The relationship between charge density wave (CDW) orders and superconductivity in arsenide superconductor SrPt$_2$As$_2$ with $T_c$ = 5.2 K which crystallizes in the CaBe$_2$Ge$_2$-type structure was studied by $^{75}$As nuclear magnetic resonance (NMR) measurements up to 520 K, and $^{75}$As nuclear quadrupole resonance (NQR) and $^{195}$Pt-NMR measurements down to 1.5 K. At high temperature, $^{75}$As-NMR spectrum and nuclear spin relaxation rate ($1/T_1$) have revealed two distinct CDW orders, one realized in the As-Pt-As layer below $T_{rm CDW}^{rm As(1)}$ $=$ 410 K and the other in the Pt-As-Pt layer below $T_{rm CDW}^{rm As(2)}$ $=$ 255 K. The $1/T_1$ measured by $^{75}$As-NQR shows a clear Hebel-Slichter peak just below $T_c$ and decreases exponentially well below $T_c$. Concomitantly, $^{195}$Pt Knight shift decreases below $T_c$. Our results indicate that superconductivity in SrPt$_2$As$_2$ is in the spin-singlet state with an $s$-wave gap and is robust under the two distinct CDW orders in different layers.
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