by two

Means: is less than n divided by two

3 occurrences in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

24 occurrences in this chapter

Equation form expr-1b63d6a711b2030d

s2≠r2s_2 \neq r_2

Read as: s sub two is not equal to r sub two

Means: s sub two is not equal to r sub two

1 occurrence in this chapter

Equation form expr-1c08653ec43246dd

s1∘s2s_1 \circ s_2

Read as: s sub one circle s sub two

Means: s sub one circle s sub two

3 occurrences in this chapter

Equation form expr-1f97d653b7ed2d2b

n+1n+1

Read as: n plus one

Means: n plus one

1 occurrence in this chapter

Equation form expr-2099e855c842be45

t1⊑s1t_1 \sqsubseteq s_1

Read as: t sub one is a subterm of s sub one

Means: t sub one is a subterm of s sub one

2 occurrences in this chapter

Equation form expr-21fedb4649f74a97

s0=0sn+1=sn+(n+1)s_0 & = 0\\ s_{n+1} & = s_n + (n+1)

Read as: s sub zero equals zero; and s sub n plus one equals s sub n plus the quantity n plus one

Means: s sub zero equals zero; and s sub n plus one equals s sub n plus the quantity n plus one

1 occurrence in this chapter

Equation form expr-22891d282d74d3ae

a⊑[b∘a]\mathrm{a} \sqsubseteq [\mathrm{b} \circ \mathrm{a}]

Read as: the letter a is a subterm of the bracketed term b circle a

Means: the letter a is a subterm of the bracketed term b circle a

2 occurrences in this chapter

Equation form expr-23873bf1e83a235d

P(n)P(n)

Read as: capital P of n

Means: capital P of n

6 occurrences in this chapter

Equation form expr-24441d9119362413

a∘b\mathrm{a} \circ \mathrm{b}

Read as: the letter a circle the letter b

Means: the letter a circle the letter b

1 occurrence in this chapter

Equation form expr-245843abef9e72e7

[[

Read as: an opening square bracket

Means: an opening square bracket

34 occurrences in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

1 occurrence in this chapter

Equation form expr-25eb3db4aac72c07

[s1[s_1

Read as: the prefix consisting of an opening square bracket followed by s sub one

Means: the prefix consisting of an opening square bracket followed by s sub one

2 occurrences in this chapter

Equation form expr-265a8fc8de551de0

<l/2< l/2

Read as: is less than l divided by two

Means: is less than l divided by two

2 occurrences in this chapter

Equation form expr-2829d5b493261acf

[s][s]

Read as: the bracketed term s

Means: the bracketed term s

1 occurrence in this chapter

Equation form expr-2b9d36a43d7551bd

a,b,c,d\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}

Read as: the letters a, b, c, and d

Means: the letters a, b, c, and d

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

1 occurrence in this chapter

Equation form expr-311cae5487839809

d\mathrm{d}

Read as: the letter d

Means: the letter d

3 occurrences in this chapter

Equation form expr-32653a8af9c29cb0

s1=a ands2=b∘c∘d,or as r1∘r2 withr1=a∘b andr2=c∘d.s_1 & = \mathrm{a} \text{ and} & s_2 & = \mathrm{b} \circ \mathrm{c} \circ \mathrm{d}, \intertext{or as $r_1 \circ r_2$ with} r_1 & = \mathrm{a} \circ b \text{ and} & r_2 &= \mathrm{c} \circ \mathrm{d}.

Read as: first decomposition: s sub one equals the letter a, and s sub two equals the string b circle c circle d; or, as r sub one circle r sub two, r sub one equals a circle b, and r sub two equals c circle d

Means: first decomposition: s sub one equals the letter a, and s sub two equals the string b circle c circle d; or, as r sub one circle r sub two, r sub one equals a circle b, and r sub two equals c circle d

1 occurrence in this chapter

Equation form expr-34c14001ab1c2005

k=0k=0

Read as: k equals zero

Means: k equals zero

1 occurrence in this chapter

Equation form expr-3a0cb9f5883dc8df

l1<kl_1 < k

Read as: l sub one is less than k

Means: l sub one is less than k

1 occurrence in this chapter

Equation form expr-3a58361c5706305a

a⊑b\mathrm{a} \sqsubseteq \mathrm{b}

Read as: the letter a is a subterm of the letter b

Means: the letter a is a subterm of the letter b

1 occurrence in this chapter

Equation form expr-3a71d3d8b32f0353

l<kl<k

Read as: l is less than k

Means: l is less than k

7 occurrences in this chapter

Equation form expr-42eddd7fd50115a0

f([a∘b])=max(f(a),f(b))+1==max(0,0)+1=1, andf([[a∘b]∘c])=max(f([a∘b]),f(c))+1==max(1,0)+1=2.f([\mathrm{a} \circ \mathrm{b}]) & = \max(f(\mathrm{a}),f(\mathrm{b})) + 1 = \\ &= \max(0, 0) + 1 = 1, \text{ and}\\ f([[\mathrm{a} \circ \mathrm{b}] \circ \mathrm{c}]) & = \max(f([\mathrm{a} \circ \mathrm{b}]), f(\mathrm{c})) + 1 = \\ & = \max(1,0) + 1 = 2.

Read as: f of the bracketed term a circle b equals the maximum of f of a and f of b, plus one, which equals the maximum of zero and zero plus one, which equals one; and f of the bracketed term whose left component is the bracketed term a circle b and whose right component is c equals the maximum of f of the bracketed term a circle b and f of c, plus one, which equals the maximum of one and zero plus one, which equals two

Means: f of the bracketed term a circle b equals the maximum of f of a and f of b, plus one, which equals the maximum of zero and zero plus one, which equals one; and f of the bracketed term whose left component is the bracketed term a circle b and whose right component is c equals the maximum of f of the bracketed term a circle b and f of c, plus one, which equals the maximum of one and zero plus one, which equals two

1 occurrence in this chapter

Equation form expr-454349e422f05297

rr

Read as: r

Means: r

7 occurrences in this chapter

Equation form expr-468d2721e090149e

5n+15n+1

Read as: five n plus one

Means: five n plus one

4 occurrences in this chapter

Equation form expr-46eee4231511cee3

t1=t2t_1 = t_2

Read as: t sub one equals t sub two

Means: t sub one equals t sub two

3 occurrences in this chapter

Equation form expr-47d04e8bce10b69f

c\mathrm{c}

Read as: the letter c

Means: the letter c

3 occurrences in this chapter

Equation form expr-48a18b5a50d9c7e5

P(1)P(1)

Read as: capital P of one

Means: capital P of one

7 occurrences in this chapter

Equation form expr-4afa70a401acfc39

≤n/2+1\le n/2 +1

Read as: is less than or equal to n divided by two, plus one

Means: is less than or equal to n divided by two, plus one

1 occurrence in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: three

1 occurrence in this chapter

Equation form expr-4ec9599fc203d176

1818

Read as: eighteen

Means: eighteen

1 occurrence in this chapter

Equation form expr-4f7e064bb6c26ad0

m2<l2/2m_2 < l_2/2

Read as: m sub two is less than l sub two divided by two

Means: m sub two is less than l sub two divided by two

1 occurrence in this chapter

Equation form expr-4fc82b26aecb47d2

1111

Read as: eleven

Means: eleven

2 occurrences in this chapter

Equation form expr-536b61d427e74677

[[c∘d][[[\mathrm{c} \circ \mathrm{d}][

Read as: the malformed string opening bracket, opening bracket, c circle d, closing bracket, opening bracket

Means: the malformed string opening bracket, opening bracket, c circle d, closing bracket, opening bracket

1 occurrence in this chapter

Equation form expr-549ae8b0f114acc4

s2s_2

Read as: s sub two

Means: s sub two

23 occurrences in this chapter

Equation form expr-54f0b300c9515311

m1m_1

Read as: m sub one

Means: m sub one

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

2 occurrences in this chapter

Equation form expr-57028f630a0b99c4

k=1k = 1

Read as: k equals one

Means: k equals one

1 occurrence in this chapter

Equation form expr-574a44731ad321fe

o(s1,s2)=[s1∘s2]o(s_1, s_2) = & [s_1 \circ s_2]

Read as: o of s sub one and s sub two equals the bracketed term s sub one circle s sub two

Means: o of s sub one and s sub two equals the bracketed term s sub one circle s sub two

1 occurrence in this chapter

Equation form expr-5753d71eaf8477f0

[a∘a][\mathrm{a} \circ \mathrm{a}]

Read as: the bracketed term a circle a

Means: the bracketed term a circle a

1 occurrence in this chapter

Equation form expr-58ba40dc1cbbac10

g(t)={0 if t is a lettermax(g(s1),g(s2))+1 if t=s1∘s2.g(t) = \begin{cases} 0 & \text{ if $t$ is a letter}\\ \max(g(s_1), g(s_2)) + 1 & \text{ if $t = s_1 \circ s_2$.} \end{cases}

Read as: g of t is defined by cases: zero if t is a letter; and the maximum of g of s sub one and g of s sub two, plus one, if t equals s sub one circle s sub two; end cases

Means: g of t is defined by cases: zero if t is a letter; and the maximum of g of s sub one and g of s sub two, plus one, if t equals s sub one circle s sub two; end cases

1 occurrence in this chapter

Equation form expr-5afa20230f875253

n∈ℕn \in \Nat

Read as: n is a natural number

Means: n is a natural number

3 occurrences in this chapter

Equation form expr-5b0f21507ed45dee

a=[b∘a]\mathrm{a} = [\mathrm{b} \circ \mathrm{a}]

Read as: the letter a equals the bracketed term b circle a

Means: the letter a equals the bracketed term b circle a

1 occurrence in this chapter

Equation form expr-5c62e091b8c0565f

PP

Read as: capital P

Means: capital P

7 occurrences in this chapter

Equation form expr-5d08324acb961fcf

t2t_2

Read as: t sub two

Means: t sub two

7 occurrences in this chapter

Equation form expr-5db105ed9849c38b

g(s1∘s2)=max(g(a),g(b∘c∘d))+1==max(0,2)+1=3while according to the other reading we getg(r1∘r2)=max(g(a∘b),g(c∘d))+1==max(1,1)+1=2g(s_1 \circ s_2) & = \max(g(\mathrm{a}), g(\mathrm{b} \circ \mathrm{c} \circ \mathrm{d})) + 1 =\\ & = \max(0,2) + 1 = 3 \intertext{while according to the other reading we get} g(r_1 \circ r_2) & = \max(g(\mathrm{a} \circ \mathrm{b}), g(\mathrm{c} \circ \mathrm{d})) + 1=\\ & = \max(1,1) + 1 = 2

Read as: under the first decomposition, g of s sub one circle s sub two equals the maximum of g of a and g of b circle c circle d, plus one, which equals the maximum of zero and two plus one, which equals three; while under the other decomposition, g of r sub one circle r sub two equals the maximum of g of a circle b and g of c circle d, plus one, which equals the maximum of one and one plus one, which equals two

Means: under the first decomposition, g of s sub one circle s sub two equals the maximum of g of a and g of b circle c circle d, plus one, which equals the maximum of zero and two plus one, which equals three; while under the other decomposition, g of r sub one circle r sub two equals the maximum of g of a circle b and g of c circle d, plus one, which equals the maximum of one and one plus one, which equals two

1 occurrence in this chapter

Equation form expr-5f716e5fe7e8a41d

b∘a∘b\mathrm{b} \circ \mathrm{a} \circ \mathrm{b}

Read as: the string b circle a circle b

Means: the string b circle a circle b

3 occurrences in this chapter

Equation form expr-5fafb923423d19e4

l<kl < k

Read as: l is less than k

Means: l is less than k

2 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

10 occurrences in this chapter

Equation form expr-614ba83ed1666f81

sk+1=sk+(k+1)s_{k+1} = s_k + (k+1)

Read as: s sub k plus one equals s sub k plus the quantity k plus one

Means: s sub k plus one equals s sub k plus the quantity k plus one

1 occurrence in this chapter

Equation form expr-61bbebac06d64589

sk=k(k+1)/2s_k = k(k+1)/2

Read as: s sub k equals the product of k and k plus one, divided by two

Means: s sub k equals the product of k and k plus one, divided by two

3 occurrences in this chapter

Equation form expr-63436afa5c02bbb0

b\mathrm{b}

Read as: the letter b

Means: the letter b

3 occurrences in this chapter

Equation form expr-6395f8eed72a7c79

l<0l<0

Read as: l is less than zero

Means: l is less than zero

2 occurrences in this chapter

Equation form expr-651ec01c3f6e67cb

m2m_2

Read as: m sub two

Means: m sub two

1 occurrence in this chapter

Equation form expr-65c74c15a686187b

oo

Read as: o

Means: o

2 occurrences in this chapter

Equation form expr-684caa4665092946

P(0+1)P(0+1)

Read as: capital P of zero plus one

Means: capital P of zero plus one

1 occurrence in this chapter

Equation form expr-6960c28d003ad4b0

s1≠r1s_1 \neq r_1

Read as: s sub one is not equal to r sub one

Means: s sub one is not equal to r sub one

1 occurrence in this chapter

Equation form expr-6acf80af558aee51

6k+26k+2

Read as: six k plus two

Means: six k plus two

1 occurrence in this chapter

Equation form expr-6b51d431df5d7f14

1212

Read as: twelve

Means: twelve

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

14 occurrences in this chapter

Equation form expr-6cb344fbbfb3800e

[r1[r_1

Read as: the prefix consisting of an opening square bracket followed by r sub one

Means: the prefix consisting of an opening square bracket followed by r sub one

2 occurrences in this chapter

Equation form expr-6cfcdd4c378d7feb

a=b\mathrm{a} = \mathrm{b}

Read as: the letter a equals the letter b

Means: the letter a equals the letter b

1 occurrence in this chapter

Equation form expr-6d6362ab00b61fc6

r2r_2

Read as: r sub two

Means: r sub two

3 occurrences in this chapter

Equation form expr-6e663cff01f19686

f(t)f(t)

Read as: f of t

Means: f of t

2 occurrences in this chapter

Equation form expr-6eeeedda664b846e

(k+1)(k+1)

Read as: the quantity k plus one

Means: the quantity k plus one

3 occurrences in this chapter

Equation form expr-6fdaf307c109495a

[s1∘s2][s_1 \circ s_2]

Read as: the bracketed term s sub one circle s sub two

Means: the bracketed term s sub one circle s sub two

11 occurrences in this chapter

Equation form expr-70fca410aad55b12

[a∘b][a \circ b]

Read as: the bracketed string a circle b

Means: the bracketed string a circle b

2 occurrences in this chapter

Equation form expr-738da5a2c220983b

[a∘a],[a∘b],[b∘a],…,[d∘d][\mathrm{a} \circ \mathrm{a}], [\mathrm{a} \circ \mathrm{b}], [\mathrm{b} \circ \mathrm{a}], \dots, [\mathrm{d} \circ \mathrm{d}]

Read as: the bracketed terms a circle a, a circle b, b circle a, and so on through d circle d

Means: the bracketed terms a circle a, a circle b, b circle a, and so on through d circle d

1 occurrence in this chapter

Equation form expr-73d429e3585e8a85

s0=0,s1=s0+1=1,s2=s1+2=1+2=3s3=s2+3=1+2+3=6, etc.s_0 & = 0,\\ s_1 & = s_0 + 1 && = 1,\\ s_2 & = s_1 + 2 && = 1 + 2 = 3\\ s_3 & = s_2 + 3 && = 1 + 2 + 3 = 6, \text{ etc.}

Read as: s sub zero equals zero; s sub one equals s sub zero plus one, which equals one; s sub two equals s sub one plus two, which equals one plus two, which equals three; s sub three equals s sub two plus three, which equals one plus two plus three, which equals six; and so on

Means: s sub zero equals zero; s sub one equals s sub zero plus one, which equals one; s sub two equals s sub one plus two, which equals one plus two, which equals three; s sub three equals s sub two plus three, which equals one plus two plus three, which equals six; and so on

1 occurrence in this chapter

Equation form expr-73d7c8e9de5e0c76

sk+1=k(k+1)2+(k+1)==k(k+1)2+2(k+1)2==k(k+1)+2(k+1)2==(k+2)(k+1)2.s_{k+1} & = \frac{k(k+1)}{2} + (k+1) = {}\\ & = \frac{k(k+1)}{2} + \frac{2(k+1)}{2} = {}\\ & = \frac{k(k+1) + 2(k+1)}{2} = {}\\ & = \frac{(k+2)(k+1)}{2}.

Read as: s sub k plus one equals k times k plus one over two, plus k plus one; this equals k times k plus one over two, plus two times k plus one over two; this equals the sum of k times k plus one and two times k plus one, all over two; this equals the product of k plus two and k plus one, divided by two

Means: s sub k plus one equals k times k plus one over two, plus k plus one; this equals k times k plus one over two, plus two times k plus one over two; this equals the sum of k times k plus one and two times k plus one, all over two; this equals the product of k plus two and k plus one, divided by two

1 occurrence in this chapter

Equation form expr-77724be1cbe5e735

m1+m2+1m_1 + m_2 + 1

Read as: m sub one plus m sub two plus one

Means: m sub one plus m sub two plus one

1 occurrence in this chapter

Equation form expr-77ceec1a9c3f6d4a

s1s_1

Read as: s sub one

Means: s sub one

28 occurrences in this chapter

Equation form expr-79d4c7f9c8579543

ℕ\Nat

Read as: the natural numbers

Means: the natural numbers

2 occurrences in this chapter

Equation form expr-7c35c5a1785d2070

k−1k-1

Read as: k minus one

Means: k minus one

2 occurrences in this chapter

Equation form expr-7c77127f7fae6fe1

r1r_1

Read as: r sub one

Means: r sub one

5 occurrences in this chapter

Equation form expr-7d579cff9643355d

=[r1∘r2]= [r_1 \circ r_2]

Read as: equals the bracketed term r sub one circle r sub two

Means: equals the bracketed term r sub one circle r sub two

1 occurrence in this chapter

Equation form expr-8072feb3a06ed4bc

[…][\dots]

Read as: an opening square bracket, an ellipsis, and a closing square bracket

Means: an opening square bracket, an ellipsis, and a closing square bracket

1 occurrence in this chapter

Equation form expr-80cd753241d12deb

o(s1,s2)o(s_1, s_2)

Read as: o of s sub one and s sub two

Means: o of s sub one and s sub two

3 occurrences in this chapter

Equation form expr-81aa5a490db29cc9

A(x)!A(x)

Read as: capital A of x

Means: capital A of x

1 occurrence in this chapter

Equation form expr-822ae77916e66920

sn=n(n+1)/2s_n = n(n+1)/2

Read as: s sub n equals the product of n and n plus one, divided by two

Means: s sub n equals the product of n and n plus one, divided by two

2 occurrences in this chapter

Equation form expr-8254c329a92850f6

kk

Read as: k

Means: k

21 occurrences in this chapter

Equation form expr-831260d605584b58

a=a\mathrm{a} = \mathrm{a}

Read as: the letter a equals the letter a

Means: the letter a equals the letter a

1 occurrence in this chapter

Equation form expr-87fe73f0944bd82a

l<2l<2

Read as: l is less than two

Means: l is less than two

2 occurrences in this chapter

Equation form expr-8950d52704165b73

6n6n

Read as: six n

Means: six n

3 occurrences in this chapter

Equation form expr-89bf8665d46c7202

t1⊑t2t_1 \sqsubseteq t_2

Read as: t sub one is a subterm of t sub two

Means: t sub one is a subterm of t sub two

7 occurrences in this chapter

Equation form expr-8a56cf81870584b5

(n+1)(n+1)

Read as: the quantity n plus one

Means: the quantity n plus one

1 occurrence in this chapter

Equation form expr-8a81608b031d678e

sk+1s_{k+1}

Read as: s sub k plus one

Means: s sub k plus one

1 occurrence in this chapter

Equation form expr-8f2f7c0bc1284ad8

<k/2< k/2

Read as: is less than k divided by two

Means: is less than k divided by two

2 occurrences in this chapter

Equation form expr-904677e5a7d1bb48

∀l(l<k→P(l))\lforall[l][(l < k \lif P(l))]

Read as: for every l, if l is less than k, then capital P of l

Means: for every l, if l is less than k, then capital P of l

1 occurrence in this chapter

Equation form expr-91ec47101c58ab98

sns_n

Read as: s sub n

Means: s sub n

1 occurrence in this chapter

Equation form expr-92b4cebba10c42ab

m1+m2+1<l12+l22+1=l1+l2+22<l1+l2+32=k/2.m_1 + m_2 + 1 < \frac{l_1}{2} + \frac{l_2}{2} + 1 = \frac{l_1+l_2+2}{2} < \frac{l_1+l_2+3}{2} = k/2.

Read as: m sub one plus m sub two plus one is less than l sub one divided by two, plus l sub two divided by two, plus one, which equals l sub one plus l sub two plus two, all divided by two, which is less than l sub one plus l sub two plus three, all divided by two, which equals k divided by two

Means: m sub one plus m sub two plus one is less than l sub one divided by two, plus l sub two divided by two, plus one, which equals l sub one plus l sub two plus two, all divided by two, which is less than l sub one plus l sub two plus three, all divided by two, which equals k divided by two

1 occurrence in this chapter

Equation form expr-9380745c7b4dfc7e

6k6k

Read as: six k

Means: six k

1 occurrence in this chapter

Equation form expr-94eaffe04c78b7f0

P(2)P(2)

Read as: capital P of two

Means: capital P of two

4 occurrences in this chapter

Equation form expr-95663f47c5513702

l1+l2+3l_1+l_2+3

Read as: l sub one plus l sub two plus three

Means: l sub one plus l sub two plus three

2 occurrences in this chapter

Equation form expr-966bae64f90f0784

∀x(A(x)→B(x))\lforall[x][(!A(x) \lif !B(x))]

Read as: for every x, if capital A of x, then capital B of x

Means: for every x, if capital A of x, then capital B of x

1 occurrence in this chapter

Equation form expr-9894f47e58b3e7f3

a∘b∘c∘d\mathrm{a} \circ \mathrm{b} \circ \mathrm{c} \circ \mathrm{d}

Read as: the string a circle b circle c circle d

Means: the string a circle b circle c circle d

1 occurrence in this chapter

Equation form expr-9acaeeb2faa6c276

P(k+1)P(k+1)

Read as: capital P of k plus one

Means: capital P of k plus one

7 occurrences in this chapter

Equation form expr-9b19467654aed1a2

k=1k=1

Read as: k equals one

Means: k equals one

1 occurrence in this chapter

Equation form expr-9b24c2ab534f3bbd

a\mathrm{a}

Read as: the letter a

Means: the letter a

7 occurrences in this chapter

Equation form expr-9c8830765f22b17f

P(l)P(l)

Read as: capital P of l

Means: capital P of l

6 occurrences in this chapter

Equation form expr-a269f7e0d4a25436

n=1n=1

Read as: n equals one

Means: n equals one

1 occurrence in this chapter

Equation form expr-a28d2de704babc9f

[[a∘b]∘d][[\mathrm{a} \circ \mathrm{b}]\circ \mathrm{d}]

Read as: the bracketed term whose left component is the bracketed term a circle b and whose right component is d

Means: the bracketed term whose left component is the bracketed term a circle b and whose right component is d

1 occurrence in this chapter

Equation form expr-a603afd1102e637b

f(t)<l(t)f(t) < l(t)

Read as: f of t is less than l of t

Means: f of t is less than l of t

1 occurrence in this chapter

Equation form expr-a821b76169533805

[t∘s][t \circ s]

Read as: the bracketed term t circle s

Means: the bracketed term t circle s

1 occurrence in this chapter

Equation form expr-aa35eb14e534c741

t=[r1∘r2]t = [r_1 \circ r_2]

Read as: t equals the bracketed term r sub one circle r sub two

Means: t equals the bracketed term r sub one circle r sub two

2 occurrences in this chapter

Equation form expr-ab69983b9cb072f8

x+1x + 1

Read as: x plus one

Means: x plus one

1 occurrence in this chapter

Equation form expr-acac86c0e609ca90

ll

Read as: l

Means: l

3 occurrences in this chapter

Equation form expr-acff89a8826b53a8

[s1∘r2[s_1 \circ r_2

Read as: the prefix consisting of an opening square bracket, s sub one, circle, and r sub two

Means: the prefix consisting of an opening square bracket, s sub one, circle, and r sub two

2 occurrences in this chapter

Equation form expr-ae7e960a10be9432

s1=b ands2=a∘b.It is also of the form r1∘r2 withr1=b∘a andr2=b.s_1 & = \mathrm{b} \text{ and} & s_2 & = \mathrm{a} \circ \mathrm{b}. \intertext{It is also of the form $r_1 \circ r_2$ with} r_1 & = \mathrm{b} \circ \mathrm{a} \text{ and} & r_2 &= \mathrm{b}.

Read as: first decomposition: s sub one equals the letter b, and s sub two equals a circle b; it is also of the form r sub one circle r sub two, where r sub one equals b circle a and r sub two equals b

Means: first decomposition: s sub one equals the letter b, and s sub two equals a circle b; it is also of the form r sub one circle r sub two, where r sub one equals b circle a and r sub two equals b

1 occurrence in this chapter

Equation form expr-b112e434482873a3

s1=r1s_1 = r_1

Read as: s sub one equals r sub one

Means: s sub one equals r sub one

3 occurrences in this chapter

Equation form expr-b1609ebe03af3a9a

[a[]∘][\mathrm{a}[]\circ]

Read as: the malformed string opening bracket, a, opening bracket, closing bracket, circle, closing bracket

Means: the malformed string opening bracket, a, opening bracket, closing bracket, circle, closing bracket

1 occurrence in this chapter

Equation form expr-b17ef6d19c7a5b1e

1616

Read as: sixteen

Means: sixteen

2 occurrences in this chapter

Equation form expr-b23cc400b399e6d0

6k+66k+6

Read as: six k plus six

Means: six k plus six

1 occurrence in this chapter

Equation form expr-b36257c92a3fa57e

P(0)P(0)

Read as: capital P of zero

Means: capital P of zero

9 occurrences in this chapter

Equation form expr-b5ca75f70927a257

t1⊑s2t_1 \sqsubseteq s_2

Read as: t sub one is a subterm of s sub two

Means: t sub one is a subterm of s sub two

2 occurrences in this chapter

Equation form expr-b82351c465a5bd9f

[b∘a][\mathrm{b} \circ \mathrm{a}]

Read as: the bracketed term b circle a

Means: the bracketed term b circle a

1 occurrence in this chapter

Equation form expr-b9f14c6be4f64429

6(k+1)6(k+1)

Read as: six times the quantity k plus one

Means: six times the quantity k plus one

1 occurrence in this chapter

Equation form expr-be8dcefb61dddd9f

t1t_1

Read as: t sub one

Means: t sub one

2 occurrences in this chapter

Equation form expr-bf325f9b50dd56e4

6k+16k+1

Read as: six k plus one

Means: six k plus one

1 occurrence in this chapter

Equation form expr-c0afe1d7e219bc4e

0<1/20 < 1/2

Read as: zero is less than one half

Means: zero is less than one half

1 occurrence in this chapter

Equation form expr-c2eb1f5cfb359787

[b∘c][\mathrm{b} \circ \mathrm{c}]

Read as: the bracketed term b circle c

Means: the bracketed term b circle c

1 occurrence in this chapter

Equation form expr-cb17bc521055938a

o(s1,s2)o(s_1,s_2)

Read as: o of s sub one and s sub two

Means: o of s sub one and s sub two

1 occurrence in this chapter

Equation form expr-cd0aa9856147b6c5

gg

Read as: g

Means: g

2 occurrences in this chapter

Equation form expr-cf917227556fc1f4

l2l_2

Read as: l sub two

Means: l sub two

1 occurrence in this chapter

Equation form expr-cfae0d4248f7142f

]]

Read as: a closing square bracket

Means: a closing square bracket

20 occurrences in this chapter

Equation form expr-d3fae59c94d1cca7

[r1∘r2][r_1 \circ r_2]

Read as: the bracketed term r sub one circle r sub two

Means: the bracketed term r sub one circle r sub two

1 occurrence in this chapter

Equation form expr-d438e055e2cd52a9

[d∘b][\mathrm{d} \circ \mathrm{b}]

Read as: the bracketed term d circle b

Means: the bracketed term d circle b

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

2 occurrences in this chapter

Equation form expr-d7d06d5e24fe3bbf

[a∘[a \circ {}

Read as: the prefix consisting of an opening square bracket, a, then a circle

Means: the prefix consisting of an opening square bracket, a, then a circle

1 occurrence in this chapter

Equation form expr-dc1f2f999a868486

t2=s1∘s2t_2 = s_1 \circ s_2

Read as: t sub two equals s sub one circle s sub two

Means: t sub two equals s sub one circle s sub two

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

1 occurrence in this chapter

Equation form expr-e3b98a4da31a127d

tt

Read as: t

Means: t

35 occurrences in this chapter

Equation form expr-e62ba7287166c75f

P(k−1)P(k-1)

Read as: capital P of k minus one

Means: capital P of k minus one

1 occurrence in this chapter

Equation form expr-e70a8203bb13d4ae

a⊑b\mathrm{a} \sqsubseteq b

Read as: the letter a is a subterm of b

Means: the letter a is a subterm of b

1 occurrence in this chapter

Equation form expr-e7f6c011776e8db7

66

Read as: six

Means: six

9 occurrences in this chapter

Equation form expr-e81eb7da8817dca6

s2=r2s_2 = r_2

Read as: s sub two equals r sub two

Means: s sub two equals r sub two

3 occurrences in this chapter

Equation form expr-ed0873d7cbb04ee0

f(t)={0 if t is a lettermax(f(s1),f(s2))+1 if t=[s1∘s2].f(t) = \begin{cases} 0 & \text{ if $t$ is a letter}\\ \max(f(s_1), f(s_2)) + 1 & \text{ if $t = [s_1 \circ s_2]$.} \end{cases}

Read as: f of t is defined by cases: zero if t is a letter; and the maximum of f of s sub one and f of s sub two, plus one, if t is the bracketed term s sub one circle s sub two; end cases

Means: f of t is defined by cases: zero if t is a letter; and the maximum of f of s sub one and f of s sub two, plus one, if t is the bracketed term s sub one circle s sub two; end cases

1 occurrence in this chapter

Equation form expr-ed18e04f44fffb19

s0=0·(0+1)/2s_0 = 0\cdot(0 + 1)/2

Read as: s sub zero equals zero times zero plus one, divided by two

Means: s sub zero equals zero times zero plus one, divided by two

1 occurrence in this chapter

Equation form expr-f498525f5f06a0b0

[s1∘s2[s_1 \circ s_2

Read as: the prefix consisting of an opening square bracket, s sub one, circle, and s sub two

Means: the prefix consisting of an opening square bracket, s sub one, circle, and s sub two

2 occurrences in this chapter

Equation form expr-f4a3a467205da1ad

[[b∘c]∘[d∘b]][[\mathrm{b} \circ \mathrm{c}] \circ [\mathrm{d} \circ \mathrm{b}]]

Read as: the bracketed term whose left component is the bracketed term b circle c and whose right component is the bracketed term d circle b

Means: the bracketed term whose left component is the bracketed term b circle c and whose right component is the bracketed term d circle b

1 occurrence in this chapter

Equation form expr-f674d1fd3c344731

<l1/2< l_1/2

Read as: is less than l sub one divided by two

Means: is less than l sub one divided by two

1 occurrence in this chapter

Equation form expr-f8b8b228b71d16e6

t=[s1∘s2]t = [s_1 \circ s_2]

Read as: t equals the bracketed term s sub one circle s sub two

Means: t equals the bracketed term s sub one circle s sub two

5 occurrences in this chapter

Equation form expr-f8ecb6921f19dc36

b∘d\mathrm{b} \circ \mathrm{d}

Read as: the string b circle d

Means: the string b circle d

1 occurrence in this chapter

Equation form expr-f8ee3fdf5d4f6514

[a∘[a∘a]][\mathrm{a} \circ [\mathrm{a} \circ \mathrm{a}]]

Read as: the bracketed term whose left component is a and whose right component is the bracketed term a circle a

Means: the bracketed term whose left component is a and whose right component is the bracketed term a circle a

1 occurrence in this chapter

Equation form expr-fcc886982cee8077

∘\circ

Read as: the circle operator

Means: the circle operator

6 occurrences in this chapter

The range of totals obtainable with n dice

The theorem states that with n dice one can throw every one of the five n plus one integer values from n through six n. The source proof uses induction beginning with one die.

Source

Recursive definition of the partial sums

Two equations define the sequence. The base equation sets s sub zero to zero. The recursion equation sets s sub n plus one to s sub n plus the quantity n plus one.

Source

First four values of the partial-sum sequence

The aligned calculation gives s sub zero as zero, s sub one as one, s sub two as one plus two, which is three, and s sub three as one plus two plus three, which is six, followed by and so on.

Source

Closed formula for the partial sums

The proposition states that s sub n equals the product of n and n plus one, divided by two.

Source

Induction-step arithmetic for the partial sums

Starting from s sub k plus one, the calculation substitutes the inductive hypothesis, expresses k plus one with denominator two, combines the numerators, and factors the result as the product of k plus two and k plus one, divided by two.

Source

Inductive definition of nice terms

The definition declares each of the letters a through d to be a nice term. If s sub one and s sub two are nice terms, their bracketed composition with the circle operator is a nice term. Nothing else is a nice term.

Source

Opening-bracket bound for nice terms

The proposition states that, for every n, the number of opening square brackets in a nice term of length n is less than n divided by two. Its source proof uses strong induction and retains the strict bound throughout.

Source

Exercise defining and bounding supernice terms

The exercise inductively defines supernice terms from four letters, unary bracketing, binary circle composition inside brackets, and an exclusion clause. It asks for a proof that a supernice term of length n has at most n divided by two plus one opening brackets. The exercise remains unsolved.

Source

Binary constructor for nice terms

The display defines o of s sub one and s sub two as the bracketed term s sub one circle s sub two.

Source

Balanced brackets in every nice term

The proposition states that the number of opening square brackets equals the number of closing square brackets in every nice term t. The source proves it by structural induction over letters and the binary constructor.

Source

Exercise on the first symbol of a nice term

The exercise asks for a structural-induction proof that no nice term begins with a closing square bracket. It remains unsolved.

Source

Proper initial segments contain more opening brackets

The proposition states that every proper initial segment of a nice term has more opening than closing square brackets. The source proof uses structural induction and enumerates all prefix positions in a binary composite.

Source

Inductive definition of the subterm relation

The definition proceeds by the structure of t sub two. For a letter, t sub one is a subterm exactly when the terms are equal. For a bracketed binary composite, t sub one is a subterm exactly when it equals the whole term or is a subterm of either immediate component.

Source

Inductive definition of bracketless terms

Every letter is a bracketless term. The circle composition of any two bracketless terms is a bracketless term, and nothing else is. The following source uses this definition to demonstrate ambiguous decomposition.

Source

Two decompositions of a bracketless term

The display decomposes b circle a circle b first as b followed by a circle b, then as b circle a followed by b. These are two different binary readings of the same unbracketed string.

Source

Unique readability of nice terms

The proposition states that a nice term is either a letter or has uniquely determined nice terms s sub one and s sub two whose bracketed circle composition is the term. The proof cites the earlier proper-initial-segment proposition.

Source

Inductive definition of depth

The depth f of a nice term t is zero when t is a letter. When t is the bracketed composition of s sub one and s sub two, its depth is the maximum of their depths plus one.

Source

Two calculations of nice-term depth

The first calculation gives depth one to the bracketed term a circle b. The second gives depth two to a term that composes that bracketed term with c.

Source

Two readings of a four-letter bracketless term

The display reads a circle b circle c circle d first as a followed by b circle c circle d, and then as a circle b followed by c circle d. The source prints the b in the second left component without the roman-letter formatting used elsewhere; its mathematical letter remains b.

Source

Conflicting values from the two bracketless readings

Under the first decomposition, the proposed depth function g returns three. Under the second decomposition, it returns two. The display demonstrates why this recursive clause does not define a function on ambiguously parsed bracketless terms.

Source

Exercise defining the length of a nice term

The exercise asks for an inductive definition of l of t, the number of symbols in the nice term t. It remains unsolved.

Source

Exercise comparing depth and length

The exercise asks for a structural-induction proof that the depth f of a nice term t is less than its length l of t, using the referenced definition of depth. It remains unsolved.

Source

Cross-reference reference-001660

the proposition on proper initial segments of nice terms

Source occurrence

Cross-reference reference-001661

the definition of the depth of a nice term

Source occurrence

Source disclosures