Geodesic
An equation calculus for primitive recursive rational-valued functions
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part V, Tome 32 (1972), pp. 116-120

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An equation calculus $\mathbb M$ primitive recursive functions of the positive rational argument is constructed in this paper. Among the axioms and inference roles of $\mathbb M$ there are the postulates of the primitive recursive arithmetic (ERA) by Goodstein. Logical constants $\,\vee,\rceil,\to,\leftrightarrow,\forall_{\leq},\exists_{\leq}$ can be defined in $\mathbb M$. It is proved that $\mathbb M$ is a conservative extension of PRA. 
@article{ZNSL_1972_32_a16,
     author = {M. Kh. Fakhmi},
     title = {An equation calculus for primitive recursive rational-valued functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {116--120},
     publisher = {mathdoc},
     volume = {32},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1972_32_a16/}
}
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M. Kh. Fakhmi. An equation calculus for primitive recursive rational-valued functions. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part V, Tome 32 (1972), pp. 116-120. http://geodesic.mathdoc.fr/item/ZNSL_1972_32_a16/