Geodesic
Grzegorczyk logic with arithmetical proof operators
Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 2, pp. 483-499.

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Logics with the modal operator “\ldots is true and provable” together with the modal proof operators “p is a proof of \ldots” are axiomatized. Kripke-style completeness, decidability and arithmetical completeness of these logics are established. 
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     author = {E. Yu. Nogina},
     title = {Grzegorczyk logic with arithmetical proof operators},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {483--499},
     publisher = {mathdoc},
     volume = {2},
     number = {2},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1996_2_2_a5/}
}
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E. Yu. Nogina. Grzegorczyk logic with arithmetical proof operators. Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 2, pp. 483-499. http://geodesic.mathdoc.fr/item/FPM_1996_2_2_a5/