This was (mostly) me being bored in a Monday evening and in an attempt to formalise in Rocq the confluence and normalisation results that Delia Kesner and Shane Ó Conchúir obtain for the lambda calculus of Milner with partial substitutions which is available as arXiv:2312.13270.

Graphs

The dependency graphs are drawn in black and white and split by milestone. Every edge runs from a dependency to the result that uses it. A box is a result named after a paper statement, an ellipse is a supporting result, and a dashed box labelled with a milestone is a result defined in another diagram. The overview gives the shape of the whole development:

The details are then one diagram per milestone:

M1, binding and reduction

M2, metatheory and checks

M3, subsystem

M4, parallel reduction

M5, metaterms and the named calculus

The rule sketch gives the transition system generated by the rules B and Gc and R:

The reduction graph gives the bounded reduct set of the term that applies the duplicating identity to a variable with every edge labelled by its rule and with the normal form marked: