This paper presents a methodology for reasoning about the computational complexity of functional programs. We introduce a first order arithmetic $\StrictTa$ which is a syntactic restriction of Peano arithmetic. We establish that the set of functions which are provably total in $\StrictTa$, is exactly the set of polynomial time functions.The cut-elimination process is polynomial time computable. Compared to others feasible arithmetics, $\StrictTa$ is conceptually simpler. The main feature of $\StrictTa$ concerns the treatment of the quantification. The range of quantifiers is restricted to the set of {\em actual terms} which is the set of constructor terms with variables. The inductive formulas are restricted to conjunctions of atomic formulas.