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 greyscale 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:

<p align="center"><img src="graphs/dependency-overview.svg" alt="Theorem dependency by milestone"></p>

The details are then one diagram per milestone:

### M1, binding and reduction

<p align="center"><img src="graphs/dependency-M1.svg" alt="M1 dependencies"></p>

### M2, metatheory and checks

<p align="center"><img src="graphs/dependency-M2.svg" alt="M2 dependencies"></p>

### M3, subsystem

<p align="center"><img src="graphs/dependency-M3.svg" alt="M3 dependencies"></p>

### M4, parallel reduction

<p align="center"><img src="graphs/dependency-M4.svg" alt="M4 dependencies"></p>

### M5, metaterms and the named calculus

<p align="center"><img src="graphs/dependency-M5.svg" alt="M5 dependencies"></p>

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

<p align="center"><img src="graphs/rules.svg" alt="Reduction rules"></p>

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:

<p align="center"><img src="graphs/reduction.svg" alt="Reduction graph of the duplicating identity"></p>