13

I want to know how could I do these three things (in red):

enter image description here

For the last one, I'm curious about the brackets.

Probably they are not too dificult to "write" in this languege but I've no clue. I hope you can help me.

19

The double brackets are defined in stmaryrd as \llbracket and \rrbracket.

For the other symbols, provided they have to be used only in display mode, you can define

\DeclareMathOperator*{\bigdoublewedge}{\bigwedge\mkern-15mu\bigwedge}
\DeclareMathOperator*{\bigdoublevee}{\bigvee\mkern-15mu\bigvee}

MWE

\documentclass{article}
\usepackage{amsmath,stmaryrd}

\DeclareMathOperator*{\bigdoublewedge}{\bigwedge\mkern-15mu\bigwedge}
\DeclareMathOperator*{\bigdoublevee}{\bigvee\mkern-15mu\bigvee}

\begin{document}
\[\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \llbracket p_{1} \rrbracket_{v}\]
\end{document} 

enter image description here


EDIT

If \bigdoublewedge and \bigdoublevee have to be used not only in display mode, the following definitions are more suitable

\newcommand{\bigdoublewedge}{%
  \mathop{
    \mathchoice{\bigwedge\mkern-15mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-11mu\bigwedge}
    }
}

\newcommand{\bigdoublevee}{%
  \mathop{
    \mathchoice{\bigvee\mkern-15mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-11mu\bigvee}
    }
}

MWE:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\bigdoublewedge}{%
  \mathop{
    \mathchoice{\bigwedge\mkern-15mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-11mu\bigwedge}
    }
}

\newcommand{\bigdoublevee}{%
  \mathop{
    \mathchoice{\bigvee\mkern-15mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-11mu\bigvee}
    }
}

\begin{document}
\[\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\textstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\scriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\scriptscriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]
\end{document} 

enter image description here

5

A variant of karlkoeller's method that doesn't require guessing the amount of the back up. The relative position of the two symbols can be adjusted by changing the factor in

\makebox[1.35\wd\z@][s]{$\m@th#1#2\hss#2$}%

which in this example is set to 1.35.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\bigdoublevee}{\big@doubleop{\bigvee}}
\newcommand{\bigdoublewedge}{\big@doubleop{\bigwedge}}
\newcommand{\big@doubleop}[1]{%
  \DOTSB\mathop{\mathpalette\big@doubleop@aux{#1}}\slimits@
}

\newcommand\big@doubleop@aux[2]{%
  \sbox\z@{$\m@th#1#2$}%
  \makebox[1.35\wd\z@][s]{$\m@th#1#2\hss#2$}%
}
\makeatother

\begin{document}
\begin{gather*}
\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\textstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\scriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\scriptscriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\end{gather*}
\end{document} 

enter image description here

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