\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.4003 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| fong-spivak-invitation_FO0001 | 17 | 0.998 | a | | |
| fong-spivak-invitation_FO0002 | 17 | 0.998 | b | | |
| fong-spivak-invitation_FO0003 | 53 | 1.000 | \mathcal{V} | | |
| fong-spivak-invitation_FO0004 | 15 | 1.000 | f: X \rightarrow Y | | |
| fong-spivak-invitation_FO0005 | 15 | 1.000 | X | | |
| fong-spivak-invitation_FO0006 | 15 | 1.000 | Y | | |
| fong-spivak-invitation_FO0007 | 15 | 1.000 | f | | |
| fong-spivak-invitation_FO0008 | 15 | 1.000 | \mathbb{R} | | |
| fong-spivak-invitation_FO0009 | 15 | 0.980 | x | | |
| fong-spivak-invitation_FO0010 | 15 | 0.980 | f(x) | | |
| fong-spivak-invitation_FO0011 | 16 | 1.000 | f: \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO0012 | 16 | 0.412 | x \leq y | | |
| fong-spivak-invitation_FO0013 | 16 | 0.412 | f(x) \leq f(y) | | |
| fong-spivak-invitation_FO0014 | 16 | 0.412 | x, y \in \mathbb{R} ;^{1} | | |
| fong-spivak-invitation_FO0015 | 16 | 1.000 | |x-y|=|f(x)-f(y)| | | |
| fong-spivak-invitation_FO0016 | 16 | 1.000 | f(x+y)=f(x)+f(y) | | |
| fong-spivak-invitation_FO0018 | 16 | 0.942 | (x, y) | | |
| fong-spivak-invitation_FO0019 | 16 | 1.000 | y | | |
| fong-spivak-invitation_FO0020 | 16 | 1.000 | f(x)=x^{2} | | |
| fong-spivak-invitation_FO0021 | 16 | 1.000 | x, y \in \mathbb{R} | | |
| fong-spivak-invitation_FO0022 | 16 | 1.000 | x^{2} \leq y^{2} | | |
| fong-spivak-invitation_FO0023 | 17 | 1.000 | c | | |
| fong-spivak-invitation_FO0024 | 17 | 1.000 | a, b | | |
| fong-spivak-invitation_FO0025 | 17 | 1.000 | \Phi | | |
| fong-spivak-invitation_FO0026 | 17 | 1.000 | A | | |
| fong-spivak-invitation_FO0027 | 17 | 1.000 | B | | |
| fong-spivak-invitation_FO0028 | 17 | 1.000 | A \vee B | | |
| fong-spivak-invitation_FO0029 | 17 | 1.000 | z_{1}, \ldots, z_{n} | | |
| fong-spivak-invitation_FO0030 | 17 | 0.999 | B: x | | |
| fong-spivak-invitation_FO0031 | 17 | 0.999 | z_{1}, z_{i} | | |
| fong-spivak-invitation_FO0032 | 17 | 0.999 | z_{i+1} | | |
| fong-spivak-invitation_FO0033 | 17 | 0.999 | z_{n} | | |
| fong-spivak-invitation_FO0034 | 18 | 0.493 | \Phi\left({ }^{\odot}\right)=\Phi\left({ }^{\odot} 6\right)= | | |
| fong-spivak-invitation_FO0035 | 18 | 0.999 | * | | |
| fong-spivak-invitation_FO0036 | 18 | 0.444 | \Phi\left(\oplus^{\odot} \vee\right)=\Phi(\cdot \oplus)= | | |
| fong-spivak-invitation_FO0037 | 18 | 0.444 | \bullet | | |
| fong-spivak-invitation_FO0038 | 19 | 1.000 | A \leq B | | |
| fong-spivak-invitation_FO0039 | 19 | 1.000 | \leq | | |
| fong-spivak-invitation_FO0040 | 19 | 1.000 | A \leq(A \vee B) | | |
| fong-spivak-invitation_FO0041 | 19 | 1.000 | B \leq(A \vee B) | | |
| fong-spivak-invitation_FO0042 | 19 | 0.996 | C | | |
| fong-spivak-invitation_FO0043 | 19 | 0.996 | A \leq C | | |
| fong-spivak-invitation_FO0044 | 19 | 0.996 | B \leq C | | |
| fong-spivak-invitation_FO0045 | 19 | 0.996 | (A \vee B) \leq C | | |
| fong-spivak-invitation_FO0046 | 19 | 0.999 | \{\bullet, *\} | | |
| fong-spivak-invitation_FO0047 | 20 | 0.883 | \mathbb{B}=\{ | | |
| fong-spivak-invitation_FO0048 | 20 | 0.883 | \} | | |
| fong-spivak-invitation_FO0049 | 20 | 0.997 | \not \leq | | |
| fong-spivak-invitation_FO0050 | 20 | 0.999 | B .^{2} | | |
| fong-spivak-invitation_FO0051 | 20 | 1.000 | A, B | | |
| fong-spivak-invitation_FO0052 | 20 | 1.000 | \mathbb{B} | | |
| fong-spivak-invitation_FO0053 | 20 | 1.000 | \Phi(A)=\Phi(B)= | | |
| fong-spivak-invitation_FO0054 | 20 | 1.000 | \Phi(A) \vee \Phi(B)= | | |
| fong-spivak-invitation_FO0055 | 20 | 1.000 | \Phi(A \vee B)= | | |
| fong-spivak-invitation_FO0056 | 20 | 1.000 | P | | |
| fong-spivak-invitation_FO0057 | 20 | 1.000 | Q | | |
| fong-spivak-invitation_FO0058 | 21 | 1.000 | a, b, c | | |
| fong-spivak-invitation_FO0059 | 21 | 1.000 | A=\{h, 1\} | | |
| fong-spivak-invitation_FO0060 | 21 | 0.935 | h | | |
| fong-spivak-invitation_FO0061 | 21 | 0.935 | \{h, h, 1, h, 1\} | | |
| fong-spivak-invitation_FO0062 | 21 | 1.000 | x \in X | | |
| fong-spivak-invitation_FO0063 | 21 | 1.000 | h \in A | | |
| fong-spivak-invitation_FO0064 | 21 | 1.000 | 1 \in A | | |
| fong-spivak-invitation_FO0065 | 21 | 1.000 | 0 \notin A | | |
| fong-spivak-invitation_FO0066 | 21 | 1.000 | \varnothing | | |
| fong-spivak-invitation_FO0067 | 21 | 1.000 | \mathbb{N} | | |
| fong-spivak-invitation_FO0068 | 21 | 1.000 | 0,1,2,3, \ldots, 90^{717}, \ldots | | |
| fong-spivak-invitation_FO0069 | 21 | 0.996 | \underline{n} | | |
| fong-spivak-invitation_FO0070 | 21 | 0.996 | n \in \mathbb{N} | | |
| fong-spivak-invitation_FO0071 | 21 | 0.996 | n | | |
| fong-spivak-invitation_FO0072 | 21 | 0.996 | 1,2, \ldots, n | | |
| fong-spivak-invitation_FO0073 | 21 | 0.999 | \underline{0}=\varnothing, \underline{1}=\{1\} | | |
| fong-spivak-invitation_FO0074 | 21 | 0.999 | \underline{5}=\{1,2,3,4,5\} | | |
| fong-spivak-invitation_FO0075 | 21 | 0.398 | \mathbb{Z} | | |
| fong-spivak-invitation_FO0076 | 21 | 0.398 | 90^{717}, \ldots | | |
| fong-spivak-invitation_FO0077 | 21 | 1.000 | \pi, 3.14,5 * \sqrt{2}, e, e^{2},-1457 | | |
| fong-spivak-invitation_FO0078 | 21 | 0.977 | 90^{717} | | |
| fong-spivak-invitation_FO0079 | 21 | 1.000 | X \subseteq Y | | |
| fong-spivak-invitation_FO0080 | 21 | 1.000 | \{h\} \subseteq A | | |
| fong-spivak-invitation_FO0081 | 21 | 1.000 | \varnothing:=\{ \} | | |
| fong-spivak-invitation_FO0082 | 21 | 1.000 | \{y \in Y \mid P(y)\} | | |
| fong-spivak-invitation_FO0083 | 21 | 1.000 | \mathbb{N}=\{n \in \mathbb{Z} \mid n \geq 0\} | | |
| fong-spivak-invitation_FO0084 | 21 | 0.741 | h, 1 | | |
| fong-spivak-invitation_FO0085 | 21 | 0.741 | h, h, 1, h, 1 | | |
| fong-spivak-invitation_FO0086 | 21 | 0.995 | Z:= | | |
| fong-spivak-invitation_FO0087 | 21 | 0.995 | Z | | |
| fong-spivak-invitation_FO0088 | 21 | 0.995 | Z= | | |
| fong-spivak-invitation_FO0089 | 21 | 1.000 | 3+2:=5 | | |
| fong-spivak-invitation_FO0090 | 21 | 1.000 | \}:=\varnothing | | |
| fong-spivak-invitation_FO0091 | 22 | 1.000 | \mathbb{N}=\{n \in \mathbb{Z} \mid n \geq 1\} | | |
| fong-spivak-invitation_FO0092 | 22 | 1.000 | \varnothing=\{n \in \mathbb{Z} \mid 1<n<2\} | | |
| fong-spivak-invitation_FO0093 | 22 | 1.000 | X_{1} | | |
| fong-spivak-invitation_FO0094 | 22 | 1.000 | X_{2} | | |
| fong-spivak-invitation_FO0095 | 22 | 1.000 | X_{1} \cup X_{2} | | |
| fong-spivak-invitation_FO0096 | 22 | 1.000 | Y=\{1,2,3,4\} | | |
| fong-spivak-invitation_FO0097 | 22 | 1.000 | X_{1}=\{1,2\} | | |
| fong-spivak-invitation_FO0098 | 22 | 1.000 | X_{2}=\{2,4\} | | |
| fong-spivak-invitation_FO0099 | 22 | 1.000 | X_{1} \cup X_{2}=\{1,2,4\} | | |
| fong-spivak-invitation_FO0100 | 22 | 1.000 | \varnothing \cup X=X | | |
| fong-spivak-invitation_FO0101 | 22 | 1.000 | X_{1} \cap X_{2} | | |
| fong-spivak-invitation_FO0102 | 22 | 1.000 | \{1,2,3\} \cap\{2,5\}=\{2\} | | |
| fong-spivak-invitation_FO0103 | 22 | 1.000 | X_{0}=\varnothing, X_{1}=\{1\}, X_{2}=\{1,2\} | | |
| fong-spivak-invitation_FO0104 | 22 | 1.000 | \bigcup_{n \in \mathbb{N}} X_{n} | | |
| fong-spivak-invitation_FO0105 | 22 | 0.994 | X_{n} | | |
| fong-spivak-invitation_FO0106 | 22 | 0.994 | \bigcup_{n \in \mathbb{N}} X_{n}=\{n \in \mathbb{N} \mid | | |
| fong-spivak-invitation_FO0107 | 22 | 1.000 | n \geq 1\} | | |
| fong-spivak-invitation_FO0108 | 22 | 1.000 | \bigcap_{n \in \mathbb{N}} X_{n} | | |
| fong-spivak-invitation_FO0109 | 22 | 0.998 | X \times Y | | |
| fong-spivak-invitation_FO0110 | 22 | 1.000 | y \in Y | | |
| fong-spivak-invitation_FO0111 | 22 | 1.000 | X \sqcup Y | | |
| fong-spivak-invitation_FO0112 | 22 | 1.000 | (x, 1) | | |
| fong-spivak-invitation_FO0113 | 22 | 1.000 | (y, 2) | | |
| fong-spivak-invitation_FO0114 | 22 | 1.000 | A:=\{h, 1\} | | |
| fong-spivak-invitation_FO0115 | 22 | 1.000 | B:=\{1,2,3\} | | |
| fong-spivak-invitation_FO0116 | 22 | 1.000 | A \times B | | |
| fong-spivak-invitation_FO0117 | 22 | 1.000 | A \sqcup B | | |
| fong-spivak-invitation_FO0118 | 22 | 0.711 | \{h, 1,2,3\} | | |
| fong-spivak-invitation_FO0119 | 22 | 1.000 | A \cup B | | |
| fong-spivak-invitation_FO0120 | 22 | 0.998 | R \subseteq X \times Y | | |
| fong-spivak-invitation_FO0121 | 22 | 1.000 | R \subseteq X \times X | | |
| fong-spivak-invitation_FO0122 | 22 | 1.000 | R \subseteq A \times A | | |
| fong-spivak-invitation_FO0123 | 22 | 0.992 | a \star b | | |
| fong-spivak-invitation_FO0124 | 22 | 0.992 | (a, b) \in R | | |
| fong-spivak-invitation_FO0125 | 22 | 1.000 | (5,6) \in R | | |
| fong-spivak-invitation_FO0126 | 22 | 1.000 | 5 \leq 6 | | |
| fong-spivak-invitation_FO0127 | 23 | 0.997 | =, \approx,<,> | | |
| fong-spivak-invitation_FO0128 | 23 | 1.000 | p \in P | | |
| fong-spivak-invitation_FO0129 | 23 | 1.000 | A_{p} \subseteq A | | |
| fong-spivak-invitation_FO0130 | 23 | 1.000 | \left\{A_{p}\right\}_{p \in P} | | |
| fong-spivak-invitation_FO0131 | 23 | 1.000 | A_{p} | | |
| fong-spivak-invitation_FO0132 | 23 | 1.000 | p | | |
| fong-spivak-invitation_FO0133 | 23 | 1.000 | a \in A | | |
| fong-spivak-invitation_FO0134 | 23 | 1.000 | \left\{A_{p^{\prime}}^{\prime}\right\}_{p^{\prime} \in P^{\prime}} | | |
| fong-spivak-invitation_FO0135 | 23 | 1.000 | p^{\prime} \in P^{\prime} | | |
| fong-spivak-invitation_FO0136 | 23 | 1.000 | A_{p}=A_{p^{\prime}}^{\prime} | | |
| fong-spivak-invitation_FO0137 | 23 | 1.000 | a, b \in\{11,12,13,21,22,23\} | | |
| fong-spivak-invitation_FO0138 | 23 | 1.000 | a \sim b | | |
| fong-spivak-invitation_FO0139 | 23 | 0.923 | (a, b) | | |
| fong-spivak-invitation_FO0140 | 23 | 0.923 | 10 .^{5} | | |
| fong-spivak-invitation_FO0141 | 24 | 0.998 | \sim | | |
| fong-spivak-invitation_FO0142 | 24 | 1.000 | a \sim a | | |
| fong-spivak-invitation_FO0143 | 24 | 1.000 | { }^{6} b \sim a | | |
| fong-spivak-invitation_FO0144 | 24 | 1.000 | a, b \in A | | |
| fong-spivak-invitation_FO0145 | 24 | 1.000 | b \sim c | | |
| fong-spivak-invitation_FO0146 | 24 | 1.000 | a \sim c | | |
| fong-spivak-invitation_FO0147 | 24 | 1.000 | a, b, c \in A | | |
| fong-spivak-invitation_FO0148 | 24 | 1.000 | a \in A_{p} | | |
| fong-spivak-invitation_FO0149 | 24 | 1.000 | b \in A_{p} | | |
| fong-spivak-invitation_FO0150 | 24 | 0.945 | X \subseteq A | | |
| fong-spivak-invitation_FO0151 | 24 | 0.980 | x^{\prime} \sim x | | |
| fong-spivak-invitation_FO0152 | 24 | 0.980 | x^{\prime} \in X | | |
| fong-spivak-invitation_FO0153 | 24 | 0.840 | x \sim y | | |
| fong-spivak-invitation_FO0154 | 24 | 0.840 | x, y \in X | | |
| fong-spivak-invitation_FO0155 | 24 | 0.998 | (\sim) | | |
| fong-spivak-invitation_FO0156 | 24 | 1.000 | p \neq q | | |
| fong-spivak-invitation_FO0157 | 24 | 1.000 | A_{q} | | |
| fong-spivak-invitation_FO0158 | 24 | 1.000 | A_{p} \cap A_{q}=\varnothing | | |
| fong-spivak-invitation_FO0159 | 24 | 1.000 | A=\bigcup_{p \in P} A_{p} | | |
| fong-spivak-invitation_FO0160 | 24 | 1.000 | A / \sim | | |
| fong-spivak-invitation_FO0161 | 25 | 1.000 | S | | |
| fong-spivak-invitation_FO0162 | 25 | 1.000 | T | | |
| fong-spivak-invitation_FO0163 | 25 | 1.000 | F \subseteq S \times T | | |
| fong-spivak-invitation_FO0164 | 25 | 1.000 | s \in S | | |
| fong-spivak-invitation_FO0165 | 25 | 1.000 | t \in T | | |
| fong-spivak-invitation_FO0166 | 25 | 1.000 | (s, t) \in F | | |
| fong-spivak-invitation_FO0167 | 25 | 1.000 | F | | |
| fong-spivak-invitation_FO0168 | 25 | 1.000 | F: S \rightarrow T | | |
| fong-spivak-invitation_FO0169 | 25 | 1.000 | F(s)=t | | |
| fong-spivak-invitation_FO0170 | 25 | 1.000 | s \mapsto t | | |
| fong-spivak-invitation_FO0171 | 25 | 1.000 | t | | |
| fong-spivak-invitation_FO0172 | 25 | 1.000 | f^{-1}(t):=\{s \in S \mid F(s)=t\} | | |
| fong-spivak-invitation_FO0173 | 25 | 1.000 | s_{1}, s_{2} \in S | | |
| fong-spivak-invitation_FO0174 | 25 | 1.000 | F\left(s_{1}\right)=t | | |
| fong-spivak-invitation_FO0175 | 25 | 1.000 | F\left(s_{2}\right)=t | | |
| fong-spivak-invitation_FO0176 | 25 | 1.000 | s_{1}=s_{2} | | |
| fong-spivak-invitation_FO0177 | 25 | 0.918 | \rightarrow, \rightarrow, \succ, \stackrel{\cong}{\rightrightarrows} | | |
| fong-spivak-invitation_FO0178 | 25 | 0.680 | \mathrm{id}_{X} | | |
| fong-spivak-invitation_FO0179 | 25 | 0.680 | \mathrm{id}_{X}(x)=x | | |
| fong-spivak-invitation_FO0180 | 25 | 1.000 | f: A \rightarrow B | | |
| fong-spivak-invitation_FO0181 | 26 | 1.000 | f: A \rightarrow \varnothing | | |
| fong-spivak-invitation_FO0182 | 26 | 1.000 | f: A \rightarrow P | | |
| fong-spivak-invitation_FO0183 | 26 | 1.000 | f^{-1}(p) \subseteq A | | |
| fong-spivak-invitation_FO0184 | 26 | 1.000 | S:=\{11,12,13,21,22,23\} | | |
| fong-spivak-invitation_FO0185 | 26 | 1.000 | P=\{a, b, c, d\} | | |
| fong-spivak-invitation_FO0186 | 26 | 1.000 | f: S \rightarrow P | | |
| fong-spivak-invitation_FO0187 | 26 | 1.000 | F: X \rightarrow Y | | |
| fong-spivak-invitation_FO0188 | 26 | 1.000 | G: Y \rightarrow Z | | |
| fong-spivak-invitation_FO0189 | 26 | 0.999 | X \rightarrow Z | | |
| fong-spivak-invitation_FO0190 | 26 | 0.999 | G(F(x)) | | |
| fong-spivak-invitation_FO0191 | 26 | 0.996 | G \circ F | | |
| fong-spivak-invitation_FO0192 | 26 | 0.996 | F ; G | | |
| fong-spivak-invitation_FO0193 | 26 | 1.000 | F(x) \in Y | | |
| fong-spivak-invitation_FO0194 | 26 | 1.000 | G | | |
| fong-spivak-invitation_FO0195 | 26 | 0.997 | \{1\} \rightarrow X | | |
| fong-spivak-invitation_FO0196 | 26 | 0.997 | \{1\} \rightarrow | | |
| fong-spivak-invitation_FO0197 | 27 | 1.000 | x:\{1\} \rightarrow X | | |
| fong-spivak-invitation_FO0198 | 27 | 1.000 | F(x)=x ; F | | |
| fong-spivak-invitation_FO0199 | 27 | 1.000 | x \leq x | | |
| fong-spivak-invitation_FO0200 | 27 | 1.000 | y \leq z | | |
| fong-spivak-invitation_FO0201 | 27 | 1.000 | x \leq z | | |
| fong-spivak-invitation_FO0202 | 27 | 0.837 | y \leq x | | |
| fong-spivak-invitation_FO0203 | 27 | 0.837 | x \cong y | | |
| fong-spivak-invitation_FO0204 | 27 | 0.837 | X, \leq | | |
| fong-spivak-invitation_FO0205 | 27 | 0.981 | (X,=) | | |
| fong-spivak-invitation_FO0206 | 27 | 1.000 | x \neq y | | |
| fong-spivak-invitation_FO0207 | 27 | 0.931 | X \times X \subseteq X \times X | | |
| fong-spivak-invitation_FO0208 | 28 | 0.999 | x=y | | |
| fong-spivak-invitation_FO0209 | 28 | 1.000 | x, y | | |
| fong-spivak-invitation_FO0210 | 28 | 1.000 | G=(V, A, s, t) | | |
| fong-spivak-invitation_FO0211 | 28 | 1.000 | V | | |
| fong-spivak-invitation_FO0212 | 28 | 1.000 | s, t: A \rightarrow V | | |
| fong-spivak-invitation_FO0213 | 28 | 1.000 | s(a)=v | | |
| fong-spivak-invitation_FO0214 | 28 | 1.000 | t(a)=w | | |
| fong-spivak-invitation_FO0215 | 28 | 1.000 | v | | |
| fong-spivak-invitation_FO0216 | 28 | 1.000 | w | | |
| fong-spivak-invitation_FO0217 | 28 | 1.000 | V=\{1,2,3,4\} | | |
| fong-spivak-invitation_FO0218 | 28 | 1.000 | A=\{a, b, c, d, e\} | | |
| fong-spivak-invitation_FO0221 | 120 | 0.999 | e | | |
| fong-spivak-invitation_FO0222 | 29 | 0.928 | 1 \rightarrow 2 | | |
| fong-spivak-invitation_FO0223 | 29 | 1.000 | d | | |
| fong-spivak-invitation_FO0224 | 29 | 0.531 | P, \leq | | |
| fong-spivak-invitation_FO0225 | 29 | 1.000 | v \leq w | | |
| fong-spivak-invitation_FO0226 | 29 | 1.000 | v \rightarrow w | | |
| fong-spivak-invitation_FO0227 | 29 | 1.000 | v \rightarrow v | | |
| fong-spivak-invitation_FO0228 | 29 | 1.000 | u \rightarrow v | | |
| fong-spivak-invitation_FO0229 | 29 | 1.000 | u \rightarrow w | | |
| fong-spivak-invitation_FO0230 | 29 | 0.998 | (P, \leq) | | |
| fong-spivak-invitation_FO0231 | 29 | 0.898 | \left(p_{1}, p_{2}\right) | | |
| fong-spivak-invitation_FO0232 | 29 | 1.000 | p_{1} \leq p_{2} | | |
| fong-spivak-invitation_FO0233 | 29 | 1.000 | \{\bullet, \circ, *\} | | |
| fong-spivak-invitation_FO0234 | 29 | 1.000 | \mathbb{N}:=\{0,1,2,3, \ldots\} | | |
| fong-spivak-invitation_FO0235 | 29 | 1.000 | 0 \leq 1 | | |
| fong-spivak-invitation_FO0236 | 29 | 1.000 | 5 \leq 100 | | |
| fong-spivak-invitation_FO0237 | 29 | 1.000 | m \leq n | | |
| fong-spivak-invitation_FO0238 | 29 | 1.000 | n \leq m | | |
| fong-spivak-invitation_FO0239 | 29 | 1.000 | m, n | | |
| fong-spivak-invitation_FO0240 | 30 | 0.989 | a \leq b | | |
| fong-spivak-invitation_FO0241 | 30 | 0.989 | b \leq a | | |
| fong-spivak-invitation_FO0242 | 30 | 1.000 | n=m | | |
| fong-spivak-invitation_FO0243 | 30 | 1.000 | 5 \leq 3 | | |
| fong-spivak-invitation_FO0244 | 30 | 1.000 | 3 \leq 2 | | |
| fong-spivak-invitation_FO0245 | 30 | 1.000 | m | | |
| fong-spivak-invitation_FO0246 | 30 | 1.000 | 2 \leq 4 | | |
| fong-spivak-invitation_FO0247 | 30 | 1.000 | 2 \not \leq 3 | | |
| fong-spivak-invitation_FO0248 | 30 | 0.994 | 1,2, \ldots, 10 | | |
| fong-spivak-invitation_FO0249 | 30 | 0.994 | a \rightarrow b | | |
| fong-spivak-invitation_FO0250 | 30 | 0.999 | -500 \leq-499 \leq 0 \leq \sqrt{2} \leq 100 / 3 | | |
| fong-spivak-invitation_FO0251 | 30 | 1.000 | \mathrm{P}(X) | | |
| fong-spivak-invitation_FO0252 | 31 | 1.000 | X=\{0,1,2\} | | |
| fong-spivak-invitation_FO0253 | 31 | 0.933 | \mathrm{P}\{1,2, \ldots, n\},{ }^{7} | | |
| fong-spivak-invitation_FO0254 | 31 | 0.998 | \mathrm{P}(\varnothing), \mathrm{P}\{1\} | | |
| fong-spivak-invitation_FO0255 | 31 | 0.998 | \mathrm{P}\{1,2\} | | |
| fong-spivak-invitation_FO0256 | 31 | 0.981 | \operatorname{Prt}(A) | | |
| fong-spivak-invitation_FO0257 | 31 | 0.999 | q \in Q | | |
| fong-spivak-invitation_FO0258 | 31 | 1.000 | A_{p} \subseteq A_{q} | | |
| fong-spivak-invitation_FO0259 | 31 | 1.000 | g: A \rightarrow Q | | |
| fong-spivak-invitation_FO0260 | 31 | 1.000 | h: P \rightarrow Q | | |
| fong-spivak-invitation_FO0261 | 31 | 1.000 | f ; h=g | | |
| fong-spivak-invitation_FO0262 | 31 | 0.519 | U | | |
| fong-spivak-invitation_FO0263 | 31 | 1.000 | p \in U | | |
| fong-spivak-invitation_FO0264 | 31 | 1.000 | p \leq q | | |
| fong-spivak-invitation_FO0265 | 31 | 1.000 | q \in U | | |
| fong-spivak-invitation_FO0266 | 31 | 0.913 | U(P) | | |
| fong-spivak-invitation_FO0267 | 31 | 0.906 | U \leq V | | |
| fong-spivak-invitation_FO0268 | 31 | 0.973 | (\mathbb{B}, \leq) | | |
| fong-spivak-invitation_FO0269 | 31 | 0.997 | U(\mathbb{B}) | | |
| fong-spivak-invitation_FO0270 | 31 | 1.000 | \mathrm{P} X | | |
| fong-spivak-invitation_FO0271 | 32 | 0.845 | \subseteq \mathbb{B} | | |
| fong-spivak-invitation_FO0272 | 32 | 0.845 | \notin | | |
| fong-spivak-invitation_FO0273 | 32 | 0.996 | (Q, \leq) | | |
| fong-spivak-invitation_FO0274 | 32 | 1.000 | P \times Q | | |
| fong-spivak-invitation_FO0275 | 32 | 1.000 | (p, q) \leq\left(p^{\prime}, q^{\prime}\right) | | |
| fong-spivak-invitation_FO0276 | 32 | 1.000 | p \leq p^{\prime} | | |
| fong-spivak-invitation_FO0277 | 32 | 1.000 | q \leq q^{\prime} | | |
| fong-spivak-invitation_FO0278 | 32 | 0.763 | \left(P, \leq^{\mathrm{op}}\right) | | |
| fong-spivak-invitation_FO0279 | 32 | 0.763 | p \leq^{\mathrm{op}} q | | |
| fong-spivak-invitation_FO0280 | 32 | 1.000 | q \leq p | | |
| fong-spivak-invitation_FO0281 | 32 | 0.999 | A, \leq_{A} | | |
| fong-spivak-invitation_FO0282 | 32 | 0.999 | B, \leq_{B} | | |
| fong-spivak-invitation_FO0283 | 32 | 0.993 | x, y \in A | | |
| fong-spivak-invitation_FO0284 | 32 | 0.993 | x \leq_{A} y | | |
| fong-spivak-invitation_FO0285 | 32 | 0.993 | f(x) \leq_{B} f(y) | | |
| fong-spivak-invitation_FO0286 | 32 | 1.000 | A \rightarrow B | | |
| fong-spivak-invitation_FO0287 | 33 | 1.000 | x \in A | | |
| fong-spivak-invitation_FO0288 | 33 | 1.000 | f(x) \in B | | |
| fong-spivak-invitation_FO0289 | 33 | 1.000 | f(y) | | |
| fong-spivak-invitation_FO0290 | 33 | 1.000 | \mathbb{B} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO0291 | 34 | 1.000 | x \rightarrow y | | |
| fong-spivak-invitation_FO0292 | 34 | 1.000 | F(x) \rightarrow F(y) | | |
| fong-spivak-invitation_FO0293 | 34 | 0.998 | |\cdot|: \mathrm{P}(X) \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO0294 | 34 | 0.996 | |S| | | |
| fong-spivak-invitation_FO0295 | 34 | 1.000 | 0 \leq 1 \leq 2 \leq 3 | | |
| fong-spivak-invitation_FO0296 | 34 | 0.948 | |\cdot| | | |
| fong-spivak-invitation_FO0297 | 34 | 0.993 | \mathrm{U}(P) \rightarrow \mathrm{P}(P) | | |
| fong-spivak-invitation_FO0298 | 34 | 0.998 | \mathrm{P}(\mathbb{B}) | | |
| fong-spivak-invitation_FO0299 | 34 | 1.000 | \uparrow p:=\left\{p^{\prime} \in P \mid p \leq p^{\prime}\right\} | | |
| fong-spivak-invitation_FO0300 | 34 | 0.801 | \uparrow: P^{\mathrm{op}} \rightarrow \mathrm{U}(P) | | |
| fong-spivak-invitation_FO0301 | 34 | 0.999 | \uparrow\left(p^{\prime}\right) \subseteq \uparrow(p) | | |
| fong-spivak-invitation_FO0302 | 34 | 0.952 | b \geq a \leq c | | |
| fong-spivak-invitation_FO0303 | 34 | 1.000 | f:\left(P, \leq_{P}\right) \rightarrow\left(Q, \leq_{Q}\right) | | |
| fong-spivak-invitation_FO0304 | 34 | 0.999 | f: P \rightarrow Q | | |
| fong-spivak-invitation_FO0305 | 34 | 0.999 | \left(P, \leq_{P}\right) | | |
| fong-spivak-invitation_FO0306 | 34 | 1.000 | P \rightarrow Q | | |
| fong-spivak-invitation_FO0307 | 34 | 1.000 | \leq_{Q} | | |
| fong-spivak-invitation_FO0308 | 35 | 1.000 | \operatorname{Prt}(X) | | |
| fong-spivak-invitation_FO0309 | 35 | 1.000 | s: X \rightarrow P | | |
| fong-spivak-invitation_FO0310 | 35 | 1.000 | f^{*}: \operatorname{Prt}(Y) \rightarrow \operatorname{Prt}(X) | | |
| fong-spivak-invitation_FO0311 | 35 | 1.000 | s: Y \rightarrow P | | |
| fong-spivak-invitation_FO0312 | 35 | 0.727 | s | | |
| fong-spivak-invitation_FO0313 | 35 | 0.727 | X \rightarrow P | | |
| fong-spivak-invitation_FO0314 | 35 | 1.000 | f^{*}(P) | | |
| fong-spivak-invitation_FO0315 | 35 | 1.000 | f^{*}(Q) | | |
| fong-spivak-invitation_FO0316 | 35 | 0.865 | P, \leq_{P} | | |
| fong-spivak-invitation_FO0317 | 35 | 0.794 | \left(Q, \leq_{Q}\right) | | |
| fong-spivak-invitation_FO0318 | 35 | 0.794 | \left(R, \leq_{R}\right) | | |
| fong-spivak-invitation_FO0319 | 35 | 0.794 | g: Q \rightarrow R | | |
| fong-spivak-invitation_FO0320 | 35 | 0.991 | (f ; g): P \rightarrow R | | |
| fong-spivak-invitation_FO0321 | 35 | 0.514 | \operatorname{id}_{P}: P \rightarrow P | | |
| fong-spivak-invitation_FO0322 | 35 | 0.514 | (P, \leq) \rightarrow\left(P, \leq^{\text {op }}\right) | | |
| fong-spivak-invitation_FO0323 | 35 | 1.000 | p, q \in P | | |
| fong-spivak-invitation_FO0324 | 35 | 0.795 | p_{1}=p_{2} | | |
| fong-spivak-invitation_FO0325 | 36 | 1.000 | g: Q \rightarrow P | | |
| fong-spivak-invitation_FO0326 | 36 | 0.957 | f ; g=\operatorname{id}_{P} | | |
| fong-spivak-invitation_FO0327 | 36 | 0.957 | g ; f=\operatorname{id}_{Q} | | |
| fong-spivak-invitation_FO0328 | 36 | 1.000 | g | | |
| fong-spivak-invitation_FO0329 | 36 | 1.000 | P, Q | | |
| fong-spivak-invitation_FO0330 | 36 | 1.000 | R | | |
| fong-spivak-invitation_FO0331 | 36 | 1.000 | f(a)=v, f(b)=w, f(c)=x, f(d)=y | | |
| fong-spivak-invitation_FO0332 | 36 | 1.000 | f(e)=z | | |
| fong-spivak-invitation_FO0333 | 36 | 0.743 | x \rightarrow z | | |
| fong-spivak-invitation_FO0334 | 36 | 1.000 | v \rightarrow y | | |
| fong-spivak-invitation_FO0335 | 36 | 0.657 | \operatorname{Prt}(\{*, \bullet, \circ\}) \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0336 | 36 | 1.000 | P \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0337 | 37 | 1.000 | f: P \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0338 | 37 | 1.000 | f^{-1} | | |
| fong-spivak-invitation_FO0339 | 37 | 1.000 | \subseteq P | | |
| fong-spivak-invitation_FO0340 | 37 | 1.000 | p \in f^{-1} | | |
| fong-spivak-invitation_FO0341 | 37 | 0.617 | =f(p) \leq f(q) | | |
| fong-spivak-invitation_FO0342 | 37 | 0.617 | \leq f(q) | | |
| fong-spivak-invitation_FO0343 | 37 | 0.617 | =f(q) | | |
| fong-spivak-invitation_FO0344 | 37 | 1.000 | q \in f^{-1} | | |
| fong-spivak-invitation_FO0345 | 37 | 0.522 | f_{U}: P \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0346 | 37 | 0.522 | f_{U}(p)=\operatorname{true} | | |
| fong-spivak-invitation_FO0347 | 37 | 1.000 | f_{U}(p)= | | |
| fong-spivak-invitation_FO0348 | 37 | 1.000 | p \notin U | | |
| fong-spivak-invitation_FO0349 | 37 | 1.000 | f_{U}(p)=\operatorname{true}=f(q) | | |
| fong-spivak-invitation_FO0350 | 37 | 1.000 | \leq f_{U}(q) | | |
| fong-spivak-invitation_FO0351 | 37 | 0.890 | f^{*}: \mathrm{U}(Q) \rightarrow \mathrm{U}(P) | | |
| fong-spivak-invitation_FO0352 | 37 | 1.000 | U \subseteq Q | | |
| fong-spivak-invitation_FO0353 | 37 | 1.000 | f^{-1}(U) \subseteq P | | |
| fong-spivak-invitation_FO0354 | 37 | 1.000 | f^{*} | | |
| fong-spivak-invitation_FO0355 | 37 | 1.000 | u: Q \rightarrow | | |
| fong-spivak-invitation_FO0356 | 37 | 1.000 | (f ; u): P \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0357 | 37 | 0.798 | (\mathbb{R}, \leq) | | |
| fong-spivak-invitation_FO0358 | 37 | 1.000 | \mathbb{N} \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO0359 | 37 | 1.000 | \left\{\left.\frac{1}{n+1} \right\rvert\, n \in \mathbb{N}\right\} \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO0360 | 38 | 0.752 | A \subseteq P | | |
| fong-spivak-invitation_FO0361 | 38 | 1.000 | p \leq a | | |
| fong-spivak-invitation_FO0362 | 38 | 1.000 | q | | |
| fong-spivak-invitation_FO0363 | 38 | 1.000 | q \leq a | | |
| fong-spivak-invitation_FO0364 | 38 | 0.998 | p=\wedge A, p=\wedge_{a \in A} a | | |
| fong-spivak-invitation_FO0365 | 38 | 0.999 | p=\wedge_{A} a | | |
| fong-spivak-invitation_FO0366 | 38 | 0.999 | A=\{a, b\} | | |
| fong-spivak-invitation_FO0367 | 38 | 0.999 | \wedge A | | |
| fong-spivak-invitation_FO0368 | 38 | 1.000 | a \wedge b | | |
| fong-spivak-invitation_FO0369 | 38 | 1.000 | a \leq p | | |
| fong-spivak-invitation_FO0370 | 38 | 1.000 | a \leq q | | |
| fong-spivak-invitation_FO0371 | 38 | 0.777 | p=\bigvee A | | |
| fong-spivak-invitation_FO0372 | 38 | 0.777 | p=\bigvee_{a \in A} a | | |
| fong-spivak-invitation_FO0373 | 38 | 1.000 | p=a \vee b | | |
| fong-spivak-invitation_FO0374 | 38 | 1.000 | p=\wedge A | | |
| fong-spivak-invitation_FO0375 | 38 | 1.000 | q=\wedge A | | |
| fong-spivak-invitation_FO0376 | 38 | 1.000 | p, q | | |
| fong-spivak-invitation_FO0377 | 38 | 1.000 | p \leq | | |
| fong-spivak-invitation_FO0378 | 38 | 1.000 | p \cong q | | |
| fong-spivak-invitation_FO0379 | 38 | 1.000 | x \in P | | |
| fong-spivak-invitation_FO0380 | 38 | 1.000 | p \leq x | | |
| fong-spivak-invitation_FO0381 | 38 | 1.000 | q \leq x | | |
| fong-spivak-invitation_FO0382 | 38 | 1.000 | x \leq p | | |
| fong-spivak-invitation_FO0383 | 38 | 1.000 | x \leq q | | |
| fong-spivak-invitation_FO0384 | 38 | 1.000 | P=\{p, q, r\} | | |
| fong-spivak-invitation_FO0385 | 38 | 1.000 | A=\{p, q\} | | |
| fong-spivak-invitation_FO0386 | 39 | 0.630 | a, b\} | | |
| fong-spivak-invitation_FO0387 | 39 | 0.992 | c \leq d | | |
| fong-spivak-invitation_FO0388 | 39 | 0.992 | d \leq c | | |
| fong-spivak-invitation_FO0389 | 39 | 0.992 | c \cong d | | |
| fong-spivak-invitation_FO0390 | 39 | 1.000 | A=\{p\} | | |
| fong-spivak-invitation_FO0391 | 39 | 1.000 | \wedge A \cong p | | |
| fong-spivak-invitation_FO0392 | 39 | 1.000 | \wedge A=p | | |
| fong-spivak-invitation_FO0393 | 39 | 0.994 | \wedge | | |
| fong-spivak-invitation_FO0394 | 39 | 0.994 | \vee | | |
| fong-spivak-invitation_FO0395 | 39 | 1.000 | p \vee p=p \wedge p=p | | |
| fong-spivak-invitation_FO0396 | 39 | 0.996 | p \vee p | | |
| fong-spivak-invitation_FO0397 | 39 | 0.996 | \bigvee\{p, p\} | | |
| fong-spivak-invitation_FO0398 | 39 | 0.996 | \{p, p\}=\{p\} | | |
| fong-spivak-invitation_FO0399 | 39 | 1.000 | p \vee p=p | | |
| fong-spivak-invitation_FO0400 | 39 | 1.000 | A, B \subseteq X | | |
| fong-spivak-invitation_FO0401 | 39 | 1.000 | A \wedge B=A \cap B | | |
| fong-spivak-invitation_FO0402 | 39 | 1.000 | A \vee B= | | |
| fong-spivak-invitation_FO0403 | 40 | 1.000 | n \mid m | | |
| fong-spivak-invitation_FO0404 | 40 | 0.924 | A \subseteq B \subseteq P | | |
| fong-spivak-invitation_FO0405 | 40 | 1.000 | \wedge B \leq \wedge A | | |
| fong-spivak-invitation_FO0406 | 40 | 1.000 | \bigvee A \leq \bigvee B | | |
| fong-spivak-invitation_FO0407 | 40 | 1.000 | m=\wedge A | | |
| fong-spivak-invitation_FO0408 | 40 | 1.000 | n=\wedge B | | |
| fong-spivak-invitation_FO0409 | 40 | 1.000 | a \in B | | |
| fong-spivak-invitation_FO0410 | 40 | 1.000 | n \leq a | | |
| fong-spivak-invitation_FO0411 | 40 | 1.000 | \Phi: P \rightarrow Q | | |
| fong-spivak-invitation_FO0412 | 40 | 1.000 | f(a \wedge | | |
| fong-spivak-invitation_FO0413 | 40 | 1.000 | b) \cong f(a) \wedge f(b) | | |
| fong-spivak-invitation_FO0414 | 40 | 1.000 | a, b \in P | | |
| fong-spivak-invitation_FO0415 | 40 | 1.000 | f(a \vee b) \cong | | |
| fong-spivak-invitation_FO0416 | 40 | 1.000 | f(a) \vee f(b) | | |
| fong-spivak-invitation_FO0417 | 40 | 1.000 | f(a \vee b) | | |
| fong-spivak-invitation_FO0418 | 40 | 1.000 | a \vee b | | |
| fong-spivak-invitation_FO0419 | 41 | 1.000 | f(a \vee | | |
| fong-spivak-invitation_FO0420 | 41 | 1.000 | b) \neq f(a) \vee f(b) | | |
| fong-spivak-invitation_FO0421 | 41 | 1.000 | f(a), f(b) \in Q | | |
| fong-spivak-invitation_FO0422 | 41 | 1.000 | f(a) \vee f(b) \leq f(a \vee b) | | |
| fong-spivak-invitation_FO0423 | 41 | 0.927 | (3 \times-): \mathbb{Z} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO0424 | 41 | 0.927 | x \in \mathbb{Z} | | |
| fong-spivak-invitation_FO0425 | 41 | 0.927 | 3 x | | |
| fong-spivak-invitation_FO0426 | 41 | 1.000 | 3 x \in \mathbb{Z} \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO0427 | 42 | 0.509 | \lceil\mathrm{z}\rceil | | |
| fong-spivak-invitation_FO0428 | 42 | 0.509 | \mathrm{z} \in \mathbb{R} | | |
| fong-spivak-invitation_FO0429 | 42 | 0.509 | \lfloor z\rfloor | | |
| fong-spivak-invitation_FO0430 | 42 | 0.937 | z \in \mathbb{R} | | |
| fong-spivak-invitation_FO0431 | 42 | 0.937 | \lceil 3.14\rceil=4 | | |
| fong-spivak-invitation_FO0432 | 42 | 0.937 | \lfloor 3.14\rfloor=3 .{ }^{9} | | |
| fong-spivak-invitation_FO0433 | 42 | 0.937 | \mathbb{R} \rightarrow \mathbb{Z} | | |
| fong-spivak-invitation_FO0434 | 42 | 0.994 | \lceil-/ 3\rceil | | |
| fong-spivak-invitation_FO0435 | 42 | 0.651 | (3 \times-) | | |
| fong-spivak-invitation_FO0436 | 42 | 1.000 | P=Q=\underline{3} | | |
| fong-spivak-invitation_FO0437 | 42 | 1.000 | f, g | | |
| fong-spivak-invitation_FO0438 | 42 | 1.000 | 1 \leq p, q \leq 3 | | |
| fong-spivak-invitation_FO0439 | 42 | 1.000 | f(p) \leq q | | |
| fong-spivak-invitation_FO0440 | 42 | 1.000 | p \leq g(q) | | |
| fong-spivak-invitation_FO0441 | 42 | 1.000 | L: \mathbb{Z} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO0442 | 42 | 1.000 | \lfloor 3\rfloor=3=\lceil 3\rceil | | |
| fong-spivak-invitation_FO0443 | 43 | 1.000 | \operatorname{Prt}(S) | | |
| fong-spivak-invitation_FO0444 | 43 | 1.000 | g: S \rightarrow T | | |
| fong-spivak-invitation_FO0445 | 43 | 0.882 | g_{!}: \operatorname{Prt}(S) \leftrightarrows \operatorname{Prt}(T): g^{*} | | |
| fong-spivak-invitation_FO0446 | 43 | 0.993 | g_{!}: \operatorname{Prt}(S) \rightarrow \operatorname{Prt}(T) | | |
| fong-spivak-invitation_FO0447 | 43 | 1.000 | \sim_{S} | | |
| fong-spivak-invitation_FO0448 | 43 | 1.000 | \sim_{T} | | |
| fong-spivak-invitation_FO0449 | 43 | 0.983 | t_{1}, t_{2} \in T | | |
| fong-spivak-invitation_FO0450 | 43 | 0.983 | t_{1} \sim_{T} t_{2} | | |
| fong-spivak-invitation_FO0451 | 43 | 0.997 | s_{1} \sim_{S} s_{2} | | |
| fong-spivak-invitation_FO0452 | 43 | 0.997 | g\left(s_{1}\right)=t_{1} | | |
| fong-spivak-invitation_FO0453 | 43 | 0.997 | g\left(s_{2}\right)=t_{2} | | |
| fong-spivak-invitation_FO0454 | 43 | 0.769 | t_{2} \sim_{T} t_{3} | | |
| fong-spivak-invitation_FO0455 | 43 | 0.769 | t_{1} \sim_{T}^{?} t_{3}- | | |
| fong-spivak-invitation_FO0456 | 43 | 1.000 | g_{!}\left(\sim_{S}\right):=\sim_{T} | | |
| fong-spivak-invitation_FO0457 | 43 | 0.998 | g^{*} | | |
| fong-spivak-invitation_FO0458 | 43 | 0.998 | \operatorname{Prt}(T) \rightarrow \operatorname{Prt}(S) | | |
| fong-spivak-invitation_FO0459 | 43 | 1.000 | g\left(s_{1}\right) \sim_{T} | | |
| fong-spivak-invitation_FO0460 | 43 | 1.000 | g\left(s_{2}\right) | | |
| fong-spivak-invitation_FO0461 | 43 | 1.000 | g^{*}\left(\sim_{T}\right):=\sim_{S} | | |
| fong-spivak-invitation_FO0462 | 43 | 1.000 | S=\{1,2,3,4\}, T=\{12,3,4\} | | |
| fong-spivak-invitation_FO0463 | 43 | 1.000 | g(1):= | | |
| fong-spivak-invitation_FO0464 | 43 | 1.000 | g(2):=12, g(3):=3 | | |
| fong-spivak-invitation_FO0465 | 43 | 1.000 | g(4):=4 | | |
| fong-spivak-invitation_FO0466 | 44 | 0.872 | c: S \rightarrow P | | |
| fong-spivak-invitation_FO0467 | 44 | 0.872 | g_{!}(c) | | |
| fong-spivak-invitation_FO0468 | 44 | 0.872 | S, T | | |
| fong-spivak-invitation_FO0469 | 44 | 1.000 | T= | | |
| fong-spivak-invitation_FO0470 | 44 | 0.746 | \{12,3,4\} | | |
| fong-spivak-invitation_FO0471 | 44 | 1.000 | g^{*}(c) | | |
| fong-spivak-invitation_FO0472 | 44 | 1.000 | c: T \rightarrow P | | |
| fong-spivak-invitation_FO0473 | 44 | 1.000 | g^{*}: \operatorname{Prt}(T) \rightarrow \operatorname{Prt}(S) | | |
| fong-spivak-invitation_FO0474 | 44 | 1.000 | d: T \rightarrow P^{\prime} | | |
| fong-spivak-invitation_FO0475 | 44 | 1.000 | g_{!}(c) \leq d | | |
| fong-spivak-invitation_FO0476 | 44 | 0.999 | e: T \rightarrow Q | | |
| fong-spivak-invitation_FO0477 | 44 | 0.999 | g_{!}(c) \not \leq e | | |
| fong-spivak-invitation_FO0478 | 44 | 1.000 | g^{*}(d) | | |
| fong-spivak-invitation_FO0479 | 44 | 1.000 | g^{*}(e) | | |
| fong-spivak-invitation_FO0480 | 44 | 1.000 | c \leq g^{*}(d) | | |
| fong-spivak-invitation_FO0481 | 44 | 1.000 | c \not \leq g^{*}(e) | | |
| fong-spivak-invitation_FO0482 | 45 | 1.000 | q:=f(p) | | |
| fong-spivak-invitation_FO0483 | 45 | 1.000 | p \leq g(f(p)) | | |
| fong-spivak-invitation_FO0484 | 45 | 1.000 | f(g(q)) \leq q | | |
| fong-spivak-invitation_FO0485 | 45 | 1.000 | g(f(p)) \leq | | |
| fong-spivak-invitation_FO0486 | 45 | 1.000 | g(q) | | |
| fong-spivak-invitation_FO0487 | 45 | 1.000 | g^{\prime} | | |
| fong-spivak-invitation_FO0488 | 45 | 1.000 | g(q) \cong g^{\prime}(q) | | |
| fong-spivak-invitation_FO0489 | 45 | 1.000 | A \subseteq Q | | |
| fong-spivak-invitation_FO0490 | 45 | 1.000 | g(A):=\{g(a) \mid a \in A\} | | |
| fong-spivak-invitation_FO0491 | 45 | 1.000 | \wedge A \in Q | | |
| fong-spivak-invitation_FO0492 | 45 | 1.000 | g(A) | | |
| fong-spivak-invitation_FO0493 | 45 | 1.000 | \wedge g(A) | | |
| fong-spivak-invitation_FO0494 | 45 | 1.000 | \bigvee A \in P | | |
| fong-spivak-invitation_FO0495 | 45 | 1.000 | f(A) | | |
| fong-spivak-invitation_FO0496 | 45 | 1.000 | \bigvee f(A) | | |
| fong-spivak-invitation_FO0497 | 45 | 1.000 | m:=\wedge A | | |
| fong-spivak-invitation_FO0498 | 45 | 1.000 | g(m) \leq g(a) | | |
| fong-spivak-invitation_FO0499 | 45 | 1.000 | g(m) | | |
| fong-spivak-invitation_FO0500 | 45 | 1.000 | b \leq g(a) | | |
| fong-spivak-invitation_FO0501 | 45 | 1.000 | b \leq g(m) | | |
| fong-spivak-invitation_FO0502 | 45 | 0.999 | f(b) \leq a | | |
| fong-spivak-invitation_FO0503 | 45 | 0.999 | f(b) | | |
| fong-spivak-invitation_FO0504 | 45 | 1.000 | f(b) \leq m | | |
| fong-spivak-invitation_FO0505 | 46 | 1.000 | f(3.9):=4 | | |
| fong-spivak-invitation_FO0506 | 46 | 1.000 | 1 \vee 2=4 | | |
| fong-spivak-invitation_FO0507 | 46 | 1.000 | g(1) \vee g(2)=1 \vee 2=3.9 \neq 4=g(4) | | |
| fong-spivak-invitation_FO0508 | 47 | 1.000 | \left\{q^{\prime} \in Q \mid p^{\prime} \leq g\left(q^{\prime}\right)\right\} \subseteq | | |
| fong-spivak-invitation_FO0509 | 47 | 1.000 | \{q \in Q \mid p \leq g(q)\} | | |
| fong-spivak-invitation_FO0510 | 47 | 1.000 | f(p) \leq f\left(p^{\prime}\right) | | |
| fong-spivak-invitation_FO0511 | 47 | 1.000 | p_{0} \leq g\left(f\left(p_{0}\right)\right) | | |
| fong-spivak-invitation_FO0512 | 47 | 1.000 | f\left(g\left(q_{0}\right)\right) \leq q_{0} | | |
| fong-spivak-invitation_FO0513 | 47 | 1.000 | p_{0} \in P | | |
| fong-spivak-invitation_FO0514 | 47 | 1.000 | q_{0} \in Q | | |
| fong-spivak-invitation_FO0515 | 47 | 1.000 | p_{0} | | |
| fong-spivak-invitation_FO0516 | 47 | 1.000 | \left\{q_{0}\right\} \subseteq\left\{q \in Q \mid g\left(q_{0}\right) \leq\right. | | |
| fong-spivak-invitation_FO0517 | 47 | 0.990 | g(q)\} | | |
| fong-spivak-invitation_FO0518 | 47 | 0.990 | \wedge\left\{q_{0}\right\}=q_{0} | | |
| fong-spivak-invitation_FO0519 | 47 | 0.999 | f^{*}: \mathrm{P}(B) \rightarrow \mathrm{P}(A) | | |
| fong-spivak-invitation_FO0520 | 47 | 1.000 | B^{\prime} \subseteq B | | |
| fong-spivak-invitation_FO0521 | 47 | 1.000 | f^{-1}\left(B^{\prime}\right) \subseteq A | | |
| fong-spivak-invitation_FO0522 | 47 | 1.000 | B^{\prime} | | |
| fong-spivak-invitation_FO0523 | 47 | 1.000 | f_{!}(A) | | |
| fong-spivak-invitation_FO0524 | 47 | 1.000 | A^{\prime} \subseteq A | | |
| fong-spivak-invitation_FO0525 | 47 | 1.000 | A^{\prime} | | |
| fong-spivak-invitation_FO0526 | 47 | 1.000 | f_{*} | | |
| fong-spivak-invitation_FO0527 | 47 | 0.923 | f^{*}, f_{!} | | |
| fong-spivak-invitation_FO0528 | 48 | 1.000 | B_{1}, B_{2} \subseteq Y | | |
| fong-spivak-invitation_FO0529 | 48 | 1.000 | f^{*}\left(B_{1}\right) | | |
| fong-spivak-invitation_FO0530 | 48 | 1.000 | f^{*}\left(B_{2}\right) | | |
| fong-spivak-invitation_FO0531 | 48 | 1.000 | A_{1}, A_{2} \subseteq X | | |
| fong-spivak-invitation_FO0532 | 48 | 1.000 | f_{!}\left(A_{1}\right) | | |
| fong-spivak-invitation_FO0533 | 48 | 1.000 | f_{!}\left(A_{2}\right) | | |
| fong-spivak-invitation_FO0534 | 48 | 1.000 | f_{*}\left(A_{1}\right) | | |
| fong-spivak-invitation_FO0535 | 48 | 1.000 | f_{*}\left(A_{2}\right) | | |
| fong-spivak-invitation_FO0536 | 48 | 1.000 | f ; g: P \rightarrow P | | |
| fong-spivak-invitation_FO0537 | 48 | 0.999 | p \leq(f \circ g)(p) | | |
| fong-spivak-invitation_FO0538 | 48 | 0.999 | (f \circ g \circ f \circ g)(p) \cong(f \circ g)(p) | | |
| fong-spivak-invitation_FO0539 | 48 | 1.000 | (f ; g ; f ; g)(p) \cong(f ; g)(p) | | |
| fong-spivak-invitation_FO0540 | 48 | 1.000 | \geq | | |
| fong-spivak-invitation_FO0541 | 48 | 1.000 | j: P \rightarrow P | | |
| fong-spivak-invitation_FO0542 | 48 | 0.999 | p \leq j(p) | | |
| fong-spivak-invitation_FO0543 | 48 | 1.000 | j(j(p)) \cong j(p) | | |
| fong-spivak-invitation_FO0544 | 48 | 1.000 | 7+2+3 | | |
| fong-spivak-invitation_FO0545 | 48 | 0.936 | j: \exp \rightarrow \exp | | |
| fong-spivak-invitation_FO0546 | 48 | 1.000 | j(x) | | |
| fong-spivak-invitation_FO0547 | 48 | 1.000 | j(y) | | |
| fong-spivak-invitation_FO0548 | 48 | 1.000 | x \leq j(x) | | |
| fong-spivak-invitation_FO0549 | 48 | 1.000 | j | | |
| fong-spivak-invitation_FO0550 | 48 | 1.000 | j(j(x))=j(x) | | |
| fong-spivak-invitation_FO0551 | 48 | 0.906 | (g \circ f)(q) \leq q | | |
| fong-spivak-invitation_FO0552 | 48 | 0.906 | (g \circ f \circ g \circ f)(q) \cong(g \circ f)(q) | | |
| fong-spivak-invitation_FO0553 | 49 | 0.673 | _{j} | | |
| fong-spivak-invitation_FO0554 | 49 | 0.506 | \mathrm{fix}_{j} | | |
| fong-spivak-invitation_FO0555 | 49 | 1.000 | j(p) | | |
| fong-spivak-invitation_FO0556 | 49 | 1.000 | j: P \rightarrow \mathrm{fix}_{j} | | |
| fong-spivak-invitation_FO0557 | 49 | 0.888 | g: | | |
| fong-spivak-invitation_FO0558 | 49 | 0.888 | _{j} \rightarrow P | | |
| fong-spivak-invitation_FO0559 | 49 | 0.979 | q \in \operatorname{fix}_{j} | | |
| fong-spivak-invitation_FO0560 | 49 | 0.979 | j(p) \leq q | | |
| fong-spivak-invitation_FO0561 | 49 | 1.000 | j(p) \leq j(q) \cong q | | |
| fong-spivak-invitation_FO0562 | 49 | 1.000 | p \Rightarrow q | | |
| fong-spivak-invitation_FO0563 | 49 | 1.000 | p \Rightarrow p | | |
| fong-spivak-invitation_FO0564 | 49 | 1.000 | q \Rightarrow r | | |
| fong-spivak-invitation_FO0565 | 49 | 1.000 | p \Rightarrow r | | |
| fong-spivak-invitation_FO0566 | 49 | 1.000 | p \Rightarrow j(p) | | |
| fong-spivak-invitation_FO0567 | 49 | 1.000 | j(j(p))=j(p) | | |
| fong-spivak-invitation_FO0568 | 49 | 1.000 | B \Rightarrow p | | |
| fong-spivak-invitation_FO0569 | 49 | 1.000 | B \Rightarrow- | | |
| fong-spivak-invitation_FO0570 | 50 | 0.951 | \mathbf{R e l}(S) | | |
| fong-spivak-invitation_FO0571 | 50 | 0.951 | R \in \mathbf{R e l}(S) | | |
| fong-spivak-invitation_FO0572 | 50 | 0.958 | R \subseteq S \times S | | |
| fong-spivak-invitation_FO0573 | 50 | 0.958 | \boldsymbol{\operatorname { R e l }}(S) | | |
| fong-spivak-invitation_FO0574 | 50 | 1.000 | R \subseteq R^{\prime} | | |
| fong-spivak-invitation_FO0575 | 50 | 1.000 | R\left(s_{1}, s_{2}\right) | | |
| fong-spivak-invitation_FO0576 | 50 | 1.000 | R^{\prime}\left(s_{1}, s_{2}\right) | | |
| fong-spivak-invitation_FO0577 | 50 | 0.999 | \boldsymbol{\operatorname { R e l }}(\{1\}) | | |
| fong-spivak-invitation_FO0578 | 50 | 0.948 | \boldsymbol{\operatorname { R e l }}(\{1,2\}) | | |
| fong-spivak-invitation_FO0579 | 50 | 1.000 | \boldsymbol{\operatorname { P o s }}(S) | | |
| fong-spivak-invitation_FO0580 | 50 | 0.936 | \sqsubseteq | | |
| fong-spivak-invitation_FO0581 | 50 | 0.936 | \leq^{\prime} | | |
| fong-spivak-invitation_FO0582 | 50 | 0.989 | \leq \sqsubseteq \leq^{\prime} | | |
| fong-spivak-invitation_FO0583 | 50 | 0.989 | a \leq^{\prime} b | | |
| fong-spivak-invitation_FO0584 | 50 | 0.989 | a, b \in S | | |
| fong-spivak-invitation_FO0585 | 50 | 1.000 | \boldsymbol{\operatorname { P o s }}(S) \rightarrow \boldsymbol{\operatorname { R e l }}(S) | | |
| fong-spivak-invitation_FO0586 | 50 | 0.995 | \mathrm{Cl}: \boldsymbol{\operatorname { R e l }}(S) \rightarrow \boldsymbol{\operatorname { P o s }}(S) | | |
| fong-spivak-invitation_FO0587 | 50 | 1.000 | s \leq s | | |
| fong-spivak-invitation_FO0588 | 50 | 1.000 | s \leq u | | |
| fong-spivak-invitation_FO0589 | 50 | 1.000 | s \leq t | | |
| fong-spivak-invitation_FO0590 | 50 | 1.000 | t \leq u | | |
| fong-spivak-invitation_FO0591 | 50 | 1.000 | S=\{1,2,3\} | | |
| fong-spivak-invitation_FO0592 | 50 | 1.000 | U(\leq) | | |
| fong-spivak-invitation_FO0593 | 50 | 1.000 | U(\leq):=\left\{\left(s_{1}, s_{2}\right) \mid s_{1} \leq s_{2}\right\} \subseteq S \times S | | |
| fong-spivak-invitation_FO0594 | 50 | 1.000 | Q \subseteq S \times S | | |
| fong-spivak-invitation_FO0595 | 50 | 1.000 | Q^{\prime} \subseteq S \times S | | |
| fong-spivak-invitation_FO0596 | 50 | 1.000 | Q \subseteq U(\leq) | | |
| fong-spivak-invitation_FO0597 | 50 | 1.000 | Q^{\prime} \nsubseteq U(\leq) | | |
| fong-spivak-invitation_FO0598 | 50 | 1.000 | Q, Q^{\prime} | | |
| fong-spivak-invitation_FO0599 | 50 | 0.825 | \mathrm{Cl}(Q) \sqsubseteq \leq | | |
| fong-spivak-invitation_FO0600 | 50 | 0.825 | \operatorname{Pos}(S) | | |
| fong-spivak-invitation_FO0601 | 50 | 0.508 | \mathrm{Cl}\left(Q^{\prime}\right) \nsubseteq \leq | | |
| fong-spivak-invitation_FO0602 | 51 | 1.000 | a \otimes b | | |
| fong-spivak-invitation_FO0605 | 53 | 1.000 | d \in[0, \infty] | | |
| fong-spivak-invitation_FO0606 | 54 | 0.994 | (X, \leq) | | |
| fong-spivak-invitation_FO0607 | 54 | 1.000 | I \in X | | |
| fong-spivak-invitation_FO0608 | 54 | 1.000 | \otimes: X \times X \rightarrow X | | |
| fong-spivak-invitation_FO0609 | 54 | 0.974 | \otimes\left(x_{1}, x_{2}\right)= | | |
| fong-spivak-invitation_FO0610 | 54 | 1.000 | x_{1} \otimes x_{2} | | |
| fong-spivak-invitation_FO0611 | 54 | 1.000 | x_{1}, x_{2}, y_{1}, y_{2} \in X | | |
| fong-spivak-invitation_FO0612 | 54 | 1.000 | x_{1} \leq y_{1} | | |
| fong-spivak-invitation_FO0613 | 54 | 1.000 | x_{2} \leq y_{2} | | |
| fong-spivak-invitation_FO0614 | 54 | 1.000 | x_{1} \otimes x_{2} \leq y_{1} \otimes y_{2} | | |
| fong-spivak-invitation_FO0615 | 54 | 1.000 | I \otimes x=x | | |
| fong-spivak-invitation_FO0616 | 54 | 1.000 | x \otimes I=x | | |
| fong-spivak-invitation_FO0617 | 54 | 1.000 | x, y, z \in X | | |
| fong-spivak-invitation_FO0618 | 54 | 1.000 | (x \otimes y) \otimes z=x \otimes(y \otimes z) | | |
| fong-spivak-invitation_FO0619 | 54 | 1.000 | x \otimes y=y \otimes x | | |
| fong-spivak-invitation_FO0620 | 54 | 0.828 | X, \leq, I, \otimes | | |
| fong-spivak-invitation_FO0621 | 55 | 1.000 | I | | |
| fong-spivak-invitation_FO0622 | 55 | 0.487 | \cong | | |
| fong-spivak-invitation_FO0623 | 55 | 0.962 | \{*\} | | |
| fong-spivak-invitation_FO0624 | 55 | 0.572 | +, *, \wedge, \vee | | |
| fong-spivak-invitation_FO0625 | 55 | 0.572 | \times | | |
| fong-spivak-invitation_FO0626 | 55 | 1.000 | -5 \leq \sqrt{2} | | |
| fong-spivak-invitation_FO0627 | 55 | 1.000 | +: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO0628 | 55 | 0.887 | \mathbb{R}, \leq, 0,+ | | |
| fong-spivak-invitation_FO0629 | 55 | 1.000 | x_{1}+x_{2} \leq y_{1}+y_{2} | | |
| fong-spivak-invitation_FO0630 | 55 | 1.000 | 0+x=x | | |
| fong-spivak-invitation_FO0631 | 55 | 0.988 | x+0=x | | |
| fong-spivak-invitation_FO0632 | 55 | 0.988 | (x+y)+z=x+(y+z) | | |
| fong-spivak-invitation_FO0633 | 55 | 0.988 | x+y=y+x | | |
| fong-spivak-invitation_FO0634 | 55 | 0.988 | (\mathbb{R}, \leq, 0,+) | | |
| fong-spivak-invitation_FO0635 | 55 | 0.842 | \mathbb{R}, \leq | | |
| fong-spivak-invitation_FO0636 | 55 | 0.975 | M | | |
| fong-spivak-invitation_FO0637 | 55 | 0.975 | *: M \times M \rightarrow M | | |
| fong-spivak-invitation_FO0638 | 55 | 1.000 | e \in M | | |
| fong-spivak-invitation_FO0639 | 55 | 1.000 | *(m, n) | | |
| fong-spivak-invitation_FO0640 | 55 | 1.000 | m * n | | |
| fong-spivak-invitation_FO0641 | 55 | 1.000 | m, n, p \in M | | |
| fong-spivak-invitation_FO0642 | 55 | 1.000 | m * n=n * m | | |
| fong-spivak-invitation_FO0643 | 55 | 0.481 | _{S} | | |
| fong-spivak-invitation_FO0644 | 55 | 0.481 | m=n | | |
| fong-spivak-invitation_FO0645 | 55 | 0.769 | (M, e, *) | | |
| fong-spivak-invitation_FO0646 | 55 | 0.570 | \left(\right. | | |
| fong-spivak-invitation_FO0647 | 55 | 0.570 | \left._{M},=, e, *\right) | | |
| fong-spivak-invitation_FO0648 | 55 | 0.932 | (M, *, e) | | |
| fong-spivak-invitation_FO0649 | 55 | 0.366 | \left(\boldsymbol{D i s c}_{M},=, *, e\right) | | |
| fong-spivak-invitation_FO0650 | 55 | 1.000 | H | | |
| fong-spivak-invitation_FO0651 | 56 | 1.000 | h \leq h^{\prime} | | |
| fong-spivak-invitation_FO0652 | 56 | 1.000 | h^{\prime} | | |
| fong-spivak-invitation_FO0653 | 56 | 1.000 | \otimes: H \times H \rightarrow H | | |
| fong-spivak-invitation_FO0654 | 56 | 1.000 | h_{1} \otimes h_{2} | | |
| fong-spivak-invitation_FO0655 | 56 | 1.000 | h_{1} | | |
| fong-spivak-invitation_FO0656 | 56 | 1.000 | h_{2} | | |
| fong-spivak-invitation_FO0657 | 56 | 0.179 | \{2 \odot, 5 \curvearrowright, 6 \odot, 6 \infty, 7 \rightarrow\}=\{2 \odot, 3 \odot, 4 \curvearrowright, 5 \curvearrowright, 6 \bigodot\} | | |
| fong-spivak-invitation_FO0658 | 56 | 1.000 | h_{1} \leq i_{1} | | |
| fong-spivak-invitation_FO0659 | 56 | 1.000 | h_{2} \leq i_{2} | | |
| fong-spivak-invitation_FO0660 | 56 | 1.000 | h_{1} \otimes h_{2} \leq i_{1} \otimes i_{2} | | |
| fong-spivak-invitation_FO0661 | 56 | 0.201 | h_{1} \otimes h_{2}=\{10 \wedge, \mathrm{~J} \in, \mathrm{Q}, \mathrm{K}, \mathrm{A}\} | | |
| fong-spivak-invitation_FO0662 | 56 | 0.409 | i_{1} \otimes i_{2}=\{5 \bullet, 5 \oslash, 6 \circlearrowright, 6 \diamond, Q \diamond\} | | |
| fong-spivak-invitation_FO0663 | 57 | 0.689 | (x \in X) | | |
| fong-spivak-invitation_FO0664 | 57 | 0.689 | (\leq) | | |
| fong-spivak-invitation_FO0665 | 57 | 0.921 | (X, \leq I, \otimes) | | |
| fong-spivak-invitation_FO0666 | 57 | 1.000 | x \otimes y | | |
| fong-spivak-invitation_FO0667 | 58 | 0.999 | x_{1} \otimes x_{2} \leq y_{1} \otimes y_{2} \otimes y_{3} | | |
| fong-spivak-invitation_FO0668 | 58 | 0.933 | 4 \leq 7 | | |
| fong-spivak-invitation_FO0669 | 58 | 1.000 | 2+5 \leq-1+5+3 | | |
| fong-spivak-invitation_FO0670 | 58 | 1.000 | x_{1} \otimes x_{2} \leq | | |
| fong-spivak-invitation_FO0671 | 58 | 1.000 | y_{1} \otimes y_{2} | | |
| fong-spivak-invitation_FO0672 | 59 | 0.977 | \mathcal{X}=(X, \leq, I, \otimes) | | |
| fong-spivak-invitation_FO0673 | 59 | 0.973 | \mathcal{X} | | |
| fong-spivak-invitation_FO0674 | 60 | 1.000 | v+w+u | | |
| fong-spivak-invitation_FO0675 | 60 | 1.000 | (v+w)+u | | |
| fong-spivak-invitation_FO0676 | 60 | 1.000 | v+(w+u) | | |
| fong-spivak-invitation_FO0677 | 61 | 1.000 | 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{Na} | | |
| fong-spivak-invitation_FO0678 | 61 | 1.000 | 2 \mathrm{NaOH}+\mathrm{H}_{2} | | |
| fong-spivak-invitation_FO0679 | 61 | 0.688 | \mathrm{NaCl} \in \mathrm{Mat} | | |
| fong-spivak-invitation_FO0680 | 61 | 0.484 | 4 \mathrm{H}_{2} \mathrm{O}+6 \mathrm{Ne} \in | | |
| fong-spivak-invitation_FO0681 | 62 | 0.725 | \otimes | | |
| fong-spivak-invitation_FO0682 | 62 | 1.000 | k | | |
| fong-spivak-invitation_FO0683 | 62 | 1.000 | x+y \rightarrow z | | |
| fong-spivak-invitation_FO0684 | 62 | 1.000 | x+y+k \rightarrow z+k | | |
| fong-spivak-invitation_FO0685 | 63 | 1.000 | x \leq I | | |
| fong-spivak-invitation_FO0686 | 63 | 1.000 | x \rightarrow 0 | | |
| fong-spivak-invitation_FO0687 | 63 | 1.000 | 10^{23} | | |
| fong-spivak-invitation_FO0688 | 64 | 1.000 | x \leq x+x | | |
| fong-spivak-invitation_FO0689 | 65 | 0.953 | =0 | | |
| fong-spivak-invitation_FO0690 | 65 | 0.953 | \operatorname{true}=1 | | |
| fong-spivak-invitation_FO0691 | 65 | 0.905 | =(\mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0692 | 65 | 0.905 | \wedge) | | |
| fong-spivak-invitation_FO0693 | 65 | 0.975 | \mathcal{V}=(V, \leq, I, \otimes) | | |
| fong-spivak-invitation_FO0694 | 66 | 0.921 | \mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0695 | 66 | 0.850 | (\mathbb{N}, \leq) | | |
| fong-spivak-invitation_FO0696 | 66 | 0.850 | 5 \leq 16 | | |
| fong-spivak-invitation_FO0697 | 66 | 0.999 | 6+4=10 | | |
| fong-spivak-invitation_FO0698 | 66 | 1.000 | 6 * 4=24 | | |
| fong-spivak-invitation_FO0699 | 66 | 1.000 | m \mid n | | |
| fong-spivak-invitation_FO0700 | 66 | 0.999 | 1 \mid m | | |
| fong-spivak-invitation_FO0701 | 66 | 0.999 | 4 \mid 12 | | |
| fong-spivak-invitation_FO0702 | 66 | 0.970 | (\mathbb{N}, \mid) | | |
| fong-spivak-invitation_FO0703 | 66 | 1.000 | x_{1} \mid y_{1} | | |
| fong-spivak-invitation_FO0704 | 66 | 1.000 | x_{2} \mid y_{2} | | |
| fong-spivak-invitation_FO0705 | 66 | 1.000 | \left(x_{1} * x_{2}\right) \mid\left(y_{1} * y_{2}\right) | | |
| fong-spivak-invitation_FO0706 | 66 | 1.000 | p_{1}, p_{2} \in \mathbb{N} | | |
| fong-spivak-invitation_FO0707 | 66 | 1.000 | x_{1} * p_{1}=y_{1} | | |
| fong-spivak-invitation_FO0708 | 66 | 1.000 | x_{2} * p_{2}=y_{2} | | |
| fong-spivak-invitation_FO0709 | 66 | 1.000 | \left(p_{1} * p_{2}\right) *\left(x_{1} * x_{2}\right)=y_{1} * y_{2} | | |
| fong-spivak-invitation_FO0710 | 67 | 1.000 | \mathrm{P}(S) | | |
| fong-spivak-invitation_FO0711 | 67 | 1.000 | \varnothing \subseteq S | | |
| fong-spivak-invitation_FO0712 | 67 | 1.000 | S \subseteq S | | |
| fong-spivak-invitation_FO0713 | 67 | 1.000 | A \subseteq B | | |
| fong-spivak-invitation_FO0714 | 67 | 1.000 | A \cap B | | |
| fong-spivak-invitation_FO0715 | 67 | 0.958 | { }^{\mathbb{N}} | | |
| fong-spivak-invitation_FO0716 | 67 | 0.422 | \operatorname{Prop}^{\mathbb{N}} | | |
| fong-spivak-invitation_FO0717 | 67 | 1.000 | n=2 | | |
| fong-spivak-invitation_FO0718 | 67 | 1.000 | n \geq 11 | | |
| fong-spivak-invitation_FO0719 | 67 | 1.000 | n+2=5 | | |
| fong-spivak-invitation_FO0720 | 67 | 0.998 | P, Q \in \operatorname{Prop}^{\mathbb{N}} | | |
| fong-spivak-invitation_FO0721 | 67 | 0.998 | P \leq Q | | |
| fong-spivak-invitation_FO0722 | 67 | 1.000 | P(n) | | |
| fong-spivak-invitation_FO0723 | 67 | 1.000 | Q(n) | | |
| fong-spivak-invitation_FO0724 | 67 | 1.000 | [0, \infty] | | |
| fong-spivak-invitation_FO0725 | 67 | 0.820 | 0,1,15.33 \overline{3} | | |
| fong-spivak-invitation_FO0726 | 67 | 0.820 | 2 \pi | | |
| fong-spivak-invitation_FO0727 | 67 | 0.820 | \infty | | |
| fong-spivak-invitation_FO0728 | 67 | 0.972 | [0, \infty], \geq | | |
| fong-spivak-invitation_FO0729 | 67 | 0.972 | \infty \geq x | | |
| fong-spivak-invitation_FO0730 | 67 | 1.000 | x \in[0, \infty] | | |
| fong-spivak-invitation_FO0731 | 67 | 0.993 | x+\infty=\infty | | |
| fong-spivak-invitation_FO0732 | 67 | 1.000 | (X, \leq)^{\mathrm{op}}:=(X, \geq) | | |
| fong-spivak-invitation_FO0733 | 68 | 0.514 | \mathcal{X}=(X, \leq) | | |
| fong-spivak-invitation_FO0734 | 68 | 0.514 | X^{\mathrm{op}}=(X, \geq) | | |
| fong-spivak-invitation_FO0735 | 68 | 0.840 | (X, \leq, I, \otimes) | | |
| fong-spivak-invitation_FO0736 | 68 | 1.000 | (X, \geq, I, \otimes) | | |
| fong-spivak-invitation_FO0737 | 68 | 0.481 | x_{1} \geq y_{1} | | |
| fong-spivak-invitation_FO0738 | 68 | 0.481 | x_{2} \geq y_{2} | | |
| fong-spivak-invitation_FO0739 | 68 | 0.481 | \mathcal{X}^{\text {op }} | | |
| fong-spivak-invitation_FO0740 | 68 | 1.000 | x_{1} \otimes x_{2} \geq y_{1} \otimes y_{2} | | |
| fong-spivak-invitation_FO0741 | 68 | 1.000 | y_{1} \leq x_{1} | | |
| fong-spivak-invitation_FO0742 | 68 | 0.925 | y_{2} \leq x_{2} | | |
| fong-spivak-invitation_FO0743 | 68 | 0.925 | y_{1} \otimes y_{2} \leq x_{1} \otimes x_{2} | | |
| fong-spivak-invitation_FO0744 | 68 | 0.989 | { }^{\mathrm{op}} | | |
| fong-spivak-invitation_FO0745 | 68 | 1.000 | x \cong x^{\prime} | | |
| fong-spivak-invitation_FO0746 | 68 | 1.000 | x \leq x^{\prime} | | |
| fong-spivak-invitation_FO0747 | 68 | 1.000 | x^{\prime} \leq x | | |
| fong-spivak-invitation_FO0748 | 68 | 0.904 | \mathcal{P}=\left(P, \leq_{P}, I_{P}, \otimes_{P}\right) | | |
| fong-spivak-invitation_FO0749 | 68 | 0.904 | Q=\left(Q, \leq_{Q}, I_{Q}, \otimes_{Q}\right) | | |
| fong-spivak-invitation_FO0750 | 68 | 0.945 | \mathcal{P} | | |
| fong-spivak-invitation_FO0751 | 68 | 0.945 | f:\left(P, \leq_{P}\right) \rightarrow | | |
| fong-spivak-invitation_FO0752 | 68 | 1.000 | I_{Q} \leq_{Q} f\left(I_{P}\right) | | |
| fong-spivak-invitation_FO0753 | 68 | 1.000 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right) \leq_{Q} f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-invitation_FO0754 | 68 | 1.000 | p_{1}, p_{2} \in P | | |
| fong-spivak-invitation_FO0755 | 68 | 0.970 | I_{Q} \cong f\left(I_{P}\right) | | |
| fong-spivak-invitation_FO0756 | 68 | 1.000 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right) \cong f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-invitation_FO0757 | 68 | 0.507 | \left(\mathrm{a}^{\prime \prime}\right) I_{Q}=f\left(I_{P}\right) | | |
| fong-spivak-invitation_FO0758 | 68 | 0.998 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right)=f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-invitation_FO0759 | 69 | 1.000 | i:(\mathbb{Z}, \leq, 0,+) \rightarrow(\mathbb{R}, \leq, 0,+) | | |
| fong-spivak-invitation_FO0760 | 69 | 1.000 | i(n)=n | | |
| fong-spivak-invitation_FO0761 | 69 | 1.000 | n \in \mathbb{Z} | | |
| fong-spivak-invitation_FO0762 | 69 | 1.000 | i(m) \leq i(n) | | |
| fong-spivak-invitation_FO0763 | 69 | 0.997 | i(0)=0 | | |
| fong-spivak-invitation_FO0764 | 69 | 0.997 | i(m+n)=i(m)+i(n) | | |
| fong-spivak-invitation_FO0765 | 69 | 1.000 | f:(\mathbb{R}, \leq, 0,+) \rightarrow(\mathbb{Z}, \leq, 0,+) | | |
| fong-spivak-invitation_FO0766 | 69 | 1.000 | f(x):=\lfloor x\rfloor | | |
| fong-spivak-invitation_FO0767 | 69 | 1.000 | f(3.14)=3 | | |
| fong-spivak-invitation_FO0768 | 69 | 1.000 | f(0)=0 | | |
| fong-spivak-invitation_FO0769 | 69 | 1.000 | f(x)+f(y) \leq f(x+y) | | |
| fong-spivak-invitation_FO0770 | 69 | 1.000 | f(0.5)+f(0.5) \neq | | |
| fong-spivak-invitation_FO0771 | 69 | 1.000 | f(0.5+0.5) | | |
| fong-spivak-invitation_FO0772 | 69 | 0.970 | =([0, \infty], \geq, 0,+) | | |
| fong-spivak-invitation_FO0773 | 69 | 0.835 | :=\infty | | |
| fong-spivak-invitation_FO0774 | 69 | 0.835 | :=0 | | |
| fong-spivak-invitation_FO0775 | 69 | 0.483 | g:(\mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0776 | 69 | 0.483 | \wedge) \rightarrow([0, \infty], \geq, 0,+) | | |
| fong-spivak-invitation_FO0777 | 69 | 1.000 | d, u:[0, \infty] \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO0778 | 69 | 1.000 | u | | |
| fong-spivak-invitation_FO0779 | 69 | 0.860 | (\mathbb{N}, \leq, 1, *) | | |
| fong-spivak-invitation_FO0780 | 69 | 1.000 | (\mathbb{N}, \leq, 0,+) \rightarrow | | |
| fong-spivak-invitation_FO0781 | 69 | 0.988 | \mathbb{Z}, \leq, 1, * | | |
| fong-spivak-invitation_FO0782 | 70 | 0.763 | \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO0783 | 70 | 1.000 | \mathcal{X}(x, y) \in V | | |
| fong-spivak-invitation_FO0784 | 70 | 0.622 | x \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO0785 | 70 | 0.622 | I \leq \mathfrak{X}(x, x) | | |
| fong-spivak-invitation_FO0786 | 70 | 0.288 | x, y, z \in \operatorname{Ob}(X) | | |
| fong-spivak-invitation_FO0787 | 70 | 0.288 | \mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \leq \mathfrak{X}(x, z) | | |
| fong-spivak-invitation_FO0788 | 70 | 0.848 | \mathfrak{X} | | |
| fong-spivak-invitation_FO0789 | 70 | 0.751 | \operatorname{Ob}(\mathcal{X})=\{p, q, r, s, t\} | | |
| fong-spivak-invitation_FO0790 | 70 | 0.940 | \mathcal{X}(x, y) | | |
| fong-spivak-invitation_FO0791 | 71 | 0.998 | t \not \leq s | | |
| fong-spivak-invitation_FO0792 | 71 | 0.998 | \mathcal{X}(s, t)= | | |
| fong-spivak-invitation_FO0793 | 71 | 0.998 | \mathcal{X}(t, s)= | | |
| fong-spivak-invitation_FO0794 | 132 | 1.000 | r | | |
| fong-spivak-invitation_FO0795 | 71 | 0.538 | \operatorname{Ob}(\mathfrak{X}) | | |
| fong-spivak-invitation_FO0796 | 71 | 0.656 | x, y \in \mathrm{Ob}(\mathfrak{X}) | | |
| fong-spivak-invitation_FO0797 | 71 | 0.656 | \mathcal{X}(x, y) \in \mathbb{B} | | |
| fong-spivak-invitation_FO0798 | 71 | 0.510 | X:=\operatorname{Ob}(\mathfrak{X}) | | |
| fong-spivak-invitation_FO0799 | 71 | 0.932 | \mathcal{X}(x, y)= | | |
| fong-spivak-invitation_FO0800 | 72 | 0.767 | \leq \mathcal{X}(x, x) | | |
| fong-spivak-invitation_FO0801 | 72 | 0.835 | \mathcal{X}(x, y) \wedge \mathcal{X}(y, z) \leq \mathcal{X}(x, z) | | |
| fong-spivak-invitation_FO0802 | 72 | 0.943 | b \in | | |
| fong-spivak-invitation_FO0803 | 72 | 0.943 | \leq b | | |
| fong-spivak-invitation_FO0804 | 72 | 0.943 | b= | | |
| fong-spivak-invitation_FO0805 | 72 | 0.943 | \mathcal{X}(x, x)= | | |
| fong-spivak-invitation_FO0806 | 72 | 0.555 | b \in \mathbb{B} | | |
| fong-spivak-invitation_FO0807 | 72 | 0.555 | \mathcal{X}(x, y)=\operatorname{tr} | | |
| fong-spivak-invitation_FO0808 | 72 | 0.994 | \mathcal{X}(y, z)= | | |
| fong-spivak-invitation_FO0809 | 72 | 0.994 | \mathcal{X}(x, z)= | | |
| fong-spivak-invitation_FO0810 | 72 | 0.777 | X, d | | |
| fong-spivak-invitation_FO0811 | 72 | 1.000 | d: X \times X \rightarrow \mathbb{R}_{\geq 0} | | |
| fong-spivak-invitation_FO0812 | 72 | 1.000 | d(x, y) | | |
| fong-spivak-invitation_FO0813 | 72 | 1.000 | d(x, x)=0 | | |
| fong-spivak-invitation_FO0814 | 72 | 1.000 | d(x, y)=0 | | |
| fong-spivak-invitation_FO0815 | 72 | 1.000 | d(x, y)=d(y, x) | | |
| fong-spivak-invitation_FO0816 | 72 | 1.000 | d(x, y)+d(y, z) \geq d(x, z) | | |
| fong-spivak-invitation_FO0817 | 72 | 1.000 | d: X \times X \rightarrow[0, \infty]=\mathbb{R}_{\geq 0} \cup\{\infty\} | | |
| fong-spivak-invitation_FO0818 | 72 | 0.672 | (X, d) | | |
| fong-spivak-invitation_FO0819 | 73 | 1.000 | z | | |
| fong-spivak-invitation_FO0820 | 73 | 1.000 | x \rightarrow y \rightarrow z | | |
| fong-spivak-invitation_FO0821 | 73 | 1.000 | 3+5 \geq 7.2 | | |
| fong-spivak-invitation_FO0822 | 73 | 0.997 | 5+7.2 \geq 3 | | |
| fong-spivak-invitation_FO0823 | 73 | 0.997 | 3+7.2 \geq 5 | | |
| fong-spivak-invitation_FO0824 | 73 | 0.997 | 5+3 \geq 7.2,7.2+5 \geq 3 | | |
| fong-spivak-invitation_FO0825 | 73 | 0.997 | 7.2+3 \geq 5 | | |
| fong-spivak-invitation_FO0826 | 73 | 0.528 | d(x, y)=? d(y, x) | | |
| fong-spivak-invitation_FO0827 | 73 | 0.358 | d( | | |
| fong-spivak-invitation_FO0828 | 73 | 0.374 | ) \neq 0 | | |
| fong-spivak-invitation_FO0829 | 73 | 0.982 | U \subseteq X | | |
| fong-spivak-invitation_FO0830 | 73 | 1.000 | d(U, V):=\max \left(d_{L}(U, V), d_{L}(V, U)\right) | | |
| fong-spivak-invitation_FO0831 | 74 | 0.958 | \{r \in \mathbb{R} \mid r<0\} | | |
| fong-spivak-invitation_FO0832 | 74 | 0.958 | \{0\} | | |
| fong-spivak-invitation_FO0833 | 74 | 0.725 | (\mathbb{R}, d) | | |
| fong-spivak-invitation_FO0834 | 74 | 0.578 | x, y \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO0835 | 74 | 0.578 | \mathcal{X}(x, y) \in[0, \infty] | | |
| fong-spivak-invitation_FO0836 | 74 | 1.000 | 0 \geq d(x, x) | | |
| fong-spivak-invitation_FO0837 | 74 | 1.000 | d(x, x) \in[0, \infty] | | |
| fong-spivak-invitation_FO0838 | 74 | 1.000 | d(x, y):=|y-x| | | |
| fong-spivak-invitation_FO0839 | 74 | 1.000 | d(3,7)=4 | | |
| fong-spivak-invitation_FO0840 | 74 | 0.953 | \mathbb{R}_{\geq 0}, \geq, 0,+ | | |
| fong-spivak-invitation_FO0841 | 74 | 0.994 | \left(\mathbb{R}_{\geq 0}, \geq, 0,+\right) | | |
| fong-spivak-invitation_FO0842 | 74 | 0.998 | w \geq 0 | | |
| fong-spivak-invitation_FO0843 | 75 | 1.000 | d_{X} | | |
| fong-spivak-invitation_FO0844 | 75 | 1.000 | d(p, q) | | |
| fong-spivak-invitation_FO0846 | 106 | 1.000 | D | | |
| fong-spivak-invitation_FO0847 | 75 | 1.000 | M_{G} | | |
| fong-spivak-invitation_FO0848 | 76 | 1.000 | d_{Y} | | |
| fong-spivak-invitation_FO0849 | 76 | 1.000 | M_{Y} | | |
| fong-spivak-invitation_FO0850 | 76 | 1.000 | M_{X} | | |
| fong-spivak-invitation_FO0852 | 76 | 0.731 | \mathcal{M}:=(\mathrm{P}(M), \subseteq, M, \cap) | | |
| fong-spivak-invitation_FO0853 | 76 | 1.000 | \mathcal{M} | | |
| fong-spivak-invitation_FO0854 | 76 | 1.000 | a, b \in | | |
| fong-spivak-invitation_FO0855 | 77 | 0.981 | M= | | |
| fong-spivak-invitation_FO0856 | 77 | 0.925 | \mathcal{W}:=(\mathbb{N} \cup\{\infty\}, \leq, \infty, \min ) | | |
| fong-spivak-invitation_FO0857 | 77 | 1.000 | \mathbb{N} \cup\{\infty\} | | |
| fong-spivak-invitation_FO0858 | 77 | 1.000 | \mathcal{W} | | |
| fong-spivak-invitation_FO0859 | 77 | 0.998 | f: \mathcal{V} \rightarrow \mathcal{W} | | |
| fong-spivak-invitation_FO0860 | 77 | 1.000 | \mathcal{V} \rightarrow \mathcal{W} | | |
| fong-spivak-invitation_FO0861 | 77 | 1.000 | \mathcal{C} | | |
| fong-spivak-invitation_FO0862 | 77 | 1.000 | \mathcal{C}_{f} | | |
| fong-spivak-invitation_FO0863 | 77 | 0.999 | \operatorname{Ob}\left(\mathcal{C}_{f}\right):=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO0864 | 77 | 0.979 | c, d \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO0865 | 77 | 0.979 | \mathcal{C}_{f}(c, d):=f(\mathcal{C}(c, d)) | | |
| fong-spivak-invitation_FO0866 | 78 | 1.000 | c \in \mathcal{C} | | |
| fong-spivak-invitation_FO0867 | 78 | 0.945 | c, d, e \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO0868 | 78 | 0.998 | f:[0, \infty] \rightarrow\{ | | |
| fong-spivak-invitation_FO0869 | 78 | 0.831 | x_{f} \neq x_{g} | | |
| fong-spivak-invitation_FO0870 | 79 | 0.703 | \mathcal{Y} | | |
| fong-spivak-invitation_FO0871 | 79 | 0.703 | F: \mathfrak{X} \rightarrow y | | |
| fong-spivak-invitation_FO0872 | 79 | 0.752 | F: \operatorname{Ob}(\mathcal{X}) \rightarrow \operatorname{Ob}(\mathcal{Y}) | | |
| fong-spivak-invitation_FO0873 | 79 | 0.907 | x_{1}, x_{2} \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO0874 | 79 | 0.907 | \mathcal{X}\left(x_{1}, x_{2}\right) \leq y\left(F\left(x_{1}\right), F\left(x_{2}\right)\right) | | |
| fong-spivak-invitation_FO0875 | 79 | 0.931 | \mathcal{X}\left(x_{1}, x_{2}\right)= | | |
| fong-spivak-invitation_FO0876 | 79 | 0.931 | x_{1} \leq x_{2} | | |
| fong-spivak-invitation_FO0877 | 79 | 0.968 | \left(X, \leq_{X}\right) \rightarrow\left(Y, \leq_{Y}\right) | | |
| fong-spivak-invitation_FO0878 | 79 | 0.998 | x_{1}, x_{2} \in X | | |
| fong-spivak-invitation_FO0879 | 79 | 0.998 | x_{1} \leq_{X} x_{2} | | |
| fong-spivak-invitation_FO0880 | 79 | 0.998 | F\left(x_{1}\right) \leq_{Y} F\left(x_{2}\right) | | |
| fong-spivak-invitation_FO0881 | 79 | 0.803 | \mathcal{V}=\mathbf{B o o l} | | |
| fong-spivak-invitation_FO0882 | 79 | 0.999 | \left(X, d_{X}\right) | | |
| fong-spivak-invitation_FO0883 | 79 | 0.999 | \left(Y, d_{Y}\right) | | |
| fong-spivak-invitation_FO0884 | 79 | 0.999 | d_{X}\left(x_{1}, x_{2}\right) \geq d_{Y}\left(F\left(x_{1}\right), F\left(x_{2}\right)\right) | | |
| fong-spivak-invitation_FO0885 | 79 | 0.888 | \mathcal{X}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO0886 | 79 | 0.625 | \operatorname{Ob}\left(\mathcal{X}^{\text {op }}\right):=\operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO0887 | 79 | 0.688 | x, y \in \mathcal{X} | | |
| fong-spivak-invitation_FO0888 | 79 | 0.688 | \mathcal{X}^{\mathrm{op}}(x, y):=\mathcal{X}(y, x) | | |
| fong-spivak-invitation_FO0889 | 79 | 1.000 | \dagger: \mathfrak{X} \rightarrow | | |
| fong-spivak-invitation_FO0890 | 79 | 0.154 | \mathcal{X}^{\text {Op }} | | |
| fong-spivak-invitation_FO0891 | 79 | 0.154 | I \leq \mathfrak{X}(x, y) | | |
| fong-spivak-invitation_FO0892 | 79 | 0.154 | I \leq \mathfrak{X}(y, x) | | |
| fong-spivak-invitation_FO0893 | 80 | 0.797 | \mathcal{X} \times \mathcal{Y} | | |
| fong-spivak-invitation_FO0894 | 80 | 0.435 | \operatorname{Ob}(\mathcal{X} \times \mathcal{Y}):=\operatorname{Ob}(\mathcal{X}) \times \operatorname{Ob}(\mathscr{Y}) | | |
| fong-spivak-invitation_FO0895 | 80 | 0.978 | (\mathcal{X} \times \mathcal{Y})\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right):=\mathcal{X}\left(x, x^{\prime}\right) \otimes y\left(y, y^{\prime}\right) | | |
| fong-spivak-invitation_FO0896 | 80 | 0.977 | \left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-invitation_FO0897 | 80 | 0.977 | \operatorname{Ob}(\mathcal{X} \times \mathcal{Y}) | | |
| fong-spivak-invitation_FO0898 | 80 | 0.722 | (x, y) \in \operatorname{Ob}(\mathcal{X} \times \mathcal{Y}) | | |
| fong-spivak-invitation_FO0899 | 80 | 0.722 | I \leq(\mathcal{X} \times \mathcal{Y})((x, y),(x, y)) | | |
| fong-spivak-invitation_FO0900 | 80 | 1.000 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) | | |
| fong-spivak-invitation_FO0901 | 80 | 1.000 | \left(x_{3}, y_{3}\right) | | |
| fong-spivak-invitation_FO0902 | 80 | 1.000 | \left(P, \leq_{P}\right) \times\left(Q, \leq_{Q}\right) | | |
| fong-spivak-invitation_FO0903 | 80 | 0.879 | \left(p_{1}, q_{1}\right) \leq\left(p_{2}, q_{2}\right) | | |
| fong-spivak-invitation_FO0904 | 80 | 0.879 | q_{1} \leq q_{2} | | |
| fong-spivak-invitation_FO0905 | 81 | 0.996 | d_{X \times Y}\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right) | | |
| fong-spivak-invitation_FO0906 | 81 | 1.000 | d_{X}\left(x, x^{\prime}\right)+d_{Y}\left(y, y^{\prime}\right) | | |
| fong-spivak-invitation_FO0907 | 81 | 0.999 | \mathbb{R} \times \mathbb{R} | | |
| fong-spivak-invitation_FO0908 | 81 | 0.999 | (5,6) | | |
| fong-spivak-invitation_FO0909 | 81 | 0.991 | (-1,4) | | |
| fong-spivak-invitation_FO0910 | 81 | 0.991 | \sqrt{40} | | |
| fong-spivak-invitation_FO0911 | 125 | 1.000 | x \times y | | |
| fong-spivak-invitation_FO0921 | 82 | 0.788 | \mathcal{V}=(V, \leq I, \otimes) | | |
| fong-spivak-invitation_FO0922 | 82 | 1.000 | v, w \in V | | |
| fong-spivak-invitation_FO0923 | 82 | 0.984 | v \multimap w | | |
| fong-spivak-invitation_FO0924 | 82 | 1.000 | a, v, w \in V | | |
| fong-spivak-invitation_FO0925 | 82 | 1.000 | v \in V | | |
| fong-spivak-invitation_FO0926 | 82 | 1.000 | (-\otimes v): V \rightarrow V | | |
| fong-spivak-invitation_FO0927 | 82 | 0.756 | (v-o-): V \rightarrow V | | |
| fong-spivak-invitation_FO0928 | 82 | 1.000 | (-\otimes v) | | |
| fong-spivak-invitation_FO0929 | 82 | 1.000 | ((v \multimap w) \otimes v) \leq w | | |
| fong-spivak-invitation_FO0930 | 82 | 0.771 | (v-o-) | | |
| fong-spivak-invitation_FO0931 | 83 | 1.000 | x, y \in[0, \infty] | | |
| fong-spivak-invitation_FO0932 | 83 | 1.000 | x \multimap y:=\max (0, y-x) | | |
| fong-spivak-invitation_FO0933 | 83 | 1.000 | a, x, y \in | | |
| fong-spivak-invitation_FO0934 | 83 | 0.543 | (\mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0935 | 83 | 0.543 | \vee) | | |
| fong-spivak-invitation_FO0936 | 83 | 0.812 | \leq p \multimap q | | |
| fong-spivak-invitation_FO0937 | 83 | 0.812 | \longrightarrow | | |
| fong-spivak-invitation_FO0938 | 83 | 1.000 | a= | | |
| fong-spivak-invitation_FO0939 | 83 | 1.000 | p= | | |
| fong-spivak-invitation_FO0940 | 83 | 0.927 | q= | | |
| fong-spivak-invitation_FO0941 | 83 | 0.927 | (a \vee p) \not \leq q | | |
| fong-spivak-invitation_FO0942 | 83 | 0.927 | a \leq(p \multimap q) | | |
| fong-spivak-invitation_FO0943 | 83 | 0.969 | c, d | | |
| fong-spivak-invitation_FO0944 | 83 | 0.969 | c \rightarrow d | | |
| fong-spivak-invitation_FO0945 | 83 | 1.000 | 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{Na} \rightarrow 2 \mathrm{NaOH}+\mathrm{H}_{2} | | |
| fong-spivak-invitation_FO0946 | 83 | 0.975 | 2 \mathrm{Na} \multimap\left(2 \mathrm{NaOH}+\mathrm{H}_{2}\right) | | |
| fong-spivak-invitation_FO0947 | 83 | 0.977 | \mathcal{V}=(V, \leq, I, \otimes, \multimap) | | |
| fong-spivak-invitation_FO0948 | 83 | 1.000 | -\otimes v:(V, \leq) \rightarrow(V, \leq) | | |
| fong-spivak-invitation_FO0949 | 83 | 0.779 | v \rightarrow-:(V, \leq) \rightarrow(V, \leq) | | |
| fong-spivak-invitation_FO0950 | 84 | 1.000 | A \subseteq V | | |
| fong-spivak-invitation_FO0951 | 84 | 1.000 | \bigvee_{a \in A} a | | |
| fong-spivak-invitation_FO0952 | 84 | 1.000 | \bigvee_{a \in A} v \otimes a | | |
| fong-spivak-invitation_FO0953 | 84 | 1.000 | v \otimes(v \multimap w) \leq w | | |
| fong-spivak-invitation_FO0954 | 84 | 1.000 | v \cong(I \multimap v) | | |
| fong-spivak-invitation_FO0955 | 84 | 1.000 | u, v, w \in V | | |
| fong-spivak-invitation_FO0956 | 84 | 1.000 | (u \multimap v) \otimes(v \multimap w) \leq(u \multimap w) | | |
| fong-spivak-invitation_FO0957 | 84 | 1.000 | (v \multimap w) \leq(v \multimap w) | | |
| fong-spivak-invitation_FO0958 | 84 | 1.000 | (v \multimap w) \otimes v \leq w | | |
| fong-spivak-invitation_FO0959 | 84 | 1.000 | v \otimes(v \multimap w)=(v \multimap w) \otimes v | | |
| fong-spivak-invitation_FO0960 | 84 | 1.000 | v \otimes I=v \leq v | | |
| fong-spivak-invitation_FO0961 | 84 | 1.000 | v \leq(I \multimap v) | | |
| fong-spivak-invitation_FO0962 | 84 | 1.000 | (I \multimap v)=I \otimes(I \multimap v) \leq v | | |
| fong-spivak-invitation_FO0963 | 84 | 1.000 | u \otimes(u \multimap v) \otimes(v \multimap w) \leq w | | |
| fong-spivak-invitation_FO0964 | 84 | 1.000 | v, w \in | | |
| fong-spivak-invitation_FO0965 | 84 | 0.999 | \operatorname{Ob}(\mathcal{V}) | | |
| fong-spivak-invitation_FO0966 | 84 | 0.999 | \mathcal{V}(v, w):=(v \multimap w) \in \mathcal{V} | | |
| fong-spivak-invitation_FO0967 | 84 | 1.000 | I \leq | | |
| fong-spivak-invitation_FO0968 | 84 | 0.688 | \mathcal{X}(x, x)=(x \multimap x) | | |
| fong-spivak-invitation_FO0969 | 84 | 0.688 | I \otimes x \leq x | | |
| fong-spivak-invitation_FO0970 | 84 | 0.901 | \mathcal{V}=(V, \leq I, \otimes, \multimap) | | |
| fong-spivak-invitation_FO0971 | 84 | 0.901 | \bigvee A | | |
| fong-spivak-invitation_FO0972 | 84 | 0.928 | 0:=\bigvee \varnothing | | |
| fong-spivak-invitation_FO0973 | 85 | 1.000 | A \subseteq[0, \infty] | | |
| fong-spivak-invitation_FO0974 | 85 | 0.999 | a \geq \bigvee A | | |
| fong-spivak-invitation_FO0975 | 85 | 1.000 | b \in[0, \infty] | | |
| fong-spivak-invitation_FO0976 | 85 | 1.000 | a \geq b | | |
| fong-spivak-invitation_FO0977 | 85 | 1.000 | \bigvee A \geq b | | |
| fong-spivak-invitation_FO0978 | 85 | 0.544 | A .{ }^{5} | | |
| fong-spivak-invitation_FO0979 | 85 | 0.544 | A=\{2,3\} | | |
| fong-spivak-invitation_FO0980 | 85 | 0.544 | \bigvee A=2 | | |
| fong-spivak-invitation_FO0981 | 85 | 0.996 | B=\{2.5,2.05,2.005, \ldots\} | | |
| fong-spivak-invitation_FO0982 | 85 | 0.997 | ([0, \infty], \geq) | | |
| fong-spivak-invitation_FO0983 | 85 | 0.997 | A=\varnothing | | |
| fong-spivak-invitation_FO0984 | 85 | 0.999 | \bigvee \varnothing \geq b | | |
| fong-spivak-invitation_FO0985 | 85 | 0.999 | \bigvee \varnothing=\infty | | |
| fong-spivak-invitation_FO0986 | 85 | 0.934 | \bigvee \varnothing | | |
| fong-spivak-invitation_FO0987 | 85 | 0.921 | \mathcal{V}=\mathbf{B o o l}=(\mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0988 | 85 | 0.735 | \mathcal{V}=\boldsymbol{\operatorname { C o s t }}=([0, \infty], \geq, 0,+) | | |
| fong-spivak-invitation_FO0989 | 85 | 1.000 | x \vee y | | |
| fong-spivak-invitation_FO0990 | 85 | 0.802 | \mathcal{V}= | | |
| fong-spivak-invitation_FO0991 | 85 | 0.802 | x, y \in \mathbb{B} | | |
| fong-spivak-invitation_FO0992 | 85 | 0.422 | \mathbf{B o o l}=(\mathbb{B}, \leq | | |
| fong-spivak-invitation_FO0993 | 85 | 0.737 | \mathrm{P}(S), \subseteq, S, \cap | | |
| fong-spivak-invitation_FO0994 | 85 | 0.720 | \{3.1,3.01,3.001, \ldots\} | | |
| fong-spivak-invitation_FO0995 | 86 | 1.000 | \mathcal{P}=(P, \leq) | | |
| fong-spivak-invitation_FO0996 | 86 | 0.931 | \mathcal{P}{ }^{\mathrm{op}} | | |
| fong-spivak-invitation_FO0997 | 86 | 1.000 | A \subseteq \mathcal{P} | | |
| fong-spivak-invitation_FO0998 | 86 | 1.000 | M_{A}:=\{p \in P \mid p \leq a | | |
| fong-spivak-invitation_FO0999 | 86 | 1.000 | a \in A\} | | |
| fong-spivak-invitation_FO1000 | 86 | 1.000 | m_{A}:=\bigvee_{p \in M_{A}} p | | |
| fong-spivak-invitation_FO1001 | 86 | 1.000 | m_{A} | | |
| fong-spivak-invitation_FO1002 | 86 | 1.000 | m_{A} \leq a | | |
| fong-spivak-invitation_FO1003 | 86 | 1.000 | p \in M_{A} | | |
| fong-spivak-invitation_FO1004 | 86 | 1.000 | m^{\prime} \in P | | |
| fong-spivak-invitation_FO1005 | 86 | 1.000 | m^{\prime} \leq a | | |
| fong-spivak-invitation_FO1006 | 86 | 1.000 | m^{\prime} \leq m | | |
| fong-spivak-invitation_FO1007 | 86 | 1.000 | m^{\prime} | | |
| fong-spivak-invitation_FO1008 | 86 | 1.000 | M_{A} | | |
| fong-spivak-invitation_FO1009 | 86 | 1.000 | V \subseteq X | | |
| fong-spivak-invitation_FO1010 | 86 | 0.877 | u \in U | | |
| fong-spivak-invitation_FO1011 | 86 | 0.877 | \forall_{u \in U} . \exists_{v \in V} . u \leq v .{ }^{6} | | |
| fong-spivak-invitation_FO1012 | 86 | 1.000 | \mathcal{V}=\mathrm{P}(M) | | |
| fong-spivak-invitation_FO1013 | 86 | 1.000 | d(U, V) \in \mathrm{P}(M) | | |
| fong-spivak-invitation_FO1014 | 86 | 1.000 | -\otimes v: V \rightarrow V | | |
| fong-spivak-invitation_FO1015 | 86 | 0.999 | v \multimap- | | |
| fong-spivak-invitation_FO1016 | 86 | 1.000 | \forall | | |
| fong-spivak-invitation_FO1017 | 86 | 1.000 | \exists | | |
| fong-spivak-invitation_FO1018 | 87 | 0.959 | \sum | | |
| fong-spivak-invitation_FO1019 | 87 | 1.000 | \mathcal{V}=(V, \leq, \otimes, I) | | |
| fong-spivak-invitation_FO1020 | 87 | 1.000 | M: X \times Y \rightarrow V | | |
| fong-spivak-invitation_FO1021 | 87 | 0.991 | M(x, y) | | |
| fong-spivak-invitation_FO1022 | 87 | 1.000 | N: Y \times Z \rightarrow V | | |
| fong-spivak-invitation_FO1023 | 87 | 1.000 | (M * N): X \times Z \rightarrow V | | |
| fong-spivak-invitation_FO1024 | 87 | 0.999 | M * N | | |
| fong-spivak-invitation_FO1025 | 87 | 1.000 | X=\{1,2,3\}, Y=\{1,2\} | | |
| fong-spivak-invitation_FO1026 | 87 | 1.000 | Z=\{1,2,3\} | | |
| fong-spivak-invitation_FO1027 | 87 | 1.000 | M: X \times Y \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO1028 | 87 | 1.000 | N: Y \times Z \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO1029 | 87 | 1.000 | I_{X}: X \times X \rightarrow V | | |
| fong-spivak-invitation_FO1030 | 87 | 0.984 | (2 \times 2) | | |
| fong-spivak-invitation_FO1031 | 88 | 1.000 | I_{X} * M=M | | |
| fong-spivak-invitation_FO1032 | 88 | 1.000 | M: W \times X \rightarrow V, N: X \times Y \rightarrow V | | |
| fong-spivak-invitation_FO1033 | 88 | 1.000 | P: Y \times Z \rightarrow V | | |
| fong-spivak-invitation_FO1034 | 88 | 1.000 | (M * N) * P=M *(N * P) | | |
| fong-spivak-invitation_FO1035 | 88 | 1.000 | M_{Y}^{2} | | |
| fong-spivak-invitation_FO1036 | 88 | 1.000 | M_{Y}^{3} | | |
| fong-spivak-invitation_FO1039 | 88 | 1.000 | M_{Y}^{2}=M_{Y}^{3} | | |
| fong-spivak-invitation_FO1040 | 88 | 0.999 | M_{Y}^{n} | | |
| fong-spivak-invitation_FO1041 | 88 | 1.000 | M_{X}^{2}, M_{X}^{3} | | |
| fong-spivak-invitation_FO1042 | 88 | 1.000 | M_{X}^{4} | | |
| fong-spivak-invitation_FO1043 | 91 | 1.000 | \$ 2.5 | | |
| fong-spivak-invitation_FO1044 | 93 | 1.000 | \mathcal{C} \rightarrow | | |
| fong-spivak-invitation_FO1045 | 93 | 0.999 | \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-invitation_FO1046 | 95 | 1.000 | { }^{2} \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1047 | 95 | 0.959 | \mathcal{C}(c, d),{ }^{3} | | |
| fong-spivak-invitation_FO1048 | 95 | 0.901 | c \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1049 | 95 | 0.901 | \mathrm{id}_{c} \in \mathcal{C}(c, c) | | |
| fong-spivak-invitation_FO1050 | 95 | 0.791 | f \in \mathcal{C}(c, d) | | |
| fong-spivak-invitation_FO1051 | 95 | 0.791 | g \in \mathcal{C}(d, e) | | |
| fong-spivak-invitation_FO1052 | 95 | 0.983 | f ; g \in \mathcal{C}(c, e) | | |
| fong-spivak-invitation_FO1053 | 95 | 0.843 | c \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1054 | 95 | 0.874 | f: c \rightarrow d | | |
| fong-spivak-invitation_FO1055 | 95 | 0.664 | \mathrm{id}_{c} ; f=f | | |
| fong-spivak-invitation_FO1056 | 95 | 0.664 | f ; \mathrm{id}_{d}=f | | |
| fong-spivak-invitation_FO1057 | 95 | 1.000 | f: c_{0} \rightarrow c_{1}, g: c_{1} \rightarrow c_{2} | | |
| fong-spivak-invitation_FO1058 | 95 | 1.000 | h: c_{2} \rightarrow c_{3} | | |
| fong-spivak-invitation_FO1059 | 95 | 0.998 | (f ; g) ; h=f ;(g ; h) | | |
| fong-spivak-invitation_FO1060 | 95 | 0.957 | \mathcal{C}(c, d) | | |
| fong-spivak-invitation_FO1061 | 95 | 0.957 | \operatorname{Hom}_{\mathcal{C}}(c, d) | | |
| fong-spivak-invitation_FO1062 | 96 | 1.000 | p: v \rightarrow w | | |
| fong-spivak-invitation_FO1063 | 96 | 0.999 | \mathrm{id}_{v} | | |
| fong-spivak-invitation_FO1064 | 96 | 1.000 | q: w \rightarrow x | | |
| fong-spivak-invitation_FO1065 | 96 | 1.000 | p ; q | | |
| fong-spivak-invitation_FO1066 | 96 | 1.000 | v \rightarrow x | | |
| fong-spivak-invitation_FO1067 | 96 | 0.658 | (V, \leq) | | |
| fong-spivak-invitation_FO1068 | 96 | 0.977 | \operatorname{Free}(G) | | |
| fong-spivak-invitation_FO1069 | 96 | 0.651 | v_{1} | | |
| fong-spivak-invitation_FO1070 | 96 | 0.651 | v_{2} | | |
| fong-spivak-invitation_FO1071 | 96 | 0.651 | \mathrm{id}_{v_{1}}: v_{1} \rightarrow v_{1}, f_{1}: v_{1} \rightarrow v_{2} | | |
| fong-spivak-invitation_FO1072 | 96 | 0.992 | \mathrm{id}_{v_{2}}: v_{2} \rightarrow v_{2} | | |
| fong-spivak-invitation_FO1073 | 96 | 0.992 | \mathrm{id}_{v_{1}} | | |
| fong-spivak-invitation_FO1074 | 96 | 0.992 | v_{1}, f_{1} | | |
| fong-spivak-invitation_FO1075 | 96 | 0.999 | f_{1} | | |
| fong-spivak-invitation_FO1076 | 96 | 0.999 | \mathrm{id}_{v_{2}} | | |
| fong-spivak-invitation_FO1077 | 96 | 0.424 | f_{1}=f_{1} | | |
| fong-spivak-invitation_FO1078 | 96 | 0.596 | \mathrm{id}_{v_{2}}{ }_{9}^{\circ} \mathrm{id}_{v_{2}}=\mathrm{id}_{v_{2}} | | |
| fong-spivak-invitation_FO1079 | 96 | 0.821 | \boldsymbol{\operatorname { F r e e }}(G) | | |
| fong-spivak-invitation_FO1080 | 97 | 0.850 | \mathbf{0} | | |
| fong-spivak-invitation_FO1081 | 97 | 0.994 | \mathbf{n} | | |
| fong-spivak-invitation_FO1082 | 97 | 0.337 | =\left\{z, s,\left(s_{9}^{\circ}\right.\right. | | |
| fong-spivak-invitation_FO1083 | 97 | 0.999 | \mathrm{id}_{\mathrm{z}} | | |
| fong-spivak-invitation_FO1084 | 97 | 0.999 | \mathbb{N}= | | |
| fong-spivak-invitation_FO1085 | 97 | 0.567 | \{0,1,2,3, \ldots\} | | |
| fong-spivak-invitation_FO1086 | 97 | 1.000 | m * 1=m=1 * m | | |
| fong-spivak-invitation_FO1087 | 97 | 1.000 | (m * n) * p=m *(n * p) | | |
| fong-spivak-invitation_FO1088 | 97 | 0.996 | M:=\mathcal{C}(\bullet, \bullet) | | |
| fong-spivak-invitation_FO1089 | 97 | 0.996 | e=\mathrm{id}_{\bullet} | | |
| fong-spivak-invitation_FO1090 | 97 | 0.896 | *: \mathcal{C}(\bullet, \bullet) \times \mathcal{C}(\bullet, \bullet) \rightarrow \mathcal{C}(\bullet, \bullet) | | |
| fong-spivak-invitation_FO1091 | 97 | 0.896 | *={ }_{9} | | |
| fong-spivak-invitation_FO1092 | 97 | 1.000 | m, n \in \mathbb{N} | | |
| fong-spivak-invitation_FO1093 | 98 | 0.170 | i | | |
| fong-spivak-invitation_FO1094 | 98 | 0.251 | g: i | | |
| fong-spivak-invitation_FO1095 | 98 | 0.251 | g \circ i | | |
| fong-spivak-invitation_FO1096 | 98 | 0.696 | f ; h | | |
| fong-spivak-invitation_FO1097 | 98 | 0.696 | \operatorname{id}_{A} | | |
| fong-spivak-invitation_FO1098 | 99 | 0.255 | s ; s=z | | |
| fong-spivak-invitation_FO1099 | 99 | 0.255 | s ; s ; s=z | | |
| fong-spivak-invitation_FO1100 | 99 | 0.153 | s=z | | |
| fong-spivak-invitation_FO1101 | 99 | 0.153 | z=z | | |
| fong-spivak-invitation_FO1102 | 99 | 0.153 | \{z, s\} | | |
| fong-spivak-invitation_FO1103 | 99 | 0.217 | =z | | |
| fong-spivak-invitation_FO1104 | 99 | 0.217 | s=s | | |
| fong-spivak-invitation_FO1105 | 99 | 0.217 | z=s | | |
| fong-spivak-invitation_FO1106 | 99 | 1.000 | \mathbb{Z} / 2 \mathbb{Z} | | |
| fong-spivak-invitation_FO1107 | 99 | 1.000 | \mathcal{D} | | |
| fong-spivak-invitation_FO1108 | 99 | 0.991 | \operatorname{Ob}(\mathcal{P})=P | | |
| fong-spivak-invitation_FO1109 | 99 | 1.000 | p \rightarrow q | | |
| fong-spivak-invitation_FO1110 | 99 | 1.000 | p \not \leq q | | |
| fong-spivak-invitation_FO1111 | 100 | 0.921 | (C, \leq) | | |
| fong-spivak-invitation_FO1112 | 100 | 0.931 | C:=\mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1113 | 100 | 0.931 | c_{1} \leq c_{2} | | |
| fong-spivak-invitation_FO1114 | 100 | 0.999 | \mathcal{C}\left(c_{1}, c_{2}\right) \neq \varnothing | | |
| fong-spivak-invitation_FO1115 | 100 | 0.999 | c_{1} \rightarrow c_{2} | | |
| fong-spivak-invitation_FO1116 | 101 | 0.878 | \boldsymbol{\operatorname { S e t }}(S, T)=\{f: S \rightarrow T \mid f | | |
| fong-spivak-invitation_FO1117 | 101 | 1.000 | \mathrm{id}_{S}: S \rightarrow S | | |
| fong-spivak-invitation_FO1118 | 101 | 0.979 | \operatorname{id}_{S}(s):=s | | |
| fong-spivak-invitation_FO1119 | 101 | 1.000 | f: S \rightarrow T | | |
| fong-spivak-invitation_FO1120 | 101 | 1.000 | g: T \rightarrow U | | |
| fong-spivak-invitation_FO1121 | 101 | 1.000 | f ; g: S \rightarrow U | | |
| fong-spivak-invitation_FO1122 | 101 | 1.000 | (f ; g)(s):=g(f(s)) | | |
| fong-spivak-invitation_FO1123 | 101 | 1.000 | \underline{2}=\{1,2\} | | |
| fong-spivak-invitation_FO1124 | 101 | 1.000 | \underline{3}=\{1,2,3\} | | |
| fong-spivak-invitation_FO1125 | 102 | 0.881 | \mathcal{C}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO1126 | 102 | 0.488 | \operatorname{Ob}\left(\mathcal{C}{ }^{\text {op }}\right):=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1127 | 102 | 0.968 | \mathcal{C}^{\mathrm{op}}(c, d):=\mathcal{C}(d, c) | | |
| fong-spivak-invitation_FO1128 | 102 | 0.107 | \{\mathbf{\boxed { , ~ }} \square\} | | |
| fong-spivak-invitation_FO1129 | 102 | 1.000 | a \cong b | | |
| fong-spivak-invitation_FO1130 | 102 | 1.000 | c \leq a | | |
| fong-spivak-invitation_FO1131 | 102 | 1.000 | c \leq b | | |
| fong-spivak-invitation_FO1132 | 102 | 1.000 | c \geq a | | |
| fong-spivak-invitation_FO1133 | 102 | 1.000 | c \geq b | | |
| fong-spivak-invitation_FO1134 | 102 | 0.456 | g: B \rightarrow A | | |
| fong-spivak-invitation_FO1135 | 102 | 0.456 | f ; g=\operatorname{id}_{A} | | |
| fong-spivak-invitation_FO1136 | 102 | 0.456 | g ; f=\operatorname{id}_{B} | | |
| fong-spivak-invitation_FO1137 | 102 | 1.000 | g=f^{-1} | | |
| fong-spivak-invitation_FO1138 | 102 | 1.000 | f=g^{-1} | | |
| fong-spivak-invitation_FO1139 | 102 | 1.000 | A:=\{a, b, c\} | | |
| fong-spivak-invitation_FO1140 | 102 | 1.000 | f: A \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO1141 | 102 | 1.000 | f(a)=2, f(b)=1, f(c)=3 | | |
| fong-spivak-invitation_FO1142 | 103 | 0.979 | A \in | | |
| fong-spivak-invitation_FO1143 | 103 | 1.000 | A \cong \underline{n} | | |
| fong-spivak-invitation_FO1144 | 103 | 1.000 | f^{-1}: \underline{3} \rightarrow A | | |
| fong-spivak-invitation_FO1145 | 103 | 1.000 | A \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO1146 | 103 | 0.998 | \mathrm{id}_{c} | | |
| fong-spivak-invitation_FO1147 | 103 | 1.000 | f: \underline{2} \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO1148 | 103 | 1.000 | g: \underline{3} \rightarrow \underline{2} | | |
| fong-spivak-invitation_FO1149 | 103 | 0.180 | f ; g=\mathrm{id}_{\underline{2}} | | |
| fong-spivak-invitation_FO1150 | 103 | 0.180 | g: f \neq \mathrm{id}_{\underline{3}} | | |
| fong-spivak-invitation_FO1151 | 105 | 1.000 | F: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-invitation_FO1152 | 105 | 0.802 | F(c) \in \mathrm{Ob}(\mathcal{D}) | | |
| fong-spivak-invitation_FO1153 | 105 | 1.000 | f: c_{1} \rightarrow c_{2} | | |
| fong-spivak-invitation_FO1154 | 105 | 1.000 | F(f): F\left(c_{1}\right) \rightarrow | | |
| fong-spivak-invitation_FO1155 | 105 | 1.000 | F\left(c_{2}\right) | | |
| fong-spivak-invitation_FO1156 | 105 | 0.306 | F\left(\mathrm{id}_{c}\right)=\mathrm{id}_{F(c)} | | |
| fong-spivak-invitation_FO1157 | 105 | 0.620 | c_{1}, c_{2}, c_{3} \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1158 | 105 | 0.620 | f \in \mathcal{C}\left(c_{1}, c_{2}\right), g \in | | |
| fong-spivak-invitation_FO1159 | 105 | 1.000 | \mathcal{C}\left(c_{2}, c_{3}\right) | | |
| fong-spivak-invitation_FO1160 | 105 | 1.000 | F(f ; g)=F(f) ; F(g) | | |
| fong-spivak-invitation_FO1161 | 105 | 0.979 | F: \mathbf{2} \rightarrow \mathbf{3} | | |
| fong-spivak-invitation_FO1162 | 106 | 1.000 | n_{0} | | |
| fong-spivak-invitation_FO1163 | 106 | 1.000 | F\left(m_{0}\right)=n_{0} | | |
| fong-spivak-invitation_FO1164 | 106 | 1.000 | F\left(f_{1}\right)=g_{1} | | |
| fong-spivak-invitation_FO1165 | 106 | 1.000 | F\left(m_{1}\right)=n_{1} | | |
| fong-spivak-invitation_FO1166 | 106 | 1.000 | F\left(m_{0}\right)=n_{0}, F\left(m_{1}\right)=n_{2} | | |
| fong-spivak-invitation_FO1167 | 106 | 0.917 | g_{2} | | |
| fong-spivak-invitation_FO1168 | 106 | 0.974 | 2 \rightarrow 3 | | |
| fong-spivak-invitation_FO1169 | 106 | 0.915 | \mathbf{2} \rightarrow \mathbf{3} | | |
| fong-spivak-invitation_FO1170 | 106 | 1.000 | \mathcal{F} | | |
| fong-spivak-invitation_FO1171 | 106 | 1.000 | F: \mathcal{F} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1172 | 106 | 1.000 | A, B^{\prime} | | |
| fong-spivak-invitation_FO1173 | 106 | 1.000 | B, C^{\prime} | | |
| fong-spivak-invitation_FO1174 | 106 | 1.000 | D^{\prime} | | |
| fong-spivak-invitation_FO1175 | 106 | 0.851 | \mathcal{C}=\bullet \rightarrow \bullet | | |
| fong-spivak-invitation_FO1176 | 106 | 0.851 | \mathcal{D}=\bullet \rightrightarrows \bullet | | |
| fong-spivak-invitation_FO1177 | 106 | 1.000 | F, G: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-invitation_FO1178 | 106 | 1.000 | A^{\prime}, B | | |
| fong-spivak-invitation_FO1179 | 106 | 1.000 | B^{\prime}, C | | |
| fong-spivak-invitation_FO1180 | 106 | 1.000 | C^{\prime} | | |
| fong-spivak-invitation_FO1181 | 106 | 0.961 | f ; h=g ; i | | |
| fong-spivak-invitation_FO1182 | 107 | 1.000 | F(f ; h)=F(g ; i) | | |
| fong-spivak-invitation_FO1183 | 107 | 0.600 | f^{\prime} ; h^{\prime}=g^{\prime} ; i^{\prime} | | |
| fong-spivak-invitation_FO1184 | 107 | 0.998 | \mathcal{N} | | |
| fong-spivak-invitation_FO1185 | 107 | 0.998 | \operatorname{Ob}(\mathcal{N})=\mathbb{N} | | |
| fong-spivak-invitation_FO1186 | 107 | 0.998 | m \rightarrow n | | |
| fong-spivak-invitation_FO1187 | 107 | 0.998 | F: \mathcal{N} \rightarrow \mathcal{N} | | |
| fong-spivak-invitation_FO1188 | 107 | 1.000 | F(n) \in \mathbb{N} | | |
| fong-spivak-invitation_FO1189 | 107 | 1.000 | F(m) \rightarrow F(n) | | |
| fong-spivak-invitation_FO1190 | 107 | 1.000 | F(m) \leq F(n) | | |
| fong-spivak-invitation_FO1191 | 107 | 1.000 | \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1192 | 107 | 0.908 | \mathrm{id}_{\mathcal{C}}: \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1193 | 107 | 1.000 | \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1194 | 107 | 1.000 | \operatorname{Ob}(\mathcal{D}) | | |
| fong-spivak-invitation_FO1195 | 107 | 0.965 | c_{1}, c_{2} \in \mathcal{C} | | |
| fong-spivak-invitation_FO1196 | 107 | 0.965 | \mathcal{C}\left(c_{1}, c_{2}\right) | | |
| fong-spivak-invitation_FO1197 | 107 | 0.965 | \mathcal{D}\left(F\left(c_{1}\right), F\left(c_{2}\right)\right) | | |
| fong-spivak-invitation_FO1198 | 107 | 0.907 | G: \mathcal{D} \rightarrow \mathcal{E} | | |
| fong-spivak-invitation_FO1199 | 107 | 1.000 | G): \mathcal{C} \rightarrow \mathcal{E} | | |
| fong-spivak-invitation_FO1200 | 107 | 1.000 | F: \mathcal{C} \rightarrow | | |
| fong-spivak-invitation_FO1201 | 107 | 0.817 | I: \mathcal{C} \rightarrow | | |
| fong-spivak-invitation_FO1202 | 108 | 1.000 | c=\stackrel{\text { string }}{\circ} | | |
| fong-spivak-invitation_FO1203 | 108 | 0.694 | ^{5} | | |
| fong-spivak-invitation_FO1204 | 108 | 0.991 | \mathrm{id}_{1} | | |
| fong-spivak-invitation_FO1205 | 108 | 0.991 | F: \mathbf{1} \rightarrow | | |
| fong-spivak-invitation_FO1206 | 108 | 0.941 | F(1) | | |
| fong-spivak-invitation_FO1207 | 108 | 0.941 | F_{S}: \mathbf{1} \rightarrow | | |
| fong-spivak-invitation_FO1208 | 108 | 0.941 | F_{S}(1)=S | | |
| fong-spivak-invitation_FO1209 | 108 | 0.998 | Z:=F(z) | | |
| fong-spivak-invitation_FO1210 | 108 | 0.993 | S:=F(s): Z \rightarrow Z | | |
| fong-spivak-invitation_FO1211 | 108 | 0.993 | S ; S=S | | |
| fong-spivak-invitation_FO1212 | 108 | 0.934 | Z=\mathbb{N} | | |
| fong-spivak-invitation_FO1213 | 108 | 0.255 | 13+(2 * 4) | | |
| fong-spivak-invitation_FO1214 | 108 | 1.000 | Z=\mathbb{N}_{\geq 2} | | |
| fong-spivak-invitation_FO1215 | 108 | 1.000 | r \in I(c) | | |
| fong-spivak-invitation_FO1217 | 110 | 0.997 | \alpha: F \Rightarrow G | | |
| fong-spivak-invitation_FO1218 | 110 | 0.999 | \alpha_{c}: F(c) \rightarrow G(c) | | |
| fong-spivak-invitation_FO1219 | 110 | 1.000 | \alpha | | |
| fong-spivak-invitation_FO1220 | 110 | 1.000 | \alpha: F \rightarrow G | | |
| fong-spivak-invitation_FO1221 | 110 | 0.986 | \alpha_{c} | | |
| fong-spivak-invitation_FO1222 | 110 | 0.854 | D: \mathcal{J} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1223 | 110 | 0.854 | \mathcal{J} | | |
| fong-spivak-invitation_FO1224 | 110 | 1.000 | D(f)=D\left(f^{\prime}\right) | | |
| fong-spivak-invitation_FO1225 | 110 | 0.961 | f, f^{\prime}: a \rightarrow b | | |
| fong-spivak-invitation_FO1226 | 110 | 0.961 | \mathcal{J}^{7} | | |
| fong-spivak-invitation_FO1227 | 110 | 1.000 | F(c) \rightrightarrows G(d) | | |
| fong-spivak-invitation_FO1228 | 110 | 0.920 | F(f){ }^{\circ} \alpha_{d}=\alpha_{c}{ }^{\circ} G(f) | | |
| fong-spivak-invitation_FO1229 | 111 | 1.000 | \alpha_{1}: v \rightarrow x | | |
| fong-spivak-invitation_FO1230 | 111 | 1.000 | \alpha_{2}: w \rightarrow y | | |
| fong-spivak-invitation_FO1231 | 111 | 1.000 | \alpha_{1}=c | | |
| fong-spivak-invitation_FO1232 | 111 | 0.998 | \alpha_{2}=g | | |
| fong-spivak-invitation_FO1233 | 111 | 0.476 | \mathcal{C} \xrightarrow[G]{F} \mathcal{D} | | |
| fong-spivak-invitation_FO1234 | 111 | 1.000 | A, B: \mathbf{1} \rightarrow | | |
| fong-spivak-invitation_FO1235 | 111 | 0.852 | \mathcal{D}^{\mathcal{C}} | | |
| fong-spivak-invitation_FO1236 | 111 | 0.997 | \mathcal{D}^{\mathcal{C}}(F, G) | | |
| fong-spivak-invitation_FO1237 | 111 | 0.998 | F \in \mathcal{D}^{\mathcal{P}} | | |
| fong-spivak-invitation_FO1238 | 111 | 0.748 | ^{\mathbf{1}} | | |
| fong-spivak-invitation_FO1239 | 111 | 0.701 | F_{0}, F_{1}, F_{2}, \ldots | | |
| fong-spivak-invitation_FO1240 | 111 | 0.701 | F_{0} \leq F_{1} \leq F_{2} \leq \cdots | | |
| fong-spivak-invitation_FO1241 | 111 | 1.000 | \alpha_{n}: F_{n} \rightarrow G_{n} | | |
| fong-spivak-invitation_FO1242 | 111 | 1.000 | F_{n} \leq G_{n} | | |
| fong-spivak-invitation_FO1243 | 112 | 1.000 | F \Rightarrow G | | |
| fong-spivak-invitation_FO1244 | 112 | 1.000 | \mathcal{N}^{\mathcal{N}} | | |
| fong-spivak-invitation_FO1245 | 112 | 1.000 | F, G: \mathcal{C} \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO1246 | 112 | 1.000 | F \rightarrow G | | |
| fong-spivak-invitation_FO1247 | 112 | 1.000 | F, G: \mathcal{P} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1248 | 112 | 0.689 | F: | | |
| fong-spivak-invitation_FO1249 | 112 | 0.795 | G \cong \mathrm{id}_{\text {Pro }} | | |
| fong-spivak-invitation_FO1250 | 112 | 0.563 | G: F \cong \mathrm{id}_{\text {Bool-Cat }} | | |
| fong-spivak-invitation_FO1251 | 112 | 1.000 | I, J: \mathcal{C} \rightarrow | | |
| fong-spivak-invitation_FO1252 | 112 | 0.949 | \alpha: I \rightarrow J | | |
| fong-spivak-invitation_FO1253 | 112 | 0.949 | := | | |
| fong-spivak-invitation_FO1254 | 112 | 0.949 | ^{\mathcal{C}} | | |
| fong-spivak-invitation_FO1255 | 113 | 0.993 | I: \mathbf{G r} \rightarrow | | |
| fong-spivak-invitation_FO1256 | 113 | 0.508 | =\{a, b, c, d, e\} | | |
| fong-spivak-invitation_FO1257 | 113 | 0.508 | =\{1,2,3,4\} | | |
| fong-spivak-invitation_FO1258 | 113 | 0.987 | G, H: \mathbf{G r} \rightarrow \mathbf{S e t} | | |
| fong-spivak-invitation_FO1259 | 113 | 0.990 | G, H \in \mathbf{G r} | | |
| fong-spivak-invitation_FO1260 | 113 | 1.000 | G \rightarrow H | | |
| fong-spivak-invitation_FO1261 | 113 | 1.000 | \alpha: G \rightarrow H | | |
| fong-spivak-invitation_FO1262 | 113 | 0.981 | \mathbf{G r} | | |
| fong-spivak-invitation_FO1263 | 114 | 1.000 | \alpha_{\text {Vertex }} | | |
| fong-spivak-invitation_FO1264 | 114 | 1.000 | \alpha_{\text {Arrow }} | | |
| fong-spivak-invitation_FO1265 | 114 | 1.000 | G, H | | |
| fong-spivak-invitation_FO1266 | 114 | 0.991 | \alpha_{\text {Arrow }}(a)=d | | |
| fong-spivak-invitation_FO1267 | 115 | 0.994 | F: \mathbf{G r} \rightarrow | | |
| fong-spivak-invitation_FO1268 | 116 | 0.156 | I: | | |
| fong-spivak-invitation_FO1269 | 116 | 0.156 | \rightarrow | | |
| fong-spivak-invitation_FO1270 | 116 | 0.156 | F{ }_{;}^{;} I | | |
| fong-spivak-invitation_FO1271 | 117 | 0.649 | G: \mathbf{G r} \rightarrow \mathbf{D D S} | | |
| fong-spivak-invitation_FO1272 | 117 | 0.858 | I: \mathcal{D} \rightarrow | | |
| fong-spivak-invitation_FO1273 | 117 | 0.858 | F: I: \mathcal{C} \rightarrow | | |
| fong-spivak-invitation_FO1274 | 117 | 0.957 | \alpha: I \Rightarrow J | | |
| fong-spivak-invitation_FO1275 | 117 | 0.587 | \alpha_{F}: F ; I \Rightarrow F ; J | | |
| fong-spivak-invitation_FO1276 | 117 | 0.587 | (F ; I)(c) \rightarrow(F ; J)(c) | | |
| fong-spivak-invitation_FO1277 | 117 | 0.677 | \left(\alpha_{F}\right)_{c}:=\alpha_{F c} | | |
| fong-spivak-invitation_FO1278 | 117 | 0.970 | \Delta_{F}: \mathcal{D} | | |
| fong-spivak-invitation_FO1279 | 117 | 1.000 | \Delta | | |
| fong-spivak-invitation_FO1280 | 117 | 1.000 | \Sigma | | |
| fong-spivak-invitation_FO1281 | 117 | 1.000 | \Pi | | |
| fong-spivak-invitation_FO1282 | 117 | 1.000 | P(a, b) | | |
| fong-spivak-invitation_FO1283 | 117 | 1.000 | Q(f(p), q) \cong P(p, g(q)) | | |
| fong-spivak-invitation_FO1284 | 118 | 1.000 | L: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-invitation_FO1285 | 118 | 1.000 | R: \mathcal{D} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1286 | 118 | 1.000 | L | | |
| fong-spivak-invitation_FO1287 | 118 | 0.990 | d \in \mathcal{D} | | |
| fong-spivak-invitation_FO1288 | 118 | 0.798 | d .^{8} | | |
| fong-spivak-invitation_FO1289 | 118 | 1.000 | f: c \rightarrow R(d) | | |
| fong-spivak-invitation_FO1290 | 118 | 1.000 | g:=\alpha_{c, d}(f) | | |
| fong-spivak-invitation_FO1291 | 118 | 1.000 | g: L(c) \rightarrow d | | |
| fong-spivak-invitation_FO1292 | 118 | 1.000 | L \dashv R | | |
| fong-spivak-invitation_FO1293 | 118 | 0.528 | B \in \operatorname{Ob} | | |
| fong-spivak-invitation_FO1294 | 118 | 1.000 | A, C | | |
| fong-spivak-invitation_FO1295 | 118 | 0.997 | B \rightarrow C | | |
| fong-spivak-invitation_FO1296 | 118 | 0.997 | C^{B} | | |
| fong-spivak-invitation_FO1297 | 118 | 1.000 | 10^{3} | | |
| fong-spivak-invitation_FO1298 | 118 | 1.000 | \left|C^{B}\right|=|C|^{|B|} | | |
| fong-spivak-invitation_FO1299 | 118 | 1.000 | (A \times B) \rightarrow C | | |
| fong-spivak-invitation_FO1300 | 118 | 1.000 | A \rightarrow C^{B} | | |
| fong-spivak-invitation_FO1301 | 118 | 0.522 | \mathcal{C}^{\mathrm{op}} \times \mathcal{D} \rightarrow | | |
| fong-spivak-invitation_FO1302 | 118 | 0.522 | f: c^{\prime} \rightarrow c | | |
| fong-spivak-invitation_FO1303 | 118 | 1.000 | g: d \rightarrow d^{\prime} | | |
| fong-spivak-invitation_FO1304 | 119 | 1.000 | b \multimap c | | |
| fong-spivak-invitation_FO1305 | 119 | 1.000 | -\times B | | |
| fong-spivak-invitation_FO1306 | 119 | 1.000 | (-)^{B} | | |
| fong-spivak-invitation_FO1307 | 119 | 1.000 | X \times B | | |
| fong-spivak-invitation_FO1308 | 119 | 1.000 | X^{B} | | |
| fong-spivak-invitation_FO1309 | 119 | 1.000 | -\times B: X \times B \rightarrow Y \times B | | |
| fong-spivak-invitation_FO1310 | 119 | 1.000 | (-)^{B}: X^{B} \rightarrow Y^{B} | | |
| fong-spivak-invitation_FO1311 | 119 | 1.000 | +: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1312 | 119 | 1.000 | (a, b) \mapsto a+b | | |
| fong-spivak-invitation_FO1313 | 119 | 1.000 | p: \mathbb{N} \rightarrow \mathbb{N}^{\mathbb{N}} | | |
| fong-spivak-invitation_FO1314 | 119 | 1.000 | p(3) | | |
| fong-spivak-invitation_FO1315 | 119 | 1.000 | \Delta_{F} | | |
| fong-spivak-invitation_FO1316 | 120 | 1.000 | \Sigma_{F} | | |
| fong-spivak-invitation_FO1317 | 120 | 1.000 | \Pi_{F} | | |
| fong-spivak-invitation_FO1318 | 120 | 1.000 | J: \mathbf{G r} \rightarrow | | |
| fong-spivak-invitation_FO1319 | 120 | 1.000 | \Sigma_{F}(J) | | |
| fong-spivak-invitation_FO1320 | 120 | 1.000 | \Pi_{F}(J) | | |
| fong-spivak-invitation_FO1321 | 120 | 0.999 | (e, f) | | |
| fong-spivak-invitation_FO1322 | 121 | 1.000 | \Pi_{F}(I) | | |
| fong-spivak-invitation_FO1323 | 121 | 1.000 | \Delta_{F}, \Sigma_{F} | | |
| fong-spivak-invitation_FO1324 | 121 | 0.989 | \mathcal{C} \rightarrow \mathbf{1} | | |
| fong-spivak-invitation_FO1325 | 121 | 0.856 | \Sigma_{!} | | |
| fong-spivak-invitation_FO1326 | 121 | 0.451 | \Pi_{!}: \mathcal{C} | | |
| fong-spivak-invitation_FO1327 | 121 | 0.772 | \Sigma_{!}(I) | | |
| fong-spivak-invitation_FO1328 | 121 | 0.772 | \Pi_{!}(I) | | |
| fong-spivak-invitation_FO1329 | 121 | 1.000 | I: \mathcal{G} \rightarrow | | |
| fong-spivak-invitation_FO1330 | 122 | 0.969 | \mathcal{G} | | |
| fong-spivak-invitation_FO1331 | 122 | 0.952 | !: \mathcal{G} \rightarrow \mathbf{1} | | |
| fong-spivak-invitation_FO1332 | 122 | 0.989 | \Pi_{!} | | |
| fong-spivak-invitation_FO1333 | 123 | 0.578 | !: C \rightarrow Z | | |
| fong-spivak-invitation_FO1334 | 123 | 0.900 | C \rightarrow\{\bullet\} | | |
| fong-spivak-invitation_FO1335 | 123 | 1.000 | c \in C | | |
| fong-spivak-invitation_FO1336 | 123 | 1.000 | \bullet \in\{\bullet\} | | |
| fong-spivak-invitation_FO1337 | 123 | 0.825 | z \in P | | |
| fong-spivak-invitation_FO1338 | 123 | 1.000 | c \in P | | |
| fong-spivak-invitation_FO1339 | 123 | 1.000 | c \leq z | | |
| fong-spivak-invitation_FO1340 | 123 | 1.000 | Z^{\prime} | | |
| fong-spivak-invitation_FO1341 | 123 | 1.000 | a: Z \rightarrow Z^{\prime} | | |
| fong-spivak-invitation_FO1342 | 123 | 1.000 | b: Z^{\prime} \rightarrow Z | | |
| fong-spivak-invitation_FO1343 | 123 | 1.000 | a ; b: Z \rightarrow Z | | |
| fong-spivak-invitation_FO1344 | 123 | 0.992 | Z \rightarrow Z | | |
| fong-spivak-invitation_FO1345 | 123 | 0.992 | \mathrm{id}_{Z} | | |
| fong-spivak-invitation_FO1346 | 123 | 0.999 | a ; b=\mathrm{id}_{\mathrm{Z}} | | |
| fong-spivak-invitation_FO1347 | 123 | 0.999 | b ; a=\mathrm{id}_{\mathrm{Z}^{\prime}} | | |
| fong-spivak-invitation_FO1348 | 124 | 1.000 | X, Y | | |
| fong-spivak-invitation_FO1349 | 124 | 1.000 | p_{X}: X \times Y \rightarrow X | | |
| fong-spivak-invitation_FO1350 | 124 | 1.000 | p_{Y}: X \times Y \rightarrow Y | | |
| fong-spivak-invitation_FO1351 | 124 | 1.000 | f: C \rightarrow X | | |
| fong-spivak-invitation_FO1352 | 124 | 1.000 | g: C \rightarrow Y | | |
| fong-spivak-invitation_FO1353 | 124 | 1.000 | C \rightarrow X \times Y | | |
| fong-spivak-invitation_FO1354 | 124 | 1.000 | \langle f, g\rangle | | |
| fong-spivak-invitation_FO1355 | 124 | 1.000 | p_{X}(x, y):=x | | |
| fong-spivak-invitation_FO1356 | 124 | 1.000 | p_{Y}(x, y):=y | | |
| fong-spivak-invitation_FO1357 | 124 | 1.000 | \langle f, g\rangle(c):=(f(c), g(c)) | | |
| fong-spivak-invitation_FO1358 | 124 | 0.999 | \underline{6} \times \underline{4} | | |
| fong-spivak-invitation_FO1359 | 124 | 0.999 | \underline{6} | | |
| fong-spivak-invitation_FO1360 | 124 | 0.999 | \underline{4} | | |
| fong-spivak-invitation_FO1361 | 125 | 0.906 | x, y \in P | | |
| fong-spivak-invitation_FO1362 | 125 | 1.000 | x \wedge y | | |
| fong-spivak-invitation_FO1363 | 125 | 1.000 | \mathcal{C} \times \mathcal{D} | | |
| fong-spivak-invitation_FO1364 | 125 | 0.953 | (c, d) | | |
| fong-spivak-invitation_FO1365 | 125 | 0.999 | (c, d) \rightarrow\left(c^{\prime}, d^{\prime}\right) | | |
| fong-spivak-invitation_FO1366 | 125 | 0.999 | (f, g) | | |
| fong-spivak-invitation_FO1367 | 125 | 0.999 | f: c \rightarrow c^{\prime} | | |
| fong-spivak-invitation_FO1368 | 125 | 0.487 | (f, g) ;\left(f^{\prime}, g^{\prime}\right)= | | |
| fong-spivak-invitation_FO1369 | 125 | 0.807 | \left(f \circ f^{\prime}, g \circ g^{\prime}\right) | | |
| fong-spivak-invitation_FO1370 | 125 | 1.000 | \mathbf{1} \times \mathbf{2} | | |
| fong-spivak-invitation_FO1371 | 125 | 0.937 | \mathcal{P} \times Q | | |
| fong-spivak-invitation_FO1372 | 125 | 1.000 | X \stackrel{f}{\leftarrow} C \xrightarrow{g} Y | | |
| fong-spivak-invitation_FO1373 | 125 | 0.991 | (X, Y) | | |
| fong-spivak-invitation_FO1374 | 125 | 0.644 | X \stackrel{p_{X}}{\longleftrightarrow} X \times Y \xrightarrow{p_{Y}} Y | | |
| fong-spivak-invitation_FO1375 | 125 | 0.644 | \boldsymbol{\operatorname { C o n e }}(X, Y) | | |
| fong-spivak-invitation_FO1376 | 125 | 0.849 | X \stackrel{f^{\prime}}{\leftarrow} C^{\prime} \xrightarrow{g^{\prime}} Y | | |
| fong-spivak-invitation_FO1377 | 125 | 0.849 | a: C \rightarrow C^{\prime} | | |
| fong-spivak-invitation_FO1378 | 125 | 0.622 | X \stackrel{p_{X}}{\longleftarrow} X \times Y \xrightarrow{p_{Y}} Y | | |
| fong-spivak-invitation_FO1379 | 126 | 0.994 | \left(C, c_{*}\right) | | |
| fong-spivak-invitation_FO1380 | 126 | 1.000 | C \in \mathcal{C} | | |
| fong-spivak-invitation_FO1381 | 126 | 0.999 | j \in \mathcal{J} | | |
| fong-spivak-invitation_FO1382 | 126 | 0.999 | c_{j}: C \rightarrow D(j) | | |
| fong-spivak-invitation_FO1383 | 126 | 0.812 | f: j \rightarrow k | | |
| fong-spivak-invitation_FO1384 | 126 | 0.812 | c_{k}=c_{j} | | |
| fong-spivak-invitation_FO1385 | 126 | 0.812 | D(f) | | |
| fong-spivak-invitation_FO1386 | 126 | 1.000 | \left(C, c_{*}\right) \rightarrow\left(C^{\prime}, c_{*}^{\prime}\right) | | |
| fong-spivak-invitation_FO1387 | 126 | 0.918 | c_{j}=a{ }_{9} c_{j}^{\prime} | | |
| fong-spivak-invitation_FO1388 | 126 | 0.997 | (D) | | |
| fong-spivak-invitation_FO1389 | 126 | 0.480 | \lim (D) | | |
| fong-spivak-invitation_FO1390 | 126 | 0.480 | \boldsymbol{\operatorname { C o n e }}(D) | | |
| fong-spivak-invitation_FO1391 | 126 | 0.991 | \lim (D)=\left(C, c_{*}\right) | | |
| fong-spivak-invitation_FO1392 | 126 | 0.991 | c_{j} | | |
| fong-spivak-invitation_FO1393 | 126 | 0.854 | j \in \mathfrak{J} | | |
| fong-spivak-invitation_FO1394 | 126 | 1.000 | D: \mathcal{J} \rightarrow | | |
| fong-spivak-invitation_FO1395 | 126 | 0.950 | \mathcal{J}=\varnothing | | |
| fong-spivak-invitation_FO1396 | 127 | 0.159 | v, w | | |
| fong-spivak-invitation_FO1397 | 127 | 0.159 | \mathfrak{d}=\stackrel{v}{\bullet} \quad \stackrel{w}{\bullet} | | |
| fong-spivak-invitation_FO1398 | 127 | 0.762 | V, A, s, t | | |
| fong-spivak-invitation_FO1399 | 127 | 0.989 | V= | | |
| fong-spivak-invitation_FO1400 | 127 | 1.000 | \left\{v_{1}, \ldots, v_{n}\right\} | | |
| fong-spivak-invitation_FO1401 | 127 | 0.999 | p_{i}:\left(\lim _{\mathcal{J}} D\right) \rightarrow D\left(v_{i}\right) | | |
| fong-spivak-invitation_FO1402 | 127 | 0.999 | p_{i}\left(d_{1}, \ldots, d_{n}\right):= | | |
| fong-spivak-invitation_FO1403 | 127 | 1.000 | d_{i} | | |
| fong-spivak-invitation_FO1404 | 127 | 0.958 | J | | |
| fong-spivak-invitation_FO1405 | 127 | 0.958 | n=0 | | |
| fong-spivak-invitation_FO1406 | 127 | 1.000 | D: \mathbf{1} \rightarrow | | |
| fong-spivak-invitation_FO1407 | 127 | 0.997 | \left(d_{1}, \ldots, d_{n}\right) | | |
| fong-spivak-invitation_FO1408 | 127 | 0.942 | D(a)\left(d_{i}\right)=d_{j} | | |
| fong-spivak-invitation_FO1409 | 127 | 0.942 | a: i \rightarrow j | | |
| fong-spivak-invitation_FO1410 | 127 | 0.942 | \mathfrak{J} | | |
| fong-spivak-invitation_FO1411 | 128 | 0.282 | \stackrel{x}{\rightarrow} \xrightarrow{a} \xrightarrow{g} | | |
| fong-spivak-invitation_FO1412 | 128 | 1.000 | X \xrightarrow{f} A \stackrel{g}{\leftarrow} Y | | |
| fong-spivak-invitation_FO1413 | 128 | 1.000 | c_{a} | | |
| fong-spivak-invitation_FO1414 | 128 | 0.492 | X \times_{A} Y | | |
| fong-spivak-invitation_FO1415 | 128 | 1.000 | X=\underline{6}, Y=\underline{4} | | |
| fong-spivak-invitation_FO1416 | 128 | 1.000 | f: \underline{6} \rightarrow A | | |
| fong-spivak-invitation_FO1417 | 128 | 1.000 | g: \underline{4} \rightarrow A | | |
| fong-spivak-invitation_FO1418 | 128 | 1.000 | g(2)=g(4)= | | |
| fong-spivak-invitation_FO1419 | 128 | 1.000 | f(5)= | | |
| fong-spivak-invitation_FO1420 | 128 | 1.000 | (i, j) \in \underline{6} \times \underline{4} | | |
| fong-spivak-invitation_FO1421 | 128 | 1.000 | f(i) | | |
| fong-spivak-invitation_FO1422 | 128 | 1.000 | g(j) | | |
| fong-spivak-invitation_FO1423 | 129 | 0.989 | F^{\text {op }} | | |
| fong-spivak-invitation_FO1424 | 129 | 0.989 | \mathcal{C}^{\text {op }} \rightarrow \mathcal{D}^{\text {op }} | | |
| fong-spivak-invitation_FO1425 | 129 | 0.999 | D^{\mathrm{op}}: \mathcal{J}^{\mathrm{op}} \rightarrow \mathcal{C}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO1426 | 130 | 0.988 | \Delta, \Sigma, \Pi | | |
| fong-spivak-invitation_FO1427 | 132 | 0.699 | \leq \mathrm{s} | | |
| fong-spivak-invitation_FO1428 | 132 | 1.000 | \$ 10 \leq \$ 20,5 \mathrm{~W} \leq 6 \mathrm{~W} | | |
| fong-spivak-invitation_FO1429 | 132 | 1.000 | (p, r) \in P \times R | | |
| fong-spivak-invitation_FO1430 | 132 | 1.000 | \Phi: P \times R \rightarrow | | |
| fong-spivak-invitation_FO1431 | 132 | 1.000 | \Phi(p, r)= | | |
| fong-spivak-invitation_FO1432 | 132 | 1.000 | p^{\prime} \leq p | | |
| fong-spivak-invitation_FO1433 | 132 | 1.000 | \Phi\left(p^{\prime}, r\right)= | | |
| fong-spivak-invitation_FO1434 | 132 | 0.631 | r \leq r^{\prime} | | |
| fong-spivak-invitation_FO1435 | 132 | 0.631 | \Phi\left(p, r^{\prime}\right)= | | |
| fong-spivak-invitation_FO1436 | 133 | 1.000 | r^{\prime} \geq r | | |
| fong-spivak-invitation_FO1437 | 133 | 0.600 | P^{\mathrm{op}} \times R \rightarrow | | |
| fong-spivak-invitation_FO1438 | 133 | 0.606 | \$ | | |
| fong-spivak-invitation_FO1439 | 133 | 0.712 | 5 \mathrm{~W} \leq 10 \mathrm{~W} | | |
| fong-spivak-invitation_FO1440 | 134 | 0.792 | \mathcal{X}^{\mathrm{op}}=(X, \geq) | | |
| fong-spivak-invitation_FO1441 | 134 | 0.792 | x \geq y | | |
| fong-spivak-invitation_FO1442 | 134 | 0.940 | \mathcal{X}=\left(X, \leq_{X}\right) | | |
| fong-spivak-invitation_FO1443 | 134 | 0.940 | \mathcal{Y}=\left(Y, \leq_{Y}\right) | | |
| fong-spivak-invitation_FO1444 | 134 | 0.616 | \Phi: \mathcal{X} \rightarrow y | | |
| fong-spivak-invitation_FO1445 | 134 | 1.000 | \Phi(x, y)= | | |
| fong-spivak-invitation_FO1446 | 134 | 0.832 | x^{\prime} \leq_{X} x | | |
| fong-spivak-invitation_FO1447 | 134 | 0.832 | y \leq_{Y} y^{\prime} | | |
| fong-spivak-invitation_FO1448 | 134 | 0.832 | \Phi(x, y) \leq_{\text {Bool }} \Phi\left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-invitation_FO1449 | 134 | 1.000 | x^{\prime} | | |
| fong-spivak-invitation_FO1450 | 134 | 1.000 | y^{\prime} | | |
| fong-spivak-invitation_FO1451 | 134 | 0.387 | \mathcal{X}^{\mathrm{op}} \times \mathcal{Y} | | |
| fong-spivak-invitation_FO1452 | 134 | 0.710 | \Lambda: \mathcal{X} \rightarrow y | | |
| fong-spivak-invitation_FO1453 | 134 | 0.710 | \Lambda(x, y)= | | |
| fong-spivak-invitation_FO1454 | 134 | 0.335 | \mathscr{X}^{\mathrm{op}} \times \mathcal{Y} | | |
| fong-spivak-invitation_FO1455 | 134 | 1.000 | \Rightarrow: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO1456 | 134 | 1.000 | b, c, d \in \mathbb{B} | | |
| fong-spivak-invitation_FO1458 | 135 | 0.311 | \Phi: \mathcal{X} \mapsto \psi | | |
| fong-spivak-invitation_FO1459 | 135 | 0.322 | \Phi: \operatorname{Ob}(\mathcal{X}) \times \operatorname{Ob}(\mathscr{Y}) \rightarrow V | | |
| fong-spivak-invitation_FO1460 | 135 | 0.322 | x, x^{\prime} \in \mathcal{X} | | |
| fong-spivak-invitation_FO1461 | 135 | 0.322 | y, y^{\prime} \in \mathcal{Y} | | |
| fong-spivak-invitation_FO1462 | 135 | 0.698 | \Phi: X \rightarrow Y | | |
| fong-spivak-invitation_FO1463 | 136 | 1.000 | W \leq N | | |
| fong-spivak-invitation_FO1464 | 136 | 0.837 | N | | |
| fong-spivak-invitation_FO1465 | 136 | 0.837 | E | | |
| fong-spivak-invitation_FO1466 | 136 | 0.837 | \Phi(N, e)= | | |
| fong-spivak-invitation_FO1467 | 136 | 0.837 | \Phi(E, a)= | | |
| fong-spivak-invitation_FO1468 | 136 | 1.000 | W | | |
| fong-spivak-invitation_FO1469 | 136 | 1.000 | \Phi(W, d)= | | |
| fong-spivak-invitation_FO1470 | 136 | 1.000 | W \leq N \leq c \leq a | | |
| fong-spivak-invitation_FO1471 | 136 | 1.000 | \Phi(m, n) \in \mathbb{B} | | |
| fong-spivak-invitation_FO1472 | 137 | 0.722 | \Phi: X \mapsto Y | | |
| fong-spivak-invitation_FO1474 | 137 | 1.000 | d_{X} * M_{\Phi} * d_{Y} | | |
| fong-spivak-invitation_FO1475 | 137 | 1.000 | \Phi=M_{X}^{3} * M_{\Phi} * M_{Y}^{2} | | |
| fong-spivak-invitation_FO1476 | 138 | 1.000 | M_{X}^{3} * M_{\Phi} * M_{Y}^{2} | | |
| fong-spivak-invitation_FO1477 | 138 | 0.484 | [0, \infty], \leq | | |
| fong-spivak-invitation_FO1478 | 138 | --- | 100, more_than_\ | --- | |
| fong-spivak-invitation_FO1479 | 138 | 0.941 | \Phi:(T \times E) \mapsto \$ | | |
| fong-spivak-invitation_FO1480 | 138 | 0.626 | T, E | | |
| fong-spivak-invitation_FO1481 | 139 | --- | 500K given \ | --- | |
| fong-spivak-invitation_FO1482 | 139 | 0.997 | \$ 500 \mathrm{~K} | | |
| fong-spivak-invitation_FO1483 | 139 | 0.988 | \mathcal{P}, Q | | |
| fong-spivak-invitation_FO1484 | 139 | 0.988 | \mathcal{R} | | |
| fong-spivak-invitation_FO1485 | 139 | 0.936 | \Phi: \mathcal{P} \mapsto Q | | |
| fong-spivak-invitation_FO1486 | 139 | 0.586 | \Psi: Q \mapsto \mathcal{R} | | |
| fong-spivak-invitation_FO1487 | 140 | 1.000 | \Psi | | |
| fong-spivak-invitation_FO1488 | 140 | 1.000 | \Phi ; \Psi | | |
| fong-spivak-invitation_FO1489 | 140 | 0.844 | p \in \mathcal{P} | | |
| fong-spivak-invitation_FO1490 | 140 | 0.844 | r \in \mathcal{R} | | |
| fong-spivak-invitation_FO1492 | 141 | 0.776 | \mathcal{X}, \mathcal{Y} | | |
| fong-spivak-invitation_FO1493 | 141 | 0.776 | \mathcal{Z} | | |
| fong-spivak-invitation_FO1494 | 141 | 0.521 | \Psi: y \rightarrow z | | |
| fong-spivak-invitation_FO1495 | 141 | 0.521 | \Phi: \Psi: \mathcal{X} \mapsto z | | |
| fong-spivak-invitation_FO1496 | 141 | 0.496 | _{\mathcal{V}} | | |
| fong-spivak-invitation_FO1497 | 141 | 0.411 | \mathcal{X} \mapsto 1 | | |
| fong-spivak-invitation_FO1498 | 141 | 0.877 | _{\text {Bool }} | | |
| fong-spivak-invitation_FO1499 | 142 | 0.661 | \mathrm{U}_{X}: \mathcal{X} \mapsto X | | |
| fong-spivak-invitation_FO1500 | 142 | 0.798 | \mathcal{X}(x, y) \in \mathcal{V} | | |
| fong-spivak-invitation_FO1501 | 142 | 0.798 | \mathrm{U}_{X} | | |
| fong-spivak-invitation_FO1502 | 142 | 0.582 | U_{\mathcal{X}}: \mathcal{X} \rightarrow \mathcal{X} | | |
| fong-spivak-invitation_FO1503 | 142 | 0.752 | \mathrm{U}_{\mathcal{P}} | | |
| fong-spivak-invitation_FO1504 | 142 | 0.412 | U_{Q} | | |
| fong-spivak-invitation_FO1505 | 142 | 0.524 | U_{\mathcal{P}} ; \Phi=\Phi | | |
| fong-spivak-invitation_FO1506 | 142 | 0.524 | \Phi=\Phi ; U_{Q} | | |
| fong-spivak-invitation_FO1507 | 142 | 0.998 | \Phi \leq \mathrm{U}_{\mathcal{P}} ; \Phi | | |
| fong-spivak-invitation_FO1508 | 142 | 0.519 | U_{\mathcal{P}} ; \Phi \leq \Phi | | |
| fong-spivak-invitation_FO1509 | 142 | 1.000 | \bigvee_{p_{1} \in P}\left(\mathcal{P}\left(p, p_{1}\right) \otimes \Phi\left(p_{1}, q\right)\right) \leq \Phi(p, q) | | |
| fong-spivak-invitation_FO1510 | 142 | 1.000 | \mathcal{P}\left(p, p_{1}\right) \otimes \Phi\left(p_{1}, q\right) \leq \Phi(p, q) | | |
| fong-spivak-invitation_FO1511 | 142 | 1.000 | p_{1} \in \mathcal{P} | | |
| fong-spivak-invitation_FO1512 | 143 | 0.676 | (=, \leq, \leq,=) | | |
| fong-spivak-invitation_FO1513 | 143 | 0.915 | \Phi: \mathcal{P} \rightarrow Q, \Psi: Q \rightarrow \mathcal{R} | | |
| fong-spivak-invitation_FO1514 | 143 | 0.915 | \Upsilon: \mathcal{R} \rightarrow \mathcal{S} | | |
| fong-spivak-invitation_FO1515 | 144 | 0.744 | _{\mathrm{C}} | | |
| fong-spivak-invitation_FO1516 | 144 | 1.000 | F: \mathcal{P} \rightarrow Q | | |
| fong-spivak-invitation_FO1517 | 144 | 0.592 | \widehat{F}: \mathcal{P} \mapsto Q | | |
| fong-spivak-invitation_FO1518 | 144 | 0.592 | \check{F}: Q \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO1519 | 144 | 1.000 | F(p) \in Q | | |
| fong-spivak-invitation_FO1520 | 144 | 1.000 | \widehat{F} | | |
| fong-spivak-invitation_FO1521 | 144 | 0.866 | Q \mapsto \mathcal{P} | | |
| fong-spivak-invitation_FO1522 | 144 | 0.866 | \check{F} | | |
| fong-spivak-invitation_FO1523 | 144 | 0.999 | \mathcal{P} \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO1524 | 144 | 0.993 | \widehat{\mathrm{id}}=\widetilde{\mathrm{id}}: \mathcal{P} \mapsto \mathcal{P} | | |
| fong-spivak-invitation_FO1525 | 145 | 0.951 | \widehat{\mathrm{id}} | | |
| fong-spivak-invitation_FO1526 | 145 | 0.895 | +: \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO1527 | 145 | 0.969 | a+b+c \in \mathbb{R} | | |
| fong-spivak-invitation_FO1528 | 145 | 0.969 | (a, b, c) \leq | | |
| fong-spivak-invitation_FO1529 | 145 | 0.982 | a^{\prime}, b^{\prime}, c^{\prime} | | |
| fong-spivak-invitation_FO1530 | 145 | 0.982 | a \leq a^{\prime}, b \leq b^{\prime} | | |
| fong-spivak-invitation_FO1531 | 145 | 0.982 | c \leq c^{\prime} | | |
| fong-spivak-invitation_FO1532 | 145 | 0.982 | a+b+c \leq a^{\prime}+b^{\prime}+c^{\prime} | | |
| fong-spivak-invitation_FO1533 | 145 | 0.513 | \widehat{+}:(\mathbb{R} \times \mathbb{R} \times \mathbb{R}) \dashv \mathbb{R} | | |
| fong-spivak-invitation_FO1534 | 145 | 0.513 | a, b, c, d | | |
| fong-spivak-invitation_FO1535 | 145 | 1.000 | a+b+c \leq d | | |
| fong-spivak-invitation_FO1536 | 145 | 0.472 | \check{+} | | |
| fong-spivak-invitation_FO1537 | 145 | 0.964 | \mathcal{Q} | | |
| fong-spivak-invitation_FO1538 | 145 | 0.706 | G: Q \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO1539 | 145 | 1.000 | \widehat{F}=\widetilde{G} | | |
| fong-spivak-invitation_FO1540 | 145 | 1.000 | \widehat{\mathrm{id}}=\widetilde{\mathrm{id}} | | |
| fong-spivak-invitation_FO1541 | 145 | 0.611 | \Phi: \mathfrak{X} \rightarrow y | | |
| fong-spivak-invitation_FO1542 | 145 | 0.969 | \varnothing:=\vee \varnothing | | |
| fong-spivak-invitation_FO1543 | 145 | 0.969 | \mathbf{C o l}(\Phi) | | |
| fong-spivak-invitation_FO1544 | 145 | 0.778 | \operatorname{Ob}(\operatorname{Col}(\Phi)):=\operatorname{Ob}(\mathcal{X}) \sqcup \operatorname{Ob}(\mathcal{Y}) | | |
| fong-spivak-invitation_FO1545 | 145 | 0.964 | a, b \in \operatorname{Ob}(\operatorname{Col}(\Phi)) | | |
| fong-spivak-invitation_FO1546 | 145 | 0.964 | \operatorname{Col}(\Phi)(a, b) \in \mathcal{V} | | |
| fong-spivak-invitation_FO1547 | 146 | 0.660 | i_{x}: \mathcal{X} \rightarrow \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-invitation_FO1548 | 146 | 0.660 | i_{y}: y \rightarrow \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-invitation_FO1549 | 146 | 0.311 | \Phi: \mathcal{X} \rightarrow | | |
| fong-spivak-invitation_FO1551 | 146 | 0.999 | \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-invitation_FO1553 | 147 | 1.000 | 5+3=8 | | |
| fong-spivak-invitation_FO1554 | 147 | 0.840 | \overline{5}= | | |
| fong-spivak-invitation_FO1555 | 147 | 0.539 | \overline{3}= | | |
| fong-spivak-invitation_FO1556 | 147 | 0.539 | \overline{8}=\{ | | |
| fong-spivak-invitation_FO1557 | 147 | 1.000 | \overline{5} \sqcup \overline{3} \cong \overline{8} | | |
| fong-spivak-invitation_FO1558 | 147 | 1.000 | \overline{5} \sqcup \overline{3} | | |
| fong-spivak-invitation_FO1559 | 147 | 1.000 | \overline{8} | | |
| fong-spivak-invitation_FO1560 | 148 | 0.964 | \mathcal{P}(a, b)= | | |
| fong-spivak-invitation_FO1561 | 148 | 1.000 | x_{0} \leq x_{1} | | |
| fong-spivak-invitation_FO1562 | 148 | 1.000 | x_{0} \leq x_{3} | | |
| fong-spivak-invitation_FO1563 | 148 | 1.000 | x_{0} \leq x_{1}, x_{1} \leq x_{2} | | |
| fong-spivak-invitation_FO1564 | 148 | 1.000 | x_{2} \leq x_{3} | | |
| fong-spivak-invitation_FO1565 | 148 | 1.000 | g: B \rightarrow C | | |
| fong-spivak-invitation_FO1566 | 148 | 1.000 | h: C \rightarrow D | | |
| fong-spivak-invitation_FO1567 | 148 | 0.305 | f ;(g ; h) | | |
| fong-spivak-invitation_FO1568 | 148 | 0.305 | (f ; g){ }_{9}^{\circ} h | | |
| fong-spivak-invitation_FO1569 | 149 | 0.440 | f ;(g ; h)=(f ; g) ; h | | |
| fong-spivak-invitation_FO1570 | 149 | 0.994 | \mathrm{id}_{x} | | |
| fong-spivak-invitation_FO1571 | 149 | 0.486 | \mathrm{id}_{x}{ }_{9} f=f=f ; \mathrm{id}_{y} | | |
| fong-spivak-invitation_FO1572 | 149 | 1.000 | f: x \rightarrow y | | |
| fong-spivak-invitation_FO1573 | 150 | 1.000 | \{1\}, \times | | |
| fong-spivak-invitation_FO1574 | 150 | 1.000 | S \times T=\{(s, t) \mid s \in S, t \in T\} | | |
| fong-spivak-invitation_FO1575 | 150 | 0.997 | (p \otimes q) \otimes r=p \otimes(q \otimes r) | | |
| fong-spivak-invitation_FO1576 | 150 | 1.000 | T \times U | | |
| fong-spivak-invitation_FO1577 | 150 | 1.000 | S \times T | | |
| fong-spivak-invitation_FO1578 | 150 | 1.000 | \alpha_{s, t, u} | | |
| fong-spivak-invitation_FO1579 | 150 | 0.913 | \alpha \mathrm{s} | | |
| fong-spivak-invitation_FO1580 | 150 | 0.907 | I \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1581 | 150 | 1.000 | \otimes: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1582 | 151 | 0.863 | \lambda_{c}: I \otimes c \cong c | | |
| fong-spivak-invitation_FO1583 | 151 | 0.997 | \rho_{c}: c \otimes I \cong c | | |
| fong-spivak-invitation_FO1584 | 151 | 0.984 | \alpha_{c, d, e}:(c \otimes d) \otimes e \cong c \otimes(d \otimes e) | | |
| fong-spivak-invitation_FO1585 | 151 | 0.999 | \sigma_{c, d}: c \otimes d \cong d \otimes c | | |
| fong-spivak-invitation_FO1586 | 151 | 0.999 | c, d \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1587 | 151 | 1.000 | \sigma \circ \sigma=\mathrm{id} | | |
| fong-spivak-invitation_FO1588 | 151 | 0.998 | (\mathcal{P}, I, \otimes) | | |
| fong-spivak-invitation_FO1589 | 151 | 0.998 | p, q \in \mathcal{P} | | |
| fong-spivak-invitation_FO1590 | 151 | 1.000 | \mathcal{P}(p, q) | | |
| fong-spivak-invitation_FO1591 | 151 | 1.000 | I:=\{1\} | | |
| fong-spivak-invitation_FO1592 | 151 | 0.990 | \otimes:=\times | | |
| fong-spivak-invitation_FO1593 | 151 | 0.600 | (S, T) \in \operatorname{Ob}( | | |
| fong-spivak-invitation_FO1594 | 151 | 0.600 | ) | | |
| fong-spivak-invitation_FO1595 | 151 | 0.600 | (S \times T) \in | | |
| fong-spivak-invitation_FO1596 | 151 | 0.992 | \{(s, t) \mid s \in S, t \in T\} | | |
| fong-spivak-invitation_FO1597 | 151 | 1.000 | f: S \rightarrow S^{\prime} | | |
| fong-spivak-invitation_FO1598 | 151 | 1.000 | g: T \rightarrow T^{\prime} | | |
| fong-spivak-invitation_FO1599 | 151 | 1.000 | (f \times g):(S \times T) \rightarrow\left(S^{\prime} \times T^{\prime}\right) | | |
| fong-spivak-invitation_FO1600 | 151 | 1.000 | (f \times g)(s, t):= | | |
| fong-spivak-invitation_FO1601 | 151 | 0.991 | (f(s), g(t)) | | |
| fong-spivak-invitation_FO1602 | 151 | 0.975 | \mathrm{id}_{S} \times \mathrm{id}_{T}=\mathrm{id}_{S \times T} | | |
| fong-spivak-invitation_FO1603 | 152 | 1.000 | S \xrightarrow{f} S^{\prime} \xrightarrow{f^{\prime}} S^{\prime \prime} | | |
| fong-spivak-invitation_FO1604 | 152 | 1.000 | T \xrightarrow{g} | | |
| fong-spivak-invitation_FO1605 | 152 | 1.000 | T^{\prime} \xrightarrow{g^{\prime}} T^{\prime \prime} | | |
| fong-spivak-invitation_FO1606 | 152 | 1.000 | \{1\} \times S \cong S | | |
| fong-spivak-invitation_FO1607 | 152 | 1.000 | \{(1, s) \mid s \in S\} \rightarrow S | | |
| fong-spivak-invitation_FO1608 | 152 | 1.000 | (1, s) | | |
| fong-spivak-invitation_FO1609 | 152 | 1.000 | 1, \times | | |
| fong-spivak-invitation_FO1610 | 152 | 0.740 | A=B=C=D=F=G=\mathbb{Z} | | |
| fong-spivak-invitation_FO1611 | 152 | 0.740 | E=\mathbb{B}= | | |
| fong-spivak-invitation_FO1612 | 152 | 0.766 | f_{C}(a)=|a|, f_{D}(a)=a * 5, g_{E}(d, b)=" d \leq b, " g_{F}(d, b)=d-b | | |
| fong-spivak-invitation_FO1613 | 152 | 0.999 | h(c, e)= | | |
| fong-spivak-invitation_FO1614 | 152 | 0.999 | 1-c | | |
| fong-spivak-invitation_FO1615 | 152 | 1.000 | g_{E}(5,3) | | |
| fong-spivak-invitation_FO1616 | 152 | 1.000 | g_{F}(5,3) | | |
| fong-spivak-invitation_FO1617 | 152 | 1.000 | g_{E}(3,5) | | |
| fong-spivak-invitation_FO1618 | 152 | 1.000 | g_{F}(3,5) | | |
| fong-spivak-invitation_FO1619 | 152 | 0.997 | h(5 | | |
| fong-spivak-invitation_FO1620 | 152 | 0.930 | h(-5 | | |
| fong-spivak-invitation_FO1621 | 152 | 1.000 | A \times B \rightarrow G \times F | | |
| fong-spivak-invitation_FO1622 | 152 | 1.000 | q_{G}(-2,3) | | |
| fong-spivak-invitation_FO1623 | 152 | 1.000 | q_{F}(-2,3) | | |
| fong-spivak-invitation_FO1624 | 152 | 1.000 | q_{G}(2,3) | | |
| fong-spivak-invitation_FO1625 | 152 | 1.000 | q_{F}(2,3) | | |
| fong-spivak-invitation_FO1626 | 153 | 0.298 | x, y \in \mathrm{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO1627 | 153 | 0.420 | \mathrm{id}_{x}: I \rightarrow \mathcal{X}(x, x) | | |
| fong-spivak-invitation_FO1628 | 153 | 0.596 | x, y, z \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-invitation_FO1629 | 153 | 0.596 | { }_{9}: \mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \rightarrow | | |
| fong-spivak-invitation_FO1630 | 153 | 0.809 | \mathcal{X}(x, z) | | |
| fong-spivak-invitation_FO1631 | 153 | 0.922 | \mathcal{V}=( | | |
| fong-spivak-invitation_FO1632 | 153 | 0.922 | ,\{1\}, \times) | | |
| fong-spivak-invitation_FO1633 | 153 | 0.496 | \operatorname{id}_{x}: I \rightarrow \mathcal{X}(x, x) | | |
| fong-spivak-invitation_FO1634 | 155 | 0.650 | (\mathcal{C}, I, \otimes) | | |
| fong-spivak-invitation_FO1635 | 155 | 0.999 | c^{*} \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1636 | 155 | 1.000 | \eta_{c}: I \rightarrow c^{*} \otimes c | | |
| fong-spivak-invitation_FO1637 | 155 | 1.000 | \epsilon_{c}: c \otimes c^{*} \rightarrow I | | |
| fong-spivak-invitation_FO1638 | 155 | 0.881 | c^{*} | | |
| fong-spivak-invitation_FO1639 | 155 | 1.000 | \xrightarrow{c} | | |
| fong-spivak-invitation_FO1640 | 155 | 1.000 | \stackrel{c^{*}}{\leftarrow} | | |
| fong-spivak-invitation_FO1641 | 156 | 0.844 | \epsilon_{\text {sound }} | | |
| fong-spivak-invitation_FO1642 | 156 | 0.844 | \eta_{\text {sound* }} | | |
| fong-spivak-invitation_FO1643 | 156 | 0.738 | c \multimap d | | |
| fong-spivak-invitation_FO1644 | 156 | 0.738 | \mathcal{C}(b \otimes c, d) \cong \mathcal{C}(b, c \multimap d) | | |
| fong-spivak-invitation_FO1645 | 156 | 1.000 | c^{\prime} | | |
| fong-spivak-invitation_FO1646 | 156 | 1.000 | c^{*} \cong c^{\prime} | | |
| fong-spivak-invitation_FO1647 | 156 | 1.000 | c \cong c^{* *} | | |
| fong-spivak-invitation_FO1648 | 156 | 0.668 | c^{*} \otimes d | | |
| fong-spivak-invitation_FO1649 | 156 | 0.959 | \mathrm{id}_{b} \otimes \eta_{c} | | |
| fong-spivak-invitation_FO1650 | 157 | 1.000 | \alpha: A \rightarrow B | | |
| fong-spivak-invitation_FO1651 | 157 | 1.000 | \beta: B \rightarrow C | | |
| fong-spivak-invitation_FO1652 | 157 | 0.999 | \beta \circ \alpha | | |
| fong-spivak-invitation_FO1653 | 157 | 1.000 | \beta | | |
| fong-spivak-invitation_FO1654 | 157 | 1.000 | A \sqcup A:= | | |
| fong-spivak-invitation_FO1655 | 157 | 0.998 | \{(a, 1),(a, 2) \mid a \in A\} | | |
| fong-spivak-invitation_FO1656 | 157 | 0.998 | (a, 1) | | |
| fong-spivak-invitation_FO1657 | 157 | 1.000 | (a, 2) | | |
| fong-spivak-invitation_FO1658 | 157 | 1.000 | \eta_{A}: \varnothing \rightarrow A \sqcup A | | |
| fong-spivak-invitation_FO1659 | 157 | 1.000 | \epsilon_{A}: A \sqcup A \rightarrow \varnothing | | |
| fong-spivak-invitation_FO1660 | 157 | 0.992 | \varnothing \rightarrow \underline{3} \sqcup \underline{3} | | |
| fong-spivak-invitation_FO1661 | 157 | 0.989 | \underline{3} \sqcup \underline{3} \rightarrow \varnothing | | |
| fong-spivak-invitation_FO1662 | 157 | 0.907 | \alpha ; \beta | | |
| fong-spivak-invitation_FO1663 | 157 | 1.000 | A \sqcup C | | |
| fong-spivak-invitation_FO1664 | 157 | 1.000 | A \sqcup B \sqcup C | | |
| fong-spivak-invitation_FO1665 | 157 | 1.000 | \alpha \subseteq(A \sqcup B) \times(A \sqcup B) | | |
| fong-spivak-invitation_FO1666 | 157 | 1.000 | \beta \subseteq(B \sqcup C) \times(B \sqcup C) | | |
| fong-spivak-invitation_FO1667 | 157 | 1.000 | \iota_{A \sqcup B}: A \sqcup B \rightarrow A \sqcup B \sqcup C, \iota_{B \sqcup C}: B \sqcup C \rightarrow A \sqcup B \sqcup C | | |
| fong-spivak-invitation_FO1668 | 157 | 0.778 | { }^{\iota} A \sqcup C: A \sqcup C \rightarrow A \sqcup B \sqcup C | | |
| fong-spivak-invitation_FO1669 | 158 | 0.544 | { }_{\text {Bool }} | | |
| fong-spivak-invitation_FO1670 | 158 | 0.992 | (\mathcal{V}, I, \otimes) | | |
| fong-spivak-invitation_FO1671 | 158 | 0.529 | \operatorname{Prof}_{\mathcal{V}} | | |
| fong-spivak-invitation_FO1672 | 158 | 0.583 | { }_{\mathcal{V}} | | |
| fong-spivak-invitation_FO1673 | 158 | 0.843 | \Phi: \mathfrak{X}_{1} \mapsto X_{2} | | |
| fong-spivak-invitation_FO1674 | 158 | 0.843 | \Psi: y_{1} \mapsto y_{2} | | |
| fong-spivak-invitation_FO1675 | 158 | 0.787 | (\Phi \times \Psi): \mathcal{X}_{1} \times y_{1} \rightarrow \mathcal{X}_{2} \times y_{2} | | |
| fong-spivak-invitation_FO1676 | 159 | 0.977 | \Phi \times \Psi | | |
| fong-spivak-invitation_FO1677 | 159 | 0.665 | (1,1)=I | | |
| fong-spivak-invitation_FO1678 | 159 | 0.400 | X \times \mathbf{1} \mapsto X | | |
| fong-spivak-invitation_FO1679 | 159 | 0.400 | \mathbf{1} \times \mathcal{X} \mapsto X | | |
| fong-spivak-invitation_FO1680 | 159 | 0.465 | \eta_{X}: \mathbf{1} \mapsto \mathscr{X}^{\mathrm{op}} \times \mathcal{X} | | |
| fong-spivak-invitation_FO1681 | 159 | 0.973 | \eta_{\mathcal{X}}\left(1, x, x^{\prime}\right):=\mathcal{X}\left(x, x^{\prime}\right) | | |
| fong-spivak-invitation_FO1682 | 159 | 0.973 | \epsilon_{\mathcal{X}}:(\mathcal{X} \times | | |
| fong-spivak-invitation_FO1683 | 159 | 0.239 | \left.\mathcal{X}^{\text {Op }}\right) \rightarrow \mathbf{1} | | |
| fong-spivak-invitation_FO1684 | 159 | 0.239 | \epsilon_{\mathcal{X}}\left(x, x^{\prime}, 1\right):=\mathcal{X}\left(x, x^{\prime}\right) | | |
| fong-spivak-invitation_FO1685 | 162 | 1.000 | u=15 x | | |
| fong-spivak-invitation_FO1686 | 162 | 1.000 | v=3 x+21 y | | |
| fong-spivak-invitation_FO1687 | 162 | 1.000 | x+7 y | | |
| fong-spivak-invitation_FO1688 | 163 | 0.608 | (\mathcal{C}, 0,+) | | |
| fong-spivak-invitation_FO1689 | 163 | 0.876 | \operatorname{Ob}(\mathcal{C})=\mathbb{N} | | |
| fong-spivak-invitation_FO1690 | 163 | 0.876 | 0 \in \mathbb{N} | | |
| fong-spivak-invitation_FO1691 | 163 | 1.000 | \mathcal{C}(m, n) | | |
| fong-spivak-invitation_FO1692 | 163 | 0.341 | \operatorname{map}_{\mathrm{id}}: n \rightarrow n | | |
| fong-spivak-invitation_FO1693 | 163 | 1.000 | \sigma_{m, n}: m+n \rightarrow n+m | | |
| fong-spivak-invitation_FO1694 | 163 | 1.000 | f: m \rightarrow n | | |
| fong-spivak-invitation_FO1695 | 163 | 1.000 | g: n \rightarrow p | | |
| fong-spivak-invitation_FO1696 | 163 | 1.000 | (f ; g): m \rightarrow p | | |
| fong-spivak-invitation_FO1697 | 163 | 1.000 | f: m \rightarrow m^{\prime} | | |
| fong-spivak-invitation_FO1698 | 163 | 1.000 | g: n \rightarrow n^{\prime} | | |
| fong-spivak-invitation_FO1699 | 163 | 0.987 | (f+g): m+n \rightarrow m^{\prime}+n^{\prime} | | |
| fong-spivak-invitation_FO1700 | 163 | 1.000 | \underline{m}=\{1, \ldots, m\} | | |
| fong-spivak-invitation_FO1701 | 163 | 1.000 | \underline{n}=\{1, \ldots, n\} | | |
| fong-spivak-invitation_FO1702 | 163 | 1.000 | f+g: m+n \longrightarrow m^{\prime}+n^{\prime} | | |
| fong-spivak-invitation_FO1703 | 164 | 1.000 | f: 3 \rightarrow 2 | | |
| fong-spivak-invitation_FO1704 | 164 | 1.000 | g: 2 \rightarrow 4 | | |
| fong-spivak-invitation_FO1705 | 164 | 1.000 | f+g | | |
| fong-spivak-invitation_FO1706 | 164 | 0.997 | \sigma_{m, n} | | |
| fong-spivak-invitation_FO1707 | 164 | 0.995 | \underline{m} \rightarrow \underline{n} | | |
| fong-spivak-invitation_FO1708 | 164 | 0.995 | m \neq n | | |
| fong-spivak-invitation_FO1709 | 164 | 0.762 | \mathbf{B i j}(m, n) | | |
| fong-spivak-invitation_FO1710 | 164 | 0.762 | \mathbf{B i j} | | |
| fong-spivak-invitation_FO1711 | 164 | 1.000 | \underline{m} \sqcup \underline{n} | | |
| fong-spivak-invitation_FO1712 | 164 | 0.994 | R \subseteq \underline{m} \times \underline{n} | | |
| fong-spivak-invitation_FO1713 | 164 | 1.000 | S \subseteq \underline{n} \times \underline{p} | | |
| fong-spivak-invitation_FO1714 | 164 | 0.859 | (\mathbb{N}, \preceq) | | |
| fong-spivak-invitation_FO1715 | 164 | 0.859 | \preceq | | |
| fong-spivak-invitation_FO1716 | 164 | 0.999 | R_{1}+R_{2} | | |
| fong-spivak-invitation_FO1717 | 164 | 0.999 | R_{1} \subseteq \underline{m_{1}} \times \underline{n_{1}} | | |
| fong-spivak-invitation_FO1718 | 164 | 0.999 | R_{2} \subseteq \underline{m_{2}} \times \underline{n_{2}} | | |
| fong-spivak-invitation_FO1719 | 164 | 0.226 | R_{1} \sqcup R_{2} \subseteq\left(\underline{m_{1}} \times \underline{n_{1}}\right) \sqcup\left(\underline{m_{2}} \times \underline{n_{2}}\right) \subseteq\left(\underline{m_{1} \sqcup \underline{m_{2}}}\right) \times\left(\underline{n_{1} \sqcup \underline{n_{2}}}\right) | | |
| fong-spivak-invitation_FO1720 | 165 | 0.727 | F(n)=n | | |
| fong-spivak-invitation_FO1721 | 165 | 0.727 | n \in \operatorname{Ob}(\mathfrak{C})=\operatorname{Ob}(\mathcal{D})=\mathbb{N} | | |
| fong-spivak-invitation_FO1722 | 165 | 1.000 | f: m_{1} \rightarrow m_{2} | | |
| fong-spivak-invitation_FO1723 | 165 | 1.000 | g: n_{1} \rightarrow n_{2} | | |
| fong-spivak-invitation_FO1724 | 165 | 1.000 | F(f)+F(g)=F(f+g) | | |
| fong-spivak-invitation_FO1725 | 165 | 0.943 | i: \mathbf{B i j} \rightarrow \mathbf{F i n S e t} | | |
| fong-spivak-invitation_FO1726 | 165 | 0.262 | f: \underline{m} \rightarrow \underline{n} | | |
| fong-spivak-invitation_FO1727 | 165 | 1.000 | F(f):=\{(i, j) \mid f(i)=j\} \subseteq \underline{m} \times \underline{n} | | |
| fong-spivak-invitation_FO1728 | 165 | 0.991 | \iota | | |
| fong-spivak-invitation_FO1729 | 165 | 0.896 | V \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1730 | 165 | 0.896 | \operatorname{in}(v) | | |
| fong-spivak-invitation_FO1731 | 165 | 0.896 | \operatorname{out}(v) | | |
| fong-spivak-invitation_FO1732 | 165 | 0.959 | \iota: \underline{m} \sqcup O \xrightarrow{\cong} I \sqcup \underline{n} | | |
| fong-spivak-invitation_FO1733 | 165 | 0.959 | I=\{(v, i) \mid v \in V, 1 \leq i \leq \operatorname{in}(v)\} | | |
| fong-spivak-invitation_FO1734 | 165 | 1.000 | O=\{(v, i) \mid v \in V, 1 \leq i \leq \operatorname{out}(v)\} | | |
| fong-spivak-invitation_FO1735 | 165 | 1.000 | e_{v, j}^{u, i}: u \rightarrow v | | |
| fong-spivak-invitation_FO1736 | 165 | 1.000 | i, j \in \mathbb{N} | | |
| fong-spivak-invitation_FO1737 | 165 | 1.000 | \iota(u, i)=(v, j) | | |
| fong-spivak-invitation_FO1738 | 165 | 0.997 | m=2 | | |
| fong-spivak-invitation_FO1739 | 165 | 0.997 | n=3 | | |
| fong-spivak-invitation_FO1740 | 166 | 1.000 | V=\{a, b, c\} | | |
| fong-spivak-invitation_FO1741 | 166 | 1.000 | \operatorname{in}(a)=1 | | |
| fong-spivak-invitation_FO1742 | 166 | 1.000 | \operatorname{out}(a)=3 | | |
| fong-spivak-invitation_FO1743 | 166 | 0.359 | (m, n) | | |
| fong-spivak-invitation_FO1744 | 166 | 0.359 | V, i n | | |
| fong-spivak-invitation_FO1745 | 166 | 0.359 | n, p | | |
| fong-spivak-invitation_FO1746 | 166 | 0.359 | V^{\prime}, i n^{\prime} | | |
| fong-spivak-invitation_FO1747 | 166 | 0.359 | \iota^{\prime} | | |
| fong-spivak-invitation_FO1748 | 166 | 0.666 | (m, p) | | |
| fong-spivak-invitation_FO1749 | 166 | 0.666 | \left(V \sqcup V^{\prime},\left[\right.\right. | | |
| fong-spivak-invitation_FO1750 | 166 | 0.666 | \left.\iota^{\prime \prime}\right) | | |
| fong-spivak-invitation_FO1751 | 166 | 0.623 | V \sqcup V^{\prime} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1752 | 166 | 0.907 | V^{\prime} | | |
| fong-spivak-invitation_FO1753 | 166 | 0.907 | ^{\prime} | | |
| fong-spivak-invitation_FO1754 | 166 | 0.991 | \iota^{\prime \prime}: \underline{m} \sqcup O \sqcup O^{\prime} \rightarrow I \sqcup I^{\prime} \sqcup \underline{p} | | |
| fong-spivak-invitation_FO1755 | 166 | 0.992 | \operatorname{Ob}(\mathbf{P G})=\mathbb{N} | | |
| fong-spivak-invitation_FO1756 | 166 | 0.682 | \mathbf{P G}(m, n)=\{(V | | |
| fong-spivak-invitation_FO1757 | 166 | 0.682 | \iota) \mid | | |
| fong-spivak-invitation_FO1758 | 167 | 0.499 | (n, n) | | |
| fong-spivak-invitation_FO1759 | 167 | 0.983 | \left(\varnothing,!,!, \mathrm{id}_{\underline{n}}\right) | | |
| fong-spivak-invitation_FO1760 | 167 | 0.983 | \varnothing \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1761 | 167 | 0.999 | G:= | | |
| fong-spivak-invitation_FO1762 | 167 | 0.409 | (V | | |
| fong-spivak-invitation_FO1763 | 167 | 0.409 | \iota) | | |
| fong-spivak-invitation_FO1764 | 167 | 0.409 | G^{\prime}:=\left(V^{\prime}\right. | | |
| fong-spivak-invitation_FO1765 | 167 | 0.409 | \left.\iota^{\prime}\right) | | |
| fong-spivak-invitation_FO1766 | 167 | 0.956 | \varnothing,!,!,!) | | |
| fong-spivak-invitation_FO1767 | 167 | 1.000 | R \subseteq P \times P | | |
| fong-spivak-invitation_FO1768 | 167 | 0.696 | (V, A, s, t) | | |
| fong-spivak-invitation_FO1769 | 167 | 0.696 | \{(s(a), t(a)) \mid a \in A\} \subseteq V \times V | | |
| fong-spivak-invitation_FO1770 | 168 | 0.884 | Q, \leq_{Q} | | |
| fong-spivak-invitation_FO1771 | 168 | 1.000 | R(x, y) | | |
| fong-spivak-invitation_FO1772 | 168 | 1.000 | f(x) \leq_{Q} f(y) | | |
| fong-spivak-invitation_FO1773 | 168 | 1.000 | P, Q, R | | |
| fong-spivak-invitation_FO1774 | 168 | 1.000 | a, b \in Q | | |
| fong-spivak-invitation_FO1775 | 168 | 1.000 | (g(a), g(b)) \in R | | |
| fong-spivak-invitation_FO1776 | 168 | 1.000 | g:\left(Q, \leq_{Q}\right) \rightarrow\left(P, \leq_{P}\right) | | |
| fong-spivak-invitation_FO1777 | 168 | 1.000 | \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1778 | 168 | 1.000 | \operatorname{Mor}(\mathcal{C}) \rightarrow \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1779 | 168 | 1.000 | \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1780 | 168 | 1.000 | f: V \rightarrow \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1781 | 168 | 1.000 | g: A \rightarrow \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-invitation_FO1782 | 168 | 1.000 | \operatorname{dom}(g(a))=f(s(a)) | | |
| fong-spivak-invitation_FO1783 | 168 | 1.000 | \operatorname{cod}(g(a))=f(t(a)) | | |
| fong-spivak-invitation_FO1784 | 169 | 1.000 | \boldsymbol{\operatorname { L o o p }}(A) | | |
| fong-spivak-invitation_FO1785 | 169 | 1.000 | A=\{a\} | | |
| fong-spivak-invitation_FO1786 | 169 | 1.000 | \{a\} | | |
| fong-spivak-invitation_FO1787 | 169 | 0.993 | G, s, t | | |
| fong-spivak-invitation_FO1788 | 169 | 0.993 | s, t: G \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1789 | 169 | 1.000 | g \in G | | |
| fong-spivak-invitation_FO1790 | 169 | 1.000 | s(g), t(g) \in \mathbb{N} | | |
| fong-spivak-invitation_FO1791 | 169 | 0.844 | s, t | | |
| fong-spivak-invitation_FO1792 | 169 | 0.990 | \Gamma=(V | | |
| fong-spivak-invitation_FO1793 | 169 | 0.990 | \ell: V \rightarrow G | | |
| fong-spivak-invitation_FO1794 | 169 | 0.982 | s(\ell(v))=\operatorname{in}(v) | | |
| fong-spivak-invitation_FO1795 | 169 | 0.982 | t(\ell(v))=\operatorname{out}(v) | | |
| fong-spivak-invitation_FO1796 | 169 | 1.000 | \ell | | |
| fong-spivak-invitation_FO1797 | 169 | 1.000 | (m, n) \in \mathbb{N} \times \mathbb{N} | | |
| fong-spivak-invitation_FO1798 | 170 | 1.000 | G=\{f: 1 \rightarrow 1, g: 2 \rightarrow 2, h: 2 \rightarrow 1\} | | |
| fong-spivak-invitation_FO1799 | 170 | 1.000 | 3 \rightarrow 2 | | |
| fong-spivak-invitation_FO1800 | 170 | 1.000 | V \rightarrow \varnothing | | |
| fong-spivak-invitation_FO1801 | 170 | 1.000 | (n, m) | | |
| fong-spivak-invitation_FO1802 | 170 | 0.992 | (\varnothing,!,!, \sigma) | | |
| fong-spivak-invitation_FO1803 | 170 | 0.992 | \sigma: n \rightarrow m | | |
| fong-spivak-invitation_FO1804 | 170 | 0.959 | (m, n) \in \mathbb{N}^{2} | | |
| fong-spivak-invitation_FO1805 | 170 | 0.860 | (G, s, t) | | |
| fong-spivak-invitation_FO1806 | 170 | 0.974 | \operatorname{Free}(G) \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1807 | 170 | 1.000 | G \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1808 | 170 | 1.000 | s(g) \rightarrow t(g) | | |
| fong-spivak-invitation_FO1810 | 171 | 1.000 | e: m \rightarrow n | | |
| fong-spivak-invitation_FO1811 | 171 | 1.000 | \mathrm{id}_{0}: 0 \rightarrow 0 | | |
| fong-spivak-invitation_FO1812 | 171 | 1.000 | \mathrm{id}_{1}: 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO1813 | 171 | 1.000 | \sigma: 2 \rightarrow 2 | | |
| fong-spivak-invitation_FO1814 | 171 | 1.000 | g: s(g) \rightarrow t(g) | | |
| fong-spivak-invitation_FO1815 | 171 | 1.000 | \alpha: m \rightarrow n | | |
| fong-spivak-invitation_FO1816 | 171 | 1.000 | \beta: p \rightarrow q | | |
| fong-spivak-invitation_FO1817 | 171 | 1.000 | \alpha+\beta: m+p \rightarrow n+q | | |
| fong-spivak-invitation_FO1818 | 171 | 0.988 | \beta: n \rightarrow p | | |
| fong-spivak-invitation_FO1819 | 171 | 0.988 | \alpha ; \beta: m \rightarrow p | | |
| fong-spivak-invitation_FO1820 | 171 | 1.000 | \operatorname{Expr}(G) | | |
| fong-spivak-invitation_FO1821 | 171 | 1.000 | f: 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO1822 | 171 | 0.988 | f: \mathrm{id}_{1}: 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO1823 | 171 | 0.999 | h+\mathrm{id}_{1}: 3 \rightarrow 2 | | |
| fong-spivak-invitation_FO1824 | 171 | 0.866 | \left(h+\mathrm{id}_{1}\right) ; \sigma ; g ; \sigma: 3 \rightarrow 2 | | |
| fong-spivak-invitation_FO1825 | 171 | 0.678 | f{ }_{9}^{\circ} \mathrm{id}_{1} | | |
| fong-spivak-invitation_FO1826 | 171 | 0.953 | f ; \mathrm{id}_{1}=f | | |
| fong-spivak-invitation_FO1827 | 171 | 0.997 | \sigma | | |
| fong-spivak-invitation_FO1828 | 171 | 0.997 | \mathrm{X}: 2 \rightarrow 2 | | |
| fong-spivak-invitation_FO1829 | 172 | 0.616 | \left(g+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+\sigma\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+g\right) \stackrel{\circ}{9}\left(h+\mathrm{id}_{1}\right) | | |
| fong-spivak-invitation_FO1830 | 172 | 1.000 | G=\{f: 1 \rightarrow 1, g: 2 \rightarrow | | |
| fong-spivak-invitation_FO1831 | 172 | 0.605 | h: 2 \rightarrow 1\} | | |
| fong-spivak-invitation_FO1832 | 172 | 0.605 | \left(f+\mathrm{id}_{1}+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\sigma+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+h\right) \stackrel{\circ}{9} \sigma \stackrel{\circ}{g} | | |
| fong-spivak-invitation_FO1833 | 172 | 0.666 | (G, s, t, E) | | |
| fong-spivak-invitation_FO1834 | 172 | 1.000 | E \subseteq \operatorname{Expr}(G) \times \operatorname{Expr}(G) | | |
| fong-spivak-invitation_FO1835 | 172 | 1.000 | e_{1} | | |
| fong-spivak-invitation_FO1836 | 172 | 1.000 | e_{2} | | |
| fong-spivak-invitation_FO1837 | 172 | 1.000 | \left(e_{1}, e_{2}\right) \in E | | |
| fong-spivak-invitation_FO1838 | 172 | 0.998 | e_{1}=e_{2} | | |
| fong-spivak-invitation_FO1839 | 173 | 1.000 | g \in G, f(g) | | |
| fong-spivak-invitation_FO1840 | 173 | 1.000 | f\left(e_{1}\right)=f\left(e_{2}\right) | | |
| fong-spivak-invitation_FO1841 | 173 | 1.000 | f(e) | | |
| fong-spivak-invitation_FO1842 | 173 | 0.658 | (G, s, t, \varnothing) | | |
| fong-spivak-invitation_FO1843 | 173 | 0.976 | R, 0,+, 1, * | | |
| fong-spivak-invitation_FO1844 | 173 | 0.976 | 0,1 \in R | | |
| fong-spivak-invitation_FO1845 | 173 | 0.986 | +, *: R \times R \rightarrow R | | |
| fong-spivak-invitation_FO1846 | 173 | 0.989 | (R,+, 0) | | |
| fong-spivak-invitation_FO1847 | 173 | 0.615 | R, *, 1 | | |
| fong-spivak-invitation_FO1848 | 173 | 1.000 | a *(b+c)=a * b+a * c | | |
| fong-spivak-invitation_FO1849 | 173 | 1.000 | (a+b) * c=a * c+b * c | | |
| fong-spivak-invitation_FO1850 | 173 | 1.000 | a, b, c \in R | | |
| fong-spivak-invitation_FO1851 | 173 | 1.000 | a * 0=0=0 * a | | |
| fong-spivak-invitation_FO1852 | 173 | 1.000 | a \in R | | |
| fong-spivak-invitation_FO1853 | 173 | 0.733 | (\mathbb{N}, 0,+, 1, *) | | |
| fong-spivak-invitation_FO1854 | 173 | 0.999 | (R, *, 1) | | |
| fong-spivak-invitation_FO1855 | 174 | 0.898 | (V, 0, \vee, I, \otimes) | | |
| fong-spivak-invitation_FO1856 | 174 | 1.000 | 0=\bigvee \varnothing | | |
| fong-spivak-invitation_FO1857 | 174 | 1.000 | \operatorname{Mat}_{n}(R) | | |
| fong-spivak-invitation_FO1858 | 174 | 1.000 | (n \times n) | | |
| fong-spivak-invitation_FO1859 | 174 | 1.000 | M \in \operatorname{Mat}_{n}(R) | | |
| fong-spivak-invitation_FO1860 | 174 | 1.000 | M: \underline{n} \times \underline{n} \rightarrow R | | |
| fong-spivak-invitation_FO1861 | 174 | 1.000 | M+N | | |
| fong-spivak-invitation_FO1862 | 174 | 1.000 | (M+N)(i, j):=M(i, j)+N(i, j) | | |
| fong-spivak-invitation_FO1863 | 174 | 0.999 | (M * N)(i, j):=\sum_{k \in \underline{n}} M(i, k) * N(k, j) | | |
| fong-spivak-invitation_FO1864 | 174 | 1.000 | 0(i, j):=0 | | |
| fong-spivak-invitation_FO1865 | 174 | 1.000 | i, j \in \underline{n} | | |
| fong-spivak-invitation_FO1866 | 174 | 1.000 | 1 \in \operatorname{Mat}_{n}(R) | | |
| fong-spivak-invitation_FO1867 | 174 | 1.000 | \operatorname{Mat}_{n}(\mathbb{N}) | | |
| fong-spivak-invitation_FO1868 | 174 | 0.998 | \mathbb{R}, 0,+, 1, * | | |
| fong-spivak-invitation_FO1869 | 174 | 1.000 | M=R | | |
| fong-spivak-invitation_FO1870 | 175 | 1.000 | R=\mathbb{N} | | |
| fong-spivak-invitation_FO1871 | 175 | 1.000 | a=15 x | | |
| fong-spivak-invitation_FO1872 | 175 | 1.000 | b=3 x+21 y | | |
| fong-spivak-invitation_FO1873 | 176 | 1.000 | 15 x | | |
| fong-spivak-invitation_FO1874 | 176 | 1.000 | 3 x=x+2 x | | |
| fong-spivak-invitation_FO1875 | 176 | 0.954 | * 7 | | |
| fong-spivak-invitation_FO1876 | 176 | 0.954 | 21 y | | |
| fong-spivak-invitation_FO1877 | 176 | 0.999 | (x, y, z) | | |
| fong-spivak-invitation_FO1878 | 177 | 1.000 | s, t: G_{R} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO1879 | 177 | 0.403 | \boldsymbol{\operatorname { F r e e }}\left(G_{R}\right) | | |
| fong-spivak-invitation_FO1880 | 177 | 0.403 | G_{R} | | |
| fong-spivak-invitation_FO1881 | 177 | 0.403 | \operatorname{SFG}_{R}:=\boldsymbol{\operatorname { F r e e }}\left(G_{R}\right) | | |
| fong-spivak-invitation_FO1882 | 177 | 0.576 | \operatorname{Free}\left(G_{R}\right) | | |
| fong-spivak-invitation_FO1883 | 178 | 0.996 | (m \times n) | | |
| fong-spivak-invitation_FO1884 | 178 | 0.996 | M:(\underline{m} \times \underline{n}) \rightarrow R | | |
| fong-spivak-invitation_FO1885 | 178 | 0.982 | (n \times p) | | |
| fong-spivak-invitation_FO1886 | 178 | 0.982 | (m \times p) | | |
| fong-spivak-invitation_FO1887 | 178 | 1.000 | a \in \underline{m} | | |
| fong-spivak-invitation_FO1888 | 178 | 1.000 | c \in \underline{p} | | |
| fong-spivak-invitation_FO1889 | 178 | 0.748 | \sum_{b \in \underline{n}} | | |
| fong-spivak-invitation_FO1890 | 178 | 1.000 | A v | | |
| fong-spivak-invitation_FO1891 | 178 | 0.997 | v ; A | | |
| fong-spivak-invitation_FO1892 | 178 | 0.918 | \operatorname{Mat}(R) | | |
| fong-spivak-invitation_FO1893 | 178 | 0.999 | A: m \rightarrow n | | |
| fong-spivak-invitation_FO1894 | 178 | 1.000 | b: p \rightarrow q | | |
| fong-spivak-invitation_FO1895 | 178 | 1.000 | A+B: m+p \rightarrow n+q | | |
| fong-spivak-invitation_FO1896 | 178 | 0.979 | m \times q | | |
| fong-spivak-invitation_FO1897 | 178 | 0.979 | n \times p | | |
| fong-spivak-invitation_FO1898 | 178 | 1.000 | A+B | | |
| fong-spivak-invitation_FO1900 | 179 | 1.000 | (1 \times 1) | | |
| fong-spivak-invitation_FO1901 | 179 | 1.000 | (a): 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO1902 | 179 | 1.000 | x \in R | | |
| fong-spivak-invitation_FO1903 | 179 | 1.000 | a * x | | |
| fong-spivak-invitation_FO1904 | 179 | 0.588 | \supset | | |
| fong-spivak-invitation_FO1905 | 179 | 0.588 | (2 \times 1) | | |
| fong-spivak-invitation_FO1906 | 179 | 0.588 | \binom{1}{1} | | |
| fong-spivak-invitation_FO1907 | 179 | 1.000 | x+y | | |
| fong-spivak-invitation_FO1908 | 222 | 0.793 | 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO1913 | 179 | 1.000 | 0 \times n | | |
| fong-spivak-invitation_FO1914 | 179 | 1.000 | n \times 0 | | |
| fong-spivak-invitation_FO1915 | 179 | 0.996 | (0 \times 3) | | |
| fong-spivak-invitation_FO1916 | 179 | 0.996 | (3 \times n) | | |
| fong-spivak-invitation_FO1917 | 179 | 1.000 | (2 \times n) | | |
| fong-spivak-invitation_FO1918 | 180 | 0.999 | S: \mathbf{S F G}_{R} \rightarrow \mathbf{M a t}(R) | | |
| fong-spivak-invitation_FO1919 | 180 | 1.000 | S(g) | | |
| fong-spivak-invitation_FO1920 | 180 | 1.000 | m \times n | | |
| fong-spivak-invitation_FO1921 | 180 | 0.999 | (i, j) | | |
| fong-spivak-invitation_FO1922 | 180 | 1.000 | \mathrm{id}_{0}: 0 \rightarrow 0, \mathrm{id}_{1}: 1 \rightarrow 1, \sigma: 2 \rightarrow 2 | | |
| fong-spivak-invitation_FO1923 | 180 | 1.000 | (\alpha+\beta):(m+p) \rightarrow(n+q) | | |
| fong-spivak-invitation_FO1924 | 180 | 1.000 | \mathrm{id}_{0}, \mathrm{id}_{1} | | |
| fong-spivak-invitation_FO1925 | 180 | 0.853 | 0 \rightarrow 0 | | |
| fong-spivak-invitation_FO1926 | 180 | 0.853 | \mathrm{id}_{1}, \sigma | | |
| fong-spivak-invitation_FO1927 | 180 | 0.607 | n=p | | |
| fong-spivak-invitation_FO1928 | 180 | 0.607 | \alpha+\beta | | |
| fong-spivak-invitation_FO1929 | 181 | 1.000 | \beta: n \rightarrow q | | |
| fong-spivak-invitation_FO1930 | 181 | 0.798 | j \in \underline{n} | | |
| fong-spivak-invitation_FO1931 | 181 | 0.627 | \alpha / | | |
| fong-spivak-invitation_FO1932 | 181 | 0.983 | S(\alpha) ; S(\beta) | | |
| fong-spivak-invitation_FO1933 | 181 | 0.959 | \boldsymbol{\operatorname { M a t }}(R) | | |
| fong-spivak-invitation_FO1934 | 181 | 0.666 | (m+i) | | |
| fong-spivak-invitation_FO1935 | 181 | 0.556 | n+j | | |
| fong-spivak-invitation_FO1936 | 181 | 1.000 | S(\alpha)+S(\beta)=S(\alpha+\beta) | | |
| fong-spivak-invitation_FO1937 | 183 | 1.000 | M \in \operatorname{Mat}(R) | | |
| fong-spivak-invitation_FO1938 | 183 | 1.000 | g \in \mathbf{S F G}_{R} | | |
| fong-spivak-invitation_FO1939 | 183 | 1.000 | S(g)=M | | |
| fong-spivak-invitation_FO1940 | 183 | 1.000 | S(g)(i, j) | | |
| fong-spivak-invitation_FO1941 | 183 | 1.000 | M(i, j) | | |
| fong-spivak-invitation_FO1942 | 183 | 0.488 | \left(\begin{array}{ccc}a & b \\ c & d\end{array}\right) | | |
| fong-spivak-invitation_FO1943 | 183 | 1.000 | \left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right) | | |
| fong-spivak-invitation_FO1944 | 183 | 1.000 | \left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right) | | |
| fong-spivak-invitation_FO1945 | 183 | 1.000 | \left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right) | | |
| fong-spivak-invitation_FO1946 | 184 | 1.000 | \mathbf{S F G}_{R} | | |
| fong-spivak-invitation_FO1947 | 184 | 1.000 | a, b \in R | | |
| fong-spivak-invitation_FO1948 | 185 | 0.974 | \binom{0}{6} | | |
| fong-spivak-invitation_FO1949 | 185 | 0.665 | R=(\mathbb{N}, 0,+, 1, *) | | |
| fong-spivak-invitation_FO1950 | 185 | 1.000 | R=\mathbb{N} / 3 \mathbb{N} | | |
| fong-spivak-invitation_FO1951 | 186 | 0.780 | (M, \mu, \eta) | | |
| fong-spivak-invitation_FO1952 | 186 | 1.000 | \mu: M \otimes M \rightarrow M | | |
| fong-spivak-invitation_FO1953 | 186 | 1.000 | \eta: I \rightarrow M | | |
| fong-spivak-invitation_FO1954 | 186 | 0.993 | (\mu \otimes \mathrm{id}) ; \mu=(\mathrm{id} \otimes \mu) ; \mu | | |
| fong-spivak-invitation_FO1955 | 186 | 1.000 | (\eta \otimes \mathrm{id}) ; \mu=\mathrm{id}=(\mathrm{id} \otimes \eta) ; \mu | | |
| fong-spivak-invitation_FO1956 | 186 | 0.528 | \sigma_{M, M}: \mu=\mu | | |
| fong-spivak-invitation_FO1957 | 186 | 1.000 | \sigma_{M, M} | | |
| fong-spivak-invitation_FO1958 | 186 | 1.000 | \mu: M \times M \rightarrow M | | |
| fong-spivak-invitation_FO1959 | 186 | 1.000 | \eta(1) \in M | | |
| fong-spivak-invitation_FO1960 | 186 | 1.000 | (a * b) * c=a *(b * c) | | |
| fong-spivak-invitation_FO1961 | 186 | 1.000 | a * e=a=e * a | | |
| fong-spivak-invitation_FO1962 | 186 | 1.000 | \mu: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO1963 | 186 | 1.000 | \mu(a, b)=a * b | | |
| fong-spivak-invitation_FO1964 | 186 | 1.000 | \eta \in \mathbb{R} | | |
| fong-spivak-invitation_FO1965 | 186 | 0.997 | \eta=1 | | |
| fong-spivak-invitation_FO1966 | 186 | 0.997 | (\mathbb{R}, *, 1) | | |
| fong-spivak-invitation_FO1967 | 186 | 0.999 | \mu(a, b)=a+b | | |
| fong-spivak-invitation_FO1968 | 186 | 0.892 | \eta=0 | | |
| fong-spivak-invitation_FO1969 | 186 | 0.892 | (\mathbb{R},+, 0) | | |
| fong-spivak-invitation_FO1970 | 186 | 0.361 | \mu=\supset- | | |
| fong-spivak-invitation_FO1971 | 186 | 0.361 | \eta=\circ | | |
| fong-spivak-invitation_FO1972 | 187 | 0.998 | U: \operatorname{Mat}(R) \rightarrow | | |
| fong-spivak-invitation_FO1973 | 187 | 1.000 | R^{n} | | |
| fong-spivak-invitation_FO1974 | 187 | 1.000 | M: m \rightarrow n | | |
| fong-spivak-invitation_FO1975 | 187 | 1.000 | R^{m} \rightarrow R^{n} | | |
| fong-spivak-invitation_FO1976 | 187 | 0.932 | \{1\} | | |
| fong-spivak-invitation_FO1977 | 187 | 0.773 | (U(M), U(\mu), U(\eta)) | | |
| fong-spivak-invitation_FO1978 | 187 | 0.537 | \operatorname{prop} \boldsymbol{\operatorname { M a t }}(R) | | |
| fong-spivak-invitation_FO1979 | 187 | 0.537 | R=\mathbb{R} | | |
| fong-spivak-invitation_FO1980 | 187 | 0.739 | (1,-\subset, \rightarrow) | | |
| fong-spivak-invitation_FO1981 | 187 | 0.428 | \operatorname{Mat}(R)^{\mathrm{op}} | | |
| fong-spivak-invitation_FO1982 | 187 | 1.000 | \mu: 2 \rightarrow 1 | | |
| fong-spivak-invitation_FO1983 | 187 | 1.000 | \eta: 0 \rightarrow 1 | | |
| fong-spivak-invitation_FO1984 | 188 | 1.000 | \mathcal{M} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO1985 | 188 | 0.503 | -\subset | | |
| fong-spivak-invitation_FO1986 | 188 | 1.000 | g: m \rightarrow n | | |
| fong-spivak-invitation_FO1987 | 188 | 1.000 | x \in R^{m} | | |
| fong-spivak-invitation_FO1988 | 188 | 1.000 | y \in R^{n} | | |
| fong-spivak-invitation_FO1989 | 188 | 0.998 | \mathrm{B}(g) \subseteq R^{m} \times R^{n} | | |
| fong-spivak-invitation_FO1990 | 188 | 1.000 | (x,(x, x)) | | |
| fong-spivak-invitation_FO1991 | 188 | 0.986 | g^{\mathrm{op}} | | |
| fong-spivak-invitation_FO1992 | 188 | 0.869 | g^{\mathrm{op}}: n \rightarrow m | | |
| fong-spivak-invitation_FO1993 | 188 | 0.869 | \mathrm{B}\left(g^{\mathrm{op}}\right) \subseteq R^{n} \times R^{m} | | |
| fong-spivak-invitation_FO1994 | 188 | 0.997 | (b, a) | | |
| fong-spivak-invitation_FO1995 | 188 | 0.473 | \mathrm{B}(-\subset) | | |
| fong-spivak-invitation_FO1996 | 188 | 0.473 | -\subset: 1 \rightarrow 2 | | |
| fong-spivak-invitation_FO1997 | 188 | 0.237 | \mathrm{B}(-) | | |
| fong-spivak-invitation_FO1998 | 188 | 0.237 | -:2 \rightarrow 1 | | |
| fong-spivak-invitation_FO1999 | 189 | 1.000 | B \subseteq R^{m} \times R^{n} | | |
| fong-spivak-invitation_FO2000 | 189 | 1.000 | N: n \rightarrow p | | |
| fong-spivak-invitation_FO2001 | 189 | 1.000 | B_{1}:=\left\{(x, M x) \mid x \in R^{m}\right\} | | |
| fong-spivak-invitation_FO2002 | 189 | 1.000 | B_{2}:=\left\{(y, N y) \mid y \in R^{n}\right\} | | |
| fong-spivak-invitation_FO2003 | 189 | 0.412 | M{ }_{9}^{\circ} N | | |
| fong-spivak-invitation_FO2004 | 189 | 0.412 | \left\{\left(x, x{ }_{9}^{\circ} M{ }_{9}^{\circ} N\right) \mid x \in R^{m}\right\} | | |
| fong-spivak-invitation_FO2005 | 189 | 1.000 | B_{1} \subseteq R^{m} \times R^{n} | | |
| fong-spivak-invitation_FO2006 | 189 | 1.000 | B_{2} \subseteq R^{n} \times R^{p} | | |
| fong-spivak-invitation_FO2007 | 189 | 0.493 | B_{1}{ }_{9} B_{2} \subseteq R^{m} \times R^{p} | | |
| fong-spivak-invitation_FO2008 | 189 | 0.999 | \mathbf{R e l}_{R} | | |
| fong-spivak-invitation_FO2009 | 189 | 1.000 | C \subseteq R^{p} \times R^{q} | | |
| fong-spivak-invitation_FO2010 | 189 | 1.000 | B+C | | |
| fong-spivak-invitation_FO2011 | 189 | 0.689 | g \in G_{R} | | |
| fong-spivak-invitation_FO2012 | 189 | 1.000 | G_{R}^{\mathrm{op}}:=\left\{g^{\mathrm{op}} \mid g \in G_{R}\right\} | | |
| fong-spivak-invitation_FO2013 | 189 | 1.000 | \mathbf{S F G}_{R}^{+} | | |
| fong-spivak-invitation_FO2014 | 190 | 1.000 | g: m \rightarrow n, h: \ell \rightarrow n | | |
| fong-spivak-invitation_FO2015 | 190 | 1.000 | h^{\mathrm{op}}: n \rightarrow \ell | | |
| fong-spivak-invitation_FO2016 | 190 | 0.882 | \left(h^{\mathrm{op}}\right) | | |
| fong-spivak-invitation_FO2017 | 190 | 0.615 | \left(h^{\mathrm{op}}\right) \subseteq R^{m} \times R^{\ell} | | |
| fong-spivak-invitation_FO2018 | 190 | 0.986 | g: m \rightarrow n, h: m \rightarrow p | | |
| fong-spivak-invitation_FO2019 | 190 | 0.986 | \left(g^{\mathrm{op}}\right): n \rightarrow | | |
| fong-spivak-invitation_FO2020 | 190 | 0.971 | g^{\mathrm{op}} ; h | | |
| fong-spivak-invitation_FO2021 | 190 | 0.864 | S(g) \in \operatorname{Mat}(R) | | |
| fong-spivak-invitation_FO2022 | 191 | 0.981 | b_{1}, b_{2} \in \mathrm{~B}(g) | | |
| fong-spivak-invitation_FO2023 | 191 | 0.981 | b_{1}+b_{2} | | |
| fong-spivak-invitation_FO2024 | 191 | 0.981 | r * b_{1} | | |
| fong-spivak-invitation_FO2025 | 191 | 0.981 | r \in R | | |
| fong-spivak-invitation_FO2026 | 191 | 0.840 | \mathbf{L i n R e l}_{R} | | |
| fong-spivak-invitation_FO2027 | 191 | 0.997 | (x, y) \in B | | |
| fong-spivak-invitation_FO2028 | 191 | 0.987 | (r * x, r * y) \in R^{m} \times R^{n} | | |
| fong-spivak-invitation_FO2029 | 191 | 0.987 | \left(x^{\prime}, y^{\prime}\right) \in B | | |
| fong-spivak-invitation_FO2030 | 191 | 0.987 | \left(x+x^{\prime}, y+y^{\prime}\right) | | |
| fong-spivak-invitation_FO2031 | 191 | 0.644 | G_{R} \sqcup G_{R}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO2032 | 192 | 1.000 | \eta_{1} | | |
| fong-spivak-invitation_FO2033 | 192 | 1.000 | \epsilon_{1} | | |
| fong-spivak-invitation_FO2034 | 192 | 1.000 | n=n^{*} | | |
| fong-spivak-invitation_FO2035 | 192 | 0.999 | \mathbb{R}\left[s, s^{-1}\right] | | |
| fong-spivak-invitation_FO2036 | 192 | 1.000 | s^{-1} | | |
| fong-spivak-invitation_FO2037 | 192 | 1.000 | s s^{-1}=s^{-1} s=1 | | |
| fong-spivak-invitation_FO2038 | 192 | 1.000 | a * u+b * v | | |
| fong-spivak-invitation_FO2039 | 192 | 1.000 | \{(F, u, v) \mid u=v\} | | |
| fong-spivak-invitation_FO2040 | 196 | 0.824 | \operatorname{Cospan}_{\mathcal{C}} | | |
| fong-spivak-invitation_FO2041 | 196 | 1.000 | s_{m, n}: m \rightarrow n | | |
| fong-spivak-invitation_FO2042 | 197 | 0.988 | \varnothing \in \mathcal{C} | | |
| fong-spivak-invitation_FO2043 | 197 | 1.000 | !_{T}: \varnothing \rightarrow T | | |
| fong-spivak-invitation_FO2044 | 198 | 0.992 | A, \leq | | |
| fong-spivak-invitation_FO2045 | 198 | 0.996 | (A, \leq) | | |
| fong-spivak-invitation_FO2046 | 198 | 1.000 | \varnothing \rightarrow T | | |
| fong-spivak-invitation_FO2047 | 198 | 0.860 | T \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO2048 | 198 | 0.860 | T=A | | |
| fong-spivak-invitation_FO2049 | 198 | 0.860 | T=B | | |
| fong-spivak-invitation_FO2050 | 198 | 0.791 | \varnothing= | | |
| fong-spivak-invitation_FO2051 | 198 | 1.000 | \varnothing={ }^{?} B | | |
| fong-spivak-invitation_FO2052 | 198 | 1.000 | B \rightarrow A | | |
| fong-spivak-invitation_FO2053 | 199 | 0.977 | \left(R, 0_{R},+_{R}, 1_{R}, *_{R}\right) | | |
| fong-spivak-invitation_FO2054 | 199 | 0.977 | \left(S, 0_{S},+_{S}, 1_{S}, *_{S}\right) | | |
| fong-spivak-invitation_FO2055 | 199 | 0.977 | f: R \rightarrow S | | |
| fong-spivak-invitation_FO2056 | 199 | 0.977 | f\left(0_{R}\right)= | | |
| fong-spivak-invitation_FO2057 | 199 | 1.000 | 0_{S}, f\left(r_{1}+{ }_{R} r_{2}\right)=f\left(r_{1}\right)+s f\left(r_{2}\right) | | |
| fong-spivak-invitation_FO2058 | 199 | 1.000 | c_{1} | | |
| fong-spivak-invitation_FO2059 | 199 | 1.000 | c_{2} | | |
| fong-spivak-invitation_FO2060 | 199 | 0.998 | \iota_{A}: A \rightarrow | | |
| fong-spivak-invitation_FO2061 | 199 | 0.900 | A+B, \iota_{B}: B \rightarrow A+B | | |
| fong-spivak-invitation_FO2062 | 199 | 0.900 | f: A \rightarrow | | |
| fong-spivak-invitation_FO2063 | 199 | 0.999 | T, g: B \rightarrow T | | |
| fong-spivak-invitation_FO2064 | 199 | 0.999 | [f, g]: A+B \rightarrow T | | |
| fong-spivak-invitation_FO2065 | 199 | 1.000 | [f, g] | | |
| fong-spivak-invitation_FO2066 | 200 | 0.997 | \iota_{A}: A \longrightarrow A \sqcup B | | |
| fong-spivak-invitation_FO2067 | 200 | 1.000 | \iota_{B}: B \rightarrow A \sqcup B | | |
| fong-spivak-invitation_FO2068 | 200 | 1.000 | f: A \rightarrow T | | |
| fong-spivak-invitation_FO2069 | 200 | 1.000 | g: B \rightarrow T | | |
| fong-spivak-invitation_FO2070 | 200 | 0.804 | [f, g]: A \sqcup B \rightarrow T | | |
| fong-spivak-invitation_FO2071 | 200 | 0.804 | \iota_{A} | | |
| fong-spivak-invitation_FO2072 | 200 | 0.804 | [f, g]=f | | |
| fong-spivak-invitation_FO2073 | 200 | 0.804 | \iota_{B} | | |
| fong-spivak-invitation_FO2074 | 200 | 0.804 | [f, g]=g | | |
| fong-spivak-invitation_FO2075 | 200 | 0.997 | x \in A \sqcup B | | |
| fong-spivak-invitation_FO2076 | 200 | 0.998 | x=\iota_{A}(a) | | |
| fong-spivak-invitation_FO2077 | 200 | 0.998 | b \in B | | |
| fong-spivak-invitation_FO2078 | 200 | 0.998 | x=\iota_{B}(b) | | |
| fong-spivak-invitation_FO2079 | 200 | 1.000 | T=\{a, b, c, \ldots, z\} | | |
| fong-spivak-invitation_FO2080 | 200 | 0.999 | f( | | |
| fong-spivak-invitation_FO2081 | 200 | 0.999 | )=a | | |
| fong-spivak-invitation_FO2082 | 200 | 0.993 | g( | | |
| fong-spivak-invitation_FO2083 | 200 | 0.993 | )=e | | |
| fong-spivak-invitation_FO2084 | 200 | 1.000 | [f, g](x) | | |
| fong-spivak-invitation_FO2085 | 200 | 1.000 | f: A \rightarrow C, g: B \rightarrow C | | |
| fong-spivak-invitation_FO2086 | 200 | 0.359 | \iota_{A} ;[f, g]=f | | |
| fong-spivak-invitation_FO2087 | 200 | 0.859 | [f, g] ; h=[f ; h, g ; h] | | |
| fong-spivak-invitation_FO2088 | 200 | 1.000 | \left[\iota_{A}, \iota_{B}\right]=\mathrm{id}_{A+B} | | |
| fong-spivak-invitation_FO2089 | 200 | 0.981 | (\mathcal{C},+, \varnothing) | | |
| fong-spivak-invitation_FO2090 | 200 | 0.807 | \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO2091 | 200 | 0.963 | \mathcal{C} \times \mathcal{C} | | |
| fong-spivak-invitation_FO2092 | 201 | 1.000 | A+\varnothing \rightarrow A | | |
| fong-spivak-invitation_FO2093 | 201 | 1.000 | \varnothing+A \rightarrow A | | |
| fong-spivak-invitation_FO2094 | 201 | 1.000 | (A+B)+C \rightarrow A+(B+C) | | |
| fong-spivak-invitation_FO2095 | 201 | 1.000 | A+B \rightarrow B+A | | |
| fong-spivak-invitation_FO2096 | 201 | 1.000 | f: A \rightarrow X | | |
| fong-spivak-invitation_FO2097 | 201 | 1.000 | g: A \rightarrow Y | | |
| fong-spivak-invitation_FO2098 | 201 | 0.996 | X+{ }_{A} Y | | |
| fong-spivak-invitation_FO2099 | 201 | 0.712 | X+_{A} Y | | |
| fong-spivak-invitation_FO2100 | 201 | 0.712 | \iota_{X}: X \rightarrow | | |
| fong-spivak-invitation_FO2101 | 201 | 0.997 | \iota_{Y}: Y \rightarrow X+{ }_{A} Y | | |
| fong-spivak-invitation_FO2102 | 201 | 0.989 | \ulcorner | | |
| fong-spivak-invitation_FO2103 | 201 | 1.000 | x: X \rightarrow T, y: Y \rightarrow T | | |
| fong-spivak-invitation_FO2104 | 201 | 0.955 | t: X+{ }_{A} Y \rightarrow T | | |
| fong-spivak-invitation_FO2105 | 202 | 1.000 | \iota_{X} | | |
| fong-spivak-invitation_FO2106 | 202 | 1.000 | \iota_{Y} | | |
| fong-spivak-invitation_FO2107 | 202 | 1.000 | B \leftarrow A \rightarrow C | | |
| fong-spivak-invitation_FO2108 | 202 | 1.000 | B \sqcup C | | |
| fong-spivak-invitation_FO2109 | 202 | 1.000 | B \vee C | | |
| fong-spivak-invitation_FO2110 | 202 | 1.000 | i: A \rightarrow A^{\prime} | | |
| fong-spivak-invitation_FO2111 | 202 | 1.000 | j: X \rightarrow X^{\prime} | | |
| fong-spivak-invitation_FO2112 | 202 | 1.000 | X^{\prime} | | |
| fong-spivak-invitation_FO2113 | 202 | 1.000 | X+{ }_{A} A^{\prime} | | |
| fong-spivak-invitation_FO2114 | 202 | 0.659 | f^{\prime}:=i^{-1}{ }_{\circ} f{ }_{\circ} j | | |
| fong-spivak-invitation_FO2115 | 202 | 0.568 | x=i | | |
| fong-spivak-invitation_FO2116 | 202 | 0.568 | x^{\prime}: X \rightarrow T | | |
| fong-spivak-invitation_FO2117 | 202 | 0.568 | x^{\prime}:=j^{-1} ; x | | |
| fong-spivak-invitation_FO2118 | 202 | 0.978 | f^{\prime} ; x^{\prime}=i^{-1} ; f ; j ; j^{-1} ; x=i^{-1} ; i ; a=a | | |
| fong-spivak-invitation_FO2119 | 203 | 1.000 | \{f(a) \sim g(a) \mid a \in A\} | | |
| fong-spivak-invitation_FO2120 | 203 | 1.000 | f(a) | | |
| fong-spivak-invitation_FO2121 | 203 | 1.000 | g(a) | | |
| fong-spivak-invitation_FO2122 | 203 | 1.000 | f: \underline{4} \rightarrow \underline{5} | | |
| fong-spivak-invitation_FO2123 | 203 | 1.000 | g: \underline{4} \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO2124 | 203 | 1.000 | X, Y \in | | |
| fong-spivak-invitation_FO2125 | 203 | 0.884 | f: \varnothing \rightarrow X | | |
| fong-spivak-invitation_FO2126 | 203 | 0.884 | g: \varnothing \rightarrow Y | | |
| fong-spivak-invitation_FO2127 | 203 | 0.923 | X \stackrel{f}{\leftarrow} \varnothing \xrightarrow{g} Y | | |
| fong-spivak-invitation_FO2128 | 203 | 1.000 | X+Y | | |
| fong-spivak-invitation_FO2129 | 203 | 1.000 | X+Y \rightarrow T | | |
| fong-spivak-invitation_FO2130 | 203 | 0.968 | { }_{2} | | |
| fong-spivak-invitation_FO2131 | 203 | 0.968 | X+\varnothing Y \cong X+Y | | |
| fong-spivak-invitation_FO2132 | 203 | 1.000 | X+\varnothing Y | | |
| fong-spivak-invitation_FO2133 | 203 | 0.997 | { }_{3} | | |
| fong-spivak-invitation_FO2134 | 204 | 1.000 | A=X=Y=\mathbb{N} | | |
| fong-spivak-invitation_FO2135 | 204 | 0.988 | f(a):=\lfloor a / 2\rfloor | | |
| fong-spivak-invitation_FO2136 | 204 | 0.988 | g(a):=\lfloor(a+1) / 2\rfloor | | |
| fong-spivak-invitation_FO2137 | 204 | 1.000 | x: X \rightarrow T | | |
| fong-spivak-invitation_FO2138 | 204 | 1.000 | y: Y \rightarrow T | | |
| fong-spivak-invitation_FO2139 | 204 | 1.000 | f \cong x=g \cong y | | |
| fong-spivak-invitation_FO2140 | 204 | 1.000 | f(1)=0 | | |
| fong-spivak-invitation_FO2141 | 204 | 1.000 | g(1)=1 | | |
| fong-spivak-invitation_FO2142 | 204 | 1.000 | t=x(0)=x(f(1))=y(g(1))=y(1) | | |
| fong-spivak-invitation_FO2143 | 204 | 1.000 | g(2)=1 | | |
| fong-spivak-invitation_FO2144 | 204 | 1.000 | f(2)=1 | | |
| fong-spivak-invitation_FO2145 | 204 | 1.000 | t=g(1)= | | |
| fong-spivak-invitation_FO2146 | 204 | 0.986 | y(g(2))=x(f(2))=x(1) | | |
| fong-spivak-invitation_FO2147 | 204 | 1.000 | X \sqcup_{A} Y \cong\{1\} | | |
| fong-spivak-invitation_FO2148 | 204 | 0.862 | j \in \operatorname{Ob} \mathcal{J} | | |
| fong-spivak-invitation_FO2149 | 204 | 0.862 | \operatorname{id}_{j} | | |
| fong-spivak-invitation_FO2150 | 205 | 0.997 | \operatorname{colim}_{\mathcal{J}} D | | |
| fong-spivak-invitation_FO2151 | 205 | 0.820 | \mathfrak{d} | | |
| fong-spivak-invitation_FO2152 | 205 | 0.963 | !: \mathbf{0} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO2153 | 206 | 0.997 | (v, d) \sim(w, e) | | |
| fong-spivak-invitation_FO2154 | 206 | 0.997 | a: v \rightarrow w | | |
| fong-spivak-invitation_FO2155 | 206 | 0.991 | D(a)(d)=e | | |
| fong-spivak-invitation_FO2156 | 206 | 0.991 | \iota_{v}: D(v) \rightarrow \operatorname{colim}_{\mathcal{J}} D | | |
| fong-spivak-invitation_FO2157 | 206 | 0.996 | d \in D(v) | | |
| fong-spivak-invitation_FO2158 | 206 | 0.508 | \mathfrak{J}=\begin{array}{|cc|}v_{1} & v_{2} \\ \bullet & \bullet\end{array} | | |
| fong-spivak-invitation_FO2159 | 206 | 0.990 | X:=D\left(v_{1}\right) | | |
| fong-spivak-invitation_FO2160 | 206 | 0.990 | Y:=D\left(v_{2}\right) | | |
| fong-spivak-invitation_FO2161 | 207 | 0.479 | \mathbf{1}=\boxed{\bullet} | | |
| fong-spivak-invitation_FO2162 | 207 | 1.000 | D(v) | | |
| fong-spivak-invitation_FO2163 | 207 | 0.995 | X: \mathbf{1} \rightarrow | | |
| fong-spivak-invitation_FO2164 | 207 | 1.000 | X \stackrel{f}{\leftarrow} N \xrightarrow{g} Y | | |
| fong-spivak-invitation_FO2165 | 207 | 0.985 | X \underset{g}{\stackrel{f}{\rightrightarrows}} Y | | |
| fong-spivak-invitation_FO2166 | 207 | 1.000 | y \sim y^{\prime} | | |
| fong-spivak-invitation_FO2167 | 207 | 1.000 | f(x)=y | | |
| fong-spivak-invitation_FO2168 | 207 | 1.000 | g(x)=y^{\prime} | | |
| fong-spivak-invitation_FO2169 | 207 | 0.883 | f, g: X \rightarrow Y | | |
| fong-spivak-invitation_FO2170 | 208 | 1.000 | A \rightarrow N \leftarrow B | | |
| fong-spivak-invitation_FO2171 | 208 | 1.000 | N \stackrel{g}{\leftarrow} B \xrightarrow{h} P | | |
| fong-spivak-invitation_FO2172 | 208 | 0.822 | C \in \mathrm{Ob} \mathcal{C} | | |
| fong-spivak-invitation_FO2173 | 208 | 0.822 | C \rightarrow C \leftarrow C | | |
| fong-spivak-invitation_FO2174 | 208 | 1.000 | A \rightarrow B \rightarrow C | | |
| fong-spivak-invitation_FO2175 | 208 | 1.000 | A \rightarrow P \leftarrow B | | |
| fong-spivak-invitation_FO2176 | 208 | 1.000 | A \rightarrow P^{\prime} \leftarrow B | | |
| fong-spivak-invitation_FO2177 | 208 | 1.000 | P \cong P^{\prime} | | |
| fong-spivak-invitation_FO2178 | 208 | 0.824 | { }_{\text {C }} | | |
| fong-spivak-invitation_FO2179 | 209 | 0.500 | _{\mathcal{C}} | | |
| fong-spivak-invitation_FO2180 | 209 | 0.500 | \operatorname{Ob}\left(\boldsymbol{\operatorname { C o s p a n }}_{\mathcal{C}}\right)=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO2181 | 209 | 0.907 | _{\mathcal{C}}, \varnothing,+ | | |
| fong-spivak-invitation_FO2182 | 209 | 1.000 | A+\varnothing \cong A | | |
| fong-spivak-invitation_FO2183 | 209 | 0.098 | \boldsymbol{C o s p a n}_{\mathcal{C}}, \varnothing,+ | | |
| fong-spivak-invitation_FO2184 | 209 | 0.749 | _{\text {FinSet }} | | |
| fong-spivak-invitation_FO2185 | 209 | 0.986 | { }_{\text {FinSet }} | | |
| fong-spivak-invitation_FO2186 | 209 | 1.000 | A \xrightarrow{f} N \stackrel{g}{\leftarrow} B | | |
| fong-spivak-invitation_FO2187 | 209 | 1.000 | g(b) | | |
| fong-spivak-invitation_FO2188 | 210 | 0.997 | A+B \rightarrow B+C | | |
| fong-spivak-invitation_FO2189 | 211 | 0.995 | (X, \mu, \eta) | | |
| fong-spivak-invitation_FO2190 | 211 | 0.531 | \mathcal{C}, I, \otimes | | |
| fong-spivak-invitation_FO2193 | 211 | 0.786 | X \otimes X | | |
| fong-spivak-invitation_FO2194 | 211 | 0.786 | (X, \delta, \epsilon) | | |
| fong-spivak-invitation_FO2195 | 211 | 1.000 | \delta: X \rightarrow X \otimes X, \epsilon: X \rightarrow I | | |
| fong-spivak-invitation_FO2196 | 211 | 1.000 | \mu, \eta, \delta | | |
| fong-spivak-invitation_FO2197 | 211 | 1.000 | \epsilon | | |
| fong-spivak-invitation_FO2198 | 211 | 1.000 | \mu | | |
| fong-spivak-invitation_FO2199 | 211 | 1.000 | \delta | | |
| fong-spivak-invitation_FO2200 | 211 | 0.986 | (\mathcal{C}, \otimes, I) | | |
| fong-spivak-invitation_FO2201 | 211 | 0.902 | \mu, \eta, \delta, \epsilon | | |
| fong-spivak-invitation_FO2202 | 211 | 0.902 | X, \mu, \eta | | |
| fong-spivak-invitation_FO2203 | 212 | 1.000 | X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-invitation_FO2204 | 212 | 1.000 | \mu, \delta | | |
| fong-spivak-invitation_FO2205 | 212 | 0.889 | X, \mu, \eta, \delta, \epsilon | | |
| fong-spivak-invitation_FO2206 | 212 | 1.000 | I, \otimes) | | |
| fong-spivak-invitation_FO2207 | 212 | 1.000 | s_{m, n}: X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-invitation_FO2208 | 212 | 0.998 | (m-1) \mu \mathrm{s} | | |
| fong-spivak-invitation_FO2209 | 212 | 0.998 | (n-1) \delta \mathrm{s} | | |
| fong-spivak-invitation_FO2210 | 212 | 1.000 | m=0 | | |
| fong-spivak-invitation_FO2211 | 212 | 0.993 | s_{m, n} | | |
| fong-spivak-invitation_FO2212 | 212 | 1.000 | \mu, \eta, \delta, \epsilon, \sigma | | |
| fong-spivak-invitation_FO2213 | 212 | 0.973 | f: X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-invitation_FO2214 | 212 | 1.000 | \sigma: X^{\otimes 2} \rightarrow X^{\otimes 2} | | |
| fong-spivak-invitation_FO2215 | 212 | 1.000 | f=s_{m, n} | | |
| fong-spivak-invitation_FO2216 | 212 | 1.000 | X \otimes X \rightarrow X \otimes X \otimes X | | |
| fong-spivak-invitation_FO2217 | 213 | 0.999 | G:=\{\mu, \eta, \delta, \epsilon\} | | |
| fong-spivak-invitation_FO2218 | 213 | 1.000 | G \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO2219 | 213 | 0.970 | (G, E) | | |
| fong-spivak-invitation_FO2220 | 215 | 0.970 | X, \mu_{X}, \delta_{X}, \eta_{X}, \epsilon_{X} | | |
| fong-spivak-invitation_FO2221 | 215 | 0.996 | \eta_{I}=\mathrm{id}_{I}=\epsilon_{I} | | |
| fong-spivak-invitation_FO2222 | 215 | 1.000 | \mu_{X}, \delta_{X}, \eta_{X}, \epsilon_{X} | | |
| fong-spivak-invitation_FO2223 | 215 | 0.532 | \operatorname{Cospan}_{\text {FinSet }} | | |
| fong-spivak-invitation_FO2224 | 215 | 0.571 | \varnothing, \sqcup) | | |
| fong-spivak-invitation_FO2225 | 215 | 1.000 | \mu_{X}: X \sqcup X \rightarrow X | | |
| fong-spivak-invitation_FO2226 | 215 | 1.000 | X \sqcup X \sqcup X | | |
| fong-spivak-invitation_FO2227 | 215 | 1.000 | \delta_{X}: X \rightarrow X \sqcup X | | |
| fong-spivak-invitation_FO2228 | 215 | 1.000 | \eta_{X}: \varnothing \rightarrow X | | |
| fong-spivak-invitation_FO2229 | 215 | 1.000 | \epsilon_{X}: X \rightarrow \varnothing | | |
| fong-spivak-invitation_FO2230 | 215 | 1.000 | X=\varnothing \sqcup X=X \sqcup \varnothing | | |
| fong-spivak-invitation_FO2231 | 216 | 1.000 | \left(X, \mu_{X}, \eta_{X}, \delta_{X}, \epsilon_{X}\right) | | |
| fong-spivak-invitation_FO2232 | 217 | 0.456 | \left(\mathcal{C}, I_{\mathcal{C}}, \otimes_{\mathcal{P}}\right) | | |
| fong-spivak-invitation_FO2233 | 217 | 0.456 | \left(\mathcal{D}, I_{\mathcal{D}}, \otimes_{\mathcal{D}}\right) | | |
| fong-spivak-invitation_FO2234 | 217 | 0.992 | (F, \varphi) | | |
| fong-spivak-invitation_FO2235 | 217 | 0.986 | \varphi_{I}: I_{\mathcal{D}} \rightarrow F\left(I_{\mathcal{C}}\right) | | |
| fong-spivak-invitation_FO2236 | 217 | 0.865 | c_{1}, c_{2} \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO2237 | 217 | 1.000 | \varphi | | |
| fong-spivak-invitation_FO2238 | 217 | 0.990 | \varphi_{I} | | |
| fong-spivak-invitation_FO2239 | 217 | 0.990 | \varphi_{c_{1}, c_{2}} | | |
| fong-spivak-invitation_FO2240 | 217 | 0.990 | c_{1}, c_{2} | | |
| fong-spivak-invitation_FO2241 | 217 | 1.000 | S \in | | |
| fong-spivak-invitation_FO2242 | 217 | 1.000 | \mathrm{P}(S):=\{R \subseteq S\} | | |
| fong-spivak-invitation_FO2243 | 217 | 0.954 | \operatorname{im}_{f}: \mathrm{P}(S) \rightarrow \mathrm{P}(T) | | |
| fong-spivak-invitation_FO2244 | 217 | 1.000 | R \subseteq S | | |
| fong-spivak-invitation_FO2245 | 217 | 1.000 | \{f(r) \mid r \in R\} \subseteq T | | |
| fong-spivak-invitation_FO2246 | 217 | 0.983 | (\{1\}, \times) | | |
| fong-spivak-invitation_FO2247 | 217 | 1.000 | \varphi_{I}:\{1\} \rightarrow | | |
| fong-spivak-invitation_FO2248 | 217 | 1.000 | \mathrm{P}(\{1\}) | | |
| fong-spivak-invitation_FO2249 | 217 | 1.000 | \varphi_{S, T}: \mathrm{P}(S) \times \mathrm{P}(T) \rightarrow \mathrm{P}(S \times T) | | |
| fong-spivak-invitation_FO2250 | 217 | 1.000 | \varphi_{I}(1) | | |
| fong-spivak-invitation_FO2251 | 217 | 1.000 | \{1\} \subseteq\{1\} | | |
| fong-spivak-invitation_FO2252 | 217 | 1.000 | A \subseteq S | | |
| fong-spivak-invitation_FO2253 | 217 | 1.000 | B \subseteq T | | |
| fong-spivak-invitation_FO2254 | 217 | 1.000 | \varphi_{S, T}(A, B) | | |
| fong-spivak-invitation_FO2255 | 217 | 1.000 | A \times B \subseteq S \times T | | |
| fong-spivak-invitation_FO2256 | 217 | 0.969 | \mathrm{P}, \varphi | | |
| fong-spivak-invitation_FO2257 | 217 | 1.000 | \varphi_{S, T} | | |
| fong-spivak-invitation_FO2258 | 218 | 1.000 | 2 \Omega | | |
| fong-spivak-invitation_FO2259 | 218 | 1.000 | A \rightarrow N | | |
| fong-spivak-invitation_FO2260 | 218 | 1.000 | B \rightarrow N | | |
| fong-spivak-invitation_FO2261 | 219 | 0.997 | (F, \varphi):(\mathcal{C},+) \longrightarrow | | |
| fong-spivak-invitation_FO2262 | 219 | 0.621 | A \xrightarrow{i} N \stackrel{o}{\leftarrow} B | | |
| fong-spivak-invitation_FO2263 | 219 | 0.621 | s \in F(N) .{ }^{6} | | |
| fong-spivak-invitation_FO2264 | 219 | 0.637 | \mathcal{C}= | | |
| fong-spivak-invitation_FO2265 | 219 | 0.637 | N \in | | |
| fong-spivak-invitation_FO2266 | 219 | 1.000 | F(N) | | |
| fong-spivak-invitation_FO2267 | 219 | 1.000 | (A \xrightarrow{f} N \stackrel{g}{\leftarrow} B, s) | | |
| fong-spivak-invitation_FO2268 | 219 | 1.000 | (B \xrightarrow{h} P \stackrel{k}{\leftarrow} C, t) | | |
| fong-spivak-invitation_FO2269 | 219 | 1.000 | B \xrightarrow{h} | | |
| fong-spivak-invitation_FO2270 | 219 | 1.000 | P \stackrel{k}{\leftarrow} C | | |
| fong-spivak-invitation_FO2271 | 219 | 1.000 | F\left(N+{ }_{B} P\right) | | |
| fong-spivak-invitation_FO2272 | 220 | 0.559 | (F, \varphi):(\mathcal{C},+) \longrightarrow( | | |
| fong-spivak-invitation_FO2273 | 220 | 0.559 | \times) | | |
| fong-spivak-invitation_FO2274 | 220 | 0.559 | _{F} | | |
| fong-spivak-invitation_FO2275 | 220 | 0.660 | \operatorname{Cospan}_{F} | | |
| fong-spivak-invitation_FO2276 | 220 | 1.000 | F(c):=\{*\} | | |
| fong-spivak-invitation_FO2277 | 220 | 1.000 | \Omega | | |
| fong-spivak-invitation_FO2278 | 220 | 0.926 | \sqcup \mathbb{R}^{+} | | |
| fong-spivak-invitation_FO2279 | 220 | 1.000 | \ell: A \rightarrow C | | |
| fong-spivak-invitation_FO2280 | 220 | 1.000 | V=\{1,2\}, A=\{e\}, s(e)=1 | | |
| fong-spivak-invitation_FO2281 | 220 | 0.808 | t(e)=2 | | |
| fong-spivak-invitation_FO2282 | 220 | 0.808 | \ell(e)=3 \Omega | | |
| fong-spivak-invitation_FO2283 | 220 | 1.000 | 3 \Omega | | |
| fong-spivak-invitation_FO2284 | 220 | 1.000 | V=\{1,2\}, A=\{e\} | | |
| fong-spivak-invitation_FO2285 | 220 | 1.000 | s(e)=2, t(e)=1 | | |
| fong-spivak-invitation_FO2286 | 221 | 0.978 | V, A, s, t, \ell | | |
| fong-spivak-invitation_FO2287 | 221 | 0.996 | \psi | | |
| fong-spivak-invitation_FO2288 | 221 | 0.995 | f: V \rightarrow V^{\prime} | | |
| fong-spivak-invitation_FO2289 | 221 | 0.989 | c \in \operatorname{Circ}(4) | | |
| fong-spivak-invitation_FO2290 | 221 | 1.000 | f: \underline{4} \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO2291 | 221 | 0.998 | \operatorname{Circ}(f)(c) | | |
| fong-spivak-invitation_FO2292 | 221 | 1.000 | \psi_{V, V^{\prime}} | | |
| fong-spivak-invitation_FO2293 | 221 | 1.000 | V, V^{\prime} | | |
| fong-spivak-invitation_FO2294 | 221 | 1.000 | V+V^{\prime} | | |
| fong-spivak-invitation_FO2295 | 221 | 0.978 | \psi_{\underline{2}, \underline{2}}(b, s) \in \operatorname{Circ}(\underline{2}+\underline{2})=\operatorname{Circ}(\underline{4}) | | |
| fong-spivak-invitation_FO2296 | 222 | 0.885 | _{\text {Circ }} | | |
| fong-spivak-invitation_FO2297 | 222 | 0.668 | \underline{2} | | |
| fong-spivak-invitation_FO2298 | 222 | 0.668 | \mathbf{C o s p a n}_{\text {Circ }} | | |
| fong-spivak-invitation_FO2299 | 222 | 0.692 | \operatorname{Cospan}_{\text {Circ }} | | |
| fong-spivak-invitation_FO2300 | 222 | 1.000 | \underline{m} \rightarrow \underline{p} \leftarrow \underline{n} | | |
| fong-spivak-invitation_FO2301 | 222 | 1.000 | \operatorname{Circ}(\underline{p}) | | |
| fong-spivak-invitation_FO2302 | 222 | 0.998 | \underline{1} \xrightarrow{i_{1}} \underline{2} \stackrel{i_{2}}{\leftarrow} \underline{1} | | |
| fong-spivak-invitation_FO2303 | 222 | 0.998 | i_{1}(1)=1 | | |
| fong-spivak-invitation_FO2304 | 222 | 0.998 | i_{2}(1)=2 | | |
| fong-spivak-invitation_FO2305 | 222 | 0.987 | (V, A, s, t, \ell) \in \operatorname{Circ}(2) | | |
| fong-spivak-invitation_FO2307 | 222 | 0.795 | \underline{1} | | |
| fong-spivak-invitation_FO2308 | 222 | 1.000 | \operatorname{Circ}\left(\left[\iota_{N}, \iota_{P}\right]\right)\left(\psi_{N, P}(s, t)\right) | | |
| fong-spivak-invitation_FO2309 | 223 | 0.971 | \underline{1}=\{\circ\} | | |
| fong-spivak-invitation_FO2310 | 223 | 1.000 | \psi_{\underline{2}, \underline{2}} | | |
| fong-spivak-invitation_FO2311 | 223 | 0.912 | \psi_{N, P}(s, t) | | |
| fong-spivak-invitation_FO2312 | 223 | 0.912 | \left[\iota_{N}, \iota_{P}\right] | | |
| fong-spivak-invitation_FO2313 | 223 | 1.000 | A \rightarrow M \leftarrow B | | |
| fong-spivak-invitation_FO2314 | 223 | 1.000 | C \rightarrow N \leftarrow D | | |
| fong-spivak-invitation_FO2315 | 223 | 1.000 | A+C \rightarrow M+N \leftarrow B+D | | |
| fong-spivak-invitation_FO2316 | 223 | 1.000 | \psi_{M, N}: \operatorname{Circ}(M) \times | | |
| fong-spivak-invitation_FO2317 | 224 | 1.000 | \operatorname{Circ}(N) \rightarrow \operatorname{Circ}(M+N) | | |
| fong-spivak-invitation_FO2318 | 224 | 1.000 | \eta: 0 \rightarrow 2 | | |
| fong-spivak-invitation_FO2319 | 224 | 0.848 | \eta: 2 \rightarrow 0 | | |
| fong-spivak-invitation_FO2320 | 224 | 0.878 | (\underline{1}, \varnothing,!,!,!) \in \operatorname{Circ}(\underline{1}) .^{7} | | |
| fong-spivak-invitation_FO2321 | 224 | 0.956 | \eta ; x ; \epsilon | | |
| fong-spivak-invitation_FO2322 | 224 | 0.956 | \boldsymbol{\operatorname { C o s p a n }}_{\text {Circ }} | | |
| fong-spivak-invitation_FO2323 | 224 | 0.956 | \underline{0} \rightarrow \underline{0} | | |
| fong-spivak-invitation_FO2324 | 225 | 0.890 | \mathcal{O} \rightarrow | | |
| fong-spivak-invitation_FO2325 | 226 | 1.000 | f, g, h, k | | |
| fong-spivak-invitation_FO2326 | 226 | 0.999 | \mathcal{M} \rightarrow | | |
| fong-spivak-invitation_FO2327 | 226 | 0.578 | \mathbf{C o s p a n}_{\text {FinSet }} | | |
| fong-spivak-invitation_FO2328 | 226 | 0.989 | _{\text {FinSet }} \rightarrow | | |
| fong-spivak-invitation_FO2329 | 226 | 1.000 | \mathcal{O} | | |
| fong-spivak-invitation_FO2330 | 226 | 0.864 | t_{1}, \ldots, t_{n}, t | | |
| fong-spivak-invitation_FO2331 | 226 | 0.864 | \mathcal{O}\left(t_{1}, \ldots, t_{n} ; t\right) | | |
| fong-spivak-invitation_FO2332 | 226 | 0.990 | t_{1}, \ldots, t_{n} ; t | | |
| fong-spivak-invitation_FO2333 | 226 | 0.882 | s_{1}, \ldots, s_{m}, t_{i} | | |
| fong-spivak-invitation_FO2334 | 227 | 0.991 | \mathrm{id}_{t} \in O(t ; t) | | |
| fong-spivak-invitation_FO2335 | 227 | 1.000 | m+n-1 | | |
| fong-spivak-invitation_FO2336 | 227 | 1.000 | n-1 | | |
| fong-spivak-invitation_FO2337 | 227 | 0.648 | \circ_{i} | | |
| fong-spivak-invitation_FO2338 | 227 | 0.648 | f \circ i g \in \mathcal{O}(m+n-1) | | |
| fong-spivak-invitation_FO2339 | 228 | 0.998 | X_{1} \times \cdots \times X_{n} \rightarrow Y | | |
| fong-spivak-invitation_FO2340 | 228 | 0.998 | X_{1}, \ldots, X_{n} ; Y | | |
| fong-spivak-invitation_FO2341 | 228 | 0.929 | f \in \operatorname{Set}\left(W_{1}, \ldots, W_{m} ; X_{i}\right), g \in \operatorname{Set}\left(X_{1}, \ldots, X_{n} ; Y\right) | | |
| fong-spivak-invitation_FO2342 | 228 | 0.929 | g \circ_{i} f | | |
| fong-spivak-invitation_FO2343 | 228 | 0.780 | \operatorname{identities} \operatorname{id}_{X} \in \operatorname{Set}(X ; X) | | |
| fong-spivak-invitation_FO2344 | 228 | 0.780 | \operatorname{id}_{X}: X \rightarrow X | | |
| fong-spivak-invitation_FO2345 | 228 | 1.000 | a \in \mathbb{N} | | |
| fong-spivak-invitation_FO2346 | 228 | 0.810 | \underline{a_{1}}+\cdots+\underline{a_{n}} \rightarrow \underline{p} \leftarrow \underline{b} | | |
| fong-spivak-invitation_FO2347 | 228 | 0.810 | \left(a_{1}, \ldots, a_{n} ; b\right) | | |
| fong-spivak-invitation_FO2348 | 228 | 0.185 | \operatorname{id}_{a} \in \operatorname{Set}(a ; a) | | |
| fong-spivak-invitation_FO2349 | 228 | 0.185 | \operatorname{cospan} \underline{a} \xrightarrow{\operatorname{id}_{a}} \underline{a} \stackrel{\operatorname{id}_{\underline{a}}}{\text { 古 }} | | |
| fong-spivak-invitation_FO2350 | 228 | 0.922 | _{\text {FinSet }}, 0,+ | | |
| fong-spivak-invitation_FO2351 | 229 | 0.536 | \underline{3}, \underline{3}, \underline{4}, \underline{2} ; \underline{3} | | |
| fong-spivak-invitation_FO2352 | 229 | 0.999 | \underline{3}+\underline{3}+\underline{4}+\underline{2} \xrightarrow{a} \underline{p} \stackrel{b}{\leftarrow} \underline{3} | | |
| fong-spivak-invitation_FO2353 | 229 | 0.999 | \underline{p} | | |
| fong-spivak-invitation_FO2354 | 229 | 0.973 | \underline{7} | | |
| fong-spivak-invitation_FO2355 | 229 | 0.972 | f \in \operatorname{Cospan}(2,2 ; 2) | | |
| fong-spivak-invitation_FO2356 | 229 | 1.000 | g \in | | |
| fong-spivak-invitation_FO2357 | 229 | 0.999 | \boldsymbol{\operatorname { C o s p a n }}(2,2,2 ; 0) | | |
| fong-spivak-invitation_FO2358 | 229 | 0.706 | g \circ_{1} f | | |
| fong-spivak-invitation_FO2359 | 229 | 0.589 | \mathcal{O}_{\mathcal{C}} | | |
| fong-spivak-invitation_FO2360 | 229 | 0.957 | \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO2361 | 229 | 0.940 | C_{1} \otimes \cdots \otimes C_{n} \rightarrow D | | |
| fong-spivak-invitation_FO2362 | 229 | 0.940 | C_{1}, \ldots, C_{n} ; D | | |
| fong-spivak-invitation_FO2363 | 230 | 0.800 | \mathrm{id}_{a} \in \mathcal{O}_{\mathcal{C}}(a ; a) | | |
| fong-spivak-invitation_FO2364 | 230 | 0.800 | \mathrm{id}_{a} | | |
| fong-spivak-invitation_FO2365 | 230 | 0.997 | F: \mathcal{O} \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO2366 | 230 | 1.000 | f: T \rightarrow U | | |
| fong-spivak-invitation_FO2367 | 230 | 1.000 | F: \mathcal{O} \rightarrow | | |
| fong-spivak-invitation_FO2368 | 230 | 1.000 | F(t) | | |
| fong-spivak-invitation_FO2369 | 230 | 1.000 | f \in \mathcal{O}\left(t_{1}, \ldots, t_{n} ; t\right) | | |
| fong-spivak-invitation_FO2370 | 230 | 1.000 | F(f) | | |
| fong-spivak-invitation_FO2371 | 230 | 1.000 | F\left(t_{i}\right) | | |
| fong-spivak-invitation_FO2372 | 230 | 1.000 | t_{1}, \ldots, t_{n} | | |
| fong-spivak-invitation_FO2373 | 230 | 0.998 | T=\mathrm{Ob} | | |
| fong-spivak-invitation_FO2374 | 230 | 0.691 | \operatorname{circ}(t) | | |
| fong-spivak-invitation_FO2375 | 231 | 0.623 | \operatorname{Circ}(\varphi) \in \operatorname{Set}(\operatorname{Circ}(2) | | |
| fong-spivak-invitation_FO2376 | 231 | 0.623 | \operatorname{Circ}(2) | | |
| fong-spivak-invitation_FO2377 | 231 | 0.490 | \operatorname{Circ}(0)) | | |
| fong-spivak-invitation_FO2378 | 233 | 1.000 | \in \mathbb{R} | | |
| fong-spivak-invitation_FO2379 | 233 | 1.000 | S^{\prime} | | |
| fong-spivak-invitation_FO2380 | 234 | 0.953 | S \geq | | |
| fong-spivak-invitation_FO2381 | 234 | 0.922 | S<\operatorname{safe} | | |
| fong-spivak-invitation_FO2382 | 234 | 0.984 | A= | | |
| fong-spivak-invitation_FO2383 | 235 | 0.911 | [a, b] | | |
| fong-spivak-invitation_FO2384 | 235 | 1.000 | [a, b] \rightarrow \mathbb{R} | | |
| fong-spivak-invitation_FO2385 | 235 | 0.998 | \left\{\left(S^{\prime}, A\right) \mid\right. | | |
| fong-spivak-invitation_FO2386 | 235 | 0.998 | \left(S^{\prime}, A\right) | | |
| fong-spivak-invitation_FO2387 | 236 | 0.929 | \in P | | |
| fong-spivak-invitation_FO2388 | 236 | 1.000 | \mathcal{E} | | |
| fong-spivak-invitation_FO2389 | 236 | 1.000 | 1 \xrightarrow{\text { true }} \Omega | | |
| fong-spivak-invitation_FO2390 | 236 | 0.996 | \mathcal{E}= | | |
| fong-spivak-invitation_FO2391 | 237 | 0.833 | \lrcorner | | |
| fong-spivak-invitation_FO2392 | 237 | 0.999 | (B, C, E, F) | | |
| fong-spivak-invitation_FO2393 | 237 | 0.876 | (A, B, D, E) | | |
| fong-spivak-invitation_FO2394 | 237 | 0.876 | (A, C, D, F) | | |
| fong-spivak-invitation_FO2395 | 237 | 0.989 | \left(B, C, B^{\prime}, C^{\prime}\right) | | |
| fong-spivak-invitation_FO2396 | 237 | 0.811 | A, B, A^{\prime}, B^{\prime} | | |
| fong-spivak-invitation_FO2397 | 237 | 0.811 | A, C, A^{\prime}, C^{\prime} | | |
| fong-spivak-invitation_FO2398 | 238 | 1.000 | i: B^{\prime} \rightarrow B | | |
| fong-spivak-invitation_FO2399 | 238 | 1.000 | i^{\prime} | | |
| fong-spivak-invitation_FO2400 | 238 | 1.000 | f^{\prime} | | |
| fong-spivak-invitation_FO2401 | 238 | 1.000 | f: C \rightarrow D | | |
| fong-spivak-invitation_FO2402 | 238 | 1.000 | \operatorname{im}(f) | | |
| fong-spivak-invitation_FO2403 | 238 | 0.811 | C \rightarrow \operatorname{im}(f) | | |
| fong-spivak-invitation_FO2404 | 238 | 0.811 | \operatorname{im}(f) \succ D | | |
| fong-spivak-invitation_FO2405 | 238 | 0.938 | \operatorname{im}(f)=\{f(c) \mid | | |
| fong-spivak-invitation_FO2406 | 238 | 1.000 | c \in C\} | | |
| fong-spivak-invitation_FO2407 | 239 | 1.000 | f: \underline{3} \rightarrow \underline{3} | | |
| fong-spivak-invitation_FO2408 | 239 | 1.000 | f=(e ; m) | | |
| fong-spivak-invitation_FO2409 | 239 | 1.000 | C, D \in \mathcal{C} | | |
| fong-spivak-invitation_FO2410 | 239 | 1.000 | D^{C} \in \mathcal{C} | | |
| fong-spivak-invitation_FO2411 | 239 | 1.000 | A \in \mathcal{C} | | |
| fong-spivak-invitation_FO2412 | 239 | 1.000 | f(-,-): A \times C \rightarrow D | | |
| fong-spivak-invitation_FO2413 | 239 | 1.000 | f(a, c) | | |
| fong-spivak-invitation_FO2414 | 239 | 1.000 | f(a,-): C \rightarrow D | | |
| fong-spivak-invitation_FO2415 | 239 | 1.000 | D^{C} | | |
| fong-spivak-invitation_FO2416 | 239 | 1.000 | v, w, x \in V | | |
| fong-spivak-invitation_FO2417 | 239 | 1.000 | v \leq I | | |
| fong-spivak-invitation_FO2418 | 239 | 1.000 | v \otimes w \leq v | | |
| fong-spivak-invitation_FO2419 | 239 | 1.000 | v \otimes w \leq w | | |
| fong-spivak-invitation_FO2420 | 239 | 1.000 | x \leq v | | |
| fong-spivak-invitation_FO2421 | 239 | 1.000 | x \leq w | | |
| fong-spivak-invitation_FO2422 | 239 | 1.000 | x \leq v \otimes w | | |
| fong-spivak-invitation_FO2423 | 240 | 0.781 | X \hookrightarrow Y | | |
| fong-spivak-invitation_FO2424 | 240 | 0.822 | \Omega \in \mathcal{E} | | |
| fong-spivak-invitation_FO2425 | 240 | 1.000 | 1 \rightarrow \Omega | | |
| fong-spivak-invitation_FO2426 | 240 | 0.496 | m: X \rightsquigarrow Y | | |
| fong-spivak-invitation_FO2427 | 240 | 0.496 | \ulcorner m\urcorner: Y \rightarrow \Omega | | |
| fong-spivak-invitation_FO2428 | 240 | 0.999 | \ulcorner m\urcorner | | |
| fong-spivak-invitation_FO2429 | 240 | 0.999 | p: Y \rightarrow \Omega | | |
| fong-spivak-invitation_FO2430 | 240 | 1.000 | Y \rightarrow \Omega | | |
| fong-spivak-invitation_FO2431 | 241 | 1.000 | 1 \rightarrow | | |
| fong-spivak-invitation_FO2432 | 241 | 0.976 | m: X \hookrightarrow Y | | |
| fong-spivak-invitation_FO2433 | 241 | 0.968 | \ulcorner m\urcorner: Y \rightarrow\{ | | |
| fong-spivak-invitation_FO2434 | 241 | 1.000 | \ulcorner m\urcorner(y) | | |
| fong-spivak-invitation_FO2435 | 241 | 1.000 | y \in X | | |
| fong-spivak-invitation_FO2436 | 241 | 1.000 | p: Y \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2437 | 241 | 1.000 | \{Y \mid p\} | | |
| fong-spivak-invitation_FO2438 | 241 | 1.000 | X=\mathbb{N}=\{0,1,2, \ldots\} | | |
| fong-spivak-invitation_FO2439 | 241 | 1.000 | Y=\mathbb{Z}=\{\ldots,-1,0,1,2, \ldots\} | | |
| fong-spivak-invitation_FO2440 | 241 | 0.995 | m: X \succ Y | | |
| fong-spivak-invitation_FO2441 | 241 | 1.000 | \ulcorner m\urcorner: Y \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2442 | 241 | 1.000 | \ulcorner m\urcorner(-5) \in \mathbb{B} | | |
| fong-spivak-invitation_FO2443 | 241 | 1.000 | \ulcorner m\urcorner(0) \in \mathbb{B} | | |
| fong-spivak-invitation_FO2444 | 241 | 0.998 | \mathrm{id}_{\mathbb{N}}: \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO2445 | 241 | 0.808 | \left\ulcorner\mathrm{id}_{\mathbb{N}}\right\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2446 | 241 | 0.808 | \left\ulcorner\mathrm{id}_{\mathbb{N}}\right\urcorner | | |
| fong-spivak-invitation_FO2447 | 241 | 1.000 | !_{\mathbb{N}}: \varnothing \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO2448 | 241 | 0.654 | \left\ulcorner!_{\mathbb{N}}\right\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2449 | 242 | 1.000 | 1 \rightarrow \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-invitation_FO2450 | 242 | 0.994 | \urcorner: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2451 | 242 | 1.000 | \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-invitation_FO2452 | 242 | 0.944 | \ulcorner( | | |
| fong-spivak-invitation_FO2453 | 242 | 0.944 | )\urcorner(P, Q) \in \mathbb{B} | | |
| fong-spivak-invitation_FO2455 | 242 | 1.000 | P \wedge Q | | |
| fong-spivak-invitation_FO2456 | 242 | 0.929 | )\urcorner=\wedge | | |
| fong-spivak-invitation_FO2457 | 242 | 0.929 | \wedge: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2458 | 242 | 1.000 | x \wedge y:=\wedge(x, y) | | |
| fong-spivak-invitation_FO2459 | 242 | 1.000 | \vee: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2461 | 242 | 1.000 | m: X \subseteq \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-invitation_FO2462 | 242 | 0.912 | \mathbb{B} \times | | |
| fong-spivak-invitation_FO2463 | 242 | 1.000 | \Omega \times \Omega | | |
| fong-spivak-invitation_FO2464 | 242 | 1.000 | \Omega \times \Omega \rightarrow \Omega | | |
| fong-spivak-invitation_FO2465 | 243 | 1.000 | \neg: \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2466 | 243 | 1.000 | P \Rightarrow Q | | |
| fong-spivak-invitation_FO2467 | 243 | 1.000 | P=(P \wedge Q) | | |
| fong-spivak-invitation_FO2468 | 243 | 1.000 | m: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2469 | 243 | 0.999 | E:=\{n \in \mathbb{N} \mid n | | |
| fong-spivak-invitation_FO2470 | 243 | 0.999 | \}, P:=\{n \in \mathbb{N} \mid n | | |
| fong-spivak-invitation_FO2471 | 243 | 1.000 | T:=\{n \in \mathbb{N} \mid n \geq 10\} | | |
| fong-spivak-invitation_FO2472 | 243 | 1.000 | \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2473 | 243 | 0.992 | \ulcorner E\urcorner(17) | | |
| fong-spivak-invitation_FO2474 | 243 | 0.995 | \ulcorner P\urcorner(17) | | |
| fong-spivak-invitation_FO2475 | 243 | 1.000 | \ulcorner T\urcorner(17) | | |
| fong-spivak-invitation_FO2476 | 243 | 1.000 | (\ulcorner E\urcorner \wedge\ulcorner P\urcorner) \vee\ulcorner T\urcorner | | |
| fong-spivak-invitation_FO2477 | 243 | 1.000 | \wedge, \vee, \Rightarrow: \Omega \times \Omega \rightarrow \Omega | | |
| fong-spivak-invitation_FO2478 | 243 | 1.000 | \neg: \Omega \rightarrow \Omega | | |
| fong-spivak-invitation_FO2479 | 244 | 1.000 | \mathbb{R}^{2} | | |
| fong-spivak-invitation_FO2480 | 244 | 0.163 | \mathcal{C}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2481 | 244 | 0.605 | P: \mathcal{C}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2482 | 244 | 0.605 | P(c) | | |
| fong-spivak-invitation_FO2483 | 244 | 1.000 | P(f): P(c) \rightarrow P\left(c^{\prime}\right) | | |
| fong-spivak-invitation_FO2484 | 244 | 1.000 | s \in P(c) | | |
| fong-spivak-invitation_FO2485 | 244 | 0.458 | \mathcal{C}^{\mathrm{Op}} | | |
| fong-spivak-invitation_FO2486 | 245 | 1.000 | P(f)(s) \in P\left(c^{\prime}\right) | | |
| fong-spivak-invitation_FO2487 | 245 | 1.000 | \left.s\right|_{f} | | |
| fong-spivak-invitation_FO2488 | 245 | 1.000 | \alpha: P \rightarrow Q | | |
| fong-spivak-invitation_FO2489 | 245 | 0.402 | I: \mathbf{A r S h p}^{\mathbf{o p}} \rightarrow | | |
| fong-spivak-invitation_FO2490 | 246 | 1.000 | \mathrm{P}(X)=\{U \subseteq X\} | | |
| fong-spivak-invitation_FO2491 | 246 | 0.999 | \mathbf{O p} \subseteq \mathrm{P}(X) | | |
| fong-spivak-invitation_FO2492 | 246 | 1.000 | X \subseteq X | | |
| fong-spivak-invitation_FO2493 | 246 | 1.000 | X \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2494 | 246 | 0.984 | U, V \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2495 | 246 | 0.984 | (U \cap V) \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2496 | 246 | 1.000 | U_{i} \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2497 | 246 | 1.000 | \left(\bigcup_{i \in I} U_{i}\right) \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2498 | 246 | 0.915 | I=\varnothing | | |
| fong-spivak-invitation_FO2499 | 246 | 0.915 | \varnothing \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2500 | 246 | 1.000 | U=\bigcup_{i \in I} U_{i} | | |
| fong-spivak-invitation_FO2501 | 246 | 1.000 | \left(U_{i}\right)_{i \in I} | | |
| fong-spivak-invitation_FO2502 | 246 | 0.932 | X, \mathbf{O p} | | |
| fong-spivak-invitation_FO2503 | 246 | 0.932 | \mathbf{O p} | | |
| fong-spivak-invitation_FO2504 | 246 | 1.000 | X, \mathbf{O p}_{X} | | |
| fong-spivak-invitation_FO2505 | 246 | 1.000 | Y, \mathbf{O p}_{Y} | | |
| fong-spivak-invitation_FO2506 | 246 | 0.999 | U \in \mathbf{O p}_{Y} | | |
| fong-spivak-invitation_FO2507 | 246 | 0.999 | f^{-1}(U) | | |
| fong-spivak-invitation_FO2508 | 246 | 1.000 | \mathbf{O p}_{X} | | |
| fong-spivak-invitation_FO2509 | 246 | 0.999 | U \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2510 | 247 | 0.936 | \epsilon \in \mathbb{R} | | |
| fong-spivak-invitation_FO2511 | 247 | 0.985 | \epsilon>0 | | |
| fong-spivak-invitation_FO2512 | 247 | 0.985 | p=(x, y) \in \mathbb{R}^{2} | | |
| fong-spivak-invitation_FO2513 | 247 | 1.000 | B(x, y ; \epsilon) | | |
| fong-spivak-invitation_FO2514 | 247 | 0.796 | U \subseteq \mathbb{R}^{2} | | |
| fong-spivak-invitation_FO2515 | 247 | 0.796 | (x, y) \in | | |
| fong-spivak-invitation_FO2516 | 247 | 1.000 | B(x, y ; \epsilon) \subseteq U | | |
| fong-spivak-invitation_FO2517 | 247 | 1.000 | d\left(x_{1}, x_{2}\right):=\left|x_{1}-x_{2}\right| | | |
| fong-spivak-invitation_FO2518 | 247 | 1.000 | x \in \mathbb{R} | | |
| fong-spivak-invitation_FO2519 | 247 | 1.000 | B(x, \epsilon) | | |
| fong-spivak-invitation_FO2520 | 247 | 1.000 | U \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO2521 | 247 | 1.000 | U_{1}, U_{2} | | |
| fong-spivak-invitation_FO2522 | 247 | 1.000 | \left(U_{i}\right)_{i \in\{1,2\}} | | |
| fong-spivak-invitation_FO2523 | 247 | 1.000 | \mathbf{O p}_{\text {crse }}=(\varnothing, X) | | |
| fong-spivak-invitation_FO2524 | 247 | 0.992 | \mathbf{O p}_{\text {fine }}=\mathrm{P}(X) | | |
| fong-spivak-invitation_FO2525 | 247 | 0.992 | (X, \mathrm{P}(X)) | | |
| fong-spivak-invitation_FO2526 | 247 | 1.000 | d\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right):=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}} | | |
| fong-spivak-invitation_FO2527 | 248 | 1.000 | \mathbf{O p}_{\text {crse }} | | |
| fong-spivak-invitation_FO2528 | 248 | 1.000 | \mathbf{O p}_{\text {fine }} | | |
| fong-spivak-invitation_FO2529 | 248 | 0.899 | \left(Y, \mathbf{O}_{Y}\right) | | |
| fong-spivak-invitation_FO2530 | 248 | 1.000 | X \rightarrow Y | | |
| fong-spivak-invitation_FO2531 | 248 | 1.000 | X=\{1,2\} | | |
| fong-spivak-invitation_FO2532 | 248 | 0.333 | \{1,2\}, \mathbf{O} \mathbf{p}_{1} | | |
| fong-spivak-invitation_FO2533 | 248 | 0.333 | \{1,2\}, \mathbf{O} \mathbf{p}_{2} | | |
| fong-spivak-invitation_FO2534 | 248 | 0.459 | (X, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2535 | 248 | 0.794 | \mathbf{O p}, \subseteq | | |
| fong-spivak-invitation_FO2536 | 248 | 0.794 | U \subseteq U | | |
| fong-spivak-invitation_FO2537 | 248 | 1.000 | U \subseteq V | | |
| fong-spivak-invitation_FO2538 | 248 | 1.000 | V \subseteq W | | |
| fong-spivak-invitation_FO2539 | 248 | 1.000 | U \subseteq W | | |
| fong-spivak-invitation_FO2540 | 248 | 0.936 | U, V | | |
| fong-spivak-invitation_FO2541 | 248 | 0.936 | \mathbf{O p}(U, V) | | |
| fong-spivak-invitation_FO2542 | 248 | 1.000 | U \nsubseteq V | | |
| fong-spivak-invitation_FO2543 | 248 | 0.623 | X, \mathbf{O p}_{1} | | |
| fong-spivak-invitation_FO2544 | 248 | 0.558 | Y \subseteq X | | |
| fong-spivak-invitation_FO2545 | 248 | 0.971 | \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-invitation_FO2546 | 248 | 0.971 | A \subseteq Y | | |
| fong-spivak-invitation_FO2547 | 248 | 0.999 | A \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-invitation_FO2548 | 248 | 0.999 | B \in \mathbf{O p} | | |
| fong-spivak-invitation_FO2549 | 248 | 0.999 | A=B \cap Y | | |
| fong-spivak-invitation_FO2550 | 248 | 1.000 | Y \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-invitation_FO2551 | 248 | 1.000 | Y \hookrightarrow X | | |
| fong-spivak-invitation_FO2552 | 248 | 0.958 | (\mathbf{O p}, \subseteq) | | |
| fong-spivak-invitation_FO2553 | 248 | 0.958 | (\mathbf{O p}, \subseteq, X, \cap) | | |
| fong-spivak-invitation_FO2554 | 248 | 0.585 | U_{i} \subseteq X | | |
| fong-spivak-invitation_FO2555 | 248 | 0.585 | \left(\bigcup_{i \in I} U_{i}\right) \cap V= | | |
| fong-spivak-invitation_FO2556 | 248 | 1.000 | \bigcup_{i \in I}\left(U_{i} \cap V\right) | | |
| fong-spivak-invitation_FO2557 | 248 | 1.000 | U, V, W \subseteq X | | |
| fong-spivak-invitation_FO2558 | 248 | 1.000 | (U \cap W) \cap(V \cap W)=(U \cap V) \cap W | | |
| fong-spivak-invitation_FO2559 | 249 | 0.945 | P: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2560 | 249 | 0.945 | P(U) | | |
| fong-spivak-invitation_FO2561 | 249 | 1.000 | V \subseteq U | | |
| fong-spivak-invitation_FO2562 | 249 | 1.000 | P(U) \rightarrow P(V) | | |
| fong-spivak-invitation_FO2563 | 249 | 1.000 | s \in P(U) | | |
| fong-spivak-invitation_FO2564 | 249 | 1.000 | \left.s\right|_{V} | | |
| fong-spivak-invitation_FO2565 | 249 | 1.000 | \left(s_{i}\right)_{i \in I} | | |
| fong-spivak-invitation_FO2566 | 249 | 1.000 | s_{i} \in P\left(U_{i}\right) | | |
| fong-spivak-invitation_FO2567 | 249 | 1.000 | i \in I | | |
| fong-spivak-invitation_FO2568 | 249 | 1.000 | i, j \in I | | |
| fong-spivak-invitation_FO2569 | 249 | 1.000 | \left.s\right|_{U_{i}}=s_{i} | | |
| fong-spivak-invitation_FO2570 | 249 | 0.997 | \boldsymbol{\operatorname { S h v }}(X, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2571 | 249 | 0.997 | \boldsymbol{\operatorname { S h v }}(X) | | |
| fong-spivak-invitation_FO2572 | 250 | 1.000 | U=\varnothing \subseteq X | | |
| fong-spivak-invitation_FO2573 | 250 | 0.948 | P(\varnothing) \cong\{()\} | | |
| fong-spivak-invitation_FO2574 | 250 | 0.948 | P(\varnothing) | | |
| fong-spivak-invitation_FO2575 | 250 | 0.984 | f\left(a_{1}\right)=f\left(a_{2}\right)=a, f\left(b_{1}\right)=b | | |
| fong-spivak-invitation_FO2576 | 250 | 1.000 | f^{-1}(y) \subseteq X | | |
| fong-spivak-invitation_FO2577 | 250 | 1.000 | f^{\prime}: X \rightarrow Y | | |
| fong-spivak-invitation_FO2578 | 251 | 0.952 | \operatorname{Sec}_{f}: \mathbf{O p}(Y)^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2579 | 251 | 1.000 | \operatorname{Sec}_{f} | | |
| fong-spivak-invitation_FO2580 | 251 | 1.000 | U \subseteq Y | | |
| fong-spivak-invitation_FO2581 | 251 | 1.000 | \operatorname{Sec}_{f}(U) | | |
| fong-spivak-invitation_FO2582 | 251 | 1.000 | s \in \operatorname{Sec}_{f}(U) | | |
| fong-spivak-invitation_FO2583 | 251 | 1.000 | U=\{a, b\} | | |
| fong-spivak-invitation_FO2584 | 251 | 0.570 | V_{1}=\{a, b, c\} | | |
| fong-spivak-invitation_FO2585 | 251 | 0.570 | \operatorname{Sec}_{f}\left(V_{1}\right) | | |
| fong-spivak-invitation_FO2586 | 251 | 0.993 | V_{2}=\{a, b, c, d\} | | |
| fong-spivak-invitation_FO2587 | 251 | 0.993 | \operatorname{Sec}_{f}\left(V_{2}\right) | | |
| fong-spivak-invitation_FO2588 | 251 | 0.990 | V_{3}=\{a, b, d, e\} | | |
| fong-spivak-invitation_FO2589 | 251 | 0.990 | \operatorname{Sec}_{f}\left(V_{3}\right) | | |
| fong-spivak-invitation_FO2590 | 251 | 1.000 | V=\{a\} | | |
| fong-spivak-invitation_FO2591 | 251 | 1.000 | \operatorname{Sec}_{f}(\{a, b, c\}) | | |
| fong-spivak-invitation_FO2592 | 251 | 1.000 | \operatorname{Sec}_{f}(\{a, c\}) | | |
| fong-spivak-invitation_FO2593 | 252 | 1.000 | U_{1}=\{a, b\} | | |
| fong-spivak-invitation_FO2594 | 252 | 1.000 | U_{2}=\{b, e\} | | |
| fong-spivak-invitation_FO2595 | 252 | 1.000 | U= | | |
| fong-spivak-invitation_FO2596 | 252 | 1.000 | \{a, b, e\}=U_{1} \cup U_{2} | | |
| fong-spivak-invitation_FO2597 | 252 | 1.000 | U_{1} | | |
| fong-spivak-invitation_FO2598 | 252 | 1.000 | U_{2} | | |
| fong-spivak-invitation_FO2599 | 252 | 1.000 | U_{1} \cap U_{2}=\{b\} | | |
| fong-spivak-invitation_FO2600 | 252 | 1.000 | s_{1} \in \operatorname{Sec}_{f}\left(U_{1}\right) | | |
| fong-spivak-invitation_FO2601 | 252 | 1.000 | s_{2} \in \operatorname{Sec}_{f}\left(U_{2}\right) | | |
| fong-spivak-invitation_FO2602 | 252 | 1.000 | g_{1} | | |
| fong-spivak-invitation_FO2603 | 252 | 1.000 | b_{2} | | |
| fong-spivak-invitation_FO2604 | 252 | 1.000 | U=\{a, b, e\} | | |
| fong-spivak-invitation_FO2605 | 252 | 0.524 | \left(s_{1}, s_{2}\right) | | |
| fong-spivak-invitation_FO2606 | 252 | 0.524 | s \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-invitation_FO2607 | 252 | 1.000 | \left.s\right|_{U_{1}}=s_{1} | | |
| fong-spivak-invitation_FO2608 | 252 | 1.000 | \left.s\right|_{U_{2}}=s_{2} | | |
| fong-spivak-invitation_FO2609 | 252 | 0.982 | h_{1} \in \operatorname{Sec}_{f}\left(U_{1}\right) | | |
| fong-spivak-invitation_FO2610 | 252 | 0.982 | h_{2} \in \operatorname{Sec}_{f}\left(U_{2}\right) | | |
| fong-spivak-invitation_FO2611 | 252 | 0.982 | d o | | |
| fong-spivak-invitation_FO2612 | 252 | 1.000 | h \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-invitation_FO2613 | 252 | 1.000 | \left.h\right|_{U_{1}}=h_{1} | | |
| fong-spivak-invitation_FO2614 | 252 | 1.000 | \left.h\right|_{U_{2}}=h_{2} | | |
| fong-spivak-invitation_FO2615 | 253 | 1.000 | f:\left(X, \mathbf{O p}_{X}\right) \rightarrow\left(Y, \mathbf{O p}_{Y}\right) | | |
| fong-spivak-invitation_FO2616 | 253 | 0.965 | \operatorname{Sec}_{f}: \mathbf{O p}_{Y}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2617 | 253 | 1.000 | \mathbf{O p}_{Y} | | |
| fong-spivak-invitation_FO2618 | 253 | 1.000 | g: U \rightarrow X | | |
| fong-spivak-invitation_FO2619 | 253 | 1.000 | T M | | |
| fong-spivak-invitation_FO2620 | 253 | 1.000 | (m, v) | | |
| fong-spivak-invitation_FO2621 | 253 | 1.000 | m \in M | | |
| fong-spivak-invitation_FO2622 | 253 | 1.000 | \pi: T M \rightarrow | | |
| fong-spivak-invitation_FO2623 | 253 | 1.000 | (m, v) \mapsto m | | |
| fong-spivak-invitation_FO2624 | 253 | 1.000 | \pi | | |
| fong-spivak-invitation_FO2625 | 253 | 0.846 | \operatorname{Sec}_{\pi} | | |
| fong-spivak-invitation_FO2626 | 254 | 1.000 | U \subseteq M | | |
| fong-spivak-invitation_FO2627 | 254 | 1.000 | v \in \operatorname{Sec}_{\pi}(U) | | |
| fong-spivak-invitation_FO2628 | 254 | 1.000 | v(u) | | |
| fong-spivak-invitation_FO2629 | 254 | 1.000 | U \neq M | | |
| fong-spivak-invitation_FO2630 | 254 | 1.000 | U \neq X | | |
| fong-spivak-invitation_FO2631 | 254 | 1.000 | U=X | | |
| fong-spivak-invitation_FO2632 | 254 | 0.997 | \boldsymbol{\operatorname { S h v }}(M) | | |
| fong-spivak-invitation_FO2633 | 254 | 0.884 | \{1,2\}, \mathbf{O p}_{1} | | |
| fong-spivak-invitation_FO2634 | 255 | 1.000 | X=\mathbf{1} | | |
| fong-spivak-invitation_FO2635 | 255 | 1.000 | \wedge, \vee, \neg, \Rightarrow, \exists, \forall | | |
| fong-spivak-invitation_FO2636 | 255 | 0.934 | \Omega=\mathbb{B} | | |
| fong-spivak-invitation_FO2637 | 255 | 0.934 | \mathcal{E}=\operatorname{Shv}(X, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2638 | 255 | 0.809 | \Omega: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2639 | 255 | 1.000 | \Omega(U) | | |
| fong-spivak-invitation_FO2640 | 255 | 1.000 | \Omega(U) \rightarrow \Omega(V) | | |
| fong-spivak-invitation_FO2641 | 256 | 1.000 | X=\{1\} | | |
| fong-spivak-invitation_FO2642 | 256 | 0.966 | \boldsymbol{\operatorname { S h v }}(1) | | |
| fong-spivak-invitation_FO2643 | 256 | 1.000 | W \subseteq V \subseteq U | | |
| fong-spivak-invitation_FO2644 | 256 | 1.000 | \Omega(V) \rightarrow \Omega(W) | | |
| fong-spivak-invitation_FO2645 | 256 | 1.000 | \Omega(U) \rightarrow \Omega(W) | | |
| fong-spivak-invitation_FO2646 | 256 | 1.000 | V_{i} \in \Omega\left(U_{i}\right) | | |
| fong-spivak-invitation_FO2647 | 256 | 1.000 | V_{i} \subseteq U_{i} | | |
| fong-spivak-invitation_FO2648 | 256 | 1.000 | V_{i} \cap U_{j}=V_{j} \cap U_{i} | | |
| fong-spivak-invitation_FO2649 | 256 | 1.000 | V:=\bigcup_{i \in I} V_{i} | | |
| fong-spivak-invitation_FO2650 | 256 | 1.000 | \left(V_{i}\right)_{i \in I} | | |
| fong-spivak-invitation_FO2651 | 256 | 1.000 | V \cap U_{j}=V_{j} | | |
| fong-spivak-invitation_FO2652 | 256 | 1.000 | j \in I | | |
| fong-spivak-invitation_FO2653 | 256 | 0.908 | \{1\} \rightarrow \Omega | | |
| fong-spivak-invitation_FO2654 | 256 | 0.908 | \{1\}: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2655 | 256 | 1.000 | \{1\}=\{1\}(U) \rightarrow \Omega(U) | | |
| fong-spivak-invitation_FO2656 | 256 | 1.000 | 1 \mapsto U | | |
| fong-spivak-invitation_FO2657 | 256 | 1.000 | U=\varnothing | | |
| fong-spivak-invitation_FO2658 | 257 | 0.944 | (V, V ; A) | | |
| fong-spivak-invitation_FO2659 | 257 | 1.000 | i: H \subseteq G | | |
| fong-spivak-invitation_FO2660 | 257 | 0.546 | \ulcorner H\urcorner: G \rightarrow \Omega | | |
| fong-spivak-invitation_FO2661 | 257 | 0.577 | (V, 0 ; 0),(0, V ; 0),(V, V ; 0) | | |
| fong-spivak-invitation_FO2662 | 257 | 0.577 | 0 ; 0 | | |
| fong-spivak-invitation_FO2663 | 257 | 1.000 | H \subseteq G | | |
| fong-spivak-invitation_FO2664 | 257 | 1.000 | U, V \in \Omega(X) | | |
| fong-spivak-invitation_FO2665 | 257 | 1.000 | \cap | | |
| fong-spivak-invitation_FO2666 | 257 | 1.000 | \cup | | |
| fong-spivak-invitation_FO2667 | 257 | 1.000 | U \Rightarrow V | | |
| fong-spivak-invitation_FO2668 | 257 | 1.000 | R \cap U \subseteq V | | |
| fong-spivak-invitation_FO2669 | 258 | 0.933 | \neg U:= | | |
| fong-spivak-invitation_FO2670 | 258 | 0.933 | U \Rightarrow | | |
| fong-spivak-invitation_FO2671 | 258 | 1.000 | R \cap U=\varnothing | | |
| fong-spivak-invitation_FO2672 | 258 | 1.000 | \neg U | | |
| fong-spivak-invitation_FO2673 | 258 | 1.000 | X=\mathbb{R} | | |
| fong-spivak-invitation_FO2674 | 258 | 1.000 | U=\{x \in \mathbb{R} \mid x<3\} | | |
| fong-spivak-invitation_FO2675 | 258 | 1.000 | V=\{x \in \mathbb{R} \mid-4<x< | | |
| fong-spivak-invitation_FO2676 | 258 | 1.000 | U=(-\infty, 3) | | |
| fong-spivak-invitation_FO2677 | 258 | 1.000 | V=(-4,4) | | |
| fong-spivak-invitation_FO2678 | 258 | 1.000 | U \wedge V=(-4,3) | | |
| fong-spivak-invitation_FO2679 | 258 | 1.000 | U \vee V=(-\infty, 4) | | |
| fong-spivak-invitation_FO2680 | 258 | 1.000 | \neg U=(3, \infty) | | |
| fong-spivak-invitation_FO2681 | 258 | 1.000 | \neg V=(-\infty,-4) \cup(4, \infty) | | |
| fong-spivak-invitation_FO2682 | 258 | 1.000 | (U \Rightarrow V)=(-4, \infty) | | |
| fong-spivak-invitation_FO2683 | 258 | 1.000 | (V \Rightarrow U)=U | | |
| fong-spivak-invitation_FO2684 | 258 | 1.000 | U=\mathbb{R}-\{0\} | | |
| fong-spivak-invitation_FO2685 | 258 | 1.000 | \neg \neg U | | |
| fong-spivak-invitation_FO2686 | 258 | 1.000 | U \subseteq \neg \neg U | | |
| fong-spivak-invitation_FO2687 | 258 | 1.000 | \neg \neg U \subseteq U | | |
| fong-spivak-invitation_FO2688 | 258 | 0.983 | \wedge U)=U | | |
| fong-spivak-invitation_FO2689 | 258 | 0.748 | \vee U)=\operatorname{true} | | |
| fong-spivak-invitation_FO2690 | 258 | 0.748 | (U \Rightarrow | | |
| fong-spivak-invitation_FO2691 | 258 | 0.840 | )= | | |
| fong-spivak-invitation_FO2692 | 258 | 0.840 | ( | | |
| fong-spivak-invitation_FO2693 | 258 | 0.840 | \Rightarrow U)=U | | |
| fong-spivak-invitation_FO2694 | 258 | 0.999 | \vee U)=U | | |
| fong-spivak-invitation_FO2695 | 258 | 0.900 | \wedge U | | |
| fong-spivak-invitation_FO2696 | 259 | 1.000 | \pi: E \rightarrow X | | |
| fong-spivak-invitation_FO2697 | 259 | 0.988 | i_{U}: U \subseteq X | | |
| fong-spivak-invitation_FO2698 | 259 | 0.691 | s: U \rightarrow E | | |
| fong-spivak-invitation_FO2699 | 259 | 0.691 | s ; \pi=i_{U} | | |
| fong-spivak-invitation_FO2700 | 259 | 1.000 | \pi: T M \rightarrow M | | |
| fong-spivak-invitation_FO2701 | 259 | 0.998 | \operatorname{VF}(U) | | |
| fong-spivak-invitation_FO2702 | 259 | 0.813 | \rightarrow \Omega | | |
| fong-spivak-invitation_FO2703 | 259 | 0.998 | \operatorname{Grad}(v) | | |
| fong-spivak-invitation_FO2704 | 259 | 0.887 | \mathcal{E}=\boldsymbol{\operatorname { S h v }}(X, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2705 | 259 | 0.887 | p: S \rightarrow \Omega | | |
| fong-spivak-invitation_FO2706 | 259 | 1.000 | S \in \mathcal{E} | | |
| fong-spivak-invitation_FO2707 | 259 | 1.000 | p(U): S(U) \rightarrow \Omega(U) | | |
| fong-spivak-invitation_FO2708 | 259 | 0.972 | \mathrm{Bob} \in S(U) | | |
| fong-spivak-invitation_FO2709 | 259 | 0.998 | s \in S(U) | | |
| fong-spivak-invitation_FO2710 | 259 | 0.998 | s= | | |
| fong-spivak-invitation_FO2711 | 260 | 1.000 | \{S \mid p\} \subseteq S | | |
| fong-spivak-invitation_FO2712 | 260 | 0.763 | \left|\Omega^{E}\right|= | | |
| fong-spivak-invitation_FO2713 | 260 | 1.000 | \mathcal{E}(S, \Omega) | | |
| fong-spivak-invitation_FO2714 | 260 | 1.000 | p, q: S \rightarrow \Omega | | |
| fong-spivak-invitation_FO2715 | 260 | 1.000 | p \leq^{S} q | | |
| fong-spivak-invitation_FO2716 | 260 | 1.000 | p(s) \subseteq U | | |
| fong-spivak-invitation_FO2717 | 260 | 1.000 | q(s) \subseteq U | | |
| fong-spivak-invitation_FO2718 | 260 | 1.000 | p(s) \subseteq q(s) | | |
| fong-spivak-invitation_FO2719 | 260 | 1.000 | \leq^{S} | | |
| fong-spivak-invitation_FO2720 | 260 | 1.000 | S=1 | | |
| fong-spivak-invitation_FO2721 | 260 | 1.000 | \left|\Omega^{S}\right| | | |
| fong-spivak-invitation_FO2722 | 260 | 1.000 | |\Omega| | | |
| fong-spivak-invitation_FO2723 | 260 | 1.000 | p \in|\Omega| | | |
| fong-spivak-invitation_FO2724 | 260 | 1.000 | p: 1 \rightarrow \Omega | | |
| fong-spivak-invitation_FO2725 | 260 | 1.000 | p=q | | |
| fong-spivak-invitation_FO2726 | 260 | 1.000 | U, V \subseteq X | | |
| fong-spivak-invitation_FO2727 | 260 | 1.000 | U=V | | |
| fong-spivak-invitation_FO2728 | 260 | 0.813 | S \in \operatorname{Shv}(X) | | |
| fong-spivak-invitation_FO2729 | 260 | 0.977 | p(s) \vdash_{s: S} q(s) | | |
| fong-spivak-invitation_FO2730 | 260 | 0.999 | \wedge, \vee, \Rightarrow, \neg | | |
| fong-spivak-invitation_FO2731 | 260 | 1.000 | (p \wedge q): S \rightarrow \Omega | | |
| fong-spivak-invitation_FO2732 | 260 | 1.000 | (p \wedge q)(s):=p(s) \wedge q(s) | | |
| fong-spivak-invitation_FO2733 | 260 | 1.000 | n+1 | | |
| fong-spivak-invitation_FO2734 | 260 | 0.979 | S, T \in \mathbf{S h v}(X, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2735 | 260 | 1.000 | p: S \times T \rightarrow \Omega | | |
| fong-spivak-invitation_FO2736 | 260 | 1.000 | t \in T(U) | | |
| fong-spivak-invitation_FO2737 | 261 | 1.000 | p(s, t) | | |
| fong-spivak-invitation_FO2738 | 261 | 0.992 | T, \ldots | | |
| fong-spivak-invitation_FO2739 | 261 | 0.998 | \exists(t: T) . p(s, t) | | |
| fong-spivak-invitation_FO2740 | 261 | 0.999 | p: \mathbb{N} \times \mathbb{Z} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2741 | 261 | 1.000 | \forall(z: \mathbb{Z}) . p(n, z) | | |
| fong-spivak-invitation_FO2742 | 261 | 1.000 | \exists(z: \mathbb{Z}) . p(n, z) | | |
| fong-spivak-invitation_FO2743 | 261 | 0.999 | z \in \mathbb{Z} | | |
| fong-spivak-invitation_FO2744 | 261 | 0.999 | \forall(n: \mathbb{N}) . p(n, z) | | |
| fong-spivak-invitation_FO2745 | 261 | 0.979 | \exists(n: \mathbb{N}) . p(n, z) | | |
| fong-spivak-invitation_FO2746 | 261 | 0.817 | \forall(t: | | |
| fong-spivak-invitation_FO2747 | 261 | 1.000 | S \rightarrow \Omega | | |
| fong-spivak-invitation_FO2748 | 261 | 1.000 | \forall(t: T) . p(s, t) | | |
| fong-spivak-invitation_FO2749 | 261 | 1.000 | V \in \Omega(U) | | |
| fong-spivak-invitation_FO2750 | 261 | 1.000 | p\left(\left.s\right|_{V}, t\right)=V | | |
| fong-spivak-invitation_FO2751 | 261 | 1.000 | t \in T(V) | | |
| fong-spivak-invitation_FO2752 | 261 | 0.998 | p(s, t)= | | |
| fong-spivak-invitation_FO2753 | 261 | 1.000 | T(V) | | |
| fong-spivak-invitation_FO2754 | 262 | 0.516 | p^{\prime}: S \rightarrow \Omega^{T} | | |
| fong-spivak-invitation_FO2755 | 262 | 0.516 | S \times T \rightarrow \Omega | | |
| fong-spivak-invitation_FO2756 | 262 | 0.516 | { }^{T} | | |
| fong-spivak-invitation_FO2757 | 262 | 0.817 | 1 \times T \stackrel{!}{\rightarrow} 1 \xrightarrow{\text { true }} \Omega | | |
| fong-spivak-invitation_FO2758 | 262 | 1.000 | V=\bigcup_{i} V_{i} | | |
| fong-spivak-invitation_FO2759 | 262 | 1.000 | V_{i} | | |
| fong-spivak-invitation_FO2760 | 262 | 1.000 | t_{i} \in | | |
| fong-spivak-invitation_FO2761 | 262 | 0.999 | T\left(V_{i}\right) | | |
| fong-spivak-invitation_FO2762 | 262 | 0.999 | p\left(\left.s\right|_{V_{i}}, t_{i}\right)=V_{i} | | |
| fong-spivak-invitation_FO2763 | 262 | 1.000 | p(t) | | |
| fong-spivak-invitation_FO2764 | 262 | 1.000 | U=\bigcup U_{i} | | |
| fong-spivak-invitation_FO2765 | 262 | 1.000 | t_{i} \in T\left(U_{i}\right) | | |
| fong-spivak-invitation_FO2766 | 262 | 1.000 | \{(s, t) \in S \times T \mid p(s, t)\} \subseteq S \times T | | |
| fong-spivak-invitation_FO2767 | 262 | 1.000 | \pi_{S}: S \times T \rightarrow S | | |
| fong-spivak-invitation_FO2768 | 262 | 0.983 | \left(\left|\Omega^{S}\right|, \leq^{S}\right) | | |
| fong-spivak-invitation_FO2769 | 262 | 0.983 | \left|\Omega^{S}\right|=\mathcal{E}(S, \Omega) | | |
| fong-spivak-invitation_FO2770 | 263 | 0.996 | j: \Omega \rightarrow \Omega | | |
| fong-spivak-invitation_FO2771 | 263 | 1.000 | p, q \in \Omega(U) | | |
| fong-spivak-invitation_FO2772 | 263 | 0.965 | (j ; j)(p) \leq j(p) | | |
| fong-spivak-invitation_FO2773 | 263 | 1.000 | j(p \wedge q)=j(p) \wedge j(q) | | |
| fong-spivak-invitation_FO2774 | 263 | 1.000 | p \in \Omega(U) | | |
| fong-spivak-invitation_FO2775 | 263 | 1.000 | q \in \Omega(U) | | |
| fong-spivak-invitation_FO2776 | 263 | 1.000 | j(j(q)) \leq | | |
| fong-spivak-invitation_FO2777 | 263 | 1.000 | j(q) | | |
| fong-spivak-invitation_FO2778 | 263 | 1.000 | j(j(q))=j(q) | | |
| fong-spivak-invitation_FO2779 | 263 | 0.999 | p, \ldots | | |
| fong-spivak-invitation_FO2780 | 263 | 1.000 | \Omega \rightarrow \Omega | | |
| fong-spivak-invitation_FO2781 | 263 | 1.000 | p \vee q | | |
| fong-spivak-invitation_FO2782 | 263 | 0.431 | q \Rightarrow p | | |
| fong-spivak-invitation_FO2783 | 263 | 1.000 | p(s) | | |
| fong-spivak-invitation_FO2784 | 263 | 0.999 | j(p(s)) | | |
| fong-spivak-invitation_FO2785 | 263 | 1.000 | p(s) \leq j(p(s)) | | |
| fong-spivak-invitation_FO2786 | 263 | 1.000 | j(j(p(s)))=j(p(s)) | | |
| fong-spivak-invitation_FO2787 | 263 | 1.000 | q: S \rightarrow \Omega | | |
| fong-spivak-invitation_FO2788 | 264 | 0.515 | t \in T(c) | | |
| fong-spivak-invitation_FO2789 | 264 | 0.515 | s \in S(c) | | |
| fong-spivak-invitation_FO2790 | 264 | 1.000 | f(s)=t | | |
| fong-spivak-invitation_FO2791 | 264 | 1.000 | t \in T(u) | | |
| fong-spivak-invitation_FO2792 | 264 | 0.999 | \left(U_{i} \subseteq U\right)_{i \in I} | | |
| fong-spivak-invitation_FO2793 | 264 | 0.999 | s_{i} \in S\left(U_{i}\right) | | |
| fong-spivak-invitation_FO2794 | 264 | 0.999 | f\left(s_{i}\right)=\left.t\right|_{u_{i}} | | |
| fong-spivak-invitation_FO2795 | 264 | 0.999 | U_{i} | | |
| fong-spivak-invitation_FO2796 | 265 | 1.000 | \mathbb{I} \mathbb{R} | | |
| fong-spivak-invitation_FO2797 | 265 | 0.680 | \boldsymbol{\operatorname { S h v }}(\mathbb{I} \mathbb{R}) | | |
| fong-spivak-invitation_FO2798 | 265 | 0.898 | a<b | | |
| fong-spivak-invitation_FO2799 | 265 | 0.898 | o_{[a, b]} | | |
| fong-spivak-invitation_FO2800 | 265 | 0.898 | o_{[a, b]}:=\{[d, u] \in \mathbb{R} \mid a<d \leq u<b\} | | |
| fong-spivak-invitation_FO2801 | 265 | 1.000 | o_{[0,5]} | | |
| fong-spivak-invitation_FO2802 | 265 | 1.000 | [d, u] | | |
| fong-spivak-invitation_FO2803 | 265 | 0.979 | \{x \in \mathbb{R} \mid 0<x<5\} | | |
| fong-spivak-invitation_FO2804 | 265 | 0.979 | o_{[4,8]} | | |
| fong-spivak-invitation_FO2805 | 265 | 0.979 | o_{[0,5]} \cup o_{[4,8]} \neq | | |
| fong-spivak-invitation_FO2806 | 265 | 0.993 | o_{[0,8]} | | |
| fong-spivak-invitation_FO2807 | 265 | 1.000 | [2,6] \in o_{[0,8]} | | |
| fong-spivak-invitation_FO2808 | 265 | 0.977 | \notin o_{[0,5]} \cup o_{[4,8]} | | |
| fong-spivak-invitation_FO2809 | 265 | 0.505 | \mathbf{B T}:=\operatorname{Shv}(\mathbb{I} \mathbb{R}, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2810 | 265 | 0.963 | \mathbb{R} \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO2811 | 265 | 1.000 | U \cap \mathbb{R} | | |
| fong-spivak-invitation_FO2812 | 265 | 0.912 | \mathbf{B T}=\mathbf{S h v}(\mathbb{I} \mathbb{R}, \mathbf{O p}) | | |
| fong-spivak-invitation_FO2813 | 266 | 0.569 | \mathbb{I R} | | |
| fong-spivak-invitation_FO2814 | 266 | 0.576 | \mathbb{I R}, \mathbf{O p} | | |
| fong-spivak-invitation_FO2815 | 266 | 0.576 | S: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO2816 | 266 | 1.000 | S(U) | | |
| fong-spivak-invitation_FO2817 | 266 | 1.000 | \left(s_{i}\right) | | |
| fong-spivak-invitation_FO2818 | 266 | 0.871 | U \subseteq \mathbb{R} \mathbb{R} | | |
| fong-spivak-invitation_FO2819 | 266 | 1.000 | S\left(o_{[a, b]}\right) | | |
| fong-spivak-invitation_FO2820 | 266 | 0.991 | \mathrm{A} \in \operatorname{Shv}(\mathbb{I} \mathbb{R}) | | |
| fong-spivak-invitation_FO2821 | 266 | 1.000 | \mathrm{A}(U):=A | | |
| fong-spivak-invitation_FO2822 | 266 | 1.000 | 7 \in \mathbb{N} | | |
| fong-spivak-invitation_FO2823 | 266 | 0.758 | F_{X} | | |
| fong-spivak-invitation_FO2824 | 266 | 0.787 | \mathbb{R} \mathbb{R} | | |
| fong-spivak-invitation_FO2825 | 266 | 0.787 | U \in \mathbf{O p}_{\mathbb{R}} | | |
| fong-spivak-invitation_FO2826 | 266 | 0.787 | F_{X}(U):= | | |
| fong-spivak-invitation_FO2827 | 266 | 0.759 | \{f: U \rightarrow X \mid f | | |
| fong-spivak-invitation_FO2828 | 266 | 0.759 | G_{X} | | |
| fong-spivak-invitation_FO2829 | 266 | 0.519 | \mathbb{R} \subseteq \mathbb{R} \mathbb{R} | | |
| fong-spivak-invitation_FO2830 | 266 | 0.519 | U \in \mathbf{O}_{\mathbb{P}_{\mathbb{R}}} | | |
| fong-spivak-invitation_FO2831 | 266 | 0.519 | G_{X}(U):= | | |
| fong-spivak-invitation_FO2832 | 266 | 0.985 | \{f: U \cap \mathbb{R} \rightarrow X \mid f | | |
| fong-spivak-invitation_FO2833 | 266 | 0.846 | \left(X, \mathbf{O p}_{X}\right) | | |
| fong-spivak-invitation_FO2834 | 266 | 0.846 | R \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO2835 | 266 | 0.846 | H_{X}(U):=\{f: U \cap | | |
| fong-spivak-invitation_FO2836 | 266 | 1.000 | R \rightarrow X \mid f | | |
| fong-spivak-invitation_FO2837 | 266 | 1.000 | H_{X} | | |
| fong-spivak-invitation_FO2838 | 267 | 0.740 | (d, d+3) | | |
| fong-spivak-invitation_FO2839 | 267 | 1.000 | L(d, d+3) | | |
| fong-spivak-invitation_FO2840 | 267 | 0.458 | \mathbb{I k}, \mathbf{O p} | | |
| fong-spivak-invitation_FO2841 | 267 | 1.000 | (0,5) | | |
| fong-spivak-invitation_FO2842 | 267 | 0.806 | (4,8) | | |
| fong-spivak-invitation_FO2843 | 267 | 1.000 | (0,8) | | |
| fong-spivak-invitation_FO2844 | 268 | 1.000 | \theta \in \mathbb{R}_{\geq 0} | | |
| fong-spivak-invitation_FO2845 | 268 | 1.000 | p(D, T) | | |
| fong-spivak-invitation_FO2846 | 268 | 0.968 | @_{t} | | |
| fong-spivak-invitation_FO2847 | 268 | 1.000 | @_{t}(q) | | |
| fong-spivak-invitation_FO2848 | 268 | 0.999 | D \in | | |
| fong-spivak-invitation_FO2849 | 268 | 1.000 | \operatorname{dials}(U) | | |
| fong-spivak-invitation_FO2850 | 268 | 1.000 | T \in \operatorname{thrusters}(U) | | |
| fong-spivak-invitation_FO2851 | 270 | 1.000 | f(x)=x+5 | | |
| fong-spivak-invitation_FO2852 | 270 | 1.000 | x+5 \leq y+5 | | |
| fong-spivak-invitation_FO2853 | 270 | 0.498 | g(x):=-x | | |
| fong-spivak-invitation_FO2854 | 270 | 0.498 | 1 \leq 2 | | |
| fong-spivak-invitation_FO2855 | 270 | 0.498 | -1 \leq ? | | |
| fong-spivak-invitation_FO2856 | 270 | 0.992 | f(x):=x+5 | | |
| fong-spivak-invitation_FO2857 | 270 | 0.992 | |x-y|=|(x+5)-(y+5)| | | |
| fong-spivak-invitation_FO2858 | 270 | 1.000 | g(x):=2 * x | | |
| fong-spivak-invitation_FO2859 | 270 | 1.000 | x=1 | | |
| fong-spivak-invitation_FO2860 | 270 | 1.000 | y=2 | | |
| fong-spivak-invitation_FO2861 | 270 | 1.000 | |x-y|=1 | | |
| fong-spivak-invitation_FO2862 | 270 | 1.000 | |2 x-2 y|=2 | | |
| fong-spivak-invitation_FO2863 | 270 | 1.000 | f(x):=3 * x | | |
| fong-spivak-invitation_FO2864 | 270 | 1.000 | 3 *(x+y)=(3 * x)+(3 * y) | | |
| fong-spivak-invitation_FO2865 | 270 | 1.000 | g(x):=x+1 | | |
| fong-spivak-invitation_FO2866 | 270 | 1.000 | x=0 | | |
| fong-spivak-invitation_FO2867 | 270 | 1.000 | y=0 | | |
| fong-spivak-invitation_FO2868 | 270 | 1.000 | g(x+y)=1 | | |
| fong-spivak-invitation_FO2869 | 270 | 1.000 | g(x)+g(y)=2 | | |
| fong-spivak-invitation_FO2870 | 271 | 0.995 | \{1,2,3,4\} | | |
| fong-spivak-invitation_FO2871 | 271 | 1.000 | A=(12)(3)(4) | | |
| fong-spivak-invitation_FO2872 | 271 | 1.000 | B=(13)(2)(4) | | |
| fong-spivak-invitation_FO2873 | 271 | 1.000 | A \vee B=(123)(4) | | |
| fong-spivak-invitation_FO2874 | 271 | 1.000 | 0 \notin\{n \in \mathbb{Z} \mid n \geq 1\} | | |
| fong-spivak-invitation_FO2875 | 271 | 0.936 | \{1,2,3\} | | |
| fong-spivak-invitation_FO2876 | 271 | 0.936 | \{1,2,3\} \cup\{1\}=\{1,2,3\} | | |
| fong-spivak-invitation_FO2877 | 271 | 0.994 | \{h, 1\} \times\{1,2,3\} | | |
| fong-spivak-invitation_FO2878 | 271 | 0.996 | \{h, 1\} \sqcup\{1,2,3\} | | |
| fong-spivak-invitation_FO2879 | 271 | 1.000 | \{h, 1\} \cup\{1,2,3\} | | |
| fong-spivak-invitation_FO2880 | 272 | 1.000 | p_{1}^{\prime}, p_{2}^{\prime} \in P^{\prime} | | |
| fong-spivak-invitation_FO2881 | 272 | 1.000 | A_{p}=A_{p_{1}^{\prime}}^{\prime} | | |
| fong-spivak-invitation_FO2882 | 272 | 1.000 | A_{p}=A_{p_{2}^{\prime}}^{\prime} | | |
| fong-spivak-invitation_FO2883 | 272 | 1.000 | A_{p_{1}^{\prime}}=A_{p_{2}^{\prime}} | | |
| fong-spivak-invitation_FO2884 | 272 | 1.000 | A_{p_{1}^{\prime}} \cap A_{p_{2}^{\prime}}=A_{p_{1}^{\prime}} | | |
| fong-spivak-invitation_FO2885 | 272 | 1.000 | A_{p_{1}^{\prime}} \neq \varnothing | | |
| fong-spivak-invitation_FO2886 | 272 | 1.000 | p_{1} \neq p_{2} | | |
| fong-spivak-invitation_FO2887 | 272 | 1.000 | A_{p_{1}^{\prime}} \cap A_{p_{2}^{\prime}}=\varnothing | | |
| fong-spivak-invitation_FO2888 | 272 | 1.000 | p_{1}^{\prime}=p_{2}^{\prime} | | |
| fong-spivak-invitation_FO2889 | 272 | 1.000 | A_{p^{\prime}}^{\prime} \neq \varnothing | | |
| fong-spivak-invitation_FO2890 | 272 | 1.000 | a \in A_{p^{\prime}}^{\prime} | | |
| fong-spivak-invitation_FO2891 | 272 | 1.000 | A_{p^{\prime}}^{\prime} \subseteq A | | |
| fong-spivak-invitation_FO2892 | 272 | 1.000 | p^{\prime \prime} \in P^{\prime} | | |
| fong-spivak-invitation_FO2893 | 272 | 1.000 | A_{p}=A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-invitation_FO2894 | 272 | 1.000 | a \in A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-invitation_FO2895 | 272 | 1.000 | a \in A_{p^{\prime}}^{\prime} \cap A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-invitation_FO2896 | 272 | 1.000 | p^{\prime}=p^{\prime \prime} | | |
| fong-spivak-invitation_FO2897 | 272 | 1.000 | A_{p}=A_{p^{\prime}} | | |
| fong-spivak-invitation_FO2898 | 272 | 1.000 | a \in A_{p} \cap A_{q} | | |
| fong-spivak-invitation_FO2899 | 272 | 1.000 | A_{p}=A_{q} | | |
| fong-spivak-invitation_FO2900 | 272 | 1.000 | a^{\prime} \in A_{p} | | |
| fong-spivak-invitation_FO2901 | 272 | 1.000 | a^{\prime} \in A_{q} | | |
| fong-spivak-invitation_FO2902 | 272 | 1.000 | a^{\prime} \in A | | |
| fong-spivak-invitation_FO2903 | 272 | 1.000 | a \sim a^{\prime} | | |
| fong-spivak-invitation_FO2904 | 272 | 1.000 | X:=\left\{a^{\prime} \in A \mid a^{\prime} \sim a\right\} | | |
| fong-spivak-invitation_FO2905 | 272 | 0.962 | a^{\prime} \in X | | |
| fong-spivak-invitation_FO2906 | 272 | 0.999 | b \sim a^{\prime} | | |
| fong-spivak-invitation_FO2907 | 272 | 0.999 | b \sim a | | |
| fong-spivak-invitation_FO2908 | 272 | 0.999 | b \in X | | |
| fong-spivak-invitation_FO2909 | 272 | 0.999 | b, c \in X | | |
| fong-spivak-invitation_FO2910 | 272 | 0.999 | c \sim a | | |
| fong-spivak-invitation_FO2911 | 272 | 0.896 | a \in X | | |
| fong-spivak-invitation_FO2912 | 272 | 1.000 | \varnothing \rightarrow\{1\} | | |
| fong-spivak-invitation_FO2913 | 272 | 1.000 | \{a, b\} \rightarrow\{1\} | | |
| fong-spivak-invitation_FO2914 | 273 | 1.000 | F \subseteq A \times \varnothing | | |
| fong-spivak-invitation_FO2915 | 273 | 0.862 | b \in \varnothing | | |
| fong-spivak-invitation_FO2916 | 273 | 0.862 | (a, b) \in F | | |
| fong-spivak-invitation_FO2917 | 273 | 0.862 | b \in^{?} \varnothing | | |
| fong-spivak-invitation_FO2918 | 273 | 0.999 | \{\bullet, *, \circ\} | | |
| fong-spivak-invitation_FO2919 | 273 | 0.962 | 1 \rightarrow 3 | | |
| fong-spivak-invitation_FO2920 | 273 | 0.582 | P:=V=\{1,2,3,4\} | | |
| fong-spivak-invitation_FO2921 | 274 | 1.000 | (\bullet)(\circ)(*) \leq(\bullet)(\circ)(*) | | |
| fong-spivak-invitation_FO2922 | 274 | 1.000 | (\bullet)(\circ)(*) \leq(\bullet \circ)(*) | | |
| fong-spivak-invitation_FO2923 | 274 | 0.982 | (\bullet)(\circ)(*) \leq(\bullet *)(\circ) | | |
| fong-spivak-invitation_FO2924 | 274 | 0.950 | (\bullet)(\circ)(*) \leq(\bullet)(\circ *) | | |
| fong-spivak-invitation_FO2925 | 274 | 0.651 | (\bullet)(\circ)(*) \leq(\bullet \circ *) | | |
| fong-spivak-invitation_FO2926 | 274 | 1.000 | (\bullet \circ)(*) \leq(\bullet \circ)(*) | | |
| fong-spivak-invitation_FO2927 | 274 | 1.000 | (\bullet \circ)(*) \leq(\bullet \circ *) | | |
| fong-spivak-invitation_FO2928 | 274 | 1.000 | (\bullet *)(\circ) \leq(\bullet *)(\circ) | | |
| fong-spivak-invitation_FO2929 | 274 | 1.000 | (\bullet *)(\circ) \leq(\bullet \circ *) | | |
| fong-spivak-invitation_FO2930 | 274 | 0.999 | (\bullet)(\circ *) \leq(\bullet)(\circ *) | | |
| fong-spivak-invitation_FO2931 | 274 | 0.896 | (\bullet)(\circ *) \leq(\bullet \circ *) | | |
| fong-spivak-invitation_FO2932 | 274 | 0.860 | (\bullet \circ *) \leq(\bullet \circ *) | | |
| fong-spivak-invitation_FO2933 | 274 | 1.000 | 4 \not \leq 6 | | |
| fong-spivak-invitation_FO2934 | 274 | 1.000 | 6 \not \leq 4 | | |
| fong-spivak-invitation_FO2935 | 274 | 1.000 | a, b \in \mathbb{R} | | |
| fong-spivak-invitation_FO2936 | 274 | 1.000 | !: S \rightarrow\{1\} | | |
| fong-spivak-invitation_FO2937 | 274 | 0.953 | \mathrm{U}(X) | | |
| fong-spivak-invitation_FO2938 | 274 | 1.000 | \mathrm{U}(X)=\mathrm{P}(X) | | |
| fong-spivak-invitation_FO2939 | 275 | 1.000 | 0 \leq \cdots \leq 3 | | |
| fong-spivak-invitation_FO2940 | 275 | 1.000 | q \in \uparrow p | | |
| fong-spivak-invitation_FO2941 | 275 | 1.000 | p \leq q^{\prime} | | |
| fong-spivak-invitation_FO2942 | 275 | 0.835 | q^{\prime} \in \uparrow p | | |
| fong-spivak-invitation_FO2943 | 275 | 0.835 | \uparrow p | | |
| fong-spivak-invitation_FO2944 | 275 | 0.966 | q \leq^{\mathrm{op}} p | | |
| fong-spivak-invitation_FO2945 | 275 | 0.966 | P^{\mathrm{op}} | | |
| fong-spivak-invitation_FO2946 | 275 | 0.966 | \uparrow q \subseteq \uparrow p | | |
| fong-spivak-invitation_FO2947 | 275 | 1.000 | q^{\prime} \in \uparrow q | | |
| fong-spivak-invitation_FO2948 | 275 | 1.000 | p \not \leq p^{\prime} | | |
| fong-spivak-invitation_FO2949 | 275 | 1.000 | \uparrow\left(p^{\prime}\right) \nsubseteq \uparrow(p) | | |
| fong-spivak-invitation_FO2950 | 275 | 1.000 | p^{\prime} \in \uparrow\left(p^{\prime}\right) | | |
| fong-spivak-invitation_FO2951 | 275 | 1.000 | p^{\prime} \notin \uparrow(p) | | |
| fong-spivak-invitation_FO2952 | 276 | 0.506 | P^{\mathrm{op}} \rightarrow \mathrm{U}(P) | | |
| fong-spivak-invitation_FO2953 | 276 | 1.000 | p_{1} \leq_{P} p_{2} | | |
| fong-spivak-invitation_FO2954 | 276 | 1.000 | f\left(p_{1}\right) \leq_{Q} f\left(p_{2}\right) | | |
| fong-spivak-invitation_FO2955 | 276 | 1.000 | f\left(p_{1}\right)=f\left(p_{2}\right) | | |
| fong-spivak-invitation_FO2956 | 276 | 1.000 | f\left(p_{1}\right) \leq f\left(p_{2}\right) | | |
| fong-spivak-invitation_FO2957 | 276 | 0.999 | X=\mathbb{Z}=\{\ldots,-2,-1,0,1, \ldots\} | | |
| fong-spivak-invitation_FO2958 | 276 | 0.999 | Y=\{n, z, p\} | | |
| fong-spivak-invitation_FO2959 | 276 | 0.999 | f: X \rightarrow | | |
| fong-spivak-invitation_FO2960 | 276 | 1.000 | P:=(n z)(p) | | |
| fong-spivak-invitation_FO2961 | 276 | 1.000 | Q:=(n p)(z) | | |
| fong-spivak-invitation_FO2962 | 276 | 1.000 | \{\{n, z\},\{p\}\} | | |
| fong-spivak-invitation_FO2963 | 276 | 1.000 | \{\{n, p\},\{z\}\} | | |
| fong-spivak-invitation_FO2964 | 276 | 1.000 | f^{*} P=(\ldots,-2,-1,0)(1,2, \ldots) | | |
| fong-spivak-invitation_FO2965 | 276 | 1.000 | f^{*} Q=(0)(\ldots,-2,-1,1,2, \ldots) | | |
| fong-spivak-invitation_FO2966 | 276 | 1.000 | \left(P, \leq_{P}\right),\left(Q, \leq_{Q}\right) | | |
| fong-spivak-invitation_FO2967 | 276 | 0.812 | \mathrm{id}_{P} | | |
| fong-spivak-invitation_FO2968 | 276 | 0.812 | \operatorname{id}_{P}\left(p_{1}\right) \leq \operatorname{id}_{P}\left(p_{2}\right) | | |
| fong-spivak-invitation_FO2969 | 276 | 1.000 | \operatorname{id}_{P}(p)=p | | |
| fong-spivak-invitation_FO2970 | 276 | 0.988 | q_{1} \leq_{Q} q_{2} | | |
| fong-spivak-invitation_FO2971 | 276 | 0.988 | g\left(q_{1}\right) \leq_{R} | | |
| fong-spivak-invitation_FO2972 | 276 | 1.000 | g\left(q_{2}\right) | | |
| fong-spivak-invitation_FO2973 | 276 | 1.000 | g\left(f\left(p_{1}\right)\right) \leq_{R} g\left(f\left(p_{2}\right)\right) | | |
| fong-spivak-invitation_FO2974 | 276 | 0.679 | (f ; g) | | |
| fong-spivak-invitation_FO2975 | 276 | 0.998 | p_{2} \leq p_{1} | | |
| fong-spivak-invitation_FO2976 | 277 | 1.000 | \Phi: \operatorname{Prt}(\{\bullet, *, \circ\}) \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2977 | 277 | 0.994 | \Phi(P)= | | |
| fong-spivak-invitation_FO2978 | 277 | 0.994 | \Phi(Q)= | | |
| fong-spivak-invitation_FO2979 | 277 | 1.000 | x \sim_{P} y | | |
| fong-spivak-invitation_FO2980 | 277 | 1.000 | x \sim_{Q} y | | |
| fong-spivak-invitation_FO2981 | 277 | 1.000 | x, y \in\{\bullet, *, \circ\} | | |
| fong-spivak-invitation_FO2982 | 277 | 0.980 | U \mapsto f^{-1}(U) | | |
| fong-spivak-invitation_FO2983 | 277 | 1.000 | u: Q \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO2984 | 277 | 1.000 | f^{*}(U) | | |
| fong-spivak-invitation_FO2985 | 277 | 0.904 | q \mapsto | | |
| fong-spivak-invitation_FO2986 | 277 | 0.904 | p \mapsto | | |
| fong-spivak-invitation_FO2987 | 277 | 0.904 | f(p) \in U | | |
| fong-spivak-invitation_FO2988 | 277 | 1.000 | \{p \in P \mid f(p) \in U\} | | |
| fong-spivak-invitation_FO2989 | 277 | 1.000 | S=\left\{\left.\frac{1}{n+1} \right\rvert\, n \in \mathbb{N}\right\} | | |
| fong-spivak-invitation_FO2990 | 277 | 1.000 | 0 \leq \frac{1}{n+1} | | |
| fong-spivak-invitation_FO2991 | 277 | 1.000 | b \leq 0 | | |
| fong-spivak-invitation_FO2992 | 277 | 1.000 | 0<b | | |
| fong-spivak-invitation_FO2993 | 277 | 1.000 | 1 / b | | |
| fong-spivak-invitation_FO2994 | 277 | 1.000 | 1 / b<n<n+1 | | |
| fong-spivak-invitation_FO2995 | 277 | 1.000 | 1<b(n+1) | | |
| fong-spivak-invitation_FO2996 | 277 | 1.000 | \frac{1}{n+1}<b | | |
| fong-spivak-invitation_FO2997 | 277 | 1.000 | a=p | | |
| fong-spivak-invitation_FO2998 | 277 | 1.000 | 4 \wedge 6=2 | | |
| fong-spivak-invitation_FO2999 | 277 | 1.000 | 4 \vee 6=12 | | |
| fong-spivak-invitation_FO3000 | 277 | 0.528 | \operatorname{gcd}(0, a) | | |
| fong-spivak-invitation_FO3001 | 277 | 1.000 | a \leq a \vee b | | |
| fong-spivak-invitation_FO3002 | 277 | 1.000 | b \leq a \vee b | | |
| fong-spivak-invitation_FO3003 | 277 | 1.000 | f(a) \leq f(a \vee b) | | |
| fong-spivak-invitation_FO3004 | 277 | 1.000 | f(b) \leq f(a \vee b) | | |
| fong-spivak-invitation_FO3005 | 277 | 0.961 | \lfloor-/ 3\rfloor | | |
| fong-spivak-invitation_FO3006 | 277 | 0.936 | r \in \mathbb{R} | | |
| fong-spivak-invitation_FO3007 | 277 | 0.936 | z \leq\lfloor r / 3\rfloor | | |
| fong-spivak-invitation_FO3008 | 277 | 0.936 | 3 * z \leq r | | |
| fong-spivak-invitation_FO3009 | 278 | 1.000 | r / 3 | | |
| fong-spivak-invitation_FO3010 | 278 | 1.000 | z^{\prime}:=\lfloor r / 3\rfloor | | |
| fong-spivak-invitation_FO3011 | 278 | 1.000 | z \leq z^{\prime} | | |
| fong-spivak-invitation_FO3012 | 278 | 1.000 | 3 * z \leq 3 * z^{\prime} \leq | | |
| fong-spivak-invitation_FO3013 | 278 | 0.988 | 3 * r / 3=r | | |
| fong-spivak-invitation_FO3014 | 278 | 1.000 | z=3 * z / 3 \leq r / 3 | | |
| fong-spivak-invitation_FO3015 | 278 | 1.000 | \lfloor r / 3\rfloor | | |
| fong-spivak-invitation_FO3016 | 278 | 1.000 | \{(p, q) \mid 1 \leq p \leq 3 | | |
| fong-spivak-invitation_FO3017 | 278 | 1.000 | 1 \leq q \leq 3\} | | |
| fong-spivak-invitation_FO3018 | 278 | 1.000 | p=q=1 | | |
| fong-spivak-invitation_FO3019 | 278 | 1.000 | f(p)=1 | | |
| fong-spivak-invitation_FO3020 | 278 | 1.000 | g(q)=2 | | |
| fong-spivak-invitation_FO3021 | 278 | 1.000 | f(p)=1 \leq 1=q | | |
| fong-spivak-invitation_FO3022 | 278 | 1.000 | p=1 \leq g(q) | | |
| fong-spivak-invitation_FO3023 | 278 | 1.000 | (p, q) | | |
| fong-spivak-invitation_FO3024 | 278 | 1.000 | (1,2),(1,3),(2,1) | | |
| fong-spivak-invitation_FO3025 | 278 | 0.991 | p=3, q=1 | | |
| fong-spivak-invitation_FO3026 | 278 | 0.991 | p=3, q=2 | | |
| fong-spivak-invitation_FO3027 | 278 | 1.000 | f(p)=3 | | |
| fong-spivak-invitation_FO3028 | 278 | 1.000 | f(p) \not \leq q | | |
| fong-spivak-invitation_FO3029 | 278 | 1.000 | p \not \leq g(q) | | |
| fong-spivak-invitation_FO3030 | 278 | 1.000 | f(2) \not \leq 1 | | |
| fong-spivak-invitation_FO3031 | 278 | 1.000 | 2 \leq g(1) | | |
| fong-spivak-invitation_FO3032 | 278 | 1.000 | C(r):=\lceil r / 3\rceil | | |
| fong-spivak-invitation_FO3033 | 278 | 1.000 | L(z) \leq r | | |
| fong-spivak-invitation_FO3034 | 278 | 1.000 | z \leq C(r) | | |
| fong-spivak-invitation_FO3035 | 278 | 1.000 | z=1 | | |
| fong-spivak-invitation_FO3036 | 278 | 1.000 | r=.01 | | |
| fong-spivak-invitation_FO3037 | 278 | 1.000 | \lceil r / 3\rceil=1 | | |
| fong-spivak-invitation_FO3038 | 278 | 1.000 | L(1) \leq 0.01 | | |
| fong-spivak-invitation_FO3039 | 278 | 1.000 | L(1) \leq r | | |
| fong-spivak-invitation_FO3040 | 278 | 1.000 | r>0 | | |
| fong-spivak-invitation_FO3041 | 278 | 1.000 | L(1) \leq 0 | | |
| fong-spivak-invitation_FO3042 | 278 | 1.000 | 1 \leq C(0)=\lceil 0 / 3\rceil=0 | | |
| fong-spivak-invitation_FO3043 | 278 | 1.000 | c_{1}, c_{2}, c_{3}, c_{4} | | |
| fong-spivak-invitation_FO3044 | 278 | 0.997 | g_{!}\left(c_{1}\right), g_{!}\left(c_{2}\right), g_{!}\left(c_{3}\right) | | |
| fong-spivak-invitation_FO3045 | 278 | 0.997 | g_{!}\left(c_{4}\right) | | |
| fong-spivak-invitation_FO3046 | 279 | 1.000 | S=\{1,2,3,4\} | | |
| fong-spivak-invitation_FO3047 | 279 | 0.980 | c \leq | | |
| fong-spivak-invitation_FO3048 | 279 | 1.000 | f: P \leftrightarrows Q: g | | |
| fong-spivak-invitation_FO3049 | 279 | 1.000 | g(q) \leq g(q) | | |
| fong-spivak-invitation_FO3050 | 279 | 1.000 | p:=g(q) | | |
| fong-spivak-invitation_FO3051 | 279 | 1.000 | f(p) \leq f(g(q)) | | |
| fong-spivak-invitation_FO3052 | 280 | 1.000 | g, g^{\prime}: Q \rightarrow P | | |
| fong-spivak-invitation_FO3053 | 280 | 1.000 | g(q) \leq g^{\prime}(q) | | |
| fong-spivak-invitation_FO3054 | 280 | 1.000 | g^{\prime}(q) \leq g(q) | | |
| fong-spivak-invitation_FO3055 | 280 | 1.000 | p \leq g^{\prime}(f(p)) | | |
| fong-spivak-invitation_FO3056 | 280 | 1.000 | j:=\bigvee A | | |
| fong-spivak-invitation_FO3057 | 280 | 1.000 | f(a) \leq f(j) | | |
| fong-spivak-invitation_FO3058 | 280 | 1.000 | f(j) | | |
| fong-spivak-invitation_FO3059 | 280 | 1.000 | f(a) \leq b | | |
| fong-spivak-invitation_FO3060 | 280 | 1.000 | a \leq g(b) | | |
| fong-spivak-invitation_FO3061 | 280 | 1.000 | j \leq g(b) | | |
| fong-spivak-invitation_FO3062 | 280 | 1.000 | f(j) \leq b | | |
| fong-spivak-invitation_FO3063 | 281 | 1.000 | f(p) | | |
| fong-spivak-invitation_FO3066 | 281 | 0.977 | B_{1}:=\{a, b\} | | |
| fong-spivak-invitation_FO3067 | 281 | 0.977 | B_{2}:=\{c\} | | |
| fong-spivak-invitation_FO3068 | 281 | 0.977 | f^{*}\left(B_{1}\right)=\left\{a_{1}\right\} | | |
| fong-spivak-invitation_FO3069 | 281 | 0.977 | f^{*}\left(B_{2}\right)=\left\{c_{1}, c_{2}\right\} | | |
| fong-spivak-invitation_FO3070 | 281 | 1.000 | A_{1}:=\varnothing | | |
| fong-spivak-invitation_FO3071 | 281 | 1.000 | A_{2}:=\left\{a_{1}, c_{1}\right\} | | |
| fong-spivak-invitation_FO3072 | 281 | 1.000 | f_{!}\left(A_{1}\right)=\varnothing | | |
| fong-spivak-invitation_FO3073 | 281 | 1.000 | f_{!}\left(A_{2}\right)=\{a, c\} | | |
| fong-spivak-invitation_FO3074 | 281 | 0.999 | A_{1} | | |
| fong-spivak-invitation_FO3075 | 281 | 0.999 | A_{2} | | |
| fong-spivak-invitation_FO3076 | 281 | 0.999 | f_{*}\left(A_{1}\right)=\{b\} | | |
| fong-spivak-invitation_FO3077 | 281 | 0.999 | f_{*}\left(A_{2}\right)=\{a, b\} | | |
| fong-spivak-invitation_FO3078 | 281 | 1.000 | g(f(p)) | | |
| fong-spivak-invitation_FO3079 | 281 | 1.000 | (f ; g)(p) | | |
| fong-spivak-invitation_FO3080 | 281 | 1.000 | g(f(g(f(p)))) \leq g(f(p)) | | |
| fong-spivak-invitation_FO3081 | 281 | 1.000 | g(f(p)) \leq g(f(g(f(p)))) | | |
| fong-spivak-invitation_FO3082 | 281 | 1.000 | p^{\prime} \leq g\left(f\left(p^{\prime}\right)\right) | | |
| fong-spivak-invitation_FO3083 | 281 | 1.000 | p^{\prime} | | |
| fong-spivak-invitation_FO3084 | 281 | 1.000 | f(g(f(p))) \leq f(p) | | |
| fong-spivak-invitation_FO3085 | 281 | 0.657 | a b | | |
| fong-spivak-invitation_FO3086 | 281 | 0.657 | \{(1,1),(1,2),(2,1)\} | | |
| fong-spivak-invitation_FO3087 | 281 | 1.000 | S:=\{1,2,3\} | | |
| fong-spivak-invitation_FO3088 | 281 | 0.997 | 1 \leq 1,2 \leq 2 | | |
| fong-spivak-invitation_FO3089 | 281 | 0.997 | 3 \leq 3 | | |
| fong-spivak-invitation_FO3090 | 281 | 0.997 | U(\leq)= | | |
| fong-spivak-invitation_FO3091 | 281 | 0.876 | \{(1,1),(1,2),(2,2),(3,3)\} | | |
| fong-spivak-invitation_FO3092 | 281 | 1.000 | Q:=\{(1,1)\} | | |
| fong-spivak-invitation_FO3093 | 281 | 1.000 | Q^{\prime}:=\{(2,1)\} | | |
| fong-spivak-invitation_FO3094 | 282 | 0.812 | \mathrm{Cl}(Q) | | |
| fong-spivak-invitation_FO3095 | 282 | 0.812 | \mathrm{Cl}(Q)= | | |
| fong-spivak-invitation_FO3096 | 282 | 1.000 | \{(1,1),(2,2),(3,3)\} | | |
| fong-spivak-invitation_FO3097 | 282 | 1.000 | \mathrm{Cl}\left(Q^{\prime}\right)=\{(1,1),(2,1),(2,2),(3,3)\} | | |
| fong-spivak-invitation_FO3098 | 282 | 0.995 | (2,1) | | |
| fong-spivak-invitation_FO3099 | 282 | 0.995 | \mathrm{Cl}\left(Q^{\prime}\right) | | |
| fong-spivak-invitation_FO3100 | 282 | 1.000 | x_{1}=-1, x_{2}=0, y_{1}=-1 | | |
| fong-spivak-invitation_FO3101 | 282 | 1.000 | y_{2}=1 | | |
| fong-spivak-invitation_FO3102 | 282 | 1.000 | -1 \leq-1 | | |
| fong-spivak-invitation_FO3103 | 282 | 1.000 | -1 * 0=0 \not \leq-1=-1 * 1 | | |
| fong-spivak-invitation_FO3104 | 282 | 0.988 | (\operatorname{Disc}(M),=, *, e) | | |
| fong-spivak-invitation_FO3105 | 282 | 1.000 | x_{1} \otimes x_{2}=x_{1} \otimes x_{2} | | |
| fong-spivak-invitation_FO3106 | 282 | 1.000 | u \leq u, v \leq v | | |
| fong-spivak-invitation_FO3107 | 282 | 1.000 | z \leq z | | |
| fong-spivak-invitation_FO3108 | 282 | 1.000 | t+u \leq y+z | | |
| fong-spivak-invitation_FO3109 | 282 | 1.000 | z \rightarrow w | | |
| fong-spivak-invitation_FO3110 | 282 | 1.000 | x+z \rightarrow y+w | | |
| fong-spivak-invitation_FO3111 | 283 | 0.954 | 1 \mid 1 | | |
| fong-spivak-invitation_FO3112 | 283 | 0.954 | 1 \mid 2 | | |
| fong-spivak-invitation_FO3113 | 283 | 0.954 | (1+1) \nmid(1+2) | | |
| fong-spivak-invitation_FO3114 | 283 | 0.964 | z \leq w | | |
| fong-spivak-invitation_FO3115 | 283 | 0.964 | \min (x, z) \leq \min (y, w) | | |
| fong-spivak-invitation_FO3116 | 283 | 0.964 | \min (x | | |
| fong-spivak-invitation_FO3117 | 283 | 0.766 | x=\min (y e s, x) | | |
| fong-spivak-invitation_FO3118 | 283 | 0.766 | \min (\min (x, y), z)=\min (x, \min (y, z)) | | |
| fong-spivak-invitation_FO3119 | 283 | 0.766 | \min (x, y)=\min (y, x) | | |
| fong-spivak-invitation_FO3120 | 283 | 0.984 | (\mathrm{P}(S), \leq, S, \cap) | | |
| fong-spivak-invitation_FO3121 | 283 | 1.000 | (P \wedge Q)(n) | | |
| fong-spivak-invitation_FO3122 | 283 | 0.601 | \left(\operatorname{Prop}^{\mathbb{N}}, \leq\right. | | |
| fong-spivak-invitation_FO3123 | 283 | 0.601 | \left.\wedge\right) | | |
| fong-spivak-invitation_FO3124 | 283 | 0.997 | n+10 \leq 1 | | |
| fong-spivak-invitation_FO3125 | 283 | 0.997 | (P \vee Q)(n) | | |
| fong-spivak-invitation_FO3126 | 283 | 0.769 | \left.\vee\right) | | |
| fong-spivak-invitation_FO3127 | 283 | 0.827 | I \otimes x=x= | | |
| fong-spivak-invitation_FO3128 | 283 | 1.000 | x \otimes I | | |
| fong-spivak-invitation_FO3129 | 283 | 0.727 | X^{\mathrm{op}} | | |
| fong-spivak-invitation_FO3130 | 283 | 0.675 | \operatorname{Cost}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO3131 | 284 | 1.000 | =\infty \geq 0=g | | |
| fong-spivak-invitation_FO3132 | 284 | 1.000 | 0 \geq 0=g | | |
| fong-spivak-invitation_FO3133 | 284 | 0.990 | f(n)=1 | | |
| fong-spivak-invitation_FO3134 | 284 | 0.990 | f:(\mathbb{N}, \leq, 0,+) \rightarrow(\mathbb{N}, \leq, 1, *) | | |
| fong-spivak-invitation_FO3135 | 284 | 0.276 | \mathbb{Z}, \leq, *, 1 | | |
| fong-spivak-invitation_FO3136 | 284 | 0.276 | \leq 0 | | |
| fong-spivak-invitation_FO3137 | 284 | 1.000 | -1) \not \leq(0 * 0) | | |
| fong-spivak-invitation_FO3138 | 284 | 0.382 | \mathcal{X}_{P} | | |
| fong-spivak-invitation_FO3139 | 284 | 0.382 | \mathfrak{X}_{P}(p, q):=\operatorname{true} | | |
| fong-spivak-invitation_FO3140 | 284 | 0.579 | \mathcal{X}_{P}(p, q):= | | |
| fong-spivak-invitation_FO3141 | 284 | 0.626 | \operatorname{Ob}\left(\mathcal{X}_{P}\right)=P | | |
| fong-spivak-invitation_FO3142 | 284 | 0.219 | X_{P}(p, q)=\operatorname{true} | | |
| fong-spivak-invitation_FO3143 | 284 | 0.694 | (\operatorname{Ob}(\mathscr{X}), \leq) | | |
| fong-spivak-invitation_FO3144 | 284 | 0.897 | \operatorname{Ob}(X) | | |
| fong-spivak-invitation_FO3145 | 284 | 0.727 | \mathcal{X}^{\prime}(x, y)= | | |
| fong-spivak-invitation_FO3146 | 284 | 0.642 | X^{\prime}(x, y)=X(x, y) | | |
| fong-spivak-invitation_FO3147 | 284 | 0.492 | d(\mathrm{US} | | |
| fong-spivak-invitation_FO3148 | 284 | 0.971 | ([0, \infty], \geq | | |
| fong-spivak-invitation_FO3149 | 284 | 0.558 | 0,+) | | |
| fong-spivak-invitation_FO3150 | 285 | 0.575 | \min (\mathcal{X}(x, y), \mathcal{X}(y, z)) \leq \mathfrak{X}(x, z) | | |
| fong-spivak-invitation_FO3151 | 285 | 0.575 | x, y, z | | |
| fong-spivak-invitation_FO3152 | 286 | 0.684 | \mathcal{X}(C, D) | | |
| fong-spivak-invitation_FO3153 | 286 | 0.684 | C \rightarrow | | |
| fong-spivak-invitation_FO3154 | 286 | 1.000 | A \rightarrow B \rightarrow D | | |
| fong-spivak-invitation_FO3155 | 286 | 1.000 | C \rightarrow D | | |
| fong-spivak-invitation_FO3156 | 286 | 0.515 | \mathcal{X}(x, y) \cap \mathcal{X}(y, z) \subseteq \mathcal{X}(x, z) | | |
| fong-spivak-invitation_FO3158 | 286 | 1.000 | \min (M(x, y), M(y, z)) \leq M(x, y) | | |
| fong-spivak-invitation_FO3159 | 286 | 1.000 | x, y, z \in\{A, B, C\} | | |
| fong-spivak-invitation_FO3160 | 286 | 0.507 | d(A, B)=d(B, A)= | | |
| fong-spivak-invitation_FO3161 | 286 | 0.501 | X_{f}=\stackrel{A}{\bullet} \quad \stackrel{B}{\bullet} | | |
| fong-spivak-invitation_FO3162 | 286 | 0.501 | X_{g}=\stackrel{A \longrightarrow}{\bullet} | | |
| fong-spivak-invitation_FO3163 | 287 | 1.000 | d(x, y) \leq d(y, x) | | |
| fong-spivak-invitation_FO3164 | 287 | 1.000 | d(y, x) \leq d(x, y) | | |
| fong-spivak-invitation_FO3165 | 287 | 1.000 | d(y, x)= | | |
| fong-spivak-invitation_FO3166 | 287 | 0.791 | (x, y) \in \mathcal{X} \times \mathcal{Y} | | |
| fong-spivak-invitation_FO3167 | 287 | 0.791 | I \leq y(y, y) | | |
| fong-spivak-invitation_FO3168 | 287 | 0.850 | I=I \otimes I \leq X(x, x) \otimes y(y, y)=(\mathcal{X} \times y)((x, y),(x, y)) | | |
| fong-spivak-invitation_FO3169 | 287 | 0.665 | y\left(y_{1}, y_{2}\right) \otimes \mathcal{X}\left(x_{2}, x_{3}\right)=\mathcal{X}\left(x_{2}, x_{3}\right) \otimes | | |
| fong-spivak-invitation_FO3170 | 287 | 0.998 | y\left(y_{1}, y_{2}\right) | | |
| fong-spivak-invitation_FO3171 | 287 | 0.993 | (\mathbb{R} \times \mathbb{R})((5,6),(-1,4))=\mathbb{R}(5,-1)+\mathbb{R}(6,4)=6+2=8 | | |
| fong-spivak-invitation_FO3172 | 287 | 0.999 | u \leq u^{\prime} | | |
| fong-spivak-invitation_FO3173 | 287 | 0.999 | u \otimes v \leq u^{\prime} \otimes v | | |
| fong-spivak-invitation_FO3174 | 287 | 0.997 | a:=(v \multimap w) | | |
| fong-spivak-invitation_FO3175 | 287 | 0.975 | (v \multimap u) \otimes v \leq u \leq u^{\prime} | | |
| fong-spivak-invitation_FO3176 | 287 | 1.000 | (v \multimap u) \leq\left(v \multimap u^{\prime}\right) | | |
| fong-spivak-invitation_FO3177 | 287 | 1.000 | (v \multimap-): V \rightarrow V | | |
| fong-spivak-invitation_FO3178 | 287 | 0.998 | (v \multimap-) | | |
| fong-spivak-invitation_FO3179 | 288 | 1.000 | x \Rightarrow y | | |
| fong-spivak-invitation_FO3180 | 288 | 0.987 | (a \wedge v) \leq w | | |
| fong-spivak-invitation_FO3181 | 288 | 0.987 | a \leq(v \Rightarrow w) | | |
| fong-spivak-invitation_FO3182 | 288 | 0.987 | v= | | |
| fong-spivak-invitation_FO3183 | 288 | 0.987 | a \wedge | | |
| fong-spivak-invitation_FO3184 | 288 | 0.570 | \Rightarrow w | | |
| fong-spivak-invitation_FO3185 | 288 | 0.672 | v=\operatorname{true} | | |
| fong-spivak-invitation_FO3186 | 288 | 0.672 | =a | | |
| fong-spivak-invitation_FO3187 | 288 | 0.672 | a \leq w | | |
| fong-spivak-invitation_FO3188 | 288 | 0.969 | \Rightarrow w)=w | | |
| fong-spivak-invitation_FO3189 | 288 | 0.672 | (\bigvee \varnothing)=f a l s e | | |
| fong-spivak-invitation_FO3190 | 288 | 0.999 | (\bigvee \varnothing)=\infty | | |
| fong-spivak-invitation_FO3191 | 288 | 1.000 | (\bigvee \varnothing)=0 | | |
| fong-spivak-invitation_FO3192 | 288 | 1.000 | \min (x, y) | | |
| fong-spivak-invitation_FO3193 | 288 | 0.764 | (\mathrm{P}(S), \subseteq, S, \cap) | | |
| fong-spivak-invitation_FO3194 | 288 | 0.764 | B \multimap C | | |
| fong-spivak-invitation_FO3195 | 288 | 1.000 | \bar{B} \cup C | | |
| fong-spivak-invitation_FO3196 | 288 | 1.000 | \bar{B} | | |
| fong-spivak-invitation_FO3197 | 288 | 1.000 | (A \cap B) \subseteq C | | |
| fong-spivak-invitation_FO3198 | 288 | 1.000 | A \subseteq(\bar{B} \cup C) | | |
| fong-spivak-invitation_FO3199 | 288 | 1.000 | v \in V, 0 \otimes v=0 | | |
| fong-spivak-invitation_FO3200 | 289 | 0.998 | \left(v, a_{1}, \ldots, a_{n}\right) | | |
| fong-spivak-invitation_FO3201 | 289 | 0.998 | s\left(a_{1}\right)=v | | |
| fong-spivak-invitation_FO3202 | 290 | 1.000 | t\left(a_{i}\right)=s\left(a_{i+1}\right) | | |
| fong-spivak-invitation_FO3203 | 290 | 1.000 | i \in\{1, \ldots, n-1\} | | |
| fong-spivak-invitation_FO3204 | 290 | 1.000 | s(p):=v | | |
| fong-spivak-invitation_FO3205 | 290 | 0.998 | p=\left(v, a_{1}, \ldots, a_{m}\right) | | |
| fong-spivak-invitation_FO3206 | 290 | 0.998 | q=\left(w, b_{1}, \ldots, b_{n}\right) | | |
| fong-spivak-invitation_FO3207 | 290 | 0.998 | t(p)=s(q) | | |
| fong-spivak-invitation_FO3208 | 290 | 0.991 | p=(v) | | |
| fong-spivak-invitation_FO3209 | 290 | 0.991 | (v) | | |
| fong-spivak-invitation_FO3210 | 290 | 0.991 | \left(w, b_{1}, \ldots, b_{n}\right) | | |
| fong-spivak-invitation_FO3211 | 290 | 1.000 | v=w | | |
| fong-spivak-invitation_FO3212 | 290 | 1.000 | \left(v, a_{1}, \ldots, a_{m}\right) | | |
| fong-spivak-invitation_FO3213 | 290 | 0.896 | (w) | | |
| fong-spivak-invitation_FO3214 | 290 | 0.896 | w=t\left(a_{m}\right) | | |
| fong-spivak-invitation_FO3215 | 290 | 0.940 | r=\left(x, c_{1}, \ldots, c_{o}\right) | | |
| fong-spivak-invitation_FO3216 | 290 | 0.379 | (p ; q){ }_{9}^{\circ} r | | |
| fong-spivak-invitation_FO3217 | 290 | 0.379 | p \circ(q ; r) | | |
| fong-spivak-invitation_FO3219 | 293 | 1.000 | f_{2} | | |
| fong-spivak-invitation_FO3223 | 290 | 1.000 | 1+2+\cdots+n | | |
| fong-spivak-invitation_FO3224 | 290 | 0.759 | n-i | | |
| fong-spivak-invitation_FO3225 | 290 | 1.000 | 0 \leq i \leq n | | |
| fong-spivak-invitation_FO3226 | 291 | 1.000 | m+n | | |
| fong-spivak-invitation_FO3227 | 291 | 0.325 | h=j=A | | |
| fong-spivak-invitation_FO3228 | 291 | 0.312 | z, s, s . s | | |
| fong-spivak-invitation_FO3229 | 291 | 0.312 | s . s . s | | |
| fong-spivak-invitation_FO3230 | 292 | 0.968 | (f(1), f(2)) | | |
| fong-spivak-invitation_FO3231 | 292 | 0.970 | n! | | |
| fong-spivak-invitation_FO3232 | 292 | 0.998 | 5 * 4 * 3 * 2 * 1=120 | | |
| fong-spivak-invitation_FO3233 | 292 | 0.998 | \{1,2,3,4,5\} | | |
| fong-spivak-invitation_FO3234 | 292 | 0.998 | \{a, b, c, d, e\} | | |
| fong-spivak-invitation_FO3235 | 292 | 0.755 | f: c \rightarrow c | | |
| fong-spivak-invitation_FO3236 | 292 | 0.755 | f ; \mathrm{id}_{c}=\mathrm{id}_{c} | | |
| fong-spivak-invitation_FO3237 | 292 | 0.755 | \mathrm{id}_{c} ; f=\mathrm{id}_{c} | | |
| fong-spivak-invitation_FO3238 | 292 | 0.755 | f=\mathrm{id}_{c} | | |
| fong-spivak-invitation_FO3239 | 292 | 1.000 | s^{n} | | |
| fong-spivak-invitation_FO3240 | 292 | 1.000 | s^{n+1} | | |
| fong-spivak-invitation_FO3241 | 292 | 1.000 | s^{0} | | |
| fong-spivak-invitation_FO3242 | 292 | 0.513 | p{ }_{9} q=\mathrm{id} | | |
| fong-spivak-invitation_FO3243 | 292 | 0.384 | q{ }_{9}^{\circ} p=\mathrm{id} | | |
| fong-spivak-invitation_FO3244 | 292 | 0.917 | \mathbf{2} \boldsymbol{\rightarrow} \mathbf{3} | | |
| fong-spivak-invitation_FO3245 | 293 | 0.979 | g^{\prime} ; i^{\prime} \mapsto f ; h | | |
| fong-spivak-invitation_FO3246 | 293 | 0.317 | g^{\prime}{ }_{9} i^{\prime} \mapsto g{ }_{9} i | | |
| fong-spivak-invitation_FO3247 | 293 | 0.317 | g{ }_{9} i=f{ }_{9}^{\circ} h | | |
| fong-spivak-invitation_FO3248 | 293 | 0.421 | F, G | | |
| fong-spivak-invitation_FO3249 | 293 | 0.421 | \stackrel{a}{\rightarrow} \xrightarrow{b} | | |
| fong-spivak-invitation_FO3250 | 293 | 0.421 | \stackrel{a^{\prime}}{\xrightarrow[f_{2}]{f_{1}}} \stackrel{b^{\prime}}{\longrightarrow} | | |
| fong-spivak-invitation_FO3251 | 293 | 0.929 | \operatorname{id}_{\mathcal{C}}: \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO3252 | 293 | 0.929 | \operatorname{id}_{\mathcal{C}}(x)=x | | |
| fong-spivak-invitation_FO3253 | 293 | 0.108 | \operatorname{id}_{\mathcal{C}}\left(\mathrm{id}_{C}\right)=\operatorname{id}_{C}=\operatorname{id}_{\operatorname{id} \mathcal{C}(c)} | | |
| fong-spivak-invitation_FO3254 | 293 | 0.166 | \operatorname{id}_{\mathcal{C}}(f ; g)=f ; g=\operatorname{id}_{\mathcal{C}}(f){ }_{\mathcal{G}} \operatorname{id}_{\mathcal{U}(g)} | | |
| fong-spivak-invitation_FO3255 | 293 | 0.468 | F{ }_{;} G | | |
| fong-spivak-invitation_FO3256 | 293 | 0.970 | (F ; G)\left(\mathrm{id}_{C}\right)= | | |
| fong-spivak-invitation_FO3257 | 293 | 0.452 | G\left(F\left(\mathrm{id}_{C}\right)\right)=G\left(\mathrm{id}_{F(c)}\right)=\mathrm{id}_{G(F(c))} | | |
| fong-spivak-invitation_FO3258 | 293 | 0.904 | \mathrm{id}_{\mathcal{C}} | | |
| fong-spivak-invitation_FO3259 | 293 | 0.987 | \mathrm{id}_{\mathcal{D}} | | |
| fong-spivak-invitation_FO3260 | 293 | 1.000 | H: \mathcal{E} \rightarrow \mathcal{F} | | |
| fong-spivak-invitation_FO3261 | 293 | 0.638 | (F ; G){ }^{\circ} H=F ;(G ; H) | | |
| fong-spivak-invitation_FO3262 | 293 | 0.638 | x \in \mathcal{C} | | |
| fong-spivak-invitation_FO3263 | 293 | 0.999 | F_{S}\left(\mathrm{id}_{1}\right)=\mathrm{id}_{S} | | |
| fong-spivak-invitation_FO3264 | 293 | 0.883 | F_{S} | | |
| fong-spivak-invitation_FO3265 | 293 | 0.883 | \mathbf{1} | | |
| fong-spivak-invitation_FO3266 | 294 | 1.000 | D(z) | | |
| fong-spivak-invitation_FO3267 | 294 | 1.000 | D(s): D(z) \rightarrow | | |
| fong-spivak-invitation_FO3268 | 294 | 0.877 | D_{Z} | | |
| fong-spivak-invitation_FO3269 | 294 | 1.000 | D(c) | | |
| fong-spivak-invitation_FO3270 | 294 | 0.884 | D(b) | | |
| fong-spivak-invitation_FO3271 | 294 | 1.000 | D(a) | | |
| fong-spivak-invitation_FO3272 | 294 | 1.000 | d(f): D(a) \rightarrow D(b) | | |
| fong-spivak-invitation_FO3273 | 294 | 1.000 | F, G, H \in \mathcal{D}^{\mathcal{C}} | | |
| fong-spivak-invitation_FO3274 | 294 | 1.000 | F, G, H: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-invitation_FO3275 | 294 | 1.000 | \beta: G \rightarrow H | | |
| fong-spivak-invitation_FO3276 | 294 | 0.983 | c \in \mathcal{C}, \alpha | | |
| fong-spivak-invitation_FO3277 | 294 | 0.983 | \beta_{c}: G(c) \rightarrow | | |
| fong-spivak-invitation_FO3278 | 294 | 0.606 | H(c) | | |
| fong-spivak-invitation_FO3279 | 294 | 0.606 | (\alpha ; \beta)_{C}:=\left(\alpha_{C} ; \beta_{C}\right) | | |
| fong-spivak-invitation_FO3280 | 294 | 1.000 | F(c) \rightarrow H(c) | | |
| fong-spivak-invitation_FO3281 | 295 | 0.997 | \operatorname{id}_{F} | | |
| fong-spivak-invitation_FO3282 | 295 | 0.599 | \left(\operatorname{id}_{F}\right)_{C}:=\operatorname{id}_{F(c)} | | |
| fong-spivak-invitation_FO3283 | 295 | 0.992 | \mathrm{id}_{F} | | |
| fong-spivak-invitation_FO3284 | 295 | 1.000 | \beta: F \rightarrow G | | |
| fong-spivak-invitation_FO3285 | 295 | 1.000 | \alpha: E \rightarrow F | | |
| fong-spivak-invitation_FO3286 | 295 | 0.093 | \left(\operatorname{id}_{F}\right)_{C} | | |
| fong-spivak-invitation_FO3287 | 295 | 0.093 | \beta_{C}=\left(\operatorname{id}_{F(c)}{ }^{\circ} \beta_{C}\right)=\beta_{C} | | |
| fong-spivak-invitation_FO3288 | 295 | 1.000 | \alpha, \beta: F \rightarrow G | | |
| fong-spivak-invitation_FO3289 | 295 | 0.981 | \alpha=\beta | | |
| fong-spivak-invitation_FO3290 | 295 | 0.981 | \alpha_{c}=\beta_{c} | | |
| fong-spivak-invitation_FO3291 | 295 | 0.728 | \beta_{C} | | |
| fong-spivak-invitation_FO3292 | 295 | 0.728 | F(c) \rightarrow G(c) | | |
| fong-spivak-invitation_FO3293 | 295 | 0.999 | \alpha_{C}=\beta_{C} | | |
| fong-spivak-invitation_FO3294 | 295 | 0.717 | \mathcal{P}:=\mathbf{1} | | |
| fong-spivak-invitation_FO3295 | 295 | 0.717 | \mathcal{C}:=\stackrel{a}{\bullet} \xrightarrow[f_{2}]{f_{1}} \stackrel{b}{\bullet} | | |
| fong-spivak-invitation_FO3296 | 295 | 0.717 | F(1):=a | | |
| fong-spivak-invitation_FO3297 | 295 | 0.717 | G(1):=b | | |
| fong-spivak-invitation_FO3298 | 295 | 0.717 | \alpha_{1}:=f_{1} | | |
| fong-spivak-invitation_FO3299 | 295 | 1.000 | \beta_{1}:=g_{2} | | |
| fong-spivak-invitation_FO3300 | 296 | 1.000 | \alpha_{\text {Arrow }}(b)=e | | |
| fong-spivak-invitation_FO3301 | 296 | 1.000 | \alpha_{\text {Vertex }}(1)=4, \alpha_{\text {Vertex }}(2)=5 | | |
| fong-spivak-invitation_FO3302 | 296 | 1.000 | \alpha_{\text {Vertex }}(3)=5 | | |
| fong-spivak-invitation_FO3303 | 297 | 1.000 | X \times B \rightarrow Y \times B | | |
| fong-spivak-invitation_FO3304 | 297 | 0.965 | f \times B | | |
| fong-spivak-invitation_FO3305 | 297 | 0.965 | (x, b) | | |
| fong-spivak-invitation_FO3306 | 297 | 0.965 | (f(x), b) | | |
| fong-spivak-invitation_FO3307 | 297 | 0.860 | \mathrm{id}_{X \times B} | | |
| fong-spivak-invitation_FO3308 | 297 | 1.000 | X^{B} \rightarrow Y^{B} | | |
| fong-spivak-invitation_FO3309 | 297 | 0.885 | f^{B} | | |
| fong-spivak-invitation_FO3310 | 297 | 0.885 | g: B \rightarrow X | | |
| fong-spivak-invitation_FO3311 | 297 | 0.885 | (g \circ f): B \rightarrow X \rightarrow Y | | |
| fong-spivak-invitation_FO3312 | 297 | 0.984 | f^{B}\left(\mathrm{id}_{X}\right)=\mathrm{id}_{X^{B}} | | |
| fong-spivak-invitation_FO3313 | 297 | 0.773 | \left(f_{1} \circ f_{2}\right): X \rightarrow Y \rightarrow Z | | |
| fong-spivak-invitation_FO3314 | 297 | 1.000 | g \in X^{B} | | |
| fong-spivak-invitation_FO3315 | 297 | 0.993 | \mathbb{N}^{\mathbb{N}} | | |
| fong-spivak-invitation_FO3316 | 297 | 0.974 | p(3): \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-invitation_FO3317 | 297 | 0.974 | p(3)(n):=n+3 | | |
| fong-spivak-invitation_FO3318 | 297 | 0.989 | !: \mathcal{C} \rightarrow \mathbf{1} | | |
| fong-spivak-invitation_FO3319 | 297 | 0.989 | 1 \in \mathbf{1} | | |
| fong-spivak-invitation_FO3320 | 297 | 0.987 | \mathrm{B}=\mathrm{Bob}, 3=\mathrm{Em} \_3 | | |
| fong-spivak-invitation_FO3321 | 297 | 1.000 | c \rightarrow z | | |
| fong-spivak-invitation_FO3322 | 298 | 0.969 | \mathcal{C} \in \mathbf{C a t} | | |
| fong-spivak-invitation_FO3323 | 298 | 0.944 | 2 V:=\bullet \bullet | | |
| fong-spivak-invitation_FO3324 | 298 | 0.944 | \mathcal{C}=\operatorname{Free}(2 V) | | |
| fong-spivak-invitation_FO3325 | 298 | 1.000 | z \in \mathcal{P} | | |
| fong-spivak-invitation_FO3326 | 298 | 1.000 | z \rightarrow x | | |
| fong-spivak-invitation_FO3327 | 298 | 1.000 | z \rightarrow y | | |
| fong-spivak-invitation_FO3328 | 298 | 1.000 | z^{\prime} | | |
| fong-spivak-invitation_FO3329 | 298 | 1.000 | z^{\prime} \rightarrow x | | |
| fong-spivak-invitation_FO3330 | 298 | 1.000 | z^{\prime} \rightarrow y | | |
| fong-spivak-invitation_FO3331 | 298 | 1.000 | z^{\prime} \rightarrow z | | |
| fong-spivak-invitation_FO3332 | 298 | 1.000 | z \leq x | | |
| fong-spivak-invitation_FO3333 | 298 | 1.000 | z \leq y | | |
| fong-spivak-invitation_FO3334 | 298 | 1.000 | z^{\prime} \leq x | | |
| fong-spivak-invitation_FO3335 | 298 | 1.000 | z^{\prime} \leq y | | |
| fong-spivak-invitation_FO3336 | 298 | 1.000 | z^{\prime} \leq z | | |
| fong-spivak-invitation_FO3337 | 298 | 1.000 | z=x \wedge y | | |
| fong-spivak-invitation_FO3338 | 298 | 0.771 | \left(\mathrm{id}_{c}, \mathrm{id}_{d}\right) | | |
| fong-spivak-invitation_FO3339 | 298 | 0.464 | \left(\left(f_{1}, g_{1}\right) \stackrel{\circ}{9}\left(f_{2}, g_{2}\right)\right) \stackrel{\circ}{9}\left(f_{3}, g_{3}\right)=\left(f_{1}, g_{1}\right) \stackrel{\circ}{9}\left(\left(f_{2}, g_{2}\right) \stackrel{\circ}{9}\left(f_{3}, g_{3}\right)\right) | | |
| fong-spivak-invitation_FO3340 | 298 | 0.380 | \left(\left(f_{1} \stackrel{\circ}{9} f_{2}\right) \stackrel{\circ}{9} f_{3},\left(g_{1} \stackrel{\circ}{9} g_{2}\right) \stackrel{\circ}{9} g_{3}\right) | | |
| fong-spivak-invitation_FO3341 | 298 | 0.380 | \left(f_{1} \stackrel{\circ}{9}\left(f_{2} \stackrel{\circ}{9}\right.\right. | | |
| fong-spivak-invitation_FO3342 | 298 | 0.222 | \left.f_{3}\right), g_{1} | | |
| fong-spivak-invitation_FO3343 | 298 | 0.222 | \left(g_{2}\right. | | |
| fong-spivak-invitation_FO3344 | 298 | 0.222 | \left.g_{3}\right) | | |
| fong-spivak-invitation_FO3345 | 298 | 0.994 | (1,1) | | |
| fong-spivak-invitation_FO3346 | 298 | 0.994 | (1,2) | | |
| fong-spivak-invitation_FO3347 | 298 | 0.962 | (1,1) \rightarrow(1,2) | | |
| fong-spivak-invitation_FO3348 | 298 | 0.962 | \mathbf{2} | | |
| fong-spivak-invitation_FO3349 | 298 | 1.000 | \mathbf{1} \times \mathcal{C} \cong \mathcal{C} | | |
| fong-spivak-invitation_FO3350 | 298 | 1.000 | X=P \times Q | | |
| fong-spivak-invitation_FO3351 | 298 | 0.983 | \mathcal{X}=\mathcal{P} \times Q | | |
| fong-spivak-invitation_FO3352 | 298 | 0.930 | X \stackrel{p_{X}}{\longleftarrow} Z \xrightarrow{p_{Y}} Y | | |
| fong-spivak-invitation_FO3353 | 298 | 0.996 | X \stackrel{p_{X}^{\prime}}{\longleftarrow} Z^{\prime} \xrightarrow{p_{Y}^{\prime}} Y | | |
| fong-spivak-invitation_FO3354 | 298 | 0.832 | f: Z^{\prime} \rightarrow Z | | |
| fong-spivak-invitation_FO3355 | 298 | 0.832 | f{ }_{9} p_{X}=p_{X}^{\prime} | | |
| fong-spivak-invitation_FO3356 | 298 | 0.832 | f{ }_{9} p_{Y}=p_{Y}^{\prime} | | |
| fong-spivak-invitation_FO3357 | 298 | 0.939 | Z \in \operatorname{Cone}(X, Y) | | |
| fong-spivak-invitation_FO3358 | 298 | 0.822 | Z^{\prime} \rightarrow Z | | |
| fong-spivak-invitation_FO3359 | 298 | 0.108 | \stackrel{v_{1}}{\bullet} | | |
| fong-spivak-invitation_FO3360 | 298 | 0.108 | { }^{v} | | |
| fong-spivak-invitation_FO3361 | 298 | 0.108 | D\left(v_{1}\right)=X | | |
| fong-spivak-invitation_FO3362 | 298 | 0.108 | D\left(v_{2}\right)=Y | | |
| fong-spivak-invitation_FO3363 | 299 | 0.998 | \lim _{\mathcal{J}} D:=\left\{\left(d_{1}, \ldots, d_{n}\right) \mid d_{i} \in D\left(v_{i}\right)\right. | | |
| fong-spivak-invitation_FO3364 | 299 | 0.998 | 1 \leq i \leq n | | |
| fong-spivak-invitation_FO3365 | 299 | 0.902 | a: v_{i} \rightarrow v_{j} \in A | | |
| fong-spivak-invitation_FO3366 | 299 | 0.902 | \left.D(a)\left(d_{i}\right)=d_{j}\right\} | | |
| fong-spivak-invitation_FO3367 | 299 | 0.720 | F^{\mathrm{op}}: \mathcal{\mathcal { C } ^ { \mathrm { op } }} \rightarrow \mathcal{D}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO3368 | 299 | 0.379 | c \in \mathrm{Ob}\left(\mathcal{C}^{\mathrm{OP}}\right)=\mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO3369 | 299 | 0.379 | F^{\mathrm{op}}(c):=F(c) | | |
| fong-spivak-invitation_FO3370 | 299 | 1.000 | f^{\prime}: c_{2} \rightarrow c_{1} | | |
| fong-spivak-invitation_FO3371 | 299 | 1.000 | F\left(f^{\prime}\right): F\left(c_{2}\right) \rightarrow F\left(c_{1}\right) | | |
| fong-spivak-invitation_FO3372 | 299 | 1.000 | F\left(f^{\prime}\right)^{\prime}: F^{\mathrm{op}}\left(c_{1}\right) \rightarrow F^{\mathrm{op}}\left(c_{2}\right) | | |
| fong-spivak-invitation_FO3373 | 299 | 1.000 | F^{\mathrm{op}}(f):=F\left(f^{\prime}\right)^{\prime} | | |
| fong-spivak-invitation_FO3374 | 299 | 0.702 | \left(-{ }^{\prime}\right) | | |
| fong-spivak-invitation_FO3375 | 299 | 0.999 | F^{\mathrm{op}} | | |
| fong-spivak-invitation_FO3376 | 299 | 0.588 | \mathscr{X}^{\mathrm{op}} \times \mathrm{y} | | |
| fong-spivak-invitation_FO3377 | 299 | 0.705 | \mathscr{X}^{\mathrm{OP}} \times y \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO3378 | 299 | 0.698 | \mathcal{X}^{\mathrm{op}} \times \mathrm{y} | | |
| fong-spivak-invitation_FO3379 | 299 | 1.000 | \Lambda | | |
| fong-spivak-invitation_FO3380 | 300 | 0.534 | \Phi: \mathscr{X}^{\mathrm{OP}} \times \mathcal{Y} \rightarrow \mathcal{V} | | |
| fong-spivak-invitation_FO3381 | 300 | 0.534 | \Phi: \operatorname{Ob}\left(\mathcal{X}^{\mathrm{OP}} \times \mathcal{Y}\right) \rightarrow | | |
| fong-spivak-invitation_FO3382 | 301 | 0.995 | \Phi:(T \times E) \longrightarrow \$ | | |
| fong-spivak-invitation_FO3383 | 301 | 0.996 | \Phi(( | | |
| fong-spivak-invitation_FO3384 | 301 | 0.996 | ), p)= | | |
| fong-spivak-invitation_FO3385 | 301 | 1.000 | p \in \$=\{\$ 100 \mathrm{~K}, \$ 500 \mathrm{~K}, \$ 1 \mathrm{M}\} | | |
| fong-spivak-invitation_FO3386 | 301 | 0.932 | (\Phi: \Psi)(-,-)= | | |
| fong-spivak-invitation_FO3387 | 301 | 0.753 | X(-, D)+9+Z(r,-) | | |
| fong-spivak-invitation_FO3389 | 302 | 0.951 | U_{\mathcal{X}} | | |
| fong-spivak-invitation_FO3390 | 302 | 0.971 | I \leq \mathcal{P}(p, p) | | |
| fong-spivak-invitation_FO3391 | 302 | 1.000 | \Phi(p, q) \leq \Phi(p, q) | | |
| fong-spivak-invitation_FO3392 | 302 | 1.000 | i_{0} \in I | | |
| fong-spivak-invitation_FO3393 | 302 | 1.000 | s_{i \in I} | | |
| fong-spivak-invitation_FO3394 | 302 | 0.998 | s_{i_{0}} \leq \bigvee_{i \in I} s | | |
| fong-spivak-invitation_FO3395 | 302 | 0.998 | I i_{0}=p | | |
| fong-spivak-invitation_FO3396 | 302 | 0.998 | s_{i}=\mathcal{P}(p, i) \otimes \Phi | | |
| fong-spivak-invitation_FO3397 | 302 | 0.558 | I=\operatorname{true} | | |
| fong-spivak-invitation_FO3398 | 302 | 0.558 | \operatorname{true} \leq \mathcal{P}(p, p) | | |
| fong-spivak-invitation_FO3399 | 302 | 0.912 | \mathcal{P}(p, p)=\operatorname{tr} | | |
| fong-spivak-invitation_FO3400 | 302 | 0.888 | \Phi(p, q) | | |
| fong-spivak-invitation_FO3401 | 302 | 0.888 | \Phi(p, q)=\operatorname{true} | | |
| fong-spivak-invitation_FO3402 | 302 | 0.449 | \Phi(p, q)= | | |
| fong-spivak-invitation_FO3403 | 302 | 0.449 | \mathcal{P}^{\mathrm{OP}} \times Q \rightarrow | | |
| fong-spivak-invitation_FO3404 | 302 | 0.987 | p \leq p_{1} | | |
| fong-spivak-invitation_FO3405 | 302 | 0.987 | \Phi\left(p_{1}, q\right) \leq \Phi(p, q) | | |
| fong-spivak-invitation_FO3406 | 302 | 0.987 | \mathcal{P}\left(p, p_{1}\right)=\operatorname{true} | | |
| fong-spivak-invitation_FO3407 | 302 | 0.987 | \Phi\left(p_{1}, q\right)= | | |
| fong-spivak-invitation_FO3408 | 302 | 0.672 | \mathcal{P}\left(p, p_{1}\right)= | | |
| fong-spivak-invitation_FO3409 | 302 | 0.775 | \bigvee_{p_{1} \in \mathcal{P}} \mathcal{P}\left(p, p_{1}\right) \wedge \Phi\left(p_{1}, q\right)=\bigvee_{p_{1} \in \mathcal{P}} | | |
| fong-spivak-invitation_FO3410 | 302 | 1.000 | \Phi(p, q)=\bigvee_{p_{1} \in \mathcal{P}} \mathcal{P}\left(p, p_{1}\right) \wedge \Phi\left(p_{1}, q\right) | | |
| fong-spivak-invitation_FO3411 | 302 | 1.000 | v \otimes I=v | | |
| fong-spivak-invitation_FO3412 | 302 | 1.000 | I \leq Q(q, q) | | |
| fong-spivak-invitation_FO3413 | 302 | 1.000 | v_{1} \leq v_{2} | | |
| fong-spivak-invitation_FO3414 | 302 | 1.000 | v \otimes v_{1} \leq v \otimes v_{2} | | |
| fong-spivak-invitation_FO3415 | 303 | 0.888 | \widehat{\mathrm{id}}: \mathcal{P} \longrightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO3416 | 303 | 0.888 | \widehat{\mathrm{id}}(p, q)=\mathcal{P}(p, q) | | |
| fong-spivak-invitation_FO3417 | 303 | 0.928 | \widehat{\operatorname{id}}(p, q):=\mathcal{P}(\operatorname{id}(p), q)=\mathcal{P}(p, q) | | |
| fong-spivak-invitation_FO3418 | 303 | 0.743 | \check{+}: \mathbb{R} \multimap \mathbb{R} \times \mathbb{R} \times \mathbb{R} | | |
| fong-spivak-invitation_FO3419 | 303 | 0.743 | \mathbb{R}(a, b+c+d) | | |
| fong-spivak-invitation_FO3420 | 303 | 0.743 | a \leq | | |
| fong-spivak-invitation_FO3421 | 303 | 1.000 | b+c+d | | |
| fong-spivak-invitation_FO3422 | 303 | 0.863 | \widehat{F}(p, q)=Q(F(p), q) | | |
| fong-spivak-invitation_FO3423 | 303 | 0.863 | \check{G}(p, q)=Q(p, G(q)) | | |
| fong-spivak-invitation_FO3424 | 303 | 0.766 | Q(F(p), q)=Q(p, G(q)) | | |
| fong-spivak-invitation_FO3425 | 303 | 0.948 | \widehat{F}=\check{G} | | |
| fong-spivak-invitation_FO3426 | 303 | 0.999 | \otimes: \mathcal{P} \times \mathcal{P} \rightarrow \mathcal{P} | | |
| fong-spivak-invitation_FO3427 | 303 | 0.999 | \left(x_{1}, x_{2}\right) \leq\left(y_{1}, y_{2}\right) | | |
| fong-spivak-invitation_FO3428 | 303 | 0.999 | \mathcal{P} \otimes \mathcal{P} | | |
| fong-spivak-invitation_FO3429 | 303 | 1.000 | 2.1 \otimes | | |
| fong-spivak-invitation_FO3430 | 303 | 0.998 | g_{E}(5,3)= | | |
| fong-spivak-invitation_FO3431 | 303 | 0.998 | g_{F}(5,3)=2 | | |
| fong-spivak-invitation_FO3432 | 303 | 0.976 | g_{E}(3,5)= | | |
| fong-spivak-invitation_FO3433 | 303 | 0.976 | g_{F}(3,5)=-2 | | |
| fong-spivak-invitation_FO3434 | 304 | 0.733 | )=5 | | |
| fong-spivak-invitation_FO3435 | 304 | 0.885 | )=-5 | | |
| fong-spivak-invitation_FO3436 | 304 | 0.693 | )=6 | | |
| fong-spivak-invitation_FO3437 | 304 | 1.000 | q_{G}(-2,3)=2, q_{F}(-2,3)=-13 | | |
| fong-spivak-invitation_FO3438 | 304 | 1.000 | q_{G}(2,3)=-1, q_{F}(2,3)=7 | | |
| fong-spivak-invitation_FO3439 | 304 | 0.970 | \mathcal{C}(x, y) | | |
| fong-spivak-invitation_FO3440 | 304 | 0.483 | \mathrm{id}_{x}: I \rightarrow \mathcal{C}(x, x) | | |
| fong-spivak-invitation_FO3441 | 304 | 0.483 | \mathrm{id}_{x}:\{1\} \rightarrow \mathcal{C}(x, x) | | |
| fong-spivak-invitation_FO3442 | 304 | 0.919 | \mathrm{id}_{x}=\mathrm{id}_{x}(1) \in \mathcal{C}(x, x) | | |
| fong-spivak-invitation_FO3443 | 304 | 0.960 | { }_{9}: \mathcal{C}(x, y) \otimes \mathcal{C}(y, z) \rightarrow \mathcal{C}(x, z) | | |
| fong-spivak-invitation_FO3444 | 304 | 0.960 | { }_{9}: \mathcal{C}(x, y) \times \mathcal{C}(y, z) \rightarrow \mathcal{C}(x, z) | | |
| fong-spivak-invitation_FO3445 | 304 | 0.662 | I \rightarrow \mathcal{X}(x, x) | | |
| fong-spivak-invitation_FO3446 | 304 | 0.662 | =([0, \infty], \geq | | |
| fong-spivak-invitation_FO3447 | 304 | 0.802 | 0 \geq \mathfrak{X}(x, x) | | |
| fong-spivak-invitation_FO3448 | 304 | 0.802 | \mathcal{X}(x, x)=0 | | |
| fong-spivak-invitation_FO3449 | 305 | 0.801 | x \in \mathcal{X} | | |
| fong-spivak-invitation_FO3450 | 305 | 0.801 | y \in \mathcal{Y} | | |
| fong-spivak-invitation_FO3451 | 305 | 0.801 | (x, y) \leq\left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-invitation_FO3452 | 305 | 0.801 | y \leq y^{\prime} | | |
| fong-spivak-invitation_FO3453 | 305 | 0.920 | \Phi: X_{1} \mapsto X_{2} | | |
| fong-spivak-invitation_FO3454 | 305 | 1.000 | y_{1} | | |
| fong-spivak-invitation_FO3455 | 305 | 1.000 | x_{1} | | |
| fong-spivak-invitation_FO3456 | 305 | 1.000 | y_{2} | | |
| fong-spivak-invitation_FO3457 | 305 | 0.627 | x_{2} | | |
| fong-spivak-invitation_FO3458 | 305 | 0.627 | \left(y_{1}, y_{2}\right) | | |
| fong-spivak-invitation_FO3459 | 305 | 0.627 | \left(x_{1}, x_{2}\right) | | |
| fong-spivak-invitation_FO3460 | 305 | 0.398 | \mathcal{X} \times \mathbf{1}-\mathfrak{X} | | |
| fong-spivak-invitation_FO3461 | 305 | 0.398 | \alpha:(\mathcal{X} \times \mathbf{1})^{\mathrm{op}} \times \mathcal{X} \rightarrow \mathcal{V} | | |
| fong-spivak-invitation_FO3462 | 305 | 0.398 | \alpha((x, 1), y):= | | |
| fong-spivak-invitation_FO3463 | 305 | 0.732 | X(x, y) | | |
| fong-spivak-invitation_FO3464 | 305 | 0.732 | \alpha^{-1}: X \mapsto X \times \mathbf{1} | | |
| fong-spivak-invitation_FO3465 | 305 | 0.732 | \alpha^{-1}(x,(y, 1)):=X(x, y) | | |
| fong-spivak-invitation_FO3466 | 305 | 0.844 | \alpha^{-1} ; \alpha=\mathrm{U}_{\mathcal{X}} | | |
| fong-spivak-invitation_FO3467 | 305 | 0.753 | \alpha ; \alpha^{-1}=\mathrm{U}_{X \times \mathbf{1}} | | |
| fong-spivak-invitation_FO3468 | 305 | 0.984 | \beta((1, x), y):=\mathcal{X}(x, y) | | |
| fong-spivak-invitation_FO3469 | 305 | 0.984 | \beta: \mathbf{1} \times | | |
| fong-spivak-invitation_FO3470 | 305 | 0.641 | x \rightarrow x | | |
| fong-spivak-invitation_FO3471 | 306 | 0.823 | \alpha^{-1} | | |
| fong-spivak-invitation_FO3472 | 306 | 0.997 | \Phi(x, y) | | |
| fong-spivak-invitation_FO3473 | 306 | 0.443 | \mathscr{X}^{\mathrm{OP}} \times \mathcal{X} | | |
| fong-spivak-invitation_FO3474 | 306 | 0.996 | \mathrm{U}_{\mathcal{X}}(a, b)=\mathcal{X}(a, b) | | |
| fong-spivak-invitation_FO3475 | 306 | 1.000 | m \rightarrow p | | |
| fong-spivak-invitation_FO3476 | 306 | 1.000 | (f ; g)(i)=g(f(i)) | | |
| fong-spivak-invitation_FO3477 | 306 | 1.000 | 1 \leq i \leq m | | |
| fong-spivak-invitation_FO3478 | 306 | 0.426 | \operatorname{id}_{m}: m \rightarrow m | | |
| fong-spivak-invitation_FO3479 | 306 | 0.426 | \operatorname{id}_{m}(i)=i | | |
| fong-spivak-invitation_FO3480 | 306 | 0.426 | \operatorname{id}_{2} | | |
| fong-spivak-invitation_FO3481 | 306 | 1.000 | \mathrm{id}_{8} | | |
| fong-spivak-invitation_FO3482 | 307 | 1.000 | \sigma_{3,5}: 8 \rightarrow 8 | | |
| fong-spivak-invitation_FO3483 | 307 | 0.996 | m \preceq n | | |
| fong-spivak-invitation_FO3484 | 307 | 0.996 | m_{1} \preceq n_{1} | | |
| fong-spivak-invitation_FO3485 | 307 | 0.996 | m_{2} \preceq n_{2} | | |
| fong-spivak-invitation_FO3486 | 307 | 1.000 | m_{1}+m_{2} \preceq n_{1}+n_{2} | | |
| fong-spivak-invitation_FO3487 | 307 | 1.000 | d \in \mathbb{N} | | |
| fong-spivak-invitation_FO3488 | 307 | 1.000 | d+m=n | | |
| fong-spivak-invitation_FO3489 | 307 | 0.999 | m=n+d | | |
| fong-spivak-invitation_FO3490 | 307 | 1.000 | q \in \mathbb{N} | | |
| fong-spivak-invitation_FO3491 | 307 | 1.000 | m * q=d | | |
| fong-spivak-invitation_FO3492 | 307 | 1.000 | 2 \preceq 4 | | |
| fong-spivak-invitation_FO3493 | 307 | 0.958 | 3 \preceq 3 | | |
| fong-spivak-invitation_FO3494 | 307 | 0.958 | 5 \preceq^{?} 7 | | |
| fong-spivak-invitation_FO3495 | 307 | 0.492 | \mathbf{B i j}(m, n):=\{f: \underline{m} \rightarrow \underline{n} \mid f | | |
| fong-spivak-invitation_FO3496 | 307 | 0.492 | \mathbf{B i j}(m, n)=\varnothing | | |
| fong-spivak-invitation_FO3497 | 307 | 1.000 | n \rightarrow n | | |
| fong-spivak-invitation_FO3498 | 307 | 1.000 | \underline{n} \rightarrow \underline{n} | | |
| fong-spivak-invitation_FO3499 | 307 | 1.000 | i \mapsto i | | |
| fong-spivak-invitation_FO3500 | 307 | 1.000 | m+n \rightarrow n+m | | |
| fong-spivak-invitation_FO3501 | 307 | 1.000 | \sigma_{m, n}: \underline{m+n} \rightarrow \underline{n+m} | | |
| fong-spivak-invitation_FO3502 | 307 | 1.000 | n \rightarrow p | | |
| fong-spivak-invitation_FO3503 | 307 | 1.000 | (f+g):(m+ | | |
| fong-spivak-invitation_FO3504 | 307 | 1.000 | n) \rightarrow\left(m^{\prime}+n^{\prime}\right) | | |
| fong-spivak-invitation_FO3505 | 307 | 1.000 | i \sim j | | |
| fong-spivak-invitation_FO3506 | 307 | 1.000 | i=j | | |
| fong-spivak-invitation_FO3507 | 308 | 0.868 | \underline{m+n+n+m} | | |
| fong-spivak-invitation_FO3508 | 308 | 1.000 | |i-j|=m+n+n | | |
| fong-spivak-invitation_FO3509 | 308 | 0.996 | m+1 \leq i \leq m+n+n | | |
| fong-spivak-invitation_FO3510 | 308 | 0.996 | m+1 \leq j \leq m+n+n | | |
| fong-spivak-invitation_FO3511 | 308 | 0.996 | |i-j|=n | | |
| fong-spivak-invitation_FO3512 | 308 | 0.808 | \underline{m+n} | | |
| fong-spivak-invitation_FO3513 | 308 | 0.808 | \sim^{\prime} | | |
| fong-spivak-invitation_FO3514 | 308 | 0.808 | \underline{m^{\prime}+n^{\prime}} | | |
| fong-spivak-invitation_FO3515 | 308 | 0.777 | \left(\sim+\sim^{\prime}\right) | | |
| fong-spivak-invitation_FO3516 | 308 | 0.777 | m+n+m^{\prime}+n^{\prime} | | |
| fong-spivak-invitation_FO3517 | 308 | 0.990 | \quad i \leq m+n | | |
| fong-spivak-invitation_FO3518 | 308 | 0.990 | j \leq m+n | | |
| fong-spivak-invitation_FO3519 | 308 | 0.793 | m+n+1 \leq i | | |
| fong-spivak-invitation_FO3520 | 308 | 0.793 | m+n+1 \leq j | | |
| fong-spivak-invitation_FO3521 | 308 | 0.793 | i \sim^{\prime} j | | |
| fong-spivak-invitation_FO3522 | 308 | 0.998 | \boldsymbol{\operatorname { R e l }}(m, n) | | |
| fong-spivak-invitation_FO3523 | 308 | 0.998 | \underline{m} \times \underline{n} | | |
| fong-spivak-invitation_FO3524 | 308 | 1.000 | \{(i, j) \in \underline{n} \times \underline{n} \mid i=j\} | | |
| fong-spivak-invitation_FO3525 | 308 | 1.000 | (i, j) \in(\underline{m+n}) \times(\underline{n+m}) | | |
| fong-spivak-invitation_FO3526 | 308 | 0.998 | \quad i \leq m | | |
| fong-spivak-invitation_FO3527 | 308 | 0.998 | m+1 \leq j | | |
| fong-spivak-invitation_FO3528 | 308 | 0.998 | i+m=j | | |
| fong-spivak-invitation_FO3529 | 308 | 0.974 | \quad m+1 \leq i | | |
| fong-spivak-invitation_FO3530 | 308 | 0.974 | j \leq m | | |
| fong-spivak-invitation_FO3531 | 308 | 0.974 | j+m=i | | |
| fong-spivak-invitation_FO3532 | 308 | 0.667 | R^{\prime} \subseteq \underline{m}^{\prime} \times \underline{n^{\prime}} | | |
| fong-spivak-invitation_FO3533 | 308 | 0.667 | \left(R+R^{\prime}\right) \subseteq | | |
| fong-spivak-invitation_FO3534 | 308 | 1.000 | \underline{m+m^{\prime}} \times \underline{n+n^{\prime}} | | |
| fong-spivak-invitation_FO3535 | 308 | 0.580 | R_{1} \sqcup R_{2} \subseteq\left(\underline{m_{1}} \times \underline{n_{1}}\right) \sqcup\left(\underline{m_{2}} \times \underline{n_{2}}\right) \subseteq\left(\underline{m_{1}} \sqcup \underline{m_{2}}\right) \times\left(\underline{n_{1} \sqcup n_{2}}\right) | | |
| fong-spivak-invitation_FO3536 | 308 | 0.818 | (n, p) | | |
| fong-spivak-invitation_FO3537 | 309 | 1.000 | \leq_{P} | | |
| fong-spivak-invitation_FO3538 | 309 | 0.999 | x \leq_{P} y | | |
| fong-spivak-invitation_FO3539 | 309 | 0.993 | x_{0}, \ldots, x_{n} | | |
| fong-spivak-invitation_FO3540 | 309 | 1.000 | x_{0}=x | | |
| fong-spivak-invitation_FO3541 | 309 | 1.000 | x_{n}=y | | |
| fong-spivak-invitation_FO3542 | 309 | 1.000 | R\left(x_{i}, x_{i+1}\right) | | |
| fong-spivak-invitation_FO3543 | 309 | 1.000 | 0 \leq i \leq n-1 | | |
| fong-spivak-invitation_FO3544 | 309 | 1.000 | f\left(x_{i}\right) \leq_{Q} f\left(x_{i+1}\right) | | |
| fong-spivak-invitation_FO3545 | 309 | 1.000 | f\left(x_{0}\right) \leq_{Q} f\left(x_{i}\right) | | |
| fong-spivak-invitation_FO3546 | 309 | 0.725 | x \leq \leq_{P} y | | |
| fong-spivak-invitation_FO3547 | 309 | 0.945 | n=1, x_{0}=x | | |
| fong-spivak-invitation_FO3548 | 309 | 0.983 | R(g(a), g(b)) | | |
| fong-spivak-invitation_FO3549 | 309 | 0.983 | a \leq_{Q} b | | |
| fong-spivak-invitation_FO3550 | 309 | 1.000 | (g(a), g(b)) \notin R | | |
| fong-spivak-invitation_FO3551 | 309 | 1.000 | Q:=\{1\} | | |
| fong-spivak-invitation_FO3552 | 309 | 1.000 | P:=\{1\} | | |
| fong-spivak-invitation_FO3553 | 309 | 1.000 | R=\varnothing | | |
| fong-spivak-invitation_FO3554 | 309 | 1.000 | (g(1), g(1)) \notin R | | |
| fong-spivak-invitation_FO3555 | 309 | 0.991 | q \in \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-invitation_FO3556 | 309 | 0.997 | q: y \rightarrow z | | |
| fong-spivak-invitation_FO3557 | 309 | 0.997 | \operatorname{cod}(q):=z | | |
| fong-spivak-invitation_FO3558 | 309 | 1.000 | F: \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO3559 | 309 | 0.990 | \mathrm{Ob}(\mathcal{G}) \rightarrow \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO3560 | 309 | 0.990 | \mathrm{Ob}(\mathcal{G})=V | | |
| fong-spivak-invitation_FO3561 | 309 | 1.000 | \operatorname{Mor}(\mathcal{G}) | | |
| fong-spivak-invitation_FO3562 | 309 | 1.000 | A \subseteq \operatorname{Mor}(\mathcal{G}) | | |
| fong-spivak-invitation_FO3563 | 310 | 1.000 | \operatorname{Mor}(\mathcal{G}) \rightarrow \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-invitation_FO3564 | 310 | 1.000 | \operatorname{dom}(F(r))=F(\operatorname{dom}(r)) | | |
| fong-spivak-invitation_FO3565 | 310 | 1.000 | \operatorname{cod}(F(r))=F(\operatorname{cod}(r)) | | |
| fong-spivak-invitation_FO3566 | 310 | 1.000 | r: w \rightarrow x | | |
| fong-spivak-invitation_FO3567 | 310 | 0.976 | r \in A | | |
| fong-spivak-invitation_FO3568 | 310 | 0.976 | \operatorname{dom}(r)=s(r) | | |
| fong-spivak-invitation_FO3569 | 310 | 0.976 | \operatorname{cod}(r)=t(r) | | |
| fong-spivak-invitation_FO3570 | 310 | 1.000 | p:=\left(v_{0}, a_{1}, a_{2}, \ldots, a_{n}\right) | | |
| fong-spivak-invitation_FO3571 | 310 | 1.000 | v_{0} \in V, a_{i} \in A, v_{0}=s\left(a_{1}\right) | | |
| fong-spivak-invitation_FO3572 | 310 | 1.000 | 1 \leq i \leq n-1 | | |
| fong-spivak-invitation_FO3573 | 310 | 1.000 | g\left(a_{i}\right) | | |
| fong-spivak-invitation_FO3574 | 310 | 1.000 | f\left(v_{0}\right) | | |
| fong-spivak-invitation_FO3575 | 310 | 1.000 | g\left(a_{i+1}\right) | | |
| fong-spivak-invitation_FO3576 | 310 | 0.442 | F(p):=\operatorname{id}_{f\left(v_{0}\right)} ; g\left(a_{1}\right) ; \cdots ; g\left(a_{n}\right) | | |
| fong-spivak-invitation_FO3577 | 310 | 1.000 | \left(f^{\prime}, g^{\prime}\right) | | |
| fong-spivak-invitation_FO3578 | 310 | 0.985 | f=f^{\prime} | | |
| fong-spivak-invitation_FO3579 | 310 | 0.985 | g=g^{\prime} | | |
| fong-spivak-invitation_FO3580 | 310 | 1.000 | F^{\prime}: \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO3581 | 310 | 1.000 | F^{\prime} | | |
| fong-spivak-invitation_FO3582 | 310 | 0.366 | \operatorname{Mor}(\mathcal{C}), \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-invitation_FO3583 | 310 | 0.366 | U(\mathcal{C}) \in | | |
| fong-spivak-invitation_FO3584 | 310 | 1.000 | a^{i} * a^{j}=a^{i+j} | | |
| fong-spivak-invitation_FO3585 | 310 | 1.000 | a^{i} \mapsto i | | |
| fong-spivak-invitation_FO3586 | 310 | 0.600 | \{a, b\} | | |
| fong-spivak-invitation_FO3587 | 310 | 0.714 | \boldsymbol{\operatorname { F r e e }}(G, s, t) | | |
| fong-spivak-invitation_FO3588 | 310 | 0.698 | (\Gamma, \ell) | | |
| fong-spivak-invitation_FO3589 | 311 | 1.000 | \ell(v) \in G | | |
| fong-spivak-invitation_FO3590 | 311 | 0.512 | \Gamma | | |
| fong-spivak-invitation_FO3591 | 311 | 0.186 | \left(f+\mathrm{id}_{1}+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\sigma+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+h\right) \stackrel{\circ}{\sigma} \circ g | | |
| fong-spivak-invitation_FO3592 | 311 | 0.990 | G=\{f: 1 \rightarrow 1, g: 2 \rightarrow 2, h: 2 \rightarrow 1\}: | | |
| fong-spivak-invitation_FO3593 | 311 | 0.999 | \{\{1\},\{2\},\{3\}\} | | |
| fong-spivak-invitation_FO3594 | 311 | 0.554 | (R, 0,+, 1, *) | | |
| fong-spivak-invitation_FO3595 | 311 | 1.000 | n=4 | | |
| fong-spivak-invitation_FO3596 | 311 | 1.000 | A, B \in \operatorname{Mat}_{2}(\mathbb{N}) | | |
| fong-spivak-invitation_FO3597 | 311 | 1.000 | A * B \neq | | |
| fong-spivak-invitation_FO3598 | 311 | 1.000 | B * A | | |
| fong-spivak-invitation_FO3599 | 311 | 0.937 | (A * | | |
| fong-spivak-invitation_FO3600 | 311 | 1.000 | B)(1,1)=0 * 0+1 * 1=1 | | |
| fong-spivak-invitation_FO3601 | 311 | 1.000 | (B * A)(1,1)=0 * 0+0 * 0=0 | | |
| fong-spivak-invitation_FO3602 | 312 | 0.997 | (16 x+4 y, x+4 y) | | |
| fong-spivak-invitation_FO3603 | 312 | 0.673 | A=\left(\begin{array}{lll}3 & 3 & 1 \\ 2 & 0 & 4\end{array}\right) | | |
| fong-spivak-invitation_FO3604 | 312 | 0.673 | B=\left(\begin{array}{llll}2 & 5 & 6 & 1\end{array}\right) | | |
| fong-spivak-invitation_FO3605 | 313 | 1.000 | c_{n} | | |
| fong-spivak-invitation_FO3606 | 313 | 1.000 | s_{a}: 1 \rightarrow 1 | | |
| fong-spivak-invitation_FO3607 | 313 | 1.000 | g_{3}: m \times n \rightarrow m \times n | | |
| fong-spivak-invitation_FO3608 | 313 | 1.000 | (i-1) m+j | | |
| fong-spivak-invitation_FO3609 | 313 | 1.000 | (j-1) n+i | | |
| fong-spivak-invitation_FO3610 | 313 | 1.000 | 1 \leq j \leq m | | |
| fong-spivak-invitation_FO3611 | 313 | 0.999 | a_{m} | | |
| fong-spivak-invitation_FO3612 | 313 | 0.528 | g=g_{1} g_{2} | | |
| fong-spivak-invitation_FO3613 | 313 | 0.735 | g_{3} | | |
| fong-spivak-invitation_FO3614 | 313 | 0.735 | g_{4}: m \rightarrow n | | |
| fong-spivak-invitation_FO3615 | 315 | 0.723 | (\mu \otimes \mathrm{id}) ; \mu=(\mathrm{id} \otimes \mu) \stackrel{\circ}{9} \mu | | |
| fong-spivak-invitation_FO3616 | 315 | 1.000 | R^{0}=\{()\} | | |
| fong-spivak-invitation_FO3617 | 315 | 0.549 | \operatorname{prop}(\operatorname{Mat}(R) | | |
| fong-spivak-invitation_FO3618 | 315 | 0.927 | R^{0} | | |
| fong-spivak-invitation_FO3619 | 315 | 0.927 | \{1\} . U | | |
| fong-spivak-invitation_FO3620 | 315 | 1.000 | R^{m} \times R^{n} \cong R^{m+n} | | |
| fong-spivak-invitation_FO3621 | 315 | 0.876 | (m, \mu, \eta) | | |
| fong-spivak-invitation_FO3622 | 315 | 0.876 | m \in \mathbb{N}, \mu: m+m \rightarrow m | | |
| fong-spivak-invitation_FO3623 | 315 | 0.876 | \eta: 0 \rightarrow m | | |
| fong-spivak-invitation_FO3624 | 315 | 0.896 | \mu(\eta, x)=x=\mu(x, \eta) | | |
| fong-spivak-invitation_FO3625 | 315 | 0.896 | \mu(x, \mu(y, z))=\mu(\mu(x, y), z) | | |
| fong-spivak-invitation_FO3626 | 315 | 1.000 | 0 \rightarrow m | | |
| fong-spivak-invitation_FO3627 | 315 | 1.000 | m \in \mathbb{N} | | |
| fong-spivak-invitation_FO3628 | 315 | 1.000 | m=2, \mu | | |
| fong-spivak-invitation_FO3629 | 315 | 1.000 | U(\eta): R^{0} \rightarrow R^{m} | | |
| fong-spivak-invitation_FO3630 | 315 | 1.000 | U(\eta)(1):= | | |
| fong-spivak-invitation_FO3631 | 315 | 0.508 | (0, \ldots, 0) | | |
| fong-spivak-invitation_FO3632 | 315 | 0.508 | U(\mu): R^{m} \times R^{m} \rightarrow R^{m} | | |
| fong-spivak-invitation_FO3633 | 315 | 1.000 | R^{m} | | |
| fong-spivak-invitation_FO3634 | 315 | 0.510 | (1, \circ,-\mathrm{c}) | | |
| fong-spivak-invitation_FO3635 | 315 | 0.510 | (5,3) \mapsto 8 | | |
| fong-spivak-invitation_FO3636 | 316 | 0.732 | \mathrm{B}(-<) | | |
| fong-spivak-invitation_FO3637 | 316 | 0.732 | -<: 1 \rightarrow 2 | | |
| fong-spivak-invitation_FO3638 | 316 | 0.732 | \left\{(x, y, z) \in R^{3} \mid\right. | | |
| fong-spivak-invitation_FO3639 | 316 | 0.982 | x=y+z\} | | |
| fong-spivak-invitation_FO3640 | 316 | 0.141 | \mathrm{B}\left(\supset^{-}\right) | | |
| fong-spivak-invitation_FO3641 | 316 | 0.141 | Э^{-}: 2 \rightarrow 1 | | |
| fong-spivak-invitation_FO3642 | 316 | 0.968 | x=y=z\} | | |
| fong-spivak-invitation_FO3643 | 316 | 1.000 | B+C \subseteq R^{m+p} \times R^{n+q} | | |
| fong-spivak-invitation_FO3644 | 316 | 0.410 | \mathrm{B}\left(g\right. | | |
| fong-spivak-invitation_FO3645 | 316 | 0.410 | \left.\left(h^{\mathrm{op}}\right)\right)=\mathrm{B}(g) ; \mathrm{B}\left(h^{\mathrm{op}}\right) | | |
| fong-spivak-invitation_FO3646 | 316 | 1.000 | S(h) | | |
| fong-spivak-invitation_FO3647 | 316 | 0.981 | h: m \rightarrow p | | |
| fong-spivak-invitation_FO3648 | 316 | 0.998 | \mathrm{B}\left(\left(g^{\mathrm{op}}\right) ; h\right)=\mathrm{B}\left(g^{\mathrm{op}}\right) ; \mathrm{B}(h) | | |
| fong-spivak-invitation_FO3649 | 316 | 0.555 | \{y \in R \mid y=0\} | | |
| fong-spivak-invitation_FO3650 | 316 | 1.000 | \left\{y \in R^{n} \mid y=0\right\} | | |
| fong-spivak-invitation_FO3651 | 316 | 1.000 | S(g) \subseteq R^{m} \times R^{n} | | |
| fong-spivak-invitation_FO3652 | 316 | 1.000 | \left\{x \in R^{m} \mid S(g)=0\right\} | | |
| fong-spivak-invitation_FO3653 | 316 | 0.868 | \{x \in R\} | | |
| fong-spivak-invitation_FO3654 | 316 | 1.000 | R^{n} \subseteq R^{n} | | |
| fong-spivak-invitation_FO3655 | 316 | 0.999 | \left\{y \in R^{n} \mid\right. | | |
| fong-spivak-invitation_FO3656 | 316 | 0.999 | \left.S(g)(x)=y\right\} | | |
| fong-spivak-invitation_FO3657 | 317 | 0.552 | \mathrm{B}(g)=\{(x, y) \mid S(g)(x)=y\} | | |
| fong-spivak-invitation_FO3658 | 317 | 1.000 | S(g)(x)=y | | |
| fong-spivak-invitation_FO3659 | 317 | 1.000 | S(g)(r x)=r y | | |
| fong-spivak-invitation_FO3660 | 317 | 1.000 | S(g)\left(x_{1}+x_{2}\right)=S(g)\left(x_{1}\right)+S(g)\left(x_{2}\right) | | |
| fong-spivak-invitation_FO3661 | 317 | 1.000 | (x, y) \in \mathrm{B}(g) | | |
| fong-spivak-invitation_FO3662 | 317 | 1.000 | (r x, r y) \in \mathrm{B}(g) | | |
| fong-spivak-invitation_FO3663 | 317 | 1.000 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \in \mathrm{B}(g) | | |
| fong-spivak-invitation_FO3664 | 317 | 1.000 | \left(x_{1}+x_{2}, y_{1}+y_{2}\right) \in \mathrm{B}(g) | | |
| fong-spivak-invitation_FO3665 | 317 | 0.999 | \mathrm{B}\left(g^{\mathrm{op}}\right) | | |
| fong-spivak-invitation_FO3666 | 317 | 0.386 | C \subseteq R^{n} \times R^{p} | | |
| fong-spivak-invitation_FO3667 | 317 | 0.386 | \left(x_{1}, z_{1}\right),\left(x_{2}, z_{2}\right) \in B \circ C | | |
| fong-spivak-invitation_FO3668 | 317 | 0.999 | y_{1}, y_{2} \in R^{n} | | |
| fong-spivak-invitation_FO3669 | 317 | 0.999 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \in B | | |
| fong-spivak-invitation_FO3670 | 317 | 0.999 | \left(y_{1}, z_{1}\right),\left(y_{2}, z_{2}\right) \in C | | |
| fong-spivak-invitation_FO3671 | 317 | 1.000 | \left(r x_{1}, r y_{1}\right) \in B | | |
| fong-spivak-invitation_FO3672 | 317 | 1.000 | \left(r y_{1}, r z_{1}\right) \in C | | |
| fong-spivak-invitation_FO3673 | 317 | 1.000 | \left(x_{1}+x_{2}, y_{1}+y_{2}\right) \in B | | |
| fong-spivak-invitation_FO3674 | 317 | 0.328 | \left(y_{1}+y_{2}, z_{1}+z_{2}\right) \in C | | |
| fong-spivak-invitation_FO3675 | 317 | 0.328 | \left(r x_{1}, r z_{1}\right) \in(B ; C) | | |
| fong-spivak-invitation_FO3676 | 317 | 0.328 | \left(x_{1}+x_{2}, z_{1}+z_{2}\right) \in(B \circ C) | | |
| fong-spivak-invitation_FO3677 | 317 | 0.902 | (B ; C) \subseteq R^{m} \times R^{p} | | |
| fong-spivak-invitation_FO3678 | 317 | 0.902 | (x, z) | | |
| fong-spivak-invitation_FO3679 | 317 | 0.929 | (y, z) \in C | | |
| fong-spivak-invitation_FO3680 | 317 | 0.929 | (B \circ C) | | |
| fong-spivak-invitation_FO3681 | 317 | 0.956 | (x, z) \in(B ; C) | | |
| fong-spivak-invitation_FO3682 | 317 | 0.905 | (r * x, r * y) \in B | | |
| fong-spivak-invitation_FO3683 | 317 | 0.905 | (r * y, r * z) \in C | | |
| fong-spivak-invitation_FO3684 | 317 | 0.905 | (r * x, r * z) \in(B ঃ C) | | |
| fong-spivak-invitation_FO3685 | 317 | 1.000 | \left(x^{\prime}, z^{\prime}\right) \in(B ; C) | | |
| fong-spivak-invitation_FO3686 | 317 | 1.000 | y^{\prime} \in R^{n} | | |
| fong-spivak-invitation_FO3687 | 317 | 0.999 | \left(y^{\prime}, z^{\prime}\right) \in C | | |
| fong-spivak-invitation_FO3688 | 317 | 0.999 | \left(x+x^{\prime}, y+y^{\prime}\right) \in B | | |
| fong-spivak-invitation_FO3689 | 317 | 0.999 | \left(y+y^{\prime}, z+z^{\prime}\right) \in C | | |
| fong-spivak-invitation_FO3690 | 317 | 0.918 | \left(x+x^{\prime}, z+z^{\prime}\right) \in(B ; C) | | |
| fong-spivak-invitation_FO3691 | 317 | 1.000 | a \not \leq b | | |
| fong-spivak-invitation_FO3692 | 317 | 1.000 | b \not \leq a | | |
| fong-spivak-invitation_FO3693 | 317 | 0.999 | f\left(1_{R}\right)=1_{S} | | |
| fong-spivak-invitation_FO3694 | 317 | 0.999 | f\left(r_{1} *_{R} r_{2}\right)=f\left(r_{1}\right) *_{S} f\left(r_{2}\right) | | |
| fong-spivak-invitation_FO3695 | 317 | 0.998 | R=\left(R, 0_{R},+_{R}, 1_{R}, *_{R}\right) | | |
| fong-spivak-invitation_FO3696 | 317 | 1.000 | f: \mathbb{N} \rightarrow R | | |
| fong-spivak-invitation_FO3697 | 318 | 0.773 | 0_{R}, 1 | | |
| fong-spivak-invitation_FO3698 | 318 | 0.773 | 1_{R} | | |
| fong-spivak-invitation_FO3699 | 318 | 0.731 | f(m+n)=f(m)+_{R} f(n) | | |
| fong-spivak-invitation_FO3700 | 318 | 1.000 | f(m * n)=f(m) *_{R} f(n) | | |
| fong-spivak-invitation_FO3701 | 318 | 1.000 | f(m * n) | | |
| fong-spivak-invitation_FO3702 | 318 | 1.000 | m n | | |
| fong-spivak-invitation_FO3703 | 318 | 1.000 | \varnothing \rightarrow c | | |
| fong-spivak-invitation_FO3704 | 318 | 1.000 | c_{1} \rightarrow c | | |
| fong-spivak-invitation_FO3705 | 318 | 1.000 | c_{2} \rightarrow c_{1} | | |
| fong-spivak-invitation_FO3706 | 318 | 0.999 | c_{1} \rightarrow c_{1} | | |
| fong-spivak-invitation_FO3707 | 318 | 0.999 | \mathrm{id}_{c_{1}} | | |
| fong-spivak-invitation_FO3708 | 318 | 0.078 | =\mathrm{id}_{c_{1}} | | |
| fong-spivak-invitation_FO3709 | 318 | 0.078 | f=\mathrm{id}_{c_{2}} | | |
| fong-spivak-invitation_FO3710 | 318 | 1.000 | p+q | | |
| fong-spivak-invitation_FO3711 | 318 | 1.000 | p \leq p+q | | |
| fong-spivak-invitation_FO3712 | 318 | 1.000 | q \leq p+q | | |
| fong-spivak-invitation_FO3713 | 318 | 1.000 | p+q \leq x | | |
| fong-spivak-invitation_FO3714 | 319 | 0.996 | o | | |
| fong-spivak-invitation_FO3715 | 319 | 0.366 | \iota_{A}{ }_{9}^{\circ}[f, g]=f | | |
| fong-spivak-invitation_FO3716 | 319 | 0.451 | \iota_{B}{ }^{\circ}[f, g]=g | | |
| fong-spivak-invitation_FO3717 | 319 | 0.428 | [f, g]: h=[f ; h, g \circ h] | | |
| fong-spivak-invitation_FO3718 | 319 | 0.998 | [f ; h, g ; h]: A+B \rightarrow D | | |
| fong-spivak-invitation_FO3719 | 319 | 0.966 | [f, g] ; h=[f ; h, g \circ h] | | |
| fong-spivak-invitation_FO3720 | 319 | 0.997 | +: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-invitation_FO3721 | 319 | 0.943 | (A, B) | | |
| fong-spivak-invitation_FO3722 | 319 | 1.000 | (f, g):(A, B) \rightarrow(C, D) | | |
| fong-spivak-invitation_FO3723 | 319 | 0.969 | f+g=\left[f{ }^{\circ} \iota_{C}, g{ }^{\circ} \iota_{D}\right]: A+B \rightarrow C+D | | |
| fong-spivak-invitation_FO3724 | 319 | 0.969 | \iota_{C}: C \rightarrow C+D | | |
| fong-spivak-invitation_FO3725 | 319 | 0.997 | \iota_{D}: D \rightarrow C+D | | |
| fong-spivak-invitation_FO3726 | 320 | 0.491 | (h, k):(C, D) \rightarrow(E, F) | | |
| fong-spivak-invitation_FO3727 | 320 | 0.491 | (f+g){ }_{\circ}(h+k)= | | |
| fong-spivak-invitation_FO3728 | 320 | 0.755 | (f ; h)+(g ; k) | | |
| fong-spivak-invitation_FO3729 | 320 | 1.000 | (f ; h)+ | | |
| fong-spivak-invitation_FO3730 | 320 | 0.158 | (g \circ k)=\left[f{ }_{9} h{ }_{9} \iota_{E}, g{ }_{9} k{ }_{9}^{\circ} \iota_{F}\right]=(f+g){ }_{9}(h+k) | | |
| fong-spivak-invitation_FO3731 | 320 | 0.998 | !_{A}: \varnothing \rightarrow A | | |
| fong-spivak-invitation_FO3732 | 320 | 0.738 | \left[\mathrm{id}_{A},!_{A}\right] | | |
| fong-spivak-invitation_FO3733 | 320 | 0.592 | { }^{\imath} A: A \rightarrow A+\varnothing | | |
| fong-spivak-invitation_FO3734 | 320 | 0.679 | \left[\mathrm{id}_{A},!_{A}\right]=\mathrm{id}_{A} | | |
| fong-spivak-invitation_FO3735 | 320 | 0.986 | \left[!_{A}, \mathrm{id}_{A}\right]: \varnothing+A \rightarrow A | | |
| fong-spivak-invitation_FO3736 | 320 | 0.644 | \left[\mathrm{id}_{A}+\iota_{B}, \iota_{C}\right]=\left[\left[\iota_{A}, \iota_{B}{ }^{\circ} \iota_{B+C}\right], \iota_{C}{ }^{\circ} \iota_{B+C}\right]:(A+B)+C \rightarrow A+(B+C) | | |
| fong-spivak-invitation_FO3737 | 320 | 0.999 | \left[\iota_{A}, \iota_{B}+\mathrm{id}_{C}\right]: A+(B+C) \rightarrow(A+B)+C | | |
| fong-spivak-invitation_FO3738 | 320 | 1.000 | \left[\iota_{A}, \iota_{B}\right]: A+B \rightarrow B+A | | |
| fong-spivak-invitation_FO3739 | 320 | 0.991 | \iota_{A}: A \rightarrow A+B | | |
| fong-spivak-invitation_FO3740 | 320 | 0.991 | \iota_{A}: A \rightarrow B+A | | |
| fong-spivak-invitation_FO3741 | 320 | 0.980 | \left[\iota_{B}, \iota_{A}\right]: B+A \rightarrow | | |
| fong-spivak-invitation_FO3742 | 320 | 1.000 | A=B=C | | |
| fong-spivak-invitation_FO3743 | 320 | 1.000 | s^{\prime} | | |
| fong-spivak-invitation_FO3744 | 320 | 1.000 | s \rightarrow s^{\prime} | | |
| fong-spivak-invitation_FO3745 | 320 | 1.000 | s=s^{\prime} | | |
| fong-spivak-invitation_FO3746 | 321 | 0.990 | \underline{5} \sqcup \underline{3} | | |
| fong-spivak-invitation_FO3747 | 321 | 0.990 | \{f(a) \sim g(a) \mid a \in \underline{4}\} | | |
| fong-spivak-invitation_FO3748 | 321 | 1.000 | \underline{5} | | |
| fong-spivak-invitation_FO3749 | 321 | 1.000 | \{1, \ldots, 5\} | | |
| fong-spivak-invitation_FO3750 | 321 | 1.000 | \underline{3} | | |
| fong-spivak-invitation_FO3751 | 321 | 1.000 | \left\{1^{\prime}, 2^{\prime}, 3^{\prime}\right\} | | |
| fong-spivak-invitation_FO3752 | 321 | 1.000 | 5 \sim 3^{\prime} | | |
| fong-spivak-invitation_FO3753 | 321 | 1.000 | \left\{1,1^{\prime}, 3\right\},\{2\},\{4\} | | |
| fong-spivak-invitation_FO3754 | 321 | 1.000 | \left\{5,2^{\prime}, 3^{\prime}\right\} | | |
| fong-spivak-invitation_FO3755 | 321 | 1.000 | \varnothing \rightarrow X+Y | | |
| fong-spivak-invitation_FO3756 | 321 | 0.635 | f{ }_{9} \iota_{X}=g{ }_{9} \iota_{Y} | | |
| fong-spivak-invitation_FO3757 | 321 | 0.479 | t: X+\varnothing Y \rightarrow T | | |
| fong-spivak-invitation_FO3758 | 321 | 0.479 | t=x | | |
| fong-spivak-invitation_FO3759 | 321 | 0.479 | t=y | | |
| fong-spivak-invitation_FO3760 | 322 | 1.000 | A, B, X, Y | | |
| fong-spivak-invitation_FO3761 | 322 | 1.000 | S \rightarrow T | | |
| fong-spivak-invitation_FO3762 | 322 | 1.000 | X \leftarrow A \rightarrow Y | | |
| fong-spivak-invitation_FO3763 | 322 | 1.000 | Q \rightarrow T | | |
| fong-spivak-invitation_FO3764 | 322 | 1.000 | Y \leftarrow B \rightarrow Z | | |
| fong-spivak-invitation_FO3765 | 322 | 1.000 | R \rightarrow T | | |
| fong-spivak-invitation_FO3766 | 322 | 1.000 | (Y, Q, R, T) | | |
| fong-spivak-invitation_FO3767 | 322 | 0.996 | X+{ }_{N} Y | | |
| fong-spivak-invitation_FO3768 | 322 | 1.000 | X \sqcup N \sqcup Y | | |
| fong-spivak-invitation_FO3769 | 322 | 1.000 | x \sim n | | |
| fong-spivak-invitation_FO3770 | 322 | 1.000 | x=f(n) | | |
| fong-spivak-invitation_FO3771 | 322 | 1.000 | y \sim n | | |
| fong-spivak-invitation_FO3772 | 322 | 1.000 | y=g(n) | | |
| fong-spivak-invitation_FO3773 | 322 | 1.000 | x \in X, y \in Y, n \in N | | |
| fong-spivak-invitation_FO3774 | 322 | 1.000 | n \in N | | |
| fong-spivak-invitation_FO3775 | 323 | 1.000 | B \rightarrow D \otimes D | | |
| fong-spivak-invitation_FO3776 | 323 | 1.000 | g: X \rightarrow Y | | |
| fong-spivak-invitation_FO3777 | 324 | 0.998 | \iota_{1}: X \rightarrow X+X | | |
| fong-spivak-invitation_FO3778 | 324 | 0.998 | X+X | | |
| fong-spivak-invitation_FO3779 | 324 | 0.446 | \iota_{1}{ }_{9}^{\circ}[\mathrm{id}, \mathrm{id}]=\mathrm{id} | | |
| fong-spivak-invitation_FO3780 | 324 | 0.080 | f=\iota_{1} | | |
| fong-spivak-invitation_FO3781 | 324 | 0.080 | { }_{9}^{\circ} f=\iota_{1} | | |
| fong-spivak-invitation_FO3782 | 324 | 0.080 | \left[\right. | | |
| fong-spivak-invitation_FO3783 | 324 | 0.080 | { }_{9}^{\circ} g=g | | |
| fong-spivak-invitation_FO3784 | 324 | 0.080 | f: X \rightarrow T | | |
| fong-spivak-invitation_FO3785 | 324 | 0.971 | X \leftarrow N \rightarrow Y | | |
| fong-spivak-invitation_FO3786 | 324 | 0.980 | F N=\{*\} | | |
| fong-spivak-invitation_FO3787 | 324 | 0.980 | F N | | |
| fong-spivak-invitation_FO3788 | 324 | 0.966 | \operatorname{Cospan}_{F}=\operatorname{Cospan}_{\mathcal{C}} | | |
| fong-spivak-invitation_FO3789 | 324 | 0.974 | \operatorname{Cospan}_{\mathcal{C}} \rightarrow \operatorname{Cospan}_{F} | | |
| fong-spivak-invitation_FO3790 | 325 | 0.848 | V=\{\mathrm{ul}, \mathrm{ur}, \mathrm{dl}, \mathrm{dr}\} | | |
| fong-spivak-invitation_FO3791 | 325 | 0.928 | A=\{\mathrm{r} 1, \mathrm{r} 2, \mathrm{r} 3, \mathrm{c} 1, \mathrm{i} 1\} | | |
| fong-spivak-invitation_FO3798 | 325 | 1.000 | \psi_{\underline{2}, \underline{2}}(b, s) | | |
| fong-spivak-invitation_FO3799 | 325 | 0.555 | 1 \xrightarrow{f} \underline{2} \stackrel{g}{\leftarrow} \underline{1} | | |
| fong-spivak-invitation_FO3800 | 325 | 0.555 | f(1)=1 | | |
| fong-spivak-invitation_FO3801 | 325 | 0.555 | g(1)=2 | | |
| fong-spivak-invitation_FO3802 | 325 | 0.923 | \underline{2},\{a\}, s, t, \ell | | |
| fong-spivak-invitation_FO3803 | 325 | 0.923 | s(a)=1, t(a)=2 | | |
| fong-spivak-invitation_FO3804 | 325 | 0.923 | \ell(a)= | | |
| fong-spivak-invitation_FO3805 | 325 | 1.000 | C:=(V, A, s, t, \ell) | | |
| fong-spivak-invitation_FO3806 | 325 | 0.940 | \underline{1} \xrightarrow{f} V \stackrel{g}{\leftarrow} \underline{2}, f(1)=\mathrm{ul}, g(1)= | | |
| fong-spivak-invitation_FO3807 | 325 | 0.940 | g(2)= | | |
| fong-spivak-invitation_FO3808 | 325 | 0.975 | \underline{2} \xrightarrow{f^{\prime}} V^{\prime} \stackrel{g^{\prime}}{\leftarrow} \underline{2} | | |
| fong-spivak-invitation_FO3809 | 325 | 1.000 | C^{\prime}:=\left(V^{\prime}, A^{\prime}, s^{\prime}, t^{\prime}, \ell^{\prime}\right) | | |
| fong-spivak-invitation_FO3810 | 325 | 1.000 | V^{\prime}=\{l, r, d\}, A^{\prime}=\left\{\mathrm{r} 1^{\prime}, \mathrm{r} 2^{\prime}\right\} | | |
| fong-spivak-invitation_FO3815 | 325 | 0.989 | V \stackrel{g}{\leftarrow} \underline{2} \xrightarrow{f^{\prime}} V^{\prime} | | |
| fong-spivak-invitation_FO3816 | 325 | 0.998 | V^{\prime \prime}=\{\mathrm{ul}, \mathrm{dl}, \mathrm{dr}, \mathrm{m}, \mathrm{r}\} | | |
| fong-spivak-invitation_FO3817 | 325 | 0.998 | \underline{1} \xrightarrow{h} V^{\prime \prime} \stackrel{k}{\leftarrow} \underline{2} | | |
| fong-spivak-invitation_FO3818 | 325 | 0.998 | h(1)= | | |
| fong-spivak-invitation_FO3819 | 325 | 0.388 | k(1)=\mathrm{r} | | |
| fong-spivak-invitation_FO3820 | 325 | 0.388 | k(2)=\mathrm{m} | | |
| fong-spivak-invitation_FO3821 | 325 | 0.388 | \left(V, \prime \prime A+A^{\prime}, s,{ }^{\prime \prime} t,{ }^{\prime \prime} \ell^{\prime \prime}\right) | | |
| fong-spivak-invitation_FO3825 | 326 | 1.000 | \eta | | |
| fong-spivak-invitation_FO3826 | 327 | 0.637 | 2,2,2,2 ; 0 | | |
| fong-spivak-invitation_FO3827 | 327 | 0.898 | \left(A, B, A^{\prime}, B^{\prime}\right) | | |
| fong-spivak-invitation_FO3828 | 327 | 0.898 | \left(A, C, A^{\prime}, C^{\prime}\right) | | |
| fong-spivak-invitation_FO3829 | 327 | 0.608 | X, p, q | | |
| fong-spivak-invitation_FO3830 | 327 | 0.274 | g^{\prime}=p | | |
| fong-spivak-invitation_FO3831 | 327 | 0.274 | h_{3} | | |
| fong-spivak-invitation_FO3832 | 327 | 0.229 | r ; h_{2}=g | | |
| fong-spivak-invitation_FO3833 | 327 | 0.229 | r ; g=p | | |
| fong-spivak-invitation_FO3834 | 327 | 1.000 | r^{\prime}: X \rightarrow A | | |
| fong-spivak-invitation_FO3835 | 327 | 0.999 | r^{\prime} ; f=r | | |
| fong-spivak-invitation_FO3836 | 327 | 0.999 | r^{\prime} ; h_{1}=q | | |
| fong-spivak-invitation_FO3837 | 328 | 1.000 | r^{\prime} ; f ; g=r ; g=p | | |
| fong-spivak-invitation_FO3838 | 328 | 1.000 | r_{0} | | |
| fong-spivak-invitation_FO3839 | 328 | 0.279 | r_{0} \stackrel{\circ}{f} f \circ g=p | | |
| fong-spivak-invitation_FO3840 | 328 | 0.279 | r_{0} \stackrel{\circ}{\circ} h_{1}=q | | |
| fong-spivak-invitation_FO3841 | 328 | 0.413 | r_{0} \stackrel{\circ}{f} f=r | | |
| fong-spivak-invitation_FO3842 | 328 | 0.413 | r^{\prime} | | |
| fong-spivak-invitation_FO3843 | 328 | 1.000 | r_{0}=r^{\prime} | | |
| fong-spivak-invitation_FO3844 | 328 | 0.823 | B, C, B^{\prime}, C^{\prime} | | |
| fong-spivak-invitation_FO3845 | 328 | 0.824 | r ; h_{2}=q ; f^{\prime} | | |
| fong-spivak-invitation_FO3846 | 328 | 0.824 | p:=r ; g | | |
| fong-spivak-invitation_FO3847 | 328 | 0.666 | r^{\prime}: X \rightarrow | | |
| fong-spivak-invitation_FO3848 | 328 | 0.427 | r^{\prime} ; f ; g=p | | |
| fong-spivak-invitation_FO3849 | 328 | 0.427 | r_{0}: h_{1}=q | | |
| fong-spivak-invitation_FO3850 | 328 | 0.104 | r_{0}:=r^{\prime}{ }_{9}^{\circ} f | | |
| fong-spivak-invitation_FO3851 | 328 | 0.104 | r_{0}{ }_{9}^{\circ} g=p | | |
| fong-spivak-invitation_FO3852 | 328 | 0.104 | r_{0}{ }_{9}^{\circ} h_{2}=q | | |
| fong-spivak-invitation_FO3853 | 328 | 1.000 | a_{1}, a_{2} \in A | | |
| fong-spivak-invitation_FO3854 | 328 | 1.000 | f\left(a_{1}\right)=f\left(a_{2}\right) | | |
| fong-spivak-invitation_FO3855 | 328 | 1.000 | a_{1}=a_{2} | | |
| fong-spivak-invitation_FO3856 | 328 | 0.908 | g_{1}, g_{2}: X \rightarrow A | | |
| fong-spivak-invitation_FO3857 | 328 | 0.908 | f=g_{2} | | |
| fong-spivak-invitation_FO3858 | 328 | 0.908 | g_{1}=g_{2} | | |
| fong-spivak-invitation_FO3859 | 329 | 1.000 | X=\{*\} | | |
| fong-spivak-invitation_FO3860 | 329 | 1.000 | g_{1}(*):=a_{1} | | |
| fong-spivak-invitation_FO3861 | 329 | 0.509 | g_{2}(*):=a_{2} | | |
| fong-spivak-invitation_FO3862 | 329 | 0.378 | f= | | |
| fong-spivak-invitation_FO3863 | 329 | 0.653 | g_{1}(x)=g_{2}(x) | | |
| fong-spivak-invitation_FO3864 | 329 | 1.000 | f\left(g_{1}(x)\right)=f\left(g_{2}(x)\right) | | |
| fong-spivak-invitation_FO3865 | 329 | 0.605 | j:=i^{-1} | | |
| fong-spivak-invitation_FO3866 | 329 | 0.605 | g:=(f: j): A \rightarrow B^{\prime} | | |
| fong-spivak-invitation_FO3867 | 329 | 0.605 | g: i=f | | |
| fong-spivak-invitation_FO3868 | 329 | 0.990 | j^{\prime}: A \rightarrow A^{\prime} | | |
| fong-spivak-invitation_FO3869 | 329 | 0.990 | j^{\prime} ; i^{\prime}=\mathrm{id}_{A} | | |
| fong-spivak-invitation_FO3870 | 329 | 0.357 | j^{\prime}{ }_{9} f^{\prime}=f{ }_{9}^{\circ} j | | |
| fong-spivak-invitation_FO3871 | 329 | 0.357 | i^{\prime}{ }_{9} j^{\prime}=\mathrm{id}_{A^{\prime}} | | |
| fong-spivak-invitation_FO3872 | 329 | 0.357 | \left(i^{\prime}{ }_{9} j^{\prime}\right){ }_{9} i^{\prime}=i^{\prime} | | |
| fong-spivak-invitation_FO3873 | 329 | 0.184 | \left(i^{\prime}{ }_{9} j^{\prime}\right){ }_{9}^{\circ} f^{\prime}=i^{\prime}{ }_{9}^{\prime} f_{9}^{\circ} j=f^{\prime}{ }_{9}^{\circ} i{ }_{9}^{\circ} j=f^{\prime} | | |
| fong-spivak-invitation_FO3874 | 329 | 0.184 | \mathrm{id}_{A^{\prime}} | | |
| fong-spivak-invitation_FO3875 | 329 | 0.184 | \mathrm{id}_{A^{\prime}}{ }^{\circ} i^{\prime}=i^{\prime} | | |
| fong-spivak-invitation_FO3876 | 329 | 0.774 | \mathrm{id}_{A^{\prime}}{ }^{\circ} f^{\prime}=f^{\prime} | | |
| fong-spivak-invitation_FO3877 | 329 | 0.774 | \left(i^{\prime} ; j^{\prime}\right)=\mathrm{id}_{A^{\prime}} | | |
| fong-spivak-invitation_FO3878 | 329 | 0.669 | g: X \rightarrow A | | |
| fong-spivak-invitation_FO3879 | 329 | 0.669 | h: X \rightarrow B | | |
| fong-spivak-invitation_FO3880 | 329 | 0.669 | f=h | | |
| fong-spivak-invitation_FO3881 | 329 | 0.669 | \mathrm{id}_{B} | | |
| fong-spivak-invitation_FO3882 | 329 | 0.650 | r: X \rightarrow A | | |
| fong-spivak-invitation_FO3883 | 329 | 0.650 | r{ }_{9}^{\circ} \mathrm{id}_{A}=g | | |
| fong-spivak-invitation_FO3884 | 329 | 0.650 | r: f=h | | |
| fong-spivak-invitation_FO3885 | 329 | 1.000 | r=g | | |
| fong-spivak-invitation_FO3886 | 330 | 1.000 | v \otimes w | | |
| fong-spivak-invitation_FO3887 | 330 | 0.958 | v \leq(w \multimap x) | | |
| fong-spivak-invitation_FO3888 | 330 | 0.958 | v \otimes W \leq x | | |
| fong-spivak-invitation_FO3889 | 330 | 0.991 | \mathbb{N}^{\mathrm{op}} \times \mathbb{N}^{\mathrm{op}} | | |
| fong-spivak-invitation_FO3890 | 330 | 0.991 | (a, b) \in \mathbb{N} \times \mathbb{N} | | |
| fong-spivak-invitation_FO3891 | 330 | 0.991 | (a, b) \leq | | |
| fong-spivak-invitation_FO3892 | 330 | 0.600 | \left(a^{\prime}, b^{\prime}\right) | | |
| fong-spivak-invitation_FO3893 | 330 | 0.600 | a^{\prime} \leq a | | |
| fong-spivak-invitation_FO3894 | 330 | 0.600 | b^{\prime} \leq b | | |
| fong-spivak-invitation_FO3895 | 330 | 0.859 | (a, b) \wedge\left(a^{\prime}, b^{\prime}\right)=\left(\max \left(a, a^{\prime}\right), \max \left(b, b^{\prime}\right)\right) | | |
| fong-spivak-invitation_FO3896 | 330 | 0.998 | x \multimap y | | |
| fong-spivak-invitation_FO3897 | 330 | 0.998 | x \multimap y=\bigvee\{w \mid w \wedge x \leq y\} | | |
| fong-spivak-invitation_FO3898 | 330 | 1.000 | y \wedge x \leq y | | |
| fong-spivak-invitation_FO3899 | 330 | 1.000 | y \leq x \multimap y | | |
| fong-spivak-invitation_FO3900 | 330 | 1.000 | x \multimap y=\bigvee\{w \mid y \leq w | | |
| fong-spivak-invitation_FO3901 | 330 | 1.000 | w \wedge x \leq y\} | | |
| fong-spivak-invitation_FO3902 | 330 | 0.543 | \mathbb{N}^{\mathrm{OP}} \times \mathbb{N}^{\mathrm{oP}} | | |
| fong-spivak-invitation_FO3903 | 330 | 0.651 | \mathbb{N}^{\mathrm{OP}} \times \mathbb{N}^{\mathrm{OP}} | | |
| fong-spivak-invitation_FO3904 | 330 | 1.000 | y=(a, b) | | |
| fong-spivak-invitation_FO3905 | 330 | 1.000 | (a+1) *(b+1) | | |
| fong-spivak-invitation_FO3906 | 331 | 0.997 | m: \mathbb{Z} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO3907 | 331 | 0.997 | \mathbb{N} \subseteq \mathbb{Z} | | |
| fong-spivak-invitation_FO3908 | 331 | 0.992 | \ulcorner m\urcorner(-5)= | | |
| fong-spivak-invitation_FO3909 | 331 | 1.000 | \ulcorner m\urcorner(0)= | | |
| fong-spivak-invitation_FO3910 | 331 | 0.615 | \ulcorner!\mathbb{N}\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO3911 | 331 | 1.000 | A \subseteq \mathbb{B} | | |
| fong-spivak-invitation_FO3912 | 331 | 0.999 | \ulcorner\Rightarrow\urcorner: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO3913 | 331 | 0.859 | \subseteq \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-invitation_FO3914 | 331 | 0.980 | \ulcorner E\urcorner,\ulcorner P\urcorner,\ulcorner T\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-invitation_FO3915 | 331 | 1.000 | P:=\{n \in \mathbb{N} \mid n | | |
| fong-spivak-invitation_FO3916 | 331 | 0.999 | \ulcorner E\urcorner(17)= | | |
| fong-spivak-invitation_FO3917 | 331 | 0.865 | \ulcorner P\urcorner(17)= | | |
| fong-spivak-invitation_FO3918 | 331 | 0.690 | \ulcorner T\urcorner(17)=\operatorname{true} | | |
| fong-spivak-invitation_FO3919 | 331 | 0.690 | 17 \geq 10 | | |
| fong-spivak-invitation_FO3920 | 331 | 0.995 | B(x, \epsilon):=\left\{x^{\prime} \in \mathbb{R} \mid\right. | | |
| fong-spivak-invitation_FO3921 | 331 | 1.000 | \left.\left|x-x^{\prime}\right|<\epsilon\right\} | | |
| fong-spivak-invitation_FO3922 | 331 | 1.000 | x \in U | | |
| fong-spivak-invitation_FO3923 | 331 | 1.000 | B(x, \epsilon) \subseteq U | | |
| fong-spivak-invitation_FO3924 | 331 | 1.000 | U_{1}:=\{x \in \mathbb{R} \mid 0<x<2\} | | |
| fong-spivak-invitation_FO3925 | 331 | 1.000 | U_{2}:=\{x \in \mathbb{R} \mid 1<x<3\} | | |
| fong-spivak-invitation_FO3926 | 331 | 1.000 | U:=U_{1} \cup U_{2}=\{x \in | | |
| fong-spivak-invitation_FO3927 | 331 | 1.000 | \mathbb{R} \mid 0<x<3\} | | |
| fong-spivak-invitation_FO3928 | 332 | 1.000 | I=\{1,2,3,4, \ldots\} | | |
| fong-spivak-invitation_FO3929 | 332 | 1.000 | U_{i}:=\left\{x \in \mathbb{R} \left\lvert\, \frac{1}{i}<x<1\right.\right\} | | |
| fong-spivak-invitation_FO3930 | 332 | 1.000 | U_{1} \subseteq U_{2} \subseteq U_{3} \subseteq \cdots | | |
| fong-spivak-invitation_FO3931 | 332 | 1.000 | U:=\bigcup_{i \in I} U_{i}=\{x \in \mathbb{R} \mid 0<x<1\} | | |
| fong-spivak-invitation_FO3932 | 332 | 1.000 | \varnothing \subseteq X | | |
| fong-spivak-invitation_FO3933 | 332 | 1.000 | \bigcup_{i \in I} A_{i} | | |
| fong-spivak-invitation_FO3934 | 332 | 1.000 | A_{i}=X | | |
| fong-spivak-invitation_FO3935 | 332 | 1.000 | A \subseteq X | | |
| fong-spivak-invitation_FO3936 | 332 | 1.000 | f^{-1}(U) \subseteq X | | |
| fong-spivak-invitation_FO3937 | 332 | 1.000 | \varnothing \rightarrow\{1\} \rightarrow\{1,2\} | | |
| fong-spivak-invitation_FO3938 | 332 | 1.000 | U_{i}=U | | |
| fong-spivak-invitation_FO3939 | 332 | 0.996 | Y=B \cap Y | | |
| fong-spivak-invitation_FO3940 | 332 | 1.000 | B=X | | |
| fong-spivak-invitation_FO3941 | 332 | 1.000 | \varnothing=\varnothing \cap Y | | |
| fong-spivak-invitation_FO3942 | 332 | 1.000 | \varnothing \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-invitation_FO3943 | 332 | 0.929 | A_{1}, A_{2} \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-invitation_FO3944 | 332 | 0.929 | B_{1}, B_{2} \in \mathbf{O p} | | |
| fong-spivak-invitation_FO3945 | 332 | 0.929 | A_{1}=B_{1} \cap Y | | |
| fong-spivak-invitation_FO3946 | 332 | 0.929 | A_{2}=B_{2} \cap Y | | |
| fong-spivak-invitation_FO3947 | 332 | 0.892 | A_{1} \cap A_{2}=\left(B_{1} \cap Y\right) \cap\left(B_{2} \cap Y\right)=\left(B_{1} \cap B_{2}\right) \cap Y | | |
| fong-spivak-invitation_FO3948 | 332 | 0.999 | B_{1} \cap B_{2} \in \mathbf{O p} | | |
| fong-spivak-invitation_FO3949 | 332 | 0.999 | A_{i} | | |
| fong-spivak-invitation_FO3950 | 332 | 1.000 | A_{i}=B_{i} \cap Y | | |
| fong-spivak-invitation_FO3951 | 332 | 1.000 | B_{i} \in \mathbf{O p} | | |
| fong-spivak-invitation_FO3952 | 332 | 1.000 | U \subseteq \mathbf{O p} | | |
| fong-spivak-invitation_FO3953 | 333 | 1.000 | \mathcal{C}(B, C)=\left(U_{3} \wedge U_{1}\right) \vee\left(U_{4} \wedge U_{2}\right) | | |
| fong-spivak-invitation_FO3954 | 333 | 1.000 | \mathcal{C}(a, b) | | |
| fong-spivak-invitation_FO3955 | 333 | 1.000 | \left\{a_{1}, a_{2}\right\} | | |
| fong-spivak-invitation_FO3956 | 333 | 1.000 | \left\{c_{1}\right\} | | |
| fong-spivak-invitation_FO3957 | 334 | 0.999 | \operatorname{Sec}_{f}\left(V_{2}\right)=\varnothing | | |
| fong-spivak-invitation_FO3958 | 334 | 0.827 | 2 * 3 * 1 * 2=12 | | |
| fong-spivak-invitation_FO3959 | 334 | 0.998 | g_{1}:=\left(a_{1}, b_{1}\right) | | |
| fong-spivak-invitation_FO3960 | 334 | 0.998 | g_{2}:=\left(b_{2}, e_{1}\right) | | |
| fong-spivak-invitation_FO3961 | 334 | 1.000 | g \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-invitation_FO3962 | 334 | 1.000 | \left.g\right|_{U_{1}}=g_{1} | | |
| fong-spivak-invitation_FO3963 | 334 | 1.000 | \left.g\right|_{U_{2}}=g_{2} | | |
| fong-spivak-invitation_FO3964 | 335 | 0.997 | F(\{1,2\}) \rightarrow F(\{1\}) \rightarrow | | |
| fong-spivak-invitation_FO3965 | 335 | 1.000 | F(\varnothing) | | |
| fong-spivak-invitation_FO3966 | 335 | 1.000 | F(\varnothing) \cong\{1\} | | |
| fong-spivak-invitation_FO3967 | 335 | 1.000 | F(\{1,2\}) \rightarrow F(\{1\}) | | |
| fong-spivak-invitation_FO3968 | 335 | 0.748 | S \in \mathbf{S h v}(X) | | |
| fong-spivak-invitation_FO3969 | 335 | 0.999 | S(\varnothing)=\{()\} | | |
| fong-spivak-invitation_FO3970 | 335 | 1.000 | S(\{1\}) | | |
| fong-spivak-invitation_FO3971 | 335 | 0.862 | \Omega: \mathbf{O p}(X)^{\mathrm{op}} \rightarrow | | |
| fong-spivak-invitation_FO3972 | 335 | 0.862 | \mathbf{S h v}(X) | | |
| fong-spivak-invitation_FO3973 | 335 | 0.945 | \Omega(\{1\}) | | |
| fong-spivak-invitation_FO3974 | 335 | 0.445 | \mathbf{O p}(\{1\}) | | |
| fong-spivak-invitation_FO3975 | 335 | 1.000 | \Omega(U):=\left\{U^{\prime} \in \mathbf{O p} \mid U^{\prime} \subseteq U\right\} | | |
| fong-spivak-invitation_FO3976 | 335 | 1.000 | U^{\prime} \mapsto U^{\prime} \cap V | | |
| fong-spivak-invitation_FO3977 | 335 | 0.996 | U^{\prime} \subseteq U | | |
| fong-spivak-invitation_FO3978 | 335 | 0.996 | \left(U^{\prime} \cap V\right) \cap W=U^{\prime} \cap W | | |
| fong-spivak-invitation_FO3979 | 335 | 1.000 | W \subseteq V | | |
| fong-spivak-invitation_FO3980 | 335 | 1.000 | U^{\prime} \cap U=U^{\prime} | | |
| fong-spivak-invitation_FO3981 | 335 | 1.000 | G^{\prime} | | |
| fong-spivak-invitation_FO3982 | 335 | 1.000 | \gamma:=\left\ulcorner G^{\prime}\right\urcorner | | |
| fong-spivak-invitation_FO3983 | 335 | 1.000 | \gamma(D)=0 | | |
| fong-spivak-invitation_FO3984 | 335 | 1.000 | A, B, C | | |
| fong-spivak-invitation_FO3985 | 335 | 1.000 | \gamma(A)=\gamma(B)=\gamma(C)=V | | |
| fong-spivak-invitation_FO3986 | 335 | 1.000 | \gamma(i)=(V, 0 ; 0) | | |
| fong-spivak-invitation_FO3987 | 335 | 1.000 | \gamma(f)=(V, V ; A) | | |
| fong-spivak-invitation_FO3988 | 336 | 1.000 | \gamma(g)=\gamma(h)=(V, V ; 0) | | |
| fong-spivak-invitation_FO3989 | 336 | 1.000 | U=\mathbb{R}-\{0\} \subseteq \mathbb{R} | | |
| fong-spivak-invitation_FO3990 | 336 | 1.000 | \mathbb{R}-U=\{0\} | | |
| fong-spivak-invitation_FO3991 | 336 | 1.000 | \neg U=\varnothing | | |
| fong-spivak-invitation_FO3992 | 336 | 1.000 | \mathbb{R}-\varnothing=\mathbb{R} | | |
| fong-spivak-invitation_FO3993 | 336 | 1.000 | \neg \neg U=\mathbb{R} | | |
| fong-spivak-invitation_FO3994 | 336 | 0.869 | \neg \neg U \subseteq{ }^{?} U | | |
| fong-spivak-invitation_FO3995 | 336 | 0.976 | V \in \mathbf{O p} | | |
| fong-spivak-invitation_FO3996 | 336 | 0.976 | \top \wedge V=V | | |
| fong-spivak-invitation_FO3997 | 336 | 0.976 | V=X | | |
| fong-spivak-invitation_FO3998 | 336 | 0.976 | \top \wedge X:=\top \cap X=X | | |
| fong-spivak-invitation_FO3999 | 336 | 1.000 | \top=\top \cap X=X | | |
| fong-spivak-invitation_FO4000 | 336 | 0.897 | (\top \vee V):=(X \cup V)=X | | |
| fong-spivak-invitation_FO4001 | 336 | 0.897 | (V \Rightarrow X)=\bigcup_{\{R \in \mathbf{O p} \mid R \cap V \subseteq X\}} R=X | | |
| fong-spivak-invitation_FO4002 | 336 | 1.000 | (X \cap V) \subseteq X | | |
| fong-spivak-invitation_FO4003 | 336 | 1.000 | (X \Rightarrow V)=V=\bigcup_{\{R \in \mathbf{O p} \mid R \cap X \leq V\}} R=Y | | |
| fong-spivak-invitation_FO4004 | 336 | 1.000 | R \cap X=R | | |
| fong-spivak-invitation_FO4005 | 336 | 0.992 | (\perp \vee V)=V | | |
| fong-spivak-invitation_FO4006 | 336 | 0.992 | V=\emptyset | | |
| fong-spivak-invitation_FO4007 | 336 | 0.992 | (\perp \vee \varnothing):=(\perp \cup \varnothing)= | | |
| fong-spivak-invitation_FO4008 | 336 | 0.679 | \perp=\perp \cup \varnothing=\varnothing | | |
| fong-spivak-invitation_FO4009 | 336 | 0.995 | (\perp \wedge V)=(\varnothing \cap V)=\varnothing | | |
| fong-spivak-invitation_FO4010 | 336 | 0.995 | (\perp \Rightarrow V)=\bigcup_{\{R \in \mathbf{O p} \mid R \cap \varnothing \subseteq V\}} R=X | | |
| fong-spivak-invitation_FO4011 | 336 | 0.835 | (X \cap \varnothing) \subseteq V | | |
| fong-spivak-invitation_FO4012 | 336 | 1.000 | \{S \mid p\} | | |
| fong-spivak-invitation_FO4013 | 336 | 0.508 | S=\mathbb{N} | | |
| fong-spivak-invitation_FO4014 | 336 | 0.508 | \leq s \leq 28 | | |
| fong-spivak-invitation_FO4015 | 336 | 0.508 | q(s) | | |
| fong-spivak-invitation_FO4016 | 336 | 1.000 | p(n, z) | | |
| fong-spivak-invitation_FO4017 | 336 | 1.000 | n \leq|z| | | |
| fong-spivak-invitation_FO4018 | 336 | 1.000 | \{0\} \subseteq \mathbb{N} | | |
| fong-spivak-invitation_FO4019 | 336 | 1.000 | \mathbb{N} \subseteq \mathbb{N} | | |
| fong-spivak-invitation_FO4020 | 336 | 1.000 | \varnothing \subseteq \mathbb{Z} | | |
| fong-spivak-invitation_FO4021 | 336 | 1.000 | \mathbb{Z} \subseteq \mathbb{Z} | | |
| fong-spivak-invitation_FO4022 | 337 | 0.532 | \forall(t: T) | | |
| fong-spivak-invitation_FO4023 | 337 | 1.000 | t_{i} \in T\left(V_{i}\right) | | |
| fong-spivak-invitation_FO4024 | 337 | 1.000 | t_{i} | | |
| fong-spivak-invitation_FO4025 | 337 | 1.000 | j(j(q)) \leq j(q) | | |
| fong-spivak-invitation_FO4026 | 337 | 0.999 | p:=j(q) | | |
| fong-spivak-invitation_FO4027 | 337 | 0.999 | j(p) \leq p | | |
| fong-spivak-invitation_FO4028 | 337 | 0.999 | p \cong j(p) | | |
| fong-spivak-invitation_FO4029 | 337 | 1.000 | p=j(p) | | |
| fong-spivak-invitation_FO4030 | 337 | 0.912 | s \in S(U), p(s) | | |
| fong-spivak-invitation_FO4031 | 338 | 0.916 | j(j(p(s))=j(p(s) | | |
| fong-spivak-invitation_FO4032 | 338 | 0.890 | 0 \leq 2 \leq 6 \leq 8 | | |
| fong-spivak-invitation_FO4033 | 338 | 0.890 | [2,6] \in O_{[0,8]} | | |
| fong-spivak-invitation_FO4034 | 338 | 0.603 | \in^{\text {? }} o_{[0,5]} \cup o_{[4,8]} | | |
| fong-spivak-invitation_FO4035 | 338 | 0.603 | \in^{\text {? }} o_{[0,5]} | | |
| fong-spivak-invitation_FO4036 | 338 | 1.000 | [2,6] \in^{?} o_{[4,8]} | | |
| fong-spivak-invitation_FO4037 | 338 | 0.984 | \mathbb{R} \subseteq \mathbb{I} \mathbb{R} | | |
| fong-spivak-invitation_FO4038 | 338 | 0.984 | U^{\prime} \subseteq \mathbb{I} \mathbb{R} | | |
| fong-spivak-invitation_FO4039 | 338 | 1.000 | U=U^{\prime} \cap \mathbb{R} | | |
| fong-spivak-invitation_FO4040 | 338 | 1.000 | U^{\prime}=\bigcup_{i \in I} o_{\left[a_{i}, b_{i}\right]} | | |
| fong-spivak-invitation_FO4041 | 338 | 1.000 | o_{\left[a_{i}, b_{i}\right]}=\left\{[d, u] \in \mathbb{R} \mid a_{i}<d<u<\right. | | |
| fong-spivak-invitation_FO4042 | 338 | 0.956 | \left.b_{i}\right\} | | |
| fong-spivak-invitation_FO4043 | 338 | 0.956 | \mathbb{R}=\{[x, x] \in \mathbb{I} \mathbb{R}\} | | |
| fong-spivak-invitation_FO4044 | 338 | 0.956 | U^{\prime} \cap \mathbb{R} | | |
| fong-spivak-invitation_FO4045 | 338 | 1.000 | B\left(m_{i}, r_{i}\right) | | |
| fong-spivak-invitation_FO4046 | 338 | 1.000 | m_{i}:=\frac{a_{i}+b_{i}}{2} | | |
| fong-spivak-invitation_FO4047 | 338 | 1.000 | r_{i}:= | | |
| fong-spivak-invitation_FO4048 | 338 | 0.500 | \frac{b_{i}-a_{i}}{2} | | |
| fong-spivak-invitation_FO4049 | 338 | 0.500 | \left(a_{i}, b_{i}\right) | | |
| fong-spivak-invitation_FO4050 | 338 | 1.000 | U=\bigcup_{j \in J} B\left(m_{j}, \epsilon_{j}\right) | | |
| fong-spivak-invitation_FO4051 | 338 | 1.000 | a_{j}:=m_{j}-\epsilon_{j} | | |
| fong-spivak-invitation_FO4052 | 338 | 1.000 | b_{j}:=m_{j}+\epsilon_{j} | | |
| fong-spivak-invitation_FO4053 | 338 | 0.991 | H_{X}(U):=\{f: U \cap R \rightarrow X \mid f | | |
| fong-spivak-invitation_FO4054 | 338 | 1.000 | f: U \cap R \rightarrow | | |
| fong-spivak-invitation_FO4055 | 338 | 1.000 | V \cap R \subseteq U \cap R | | |
| fong-spivak-invitation_FO4056 | 338 | 1.000 | U=\bigcup_{i} U_{i} | | |
| fong-spivak-invitation_FO4057 | 338 | 1.000 | f_{i}: U_{i} \cap R \rightarrow X | | |
| fong-spivak-invitation_FO4058 | 338 | 1.000 | U \cap R | | |
| fong-spivak-invitation_FO4059 | 338 | 1.000 | U \cap R=\left(\bigcup_{i} U_{i}\right) \cap R=\bigcup_{i}\left(U_{i} \cap R\right) | |