Conflict between amsthm and some other package

If the class defines either a macro or an environment called proof, add the instructions \let\proof\relax and \let\endproof\relax before loading amsthm.

The above instruction worked for me only after putting \let\proof\relax and \let\endproof\relax AFTER the \usepackage{amsthm}

Put these lines before \usepackage(s):

\documentclass[runningheads]{llncs}

\let\proof\relax
\let\endproof\relax
\let\example\relax
\let\endexample\relax

\usepackage{amsthm}
\usepackage{mathtools}