Do more cleanup

2 files changed, 0 insertions, 1222 deletions

diff --git a/backends/hol4/divDefLibExampleScript.sml b/backends/hol4/divDefLibTestScript.sml
index c4a57783..c4a57783 100644
--- a/backends/hol4/divDefLibExampleScript.sml
+++ b/backends/hol4/divDefLibTestScript.sml
diff --git a/backends/hol4/divDefProto2TestScript.sml b/backends/hol4/divDefProto2TestScript.sml
deleted file mode 100644
index bc9ea9a7..00000000
--- a/backends/hol4/divDefProto2TestScript.sml
+++ /dev/null
@@ -1,1222 +0,0 @@
-open HolKernel boolLib bossLib Parse
-open boolTheory arithmeticTheory integerTheory intLib listTheory stringTheory
-
-open primitivesArithTheory primitivesBaseTacLib ilistTheory primitivesTheory
-open primitivesLib
-open divDefProto2Theory
-
-val _ = new_theory "divDefProto2TestScript"
-
-(*======================
- * Example 1: nth
- *======================*)
-Datatype:
-  list_t =
-    ListCons 't list_t
-  | ListNil
-End
-
-(* We use this version of the body to prove that the body is valid *)
-val nth_body_def = Define ‘
-  nth_body (f : (('t list_t # u32) + 't) -> (('t list_t # u32) + 't) result)
-    (x : (('t list_t # u32) + 't)) :
-    (('t list_t # u32) + 't) result =
-    (* Destruct the input. We need this to call the proper function in case
-       of mutually recursive definitions, but also to eliminate arguments
-       which correspond to the output value (the input type is the same
-       as the output type). *)
-    case x of
-    | INL x => (
-      let (ls, i) = x in
-      case ls of
-      | ListCons x tl =>
-        if u32_to_int i = (0:int)
-        then Return (INR x)
-        else
-          do
-          i0 <- u32_sub i (int_to_u32 1);
-          r <- f (INL (tl, i0));
-          (* Eliminate the invalid outputs. This is not necessary here,
-             but it is in the case of non tail call recursive calls. *)
-          case r of
-          | INL _ => Fail Failure
-          | INR i1 => Return (INR i1)
-          od
-      | ListNil => Fail Failure)
-    | INR _ => Fail Failure
-’
-
-val dbg = ref false
-fun print_dbg s = if (!dbg) then print s else ()
-
-(* Tactic which makes progress in a proof of validity by making a case
-   disjunction (we use this to explore all the paths in a function body). *)
-fun prove_valid_case_progress
-
-(*
-val (asms, g) = top_goal ()
-*)
-
-
-(* Tactic to prove that a body is valid: perform one step. *)
-fun prove_body_is_valid_tac_step (asms, g) =
-  let
-    (* The goal has the shape:
-       {[
-         (∀g h. ... g x = ... h x) ∨
-         ∃h y. is_valid_fp_body n h ∧ ∀g. ... g x = ... od
-       ]}   
-     *)
-    (* Retrieve the scrutinee in the goal (‘x’).
-       There are two cases:
-       - either the function has the shape:
-         {[
-           (λ(y,z). ...) x
-         ]}
-         in which case we need to destruct ‘x’
-       - or we have a normal ‘case ... of’
-     *)
-    val body = (lhs o snd o strip_forall o fst o dest_disj) g
-    val scrut =
-      let
-        val (app, x) = dest_comb body
-        val (app, _) = dest_comb app
-        val {Name=name, Thy=thy, Ty = _ } = dest_thy_const app
-      in
-        if thy = "pair" andalso name = "UNCURRY" then x else failwith "not a curried argument"
-      end
-      handle HOL_ERR _ => strip_all_cases_get_scrutinee body
-    (* Retrieve the first quantified continuations from the goal (‘g’) *)
-    val qc = (hd o fst o strip_forall o fst o dest_disj) g
-    (* Check if the scrutinee is a recursive call *)
-    val (scrut_app, _) = strip_comb scrut
-    val _ = print_dbg ("prove_body_is_valid_step: Scrutinee: " ^ term_to_string scrut ^ "\n")
-    (* For the recursive calls: *)
-    fun step_rec () =
-      let
-        val _ = print_dbg ("prove_body_is_valid_step: rec call\n")
-        (* We need to instantiate the ‘h’ existantially quantified function *)
-        (* First, retrieve the body of the function: it is given by the ‘Return’ branch *)
-        val (_, _, branches) = TypeBase.dest_case body
-        (* Find the branch corresponding to the return *)
-        val ret_branch = List.find (fn (pat, _) =>
-          let
-            val {Name=name, Thy=thy, Ty = _ } = (dest_thy_const o fst o strip_comb) pat
-          in
-            thy = "primitives" andalso name = "Return"
-          end) branches
-        val var = (hd o snd o strip_comb o fst o valOf) ret_branch
-        val br = (snd o valOf) ret_branch
-        (* Abstract away the input variable introduced by the pattern and the continuation ‘g’ *)
-        val h = list_mk_abs ([qc, var], br)
-        val _ = print_dbg ("prove_body_is_valid_step: h: " ^ term_to_string h ^ "\n")
-        (* Retrieve the input parameter ‘x’ *)
-        val input = (snd o dest_comb) scrut
-        val _ = print_dbg ("prove_body_is_valid_step: y: " ^ term_to_string input ^ "\n")
-      in
-        ((* Choose the right possibility (this is a recursive call) *)
-         disj2_tac >>
-         (* Instantiate the quantifiers *)
-         qexists ‘^h’ >>
-         qexists ‘^input’ >>
-         (* Unfold the predicate once *)
-         pure_once_rewrite_tac [is_valid_fp_body_def] >>
-         (* We have two subgoals:
-            - we have to prove that ‘h’ is valid
-            - we have to finish the proof of validity for the current body
-          *)
-         conj_tac >> fs [case_result_switch_eq])
-      end
-  in
-    (* If recursive call: special treatment. Otherwise, we do a simple disjunction *)
-    (if term_eq scrut_app qc then step_rec ()
-     else (Cases_on ‘^scrut’ >> fs [case_result_switch_eq])) (asms, g)
-  end
-
-(* Tactic to prove that a body is valid *)
-fun prove_body_is_valid_tac (body_def : thm option) : tactic =
-  let val body_def_thm = case body_def of SOME th => [th] | NONE => []
-  in
-    pure_once_rewrite_tac [is_valid_fp_body_def] >> gen_tac >>
-    (* Expand *)
-    fs body_def_thm >>
-    fs [bind_def, case_result_switch_eq] >>
-    (* Explore the body *)
-    rpt prove_body_is_valid_tac_step
-  end
-
-(* TODO: move *)
-val is_valid_fp_body_tm = “is_valid_fp_body”
-
-(* Prove that a body satisfies the validity condition of the fixed point *)
-fun prove_body_is_valid (body : term) : thm =
-  let
-    (* Explore the body and count the number of occurrences of nested recursive
-       calls so that we can properly instantiate the ‘N’ argument of ‘is_valid_fp_body’.
-
-       We first retrieve the name of the continuation parameter.
-       Rem.: we generated fresh names so that, for instance, the continuation name
-       doesn't collide with other names. Because of this, we don't need to look for
-       collisions when exploring the body (and in the worst case, we would cound
-       an overapproximation of the number of recursive calls, which is perfectly
-       valid).
-     *)
-    val fcont = (hd o fst o strip_abs) body
-    val fcont_name = (fst o dest_var) fcont
-    fun max x y = if x > y then x else y
-    fun count_body_rec_calls (body : term) : int =
-      case dest_term body of
-        VAR (name, _) => if name = fcont_name then 1 else 0
-      | CONST _ => 0
-      | COMB (x, y) => max (count_body_rec_calls x) (count_body_rec_calls y)
-      | LAMB (_, x) => count_body_rec_calls x
-    val num_rec_calls = count_body_rec_calls body
-
-    (* Generate the term ‘SUC (SUC ... (SUC n))’ where ‘n’ is a fresh variable.
-
-       Remark: we first prove ‘is_valid_fp_body (SUC ... n) body’ then substitue
-       ‘n’ with ‘0’ to prevent the quantity from being rewritten to a bit
-       representation, which would prevent unfolding of the ‘is_valid_fp_body’.
-     *)
-    val nvar = genvar num_ty
-    (* Rem.: we stack num_rec_calls + 1 occurrences of ‘SUC’ (and the + 1 is important) *)
-    fun mk_n i = if i = 0 then mk_suc nvar else mk_suc (mk_n (i-1))
-    val n_tm = mk_n num_rec_calls
-
-    (* Generate the lemma statement *)
-    val is_valid_tm = list_mk_icomb (is_valid_fp_body_tm, [n_tm, body])
-    val is_valid_thm = prove (is_valid_tm, prove_body_is_valid_tac NONE)
-
-    (* Replace ‘nvar’ with ‘0’ *)
-    val is_valid_thm = INST [nvar |-> zero_num_tm] is_valid_thm
-  in
-    is_valid_thm
-  end
-
-(*
-val (asms, g) = top_goal ()
-*)
-
-(* We first prove the theorem with ‘SUC (SUC n)’ where ‘n’ is a variable
-   to prevent this quantity from being rewritten to 2 *)
-Theorem nth_body_is_valid_aux:
-  is_valid_fp_body (SUC (SUC n)) nth_body
-Proof
-  prove_body_is_valid_tac (SOME nth_body_def)
-QED
-
-Theorem nth_body_is_valid:
-  is_valid_fp_body (SUC (SUC 0)) nth_body
-Proof
-  irule nth_body_is_valid_aux
-QED
-
-val nth_raw_def = Define ‘
-  nth (ls : 't list_t) (i : u32) =
-    case fix nth_body (INL (ls, i)) of
-    | Fail e => Fail e
-    | Diverge => Diverge
-    | Return r =>
-      case r of
-      | INL _ => Fail Failure
-      | INR x => Return x
-’
-
-val fix_tm = “fix”
-
-(* Generate the raw definitions by using the grouped definition body and the
-   fixed point operator *)
-fun mk_raw_defs (in_out_tys : (hol_type * hol_type) list)
-  (def_tms : term list) (body_is_valid : thm) : thm list =
-  let
-    (* Retrieve the body *)
-    val body = (List.last o snd o strip_comb o concl) body_is_valid
-
-    (* Create the term ‘fix body’ *)
-    val fixed_body = mk_icomb (fix_tm, body)
-
-    (* For every function in the group, generate the equation that we will
-       use as definition. In particular:
-       - add the properly injected input ‘x’ to ‘fix body’ (ex.: for ‘nth ls i’
-         we add the input ‘INL (ls, i)’)
-       - wrap ‘fix body x’ into a case disjunction to extract the relevant output
-
-       For instance, in the case of ‘nth ls i’:
-       {[
-         nth (ls : 't list_t) (i : u32) =
-           case fix nth_body (INL (ls, i)) of
-           | Fail e => Fail e
-           | Diverge => Diverge
-           | Return r =>
-             case r of
-             | INL _ => Fail Failure
-             | INR x => Return x
-       ]}
-     *)
-    fun mk_def_eq (i : int, def_tm : term) : term =
-      let
-        (* Retrieve the lhs of the original definition equation, and in
-           particular the inputs *)
-        val def_lhs = lhs def_tm
-        val args = (snd o strip_comb) def_lhs
-
-        (* Inject the inputs into the param type *)
-        val input = pairSyntax.list_mk_pair args
-        val input = inject_in_param_sum in_out_tys i true input
-
-        (* Compose*)
-        val def_rhs = mk_comb (fixed_body, input)
-
-        (* Wrap in the case disjunction *)
-        val def_rhs = mk_case_select_result_sum def_rhs in_out_tys i false
-
-        (* Create the equation *)
-        val def_eq_tm = mk_eq (def_lhs, def_rhs)
-      in
-        def_eq_tm
-      end
-    val raw_def_tms = map mk_def_eq (enumerate def_tms)
-
-    (* Generate the definitions *)
-    val raw_defs = map (fn tm => Define ‘^tm’) raw_def_tms
-  in
-    raw_defs
-  end
-
-(* Tactic which makes progress in a proof by making a case disjunction (we use
-   this to explore all the paths in a function body). *)
-fun case_progress (asms, g) =
-  let
-    val scrut = (strip_all_cases_get_scrutinee o lhs) g
-  in Cases_on ‘^scrut’ (asms, g) end
-
-(* Prove the final equation, that we will use as definition. *)
-fun prove_def_eq_tac
-  (current_raw_def : thm) (all_raw_defs : thm list) (is_valid : thm)
-  (body_def : thm option) : tactic =
-  let
-    val body_def_thm = case body_def of SOME th => [th] | NONE => []
-  in
-    rpt gen_tac >>
-    (* Expand the definition *)
-    pure_once_rewrite_tac [current_raw_def] >>
-    (* Use the fixed-point equality *)
-    pure_once_rewrite_left_tac [HO_MATCH_MP fix_fixed_eq is_valid] >>
-    (* Expand the body definition *)
-    pure_rewrite_tac body_def_thm >>
-    (* Expand all the definitions from the group *)
-    pure_rewrite_tac all_raw_defs >>
-    (* Explore all the paths - maybe we can be smarter, but this is fast and really easy *)
-    fs [bind_def] >>
-    rpt (case_progress >> fs [])
-  end
-
-(* Prove the final equations that we will give to the user as definitions *)
-fun prove_def_eqs (body_is_valid : thm) (def_tms : term list) (raw_defs : thm list) : thm list=
-  let
-    val defs_tgt_raw = zip def_tms raw_defs
-    (* Substitute the function variables with the constants introduced in the raw
-       definitions *)
-    fun compute_fsubst (def_tm, raw_def) : {redex: term, residue: term} =
-      let
-        val (fvar, _) = (strip_comb o lhs) def_tm
-        val fconst = (fst o strip_comb o lhs o snd o strip_forall o concl) raw_def
-      in
-        (fvar |-> fconst)
-      end
-    val fsubst = map compute_fsubst defs_tgt_raw
-    val defs_tgt_raw = map (fn (x, y) => (subst fsubst x, y)) defs_tgt_raw
-
-    fun prove_def_eq (def_tm, raw_def) : thm =
-      let
-        (* Quantify the parameters *)
-        val (_, params) = (strip_comb o lhs) def_tm
-        val def_eq_tm = list_mk_forall (params, def_tm)
-        (* Prove *)
-        val def_eq = prove (def_eq_tm, prove_def_eq_tac raw_def raw_defs body_is_valid NONE)
-      in
-        def_eq
-      end
-    val def_eqs = map prove_def_eq defs_tgt_raw
-  in
-    def_eqs
-  end
-
-Theorem nth_def:
-  ∀ls i. nth (ls : 't list_t) (i : u32) : 't result =
-    case ls of
-    | ListCons x tl =>
-      if u32_to_int i = (0:int)
-      then (Return x)
-      else
-        do
-        i0 <- u32_sub i (int_to_u32 1);
-        nth tl i0
-        od
-    | ListNil => Fail Failure
-Proof
-  prove_def_eq_tac nth_raw_def [nth_raw_def] nth_body_is_valid nth_body_def
-QED
-
-(*======================
- * Example 2: even, odd
- *======================*)
-
-val def_qt = ‘
-  (even (i : int) : bool result =
-    if i = 0 then Return T else odd (i - 1)) /\
-  (odd (i : int) : bool result =
-    if i = 0 then Return F else even (i - 1))
-’
-
-val result_ty = “:'a result”
-val error_ty = “:error”
-val alpha_ty = “:'a”
-val num_ty = “:num”
-
-val return_tm = “Return : 'a -> 'a result”
-val fail_tm = “Fail : error -> 'a result”
-val fail_failure_tm = “Fail Failure : 'a result”
-val diverge_tm = “Diverge : 'a result”
-
-val zero_num_tm = “0:num”
-val suc_tm = “SUC”
-
-fun mk_result (ty : hol_type) : hol_type = Type.type_subst [ alpha_ty |-> ty ] result_ty
-fun dest_result (ty : hol_type) : hol_type =
-  let
-    val {Args=out_ty, Thy=thy, Tyop=tyop} = dest_thy_type ty
-  in
-    if thy = "primitives" andalso tyop = "result" then hd out_ty
-    else failwith "dest_result: not a result"
-  end
-
-fun mk_return (x : term) : term = mk_icomb (return_tm, x)
-fun mk_fail (ty : hol_type) (e : term) : term = mk_comb (inst [ alpha_ty |-> ty ] fail_tm, e)
-fun mk_fail_failure (ty : hol_type) : term = inst [ alpha_ty |-> ty ] fail_failure_tm
-fun mk_diverge (ty : hol_type) : term = inst [ alpha_ty |-> ty ] diverge_tm
-
-fun mk_suc (n : term) = mk_comb (suc_tm, n)
-
-(*
- *)
-
-(* **BODY GENERATION**:
-   ====================
-
-   When generating a recursive definition, we apply a fixed-point operator to
-   a function body. In case we define a group of mutually recursive definitions,
-   we generate *one* single body for the whole group of definitions. It works
-   as follows.
-
-   The input of the body is an enumeration: we start by branching over this input, and
-   every branch corresponds to one function in the mutually recursive group.  Also, the
-   inputs must be grouped into tuples.  Whenever we make a recursive call, we wrap the
-   input parameters into the proper variant, so as to call the proper function.
-
-   Moreover, the input of the body must have the same type as its output: we also
-   store the outputs of the functions in some variants of the enumeration.
-
-   In order to make this work, we need to shape the body so that:
-   - input values/output values are properly injected into the enumeration
-   - whenever we get an output value (which is an enumeration), we extract
-     the value from the proper variant of the enumeration
-
-   We encode the enumeration with a nested sum type, whose constructors
-   are ‘INL’ and ‘INR’.
-
-   Example:
-   ========
-   We consider the following group of mutually recursive definitions: *)
-
-val even_odd_qt = Defn.parse_quote ‘
- (even (i : int) : bool result = if i = 0 then Return T else odd (i - 1)) /\
- (odd (i : int) : bool result = if i = 0 then Return F else even (i - 1))
-’
-
-(* From those equations, we generate the following body: *)
-
-val even_odd_body_def = Define ‘
-  even_odd_body
-    (* The body takes a continuation - required by the fixed-point operator *)
-    (f : (int + bool + int + bool) -> (int + bool + int + bool) result)
-
-    (* The type of the input is:
-       input of even + output of even + input of odd + output of odd *)
-    (x : int + bool + int + bool) :
-
-    (* The output type is the same as the input type - this constraint
-       comes from limitations in the way we can define the fixed-point
-       operator inside the HOL logic *)
-    (int + bool + int + bool) result =
-
-  (* Case disjunction over the input, in order to figure out which
-     function from the group is actually called (even, or odd). *)
-  case x of
-  | INL i => (* Input of even *)
-    (* Body of even *)
-    if i = 0 then Return (INR (INL T))
-    else
-      (* Recursive calls are calls to the continuation f, wrapped
-         in the proper variant: here we call odd *)
-      (case f (INR (INR (INL (i - 1)))) of
-       | Fail e => Fail e
-       | Diverge => Diverge
-       | Return r =>
-         (* Extract the proper value from the enumeration: here, the
-            call is tail-call so this is not really necessary, but we
-            might need to retrieve the output of the call to odd, which
-            is a boolean, and do something more complex with it. *)
-         case r of
-         | INL _ => Fail Failure
-         | INR (INL _) => Fail Failure
-         | INR (INR (INL _)) => Fail Failure
-         | INR (INR (INR b)) => (* Extract the output of odd *)
-           (* Return: inject into the variant for the output of even *)
-           Return (INR (INL b))
-         )
-  | INR (INL _) => (* Output of even *)
-    (* We must ignore this one *)
-    Fail Failure
-  | INR (INR (INL i)) =>
-    (* Body of odd *)
-    if i = 0 then Return (INR (INR (INR F)))
-    else
-      (* Call to even *)
-      (case f (INL (i - 1)) of
-       | Fail e => Fail e
-       | Diverge => Diverge
-       | Return r =>
-         (* Extract the proper value from the enumeration *)
-         case r of
-         | INL _ => Fail Failure
-         | INR (INL b) => (* Extract the output of even *)
-           (* Return: inject into the variant for the output of odd *)
-           Return (INR (INR (INR b)))
-         | INR (INR (INL _)) => Fail Failure
-         | INR (INR (INR _)) => Fail Failure
-         )
-  | INR (INR (INR _)) => (* Output of odd *)
-    (* We must ignore this one *)
-    Fail Failure
-’
-
-(* Small helper to generate wrappers of the shape: ‘INL x’, ‘INR (INL x)’, etc.
-   Note that we should have: ‘length before_tys + 1 + length after tys >= 2’
-
-   Ex.:
-   ====
-   The enumeration has type: “: 'a + 'b + 'c + 'd”.
-   We want to generate the variant which injects “x:'c” into this enumeration.
-
-   We need to split the list of types into:
-   {[
-     before_tys = [“:'a”, “'b”]
-     tm = “x: 'c”
-     after_tys = [“:'d”]
-   ]}
-
-   The function will generate:
-   {[
-     INR (INR (INL x) : 'a + 'b + 'c + 'd
-   ]}
-
-   (* Debug *)
-   val before_tys = [“:'a”, “:'b”, “:'c”]
-   val tm = “x:'d”
-   val after_tys = [“:'e”, “:'f”]
-
-   val before_tys = [“:'a”, “:'b”, “:'c”]
-   val tm = “x:'d”
-   val after_tys = []
-
-   mk_inl_inr_wrapper before_tys tm after_tys
- *)
-fun list_mk_inl_inr (before_tys : hol_type list) (tm : term) (after_tys : hol_type list) :
-  term =
-  let
-    val (before_tys, pat) =
-      if after_tys = []
-      then
-        let
-          val just_before_ty = List.last before_tys
-          val before_tys = List.take (before_tys, List.length before_tys - 1)
-          val pat = sumSyntax.mk_inr (tm, just_before_ty)
-        in
-          (before_tys, pat)
-        end
-      else (before_tys, sumSyntax.mk_inl (tm, sumSyntax.list_mk_sum after_tys))
-    val pat = foldr (fn (ty, pat) => sumSyntax.mk_inr (pat, ty)) pat before_tys
-  in
-    pat
-  end
-
-
-(* This function wraps a term into the proper variant of the input/output
-   sum.
-
-   Ex.:
-   ====
-   For the input of the first function, we generate: ‘INL x’
-   For the output of the first function, we generate: ‘INR (INL x)’
-   For the input of the 2nd function, we generate: ‘INR (INR (INL x))’
-   etc.
-
-   If ‘is_input’ is true: we are wrapping an input. Otherwise we are wrapping
-   an output.
-
-   (* Debug *)
-   val tys = [(“:'a”, “:'b”), (“:'c”, “:'d”), (“:'e”, “:'f”)]
-   val j = 1
-   val tm = “x:'c”
-   val tm = “y:'d”
-   val is_input = true
- *)
-fun inject_in_param_sum (tys : (hol_type * hol_type) list) (j : int) (is_input : bool)
-  (tm : term) : term =
-  let
-    fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls)
-    val before_tys = flatten (List.take (tys, j))
-    val (input_ty, output_ty) = List.nth (tys, j)
-    val after_tys = flatten (List.drop (tys, j + 1))
-    val (before_tys, after_tys) =
-      if is_input then (before_tys, output_ty :: after_tys)
-      else (before_tys @ [input_ty], after_tys)
-  in
-    list_mk_inl_inr before_tys tm after_tys
-  end
-
-(* Remark: the order of the branches when creating matches is important.
-   For instance, in the case of ‘result’ it must be: ‘Return’, ‘Fail’, ‘Diverge’.
-
-   For the purpose of stability and maintainability, we introduce this small helper
-   which reorders the cases in a pattern before actually creating the case
-   expression.
- *)
-fun unordered_mk_case (scrut: term, pats: (term * term) list) : term =
-  let
-    (* Retrieve the constructors *)
-    val cl = TypeBase.constructors_of (type_of scrut)
-    (* Retrieve the names of the constructors *)
-    val names = map (fst o dest_const) cl
-    (* Use those to reorder the patterns *)
-    fun is_pat (name : string) (pat, _) =
-      let
-        val app = (fst o strip_comb) pat
-        val app_name = (fst o dest_const) app
-      in
-        app_name = name
-      end
-    val pats = map (fn name => valOf (List.find (is_pat name) pats)) names
-  in
-    (* Create the case *)
-    TypeBase.mk_case (scrut, pats)
-  end
-
-(* Wrap a term of type “:'a result” into a ‘case of’ which matches over
-   the result.
-
-   Ex.:
-   ====
-   {[
-     f x
-
-       ~~>
-
-     case f x of
-     | Fail e => Fail e
-     | Diverge => Diverge
-     | Return y => ... (* The branch content is generated by the continuation *)
-   ]}
-
-   ‘gen_ret_branch’ is a *continuation* which generates the content of the
-   ‘Return’ branch (i.e., the content of the ‘...’ in the example above).
-   It receives as input the value contained by the ‘Return’ (i.e., the variable
-   ‘y’ in the example above).
-
-   Remark.: the type of the term generated by ‘gen_ret_branch’ must have
-   the type ‘result’, but it can change the content of the result (i.e.,
-   if ‘scrut’ has type ‘:'a result’, we can change the type of the wrapped
-   expression to ‘:'b result’).
-
-   (* Debug *)
-   val scrut = “x: int result”
-   fun gen_ret_branch x = mk_return x
-
-   val scrut = “x: int result”
-   fun gen_ret_branch _ = “Return T”
-
-   mk_result_case scrut gen_ret_branch
- *)
-fun mk_result_case (scrut : term) (gen_ret_branch : term -> term) : term =
-  let
-    val scrut_ty = dest_result (type_of scrut)
-    (* Return branch *)
-    val ret_var = genvar scrut_ty
-    val ret_pat = mk_return ret_var
-    val ret_br = gen_ret_branch ret_var
-    val ret_ty = dest_result (type_of ret_br)
-    (* Failure branch *)
-    val fail_var = genvar error_ty
-    val fail_pat = mk_fail scrut_ty fail_var
-    val fail_br = mk_fail ret_ty fail_var
-    (* Diverge branch *)
-    val div_pat = mk_diverge scrut_ty
-    val div_br = mk_diverge ret_ty
-  in
-    unordered_mk_case (scrut, [(ret_pat, ret_br), (fail_pat, fail_br), (div_pat, div_br)])
-  end
-
-(* Generate a ‘case ... of’ over a sum type.
-
-   Ex.:
-   ====
-   If the scrutinee is: “x : 'a + 'b + 'c” (i.e., the tys list is: [“:'a”, “:b”, “:c”]),
-   we generate:
-
-   {[
-     case x of
-     | INL y0 => ... (* Branch of index 0 *)
-     | INR (INL y1) => ... (* Branch of index 1 *)
-     | INR (INR (INL y2)) => ... (* Branch of index 2 *)
-     | INR (INR (INR y3)) => ... (* Branch of index 3 *)
-   ]}
-
-   The content of the branches is generated by the ‘gen_branch’ continuation,
-   which receives as input the index of the branch as well as the variable
-   introduced by the pattern (in the example above: ‘y0’ for the branch 0,
-   ‘y1’ for the branch 1, etc.)
-
-   (* Debug *)
-   val tys = [“:'a”, “:'b”]
-   val scrut = mk_var ("x", sumSyntax.list_mk_sum tys)
-   fun gen_branch i (x : term) = “F”
-
-   val tys = [“:'a”, “:'b”, “:'c”, “:'d”]
-   val scrut = mk_var ("x", sumSyntax.list_mk_sum tys)
-   fun gen_branch i (x : term) = if type_of x = “:'c” then mk_return x else mk_fail_failure “:'c”
-
-   list_mk_sum_case scrut tys gen_branch
- *)
-(* For debugging *)
-val list_mk_sum_case_case = ref (“T”, [] : (term * term) list)
-(*
-val (scrut, [(pat1, br1), (pat2, br2)]) = !list_mk_sum_case_case
-*)
-fun list_mk_sum_case (scrut : term) (tys : hol_type list)
-  (gen_branch : int -> term -> term) : term =
-  let
-    (* Create the cases. Note that without sugar, the match actually looks like this:
-       {[
-         case x of
-         | INL y0 => ... (* Branch of index 0 *)
-         | INR x1
-           case x1 of
-           | INL y1 => ... (* Branch of index 1 *)
-           | INR x2 =>
-             case x2 of
-             | INL y2 => ... (* Branch of index 2 *)
-             | INR y3 => ... (* Branch of index 3 *)
-       ]}
-     *)
-    fun create_case (j : int) (scrut : term) (tys : hol_type list) : term =
-      let
-        val _ = print_dbg ("list_mk_sum_case: " ^
-                           String.concatWith ", " (map type_to_string tys) ^ "\n")
-      in
-        case tys of
-          [] => failwith "tys is too short"
-        | [ ty ] =>
-          (* Last element: no match to perform *)
-          gen_branch j scrut
-        | ty1 :: tys =>
-          (* Not last: we create a pattern:
-             {[
-               case scrut of
-               | INL pat_var1 => ... (* Branch of index i *)
-               | INR pat_var2 =>
-                 ... (* Generate this term recursively *)
-             ]}
-           *)
-          let
-            (* INL branch *)
-            val after_ty = sumSyntax.list_mk_sum tys
-            val pat_var1 = genvar ty1
-            val pat1 = sumSyntax.mk_inl (pat_var1, after_ty)
-            val br1 = gen_branch j pat_var1
-            (* INR branch *)
-            val pat_var2 = genvar after_ty
-            val pat2 = sumSyntax.mk_inr (pat_var2, ty1)
-            val br2 = create_case (j+1) pat_var2 tys
-            val _ = print_dbg ("list_mk_sum_case: assembling:\n" ^
-                               term_to_string scrut ^ ",\n" ^
-                               "[(" ^ term_to_string pat1 ^ ",\n  " ^ term_to_string br1 ^ "),\n\n" ^
-                               " (" ^ term_to_string pat2 ^ ",\n  " ^ term_to_string br2 ^ ")]\n\n")
-            val case_elems = (scrut, [(pat1, br1), (pat2, br2)])
-            val _ = list_mk_sum_case_case := case_elems
-          in
-            (* Put everything together *)
-            TypeBase.mk_case case_elems
-          end
-      end
-  in
-    create_case 0 scrut tys
-  end
-
-(* Generate a ‘case ... of’ to select the input/output of the ith variant of
-   the param enumeration.
-
-   Ex.:
-   ====
-   There are two functions in the group, and we select the input of the function of index 1:
-   {[
-     case x of
-     | INL _ => Fail Failure              (* Input of function of index 0 *)
-     | INR (INL _) => Fail Failure        (* Output of function of index 0 *)
-     | INR (INR (INL y)) => Return y      (* Input of the function of index 1: select this one *)
-     | INR (INR (INR _)) => Fail Failure  (* Output of the function of index 1 *)
-   ]}
-
-   (* Debug *)
-   val tys = [(“:'a”, “:'b”)]
-   val scrut = “x : 'a + 'b”
-   val fi = 0
-   val is_input = true
-
-   val tys = [(“:'a”, “:'b”), (“:'c”, “:'d”)]
-   val scrut = “x : 'a + 'b + 'c + 'd”
-   val fi = 1
-   val is_input = false
-
-   val scrut = mk_var ("x", sumSyntax.list_mk_sum (flatten tys))
-
-   list_mk_case_select scrut tys fi is_input
- *)
-fun list_mk_case_sum_select (scrut : term) (tys : (hol_type * hol_type) list)
-  (fi : int) (is_input : bool) : term =
-  let
-    (* The index of the element in the enumeration that we will select *)
-    val i = 2 * fi + (if is_input then 0 else 1)
-    (* Flatten the types and numerotate them *)
-    fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls)
-    val tys = flatten tys
-    (* Get the return type *)
-    val ret_ty = List.nth (tys, i)
-    (* The continuation which will generate the content of the branches *)
-    fun gen_branch j var = if j = i then mk_return var else mk_fail_failure ret_ty
-  in
-    (* Generate the ‘case ... of’ *)
-    list_mk_sum_case scrut tys gen_branch
-  end
-
-(* Generate a ‘case ... of’ to select the input/output of the ith variant of
-   the param enumeration.
-
-   Ex.:
-   ====
-   There are two functions in the group, and we select the input of the function of index 1:
-   {[
-     case x of
-     | Fail e => Fail e
-     | Diverge => Diverge
-     | Return r =>
-       case r of
-       | INL _ => Fail Failure              (* Input of function of index 0 *)
-       | INR (INL _) => Fail Failure        (* Output of function of index 0 *)
-       | INR (INR (INL y)) => Return y      (* Input of the function of index 1: select this one *)
-       | INR (INR (INR _)) => Fail Failure  (* Output of the function of index 1 *)
-   ]}
- *)
-fun mk_case_select_result_sum (scrut : term) (tys : (hol_type * hol_type) list)
-  (fi : int) (is_input : bool) : term =
-  (* We match over the result, then over the enumeration *)
-  mk_result_case scrut (fn x => list_mk_case_sum_select x tys fi is_input)
-
-(*
-val scrut = call
-val tys = in_out_tys
-val is_input = false
-val call = mk_case_select_result_sum call in_out_tys fi false
-*)
-
-(* TODO: move *)
-fun enumerate (ls : 'a list) : (int * 'a) list =
-  zip (List.tabulate (List.length ls, fn i => i)) ls
-
-(* Generate a body for the fixed-point operator from a quoted group of mutually
-   recursive definitions.
-
-   See TODO for detailed explanations: from the quoted equations for ‘nth’
-   (or for [‘even’, ‘odd’]) we generate the body ‘nth_body’ (or ‘even_odd_body’,
-   respectively).
- *)
-fun mk_body (fnames : string list) (in_out_tys : (hol_type * hol_type) list)
-  (def_tms : term list) : term =
-  let
-    val fnames_set = Redblackset.fromList String.compare fnames
-
-    (* Compute a map from function name to function index *)
-    val fnames_map = Redblackmap.fromList String.compare
-      (map (fn (x, y) => (y, x)) (enumerate fnames))
-
-    (* Compute the input/output type, that we dub the "parameter type" *)
-    fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls)
-    val param_type = sumSyntax.list_mk_sum (flatten in_out_tys)
-
-    (* Introduce a variable for the confinuation *)
-    val fcont = genvar (param_type --> mk_result param_type)
-
-    (* In the function equations, replace all the recursive calls with calls to the continuation.
-
-       When replacing a recursive call, we have to do two things:
-       - we need to inject the input parameters into the parameter type
-         Ex.:
-         - ‘nth tl i’ becomes ‘f (INL (tl, i))’ where ‘f’ is the continuation
-         - ‘even i’ becomes ‘f (INL i)’ where ‘f’ is the continuation
-       - we need to wrap the the call to the continuation into a ‘case ... of’
-         to extract its output (we need to make sure that the transformation
-         preserves the type of the expression!)
-         Ex.: ‘nth tl i’ becomes:
-         {[
-           case f (INL (tl, i)) of
-           | Fail e => Fail e
-           | Diverge => Diverge
-           | Return r =>
-             case r of
-             | INL _ => Fail Failure
-             | INR x => Return (INR x)
-         ]}
-     *)
-     (* For debugging *)
-     val replace_rec_calls_rec_call_tm = ref “T”
-     fun replace_rec_calls (fnames_set : string Redblackset.set) (tm : term) : term =
-       let
-         val _ = print_dbg ("replace_rec_calls: original expression:\n" ^
-                            term_to_string tm ^ "\n\n")
-         val ntm =
-           case dest_term tm of
-             VAR (name, ty) =>
-             (* Check that this is not one of the functions in the group - remark:
-                we could handle that by introducing lambdas.
-              *)
-             if Redblackset.member (fnames_set, name)
-             then failwith ("mk_body: not well-formed definition: found " ^ name ^
-                            " in an improper position")
-             else tm
-           | CONST _ => tm
-           | LAMB (x, tm) =>
-             let
-               (* The variable might shadow one of the functions *)
-               val fnames_set = Redblackset.delete (fnames_set, (fst o dest_var) x)
-               (* Update the term in the lambda *)
-               val tm = replace_rec_calls fnames_set tm
-             in
-               (* Reconstruct *)
-               mk_abs (x, tm)
-             end
-           | COMB (_, _) =>
-             let
-               (* Completely destruct the application, check if this is a recursive call *)
-               val (app, args) = strip_comb tm
-               val is_rec_call = Redblackset.member (fnames_set, (fst o dest_var) app)
-                 handle HOL_ERR _ => false
-               (* Whatever the case, apply the transformation to all the inputs *)
-               val args = map (replace_rec_calls fnames_set) args
-             in
-               (* If this is not a recursive call: apply the transformation to all the
-                  terms. Otherwise, replace. *)
-               if not is_rec_call then list_mk_comb (replace_rec_calls fnames_set app, args)
-               else
-                 (* Rec call: replace *)
-                 let
-                   val _ = replace_rec_calls_rec_call_tm := tm
-                   (* First, find the index of the function *)
-                   val fname = (fst o dest_var) app
-                   val fi = Redblackmap.find (fnames_map, fname)
-                   (* Inject the input values into the param type *)
-                   val input = pairSyntax.list_mk_pair args
-                   val input = inject_in_param_sum in_out_tys fi true input
-                   (* Create the recursive call *)
-                   val call = mk_comb (fcont, input)
-                   (* Wrap the call into a ‘case ... of’ to extract the output *)
-                   val call = mk_case_select_result_sum call in_out_tys fi false
-                 in
-                   (* Return *)
-                   call
-                 end
-             end
-         val _ = print_dbg ("replace_rec_calls: new expression:\n" ^ term_to_string ntm ^ "\n\n")
-       in
-         ntm
-       end
-       handle HOL_ERR e =>
-         let
-           val _ = print_dbg ("replace_rec_calls: failed on:\n" ^ term_to_string tm ^ "\n\n")
-         in
-           raise (HOL_ERR e)
-         end
-     fun replace_rec_calls_in_eq (eq : term) : term =
-       let
-         val (l, r) = dest_eq eq
-       in
-         mk_eq (l, replace_rec_calls fnames_set r)
-       end
-     val def_tms_with_fcont = map replace_rec_calls_in_eq def_tms
-
-     (* Wrap all the function bodies to inject their result into the param type.
-
-        We collect the function inputs at the same time, because they will be
-        grouped into a tuple that we will have to deconstruct.
-      *)
-     fun inject_body_to_enums (i : int, def_eq : term) : term list * term =
-       let
-         val (l, body) = dest_eq def_eq
-         val (_, args) = strip_comb l
-         (* We have the deconstruct the result, then, in the ‘Return’ branch,
-            properly wrap the returned value *)
-         val body = mk_result_case body (fn x => mk_return (inject_in_param_sum in_out_tys i false x))
-       in
-         (args, body)
-       end
-     val def_tms_inject = map inject_body_to_enums (enumerate def_tms_with_fcont)
-
-     (* Currify the body inputs.
-
-        For instance, if the body has inputs: ‘x’, ‘y’; we return the following:
-        {[
-          (‘z’, ‘case z of (x, y) => ... (* body *) ’)
-        ]}
-        where ‘z’ is fresh.
-
-        We return: (curried input, body).
-
-        (* Debug *)
-        val body = “(x:'a, y:'b, z:'c)”
-        val args = [“x:'a”, “y:'b”, “z:'c”]
-        currify_body_inputs (args, body)
-      *)
-     fun currify_body_inputs (args : term list, body : term) : term * term =
-       let
-         fun mk_curry (args : term list) (body : term) : term * term =
-           case args of
-             [] => failwith "no inputs"
-           | [x] => (x, body)
-           | x1 :: args =>
-             let
-               val (x2, body) = mk_curry args body
-               val scrut = genvar (pairSyntax.list_mk_prod (map type_of (x1 :: args)))
-               val pat = pairSyntax.mk_pair (x1, x2)
-               val br = body
-             in
-               (scrut, TypeBase.mk_case (scrut, [(pat, br)]))
-             end
-       in
-         mk_curry args body
-       end
-     val def_tms_currified = map currify_body_inputs def_tms_inject
-
-     (* Group all the functions into a single body, with an outer ‘case .. of’
-        which selects the appropriate body depending on the input *)
-     val param_ty = sumSyntax.list_mk_sum (flatten in_out_tys)
-     val input = genvar param_ty
-     fun mk_mut_rec_body_branch (i : int) (patvar : term) : term =
-       (* Case disjunction on whether the branch is for an input value (in
-          which case we call the proper body) or an output value (in which
-          case we return ‘Fail ...’ *)
-       if i mod 2 = 0 then
-         let
-           val fi = i div 2
-           val (x, def_tm) = List.nth (def_tms_currified, fi)
-           (* The variable in the pattern and the variable expected by the
-              body may not be the same: we introduce a let binding *)
-           val def_tm = mk_let (mk_abs (x, def_tm), patvar)
-         in
-           def_tm
-         end
-       else
-         (* Output value: fail *)
-         mk_fail_failure param_ty
-     val mut_rec_body = list_mk_sum_case input (flatten in_out_tys) mk_mut_rec_body_branch
-
-
-     (* Abstract away the parameters to produce the final body of the fixed point *)
-     val mut_rec_body = list_mk_abs ([fcont, input], mut_rec_body)
-  in
-    mut_rec_body
-  end
-
-(* For explanations about the different steps, see TODO *)
-fun DefineDiv (def_qt : term quotation) =
-  let
-    (* Parse the definitions.
-    
-       Example:
-       {[
-         (even (i : int) : bool result = if i = 0 then Return T else odd (i - 1)) /\
-         (odd (i : int) : bool result = if i = 0 then Return F else even (i - 1))
-       ]}
-     *)
-    val def_tms = (strip_conj o list_mk_conj o rev) (Defn.parse_quote def_qt)
-
-    (* Compute the names and the input/output types of the functions *)
-    fun compute_names_in_out_tys (tm : term) : string * (hol_type * hol_type) =
-      let
-        val app = lhs tm
-        val name = (fst o dest_var o fst o strip_comb) app
-        val out_ty = dest_result (type_of app)
-        val input_tys = pairSyntax.list_mk_prod (map type_of ((snd o strip_comb) app))
-      in
-        (name, (input_tys, out_ty))
-      end
-    val (fnames, in_out_tys) = unzip (map compute_names_in_out_tys def_tms)
-
-    (* Generate the body.
-
-       See the comments at the beginning of the file (lookup "BODY GENERATION").
-     *)
-    val body = mk_body fnames in_out_tys def_tms
-
-    (* Prove that the body satisfies the validity property required by the fixed point *)
-    val body_is_valid = prove_body_is_valid body
-    
-    (* Generate the definitions for the various functions by using the fixed point
-       and the body. *)
-    val raw_defs = mk_raw_defs in_out_tys def_tms body_is_valid
-
-    (* Prove the final equations *)
-    val def_eqs = prove_def_eqs body_is_valid def_tms raw_defs
-  in
-    def_eqs
-  end
-
-val [even_def, odd_def] = DefineDiv ‘
-  (even (i : int) : bool result =
-    if i = 0 then Return T else odd (i - 1)) /\
-  (odd (i : int) : bool result =
-    if i = 0 then Return F else even (i - 1))
-’
-
-val [nth_def] = DefineDiv ‘
-  nth (ls : 't list_t) (i : u32) : 't result =
-    case ls of
-    | ListCons x tl =>
-      if u32_to_int i = (0:int)
-      then (Return x)
-      else
-        do
-        i0 <- u32_sub i (int_to_u32 1);
-        nth tl i0
-        od
-    | ListNil => Fail Failure
-’
-
-val even_odd_body_def = Define ‘
-  even_odd_body
-  (* The body takes a continuation - required by the fixed-point operator *)
-  (f : (int + bool + int + bool) -> (int + bool + int + bool) result)
-  (* The type of the input/output is:
-     input of even + output of even + input of odd + output of odd
-   *)
-  (x : int + bool + int + bool) :
-  (int + bool + int + bool) result =
-  (* Case disjunction over the input, in order to figure out which
-     function from the group is actually called (even , or odd). *)
-  case x of
-  | INL i => (* Input of even *)
-    (* Body of even *)
-    if i = 0 then Return (INR (INL T))
-    else
-      (* Recursive calls are calls to the continuation f, wrapped
-         in the proper variant: here we call odd *)
-      (case f (INR (INR (INL (i - 1)))) of
-       | Fail e => Fail e
-       | Diverge => Diverge
-       | Return r =>
-         (* Eliminate the unwanted results *)
-         case r of
-         | INL _ => Fail Failure
-         | INR (INL _) => Fail Failure
-         | INR (INR (INL _)) => Fail Failure
-         | INR (INR (INR b)) => (* Extract the output of odd *)
-           (* Inject into the variant for the output of even *)
-           Return (INR (INL b))
-         )
-  | INR (INL _) => (* Output of even *)
-    (* We must ignore this one *)
-    Fail Failure
-  | INR (INR (INL i)) =>
-    (* Body of odd *)
-    if i = 0 then Return (INR (INR (INR F)))
-    else
-      (* Call to even *)
-      (case f (INL (i - 1)) of
-       | Fail e => Fail e
-       | Diverge => Diverge
-       | Return r =>
-         (* Eliminate the unwanted results *)
-         case r of
-         | INL _ => Fail Failure
-         | INR (INL b) => (* Extract the output of even *)
-           (* Inject into the variant for the output of odd *)
-           Return (INR (INR (INR b)))
-         | INR (INR (INL _)) => Fail Failure
-         | INR (INR (INR _)) => Fail Failure
-         )
-  | INR (INR (INR _)) => (* Output of odd *)
-    (* We must ignore this one *)
-    Fail Failure
-’
-
-Theorem even_odd_body_is_valid_aux:
-  is_valid_fp_body (SUC (SUC n)) even_odd_body
-Proof
-  prove_body_is_valid_tac (SOME even_odd_body_def)
-QED
-
-Theorem even_odd_body_is_valid:
-  is_valid_fp_body (SUC (SUC 0)) even_odd_body
-Proof
-  irule even_odd_body_is_valid_aux
-QED
-
-val even_raw_def = Define ‘
-  even (i : int) =
-    case fix even_odd_body (INL i) of
-    | Fail e => Fail e
-    | Diverge => Diverge
-    | Return r =>
-      case r of
-      | INL _ => Fail Failure
-      | INR (INL b) => Return b
-      | INR (INR (INL _)) => Fail Failure
-      | INR (INR (INR _)) => Fail Failure
-’
-
-val odd_raw_def = Define ‘
-  odd (i : int) =
-    case fix even_odd_body (INR (INR (INL i))) of
-    | Fail e => Fail e
-    | Diverge => Diverge
-    | Return r =>
-      case r of
-      | INL _ => Fail Failure
-      | INR (INL b) => Fail Failure
-      | INR (INR (INL _)) => Fail Failure
-      | INR (INR (INR b)) => Return b
-’
-
-Theorem even_def:
-  ∀i. even (i : int) : bool result =
-    if i = 0 then Return T else odd (i - 1)
-Proof
-  prove_def_eq_tac even_raw_def [even_raw_def, odd_raw_def] even_odd_body_is_valid even_odd_body_def
-QED
-
-Theorem odd_def:
-  ∀i. odd (i : int) : bool result =
-    if i = 0 then Return F else even (i - 1)
-Proof
-  prove_def_eq_tac odd_raw_def [even_raw_def, odd_raw_def] even_odd_body_is_valid even_odd_body_def
-QED
-
-val _ = export_theory ()