2106.07890: formula evidence

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318 rows

Inline formulas (first occurrence) (318)

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
2106.07890_FO0001	3	0.985	\mathcal{L}
2106.07890_FO0002	4	1.000	\widehat{\mathcal{L}}
2106.07890_FO0003	2	0.814	{ }^{+}
2106.07890_FO0004	2	0.814	\mathrm{B}^{+}
2106.07890_FO0005	3	0.894	\infty
2106.07890_FO0006	5	1.000	x \rightarrow y
2106.07890_FO0007	5	1.000	x
2106.07890_FO0008	5	1.000	y
2106.07890_FO0009	5	1.000	z
2106.07890_FO0010	5	0.997	h^{x}:=\operatorname{hom}(x,-)
2106.07890_FO0011	5	1.000	*
2106.07890_FO0012	5	1.000	h^{x}
2106.07890_FO0013	5	0.946	\widehat{\mathrm{C}}:=
2106.07890_FO0014	5	0.946	^{\mathrm{C}}
2106.07890_FO0015	6	0.709	h^{x} \sqcup h^{y}
2106.07890_FO0016	6	0.925	x=
2106.07890_FO0017	6	0.925	y=
2106.07890_FO0018	6	1.000	h^{\text {red }} \sqcup h^{\text {blue }}
2106.07890_FO0019	6	1.000	c
2106.07890_FO0020	6	1.000	h^{\text {red }}(c) \sqcup h^{\text {blue }}(c)
2106.07890_FO0021	6	0.963	h^{x} \times h^{y}: \mathrm{C} \rightarrow
2106.07890_FO0022	6	0.886	h^{\text {red }}(c) \times h^{\text {blue }}(c)
2106.07890_FO0023	6	1.000	h^{\text {red }} \times h^{\text {blue }}
2106.07890_FO0024	6	0.994	\widehat{\mathrm{C}}
2106.07890_FO0025	6	0.969	[]:, \widehat{\mathrm{C}} \times \widehat{\mathrm{C}} \rightarrow \widehat{\mathrm{C}}
2106.07890_FO0026	6	0.969	\widehat{\mathrm{C}}(F \times
2106.07890_FO0027	6	0.997	G, H) \cong \widehat{\mathrm{C}}(F,[G, H])
2106.07890_FO0028	6	0.997	F, G
2106.07890_FO0029	6	0.997	H
2106.07890_FO0030	6	0.877	\mathrm{A}(x, y)
2106.07890_FO0031	6	0.997	[G, H]
2106.07890_FO0032	6	1.000	x \leq y
2106.07890_FO0033	6	1.000	\wedge
2106.07890_FO0034	6	0.992	x \wedge y \leq z
2106.07890_FO0035	6	0.992	x \leq(y \Rightarrow z)
2106.07890_FO0036	6	1.000	x, y, z
2106.07890_FO0037	7	0.998	h^{c} \times h^{\text {red }}
2106.07890_FO0038	7	0.998	h^{\text {blue }}
2106.07890_FO0039	7	0.991	\left(h^{c} \times h^{\text {red }}\right)(d)
2106.07890_FO0040	7	0.991	\left(h^{\text {blue }}\right)(d)
2106.07890_FO0041	7	1.000	\left[h^{\text {red }}, h^{\text {blue }}\right](c)
2106.07890_FO0042	7	0.983	\emptyset
2106.07890_FO0043	7	0.996	h^{c} \times
2106.07890_FO0044	7	0.508	h^{\text {red }}(d)=h^{c}(d) \times h^{\text {red }}(d)
2106.07890_FO0045	7	0.508	d
2106.07890_FO0046	7	0.943	h^{\text {blue }}(d)
2106.07890_FO0047	7	0.991	\left(h^{c} \times h^{\text {red }}\right)(d)=*
2106.07890_FO0048	7	1.000	h^{\text {blue }}(d)=\emptyset
2106.07890_FO0049	7	1.000	\left(h^{c} \times h^{\text {red }}\right)(d) \rightarrow h^{\text {blue }}(d)
2106.07890_FO0050	7	0.936	\left[h^{\text {red }}, h^{\text {blue }}\right]
2106.07890_FO0051	7	0.936	=*
2106.07890_FO0052	7	0.996	\left[h^{\text {red }}, h^{\text {blue }}\right](
2106.07890_FO0053	7	0.996	)=\emptyset
2106.07890_FO0054	7	1.000	\left[h^{x}, h^{y}\right]: \mathrm{C} \rightarrow
2106.07890_FO0055	8	0.781	\mathrm{C}^{\mathrm{op}} \rightarrow
2106.07890_FO0056	8	0.999	h^{x}=\operatorname{hom}(x,-)
2106.07890_FO0057	8	1.000	\mathrm{C}^{\mathrm{op}}
2106.07890_FO0058	8	0.994	\mathrm{C}^{\mathrm{op}}(x, y):=\mathrm{C}(y, x)
2106.07890_FO0059	8	0.994	\mathrm{C}(x, y)
2106.07890_FO0060	8	0.793	[0,1]
2106.07890_FO0061	8	1.000	a
2106.07890_FO0062	8	1.000	b
2106.07890_FO0063	9	0.957	(\mathcal{V}, \leq, \otimes, 1)
2106.07890_FO0064	9	0.957	(\mathcal{V}, \leq)
2106.07890_FO0065	9	0.994	(\mathcal{V}, \otimes, 1)
2106.07890_FO0066	9	0.994	x \otimes y \leq x^{\prime} \otimes y^{\prime}
2106.07890_FO0067	9	0.994	x \leq x^{\prime}
2106.07890_FO0068	9	1.000	y \leq y^{\prime}
2106.07890_FO0069	9	0.835	\mathcal{V}, \leq, \otimes, 1
2106.07890_FO0070	9	1.000	\mathcal{V}
2106.07890_FO0071	9	1.000	\mathcal{C}
2106.07890_FO0072	9	1.000	\mathcal{C}(x, y) \in \mathcal{V}
2106.07890_FO0073	9	0.998	x, y, z \in \mathcal{C}
2106.07890_FO0074	9	1.000	[0,1]:=\{x \in \mathbb{R}: 0 \leq x \leq 1\}
2106.07890_FO0075	9	0.923	\leq
2106.07890_FO0076	9	0.653	(x, y) \mapsto \mathcal{C}(x, y)
2106.07890_FO0077	9	0.999	x, y \in \mathcal{C}
2106.07890_FO0078	9	0.999	\mathcal{C}(x, x)=1
2106.07890_FO0079	9	0.999	x \in \mathcal{C}
2106.07890_FO0080	9	0.999	\mathcal{C}(y, z) \mathcal{C}(x, y) \leq \mathcal{C}(x, z)
2106.07890_FO0081	9	1.000	[x, y] \in \mathcal{V}
2106.07890_FO0082	9	1.000	\mathcal{V}(x, y)
2106.07890_FO0083	10	1.000	[x, y]
2106.07890_FO0084	10	1.000	(x, y) \mapsto[x, y]
2106.07890_FO0085	10	1.000	1 \leq[x, x]
2106.07890_FO0086	10	1.000	[y, z] \otimes[x, y] \leq[x, z]
2106.07890_FO0087	10	1.000	1 \otimes x \leq x
2106.07890_FO0088	10	1.000	[y, z] \leq[y, z]
2106.07890_FO0089	10	1.000	[y, z] \otimes y \leq z
2106.07890_FO0090	10	1.000	[x, y] \otimes x \leq y
2106.07890_FO0091	10	1.000	[y, z] \otimes[x, y] \otimes x \leq z
2106.07890_FO0092	10	1.000	a \otimes b:=a b
2106.07890_FO0093	10	1.000	a, b \in[0,1]
2106.07890_FO0094	10	1.000	a b \leq c
2106.07890_FO0095	10	1.000	a \leq[b, c]
2106.07890_FO0096	10	1.000	c<b
2106.07890_FO0097	10	1.000	a \leq \frac{c}{b}=[b, c]
2106.07890_FO0098	10	1.000	b \leq c
2106.07890_FO0099	10	1.000	[b, c]=1
2106.07890_FO0101	10	1.000	a, b, c \in[0,1]
2106.07890_FO0102	10	0.528	\min \{b / a, 1\}
2106.07890_FO0103	10	1.000	b / 0 \geq 1
2106.07890_FO0104	10	1.000	0 \leq b \leq 1
2106.07890_FO0105	10	1.000	a b
2106.07890_FO0106	10	1.000	a \otimes b
2106.07890_FO0107	10	1.000	a \times b
2106.07890_FO0108	10	1.000	\min \{a, b\}
2106.07890_FO0109	10	0.999	a \sqcup b
2106.07890_FO0110	10	0.722	\mathrm{C}(x, y)=1
2106.07890_FO0111	10	1.000	\mathrm{C}(x, y)=0
2106.07890_FO0112	10	1.000	2:=\{0,1\}
2106.07890_FO0113	11	1.000	\pi(y \mid x)
2106.07890_FO0114	11	0.959	\pi(y \mid x)=0
2106.07890_FO0115	11	0.598	\pi(x \mid x)=
2106.07890_FO0116	11	1.000	\mathcal{D}
2106.07890_FO0117	12	0.963	\mathcal{L} \rightarrow \widehat{\mathcal{L}}
2106.07890_FO0118	12	1.000	\mathcal{L} \rightarrow \mathcal{D}
2106.07890_FO0119	12	0.379	\mathcal{D} \cdots \cdots>\widehat{\mathcal{L}}
2106.07890_FO0120	12	0.940	(\mathcal{V}, \otimes, \leq, 1)
2106.07890_FO0121	12	0.940	\mathcal{C} \rightarrow \mathcal{D}
2106.07890_FO0122	12	0.940	f: \mathcal{C} \rightarrow \mathcal{D}
2106.07890_FO0123	12	1.000	\mathcal{D}=\mathcal{V}
2106.07890_FO0124	12	1.000	f: \mathcal{C} \rightarrow \mathcal{V}
2106.07890_FO0125	12	1.000	\mathcal{C}(x, y) \leq
2106.07890_FO0126	12	1.000	\mathcal{V}(f x, f y)
2106.07890_FO0127	12	1.000	\mathcal{D}^{\mathcal{C}}
2106.07890_FO0128	12	1.000	\mathcal{D}^{\mathcal{C}}(f, g) \in \mathcal{V}
2106.07890_FO0129	12	1.000	f, g: \mathcal{C} \rightarrow \mathcal{D}
2106.07890_FO0130	13	0.940	\widehat{\mathcal{C}}:=[0,1]^{\mathcal{C}}
2106.07890_FO0131	13	1.000	f, g: \mathcal{C} \rightarrow[0,1]
2106.07890_FO0132	13	1.000	\widehat{\mathcal{C}}(f, g)=\inf _{c \in \mathcal{C}}\{1, g c / f c\}
2106.07890_FO0133	13	1.000	\mathcal{V}=[0,1]
2106.07890_FO0134	13	1.000	\mathcal{D}=[0,1]
2106.07890_FO0135	13	1.000	\mathcal{C} \rightarrow[0,1]
2106.07890_FO0136	13	1.000	c, d \in \mathcal{C}
2106.07890_FO0137	13	1.000	\mathcal{C}(c, d) \leq[\mathcal{C}(x, c), \mathcal{C}(x, d)]
2106.07890_FO0138	13	1.000	\mathcal{C}(c, d) \mathcal{C}(x, c) \leq \mathcal{C}(x, d)
2106.07890_FO0139	13	0.990	h^{x}:=
2106.07890_FO0140	13	0.999	\mathcal{C}(x,-)
2106.07890_FO0141	13	0.999	f: \mathcal{C} \rightarrow[0,1]
2106.07890_FO0142	13	1.000	f=h^{x}
2106.07890_FO0143	13	1.000	x \mapsto h^{x}
2106.07890_FO0144	13	0.991	\widehat{\mathcal{C}}\left(h^{x}, f\right)=f(x)
2106.07890_FO0145	13	1.000	f
2106.07890_FO0146	13	1.000	\widehat{\mathcal{C}}\left(h^{x}, f\right)=\inf _{c \in \mathcal{C}}\left\{\left[h^{x}(c), f c\right]\right\}
2106.07890_FO0147	13	0.996	c \in \mathcal{C}
2106.07890_FO0148	13	0.996	\widehat{\mathcal{C}}\left(h^{x}, f\right) \leq\left[h^{x}(c), f c\right]
2106.07890_FO0149	13	0.996	c=x
2106.07890_FO0150	14	0.738	\mathcal{C}(x, c) \leq
2106.07890_FO0151	14	0.971	[f x, f c]
2106.07890_FO0152	14	0.971	\mathcal{C}(x, c) \leq[f x, f c]
2106.07890_FO0153	14	1.000	\mathcal{C}(x, c) f x \leq f c
2106.07890_FO0154	14	1.000	f x \leq[\mathcal{C}(x, c), f c]
2106.07890_FO0155	14	1.000	f x \leq \inf _{c \in \mathcal{C}}\left\{\left[h^{x}(c), f c\right]\right\}=
2106.07890_FO0156	14	1.000	\widehat{\mathcal{C}}\left(h^{x}, f\right)
2106.07890_FO0157	14	0.900	\mathcal{C}(y, x)=\widehat{\mathcal{C}}\left(h^{x}, h^{y}\right)
2106.07890_FO0158	14	0.900	x, y
2106.07890_FO0159	14	1.000	f=h^{y}
2106.07890_FO0160	14	1.000	\widehat{\mathcal{C}}\left(h^{x}, h^{y}\right)=h^{y}(x)=\mathcal{C}(y, x)
2106.07890_FO0161	14	1.000	\mathcal{C}^{o p} \rightarrow \widehat{\mathcal{C}}
2106.07890_FO0162	14	1.000	\mathcal{C}^{o p}
2106.07890_FO0163	14	1.000	\widehat{\mathcal{C}}
2106.07890_FO0164	14	1.000	o p
2106.07890_FO0165	14	1.000	\mathcal{C}^{\mathrm{op}}
2106.07890_FO0166	14	1.000	Y
2106.07890_FO0167	14	0.999	\{X \rightarrow Y\}
2106.07890_FO0168	14	0.999	X
2106.07890_FO0169	14	1.000	\widehat{\mathcal{C}}=[0,1]^{\mathcal{C}}
2106.07890_FO0170	14	0.674	\widehat{\mathcal{L}}:=[0,1]^{\mathcal{L}}
2106.07890_FO0171	14	1.000	h^{x}:=\mathcal{L}(x,-)
2106.07890_FO0172	14	1.000	x \leq c
2106.07890_FO0173	15	1.000	\mathcal{L}(x, y)=a \neq 0
2106.07890_FO0174	15	1.000	h^{y}
2106.07890_FO0175	15	0.850	{ }^{\mathrm{C}}
2106.07890_FO0176	16	0.666	F: \mathrm{J} \rightarrow \mathrm{C}
2106.07890_FO0177	16	0.965	\lim F
2106.07890_FO0178	16	0.999	Z
2106.07890_FO0179	16	0.999	\mathrm{C}(Z, \lim F) \cong \operatorname{Set}^{\mathrm{J}}(*, \mathrm{C}(Z, F))
2106.07890_FO0180	16	0.647	(Z, F)
2106.07890_FO0181	16	1.000	=(i) \rightarrow \mathrm{C}(Z, F i)
2106.07890_FO0182	16	1.000	i
2106.07890_FO0183	16	0.755	Z \rightarrow F i
2106.07890_FO0184	16	0.999	W: \mathcal{J} \rightarrow \mathcal{V}
2106.07890_FO0185	16	1.000	\mathcal{J}
2106.07890_FO0186	16	1.000	\mathcal{E}
2106.07890_FO0187	16	1.000	F: \mathcal{J} \rightarrow \mathcal{E}
2106.07890_FO0188	16	1.000	F
2106.07890_FO0189	16	1.000	\lim _{W} F
2106.07890_FO0190	16	1.000	\mathcal{E} \rightarrow \mathcal{V}
2106.07890_FO0191	17	1.000	\mathcal{E}\left(Z, \lim _{W} F\right) \cong
2106.07890_FO0192	17	1.000	\mathcal{V}^{\mathcal{J}}(W, \mathcal{E}(Z, F))
2106.07890_FO0193	17	1.000	W: \mathcal{J}^{\mathrm{op}} \rightarrow \mathcal{V}
2106.07890_FO0194	17	0.997	\operatorname{colim}_{W} F
2106.07890_FO0195	17	1.000	\mathcal{E}=\widehat{\mathcal{L}}:=[0,1]^{\mathcal{L}}
2106.07890_FO0196	17	1.000	\mathcal{J}(i, j)=\delta_{i j}
2106.07890_FO0197	17	1.000	i, j \in\{1,2\}
2106.07890_FO0198	17	1.000	W: \mathcal{J} \rightarrow
2106.07890_FO0199	17	0.516	w_{1}:=W(1)
2106.07890_FO0200	17	0.516	w_{2}:=
2106.07890_FO0201	17	1.000	W(2)
2106.07890_FO0202	17	1.000	f, g: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0203	17	1.000	F: \mathcal{J} \rightarrow
2106.07890_FO0204	17	0.801	f:=F(1)
2106.07890_FO0205	17	0.801	g:=F(2)
2106.07890_FO0206	17	1.000	W
2106.07890_FO0207	17	0.996	\left(w_{1}, f\right) \times\left(w_{2}, g\right): \mathcal{L} \rightarrow[0,1]
2106.07890_FO0208	17	0.996	c \mapsto
2106.07890_FO0209	17	0.999	\min \left\{\frac{f c}{w_{1}}, \frac{g c}{w_{2}}, 1\right\}
2106.07890_FO0210	17	1.000	c \mapsto \min \left\{\frac{f c}{w_{1}}, \frac{g c}{w_{2}}, 1\right\}
2106.07890_FO0211	17	1.000	Z: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0212	17	1.000	\widehat{\mathcal{L}}(Z, F)
2106.07890_FO0213	17	1.000	J \rightarrow[0,1]
2106.07890_FO0214	17	1.000	J
2106.07890_FO0215	17	1.000	[0,1]^{\mathcal{J}}
2106.07890_FO0216	17	0.894	f=F(1)
2106.07890_FO0217	17	0.894	g=F(2)
2106.07890_FO0218	18	1.000	\mathcal{L}(c, d) \leq\left[\lim _{W} F(c), \lim _{W} F(d)\right]
2106.07890_FO0219	18	1.000	\mathcal{L}(c, d) \lim _{W} F(c) \leq
2106.07890_FO0220	18	1.000	\lim _{W} F(d)
2106.07890_FO0221	18	1.000	c, d \in \mathcal{L}
2106.07890_FO0222	18	0.967	g
2106.07890_FO0223	18	1.000	\left(w_{1}, f\right) \times\left(w_{2}, g\right)
2106.07890_FO0224	18	1.000	w_{1}, w_{2} \in[0,1]
2106.07890_FO0225	18	1.000	\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right): \mathcal{L} \rightarrow[0,1]
2106.07890_FO0226	18	1.000	\pi(c \mid x) \leq w_{1}
2106.07890_FO0227	18	0.996	\pi(c \mid y) \leq w_{2}
2106.07890_FO0228	18	1.000	w_{1} h^{x}(c) \times w_{2} h^{y}(c)
2106.07890_FO0229	18	1.000	\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right)
2106.07890_FO0230	18	1.000	w_{1}=w_{2}=1
2106.07890_FO0231	18	1.000	f \times g
2106.07890_FO0232	18	1.000	f, g, h: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0233	19	0.991	W: \mathcal{J}^{\mathrm{op}} \rightarrow[0,1]
2106.07890_FO0234	19	1.000	w_{2}:=W(2)
2106.07890_FO0235	19	1.000	F: \mathcal{J} \rightarrow \widehat{\mathcal{L}}
2106.07890_FO0236	19	0.998	\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right): \mathcal{L} \rightarrow[0,1]
2106.07890_FO0237	19	1.000	\max \left\{w_{1} f c, w_{2} g c\right\}
2106.07890_FO0238	19	0.539	\mathcal{J}=\mathcal{J}^{\mathrm{op}}
2106.07890_FO0239	19	1.000	\widehat{\mathcal{L}}(F-, Z)
2106.07890_FO0240	19	1.000	i=1,2
2106.07890_FO0241	20	0.912	c \mapsto \max \left\{w_{1} f c, w_{2} g c\right\}
2106.07890_FO0242	20	1.000	\mathcal{L}(c, d) \leq\left[\operatorname{colim}_{W} F(c), \operatorname{colim}_{W} F(d)\right]
2106.07890_FO0243	20	1.000	\mathcal{L}(c, d) \operatorname{colim}_{W} F(c) \leq \operatorname{colim}_{W} F(d)
2106.07890_FO0244	20	1.000	\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right)
2106.07890_FO0245	20	1.000	G, H: \mathrm{C} \rightarrow
2106.07890_FO0246	20	1.000	c \mapsto \widehat{\mathrm{C}}\left(h^{c} \times G, H\right)
2106.07890_FO0247	20	1.000	h^{c} \times G
2106.07890_FO0248	20	1.000	[f, g]: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0249	20	1.000	h^{c} \times f
2106.07890_FO0250	20	1.000	\mathcal{L}(c, d) \leq[[f, g](c),[f, g](d)]
2106.07890_FO0251	20	1.000	\mathcal{L}(c, d)[f, g](c) \leq[f, g](d)
2106.07890_FO0252	20	1.000	\mathcal{L}(c, d) \leq[g(c), g(d)]
2106.07890_FO0253	20	1.000	\mathcal{L}(c, d) g(c) \leq g(d)
2106.07890_FO0254	20	1.000	\mathcal{L}(c, d) \widehat{\mathcal{L}}\left(h^{c}, g\right) \leq \widehat{\mathcal{L}}\left(h^{d}, g\right)
2106.07890_FO0255	20	1.000	\mathcal{L}(c, d)\left(\widehat{\mathcal{L}}\left(h^{c}, g\right) \times \widehat{\mathcal{L}}(f, g)\right) \leq \widehat{\mathcal{L}}\left(h^{d}, g\right) \times \widehat{\mathcal{L}}(f, g)
2106.07890_FO0256	20	1.000	\mathcal{L}(c, d) \widehat{\mathcal{L}}\left(h^{c} \times f, g\right) \leq \widehat{\mathcal{L}}\left(h^{d} \times f, g\right)
2106.07890_FO0257	21	1.000	\mathcal{L}(c, d) \leq\left[\widehat{\mathcal{L}}\left(h^{c} \times f, g\right), \widehat{\mathcal{L}}\left(h^{d} \times f, g\right)\right]=[[f, g](c),[f, g](d)]
2106.07890_FO0258	21	1.000	(f \times-): \widehat{\mathcal{L}} \rightarrow \widehat{\mathcal{L}}
2106.07890_FO0259	21	1.000	[f,-]
2106.07890_FO0260	21	1.000	-\times f
2106.07890_FO0261	21	1.000	\widehat{\mathcal{L}}\left(h^{c},[f, g]\right)=[f, g](c)
2106.07890_FO0262	21	1.000	[f, g](c)=\widehat{\mathcal{L}}\left(h^{c} \times f, g\right)
2106.07890_FO0263	21	1.000	\widehat{\mathcal{L}}\left(h^{c},[f, g]\right)=\widehat{\mathcal{L}}\left(h^{c} \times f, g\right)
2106.07890_FO0264	21	1.000	h^{c}
2106.07890_FO0265	21	1.000	h
2106.07890_FO0266	21	1.000	[f, g]
2106.07890_FO0267	21	1.000	x, y \in \mathcal{L}
2106.07890_FO0268	21	0.999	\left[h^{x}, h^{y}\right](c)=
2106.07890_FO0269	21	1.000	d=c
2106.07890_FO0270	21	1.000	\pi(c \mid y)=
2106.07890_FO0271	21	1.000	\pi(c \mid c) \times \pi(c \mid x)=\pi(c \mid x) \neq 0
2106.07890_FO0272	21	1.000	\left[h^{x}, h^{y}\right](c) \neq 0
2106.07890_FO0273	21	1.000	x \Rightarrow y
2106.07890_FO0274	21	1.000	\left[h^{x}, h^{y}\right]: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0275	21	1.000	\mathcal{M}
2106.07890_FO0276	21	0.999	[0, \infty]
2106.07890_FO0277	21	1.000	a+\infty:=\infty
2106.07890_FO0278	21	1.000	\infty+a:=\infty
2106.07890_FO0279	21	0.998	b \leq a
2106.07890_FO0280	22	0.650	d_{\mathcal{M}}
2106.07890_FO0281	22	1.000	[a, b]:=\max \{b-a, 0\}
2106.07890_FO0282	22	1.000	a \times b=\max \{a, b\}
2106.07890_FO0283	22	0.997	a \sqcup b=\min \{a, b\}
2106.07890_FO0284	22	0.997	-\ln :[0,1] \rightarrow[0, \infty]
2106.07890_FO0285	22	1.000	[0, \infty] \rightarrow[0,1]
2106.07890_FO0286	22	1.000	a \mapsto \exp (-a)
2106.07890_FO0287	22	1.000	\mathcal{M}(x, y):=-\ln \mathcal{L}(x, y)
2106.07890_FO0288	22	1.000	0 \geq \mathcal{M}(x, x)
2106.07890_FO0289	22	1.000	\mathcal{M}(x, y)+\mathcal{M}(y, z) \geq \mathcal{M}(x, z)
2106.07890_FO0290	22	0.998	d_{\mathcal{M}}(x, y):=\mathcal{M}(x, y)
2106.07890_FO0291	22	1.000	\widehat{\mathcal{M}}:=[0, \infty]^{\mathcal{M}}
2106.07890_FO0292	23	1.000	f: \mathcal{M} \rightarrow[0, \infty]
2106.07890_FO0293	23	1.000	d_{M}(x, y) \geq[f x, f y]=\max \{f y-f x, 0\}
2106.07890_FO0294	23	1.000	[a, b]=\max \{b-a, 0\}
2106.07890_FO0295	23	1.000	d_{[0, \infty]}(a, b)
2106.07890_FO0296	23	1.000	d_{[0, \infty]}(f x, f y) \leq d_{M}(x, y)
2106.07890_FO0297	23	1.000	\widehat{\mathcal{M}}
2106.07890_FO0298	23	0.905	\mathcal{M}^{\mathrm{op}} \rightarrow \widehat{\mathcal{M}}
2106.07890_FO0299	23	0.905	x \mapsto d_{\mathcal{M}}(x,-)
2106.07890_FO0300	23	0.998	f, g \in \widehat{\mathcal{M}}
2106.07890_FO0301	23	0.998	w_{1}, w_{2}
2106.07890_FO0302	23	1.000	f, g: \mathcal{M} \rightarrow[0, \infty]
2106.07890_FO0303	23	1.000	w_{1}, w_{2} \in[0, \infty]
2106.07890_FO0304	23	1.000	f^{\prime}, g^{\prime}: \mathcal{L} \rightarrow[0,1]
2106.07890_FO0305	23	1.000	w_{1}^{\prime}, w_{2}^{\prime} \in[0,1]
2106.07890_FO0306	23	1.000	f^{\prime} c=\exp (-f c), g^{\prime} c=\exp (-g c)
2106.07890_FO0307	23	1.000	w_{i}^{\prime}=\exp \left(-w_{i}\right)
2106.07890_FO0308	24	1.000	d_{\mathcal{M}}(x,-)
2106.07890_FO0309	24	0.968	((-\infty, \infty], \oplus, \odot)
2106.07890_FO0310	24	1.000	([0, \infty], \oplus, \odot)
2106.07890_FO0311	24	0.999	w_{1}=w_{2}=0
2106.07890_FO0312	24	0.999	[0, \infty] \times \widehat{\mathcal{M}} \rightarrow \widehat{\mathcal{M}}
2106.07890_FO0313	24	0.999	(s, f) \mapsto s \odot f
2106.07890_FO0314	24	0.775	(s \cdot f)(x)=f(x)+s
2106.07890_FO0315	24	1.000	\left(s_{1} \oplus s_{2}\right) f(x)=\min \left\{s_{1}, s_{2}\right\} f(x)=\min \left\{f x+s_{1}, f x+s_{2}\right\}
2106.07890_FO0316	24	1.000	\left(s_{1} \odot s_{2}\right) f(x)=f x+s_{1}+s_{2}
2106.07890_FO0317	24	1.000	\left(s_{1}\right) \odot\left(s_{2} \odot f\right)(x)=
2106.07890_FO0318	24	1.000	\left(s_{2} \odot f\right)(x)+s_{1}=f x+s_{2}+s_{1}
2106.07890_FO0319	26	0.934	\sqcup

Identifier

Page

Conf.

LaTeX source

Rendered

Image

2106.07890_FO0001

0.985

\mathcal{L}

2106.07890_FO0002

1.000

\widehat{\mathcal{L}}

2106.07890_FO0003

0.814

{ }^{+}

2106.07890_FO0004

0.814

\mathrm{B}^{+}

2106.07890_FO0005

0.894

\infty

2106.07890_FO0006

1.000

x \rightarrow y

2106.07890_FO0007

1.000

2106.07890_FO0008

1.000

2106.07890_FO0009

1.000

2106.07890_FO0010

0.997

h^{x}:=\operatorname{hom}(x,-)

2106.07890_FO0011

1.000

2106.07890_FO0012

1.000

h^{x}

2106.07890_FO0013

0.946

\widehat{\mathrm{C}}:=

2106.07890_FO0014

0.946

^{\mathrm{C}}

2106.07890_FO0015

0.709

h^{x} \sqcup h^{y}

2106.07890_FO0016

0.925

2106.07890_FO0017

0.925

2106.07890_FO0018

1.000

h^{\text {red }} \sqcup h^{\text {blue }}

2106.07890_FO0019

1.000

2106.07890_FO0020

1.000

h^{\text {red }}(c) \sqcup h^{\text {blue }}(c)

2106.07890_FO0021

0.963

h^{x} \times h^{y}: \mathrm{C} \rightarrow

2106.07890_FO0022

0.886

h^{\text {red }}(c) \times h^{\text {blue }}(c)

2106.07890_FO0023

1.000

h^{\text {red }} \times h^{\text {blue }}

2106.07890_FO0024

0.994

\widehat{\mathrm{C}}

2106.07890_FO0025

0.969

[]:, \widehat{\mathrm{C}} \times \widehat{\mathrm{C}} \rightarrow \widehat{\mathrm{C}}

2106.07890_FO0026

0.969

\widehat{\mathrm{C}}(F \times

2106.07890_FO0027

0.997

G, H) \cong \widehat{\mathrm{C}}(F,[G, H])

2106.07890_FO0028

0.997

F, G

2106.07890_FO0029

0.997

2106.07890_FO0030

0.877

\mathrm{A}(x, y)

2106.07890_FO0031

0.997

[G, H]

2106.07890_FO0032

1.000

x \leq y

2106.07890_FO0033

1.000

\wedge

2106.07890_FO0034

0.992

x \wedge y \leq z

2106.07890_FO0035

0.992

x \leq(y \Rightarrow z)

2106.07890_FO0036

1.000

x, y, z

2106.07890_FO0037

0.998

h^{c} \times h^{\text {red }}

2106.07890_FO0038

0.998

h^{\text {blue }}

2106.07890_FO0039

0.991

\left(h^{c} \times h^{\text {red }}\right)(d)

2106.07890_FO0040

0.991

\left(h^{\text {blue }}\right)(d)

2106.07890_FO0041

1.000

\left[h^{\text {red }}, h^{\text {blue }}\right](c)

2106.07890_FO0042

0.983

\emptyset

2106.07890_FO0043

0.996

h^{c} \times

2106.07890_FO0044

0.508

h^{\text {red }}(d)=h^{c}(d) \times h^{\text {red }}(d)

2106.07890_FO0045

0.508

2106.07890_FO0046

0.943

h^{\text {blue }}(d)

2106.07890_FO0047

0.991

\left(h^{c} \times h^{\text {red }}\right)(d)=*

2106.07890_FO0048

1.000

h^{\text {blue }}(d)=\emptyset

2106.07890_FO0049

1.000

\left(h^{c} \times h^{\text {red }}\right)(d) \rightarrow h^{\text {blue }}(d)

2106.07890_FO0050

0.936

\left[h^{\text {red }}, h^{\text {blue }}\right]

2106.07890_FO0051

0.936

2106.07890_FO0052

0.996

\left[h^{\text {red }}, h^{\text {blue }}\right](

2106.07890_FO0053

0.996

)=\emptyset

2106.07890_FO0054

1.000

\left[h^{x}, h^{y}\right]: \mathrm{C} \rightarrow

2106.07890_FO0055

0.781

\mathrm{C}^{\mathrm{op}} \rightarrow

2106.07890_FO0056

0.999

h^{x}=\operatorname{hom}(x,-)

2106.07890_FO0057

1.000

\mathrm{C}^{\mathrm{op}}

2106.07890_FO0058

0.994

\mathrm{C}^{\mathrm{op}}(x, y):=\mathrm{C}(y, x)

2106.07890_FO0059

0.994

\mathrm{C}(x, y)

2106.07890_FO0060

0.793

[0,1]

2106.07890_FO0061

1.000

2106.07890_FO0062

1.000

2106.07890_FO0063

0.957

(\mathcal{V}, \leq, \otimes, 1)

2106.07890_FO0064

0.957

(\mathcal{V}, \leq)

2106.07890_FO0065

0.994

(\mathcal{V}, \otimes, 1)

2106.07890_FO0066

0.994

x \otimes y \leq x^{\prime} \otimes y^{\prime}

2106.07890_FO0067

0.994

x \leq x^{\prime}

2106.07890_FO0068

1.000

y \leq y^{\prime}

2106.07890_FO0069

0.835

\mathcal{V}, \leq, \otimes, 1

2106.07890_FO0070

1.000

\mathcal{V}

2106.07890_FO0071

1.000

\mathcal{C}

2106.07890_FO0072

1.000

\mathcal{C}(x, y) \in \mathcal{V}

2106.07890_FO0073

0.998

x, y, z \in \mathcal{C}

2106.07890_FO0074

1.000

[0,1]:=\{x \in \mathbb{R}: 0 \leq x \leq 1\}

2106.07890_FO0075

0.923

\leq

2106.07890_FO0076

0.653

(x, y) \mapsto \mathcal{C}(x, y)

2106.07890_FO0077

0.999

x, y \in \mathcal{C}

2106.07890_FO0078

0.999

\mathcal{C}(x, x)=1

2106.07890_FO0079

0.999

x \in \mathcal{C}

2106.07890_FO0080

0.999

\mathcal{C}(y, z) \mathcal{C}(x, y) \leq \mathcal{C}(x, z)

2106.07890_FO0081

1.000

[x, y] \in \mathcal{V}

2106.07890_FO0082

1.000

\mathcal{V}(x, y)

2106.07890_FO0083

1.000

[x, y]

2106.07890_FO0084

1.000

(x, y) \mapsto[x, y]

2106.07890_FO0085

1.000

1 \leq[x, x]

2106.07890_FO0086

1.000

[y, z] \otimes[x, y] \leq[x, z]

2106.07890_FO0087

1.000

1 \otimes x \leq x

2106.07890_FO0088

1.000

[y, z] \leq[y, z]

2106.07890_FO0089

1.000

[y, z] \otimes y \leq z

2106.07890_FO0090

1.000

[x, y] \otimes x \leq y

2106.07890_FO0091

1.000

[y, z] \otimes[x, y] \otimes x \leq z

2106.07890_FO0092

1.000

a \otimes b:=a b

2106.07890_FO0093

1.000

a, b \in[0,1]

2106.07890_FO0094

1.000

a b \leq c

2106.07890_FO0095

1.000

a \leq[b, c]

2106.07890_FO0096

1.000

c<b

2106.07890_FO0097

1.000

a \leq \frac{c}{b}=[b, c]

2106.07890_FO0098

1.000

b \leq c

2106.07890_FO0099

1.000

[b, c]=1

2106.07890_FO0101

1.000

a, b, c \in[0,1]

2106.07890_FO0102

0.528

\min \{b / a, 1\}

2106.07890_FO0103

1.000

b / 0 \geq 1

2106.07890_FO0104

1.000

0 \leq b \leq 1

2106.07890_FO0105

1.000

a b

2106.07890_FO0106

1.000

a \otimes b

2106.07890_FO0107

1.000

a \times b

2106.07890_FO0108

1.000

\min \{a, b\}

2106.07890_FO0109

0.999

a \sqcup b

2106.07890_FO0110

0.722

\mathrm{C}(x, y)=1

2106.07890_FO0111

1.000

\mathrm{C}(x, y)=0

2106.07890_FO0112

1.000

2:=\{0,1\}

2106.07890_FO0113

1.000

\pi(y \mid x)

2106.07890_FO0114

0.959

\pi(y \mid x)=0

2106.07890_FO0115

0.598

\pi(x \mid x)=

2106.07890_FO0116

1.000

\mathcal{D}

2106.07890_FO0117

0.963

\mathcal{L} \rightarrow \widehat{\mathcal{L}}

2106.07890_FO0118

1.000

\mathcal{L} \rightarrow \mathcal{D}

2106.07890_FO0119

0.379

\mathcal{D} \cdots \cdots>\widehat{\mathcal{L}}

2106.07890_FO0120

0.940

(\mathcal{V}, \otimes, \leq, 1)

2106.07890_FO0121

0.940

\mathcal{C} \rightarrow \mathcal{D}

2106.07890_FO0122

0.940

f: \mathcal{C} \rightarrow \mathcal{D}

2106.07890_FO0123

1.000

\mathcal{D}=\mathcal{V}

2106.07890_FO0124

1.000

f: \mathcal{C} \rightarrow \mathcal{V}

2106.07890_FO0125

1.000

\mathcal{C}(x, y) \leq

2106.07890_FO0126

1.000

\mathcal{V}(f x, f y)

2106.07890_FO0127

1.000

\mathcal{D}^{\mathcal{C}}

2106.07890_FO0128

1.000

\mathcal{D}^{\mathcal{C}}(f, g) \in \mathcal{V}

2106.07890_FO0129

1.000

f, g: \mathcal{C} \rightarrow \mathcal{D}

2106.07890_FO0130

0.940

\widehat{\mathcal{C}}:=[0,1]^{\mathcal{C}}

2106.07890_FO0131

1.000

f, g: \mathcal{C} \rightarrow[0,1]

2106.07890_FO0132

1.000

\widehat{\mathcal{C}}(f, g)=\inf _{c \in \mathcal{C}}\{1, g c / f c\}

2106.07890_FO0133

1.000

\mathcal{V}=[0,1]

2106.07890_FO0134

1.000

\mathcal{D}=[0,1]

2106.07890_FO0135

1.000

\mathcal{C} \rightarrow[0,1]

2106.07890_FO0136

1.000

c, d \in \mathcal{C}

2106.07890_FO0137

1.000

\mathcal{C}(c, d) \leq[\mathcal{C}(x, c), \mathcal{C}(x, d)]

2106.07890_FO0138

1.000

\mathcal{C}(c, d) \mathcal{C}(x, c) \leq \mathcal{C}(x, d)

2106.07890_FO0139

0.990

h^{x}:=

2106.07890_FO0140

0.999

\mathcal{C}(x,-)

2106.07890_FO0141

0.999

f: \mathcal{C} \rightarrow[0,1]

2106.07890_FO0142

1.000

f=h^{x}

2106.07890_FO0143

1.000

x \mapsto h^{x}

2106.07890_FO0144

0.991

\widehat{\mathcal{C}}\left(h^{x}, f\right)=f(x)

2106.07890_FO0145

1.000

2106.07890_FO0146

1.000

\widehat{\mathcal{C}}\left(h^{x}, f\right)=\inf _{c \in \mathcal{C}}\left\{\left[h^{x}(c), f c\right]\right\}

2106.07890_FO0147

0.996

c \in \mathcal{C}

2106.07890_FO0148

0.996

\widehat{\mathcal{C}}\left(h^{x}, f\right) \leq\left[h^{x}(c), f c\right]

2106.07890_FO0149

0.996

c=x

2106.07890_FO0150

0.738

\mathcal{C}(x, c) \leq

2106.07890_FO0151

0.971

[f x, f c]

2106.07890_FO0152

0.971

\mathcal{C}(x, c) \leq[f x, f c]

2106.07890_FO0153

1.000

\mathcal{C}(x, c) f x \leq f c

2106.07890_FO0154

1.000

f x \leq[\mathcal{C}(x, c), f c]

2106.07890_FO0155

1.000

f x \leq \inf _{c \in \mathcal{C}}\left\{\left[h^{x}(c), f c\right]\right\}=

2106.07890_FO0156

1.000

\widehat{\mathcal{C}}\left(h^{x}, f\right)

2106.07890_FO0157

0.900

\mathcal{C}(y, x)=\widehat{\mathcal{C}}\left(h^{x}, h^{y}\right)

2106.07890_FO0158

0.900

x, y

2106.07890_FO0159

1.000

f=h^{y}

2106.07890_FO0160

1.000

\widehat{\mathcal{C}}\left(h^{x}, h^{y}\right)=h^{y}(x)=\mathcal{C}(y, x)

2106.07890_FO0161

1.000

\mathcal{C}^{o p} \rightarrow \widehat{\mathcal{C}}

2106.07890_FO0162

1.000

\mathcal{C}^{o p}

2106.07890_FO0163

1.000

\widehat{\mathcal{C}}

2106.07890_FO0164

1.000

o p

2106.07890_FO0165

1.000

\mathcal{C}^{\mathrm{op}}

2106.07890_FO0166

1.000

2106.07890_FO0167

0.999

\{X \rightarrow Y\}

2106.07890_FO0168

0.999

2106.07890_FO0169

1.000

\widehat{\mathcal{C}}=[0,1]^{\mathcal{C}}

2106.07890_FO0170

0.674

\widehat{\mathcal{L}}:=[0,1]^{\mathcal{L}}

2106.07890_FO0171

1.000

h^{x}:=\mathcal{L}(x,-)

2106.07890_FO0172

1.000

x \leq c

2106.07890_FO0173

1.000

\mathcal{L}(x, y)=a \neq 0

2106.07890_FO0174

1.000

h^{y}

2106.07890_FO0175

0.850

{ }^{\mathrm{C}}

2106.07890_FO0176

0.666

F: \mathrm{J} \rightarrow \mathrm{C}

2106.07890_FO0177

0.965

\lim F

2106.07890_FO0178

0.999

2106.07890_FO0179

0.999

\mathrm{C}(Z, \lim F) \cong \operatorname{Set}^{\mathrm{J}}(*, \mathrm{C}(Z, F))

2106.07890_FO0180

0.647

(Z, F)

2106.07890_FO0181

1.000

*=*(i) \rightarrow \mathrm{C}(Z, F i)

2106.07890_FO0182

1.000

2106.07890_FO0183

0.755

Z \rightarrow F i

2106.07890_FO0184

0.999

W: \mathcal{J} \rightarrow \mathcal{V}

2106.07890_FO0185

1.000

\mathcal{J}

2106.07890_FO0186

1.000

\mathcal{E}

2106.07890_FO0187

1.000

F: \mathcal{J} \rightarrow \mathcal{E}

2106.07890_FO0188

1.000

2106.07890_FO0189

1.000

\lim _{W} F

2106.07890_FO0190

1.000

\mathcal{E} \rightarrow \mathcal{V}

2106.07890_FO0191

1.000

\mathcal{E}\left(Z, \lim _{W} F\right) \cong

2106.07890_FO0192

1.000

\mathcal{V}^{\mathcal{J}}(W, \mathcal{E}(Z, F))

2106.07890_FO0193

1.000

W: \mathcal{J}^{\mathrm{op}} \rightarrow \mathcal{V}

2106.07890_FO0194

0.997

\operatorname{colim}_{W} F

2106.07890_FO0195

1.000

\mathcal{E}=\widehat{\mathcal{L}}:=[0,1]^{\mathcal{L}}

2106.07890_FO0196

1.000

\mathcal{J}(i, j)=\delta_{i j}

2106.07890_FO0197

1.000

i, j \in\{1,2\}

2106.07890_FO0198

1.000

W: \mathcal{J} \rightarrow

2106.07890_FO0199

0.516

w_{1}:=W(1)

2106.07890_FO0200

0.516

w_{2}:=

2106.07890_FO0201

1.000

W(2)

2106.07890_FO0202

1.000

f, g: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0203

1.000

F: \mathcal{J} \rightarrow

2106.07890_FO0204

0.801

f:=F(1)

2106.07890_FO0205

0.801

g:=F(2)

2106.07890_FO0206

1.000

2106.07890_FO0207

0.996

\left(w_{1}, f\right) \times\left(w_{2}, g\right): \mathcal{L} \rightarrow[0,1]

2106.07890_FO0208

0.996

c \mapsto

2106.07890_FO0209

0.999

\min \left\{\frac{f c}{w_{1}}, \frac{g c}{w_{2}}, 1\right\}

2106.07890_FO0210

1.000

c \mapsto \min \left\{\frac{f c}{w_{1}}, \frac{g c}{w_{2}}, 1\right\}

2106.07890_FO0211

1.000

Z: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0212

1.000

\widehat{\mathcal{L}}(Z, F)

2106.07890_FO0213

1.000

J \rightarrow[0,1]

2106.07890_FO0214

1.000

2106.07890_FO0215

1.000

[0,1]^{\mathcal{J}}

2106.07890_FO0216

0.894

f=F(1)

2106.07890_FO0217

0.894

g=F(2)

2106.07890_FO0218

1.000

\mathcal{L}(c, d) \leq\left[\lim _{W} F(c), \lim _{W} F(d)\right]

2106.07890_FO0219

1.000

\mathcal{L}(c, d) \lim _{W} F(c) \leq

2106.07890_FO0220

1.000

\lim _{W} F(d)

2106.07890_FO0221

1.000

c, d \in \mathcal{L}

2106.07890_FO0222

0.967

2106.07890_FO0223

1.000

\left(w_{1}, f\right) \times\left(w_{2}, g\right)

2106.07890_FO0224

1.000

w_{1}, w_{2} \in[0,1]

2106.07890_FO0225

1.000

\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right): \mathcal{L} \rightarrow[0,1]

2106.07890_FO0226

1.000

\pi(c \mid x) \leq w_{1}

2106.07890_FO0227

0.996

\pi(c \mid y) \leq w_{2}

2106.07890_FO0228

1.000

w_{1} h^{x}(c) \times w_{2} h^{y}(c)

2106.07890_FO0229

1.000

\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right)

2106.07890_FO0230

1.000

w_{1}=w_{2}=1

2106.07890_FO0231

1.000

f \times g

2106.07890_FO0232

1.000

f, g, h: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0233

0.991

W: \mathcal{J}^{\mathrm{op}} \rightarrow[0,1]

2106.07890_FO0234

1.000

w_{2}:=W(2)

2106.07890_FO0235

1.000

F: \mathcal{J} \rightarrow \widehat{\mathcal{L}}

2106.07890_FO0236

0.998

\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right): \mathcal{L} \rightarrow[0,1]

2106.07890_FO0237

1.000

\max \left\{w_{1} f c, w_{2} g c\right\}

2106.07890_FO0238

0.539

\mathcal{J}=\mathcal{J}^{\mathrm{op}}

2106.07890_FO0239

1.000

\widehat{\mathcal{L}}(F-, Z)

2106.07890_FO0240

1.000

i=1,2

2106.07890_FO0241

0.912

c \mapsto \max \left\{w_{1} f c, w_{2} g c\right\}

2106.07890_FO0242

1.000

\mathcal{L}(c, d) \leq\left[\operatorname{colim}_{W} F(c), \operatorname{colim}_{W} F(d)\right]

2106.07890_FO0243

1.000

\mathcal{L}(c, d) \operatorname{colim}_{W} F(c) \leq \operatorname{colim}_{W} F(d)

2106.07890_FO0244

1.000

\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right)

2106.07890_FO0245

1.000

G, H: \mathrm{C} \rightarrow

2106.07890_FO0246

1.000

c \mapsto \widehat{\mathrm{C}}\left(h^{c} \times G, H\right)

2106.07890_FO0247

1.000

h^{c} \times G

2106.07890_FO0248

1.000

[f, g]: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0249

1.000

h^{c} \times f

2106.07890_FO0250

1.000

\mathcal{L}(c, d) \leq[[f, g](c),[f, g](d)]

2106.07890_FO0251

1.000

\mathcal{L}(c, d)[f, g](c) \leq[f, g](d)

2106.07890_FO0252

1.000

\mathcal{L}(c, d) \leq[g(c), g(d)]

2106.07890_FO0253

1.000

\mathcal{L}(c, d) g(c) \leq g(d)

2106.07890_FO0254

1.000

\mathcal{L}(c, d) \widehat{\mathcal{L}}\left(h^{c}, g\right) \leq \widehat{\mathcal{L}}\left(h^{d}, g\right)

2106.07890_FO0255

1.000

\mathcal{L}(c, d)\left(\widehat{\mathcal{L}}\left(h^{c}, g\right) \times \widehat{\mathcal{L}}(f, g)\right) \leq \widehat{\mathcal{L}}\left(h^{d}, g\right) \times \widehat{\mathcal{L}}(f, g)

2106.07890_FO0256

1.000

\mathcal{L}(c, d) \widehat{\mathcal{L}}\left(h^{c} \times f, g\right) \leq \widehat{\mathcal{L}}\left(h^{d} \times f, g\right)

2106.07890_FO0257

1.000

\mathcal{L}(c, d) \leq\left[\widehat{\mathcal{L}}\left(h^{c} \times f, g\right), \widehat{\mathcal{L}}\left(h^{d} \times f, g\right)\right]=[[f, g](c),[f, g](d)]

2106.07890_FO0258

1.000

(f \times-): \widehat{\mathcal{L}} \rightarrow \widehat{\mathcal{L}}

2106.07890_FO0259

1.000

[f,-]

2106.07890_FO0260

1.000

-\times f

2106.07890_FO0261

1.000

\widehat{\mathcal{L}}\left(h^{c},[f, g]\right)=[f, g](c)

2106.07890_FO0262

1.000

[f, g](c)=\widehat{\mathcal{L}}\left(h^{c} \times f, g\right)

2106.07890_FO0263

1.000

\widehat{\mathcal{L}}\left(h^{c},[f, g]\right)=\widehat{\mathcal{L}}\left(h^{c} \times f, g\right)

2106.07890_FO0264

1.000

h^{c}

2106.07890_FO0265

1.000

2106.07890_FO0266

1.000

[f, g]

2106.07890_FO0267

1.000

x, y \in \mathcal{L}

2106.07890_FO0268

0.999

\left[h^{x}, h^{y}\right](c)=

2106.07890_FO0269

1.000

d=c

2106.07890_FO0270

1.000

\pi(c \mid y)=

2106.07890_FO0271

1.000

\pi(c \mid c) \times \pi(c \mid x)=\pi(c \mid x) \neq 0

2106.07890_FO0272

1.000

\left[h^{x}, h^{y}\right](c) \neq 0

2106.07890_FO0273

1.000

x \Rightarrow y

2106.07890_FO0274

1.000

\left[h^{x}, h^{y}\right]: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0275

1.000

\mathcal{M}

2106.07890_FO0276

0.999

[0, \infty]

2106.07890_FO0277

1.000

a+\infty:=\infty

2106.07890_FO0278

1.000

\infty+a:=\infty

2106.07890_FO0279

0.998

b \leq a

2106.07890_FO0280

0.650

d_{\mathcal{M}}

2106.07890_FO0281

1.000

[a, b]:=\max \{b-a, 0\}

2106.07890_FO0282

1.000

a \times b=\max \{a, b\}

2106.07890_FO0283

0.997

a \sqcup b=\min \{a, b\}

2106.07890_FO0284

0.997

-\ln :[0,1] \rightarrow[0, \infty]

2106.07890_FO0285

1.000

[0, \infty] \rightarrow[0,1]

2106.07890_FO0286

1.000

a \mapsto \exp (-a)

2106.07890_FO0287

1.000

\mathcal{M}(x, y):=-\ln \mathcal{L}(x, y)

2106.07890_FO0288

1.000

0 \geq \mathcal{M}(x, x)

2106.07890_FO0289

1.000

\mathcal{M}(x, y)+\mathcal{M}(y, z) \geq \mathcal{M}(x, z)

2106.07890_FO0290

0.998

d_{\mathcal{M}}(x, y):=\mathcal{M}(x, y)

2106.07890_FO0291

1.000

\widehat{\mathcal{M}}:=[0, \infty]^{\mathcal{M}}

2106.07890_FO0292

1.000

f: \mathcal{M} \rightarrow[0, \infty]

2106.07890_FO0293

1.000

d_{M}(x, y) \geq[f x, f y]=\max \{f y-f x, 0\}

2106.07890_FO0294

1.000

[a, b]=\max \{b-a, 0\}

2106.07890_FO0295

1.000

d_{[0, \infty]}(a, b)

2106.07890_FO0296

1.000

d_{[0, \infty]}(f x, f y) \leq d_{M}(x, y)

2106.07890_FO0297

1.000

\widehat{\mathcal{M}}

2106.07890_FO0298

0.905

\mathcal{M}^{\mathrm{op}} \rightarrow \widehat{\mathcal{M}}

2106.07890_FO0299

0.905

x \mapsto d_{\mathcal{M}}(x,-)

2106.07890_FO0300

0.998

f, g \in \widehat{\mathcal{M}}

2106.07890_FO0301

0.998

w_{1}, w_{2}

2106.07890_FO0302

1.000

f, g: \mathcal{M} \rightarrow[0, \infty]

2106.07890_FO0303

1.000

w_{1}, w_{2} \in[0, \infty]

2106.07890_FO0304

1.000

f^{\prime}, g^{\prime}: \mathcal{L} \rightarrow[0,1]

2106.07890_FO0305

1.000

w_{1}^{\prime}, w_{2}^{\prime} \in[0,1]

2106.07890_FO0306

1.000

f^{\prime} c=\exp (-f c), g^{\prime} c=\exp (-g c)

2106.07890_FO0307

1.000

w_{i}^{\prime}=\exp \left(-w_{i}\right)

2106.07890_FO0308

1.000

d_{\mathcal{M}}(x,-)

2106.07890_FO0309

0.968

((-\infty, \infty], \oplus, \odot)

2106.07890_FO0310

1.000

([0, \infty], \oplus, \odot)

2106.07890_FO0311

0.999

w_{1}=w_{2}=0

2106.07890_FO0312

0.999

[0, \infty] \times \widehat{\mathcal{M}} \rightarrow \widehat{\mathcal{M}}

2106.07890_FO0313

0.999

(s, f) \mapsto s \odot f

2106.07890_FO0314

0.775

(s \cdot f)(x)=f(x)+s

2106.07890_FO0315

1.000

\left(s_{1} \oplus s_{2}\right) f(x)=\min \left\{s_{1}, s_{2}\right\} f(x)=\min \left\{f x+s_{1}, f x+s_{2}\right\}

2106.07890_FO0316

1.000

\left(s_{1} \odot s_{2}\right) f(x)=f x+s_{1}+s_{2}

2106.07890_FO0317

1.000

\left(s_{1}\right) \odot\left(s_{2} \odot f\right)(x)=

2106.07890_FO0318

1.000

\left(s_{2} \odot f\right)(x)+s_{1}=f x+s_{2}+s_{1}

2106.07890_FO0319

0.934

\sqcup