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+# Unit checking
+
+Unit checking is the process of inferring the units of Modelica variables, and ensuring the variables are used consistently in expressions and equations.
+
+It takes place after any evaluation of evaluable parameters and no evaluable parameters shall be evaluated during this process.
+(This must be true as long as tools allow the users to opt-out from unit-checking.)
+
+It is performed by using a type inference algorithm, albeit adapted to work with units.
+The algorithm proposed here is based on the Hindley-Milner algorithm.
+At a high level the algorithm collects a set of constraints on the units of expressions.
+It extracts a set of substitutions from these constraints which once applied determine whether a variable has a unit and what that unit would be.
+Any inconsistency discovered during this process is a unit error, and any unit expression that contains remaining unknowns at the end of the process is considered unknown.
+
+The document is organized as follows:
+* [Introduction of what is meant by the unit of a variable](#the-unit-of-a-modelica-variable)
+* [Description of the objects used in the inference algorithm](#unit-constraints-and-meta-expressions)
+* [Definition of unit equivalence and convertibility](#unit-equivalence-and-convertibility)
+* [Unit inference algorihtm](#unit-inference)
+* [Application to Modelica](#unit-constraints-of-a-modelica-model)
+* [Modelica operators on units](#modelica-operators-on-units)
+* [Extensions and limitations](#possible-extensions-and-current-limits)
+
+## The unit of a Modelica variable
+
+The unit of a Modelica variable is the unit associated with the variable after unit-checking has been performed, possibly remaining unknown.
+
+Note that the unit of a variable is not the same concept as the `unit`-attribute of the variable, but in a unit-consistent model, the unit will be equal to the `unit`-attribute when present.
+
+## Unit constraints and meta-expressions
+
+Unit constraints are introduced in the form of an [equivalence](#equivalence) of two unit _meta-expressions_.
+After an evaluation stage, a unit constraint of the form _`u1` is equivalent to `u2`_ will be represented by the _single-sided form_ `u1*u2^-1`.
+
+This section outlines the different constructs that comprise meta-expressions.
+
+### Variables
+The unit of the Modelica variable `var` is represented by the unit variable `var.unit`.
+These are unknowns for the inference algorithm to determine.
+
+### Literals
+
+#### `well-formed` unit
+The `well-formed` units correspond to [Modelica unit expressions](https://specification.modelica.org/master/unit-expressions.html).
+
+For the purposes of unit checking, they can be uniquely represented by their base unit factorization and scaling, up to the order of the base units.
+
+(We discuss why we ignore unit offsets later.)
+
+#### `empty` unit
+An expression with `empty` unit won't contribute to the unit of a parent expression, and will meet any constraints imposed on it.
+In practice, these are used to represent the unit of literal values.
+They provide a more flexible solution than considering any literal value to have unit `"1"`.
+They also provide a stricter solution than letting the unit of literals be inferred, which would effectively make literals wildcard from the perspective of the unit system.
+
+#### `undefined` unit
+An expression with `undefined` unit is used to represent unspecified or error cases, and will meet any constraints imposed on it.
+
+### Operators
+
+#### *
+Consider the meta-expression `e1 * e2`.
+* If both `e1` and `e2` are `well-formed`, it evaluates to a `well-formed` unit resulting from multiplying `e1` by `e2`.
+* If one of them is `empty`, it evaluates to the other one. (In effect, the `empty` unit becomes unit `"1"`)
+* If one of them is `undefined`, it evaluates to `undefined`.
+
+#### ^
+Consider the meta-expression `b ^ e`, where `e` is a rational.
+* If `b` is `well-formed`, it evaluates to a `well-formed` unit resulting from raising `b` to the power `e`.
+* If `b` is `empty`, it evaluates to `empty`.
+* If `b` is `undefined`, it evaluates to `undefined`.
+
+#### der
+Consider the meta-expression `der(e)`.
+* In a constraint where a `well-formed` unit is present, it evaluates to `e * ("s" ^ -1)`.
+* If `e` is `empty`, it evaluates to `empty`.
+* If `e` is `undefined`, it evaluates to `undefined`.
+
+This definition keeps `der(e)` in symbolic form in a model that does not contain any units.
+Otherwise, `der(x) = x` would lead to unit errors even if `x` has no declared unit.
+This also means that in `k*der(x) = -x`, the unit of `k` won't be inferred to be `"s"`.
+
+### Evaluation
+Unit variables can't be found to be `empty` or `undefined` unit.
+This means that after evaluation, a unit meta-expression is either `undefined`, `empty` or does not contain any `undefined` or `empty` unit.
+
+After evaluation a constraint will not contain both a well-formed unit and a `der` meta-expression.
+This means that all constraints can be simplified to have at most one well-formed unit.
+
+## Unit equivalence and convertibility
+
+### Equivalence
+Two well-formed units are said to be equivalent if both of the following are true:
+* Their base unit factorizations are equal.
+* Their base unit scalings are equal.
+
+Examples:
+- `"N"` and `"m/s2"` have different base unit factorizations, they are not equivalent.
+- `"s"` and `"ms"` have different base unit scaling, they are not equivalent.
+- `"K"` and `"degC"` are equivalent.
+- `"kN"` and `"W.s/mm"` are equivalent.
+
+It is a quality of implementation how equality of scalings are checked.
+
+`empty` units and `undefined` units are always equivalent to each other, themselves and any `well-formed` units.
+
+### Convertibility
+A well-formed unit `u1` is said to be convertible to another well-formed unit `u2` if:
+* Their base unit factorizations are equal.
+
+Examples:
+- `"N"` is not convertible to `"m/s2"`, since they have different base unit factorizations.
+- `"s"` is convertible to `"ms"`, since they differ by base unit scaling.
+- `"K"` is convertible to `"degC"`.
+- `"kN"` is convertible to `"W.s/mm"`.
+
+`empty` units and `undefined` units are never convertible to each other, themselves or any `well-formed` units.
+
+## Unit inference
+
+### Modified Hindley-Milner
+
+The original Hindley-Milner algorithm is designed to infer types, and consists of building a set of substitutions that returns the type of any expression of a given program.
+Applying this algorithm to perform unit inference in Modelica comes with two important differences:
+  * Not all variables must have a unit in a given model.
+  * There are operations on meta-expressions, that is, not all constraints are of the form  _`var1.unit` is equivalent to `var2.unit`_.
+
+#### Solvability of constraints
+
+After meta-expression evaluation and gathering of variables in the single-sided form, a constraint's solvability can be determined using:
+* A set `V` of unit variables present with non-zero exponent.
+* A set `D` of unit variables, present inside a `der` meta-expression.
+
+With this information, a variable `var.unit` can be solved from a given constraint if:
+* `var.unit` is present in `V`, but not in `D`.
+
+If a constraint has such a variable, the constraint is solvable.
+
+#### Conditional constraints
+
+One notion that doesn't exist in the original Hindley-Milner algorithm is that of a conditional constraint.
+In Modelica, errors that are in unused part of a simulation model should typically be ignored (e.g., removed conditional components).
+In the context of unit inference, some Modelica constructs generate constraints that are predicated on certain expressions having been evaluated or not.
+The constraint conditions are conjunctive, in other words, if even a single condition evaluates to `false` the constraint should be discarded.
+
+#### The algorithm
+
+The Hindley-Milner algorithm modified to perform unit inference in Modelica:
+* Traverse the flattened equation system and collect constraints before any evaluation has occured.
+* After evaluable parameters have been evaluated, and expressions have been reduced to values when possible, evaluate the constraints, as well as their conditions. (This propagates `empty` and `undefined` units to the top of the constraints.)
+* Discard any constraints of the form _`u` must be equivalent to empty unit_, or _`u` must be equivalent to undefined unit_, as they are trivially satisfied.
+* Discard any constraints having at least a condition that was evaluated to `false`.
+* Represent all constraints in their single-sided form.
+* Start with an empty set `S` of substitutions.
+* Loop:
+    * Select a solvable constraint or one that can be checked, if no such constraint exists exit the loop.
+    * If the constaint is in the form of just a `well-formed` unit, check that it is equivalent to `"1"`, it is an error otherwise.
+    * If it can be solved for `var.unit`:
+        * Replace the constraint by the substitution _`var.unit` => `u`_, where `var.unit` is not present in `u`.
+        * Apply the substitution to the right-hand side of the substitutions already in `S`.
+        * Add the substitution to `S`.
+        * Apply the substitution to all remaining constraints.
+* The unit of a variable for which there is no substitution in `S` is unknown.
+* For a variable `var` that has a substitution `var.unit` => `u` in `S`:
+    * If `u` is a `well-formed` unit, then this is the unit of `var`. (Given that no unit errors have been detected, a variable with a `unit`-attribute will get the given unit.)
+    * Otherwise, the unit of `var` is unknown.
+
+## Unit constraints of a Modelica model
+
+The following rules describe the unit meta-expression of a given Modelica expression, where it makes sense.
+They also provide the constraints that arise from different Modelica constructs.
+
+When it is specified that _`u1` must be equivalent to `u2`_, the corresponding constraint should be collected.
+
+In this section, the unit meta-expression of expression `e` is called `e.unit` and conditions of conditional constraints are highlighted with [].
+
+### Variables
+If `var` has a declared `unit`-attribute, `var.unit` must be equivalent to it.
+
+`var.unit` must be equivalent to the unit of the following attributes if present:
+ * `start`
+ * `min`
+ * `max`
+ * `nominal`
+
+(`var.unit` should be convertible to the `displayUnit`-attribute of `var` if it is present, but there is no corresponding constraints for the unit inference.)
+
+Consider the expression `e` of the form `var`, where `var` is a Modelica variable.
+* If `var` doesn't have a declared unit attribute, and has constant component variablity, `e.unit` is the `empty` unit.
+* Otherwise, `e.unit` is `var.unit`.
+
+### Literals
+`Real` literals have `empty` unit.
+
+### Binary operations
+Consider an expression `e` of the form `e1 op e2`.
+
+#### Additive operators
+* `e.unit` must be equivalent to `e1.unit`.
+* `e.unit` must be equivalent to `e2.unit`.
+
+Note that this requires introducing an intermediate variable to represent `e.unit`.
+This allows unit propagation when one of the operand has `empty` unit.
+
+#### Multiplication
+`e.unit` is `e1.unit * e2.unit`.
+
+#### Division
+`e.unit` is `e1.unit * (e2.unit ^ -1)`.
+
+#### Relational operators
+`e1.unit` must be equivalent to `e2.unit`
+
+#### Power operator
+* `e.unit` must be equivalent to `e1.unit ^ e2`.
+* If [`e2` wasn't evaluated or is a `Real` expression], `e1.unit` must be equivalent to `"1"`.
+
+Note that this requires introducing an intermediate variable to represent `e.unit`.
+Technically, `e2` is a Modelica expression rather than a rational number, which means that `e1.unit ^ e2` cannot be represented as a unit meta-expression.
+In practice, if the condition of the second constraint is verified, `e1.unit ^ e2` is `"1"` and if it isn't verified, then it can be represented as a meta-expression.
+
+(If the exponent was not evaluated or is a `Real`, the unit of the base should be `"1"`.)
+(It is up for discussion, when `e2` is `Real`, whether `e2.unit` should be equivalent to `"1"`. It could be handled in a similar way as transcendental functions. See below.)
+
+### Function calls
+Consider the expression `e` of the form `f(e1, e2, …, en)`.
+* If the output of the function has a declared unit, `e.unit` is the declared unit.
+* `e.unit` is `undefined` otherwise.
+
+If function input `i` has a declared unit, `ei.unit` must be equivalent to this unit.
+
+The natural generalization to functions with multiple outputs applies.
+
+### Transcendental functions
+As examples of transcendental functions, consider `sin`, `sign`, and `atan2`.
+Other transcendental functions can be handled similarly.
+
+Consider expression `e` of the form `sin(e1)`.
+* If `e1.unit` is `empty`, then so is `e.unit`.
+* If `e1.unit` is not `empty`, then it must be equivalent to `"1"` and `e.unit` is `"1"`.
+
+Note that some of these constraints can only be processed once `e1.unit` has been determined.
+This avoids introducing a unit `"1"` in the inference algorithm.
+
+Implementation note: The rule above can be implemented as making `e.unit` equivalent to `e1.unit`, and if `e1.unit` is `well-formed` after completed unit inference, then verify that it is equivalent to `"1"`.
+
+Consider an expression `e` of the form `sign(e1)`.
+* If `e1.unit` is `empty`, then so is `e.unit`.
+* If `e1.unit` is not `empty`, then `e.unit` is `"1"`.
+
+Consider an expression `e` of the form `atan2(e1, e2)`.
+* If `e1.unit` must be equivalent to `e2.unit`.
+* If `e1.unit` or `e2.unit` is `undefined`, then so is `e.unit`.
+* If `e1.unit` and `e2.unit` are `empty`, then so is `e.unit`.
+* If `e1.unit` or `e2.unit` is not `empty`, then `e.unit` is `"1"`.
+
+Implementation note: The rule above can be implemented as making `e.unit` equivalent to both `e1.unit * e1.unit^-1` and `e2.unit * e2.unit^-1`.
+
+### Operator der
+The unit of the expression `der(e)` is `der(e.unit)`.
+
+### If-expressions
+Consider the expression `e` of the form `if cond then e1 else e2`.
+If [`cond` was not evaluated]:
+* `e.unit` must be equivalent to `e1.unit`.
+* `e.unit` must be equivalent to `e2.unit`.
+
+If [`cond` was evaluated], `e.unit` must be equivalent to the unit of the selected branch.
+
+Note that this requires introducing a intermediate variable to represent `e.unit`.
+
+### Equations
+Both sides of binding equations, equality equations and connect equations must be unit equivalent.
+
+## Modelica operators on units
+
+The following operators facilitate writing unit-consistent models.
+
+### `withUnit($value$, $unit$)`
+Creates an expression with unit `$unit$`, and value `$value$`.
+Argument $value$ needs to be a `Real` expression, with `empty` unit.
+
+One application of this operator is to attach a unit to a literal.
+
+### `withoutUnit($value$, $unit$)`
+Creates an expression with empty unit, whose value is the numerical value of `$value$` expressed in `$unit$`.
+Argument `$value$` needs to be a `Real` expression, with a well-formed unit.
+The unit of `$value$` must be convertible to `$unit$`.
+
+### `inUnit($value$, $unit$)`
+Creates an expression with unit `$unit$`, whose value is the conversion of `$value$` to `$unit$`.
+Argument `$value$` needs to be a `Real` expression, with a well-formed unit.
+The unit of `$value$` must be convertible to `$unit$`.
+
+## Possible extensions and current limits
+
+### Units with offset
+The current proposal doesn't allow distinguishing `"K"` and `"degC"`.
+In order to do that one needs to keep track of whether a variable is meant to represent a difference or not.
+We have a proof of concept for such a feature, and the mechanism is similar and orthogonal to the one described here.
+
+### Builtin operators and functions
+The current state of this proposal does not cover the unit checking of all Modelica builtin operators and functions.
+
+### Arrays
+We don't try to specify how this would work on arrays, although the current proposal works straight-forwardly with element-wise operators.
+We are confident that it could be extended to matrix computations too, but haven't a proof of concept for that part.
+
+### Constants
+This proposal takes a stance regarding how the unit of constants should be handled but we recognise that this is still very much up for discussion.
+It is also somewhat orthogonal to the rest of the proposal.
+
+### Functions
+The Hindler-Milner algorithm provides a way to infer unit attributes of the inputs and outputs of functions.
+Concretely, it would imply computing the bundles of constraints from the body of each functions and adding them to the algorithm, after adequate substitutions, when processing a function call.
+
+### Variable initialisation pattern
+A common pattern used in the MSL is the following:
+```
+model M
+  Real d(unit = "s") = 12;
+  Real v(unit = "m/s") = 0.1 / (3 * d);
+end M;
+```
+where a modeler knows how to express a variable with unit as a function of another, with help of literal values that should really carry a unit.
+The way to fix that, with this proposal, is to attach a unit to the literal values using `withUnit`.
+
+### Syntax extension
+We could extend the Modelica language to add syntactic sugar for `withUnit`.
+The following variants have all been successfully test implemented:
+* `0.1[m]`
+* `0.1 'm'` (or as `0.1'm'` to show more clearly that the quoted unit belongs to the literal)
+* `0.1."m"`
+
+### unsafeUnit operator
+Another way to deal with the pattern described above would be to provide an unsafe operator that effectively drops any constraints generated in its subexpression.
+The whole expression would have `undefined` unit.