(abridged) We present the IRAM-30m observations of multiple-J CO and CI line emission in a sample of redshift ~2-4 Herschel-ATLAS SMGs. A non-negligible effect of differential lensing is found for the CO emission lines, which could have caused significant underestimations of the linewidths, hence of the dynamical masses. The CO SLEDs are found to be similar to those of the local starburst-dominated ULIRGs and of the previously studied SMGs. After correcting for lensing amplification, we derived the global properties of the bulk of molecular gas in the SMGs using non-LTE radiative transfer modelling. The gas thermal pressure is found to be correlated with star formation efficiency. Further decomposing the CO SLEDs into two excitation components, we find a low-excitation component, which is less correlated with star formation, and a high-excitation one which is tightly related to the on-going star-forming activity. Additionally, tight linear correlations between the FIR and CO line luminosities have been confirmed for the $J ge 5$ CO lines, implying that these CO lines are good tracers of star formation. The [CI](2-1) lines follow the tight linear correlation between the luminosities of the [CI](2-1) and the CO(1-0) line found in local starbursts, indicating that CI lines could serve as good total molecular gas mass tracers for high-redshift SMGs. The total mass of the molecular gas reservoir, $(1-30) times 10^{10} M_odot$, suggests a typical molecular gas depletion time ~20-100 Myr and a gas to dust mass ratio ${delta}_{rm GDR}$~30-100. The ratio between CO line luminosity and the dust mass appears to be slowly increasing with redshift for the SMGs, which need to be further confirmed. Finally, through comparing the linewidth of CO and H2O lines, we find that they agree well in almost all our SMGs, confirming that the emitting regions are co-spatially located.
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