\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

# Program extraction

Authors:

  Jean-Christophe Filliâtre and Pierre Letouzey

We present here the Coq extraction commands, used to build certified and relatively efficient functional programs, extracting them from either Coq functions or Coq proofs of specifications. The functional languages available as output are currently OCaml, Haskell and Scheme. In the following, "ML" will be used (abusively) to refer to any of the three.

Before using any of the commands or options described in this chapter, the extraction framework should first be loaded explicitly via `Require Extraction`, or via the more robust `From Coq Require Extraction`. Note that in earlier versions of Coq, these commands and options were directly available without any preliminary `Require`.

Require Extraction.
  [Loading ML file extraction_plugin.cmxs (using legacy method) ... done]

## Generating ML Code

Note

In the following, a qualified identifier `qualid` can be used to refer to any kind of Coq global "object" : constant, inductive type, inductive constructor or module name.

The next two commands are meant to be used for rapid preview of extraction. They both display extracted term(s) inside Coq.

Command Extraction qualid

  Extraction of the mentioned object in the Coq toplevel.

Command Recursive Extraction qualid+

  Recursive extraction of all the mentioned objects and all their dependencies in the Coq toplevel.

All the following commands produce real ML files. User can choose to produce one monolithic file or one file per Coq library.

Command Extraction string qualid+

  Recursive extraction of all the mentioned objects and all their dependencies in one monolithic file `string`. Global and local identifiers are renamed according to the chosen ML language to fulfill its syntactic conventions, keeping original names as much as possible.

Command Extraction Library ident

  Extraction of the whole Coq library `ident.v` to an ML module `ident.ml`. In case of name clash, identifiers are here renamed using prefixes `coq_` or `Coq_` to ensure a session-independent renaming.

Command Recursive Extraction Library ident

  Extraction of the Coq library `ident.v` and all other modules `ident.v` depends on.

Command Separate Extraction qualid+

  Recursive extraction of all the mentioned objects and all their dependencies, just as `Extraction string qualid+`, but instead of producing one monolithic file, this command splits the produced code in separate ML files, one per corresponding Coq `.v` file. This command is hence quite similar to `Recursive Extraction Library`, except that only the needed parts of Coq libraries are extracted instead of the whole. The naming convention in case of name clash is the same one as `Extraction Library`: identifiers are here renamed using prefixes `coq_` or `Coq_`.

The following command is meant to help automatic testing of the extraction, see for instance the `test-suite` directory in the Coq sources.

Command Extraction TestCompile qualid+

  All the mentioned objects and all their dependencies are extracted to a temporary OCaml file, just as in `Extraction "file"`. Then this temporary file and its signature are compiled with the same OCaml compiler used to built Coq. This command succeeds only if the extraction and the OCaml compilation succeed. It fails if the current target language of the extraction is not OCaml.

## Extraction Options

### Setting the target language

Command Extraction Language language
  language`::=`OCaml`````|`Haskell`````|`Scheme`````|`JSON``

  The ability to fix target language is the first and most important of the extraction options. Default is `OCaml`.

  The JSON output is mostly for development or debugging: it contains the raw ML term produced as an intermediary target.

### Inlining and optimizations

Since OCaml is a strict language, the extracted code has to be optimized in order to be efficient (for instance, when using induction principles we do not want to compute all the recursive calls but only the needed ones). So the extraction mechanism provides an automatic optimization routine that will be called each time the user wants to generate an OCaml program. The optimizations can be split in two groups: the type-preserving ones (essentially constant inlining and reductions) and the non-type-preserving ones (some function abstractions of dummy types are removed when it is deemed safe in order to have more elegant types). Therefore some constants may not appear in the resulting monolithic OCaml program. In the case of modular extraction, even if some inlining is done, the inlined constants are nevertheless printed, to ensure session-independent programs.

Concerning Haskell, type-preserving optimizations are less useful because of laziness. We still make some optimizations, for example in order to produce more readable code.

The type-preserving optimizations are controlled by the following Coq flags and commands:

Flag Extraction Optimize

  Default is on. This flag controls all type-preserving optimizations made on the ML terms (mostly reduction of dummy beta/iota redexes, but also simplifications on Cases, etc). Turn this flag off if you want a ML term as close as possible to the Coq term.

Flag Extraction Conservative Types

  Default is off. This flag controls the non-type-preserving optimizations made on ML terms (which try to avoid function abstraction of dummy types). Turn this flag on to make sure that `e:t` implies that `e':t'` where `e'` and `t'` are the extracted code of `e` and `t` respectively.

Flag Extraction KeepSingleton

  Default is off. Normally, when the extraction of an inductive type produces a singleton type (i.e. a type with only one constructor, and only one argument to this constructor), the inductive structure is removed and this type is seen as an alias to the inner type. The typical example is `sig`. This flag allows disabling this optimization when one wishes to preserve the inductive structure of types.

Flag Extraction AutoInline

  Default is on. The extraction mechanism inlines the bodies of some defined constants, according to some heuristics like size of bodies, uselessness of some arguments, etc. Those heuristics are not always perfect; if you want to disable this feature, turn this flag off.

Command Extraction Inline qualid+

  In addition to the automatic inline feature, the constants mentioned by this command will always be inlined during extraction.

Command Extraction NoInline qualid+

  Conversely, the constants mentioned by this command will never be inlined during extraction.

Command Print Extraction Inline

  Prints the current state of the table recording the custom inlinings declared by the two previous commands.

Command Reset Extraction Inline

  Empties the table recording the custom inlinings (see the previous commands).

Inlining and printing of a constant declaration:

The user can explicitly ask for a constant to be extracted by two means:

> - by mentioning it on the extraction command line

> - by extracting the whole Coq module of this constant.

In both cases, the declaration of this constant will be present in the produced file. But this same constant may or may not be inlined in the following terms, depending on the automatic/custom inlining mechanism.

For the constants non-explicitly required but needed for dependency reasons, there are two cases:

> - If an inlining decision is taken, whether automatically or not, all occurrences of this constant are replaced by its extracted body, and this constant is not declared in the generated file.

> - If no inlining decision is taken, the constant is normally declared in the produced file.

### Extra elimination of useless arguments

The following command provides some extra manual control on the code elimination performed during extraction, in a way which is independent but complementary to the main elimination principles of extraction (logical parts and types).

Command Extraction Implicit qualid [ ident​integer* ]

  Declares some arguments of `qualid` as implicit, meaning that they are useless in extracted code. The extracted code will omit these arguments. Here `qualid` can be any function or inductive constructor, and the `ident`s are the names of the useless arguments. Arguments can can also be identified positionally by `integer`s starting from 1.

When an actual extraction takes place, an error is normally raised if the `Extraction Implicit` declarations cannot be honored, that is if any of the implicit arguments still occurs in the final code. This behavior can be relaxed via the following flag:

Flag Extraction SafeImplicits

  Default is on. When this flag is off, a warning is emitted instead of an error if some implicit arguments still occur in the final code of an extraction. This way, the extracted code may be obtained nonetheless and reviewed manually to locate the source of the issue (in the code, some comments mark the location of these remaining implicit arguments). Note that this extracted code might not compile or run properly, depending of the use of these remaining implicit arguments.

### Realizing axioms

Extraction will fail if it encounters an informative axiom not realized. A warning will be issued if it encounters a logical axiom, to remind the user that inconsistent logical axioms may lead to incorrect or non-terminating extracted terms.

It is possible to assume some axioms while developing a proof. Since these axioms can be any kind of proposition or object or type, they may perfectly well have some computational content. But a program must be a closed term, and of course the system cannot guess the program which realizes an axiom. Therefore, it is possible to tell the system what ML term corresponds to a given axiom.

Command Extract Constant qualid stringtv* => ident​string

  Give an ML extraction for the given constant.

  `stringtv`

    If the type scheme axiom is an arity (a sequence of products followed by a sort), then some type variables have to be given (as quoted strings).

    The number of type variables is checked by the system. For example:

    Axiom Y : Set -> Set -> Set.
      Y is declared
    Extract Constant Y "'a" "'b" => " 'a * 'b ".

Command Extract Inlined Constant qualid => ident​string

  Same as the previous one, except that the given ML terms will be inlined everywhere instead of being declared via a `let`.

  Note

  This command is sugar for an `Extract Constant` followed by a `Extraction Inline`. Hence a `Reset Extraction Inline` will have an effect on the realized and inlined axiom.

Caution

It is the responsibility of the user to ensure that the ML terms given to realize the axioms do have the expected types. In fact, the strings containing realizing code are just copied to the extracted files. The extraction recognizes whether the realized axiom should become a ML type constant or a ML object declaration. For example:

Axiom X:Set.
  X is declared
Axiom x:X.
  x is declared
Extract Constant X => "int".
Extract Constant x => "0".

Realizing an axiom via `Extract Constant` is only useful in the case of an informative axiom (of sort `Type` or `Set`). A logical axiom has no computational content and hence will not appear in extracted terms. But a warning is nonetheless issued if extraction encounters a logical axiom. This warning reminds user that inconsistent logical axioms may lead to incorrect or non-terminating extracted terms.

If an informative axiom has not been realized before an extraction, a warning is also issued and the definition of the axiom is filled with an exception labeled `AXIOM TO BE REALIZED`. The user must then search these exceptions inside the extracted file and replace them by real code.

### Realizing inductive types

The system also provides a mechanism to specify ML terms for inductive types and constructors. For instance, the user may want to use the ML native boolean type instead of the Coq one. The syntax is the following:

Command Extract Inductive qualid => ident​string [ ident​string* ] stringmatch?

  Give an ML extraction for the given inductive type. You must specify extractions for the type itself (the initial `ident​string`) and all its constructors (the `[ ident​string* ]`). In this form, the ML extraction must be an ML inductive datatype, and the native pattern matching of the language will be used.

  When the initial `ident​string` matches the name of the type of characters or strings (`char` and `string` for OCaml, `Prelude.Char` and `Prelude.String` for Haskell), extraction of literals is handled in a specialized way, so as to generate literals in the target language. This feature requires the type designated by `qualid` to be registered as the standard char or string type, using the `Register` command.

  `stringmatch`

    Indicates how to perform pattern matching over this inductive type. In this form, the ML extraction could be an arbitrary type. For an inductive type with \(k\) constructors, the function used to emulate the pattern matching should expect \(k+1\) arguments, first the \(k\) branches in functional form, and then the inductive element to destruct. For instance, the match branch `| S n => foo` gives the functional form `(fun n -> foo)`. Note that a constructor with no arguments is considered to have one unit argument, in order to block early evaluation of the branch: `| O => bar` leads to the functional form `(fun () -> bar)`. For instance, when extracting `nat` into OCaml `int`, the code to be provided has type: `(unit->'a)->(int->'a)->int->'a`.

Caution

As for `Extract Constant`, this command should be used with care:

- The ML code provided by the user is currently not checked at all by extraction, even for syntax errors.

- Extracting an inductive type to a pre-existing ML inductive type is quite sound. But extracting to a general type (by providing an ad-hoc pattern matching) will often not be fully rigorously correct. For instance, when extracting `nat` to OCaml `int`, it is theoretically possible to build `nat` values that are larger than OCaml `max_int`. It is the user's responsibility to be sure that no overflow or other bad events occur in practice.

- Translating an inductive type to an arbitrary ML type does not magically improve the asymptotic complexity of functions, even if the ML type is an efficient representation. For instance, when extracting `nat` to OCaml `int`, the function `Nat.mul` stays quadratic. It might be interesting to associate this translation with some specific `Extract Constant` when primitive counterparts exist.

Typical examples are the following:

Extract Inductive unit => "unit" [ "()" ].
Extract Inductive bool => "bool" [ "true" "false" ].
Extract Inductive sumbool => "bool" [ "true" "false" ].

Note

When extracting to OCaml, if an inductive constructor or type has arity 2 and the corresponding string is enclosed by parentheses, and the string meets OCaml's lexical criteria for an infix symbol, then the rest of the string is used as an infix constructor or type.

Extract Inductive list => "list" [ "[]" "(::)" ].
Extract Inductive prod => "(*)" [ "(,)" ].

As an example of translation to a non-inductive datatype, let's turn `nat` into OCaml `int` (see caveat above):

Extract Inductive nat => int [ "0" "succ" ] "(fun fO fS n -> if n=0 then fO () else fS (n-1))".

### Avoiding conflicts with existing filenames

When using `Extraction Library`, the names of the extracted files directly depend on the names of the Coq files. It may happen that these filenames are in conflict with already existing files, either in the standard library of the target language or in other code that is meant to be linked with the extracted code. For instance the module `List` exists both in Coq and in OCaml. It is possible to instruct the extraction not to use particular filenames.

Command Extraction Blacklist ident+

  Instruct the extraction to avoid using these names as filenames for extracted code.

Command Print Extraction Blacklist

  Show the current list of filenames the extraction should avoid.

Command Reset Extraction Blacklist

  Allow the extraction to use any filename.

For OCaml, a typical use of these commands is `Extraction Blacklist String List`.

### Additional settings

Option Extraction File Comment string

  This option provides a comment that is included at the beginning of the output files.

Option Extraction Flag natural

  This option controls which optimizations are used during extraction, providing a finer-grained control than `Extraction Optimize`. The bits of `natural` are used as a bit mask. Keeping an option off keeps the extracted ML more similar to the Coq term. Values are:

  Bit | Value | Optimization (default is on unless noted otherwise)
  0 | 1 | Remove local dummy variables
  1 | 2 | Use special treatment for fixpoints
  2 | 4 | Simplify case with iota-redux
  3 | 8 | Factor case branches as functions
  4 | 16 | (not available, default false)
  5 | 32 | Simplify case as function of one argument
  6 | 64 | Simplify case by swapping case and lambda
  7 | 128 | Some case optimization
  8 | 256 | Push arguments inside a letin
  9 | 512 | Use linear let reduction (default false)
  10 | 1024 | Use linear beta reduction (default false)

Flag Extraction TypeExpand

  If this flag is set, fully expand Coq types in ML. See the Coq source code to learn more.

## Differences between Coq and ML type systems

Due to differences between Coq and ML type systems, some extracted programs are not directly typable in ML. We now solve this problem (at least in OCaml) by adding when needed some unsafe casting `Obj.magic`, which give a generic type `'a` to any term.

First, if some part of the program is very polymorphic, there may be no ML type for it. In that case the extraction to ML works alright but the generated code may be refused by the ML type checker. A very well known example is the `distr-pair` function:

Definition dp {A B:Type}(x:A)(y:B)(f:forall C:Type, C->C) := (f A x, f B y).
  dp is defined

In OCaml, for instance, the direct extracted term would be:

    let dp x y f = Pair((f () x),(f () y))

and would have type:

    dp : 'a -> 'a -> (unit -> 'a -> 'b) -> ('b,'b) prod

which is not its original type, but a restriction.

We now produce the following correct version:

    let dp x y f = Pair ((Obj.magic f () x), (Obj.magic f () y))

Secondly, some Coq definitions may have no counterpart in ML. This happens when there is a quantification over types inside the type of a constructor; for example:

Inductive anything : Type := dummy : forall A:Set, A -> anything.
  anything is defined anything_rect is defined anything_ind is defined anything_rec is defined anything_sind is defined

which corresponds to the definition of an ML dynamic type. In OCaml, we must cast any argument of the constructor dummy (no GADT are produced yet by the extraction).

Even with those unsafe castings, you should never get error like `segmentation fault`. In fact even if your program may seem ill-typed to the OCaml type checker, it can't go wrong : it comes from a Coq well-typed terms, so for example inductive types will always have the correct number of arguments, etc. Of course, when launching manually some extracted function, you should apply it to arguments of the right shape (from the Coq point-of-view).

More details about the correctness of the extracted programs can be found in [Let02].

We have to say, though, that in most "realistic" programs, these problems do not occur. For example all the programs of Coq library are accepted by the OCaml type checker without any `Obj.magic` (see examples below).

## Some examples

We present here two examples of extraction, taken from the Coq Standard Library. We choose OCaml as the target language, but everything, with slight modifications, can also be done in the other languages supported by extraction. We then indicate where to find other examples and tests of extraction.

### A detailed example: Euclidean division

The file `Euclid` contains the proof of Euclidean division. The natural numbers used here are unary, represented by the type `nat`, which is defined by two constructors `O` and `S`. This module contains a theorem `eucl_dev`, whose type is:

    forall b:nat, b > 0 -> forall a:nat, diveucl a b

where `diveucl` is a type for the pair of the quotient and the modulo, plus some logical assertions that disappear during extraction. We can now extract this program to OCaml:

Require Extraction.
Require Import Euclid Wf_nat.
Extraction Inline gt_wf_rec lt_wf_rec induction_ltof2.
Recursive Extraction eucl_dev.
  type nat = | O | S of nat type sumbool = | Left | Right (** val sub : nat -> nat -> nat **) let rec sub n m = match n with | O -> n | S k -> (match m with | O -> n | S l -> sub k l) (** val le_lt_dec : nat -> nat -> sumbool **) let rec le_lt_dec n m = match n with | O -> Left | S n0 -> (match m with | O -> Right | S n1 -> le_lt_dec n0 n1) (** val le_gt_dec : nat -> nat -> sumbool **) let le_gt_dec = le_lt_dec type diveucl = | Divex of nat * nat (** val eucl_dev : nat -> nat -> diveucl **) let rec eucl_dev n m = let s = le_gt_dec n m in (match s with | Left -> let d = let y = sub m n in eucl_dev n y in let Divex (q, r) = d in Divex ((S q), r) | Right -> Divex (O, m))

The inlining of `gt_wf_rec` and others is not mandatory. It only enhances readability of extracted code. You can then copy-paste the output to a file `euclid.ml` or let Coq do it for you with the following command:

    Extraction "euclid" eucl_dev.

Let us play the resulting program (in an OCaml toplevel):

    #use "euclid.ml";;
    type nat = O | S of nat
    type sumbool = Left | Right
    val sub : nat -> nat -> nat = <fun>
    val le_lt_dec : nat -> nat -> sumbool = <fun>
    val le_gt_dec : nat -> nat -> sumbool = <fun>
    type diveucl = Divex of nat * nat
    val eucl_dev : nat -> nat -> diveucl = <fun>

    # eucl_dev (S (S O)) (S (S (S (S (S O)))));;
    - : diveucl = Divex (S (S O), S O)

It is easier to test on OCaml integers:

    # let rec nat_of_int = function 0 -> O | n -> S (nat_of_int (n-1));;
    val nat_of_int : int -> nat = <fun>

    # let rec int_of_nat = function O -> 0 | S p -> 1+(int_of_nat p);;
    val int_of_nat : nat -> int = <fun>

    # let div a b =
      let Divex (q,r) = eucl_dev (nat_of_int b) (nat_of_int a)
      in (int_of_nat q, int_of_nat r);;
    val div : int -> int -> int * int = <fun>

    # div 173 15;;
    - : int * int = (11, 8)

Note that these `nat_of_int` and `int_of_nat` are now available via a mere `Require Import ExtrOcamlIntConv` and then adding these functions to the list of functions to extract. This file `ExtrOcamlIntConv.v` and some others in `plugins/extraction/` are meant to help building concrete program via extraction.

### Extraction's horror museum

Some pathological examples of extraction are grouped in the file `test-suite/success/extraction.v` of the sources of Coq.

### Users' Contributions

Several of the Coq Users' Contributions use extraction to produce certified programs. In particular the following ones have an automatic extraction test:

> - `additions` : https://github.com/coq-contribs/additions

> - `bdds` : https://github.com/coq-contribs/bdds

> - `canon-bdds` : https://github.com/coq-contribs/canon-bdds

> - `chinese` : https://github.com/coq-contribs/chinese

> - `continuations` : https://github.com/coq-contribs/continuations

> - `coq-in-coq` : https://github.com/coq-contribs/coq-in-coq

> - `exceptions` : https://github.com/coq-contribs/exceptions

> - `firing-squad` : https://github.com/coq-contribs/firing-squad

> - `founify` : https://github.com/coq-contribs/founify

> - `graphs` : https://github.com/coq-contribs/graphs

> - `higman-cf` : https://github.com/coq-contribs/higman-cf

> - `higman-nw` : https://github.com/coq-contribs/higman-nw

> - `hardware` : https://github.com/coq-contribs/hardware

> - `multiplier` : https://github.com/coq-contribs/multiplier

> - `search-trees` : https://github.com/coq-contribs/search-trees

> - `stalmarck` : https://github.com/coq-contribs/stalmarck

Note that `continuations` and `multiplier` are a bit particular. They are examples of developments where `Obj.magic` is needed. This is probably due to a heavy use of impredicativity. After compilation, those two examples run nonetheless, thanks to the correction of the extraction [Let02].

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