  We present new fluorine abundance estimations in two carbon enhanced
metal-poor (CEMP) stars, HE 1429-0551 and HE 1305+0007. HE 1429-0551 is also
enriched in slow neutron-capture process (s-process) elements, a CEMP-s, and HE
1305+0007 is enhanced in both, slow and rapid neutron-capture process elements,
a CEMP-s/r. The F abundances estimates are derived from the vibration-rotation
transition of the HF molecule at 23358.6 A using high-resolution infrared
spectra obtained with the Immersion Grating Infrared Spectrometer (IGRINS) at
the 4m-class Lowell Discovery Telescope. Our results include a F abundance
measurement in HE 1429-0551 of A(F) = +3.93 ([F/Fe] = +1.90) at [Fe/H] = -2.53,
and a F upper limit in HE 1305+0007 of A(F) < +3.28 ([F/Fe] < +1.00) at [Fe/H]
= -2.28. Our new derived F abundance in HE 1429-0551 makes this object the most
metal-poor star where F has been detected. We carefully compare these results
with literature values and state-of-the-art CEMP-s model predictions including
detailed AGB nucleosynthesis and binary evolution. The modelled fluorine
abundance for HE 1429-0551 is within reasonable agreement with our observed
abundance, although is slightly higher than our observed value. For HE
1429-0551, our findings support the scenario via mass transfer by a primary
companion during its thermally-pulsing phase. Our estimated upper limit in HE
1305+0007, along with data from the literature, shows large discrepancies
compared with AGB models. The discrepancy is principally due to the
simultaneous s- and r-process element enhancements which the model struggles to
reproduce.
