UniTO/tesi/gabriel/part4_unicode.org

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* Translation validation of the Pattern Matching Compiler

** Source program
Our algorithm takes as its input a source program and translates it
into an algebraic data structure which type we call /decision_tree/.

#+BEGIN_SRC 
type decision_tree =
  | Unreachable
  | Failure
  | Leaf of source_expr
  | Guard of source_blackbox * decision_tree * decision_tree
  | Switch of accessor * (constructor * decision_tree) list * decision_tree
#+END_SRC

Unreachable, Leaf of source_expr and Failure are the terminals of the three.
We distinguish
- Unreachable: statically it is known that no value can go there
- Failure: a value matching this part results in an error
- Leaf: a value matching this part results into the evaluation of a
  source black box of  code
Our algorithm doesn't support type-declaration-based analysis 
to know the list of constructors at a given type.
Let's consider some trivial examples:

| function true -> 1
is translated to
|Switch ([(true, Leaf 1)], Failure)
while
| function 
| \vert{} true -> 1 
| \vert{} false -> 2
will be translated to

|Switch ([(true, Leaf 1); (false, Leaf 2)])
It is possible to produce Unreachable examples by using
 refutation clauses (a "dot" in the right-hand-side)
|function 
|\vert{} true -> 1 
|\vert{} false -> 2 
|\vert{} _ -> .
that gets translated into
| Switch ([(true, Leaf 1); (false, Leaf 2)], Unreachable)

We trust this annotation, which is reasonable as the type-checker
verifies that it indeed holds.

Guard nodes of the tree are emitted whenever a guard is found. Guards
node contains a blackbox of  code that is never evaluated and two
branches, one that is taken in case the guard evaluates to true and
the other one that contains the path taken when the guard evaluates to
true.

\begin{comment}
[ ] Finisci con Switch
[ ] Spiega del fallback
\end{comment}

The source code of a pattern matching function has the
following form:

|match variable with
|\vert pattern₁ \to expr₁
|\vert pattern₂ when guard \to expr₂
|\vert pattern₃ as var \to expr₃ 
|⋮
|\vert pₙ \to exprₙ

Patterns could or could not be exhaustive.

Pattern matching code could also be written using the more compact form:
|function
|\vert pattern₁ \to expr₁
|\vert pattern₂ when guard \to expr₂
|\vert pattern₃ as var \to expr₃ 
|⋮
|\vert pₙ \to exprₙ


This BNF grammar describes formally the grammar of the source program:

| start ::= "match" id "with" patterns \vert{} "function" patterns
| patterns ::= (pattern0\vert{}pattern1) pattern1+
| ;; pattern0 and pattern1 are needed to distinguish the first case in which
| ;; we can avoid writing the optional vertical line
| pattern0 ::= clause 
| pattern1 ::= "\vert" clause
| clause ::= lexpr "->" rexpr
| lexpr ::= rule (ε\vert{}condition)
| rexpr ::= _code ;; arbitrary code
| rule ::= wildcard\vert{}variable\vert{}constructor_pattern\vert or_pattern ;;
| wildcard ::= "_"
| variable ::= identifier
| constructor_pattern ::= constructor (rule\vert{}ε) (assignment\vert{}ε)
| constructor ::= int\vert{}float\vert{}char\vert{}string\vert{}bool \vert{}unit\vert{}record\vert{}exn\vert{}objects\vert{}ref \vert{}list\vert{}tuple\vert{}array\vert{}variant\vert{}parameterized_variant ;;  data types
| or_pattern ::=  rule ("\vert{}" wildcard\vert{}variable\vert{}constructor_pattern)+
| condition ::= "when" bexpr 
| assignment ::= "as" id
| bexpr ::= _code ;; arbitrary code
The source program is parsed using the ocaml-compiler-libs library.
The result of parsing, when successful, results in a list of clauses
and a list of type declarations.
Every clause consists of three objects: a left-hand-side that is the
kind of pattern expressed, an option guard and a right-hand-side expression.
Patterns are encoded in the following way:
| pattern         | type        |
|-----------------+-------------|
| _               | Wildcard    |
| p₁ as x         | Assignment  |
| c(p₁,p₂,...,pₙ) | Constructor |
| (p₁\vert p₂)    | Orpat       |


Once parsed, the type declarations and the list of clauses are encoded in the form of a matrix
that is later evaluated using a matrix decomposition algorithm.

Patterns are of the form
| pattern         | type of pattern     |
|-----------------+---------------------|
| _               | wildcard            |
| x               | variable            |
| c(p₁,p₂,...,pₙ) | constructor pattern |
| (p₁\vert p₂)    | or-pattern          |

The pattern /p/ matches a value /v/, written as p ≼ v, when one of the
following rules apply

|--------------------+---+--------------------+-------------------------------------------|
| _                  | ≼ | v                  | ∀v                                        |
| x                  | ≼ | v                  | ∀v                                        |
| (p₁ \vert p₂)      | ≼ | v                  | iff p₁ ≼ v or p₂ ≼ v                      |
| c(p₁, p₂, ..., pₐ) | ≼ | c(v₁, v₂, ..., vₐ) | iff (p₁, p₂, ..., pₐ) ≼ (v₁, v₂, ..., vₐ) |
| (p₁, p₂, ..., pₐ)  | ≼ | (v₁, v₂, ..., vₐ)  | iff pᵢ ≼ vᵢ ∀i ∈ [1..a]                   |
|--------------------+---+--------------------+-------------------------------------------|

When a value /v/ matches pattern /p/ we say that /v/ is an /instance/ of /p/.


During compilation by the translators, expressions at the
right-hand-side are compiled into
Lambda code and are referred as lambda code actions lᵢ.

We define the /pattern matrix/ P as the matrix |m x n| where m bigger
or equal than the number of clauses in the source program and n is
equal to the arity of the constructor with the gratest arity.
\begin{equation*}
P =
\begin{pmatrix}
p_{1,1} & p_{1,2} & \cdots & p_{1,n}  \\
p_{2,1} & p_{2,2} & \cdots & p_{2,n}  \\
\vdots & \vdots & \ddots & \vdots  \\
p_{m,1} & p_{m,2} & \cdots & p_{m,n} )
\end{pmatrix}
\end{equation*}
every row of /P/ is called a pattern vector
 $\vec{p_i}$ = (p₁, p₂, ..., pₙ); In every instance of P pattern
vectors appear normalized on the length of the longest pattern vector.
Considering the pattern matrix P we say that the value vector
$\vec{v}$ = (v₁, v₂, ..., vᵢ) matches the pattern vector pᵢ in P  if and only if the following two
conditions are satisfied:
- p_{i,1}, p_{i,2}, \cdots, p_{i,n}  ≼ (v₁, v₂, ..., vᵢ)
- ∀j < i p_{j,1}, p_{j,2}, \cdots, p_{j,n} ⋠ (v₁, v₂, ..., vᵢ) 

We can define the following three relations with respect to patterns:
- Pattern p is less precise than pattern q, written p ≼ q, when all
  instances of q are instances of p
- Pattern p and q are equivalent, written p ≡ q, when their instances
  are the same
- Patterns p and q are compatible when they share a common instance

Wit the support of two auxiliary functions
- tail of an ordered family
  | tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ min(I)}
- first non-⊥ element of an ordered family
  | First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
  | First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥}  \quad  if ∃i, xᵢ ≠ ⊥
we now define what it means to run a pattern row against a value
vector of the same length, that is (pᵢ)ⁱ(vᵢ)ⁱ
| pᵢ                       | vᵢ                   | result_{pat}                                    |
|--------------------------+----------------------+-------------------------------------------------|
| ∅                        | (∅)                  | []                                              |
| (_, tail(pᵢ)ⁱ)           | (vᵢ)                 | tail(pᵢ)ⁱ(tail(vᵢ)ⁱ)                            |
| (x, tail(pᵢ)ⁱ)           | (vᵢ)                 | σ[x↦v₀]        if tail(pᵢ)ⁱ(tail(vᵢ)ⁱ) = σ      |
| (K(qⱼ)ʲ, tail(pᵢ)ⁱ)      | (K(v'ⱼ)ʲ,tail(vⱼ)ʲ)  | ((qⱼ)ʲ +++ tail(pᵢ)ⁱ)((v'ⱼ)ʲ +++ tail(vᵢ)ⁱ)     |
| (K(qⱼ)ʲ, tail(pᵢ)ⁱ)      | (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) | ⊥              if K ≠ K'                        |
| (q₁\vert{}q₂, tail(pᵢ)ⁱ) | (vᵢ)ⁱ                | First((q₁,tail(pᵢ)ⁱ)(vᵢ)ⁱ, (q₂,tail(pᵢ)ⁱ)(vᵢ)ⁱ) |

A source program tₛ is a collection of pattern clauses pointing to
/bb/ terms. Running a program tₛ against an input value vₛ, written
tₛ(vₛ) produces a result /r/ belonging to the following grammar:
| tₛ ::= (p → bb)^{i∈I}
| tₛ(vₛ) → r
| r ::= guard list * (Match bb \vert{} NoMatch \vert{} Absurd)
\begin{comment}
TODO: understand how to explain guard
\end{comment}
We can define what it means to run an input value vₛ against a source
program tₛ:
| tₛ(vₛ) := ⊥ \quad \quad \quad if ∀i, pᵢ(vₛ) = ⊥
| First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥}  \quad  if ∃i, xᵢ ≠ ⊥ 
| tₛ(vₛ) = Absurd if bb_min{pᵢ → bbᵢ \vert{} pᵢ(vₛ) ≠ ⊥} = /refutation clause/
| tₛ(vₛ) = Match bb_min{pᵢ → bbᵢ \vert{} pᵢ(vₛ) ≠ ⊥}

[ ... ] Big part that I think doesn't need revision from you [ ... ]

In our prototype we make use of accessors to encode stored values.

\begin{minipage}{0.6\linewidth}
\begin{verbatim}

let value =  1 :: 2 :: 3 :: []
(* that can also be written *)
let value = []
    |> List.cons 3 
    |> List.cons 2 
    |> List.cons 1
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{verbatim}


   (field 0 x) = 1
   (field 0 (field 1 x)) = 2
   (field 0 (field 1 (field 1 x)) = 3
   (field 0 (field 1 (field 1 (field 1 x)) = []
\end{verbatim}
\end{minipage}
An \emph{accessor} /a/ represents the
access path to a value that can be reached by deconstructing the
scrutinee.
| a ::= Here \vert n.a
The above example, in encoded form:
\begin{verbatim}
Here = 1
1.Here = 2
1.1.Here = 3
1.1.1.Here = []
\end{verbatim}
In our prototype the source matrix mₛ is defined as follows
| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I}

Source matrices are used to build source decision trees Cₛ.
A decision tree is defined as either a Leaf, a Failure terminal or
an intermediate node with different children sharing the same accessor /a/
and an optional fallback.
Failure is emitted only when the patterns don't cover the whole set of
possible input values /S/. The fallback is not needed when the user
doesn't use a wildcard pattern.
%%% Give example of thing
| Cₛ ::= Leaf bb \vert Switch(a, (Kᵢ → Cᵢ)^{i∈S} , C?) \vert Failure \vert Unreachable
| vₛ ::= K(vᵢ)^{i∈I} \vert n ∈ ℕ
\begin{comment}
Are K and Here clear here?
\end{comment}
We say that a translation of a source program to a decision tree
is correct when for every possible input, the source program and its
respective decision tree produces the same result

| ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ)

We define the decision tree of source programs
〚tₛ〛
in terms of the decision tree of pattern matrices
〚mₛ〛
by the following:
| 〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Here), (pᵢ → bbᵢ)^{i∈I} 〛
Decision tree computed from pattern matrices respect the following invariant:
| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I})
| v(Here) = v
| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[
\begin{comment}
TODO: EXPLAIN
\end{comment}

We proceed to show the correctness of the invariant by a case
analysys.

Base cases:
1. [| ∅, (∅ → bbᵢ)ⁱ |] ≡ Leaf bbᵢ where i := min(I), that is a
   decision tree [|ms|] defined by an empty accessor and empty
   patterns pointing to blackboxes /bbᵢ/. This respects the invariant
   because a source matrix in the case of empty rows returns
   the first expression and (Leaf bb)(v) := Match bb
2. [| (aⱼ)ʲ, ∅ |] ≡ Failure 
Regarding non base cases:
Let's first define some auxiliary functions
- The index family of a constructor
  | Idx(K) := [0; arity(K)[
- head of an ordered family (we write x for any object here, value, pattern etc.)
  | head((xᵢ)^{i ∈ I}) = x_min(I)
- tail of an ordered family
  | tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ min(I)}
- head constructor of a value or pattern
  | constr(K(xᵢ)ⁱ) = K
  | constr(_) = ⊥
  | constr(x) = ⊥
- first non-⊥ element of an ordered family
  | First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
  | First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥}  \quad  if ∃i, xᵢ ≠ ⊥
- definition of group decomposition:
| let constrs((pᵢ)^{i ∈ I}) = { K \vert{} K = constr(pᵢ), i ∈ I }
| let Groups(m) where m = ((aᵢ)ⁱ ((pᵢⱼ)ⁱ → eⱼ)ⁱʲ) =
| \quad \quad let (Kₖ)ᵏ = constrs(pᵢ₀)ⁱ in
| \quad \quad ( Kₖ →
| \quad \quad \quad \quad ((a₀.ₗ)ˡ +++ tail(aᵢ)ⁱ)
| \quad \quad \quad \quad (
| \quad \quad \quad \quad if pₒⱼ is Kₖ(qₗ) then
| \quad \quad \quad \quad \quad \quad (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  → eⱼ
| \quad \quad \quad \quad if pₒⱼ is _ then
| \quad \quad \quad \quad \quad \quad (_)ˡ +++ tail(pᵢⱼ)ⁱ   → eⱼ
| \quad \quad \quad \quad else ⊥
| \quad \quad \quad \quad )ʲ
| \quad \quad ), (
| \quad \quad \quad \quad tail(aᵢ)ⁱ, (tail(pᵢⱼ)ⁱ  → eⱼ if p₀ⱼ is _ else ⊥)ʲ
| \quad \quad  )
Groups(m) is an auxiliary function that decomposes a matrix m into
submatrices, according to the head constructor of their first pattern.
Groups(m) returns one submatrix m_r for each head constructor K that
occurs on the first row of m, plus one "wildcard submatrix" m_{wild}
that matches on all values that do not start with one of those head
constructors.
Intuitively, m is equivalent to its decomposition in the following
sense: if the first pattern of an input vector (v_i)^i starts with one
of the head constructors Kₖ, then running (v_i)^i against m is the same
as running it against the submatrix m_{Kₖ}; otherwise (its head
constructor ∉ (Kₖ)ᵏ) it is equivalent to running it against
the wildcard submatrix.

We formalize this intuition as follows
*** Lemma (Groups):
    Let /m/ be a matrix with 
    | Groups(m) = (k_r \to m_r)^k, m_{wild}
    For any value vector $(v_i)^l$ such that $v_0 = k(v'_l)^l$ for some
    constructor k,
    we have:
    | if k = kₖ \text{ for some k then} 
    | \quad    m(vᵢ)ⁱ = mₖ((v_{l}')ˡ +++ (v_{i})^{i∈I\DZ}) 
    |  \text{else} 
    |   \quad   m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\DZ}

\begin{comment}
TODO: fix \{0}
\end{comment}

*** Proof:
Let /m/ be a matrix ((aᵢ)ⁱ, ((pᵢⱼ)ⁱ → eⱼ)ʲ) with 
| Groups(m) = (Kₖ → mₖ)ᵏ, m_{wild}
Below we are going to assume that m is a simplified matrix such that
the first row does not contain an or-pattern or a binding to a
variable.

Let (vᵢ)ⁱ be an input matrix with v₀ = Kᵥ(v'_{l})ˡ for some constructor Kᵥ.
We have to show that:
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
    m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
- otherwise
    m(vᵢ)ⁱ = m_{wild}(tail(vᵢ)ⁱ)
Let us call (rₖⱼ) the j-th row of the submatrix mₖ, and rⱼ_{wild}
the j-th row of the wildcard submatrix m_{wild}.

Our goal contains same-behavior equalities between matrices, for
a fixed input vector (vᵢ)ⁱ. It suffices to show same-behavior
equalities between each row of the matrices for this input
vector. We show that for any j,
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
  | (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
- otherwise
  | (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ
In the first case (Kᵥ is Kₖ for some Kₖ ∈ constrs(p₀ⱼ)ʲ), we
have to prove that
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
By definition of mₖ we know that rₖⱼ is equal to
| if pₒⱼ is Kₖ(qₗ) then
| \quad  (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  → eⱼ
| if pₒⱼ is _ then
| \quad  (_)ˡ +++ tail(pᵢⱼ)ⁱ   → eⱼ
| else ⊥
\begin{comment}
Maybe better as a table?
| pₒⱼ    | rₖⱼ                       |
|--------+---------------------------|
| Kₖ(qₗ) | (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ |
| _      | (_)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ  |
| else   | ⊥                         |
\end{comment}
By definition of (pᵢ)ⁱ(vᵢ)ⁱ we know that (pᵢⱼ)ⁱ(vᵢ) is equal to
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K(v'ₗ)ˡ,tail(vᵢ)ⁱ)  := ((qⱼ)ʲ +++ tail(pᵢⱼ)ⁱ)((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
| (_, tail(pᵢⱼ)ⁱ)      (vᵢ)                 := tail(pᵢⱼ)ⁱ(tail(vᵢ)ⁱ)
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) := ⊥ if K ≠ K'

We prove this first case by a second case analysis on p₀ⱼ.

TODO

In the second case (Kᵥ is distinct from Kₖ for all Kₖ ∈ constrs(pₒⱼ)ʲ),
we have to prove that
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ

TODO


** Target translation

The target program of the following general form is parsed using a parser
generated by Menhir, a LR(1) parser generator for the OCaml programming language.
Menhir compiles LR(1) a grammar specification, in this case a subset
of the Lambda intermediate language, down to OCaml code. 
This is the grammar of the target language (TODO: check menhir grammar)
| start ::= sexpr
| sexpr ::= variable \vert{} string \vert{} "(" special_form ")"
| string ::= "\"" identifier "\"" ;; string between doublequotes
| variable ::= identifier 
| special_form ::=  let\vert{}catch\vert{}if\vert{}switch\vert{}switch-star\vert{}field\vert{}apply\vert{}isout
| let ::= "let" assignment sexpr ;; (assignment sexpr)+ outside of pattern match code
| assignment ::= "function" variable variable+ ;; the first variable is the identifier of the function
| field ::= "field" digit variable
| apply ::= ocaml_lambda_code ;; arbitrary code
| catch ::= "catch" sexpr with sexpr
| with ::= "with" "(" label ")"
| exit ::= "exit" label
| switch-star ::= "switch*" variable case*
| switch::= "switch" variable case* "default:" sexpr
| case ::= "case" casevar ":" sexpr
| casevar ::= ("tag"\vert{}"int") integer
| if ::= "if" bexpr sexpr sexpr
| bexpr ::= "("  ("!="\vert{}"=="\vert{}">="\vert{}"<="\vert{}">"\vert{}"<") sexpr digit \vert{} field ")"
| label ::= integer
The prototype doesn't support strings.

The AST built by the parser is traversed and evaluated by the symbolic
execution engine.
Given that the target language supports jumps in the form of "catch - exit"
blocks the engine tries to evaluate the instructions inside the blocks
and stores the result of the partial evaluation into a record.
When a jump is encountered, the information at the point allows to
finalize the evaluation of the jump block.
In the environment the engine also stores bindings to values and
functions.
Integer additions and subtractions are simplified in a second pass.
The result of the symbolic evaluation is a target decision tree Cₜ
| Cₜ ::= Leaf bb \vert Switch(a, (πᵢ → Cᵢ)^{i∈S} , C?) \vert Failure
| vₜ ::= Cell(tag ∈ ℕ, (vᵢ)^{i∈I}) \vert n ∈ ℕ
Every branch of the decision tree is "constrained" by a domain
| Domain π = { n\vert{}n∈ℕ x n\vert{}n∈Tag⊆ℕ }
Intuitively, the π condition at every branch tells us the set of
possible values that can "flow" through that path.
π conditions are refined by the engine during the evaluation; at the
root of the decision tree the domain corresponds to the set of
possible values that the type of the function can hold.
C? is the fallback node of the tree that is taken whenever the value
at that point of the execution can't flow to any other subbranch.
Intuitively, the π_{fallback} condition of the C? fallback node is
| π_{fallback} = ¬\bigcup\limits_{i∈I}πᵢ
The fallback node can be omitted in the case where the domain of the
children nodes correspond to set of possible values pointed by the
accessor at that point of the execution
| domain(vₛ(a)) = \bigcup\limits_{i∈I}πᵢ
We say that a translation of a target program to a decision tree
is correct when for every possible input, the target program and its
respective decision tree produces the same result
| ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ)



** Equivalence checking

The equivalence checking algorithm takes as input a domain of
possible values \emph{S} and a pair of source and target decision trees and
in case the two trees are not equivalent it returns a counter example.
Our algorithm respects the following correctness statement:

\begin{comment}
TODO: we have to define what \coversTEX mean for readers to understand the specifications
(or we use a simplifying lie by hiding \coversTEX in the statements).
\end{comment}

\begin{align*}
 \EquivTEX S {C_S} {C_T} \emptyqueue = \YesTEX \;\land\; \coversTEX {C_T} S
 & \implies
 \forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T)
\\
 \EquivTEX S {C_S} {C_T} \emptyqueue = \NoTEX {v_S} {v_T} \;\land\; \coversTEX {C_T} S
 & \implies
 v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T)
\end{align*}
Our equivalence-checking algorithm $\EquivTEX S {C_S} {C_T} G$ is
a exactly decision procedure for the provability of the judgment
$(\EquivTEX  S {C_S} {C_T} G)$, defined below.
\begin{mathpar}
  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{constraint trees}} \\
    C & \bnfeq & \Leaf t \\
    & \bnfor & \Failure \\
    & \bnfor & \Switch a {\Fam i {\pi_i, C_i}} \Cfb \\
    & \bnfor & \Guard t {C_0} {C_1} \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{boolean result}} \\
    b & \in & \{ 0, 1 \} \\[0.5em]
    & & \text{\emph{guard queues}} \\
    G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{input space}} \\
    S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\
  \end{array}
  \\
  \infer{ }
  {\EquivTEX \emptyset {C_S} {C_T} G}

  \infer{ }
  {\EquivTEX S \Failure \Failure \emptyqueue}

  \infer
  {\trel {t_S} {t_T}}
  {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}

  \infer
  {\forall i,\;
    \EquivTEX
      {(S \land a = K_i)}
      {C_i} {\trim {C_T} {a = K_i}} G
   \\
   \EquivTEX
     {(S \land a \notin \Fam i {K_i})}
     \Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G
  }
  {\EquivTEX S
    {\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G}

\begin{comment}
  % TODO explain somewhere why the judgment is not symmetric:
  % we avoid defining trimming on source trees, which would
  % require more expressive conditions than just constructors
\end{comment}
  \infer
  {C_S \in {\Leaf t, \Failure}
   \\
   \forall i,\; \EquivTEX  {(S \land a \in D_i)} {C_S} {C_i} G
   \\
   \EquivTEX {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G}
  {\EquivTEX S
    {C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G}

  \infer
  {\EquivTEX S {C_0} {C_T} {G, (t_S = 0)}
   \\
   \EquivTEX S {C_1} {C_T} {G, (t_S = 1)}}
  {\EquivTEX S
    {\Guard {t_S} {C_0} {C_1}} {C_T} G}

  \infer
  {\trel {t_S} {t_T}
   \\
   \EquivTEX S {C_S} {C_b} G}
  {\EquivTEX S
    {C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}}
\end{mathpar}
Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
produces as a result /r/ in the following way:
| ( 〚tₛ〛ₛ(vₛ) ≡ Cₛ(vₛ) ) → r
| tₛ(vₛ) → r
Likewise
| ( 〚tₜ〛ₜ(vₜ) ≡ Cₜ(vₜ) ) → r
| tₜ(vₜ) → r
| result r ::= guard list * (Match blackbox \vert{} NoMatch \vert{} Absurd)
| guard ::= blackbox.
Having defined equivalence between two inputs of which one is
expressed in the source language and the other in the target language,
vₛ ≃ vₜ, we can define the equivalence between a couple of programs or
a couple of decision trees 
| tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ)
| Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ)
The result of the proposed equivalence algorithm is /Yes/ or /No(vₛ,
vₜ)/. In particular, in the negative case, vₛ and vₜ are a couple of
possible counter examples for which the decision trees produce a
different result.

In the presence of guards we can say that two results are
equivalent modulo the guards queue, written /r₁ ≃gs r₂/, when:
| (gs₁, r₁) ≃gs (gs₂, r₂)  ⇔  (gs₁, r₁) = (gs₂ ++ gs, r₂)
We say that Cₜ covers the input space /S/, written
/covers(Cₜ, S)/ when every value vₛ∈S is a valid input to the
decision tree Cₜ. (TODO: rephrase)
Given an input space /S/ and a couple of decision trees, where
the target decision tree Cₜ covers the input space /S/ we can define equivalence:
| equiv(S, Cₛ, Cₜ, gs) = Yes ∧ covers(Cₜ, S) → ∀vₛ≃vₜ ∈ S, Cₛ(vₛ) ≃gs Cₜ(vₜ)
Similarly we say that a couple of decision trees in the presence of
an input space /S/ are /not/ equivalent in the following way:
| equiv(S, Cₛ, Cₜ, gs) = No(vₛ,vₜ) ∧ covers(Cₜ, S) → vₛ≃vₜ ∈ S ∧ Cₛ(vₛ) ≠gs Cₜ(vₜ)
Corollary: For a full input space /S/, that is the universe of the
target program:
| equiv(S, 〚tₛ〛ₛ, 〚tₜ〛ₜ, ∅) = Yes  ⇔  tₛ ≃ tₜ


\begin{comment}
TODO: put ^i∈I where needed
\end{comment}
\subsubsection{The trimming lemma}
The trimming lemma allows to reduce the size of a decision tree given
an accessor /a/ → π relation (TODO: expand)
| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π}(vₜ)
We prove this by induction on Cₜ:

- Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself
    | Leaf_{bb/a→π}(v) = Leaf_{bb}(v)
- The same applies to Failure terminal
    | Failure_{/a→π}(v) = Failure(v)
- When Cₜ = Switch(b, (π→Cᵢ)ⁱ)_{/a→π} then we look at the accessor
    /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming
    a switch node yields the following result:
    | Switch(b, (π→Cᵢ)^{i∈I})_{/a→π} :=  Switch(b, (π'ᵢ→C_{i/a→π})^{i∈I})
\begin{comment}
TODO: understand how to properly align lists
check that every list is aligned
\end{comment}
For the trimming lemma we have to prove that running the value vₜ against
the decision tree Cₜ is the same as running vₜ against the tree
C_{trim} that is the result of the trimming operation on Cₜ
| Cₜ(vₜ) = C_{trim}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node.
In the case where ∃k \vert{} vₜ∈(b→πₖ) then we can prove that
| C_{k/a→π}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ'
while when a = b  then πₖ'=(πₖ∩π) and vₜ∈πₖ' because:
- by the hypothesis, vₜ∈π
- we are in the case where vₜ∈πₖ
So vₜ ∈ πₖ' and by induction
| Cₖ(vₜ) = C_{k/a→π}(vₜ)
We also know that ∀vₜ∈(b→πₖ) → Cₜ(vₜ) = Cₖ(vₜ) 
By putting together the last two steps, we have proven the trimming
lemma.

\begin{comment}
TODO: what should I say about covering??? I swap π and π' 
Covering lemma:
        ∀a,π covers(Cₜ,S) → covers(C_{t/a→π}, (S∩a→π))
            Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %%


%%%%%%% Also: Should I swap π and π'  ?
\end{comment}

\subsubsection{Equivalence checking}
The equivalence checking algorithm takes as parameters an input space
/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/:
| equiv(S, Cₛ, Cₜ) → Yes \vert{} No(vₛ, vₜ)

When the algorithm returns Yes and the input space is covered by /Cₛ/
we can say that the couple of decision trees are the same for
every couple of source value /vₛ/ and target value /vₜ/ that are equivalent.
\begin{comment}
Define "covered"
Is it correct to say the same? How to correctly distinguish in words ≃ and = ?
\end{comment}
| equiv(S, Cₛ, Cₜ) = Yes and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) = Cₜ(vₜ)
In the case where the algorithm returns No we have at least a couple
of counter example values vₛ and vₜ for which the two decision trees
outputs a different result.
| equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ)
We define  the following
| Forall(Yes) = Yes
| Forall(Yes::l) = Forall(l)
| Forall(No(vₛ,vₜ)::_) = No(vₛ,vₜ)
There exists and are injective:
| int(k) ∈ ℕ (arity(k) = 0) 
| tag(k) ∈ ℕ (arity(k) > 0) 
| π(k) = {n\vert int(k) = n} x {n\vert tag(k) = n}
where k is a constructor.

\begin{comment}
TODO: explain:
∀v∈a→π, C_{/a→π}(v) = C(v)
\end{comment}

We proceed by case analysis:
\begin{comment}
I start numbering from zero to leave the numbers as they were on the blackboard, were we skipped some things
I think the unreachable case should go at the end.
\end{comment}
0. in case of unreachable: 
| Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ
1. In the case of an empty input space
   | equiv(∅, Cₛ, Cₜ) := Yes
   and that is trivial to prove because there is no pair of values (vₛ, vₜ) that could be
   tested against the decision trees.
   In the other subcases S is always non-empty.
2. When there are /Failure/ nodes at both sides the result is /Yes/:
   |equiv(S, Failure, Failure) := Yes
   Given that ∀v, Failure(v) = Failure, the statement holds.
3. When we have a Leaf or a Failure at the left side:
   | equiv(S, Failure as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
   | equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
   Our algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and
   subtree Cₜᵢ
   | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i
   or we have a counter example vₛ, vₜ for which
   | vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ)
   then because 
   | vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) ,
   | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) 
   we can say that
   | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
4. When we have a Switch on the right we define π_{fallback} as the domain of
   values not covered but the union of the constructors kᵢ
   | π_{fallback} = ¬\bigcup\limits_{i∈I}π(kᵢ)
   Our algorithm proceeds by trimming
   | equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) := 
   | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++   equiv(S∩(a→πₙ), Cₛ, C_{a→π_{fallback}}))
    The statement still holds and we show this by first analyzing the
   /Yes/ case:
    | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes
    The constructor k is either included in the set of constructors kᵢ:
    | k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ)
    We also know that 
    | (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ)      
    | (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)      
    (1) is true by induction and (2) is a consequence of the trimming lemma.
    Putting everything together:
    | Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)
            
    When the k∉(kᵢ)ⁱ [TODO]

    The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum  k,
    | equiv(Sₖ, Cₛₖ, C_{T/a→πₖ} = No(vₛ, vₜ)
    Then we can say that 
    | Cₛₖ(vₛ) ≠ C_{t/a→πₖ}(vₜ)
    that is enough for proving that 
    | Cₛₖ(vₛ) ≠ (C_{t/a→πₖ}(vₜ) = Cₜ(vₜ))