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Contemporary proof assistants such as Coq require that recursive functions be terminating and corecursive functions be productive to maintain logical consistency of their type theories, and some ensure these properties using syntactic checks. However, being syntactic, they are inherently delicate and restrictive, preventing users from easily writing obviously terminating or productive functions at their whim.
Meanwhile, there exist many sized type theories that perform type-based termination and productivity checking, including theories based on the Calculus of (Co)Inductive Constructions (CIC), the core calculus underlying Coq. These theories are more robust and compositional in comparison. So why haven’t they been adapted to Coq?
In this paper, we venture to answer this question with CIC
$\widehat{\ast}$
, a sized type theory based on CIC. It extends past work on sized types in CIC with additional Coq features such as global and local definitions. We also present a corresponding size inference algorithm and implement it within Coq’s kernel; for maximal backward compatibility with existing Coq developments, it requires no additional annotations from the user.
In our evaluation of the implementation, we find a severe performance degradation when compiling parts of the Coq standard library, inherent to the algorithm itself. We conclude that if we wish to maintain backward compatibility, using size inference as a replacement for syntactic checking is impractical in terms of performance.
Interactions between shock waves and boundary layers produce flow separations and augmented pressure/thermal loads in hypersonic flight. This study provides details of Mach 7 impinging-shock-flat-plate experiments conducted in the T4 Stalker Tube. Measurements were taken at flow conditions of Mach 7.0 (2.44 MJ kg
$^{-1}$) and Mach 7.7 (2.88 MJ kg
$^{-1}$) flight enthalpies with a range of freestream unit Reynolds numbers from 
$1.43 \times 10^{6}$ m
$^{-1}$ to 
$5.01 \times 10^{6}$ m
$^{-1}$. A shock generator at 
$12^{\circ }$ or 
$16^{\circ }$ to the freestream created an oblique shock which impinged on a boundary layer over a flat plate to induce flow separation. The flow field was examined using simultaneous measurements of wall static pressure, heat transfer and schlieren visualisation. Measured heat transfer along the flat plate without the shock impingement indicated that the boundary layer remained laminar for all flow conditions. The shock impingement flow field was successfully established within the facility test duration. The onset of separation was observed by a rise in wall pressure and a decrease in heat transfer at the location corresponding to the stem of the separation shock. Downstream of this initial rise, an increased pressure and higher heating loads were observed. The heat-transfer levels also indicated an immediate boundary layer transition due to the shock impingement. The separation data of the present work showed good agreement with our previous work on shock impingement on heated walls (Chang et al., J. Fluid Mech., vol. 908, 2021, pp. 1–13). A comparison with the previous scaling indicated that the separation also relates to the pressure ratio and the wall temperature parameter.
