Merge pull request #51 from AeneasVerif/son_merge_back2

Improve the `pspec` attribute and the `divergent` encoding

8 files changed, 1690 insertions, 774 deletions

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 6a52387d..9458c926 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -5,6 +5,7 @@ import Mathlib.Tactic.RunCmd
 import Mathlib.Tactic.Linarith
 import Base.Primitives.Base
 import Base.Arith.Base
+import Base.Diverge.ElabBase
 
 /- TODO: this is very useful, but is there more? -/
 set_option profiler true
@@ -12,6 +13,78 @@ set_option profiler.threshold 100
 
 namespace Diverge
 
+/- Auxiliary lemmas -/
+namespace Lemmas
+  -- TODO: not necessary anymore
+  def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
+    if heq: m = n then True
+    else
+      f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
+      for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
+  termination_by for_all_fin_aux n _ m h => n - m
+  decreasing_by
+    simp_wf
+    apply Nat.sub_add_lt_sub <;> try simp
+    simp_all [Arith.add_one_le_iff_le_ne]
+
+  def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
+
+  theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
+    (h : m ≤ n) →
+    for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
+    := by
+    generalize h: (n - m) = k
+    revert m
+    induction k -- TODO: induction h rather?
+    case zero =>
+      simp_all
+      intro m h1 h2
+      have h: n = m := by
+        linarith
+      unfold for_all_fin_aux; simp_all
+      simp_all
+      -- There is no i s.t. m ≤ i
+      intro i h3; cases i; simp_all
+      linarith
+    case succ k hi =>
+      intro m hk hmn
+      intro hf i hmi
+      have hne: m ≠ n := by
+        have hineq := Nat.lt_of_sub_eq_succ hk
+        linarith
+      -- m = i?
+      if heq: m = i then
+        -- Yes: simply use the `for_all_fin_aux` hyp
+        unfold for_all_fin_aux at hf
+        simp_all
+      else
+        -- No: use the induction hypothesis
+        have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
+        have hineq: m + 1 ≤ n := by
+          have hineq := Nat.lt_of_sub_eq_succ hk
+          simp [*, Nat.add_one_le_iff]
+        have heq1: n - (m + 1) = k := by
+          -- TODO: very annoying arithmetic proof
+          simp [Nat.sub_eq_iff_eq_add hineq]
+          have hineq1: m ≤ n := by linarith
+          simp [Nat.sub_eq_iff_eq_add hineq1] at hk
+          simp_arith [hk]
+        have hi := hi (m + 1) heq1 hineq
+        apply hi <;> simp_all
+        . unfold for_all_fin_aux at hf
+          simp_all
+        . simp_all [Arith.add_one_le_iff_le_ne]
+
+  -- TODO: this is not necessary anymore
+  theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
+    for_all_fin f → ∀ i, f i
+    := by
+    intro Hf i
+    apply for_all_fin_aux_imp_forall <;> try assumption
+    simp
+
+end Lemmas
+
 namespace Fix
 
   open Primitives
@@ -436,6 +509,10 @@ namespace FixI
   /- Indexed fixed-point: definitions with indexed types, convenient to use for mutually
      recursive definitions. We simply port the definitions and proofs from Fix to a more
      specific case.
+
+     Remark: the index designates the function in the mutually recursive group
+     (it should be a finite type). We make the return type depend on the input
+     type because we group the type parameters in the input type.
    -/
   open Primitives Fix
 
@@ -505,7 +582,6 @@ namespace FixI
     kk_ty id a b → kk_ty id a b
 
   abbrev in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
-  -- TODO: remove?
   abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
     in_out_ty :=
     Sigma.mk in_ty out_ty
@@ -538,73 +614,6 @@ namespace FixI
         let j: Fin tys1.length := ⟨ j, jLt ⟩
         Eq.mp (by simp) (get_fun tl j)
 
-  def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
-    if heq: m = n then True
-    else
-      f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
-      for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
-  termination_by for_all_fin_aux n _ m h => n - m
-  decreasing_by
-    simp_wf
-    apply Nat.sub_add_lt_sub <;> try simp
-    simp_all [Arith.add_one_le_iff_le_ne]
-
-  def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
-
-  theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
-    (h : m ≤ n) →
-    for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
-    := by
-    generalize h: (n - m) = k
-    revert m
-    induction k -- TODO: induction h rather?
-    case zero =>
-      simp_all
-      intro m h1 h2
-      have h: n = m := by
-        linarith
-      unfold for_all_fin_aux; simp_all
-      simp_all
-      -- There is no i s.t. m ≤ i
-      intro i h3; cases i; simp_all
-      linarith
-    case succ k hi =>
-      intro m hk hmn
-      intro hf i hmi
-      have hne: m ≠ n := by
-        have hineq := Nat.lt_of_sub_eq_succ hk
-        linarith
-      -- m = i?
-      if heq: m = i then
-        -- Yes: simply use the `for_all_fin_aux` hyp
-        unfold for_all_fin_aux at hf
-        simp_all
-      else
-        -- No: use the induction hypothesis
-        have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
-        have hineq: m + 1 ≤ n := by
-          have hineq := Nat.lt_of_sub_eq_succ hk
-          simp [*, Nat.add_one_le_iff]
-        have heq1: n - (m + 1) = k := by
-          -- TODO: very annoying arithmetic proof
-          simp [Nat.sub_eq_iff_eq_add hineq]
-          have hineq1: m ≤ n := by linarith
-          simp [Nat.sub_eq_iff_eq_add hineq1] at hk
-          simp_arith [hk]
-        have hi := hi (m + 1) heq1 hineq
-        apply hi <;> simp_all
-        . unfold for_all_fin_aux at hf
-          simp_all
-        . simp_all [Arith.add_one_le_iff_le_ne]
-
-  -- TODO: this is not necessary anymore
-  theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
-    for_all_fin f → ∀ i, f i
-    := by
-    intro Hf i
-    apply for_all_fin_aux_imp_forall <;> try assumption
-    simp
-
   /- Automating the proofs -/
   @[simp] theorem is_valid_p_same
     (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (x : Result c) :
@@ -707,6 +716,218 @@ namespace FixI
 
 end FixI
 
+namespace FixII
+  /- Similar to FixI, but we split the input arguments between the type parameters
+     and the input values.
+   -/
+  open Primitives Fix
+
+  -- The index type
+  variable {id : Type u}
+
+  -- The input/output types
+  variable {ty : id → Type v} {a : (i:id) → ty i → Type w} {b : (i:id) → ty i → Type x}
+
+  -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+  def karrow_rel (k1 k2 : (i:id) → (t:ty i) → (a i t) → Result (b i t)) : Prop :=
+    ∀ i t x, result_rel (k1 i t x) (k2 i t x)
+
+  def kk_to_gen (k : (i:id) → (t:ty i) → (x:a i t) → Result (b i t)) :
+    (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst) :=
+    λ ⟨ i, t, x ⟩ => k i t x
+
+  def kk_of_gen (k : (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) :
+    (i:id) → (t:ty i) → a i t → Result (b i t) :=
+    λ i t x => k ⟨ i, t, x ⟩
+
+  def k_to_gen (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) :
+    ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst) :=
+    λ kk => kk_to_gen (k (kk_of_gen kk))
+
+  def k_of_gen (k : ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) :
+    ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t) :=
+    λ kk => kk_of_gen (k (kk_to_gen kk))
+
+  def e_to_gen (e : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c) :
+    ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → Result c :=
+    λ k => e (kk_of_gen k)
+
+  def is_valid_p (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+    (e : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c) : Prop :=
+    Fix.is_valid_p (k_to_gen k) (e_to_gen e)
+
+  def is_valid (f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) : Prop :=
+    ∀ k i t x, is_valid_p k (λ k => f k i t x)
+
+  def fix
+    (f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) :
+    (i:id) → (t:ty i) → a i t → Result (b i t) :=
+    kk_of_gen (Fix.fix (k_to_gen f))
+
+  theorem is_valid_fix_fixed_eq
+    {{f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)}}
+    (Hvalid : is_valid f) :
+    fix f = f (fix f) := by
+    have Hvalid' : Fix.is_valid (k_to_gen f) := by
+      intro k x
+      simp only [is_valid, is_valid_p] at Hvalid
+      let ⟨ i, t, x ⟩ := x
+      have Hvalid := Hvalid (k_of_gen k) i t x
+      simp only [k_to_gen, k_of_gen, kk_to_gen, kk_of_gen] at Hvalid
+      refine Hvalid
+    have Heq := Fix.is_valid_fix_fixed_eq Hvalid'
+    simp [fix]
+    conv => lhs; rw [Heq]
+
+  /- Some utilities to define the mutually recursive functions -/
+
+  -- TODO: use more
+  abbrev kk_ty (id : Type u) (ty : id → Type v) (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :=
+    (i:id) → (t:ty i) → a i t → Result (b i t)
+  abbrev k_ty (id : Type u) (ty : id → Type v) (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :=
+    kk_ty id ty a b → kk_ty id ty a b
+
+  abbrev in_out_ty : Type (imax (u + 1) (imax (v + 1) (w + 1))) :=
+    (ty : Type u) × (ty → Type v) × (ty → Type w)
+  abbrev mk_in_out_ty (ty : Type u) (in_ty : ty → Type v) (out_ty : ty → Type w) :
+    in_out_ty :=
+    Sigma.mk ty (Prod.mk in_ty out_ty)
+
+  -- Initially, we had left out the parameters id, a and b.
+  -- However, by parameterizing Funs with those parameters, we can state
+  -- and prove lemmas like Funs.is_valid_p_is_valid_p
+  inductive Funs (id : Type u) (ty : id → Type v)
+    (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :
+    List in_out_ty.{v, w, x} → Type (max (u + 1) (max (v + 1) (max (w + 1) (x + 1)))) :=
+    | Nil : Funs id ty a b []
+    | Cons {it: Type v} {ity : it → Type w} {oty : it → Type x} {tys : List in_out_ty}
+           (f : kk_ty id ty a b → (i:it) → (x:ity i) → Result (oty i)) (tl : Funs id ty a b tys) :
+           Funs id ty a b (⟨ it, ity, oty ⟩ :: tys)
+
+  def get_fun {tys : List in_out_ty} (fl : Funs id ty a b tys) :
+    (i : Fin tys.length) → kk_ty id ty a b → (t : (tys.get i).fst) →
+    ((tys.get i).snd.fst t) → Result ((tys.get i).snd.snd t) :=
+    match fl with
+    | .Nil => λ i => by have h:= i.isLt; simp at h
+    | @Funs.Cons id ty a b it ity oty tys1 f tl =>
+      λ ⟨ i, iLt ⟩ =>
+      match i with
+      | 0 =>
+        Eq.mp (by simp [List.get]) f
+      | .succ j =>
+        have jLt: j < tys1.length := by
+          simp at iLt
+          revert iLt
+          simp_arith
+        let j: Fin tys1.length := ⟨ j, jLt ⟩
+        Eq.mp (by simp) (get_fun tl j)
+
+  /- Automating the proofs -/
+  @[simp] theorem is_valid_p_same
+    (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) (x : Result c) :
+    is_valid_p k (λ _ => x) := by
+    simp [is_valid_p, k_to_gen, e_to_gen]
+
+  @[simp] theorem is_valid_p_rec
+     (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) (i : id) (t : ty i) (x : a i t) :
+    is_valid_p k (λ k => k i t x) := by
+    simp [is_valid_p, k_to_gen, e_to_gen, kk_to_gen, kk_of_gen]
+
+  theorem is_valid_p_ite
+    (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+    (cond : Prop) [h : Decidable cond]
+    {e1 e2 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c}
+    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+    is_valid_p k (λ k => ite cond (e1 k) (e2 k)) := by
+    split <;> assumption
+
+  theorem is_valid_p_dite
+    (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+    (cond : Prop) [h : Decidable cond]
+    {e1 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → cond → Result c}
+    {e2 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Not cond → Result c}
+    (he1: ∀ x, is_valid_p k (λ k => e1 k x))
+    (he2 : ∀ x, is_valid_p k (λ k => e2 k x)) :
+    is_valid_p k (λ k => dite cond (e1 k) (e2 k)) := by
+    split <;> simp [*]
+
+  theorem is_valid_p_bind
+    {{k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)}}
+    {{g : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c}}
+    {{h : c → ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result d}}
+    (Hgvalid : is_valid_p k g)
+    (Hhvalid : ∀ y, is_valid_p k (h y)) :
+    is_valid_p k (λ k => do let y ← g k; h y k) := by
+    apply Fix.is_valid_p_bind
+    . apply Hgvalid
+    . apply Hhvalid
+
+  def Funs.is_valid_p
+    (k : k_ty id ty a b)
+    (fl : Funs id ty a b tys) :
+    Prop :=
+    match fl with
+    | .Nil => True
+    | .Cons f fl =>
+      (∀ i x, FixII.is_valid_p k (λ k => f k i x)) ∧ fl.is_valid_p k
+
+  theorem Funs.is_valid_p_Nil (k : k_ty id ty a b) :
+    Funs.is_valid_p k Funs.Nil := by simp [Funs.is_valid_p]
+
+  def Funs.is_valid_p_is_valid_p_aux
+    {k : k_ty id ty a b}
+    {tys : List in_out_ty}
+    (fl : Funs id ty a b tys) (Hvalid : is_valid_p k fl) :
+    ∀ (i : Fin tys.length) (t : (tys.get i).fst) (x : (tys.get i).snd.fst t),
+    FixII.is_valid_p k (fun k => get_fun fl i k t x) := by
+    -- Prepare the induction
+    have ⟨ n, Hn ⟩ : { n : Nat // tys.length = n } := ⟨ tys.length, by rfl ⟩
+    revert tys fl Hvalid
+    induction n
+    --
+    case zero =>
+      intro tys fl Hvalid Hlen;
+      have Heq: tys = [] := by cases tys <;> simp_all
+      intro i x
+      simp_all
+      have Hi := i.isLt
+      simp_all
+    case succ n Hn =>
+      intro tys fl Hvalid Hlen i x;
+      cases fl <;> simp at Hlen i x Hvalid
+      rename_i ity oty tys f fl
+      have ⟨ Hvf, Hvalid ⟩ := Hvalid
+      have Hvf1: is_valid_p k fl := by
+        simp [Hvalid, Funs.is_valid_p]
+      have Hn := @Hn tys fl Hvf1 (by simp [*])
+      -- Case disjunction on i
+      match i with
+      | ⟨ 0, _ ⟩ =>
+        simp at x
+        simp [get_fun]
+        apply (Hvf x)
+      | ⟨ .succ j, HiLt ⟩ =>
+        simp_arith at HiLt
+        simp at x
+        let j : Fin (List.length tys) := ⟨ j, by simp_arith [HiLt] ⟩
+        have Hn := Hn j x
+        apply Hn
+
+  def Funs.is_valid_p_is_valid_p
+    (tys : List in_out_ty)
+    (k : k_ty (Fin (List.length tys)) (λ i => (tys.get i).fst)
+              (fun i t => (List.get tys i).snd.fst t) (fun i t => (List.get tys i).snd.snd t))
+    (fl : Funs (Fin tys.length) (λ i => (tys.get i).fst)
+               (λ i t => (tys.get i).snd.fst t) (λ i t => (tys.get i).snd.snd t) tys) :
+    fl.is_valid_p k  →
+    ∀ (i : Fin tys.length) (t : (tys.get i).fst) (x : (tys.get i).snd.fst t),
+      FixII.is_valid_p k (fun k => get_fun fl i k t x)
+    := by
+    intro Hvalid
+    apply is_valid_p_is_valid_p_aux; simp [*]
+
+end FixII
+
 namespace Ex1
   /- An example of use of the fixed-point -/
   open Primitives Fix
@@ -1133,3 +1354,218 @@ namespace Ex6
     Heqix
 
 end Ex6
+
+namespace Ex7
+  /- `list_nth` again, but this time we use FixII -/
+  open Primitives FixII
+
+  @[simp] def tys.{u} : List in_out_ty :=
+    [mk_in_out_ty (Type u) (λ a => List a × Int) (λ a => a)]
+
+  @[simp] def ty (i : Fin 1) := (tys.get i).fst
+  @[simp] def input_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.fst t
+  @[simp] def output_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.snd t
+
+  def list_nth_body.{u} (k : (i:Fin 1) → (t:ty i) →  input_ty i t → Result (output_ty i t))
+    (a : Type u) (x : List a × Int) : Result a :=
+    let ⟨ ls, i ⟩ := x
+    match ls with
+    | [] => .fail .panic
+    | hd :: tl =>
+      if i = 0 then .ret hd
+      else k 0 a ⟨ tl, i - 1 ⟩
+
+  @[simp] def bodies :
+    Funs (Fin 1) ty input_ty output_ty tys :=
+    Funs.Cons list_nth_body Funs.Nil
+
+  def body (k : (i : Fin 1) → (t : ty i) → (x : input_ty i t) → Result (output_ty i t)) (i: Fin 1) :
+    (t : ty i) → (x : input_ty i t) → Result (output_ty i t) := get_fun bodies i k
+
+  theorem body_is_valid: is_valid body := by
+    -- Split the proof into proofs of validity of the individual bodies
+    rw [is_valid]
+    simp only [body]
+    intro k
+    apply (Funs.is_valid_p_is_valid_p tys)
+    simp [Funs.is_valid_p]
+    (repeat (apply And.intro)); intro x; try simp at x
+    simp only [list_nth_body]
+    -- Prove the validity of the individual bodies
+    intro k x
+    split <;> try simp
+    split <;> simp
+
+  -- Writing the proof terms explicitly
+  theorem list_nth_body_is_valid' (k : k_ty (Fin 1) ty input_ty output_ty)
+    (a : Type u) (x : List a × Int) : is_valid_p k (fun k => list_nth_body k a x) :=
+    let ⟨ ls, i ⟩ := x
+    match ls with
+    | [] => is_valid_p_same k (.fail .panic)
+    | hd :: tl =>
+      is_valid_p_ite k (Eq i 0) (is_valid_p_same k (.ret hd)) (is_valid_p_rec k 0 a ⟨tl, i-1⟩)
+
+  theorem body_is_valid' : is_valid body :=
+    fun k =>
+    Funs.is_valid_p_is_valid_p tys k bodies
+      (And.intro (list_nth_body_is_valid' k) (Funs.is_valid_p_Nil k))
+
+  def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
+    fix body 0 a ⟨ ls , i ⟩
+
+  -- The unfolding equation - diverges if `i < 0`
+  theorem list_nth_eq (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    := by
+    have Heq := is_valid_fix_fixed_eq body_is_valid
+    simp [list_nth]
+    conv => lhs; rw [Heq]
+
+  -- Write the proof term explicitly: the generation of the proof term (without tactics)
+  -- is automatable, and the proof term is actually a lot simpler and smaller when we
+  -- don't use tactics.
+  theorem list_nth_eq'.{u} {a : Type u} (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    :=
+    -- Use the fixed-point equation
+    have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+    -- Add the index
+    have Heqi := congr_fun Heq 0
+    -- Add the type parameter
+    have Heqia := congr_fun Heqi a
+    -- Add the input
+    have Heqix := congr_fun Heqia (ls, i)
+    -- Done
+    Heqix
+
+end Ex7
+
+namespace Ex8
+  /- Higher-order example, with FixII -/
+  open Primitives FixII
+
+  variable {id : Type u} {ty : id → Type v}
+  variable {a : (i:id) → ty i → Type w} {b : (i:id) → ty i → Type x}
+
+  /- An auxiliary function, which doesn't require the fixed-point -/
+  def map {a : Type y} {b : Type z} (f : a → Result b) (ls : List a) : Result (List b) :=
+    match ls with
+    | [] => .ret []
+    | hd :: tl =>
+      do
+        let hd ← f hd
+        let tl ← map f tl
+        .ret (hd :: tl)
+
+  /- The validity theorems for `map`, generic in `f` -/
+
+  -- This is not the most general lemma, but we keep it to test the `divergence` encoding on a simple case
+  @[divspec]
+  theorem map_is_valid_simple
+    (i : id) (t : ty i)
+    (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+    (ls : List (a i t)) :
+    is_valid_p k (λ k => map (k i t) ls) := by
+    induction ls <;> simp [map]
+    apply is_valid_p_bind <;> try simp_all
+    intros
+    apply is_valid_p_bind <;> try simp_all
+
+  @[divspec]
+  theorem map_is_valid
+    (d : Type y)
+    {{f : ((i:id) → (t : ty i) → a i t → Result (b i t)) → d → Result c}}
+    (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+    (Hfvalid : ∀ x1, is_valid_p k (fun kk1 => f kk1 x1))
+    (ls : List d) :
+    is_valid_p k (λ k => map (f k) ls) := by
+    induction ls <;> simp [map]
+    apply is_valid_p_bind <;> try simp_all
+    intros
+    apply is_valid_p_bind <;> try simp_all
+
+end Ex8
+
+namespace Ex9
+  /- An example which uses map -/
+  open Primitives FixII Ex8
+
+  inductive Tree (a : Type u) :=
+  | leaf (x : a)
+  | node (tl : List (Tree a))
+
+  @[simp] def tys.{u} : List in_out_ty :=
+    [mk_in_out_ty (Type u) (λ a => Tree a) (λ a => Tree a)]
+
+  @[simp] def ty (i : Fin 1) := (tys.get i).fst
+  @[simp] def input_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.fst t
+  @[simp] def output_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.snd t
+
+  def id_body.{u} (k : (i:Fin 1) → (t:ty i) →  input_ty i t → Result (output_ty i t))
+    (a : Type u) (t : Tree a) : Result (Tree a) :=
+    match t with
+    | .leaf x => .ret (.leaf x)
+    | .node tl =>
+      do
+        let tl ← map (k 0 a) tl
+        .ret (.node tl)
+
+  @[simp] def bodies :
+    Funs (Fin 1) ty input_ty output_ty tys :=
+    Funs.Cons id_body Funs.Nil
+
+  theorem id_body_is_valid :
+    ∀ (k : ((i : Fin 1) → (t : ty i) → input_ty i t → Result (output_ty i t)) → (i : Fin 1) → (t : ty i) → input_ty i t → Result (output_ty i t))
+    (a : Type u) (x : Tree a),
+    @is_valid_p (Fin 1) ty input_ty output_ty (output_ty 0 a) k (λ k => id_body k a x) := by
+    intro k a x
+    simp only [id_body]
+    split <;> try simp
+    apply is_valid_p_bind <;> try simp [*]
+    -- We have to show that `map k tl` is valid
+    -- Remark: `map_is_valid` doesn't work here, we need the specialized version
+    apply map_is_valid_simple
+
+  def body (k : (i : Fin 1) → (t : ty i) → (x : input_ty i t) → Result (output_ty i t)) (i: Fin 1) :
+    (t : ty i) → (x : input_ty i t) → Result (output_ty i t) := get_fun bodies i k
+
+  theorem body_is_valid : is_valid body :=
+    fun k =>
+    Funs.is_valid_p_is_valid_p tys k bodies
+      (And.intro (id_body_is_valid k) (Funs.is_valid_p_Nil k))
+
+  def id {a: Type u} (t : Tree a) : Result (Tree a) :=
+    fix body 0 a t
+
+  -- Writing the proof term explicitly
+  theorem id_eq' {a : Type u} (t : Tree a) :
+    id t =
+      (match t with
+       | .leaf x => .ret (.leaf x)
+       | .node tl =>
+         do
+           let tl ← map id tl
+           .ret (.node tl))
+    :=
+    -- The unfolding equation
+    have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+    -- Add the index
+    have Heqi := congr_fun Heq 0
+    -- Add the type parameter
+    have Heqia := congr_fun Heqi a
+    -- Add the input
+    have Heqix := congr_fun Heqia t
+    -- Done
+    Heqix
+
+end Ex9
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index c6628486..6115b13b 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -17,15 +17,24 @@ syntax (name := divergentDef)
 open Lean Elab Term Meta Primitives Lean.Meta
 open Utils
 
+def normalize_let_bindings := true
+
 /- The following was copied from the `wfRecursion` function. -/
 
 open WF in
 
+-- TODO: use those
+def UnitType := Expr.const ``PUnit [Level.succ .zero]
+def UnitValue := Expr.const ``PUnit.unit [Level.succ .zero]
+
+def mkProdType (x y : Expr) : MetaM Expr :=
+  mkAppM ``Prod #[x, y]
+
 def mkProd (x y : Expr) : MetaM Expr :=
   mkAppM ``Prod.mk #[x, y]
 
-def mkInOutTy (x y : Expr) : MetaM Expr :=
-  mkAppM ``FixI.mk_in_out_ty #[x, y]
+def mkInOutTy (x y z : Expr) : MetaM Expr := do
+  mkAppM ``FixII.mk_in_out_ty #[x, y, z]
 
 -- Return the `a` in `Return a`
 def getResultTy (ty : Expr) : MetaM Expr :=
@@ -47,6 +56,17 @@ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
   else
     pure (args.get! 0, args.get! 1)
 
+/- Make a sigma type.
+
+   `x` should be a variable, and `ty` and type which (might) uses `x`
+ -/
+def mkSigmaType (x : Expr) (sty : Expr) : MetaM Expr := do
+  trace[Diverge.def.sigmas] "mkSigmaType: {x} {sty}"
+  let alpha ← inferType x
+  let beta ← mkLambdaFVars #[x] sty
+  trace[Diverge.def.sigmas] "mkSigmaType: ({alpha}) ({beta})"
+  mkAppOptM ``Sigma #[some alpha, some beta]
+
 /- Generate a Sigma type from a list of *variables* (all the expressions
    must be variables).
 
@@ -60,20 +80,78 @@ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
 def mkSigmasType (xl : List Expr) : MetaM Expr :=
   match xl with
   | [] => do
-    trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
-    pure (Expr.const ``PUnit.unit [])
+    trace[Diverge.def.sigmas] "mkSigmasType: []"
+    pure (Expr.const ``PUnit [Level.succ .zero])
   | [x] => do
-    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
-    let ty ← Lean.Meta.inferType x
+    trace[Diverge.def.sigmas] "mkSigmasType: [{x}]"
+    let ty ← inferType x
     pure ty
   | x :: xl => do
-    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
-    let alpha ← Lean.Meta.inferType x
+    trace[Diverge.def.sigmas] "mkSigmasType: [{x}::{xl}]"
     let sty ← mkSigmasType xl
-    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]: alpha={alpha}, sty={sty}"
-    let beta ← mkLambdaFVars #[x] sty
-    trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
-    mkAppOptM ``Sigma #[some alpha, some beta]
+    mkSigmaType x sty
+
+/- Generate a product type from a list of *variables* (this is similar to `mkSigmas`).
+
+   Example:
+   - xl = [(ls:List a), (i:Int)]
+
+   Generates:
+   `List a × Int`
+ -/
+def mkProdsType (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.prods] "mkProdsType: []"
+    pure (Expr.const ``PUnit [Level.succ .zero])
+  | [x] => do
+    trace[Diverge.def.prods] "mkProdsType: [{x}]"
+    let ty ← inferType x
+    pure ty
+  | x :: xl => do
+    trace[Diverge.def.prods] "mkProdsType: [{x}::{xl}]"
+    let ty ← inferType x
+    let xl_ty ← mkProdsType xl
+    mkAppM ``Prod #[ty, xl_ty]
+
+/- Split the input arguments between the types and the "regular" arguments.
+
+   We do something simple: we treat an input argument as an
+   input type iff it appears in the type of the following arguments.
+
+   Note that what really matters is that we find the arguments which appear
+   in the output type.
+
+   Also, we stop at the first input that we treat as an
+   input type.
+ -/
+def splitInputArgs (in_tys : Array Expr) (out_ty : Expr) : MetaM (Array Expr × Array Expr) := do
+  -- Look for the first parameter which appears in the subsequent parameters
+  let rec splitAux (in_tys : List Expr) : MetaM (HashSet FVarId × List Expr × List Expr) :=
+    match in_tys with
+    | [] => do
+      let fvars ← getFVarIds (← inferType out_ty)
+      pure (fvars, [], [])
+    | ty :: in_tys => do
+      let (fvars, in_tys, in_args) ← splitAux in_tys
+      -- Have we already found where to split between type variables/regular
+      -- variables?
+      if ¬ in_tys.isEmpty then
+        -- The fvars set is now useless: no need to update it anymore
+        pure (fvars, ty :: in_tys, in_args)
+      else
+        -- Check if ty appears in the set of free variables:
+        let ty_id := ty.fvarId!
+        if fvars.contains ty_id then
+          -- We must split here. Note that we don't need to update the fvars
+          -- set: it is not useful anymore
+          pure (fvars, [ty], in_args)
+        else
+          -- We must split later: update the fvars set
+          let fvars := fvars.insertMany (← getFVarIds (← inferType ty))
+          pure (fvars, [], ty :: in_args)
+  let (_, in_tys, in_args) ← splitAux in_tys.data
+  pure (Array.mk in_tys, Array.mk in_args)
 
 /- Apply a lambda expression to some arguments, simplifying the lambdas -/
 def applyLambdaToArgs (e : Expr) (xs : Array Expr) : MetaM Expr := do
@@ -105,7 +183,7 @@ def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
   match xl with
   | [] => do
     trace[Diverge.def.sigmas] "mkSigmasVal: []"
-    pure (Expr.const ``PUnit.unit [])
+    pure (Expr.const ``PUnit.unit [Level.succ .zero])
   | [x] => do
     trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
     pure x
@@ -122,6 +200,17 @@ def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
     trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
     mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
 
+/- Group a list of expressions into a (non-dependent) tuple -/
+def mkProdsVal (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] =>
+    pure (Expr.const ``PUnit.unit [Level.succ .zero])
+  | [x] => do
+    pure x
+  | x :: xl => do
+    let xl ← mkProdsVal xl
+    mkAppM ``Prod.mk #[x, xl]
+
 def mkAnonymous (s : String) (i : Nat) : Name :=
   .num (.str .anonymous s) i
 
@@ -159,31 +248,31 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
   match xl with
   | [] => do
     -- This would be unexpected
-    throwError "mkSigmasMatch: empyt list of input parameters"
+    throwError "mkSigmasMatch: empty list of input parameters"
   | [x] => do
     -- In the example given for the explanations: this is the inner match case
     trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]"
     mkLambdaFVars #[x] out
   | fst :: xl => do
-    -- In the example given for the explanations: this is the outer match case
-    -- Remark: for the naming purposes, we use the same convention as for the
-    -- fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
-    -- those definitions might help)
-    --
-    -- We want to build the match expression:
-    -- ```
-    -- λ scrut =>
-    -- match scrut with
-    -- | Sigma.mk x ...  -- the hole is given by a recursive call on the tail
-    -- ```
+    /- In the example given for the explanations: this is the outer match case
+       Remark: for the naming purposes, we use the same convention as for the
+       fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
+       those definitions might help)
+
+       We want to build the match expression:
+       ```
+       λ scrut =>
+       match scrut with
+       | Sigma.mk x ...  -- the hole is given by a recursive call on the tail
+       ``` -/
     trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]"
-    let alpha ← Lean.Meta.inferType fst
+    let alpha ← inferType fst
     let snd_ty ← mkSigmasType xl
     let beta ← mkLambdaFVars #[fst] snd_ty
     let snd ← mkSigmasMatch xl out (index + 1)
     let mk ← mkLambdaFVars #[fst] snd
     -- Introduce the "scrut" variable
-    let scrut_ty ← mkSigmasType (fst :: xl)
+    let scrut_ty ← mkSigmaType fst snd_ty
     withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
     trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})"
     -- TODO: make the computation of the motive more efficient
@@ -206,6 +295,67 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
     trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}"
     pure sm
 
+/- This is similar to `mkSigmasMatch`, but with non-dependent tuples
+
+   Remark: factor out with `mkSigmasMatch`? This is extremely similar.
+-/
+partial def mkProdsMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr :=
+  match xl with
+  | [] => do
+    -- This would be unexpected
+    throwError "mkProdsMatch: empty list of input parameters"
+  | [x] => do
+    -- In the example given for the explanations: this is the inner match case
+    trace[Diverge.def.prods] "mkProdsMatch: [{x}]"
+    mkLambdaFVars #[x] out
+  | fst :: xl => do
+    trace[Diverge.def.prods] "mkProdsMatch: [{fst}::{xl}]"
+    let alpha ← inferType fst
+    let beta ← mkProdsType xl
+    let snd ← mkProdsMatch xl out (index + 1)
+    let mk ← mkLambdaFVars #[fst] snd
+    -- Introduce the "scrut" variable
+    let scrut_ty ← mkProdType alpha beta
+    withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
+    trace[Diverge.def.prods] "mkProdsMatch: scrut: ({scrut}) : ({← inferType scrut})"
+    -- TODO: make the computation of the motive more efficient
+    let motive ← do
+      let out_ty ← inferType out
+      mkLambdaFVars #[scrut] out_ty
+    -- The final expression: putting everything together
+    trace[Diverge.def.prods] "mkProdsMatch:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
+    let sm ← mkAppOptM ``Prod.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+    -- Abstracting the "scrut" variable
+    let sm ← mkLambdaFVars #[scrut] sm
+    trace[Diverge.def.prods] "mkProdsMatch: sm: {sm}"
+    pure sm
+
+/- Same as `mkSigmasMatch` but also accepts an empty list of inputs, in which case
+   it generates the expression:
+   ```
+   λ () => e
+   ``` -/
+def mkSigmasMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr :=
+  if xl.isEmpty then do
+    let scrut_ty := Expr.const ``PUnit [Level.succ .zero]
+    withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do
+    mkLambdaFVars #[scrut] out
+  else
+    mkSigmasMatch xl out
+
+/- Same as `mkProdsMatch` but also accepts an empty list of inputs, in which case
+   it generates the expression:
+   ```
+   λ () => e
+   ``` -/
+def mkProdsMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr :=
+  if xl.isEmpty then do
+    let scrut_ty := Expr.const ``PUnit [Level.succ .zero]
+    withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do
+    mkLambdaFVars #[scrut] out
+  else
+    mkProdsMatch xl out
+
 /- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/
 private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
   @Sigma.casesOn (List a)
@@ -238,6 +388,52 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
   let ofNat ← mkAppOptM ``Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat #[n_lit, i_lit]
   mkAppOptM ``OfNat.ofNat #[none, none, ofNat]
 
+/- Information about the type of a function in a declaration group.
+
+   In the comments about the fields, we take as example the
+   `list_nth (α : Type) (ls : List α) (i : Int) : Result α` function.
+ -/
+structure TypeInfo where
+  /- The total number of input arguments.
+
+     For list_nth: 3
+   -/
+  total_num_args : ℕ
+  /- The number of type parameters (they should be a prefix of the input arguments).
+
+     For `list_nth`: 1
+   -/
+  num_params : ℕ
+  /- The type of the dependent tuple grouping the type parameters.
+
+     For `list_nth`: `Type`
+   -/
+  params_ty : Expr
+  /- The type of the tuple grouping the input values. This is a function taking
+     as input a value of type `params_ty`.
+
+     For `list_nth`: `λ a => List a × Int`
+   -/
+  in_ty : Expr
+  /- The output type, without the `Return`. This is a function taking
+     as input a value of type `params_ty`.
+
+     For `list_nth`: `λ a => a`
+   -/
+  out_ty : Expr
+
+def mkInOutTyFromTypeInfo (info : TypeInfo) : MetaM Expr := do
+  mkInOutTy info.params_ty info.in_ty info.out_ty
+
+instance : Inhabited TypeInfo :=
+  { default := { total_num_args := 0, num_params := 0, params_ty := UnitType,
+                 in_ty := UnitType, out_ty := UnitType } }
+
+instance : ToMessageData TypeInfo :=
+ ⟨ λ ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩ =>
+  f!"\{ total_num_args: {total_num_args}, num_params: {num_params}, params_ty: {params_ty}, in_ty: {in_ty}, out_ty: {out_ty} }}"
+ ⟩
+
 /- Generate and declare as individual definitions the bodies for the individual funcions:
    - replace the recursive calls with calls to the continutation `k`
    - make those bodies take one single dependent tuple as input
@@ -246,15 +442,17 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
    We return the new declarations.
  -/
 def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
-  (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+  (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) :
   MetaM (Array Expr) := do
   let grSize := preDefs.size
 
-  -- Compute the map from name to (index × input type).
-  -- Remark: the continuation has an indexed type; we use the index (a finite number of
-  -- type `Fin`) to control which function we call at the recursive call site.
-  let nameToInfo : HashMap Name (Nat × Expr) :=
-    let bl := preDefs.mapIdx fun i d => (d.declName, (i.val, (inOutTys.get! i.val).fst))
+  /- Compute the map from name to (index, type info).
+
+     Remark: the continuation has an indexed type; we use the index (a finite number of
+     type `Fin`) to control which function we call at the recursive call site. -/
+  let nameToInfo : HashMap Name (Nat × TypeInfo) :=
+    let bl := preDefs.mapIdx fun i d =>
+      (d.declName, (i.val, paramInOutTys.get! i.val))
     HashMap.ofList bl.toList
 
   trace[Diverge.def.genBody] "nameToId: {nameToInfo.toList}"
@@ -262,35 +460,65 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
   -- Auxiliary function to explore the function bodies and replace the
   -- recursive calls
   let visit_e (i : Nat) (e : Expr) : MetaM Expr := do
-    trace[Diverge.def.genBody] "visiting expression (dept: {i}): {e}"
+    trace[Diverge.def.genBody.visit] "visiting expression (dept: {i}): {e}"
     let ne ← do
       match e with
       | .app .. => do
         e.withApp fun f args => do
-          trace[Diverge.def.genBody] "this is an app: {f} {args}"
+          trace[Diverge.def.genBody.visit] "this is an app: {f} {args}"
           -- Check if this is a recursive call
           if f.isConst then
             let name := f.constName!
             match nameToInfo.find? name with
             | none => pure e
-            | some (id, in_ty) =>
-              trace[Diverge.def.genBody] "this is a recursive call"
+            | some (id, type_info) =>
+              trace[Diverge.def.genBody.visit] "this is a recursive call"
               -- This is a recursive call: replace it
               -- Compute the index
               let i ← mkFinVal grSize id
-              -- Put the arguments in one big dependent tuple
-              let args ← mkSigmasVal in_ty args.toList
-              mkAppM' kk_var #[i, args]
+              -- It can happen that there are no input values given to the
+              -- recursive calls, and only type parameters.
+              let num_args := args.size
+              if num_args ≠ type_info.total_num_args ∧ num_args ≠ type_info.num_params then
+                throwError "Invalid number of arguments for the recursive call: {e}"
+              -- Split the arguments, and put them in two tuples (the first
+              -- one is a dependent tuple)
+              let (param_args, args) := args.toList.splitAt type_info.num_params
+              trace[Diverge.def.genBody.visit] "param_args: {param_args}, args: {args}"
+              let param_args ← mkSigmasVal type_info.params_ty param_args
+              -- Check if there are input values
+              if num_args = type_info.total_num_args then do
+                trace[Diverge.def.genBody.visit] "Recursive call with input values"
+                let args ← mkProdsVal args
+                mkAppM' kk_var #[i, param_args, args]
+              else do
+                trace[Diverge.def.genBody.visit] "Recursive call without input values"
+                mkAppM' kk_var #[i, param_args]
           else
             -- Not a recursive call: do nothing
             pure e
        | .const name _ =>
-         -- Sanity check: we eliminated all the recursive calls
-         if (nameToInfo.find? name).isSome then
-           throwError "mkUnaryBodies: a recursive call was not eliminated"
+         /- This might refer to the one of the top-level functions if we use
+            it without arguments (if we give it to a higher-order
+            function for instance) and there are actually no type parameters.
+          -/
+         if (nameToInfo.find? name).isSome then do
+           -- Checking the type information
+           match nameToInfo.find? name with
+           | none => pure e
+           | some (id, type_info) =>
+             trace[Diverge.def.genBody.visit] "this is a recursive call"
+             -- This is a recursive call: replace it
+             -- Compute the index
+             let i ← mkFinVal grSize id
+             -- Check that there are no type parameters
+             if type_info.num_params ≠ 0 then throwError "mkUnaryBodies: a recursive call was not eliminated"
+             -- Introduce the call to the continuation
+             let param_args ← mkSigmasVal type_info.params_ty []
+             mkAppM' kk_var #[i, param_args]
          else pure e
        | _ => pure e
-    trace[Diverge.def.genBody] "done with expression (depth: {i}): {e}"
+    trace[Diverge.def.genBody.visit] "done with expression (depth: {i}): {e}"
     pure ne
 
   -- Explore the bodies
@@ -300,13 +528,20 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
     let body ← mapVisit visit_e preDef.value
     trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}"
 
-    -- Currify the function by grouping the arguments into a dependent tuple
+    -- Currify the function by grouping the arguments into dependent tuples
     -- (over which we match to retrieve the individual arguments).
     lambdaTelescope body fun args body => do
-    let body ← mkSigmasMatch args.toList body 0
+    -- Split the arguments between the type parameters and the "regular" inputs
+    let (_, type_info) := nameToInfo.find! preDef.declName
+    let (param_args, args) := args.toList.splitAt type_info.num_params
+    let body ← mkProdsMatchOrUnit args body
+    trace[Diverge.def.genBody] "Body after mkProdsMatchOrUnit: {body}"
+    let body ← mkSigmasMatchOrUnit param_args body
+    trace[Diverge.def.genBody] "Body after mkSigmasMatchOrUnit: {body}"
 
     -- Add the declaration
     let value ← mkLambdaFVars #[kk_var] body
+    trace[Diverge.def.genBody] "Body after abstracting kk: {value}"
     let name := preDef.declName.append "body"
     let levelParams := grLvlParams
     let decl := Declaration.defnDecl {
@@ -318,41 +553,46 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
       safety := .safe
       all := [name]
     }
+    trace[Diverge.def.genBody] "About to add decl"
     addDecl decl
     trace[Diverge.def] "individual body of {preDef.declName}: {body}"
     -- Return the constant
     let body := Lean.mkConst name (levelParams.map .param)
-    -- let body ← mkAppM' body #[kk_var]
     trace[Diverge.def] "individual body (after decl): {body}"
     pure body
 
--- Generate a unique function body from the bodies of the mutually recursive group,
--- and add it as a declaration in the context.
--- We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
+/- Generate a unique function body from the bodies of the mutually recursive group,
+   and add it as a declaration in the context.
+   We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
+ -/
 def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
   (kk_var i_var : Expr)
-  (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
+  (param_ty in_ty out_ty : Expr) (paramInOutTys : Array TypeInfo)
   (bodies : Array Expr) : MetaM (Expr × Expr) := do
   -- Generate the body
   let grSize := bodies.size
   let finTypeExpr := mkFin grSize
   -- TODO: not very clean
-  let inOutTyType ← do
-    let (x, y) := inOutTys.get! 0
-    inferType (← mkInOutTy x y)
-  let rec mkFuns (inOutTys : List (Expr × Expr)) (bl : List Expr) : MetaM Expr :=
-    match inOutTys, bl with
+  let paramInOutTyType ← do
+    let info := paramInOutTys.get! 0
+    inferType (← mkInOutTyFromTypeInfo info)
+  let rec mkFuns (paramInOutTys : List TypeInfo) (bl : List Expr) : MetaM Expr :=
+    match paramInOutTys, bl with
     | [], [] =>
-      mkAppOptM ``FixI.Funs.Nil #[finTypeExpr, in_ty, out_ty]
-    | (ity, oty) :: inOutTys, b :: bl => do
+      mkAppOptM ``FixII.Funs.Nil #[finTypeExpr, param_ty, in_ty, out_ty]
+    | info :: paramInOutTys, b :: bl => do
+      let pty := info.params_ty
+      let ity := info.in_ty
+      let oty := info.out_ty
       -- Retrieving ity and oty - this is not very clean
-      let inOutTysExpr ← mkListLit inOutTyType (← inOutTys.mapM (λ (x, y) => mkInOutTy x y))
-      let fl ← mkFuns inOutTys bl
-      mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
+      let paramInOutTysExpr ← mkListLit paramInOutTyType
+        (← paramInOutTys.mapM mkInOutTyFromTypeInfo)
+      let fl ← mkFuns paramInOutTys bl
+      mkAppOptM ``FixII.Funs.Cons #[finTypeExpr, param_ty, in_ty, out_ty, pty, ity, oty, paramInOutTysExpr, b, fl]
     | _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length"
-  let bodyFuns ← mkFuns inOutTys bodies.toList
+  let bodyFuns ← mkFuns paramInOutTys.toList bodies.toList
   -- Wrap in `get_fun`
-  let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, kk_var]
+  let body ← mkAppM ``FixII.get_fun #[bodyFuns, i_var, kk_var]
   -- Add the index `i` and the continuation `k` as a variables
   let body ← mkLambdaFVars #[kk_var, i_var] body
   trace[Diverge.def] "mkDeclareMutRecBody: body: {body}"
@@ -391,11 +631,11 @@ instance : ToMessageData MatchInfo where
   -- This is not a very clean formatting, but we don't need more
   toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}"
 
--- Small helper: prove that an expression which doesn't use the continuation `kk`
--- is valid, and return the proof.
+/- Small helper: prove that an expression which doesn't use the continuation `kk`
+   is valid, and return the proof. -/
 def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
   trace[Diverge.def.valid] "proveNoKExprIsValid: {e}"
-  let eIsValid ← mkAppM ``FixI.is_valid_p_same #[k_var, e]
+  let eIsValid ← mkAppM ``FixII.is_valid_p_same #[k_var, e]
   trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}"
   pure eIsValid
 
@@ -410,7 +650,16 @@ mutual
    ```
  -/
 partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
-  trace[Diverge.def.valid] "proveValid: {e}"
+  trace[Diverge.def.valid] "proveExprIsValid: {e}"
+  -- Normalize to eliminate the lambdas - TODO: this is slightly dangerous.
+  let e ← do
+    if e.isLet ∧ normalize_let_bindings then do
+      let updt_config config :=
+        { config with transparency := .reducible, zetaNonDep := false }
+      let e ← withConfig updt_config (whnf e)
+      trace[Diverge.def.valid] "e (after normalization): {e}"
+      pure e
+    else pure e
   match e with
   | .const _ _ => throwError "Unimplemented" -- Shouldn't get there?
   | .bvar _
@@ -418,9 +667,10 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
   | .lit _
   | .mvar _
   | .sort _ => throwError "Unreachable"
-  | .lam .. => throwError "Unimplemented"
+  | .lam .. => throwError "Unimplemented" -- TODO
   | .forallE .. => throwError "Unreachable" -- Shouldn't get there
   | .letE .. => do
+    -- Remark: this branch is not taken if we normalize the expressions (above)
     -- Telescope all the let-bindings (remark: this also telescopes the lambdas)
     lambdaLetTelescope e fun xs body => do
     -- Note that we don't visit the bound values: there shouldn't be
@@ -438,164 +688,268 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
     proveNoKExprIsValid k_var e
   | .app .. =>
     e.withApp fun f args => do
-      -- There are several cases: first, check if this is a match/if
-      -- Check if the expression is a (dependent) if then else.
-      -- We treat the if then else expressions differently from the other matches,
-      -- and have dedicated theorems for them.
-      let isIte := e.isIte
-      if isIte || e.isDIte then do
-        e.withApp fun f args => do
-        trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
-        if args.size ≠ 5 then
-           throwError "Wrong number of parameters for {f}: {args}"
-        let cond := args.get! 1
-        let dec := args.get! 2
-        -- Prove that the branches are valid
-        let br0 := args.get! 3
-        let br1 := args.get! 4
-        let proveBranchValid (br : Expr) : MetaM Expr :=
-          if isIte then proveExprIsValid k_var kk_var br
-          else do
-            -- There is a lambda
-            lambdaOne br fun x br => do
-            let brValid ← proveExprIsValid k_var kk_var br
-            mkLambdaFVars #[x] brValid
-        let br0Valid ← proveBranchValid br0
-        let br1Valid ← proveBranchValid br1
-        let const := if isIte then ``FixI.is_valid_p_ite else ``FixI.is_valid_p_dite
-        let eIsValid ← mkAppOptM const #[none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid]
-        trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
-        pure eIsValid
-      -- Check if the expression is a match (this case is for when the elaborator
-      -- introduces auxiliary definitions to hide the match behind syntactic
-      -- sugar):
-      else if let some me := ← matchMatcherApp? e then do
-        trace[Diverge.def.valid]
-          "matcherApp:
-           - params: {me.params}
-           - motive: {me.motive}
-           - discrs: {me.discrs}
-           - altNumParams: {me.altNumParams}
-           - alts: {me.alts}
-           - remaining: {me.remaining}"
-        -- matchMatcherApp does all the work for us: we simply need to gather
-        -- the information and call the auxiliary helper `proveMatchIsValid`
-        if me.remaining.size ≠ 0 then
-          throwError "MatcherApp: non empty remaining array: {me.remaining}"
-        let me : MatchInfo := {
-          matcherName := me.matcherName
-          matcherLevels := me.matcherLevels
-          params := me.params
-          motive := me.motive
-          scruts := me.discrs
-          branchesNumParams := me.altNumParams
-          branches := me.alts
-        }
-        proveMatchIsValid k_var kk_var me
-      -- Check if the expression is a raw match (this case is for when the expression
-      -- is a direct call to the primitive `casesOn` function, without syntactic sugar).
-      -- We have to check this case because functions like `mkSigmasMatch`, which we
-      -- use to currify function bodies, introduce such raw matches.
-      else if ← isCasesExpr f then do
-        trace[Diverge.def.valid] "rawMatch: {e}"
-        -- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
-        --
-        -- The casesOn definition is always of the following shape:
-        -- - input parameters (implicit parameters)
-        -- - motive (implicit), -- the motive gives the return type of the match
-        -- - scrutinee (explicit)
-        -- - branches (explicit).
-        -- In particular, we notice that the scrutinee is the first *explicit*
-        -- parameter - this is how we spot it.
-        let matcherName := f.constName!
-        let matcherLevels := f.constLevels!.toArray
-        -- Find the first explicit parameter: this is the scrutinee
-        forallTelescope (← inferType f) fun xs _ => do
-        let rec findFirstExplicit (i : Nat) : MetaM Nat := do
-          if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
-          else
-            let x := xs.get! i
-            let xFVarId := x.fvarId!
-            let localDecl ← xFVarId.getDecl
-            match localDecl.binderInfo with
-            | .default => pure i
-            | _ => findFirstExplicit (i + 1)
-        let scrutIdx ← findFirstExplicit 0
-        -- Split the arguments
-        let params := args.extract 0 (scrutIdx - 1)
-        let motive := args.get! (scrutIdx - 1)
-        let scrut := args.get! scrutIdx
-        let branches := args.extract (scrutIdx + 1) args.size
-        -- Compute the number of parameters for the branches: for this we use
-        -- the type of the uninstantiated casesOn constant (we can't just
-        -- destruct the lambdas in the branch expressions because the result
-        -- of a match might be a lambda expression).
-        let branchesNumParams : Array Nat ← do
-          let env ← getEnv
-          let decl := env.constants.find! matcherName
-          let ty := decl.type
-          forallTelescope ty fun xs _ => do
-          let xs := xs.extract (scrutIdx + 1) xs.size
-          xs.mapM fun x => do
-          let xty ← inferType x
-          forallTelescope xty fun ys _ => do
-          pure ys.size
-        let me : MatchInfo := {
-          matcherName,
-          matcherLevels,
-          params,
-          motive,
-          scruts := #[scrut],
-          branchesNumParams,
-          branches,
-        }
-        proveMatchIsValid k_var kk_var me
-      -- Check if this is a monadic let-binding
-      else if f.isConstOf ``Bind.bind then do
-        trace[Diverge.def.valid] "bind:\n{args}"
-        -- We simply need to prove that the subexpressions are valid, and call
-        -- the appropriate lemma.
-        let x := args.get! 4
-        let y := args.get! 5
-        -- Prove that the subexpressions are valid
-        let xValid ← proveExprIsValid k_var kk_var x
-        trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
-        let yValid ← do
-          -- This is a lambda expression
-          lambdaOne y fun x y => do
-          trace[Diverge.def.valid] "bind: y: {y}"
-          let yValid ← proveExprIsValid k_var kk_var y
-          trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
-          trace[Diverge.def.valid] "bind: yValid: x: {x}"
-          let yValid ← mkLambdaFVars #[x] yValid
-          trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
-          pure yValid
-        -- Put everything together
-        trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
-        mkAppM ``FixI.is_valid_p_bind #[xValid, yValid]
-      -- Check if this is a recursive call, i.e., a call to the continuation `kk`
-      else if f.isFVarOf kk_var.fvarId! then do
-        trace[Diverge.def.valid] "rec: args: \n{args}"
-        if args.size ≠ 2 then throwError "Recursive call with invalid number of parameters: {args}"
-        let i_arg := args.get! 0
-        let x_arg := args.get! 1
-        let eIsValid ← mkAppM ``FixI.is_valid_p_rec #[k_var, i_arg, x_arg]
-        trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
-        pure eIsValid
+    proveAppIsValid k_var kk_var e f args
+
+partial def proveAppIsValid (k_var kk_var : Expr) (e : Expr) (f : Expr) (args : Array Expr): MetaM Expr := do
+  trace[Diverge.def.valid] "proveAppIsValid: {e}\nDecomposed: {f} {args}"
+  /- There are several cases: first, check if this is a match/if
+     Check if the expression is a (dependent) if then else.
+     We treat the if then else expressions differently from the other matches,
+     and have dedicated theorems for them. -/
+  let isIte := e.isIte
+  if isIte || e.isDIte then do
+    e.withApp fun f args => do
+    trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
+    if args.size ≠ 5 then
+       throwError "Wrong number of parameters for {f}: {args}"
+    let cond := args.get! 1
+    let dec := args.get! 2
+    -- Prove that the branches are valid
+    let br0 := args.get! 3
+    let br1 := args.get! 4
+    let proveBranchValid (br : Expr) : MetaM Expr :=
+      if isIte then proveExprIsValid k_var kk_var br
       else do
-        -- Remaining case: normal application.
-        -- It shouldn't use the continuation.
-        proveNoKExprIsValid k_var e
+        -- There is a lambda
+        lambdaOne br fun x br => do
+        let brValid ← proveExprIsValid k_var kk_var br
+        mkLambdaFVars #[x] brValid
+    let br0Valid ← proveBranchValid br0
+    let br1Valid ← proveBranchValid br1
+    let const := if isIte then ``FixII.is_valid_p_ite else ``FixII.is_valid_p_dite
+    let eIsValid ←
+      mkAppOptM const #[none, none, none, none, none,
+                        some k_var, some cond, some dec, none, none,
+                        some br0Valid, some br1Valid]
+    trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
+    pure eIsValid
+  /- Check if the expression is a match (this case is for when the elaborator
+     introduces auxiliary definitions to hide the match behind syntactic
+     sugar): -/
+  else if let some me := ← matchMatcherApp? e then do
+    trace[Diverge.def.valid]
+      "matcherApp:
+       - params: {me.params}
+       - motive: {me.motive}
+       - discrs: {me.discrs}
+       - altNumParams: {me.altNumParams}
+       - alts: {me.alts}
+       - remaining: {me.remaining}"
+    -- matchMatcherApp does all the work for us: we simply need to gather
+    -- the information and call the auxiliary helper `proveMatchIsValid`
+    if me.remaining.size ≠ 0 then
+      throwError "MatcherApp: non empty remaining array: {me.remaining}"
+    let me : MatchInfo := {
+      matcherName := me.matcherName
+      matcherLevels := me.matcherLevels
+      params := me.params
+      motive := me.motive
+      scruts := me.discrs
+      branchesNumParams := me.altNumParams
+      branches := me.alts
+    }
+    proveMatchIsValid k_var kk_var me
+  /- Check if the expression is a raw match (this case is for when the expression
+     is a direct call to the primitive `casesOn` function, without syntactic sugar).
+     We have to check this case because functions like `mkSigmasMatch`, which we
+     use to currify function bodies, introduce such raw matches. -/
+  else if ← isCasesExpr f then do
+    trace[Diverge.def.valid] "rawMatch: {e}"
+    /- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
+
+       The casesOn definition is always of the following shape:
+       - input parameters (implicit parameters)
+       - motive (implicit), -- the motive gives the return type of the match
+       - scrutinee (explicit)
+       - branches (explicit).
+       In particular, we notice that the scrutinee is the first *explicit*
+       parameter - this is how we spot it.
+     -/
+    let matcherName := f.constName!
+    let matcherLevels := f.constLevels!.toArray
+    -- Find the first explicit parameter: this is the scrutinee
+    forallTelescope (← inferType f) fun xs _ => do
+    let rec findFirstExplicit (i : Nat) : MetaM Nat := do
+      if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
+      else
+        let x := xs.get! i
+        let xFVarId := x.fvarId!
+        let localDecl ← xFVarId.getDecl
+        match localDecl.binderInfo with
+        | .default => pure i
+        | _ => findFirstExplicit (i + 1)
+    let scrutIdx ← findFirstExplicit 0
+    -- Split the arguments
+    let params := args.extract 0 (scrutIdx - 1)
+    let motive := args.get! (scrutIdx - 1)
+    let scrut := args.get! scrutIdx
+    let branches := args.extract (scrutIdx + 1) args.size
+    /- Compute the number of parameters for the branches: for this we use
+       the type of the uninstantiated casesOn constant (we can't just
+       destruct the lambdas in the branch expressions because the result
+       of a match might be a lambda expression). -/
+    let branchesNumParams : Array Nat ← do
+      let env ← getEnv
+      let decl := env.constants.find! matcherName
+      let ty := decl.type
+      forallTelescope ty fun xs _ => do
+      let xs := xs.extract (scrutIdx + 1) xs.size
+      xs.mapM fun x => do
+      let xty ← inferType x
+      forallTelescope xty fun ys _ => do
+      pure ys.size
+    let me : MatchInfo := {
+      matcherName,
+      matcherLevels,
+      params,
+      motive,
+      scruts := #[scrut],
+      branchesNumParams,
+      branches,
+    }
+    proveMatchIsValid k_var kk_var me
+  -- Check if this is a monadic let-binding
+  else if f.isConstOf ``Bind.bind then do
+    trace[Diverge.def.valid] "bind:\n{args}"
+    -- We simply need to prove that the subexpressions are valid, and call
+    -- the appropriate lemma.
+    let x := args.get! 4
+    let y := args.get! 5
+    -- Prove that the subexpressions are valid
+    let xValid ← proveExprIsValid k_var kk_var x
+    trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
+    let yValid ← do
+      -- This is a lambda expression
+      lambdaOne y fun x y => do
+      trace[Diverge.def.valid] "bind: y: {y}"
+      let yValid ← proveExprIsValid k_var kk_var y
+      trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
+      trace[Diverge.def.valid] "bind: yValid: x: {x}"
+      let yValid ← mkLambdaFVars #[x] yValid
+      trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
+      pure yValid
+    -- Put everything together
+    trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
+    mkAppM ``FixII.is_valid_p_bind #[xValid, yValid]
+  -- Check if this is a recursive call, i.e., a call to the continuation `kk`
+  else if f.isFVarOf kk_var.fvarId! then do
+    trace[Diverge.def.valid] "rec: args: \n{args}"
+    if args.size ≠ 3 then throwError "Recursive call with invalid number of parameters: {args}"
+    let i_arg := args.get! 0
+    let t_arg := args.get! 1
+    let x_arg := args.get! 2
+    let eIsValid ← mkAppM ``FixII.is_valid_p_rec #[k_var, i_arg, t_arg, x_arg]
+    trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
+    pure eIsValid
+  else do
+    /- Remaining case: normal application.
+       Check if the arguments use the continuation:
+       - if no: this is simple
+       - if yes: we have to lookup theorems in div spec database and continue -/
+    trace[Diverge.def.valid] "regular app: {e}, f: {f}, args: {args}"
+    let argsFVars ← args.mapM getFVarIds
+    let allArgsFVars := argsFVars.foldl (fun hs fvars => hs.insertMany fvars) HashSet.empty
+    trace[Diverge.def.valid] "allArgsFVars: {allArgsFVars.toList.map mkFVar}"
+    if ¬ allArgsFVars.contains kk_var.fvarId! then do
+      -- Simple case
+      trace[Diverge.def.valid] "kk doesn't appear in the arguments"
+      proveNoKExprIsValid k_var e
+    else do
+      -- Lookup in the database for suitable theorems
+      trace[Diverge.def.valid] "kk appears in the arguments"
+      let thms ← divspecAttr.find? e
+      trace[Diverge.def.valid] "Looked up theorems: {thms}"
+      -- Try the theorems one by one
+      proveAppIsValidApplyThms k_var kk_var e f args thms.toList
+
+partial def proveAppIsValidApplyThms (k_var kk_var : Expr) (e : Expr)
+  (f : Expr) (args : Array Expr) (thms : List Name) : MetaM Expr := do
+  match thms with
+  | [] => throwError "Could not prove that the following expression is valid: {e}"
+  | thName :: thms =>
+    -- Lookup the theorem itself
+    let env ← getEnv
+    let thDecl := env.constants.find! thName
+    -- Introduce fresh meta-variables for the universes
+    let ul : List (Name × Level) ←
+      thDecl.levelParams.mapM (λ x => do pure (x, ← mkFreshLevelMVar))
+    let ulMap : HashMap Name Level := HashMap.ofList ul
+    let thTy := thDecl.type.instantiateLevelParamsCore (λ x => ulMap.find! x)
+    trace[Diverge.def.valid] "Trying with theorem {thName}: {thTy}"
+    -- Introduce meta variables for the universally quantified variables
+    let (mvars, _binders, thTyBody) ← forallMetaTelescope thTy
+    let thTermToMatch := thTyBody
+    trace[Diverge.def.valid] "thTermToMatch: {thTermToMatch}"
+    -- Create the term: `is_valid_p k (λ kk => e)`
+    let termToMatch ← mkLambdaFVars #[kk_var] e
+    let termToMatch ← mkAppM ``FixII.is_valid_p #[k_var, termToMatch]
+    trace[Diverge.def.valid] "termToMatch: {termToMatch}"
+    -- Attempt to match
+    trace[Diverge.def.valid] "Matching terms:\n\n{termToMatch}\n\n{thTermToMatch}"
+    let ok ← isDefEq termToMatch thTermToMatch
+    if ¬ ok then
+      -- Failure: attempt with the other theorems
+      proveAppIsValidApplyThms k_var kk_var e f args thms
+    else do
+      /- Success: continue with this theorem
+
+         Instantiate the meta variables (some of them will not be instantiated:
+         they are new subgoals)
+       -/
+      let mvars ← mvars.mapM instantiateMVars
+      let th ← mkAppOptM thName (Array.map some mvars)
+      trace[Diverge.def.valid] "Instantiated theorem: {th}\n{← inferType th}"
+      -- Filter the instantiated meta variables
+      let mvars := mvars.filter (fun v => v.isMVar)
+      let mvars := mvars.map (fun v => v.mvarId!)
+      trace[Diverge.def.valid] "Remaining subgoals: {mvars}"
+      for mvarId in mvars do
+        -- Prove the subgoal (i.e., the precondition of the theorem)
+        let mvarDecl ← mvarId.getDecl
+        let declType ← instantiateMVars mvarDecl.type
+        -- Reduce the subgoal before diving in, if necessary
+        trace[Diverge.def.valid] "Subgoal: {declType}"
+        -- Dive in the type
+        forallTelescope declType fun forall_vars mvar_e => do
+        trace[Diverge.def.valid] "forall_vars: {forall_vars}"
+        -- `mvar_e` should have the shape `is_valid_p k (λ kk => ...)`
+        -- We need to retrieve the new `k` variable, and dive into the
+        -- `λ kk => ...`
+        mvar_e.consumeMData.withApp fun is_valid args => do
+        if is_valid.constName? ≠ ``FixII.is_valid_p ∨ args.size ≠ 7 then
+          throwError "Invalid precondition: {mvar_e}"
+        else do
+          let k_var := args.get! 5
+          let e_lam := args.get! 6
+          trace[Diverge.def.valid] "k_var: {k_var}\ne_lam: {e_lam}"
+          -- The outer lambda should be for the kk_var
+          lambdaOne e_lam.consumeMData fun kk_var e => do
+          -- Continue
+          trace[Diverge.def.valid] "kk_var: {kk_var}\ne: {e}"
+          -- We sometimes need to reduce the term - TODO: this is really dangerous
+          let e ← do
+            let updt_config config :=
+              { config with transparency := .reducible, zetaNonDep := false }
+            withConfig updt_config (whnf e)
+          trace[Diverge.def.valid] "e (after normalization): {e}"
+          let e_valid ← proveExprIsValid k_var kk_var e
+          trace[Diverge.def.valid] "e_valid (for e): {e_valid}"
+          let e_valid ← mkLambdaFVars forall_vars e_valid
+          trace[Diverge.def.valid] "e_valid (with foralls): {e_valid}"
+          let _ ← inferType e_valid -- Sanity check
+          -- Assign the meta variable
+          mvarId.assign e_valid
+      pure th
 
 -- Prove that a match expression is valid.
 partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do
   trace[Diverge.def.valid] "proveMatchIsValid: {me}"
   -- Prove the validity of the branch expressions
   let branchesValid:Array Expr ← me.branches.mapIdxM fun idx br => do
-    -- Go inside the lambdas - note that we have to be careful: some of the
-    -- binders might come from the match, and some of the binders might come
-    -- from the fact that the expression in the match is a lambda expression:
-    -- we use the branchesNumParams field for this reason
+    /- Go inside the lambdas - note that we have to be careful: some of the
+       binders might come from the match, and some of the binders might come
+       from the fact that the expression in the match is a lambda expression:
+       we use the branchesNumParams field for this reason. -/
     let numParams := me.branchesNumParams.get! idx
     lambdaTelescopeN br numParams fun xs br => do
     -- Prove that the branch expression is valid
@@ -603,13 +957,14 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     -- Reconstruct the lambda expression
     mkLambdaFVars xs brValid
   trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
-  -- Compute the motive, which has the following shape:
-  -- ```
-  -- λ scrut => is_valid_p k (λ k => match scrut with ...)
-  --                                 ^^^^^^^^^^^^^^^^^^^^
-  --                         this is the original match expression, with the
-  --                        the difference that the scrutinee(s) is a variable
-  -- ```
+  /- Compute the motive, which has the following shape:
+     ```
+     λ scrut => is_valid_p k (λ k => match scrut with ...)
+                                     ^^^^^^^^^^^^^^^^^^^^
+                             this is the original match expression, with the
+                            the difference that the scrutinee(s) is a variable
+     ```
+   -/
   let validMotive : Expr ← do
     -- The motive is a function of the scrutinees (i.e., a lambda expression):
     -- introduce binders for the scrutinees
@@ -628,7 +983,7 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     let matchE ← mkAppOptM me.matcherName args
     -- Wrap in the `is_valid_p` predicate
     let matchE ← mkLambdaFVars #[kk_var] matchE
-    let validMotive ← mkAppM ``FixI.is_valid_p #[k_var, matchE]
+    let validMotive ← mkAppM ``FixII.is_valid_p #[k_var, matchE]
     -- Abstract away the scrutinee variables
     mkLambdaFVars scrutVars validMotive
   trace[Diverge.def.valid] "valid motive: {validMotive}"
@@ -646,10 +1001,10 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
 
 end
 
--- Prove that a single body (in the mutually recursive group) is valid.
---
--- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
--- we prove that `is_even.body` and `is_odd.body` are valid.
+/- Prove that a single body (in the mutually recursive group) is valid.
+
+   For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+   we prove that `is_even.body` and `is_odd.body` are valid. -/
 partial def proveSingleBodyIsValid
   (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
   MetaM Expr := do
@@ -661,24 +1016,29 @@ partial def proveSingleBodyIsValid
   let body := (env.constants.find! name).value!
   trace[Diverge.def.valid] "body: {body}"
   lambdaTelescope body fun xs body => do
-  assert! xs.size = 2
+  trace[Diverge.def.valid] "xs: {xs}"
+  if xs.size ≠ 3 then throwError "Invalid number of lambdas: {xs} (expected 3)"
   let kk_var := xs.get! 0
-  let x_var := xs.get! 1
+  let t_var := xs.get! 1
+  let x_var := xs.get! 2
   -- State the type of the theorem to prove
-  let thmTy ← mkAppM ``FixI.is_valid_p
-    #[k_var, ← mkLambdaFVars #[kk_var] (← mkAppM' bodyConst #[kk_var, x_var])]
+  trace[Diverge.def.valid] "bodyConst: {bodyConst} : {← inferType bodyConst}"
+  let bodyApp ← mkAppOptM' bodyConst #[.some kk_var, .some t_var, .some x_var]
+  trace[Diverge.def.valid] "bodyApp: {bodyApp}"
+  let bodyApp ← mkLambdaFVars #[kk_var] bodyApp
+  trace[Diverge.def.valid] "bodyApp: {bodyApp}"
+  let thmTy ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp]
   trace[Diverge.def.valid] "thmTy: {thmTy}"
   -- Prove that the body is valid
+  trace[Diverge.def.valid] "body: {body}"
   let proof ← proveExprIsValid k_var kk_var body
-  let proof ← mkLambdaFVars #[k_var, x_var] proof
+  let proof ← mkLambdaFVars #[k_var, t_var, x_var] proof
   trace[Diverge.def.valid] "proveSingleBodyIsValid: proof:\n{proof}:\n{← inferType proof}"
   -- The target type (we don't have to do this: this is simply a sanity check,
   -- and this allows a nicer debugging output)
   let thmTy ← do
-    let body ← mkAppM' bodyConst #[kk_var, x_var]
-    let body ← mkLambdaFVars #[kk_var] body
-    let ty ← mkAppM ``FixI.is_valid_p #[k_var, body]
-    mkForallFVars #[k_var, x_var] ty
+    let ty ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp]
+    mkForallFVars #[k_var, t_var, x_var] ty
   trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}"
   -- Save the theorem
   let name := preDef.declName ++ "body_is_valid"
@@ -694,18 +1054,18 @@ partial def proveSingleBodyIsValid
   -- Return the theorem
   pure (Expr.const name (preDef.levelParams.map .param))
 
--- Prove that the list of bodies are valid.
---
--- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
--- we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
--- valid.
-partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
+/- Prove that the list of bodies are valid.
+
+  For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+  we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
+  valid. -/
+partial def proveFunsBodyIsValid (paramInOutTys: Expr) (bodyFuns : Expr)
   (k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do
   -- Create the big "and" expression, which groups the validity proof of the individual bodies
   let rec mkValidConj (i : Nat) : MetaM Expr := do
     if i = bodiesValid.size then
       -- We reached the end
-      mkAppM ``FixI.Funs.is_valid_p_Nil #[k_var]
+      mkAppM ``FixII.Funs.is_valid_p_Nil #[k_var]
     else do
       -- We haven't reached the end: introduce a conjunction
       let valid := bodiesValid.get! i
@@ -713,20 +1073,20 @@ partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
       mkAppM ``And.intro #[valid, ← mkValidConj (i + 1)]
   let andExpr ← mkValidConj 0
   -- Wrap in the `is_valid_p_is_valid_p` theorem, and abstract the continuation
-  let isValid ← mkAppM ``FixI.Funs.is_valid_p_is_valid_p #[inOutTys, k_var, bodyFuns, andExpr]
+  let isValid ← mkAppM ``FixII.Funs.is_valid_p_is_valid_p #[paramInOutTys, k_var, bodyFuns, andExpr]
   mkLambdaFVars #[k_var] isValid
 
--- Prove that the mut rec body (i.e., the unary body which groups the bodies
--- of all the functions in the mutually recursive group and on which we will
--- apply the fixed-point operator) is valid.
---
--- We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
--- which we return.
---
--- TODO: maybe this function should introduce k_var itself
+/- Prove that the mut rec body (i.e., the unary body which groups the bodies
+   of all the functions in the mutually recursive group and on which we will
+   apply the fixed-point operator) is valid.
+
+   We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
+   which we return.
+
+   TODO: maybe this function should introduce k_var itself -/
 def proveMutRecIsValid
   (grName : Name) (grLvlParams : List Name)
-  (inOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
+  (paramInOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
   (k_var : Expr) (preDefs : Array PreDefinition)
   (bodies : Array Expr) : MetaM Expr := do
   -- First prove that the individual bodies are valid
@@ -737,9 +1097,10 @@ def proveMutRecIsValid
       proveSingleBodyIsValid k_var preDef body
   -- Then prove that the mut rec body is valid
   trace[Diverge.def.valid] "## Proving that the 'Funs' body is valid"
-  let isValid ← proveFunsBodyIsValid inOutTys bodyFuns k_var bodiesValid
+  let isValid ← proveFunsBodyIsValid paramInOutTys bodyFuns k_var bodiesValid
+  trace[Diverge.def.valid] "Generated the term: {isValid}"
   -- Save the theorem
-  let thmTy ← mkAppM ``FixI.is_valid #[mutRecBodyConst]
+  let thmTy ← mkAppM ``FixII.is_valid #[mutRecBodyConst]
   let name := grName ++ "mut_rec_body_is_valid"
   let decl := Declaration.thmDecl {
     name
@@ -753,26 +1114,29 @@ def proveMutRecIsValid
   -- Return the theorem
   pure (Expr.const name (grLvlParams.map .param))
 
--- Generate the final definions by using the mutual body and the fixed point operator.
---
--- For instance:
--- ```
--- def is_even (i : Int) : Result Bool := mut_rec_body 0 i
--- def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
--- ```
-def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+/- Generate the final definions by using the mutual body and the fixed point operator.
+
+   For instance:
+   ```
+   def is_even (i : Int) : Result Bool := mut_rec_body 0 i
+   def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
+   ```
+ -/
+def mkDeclareFixDefs (mutRecBody : Expr) (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) :
   TermElabM (Array Name) := do
   let grSize := preDefs.size
   let defs ← preDefs.mapIdxM fun idx preDef => do
     lambdaTelescope preDef.value fun xs _ => do
-    -- Retrieve the input type
-    let in_ty := (inOutTys.get! idx.val).fst
+    -- Retrieve the parameters info
+    let type_info := paramInOutTys.get! idx.val
     -- Create the index
     let idx ← mkFinVal grSize idx.val
-    -- Group the inputs into a dependent tuple
-    let input ← mkSigmasVal in_ty xs.toList
+    -- Group the inputs into two tuples
+    let (params_args, input_args) := xs.toList.splitAt type_info.num_params
+    let params ← mkSigmasVal type_info.params_ty params_args
+    let input ← mkProdsVal input_args
     -- Apply the fixed point
-    let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
+    let fixedBody ← mkAppM ``FixII.fix #[mutRecBody, idx, params, input]
     let fixedBody ← mkLambdaFVars xs fixedBody
     -- Create the declaration
     let name := preDef.declName
@@ -790,7 +1154,8 @@ def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preD
   pure defs
 
 -- Prove the equations that we will use as unfolding theorems
-partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Expr))
+partial def proveUnfoldingThms (isValidThm : Expr)
+  (paramInOutTys : Array TypeInfo)
   (preDefs : Array PreDefinition) (decls : Array Name) : MetaM Unit := do
   let grSize := preDefs.size
   let proveIdx (i : Nat) : MetaM Unit := do
@@ -810,14 +1175,18 @@ partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Ex
     trace[Diverge.def.unfold] "proveUnfoldingThms: thm statement: {thmTy}"
     -- The proof
     -- Use the fixed-point equation
-    let proof ← mkAppM ``FixI.is_valid_fix_fixed_eq #[isValidThm]
+    let proof ← mkAppM ``FixII.is_valid_fix_fixed_eq #[isValidThm]
     -- Add the index
     let idx ← mkFinVal grSize i
     let proof ← mkAppM ``congr_fun #[proof, idx]
-    -- Add the input argument
-    let arg ← mkSigmasVal (inOutTys.get! i).fst xs.toList
-    let proof ← mkAppM ``congr_fun #[proof, arg]
-    -- Abstract the arguments away
+    -- Add the input arguments
+    let type_info := paramInOutTys.get! i
+    let (params, args) := xs.toList.splitAt type_info.num_params
+    let params ← mkSigmasVal type_info.params_ty params
+    let args ← mkProdsVal args
+    let proof ← mkAppM ``congr_fun #[proof, params]
+    let proof ← mkAppM ``congr_fun #[proof, args]
+    -- Abstract all the arguments away
     let proof ← mkLambdaFVars xs proof
     trace[Diverge.def.unfold] "proveUnfoldingThms: proof: {proof}:\n{← inferType proof}"
     -- Declare the theorem
@@ -845,7 +1214,9 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
   trace[Diverge.def] ("divRecursion: defs:\n" ++ msg)
 
-  -- TODO: what is this?
+  -- Apply all the "attribute" functions (for instance, the function which
+  -- registers the theorem in the simp database if there is the `simp` attribute,
+  -- etc.)
   for preDef in preDefs do
     applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
 
@@ -859,40 +1230,53 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let grLvlParams := def0.levelParams
   trace[Diverge.def] "def0 universe levels: {def0.levelParams}"
 
-  -- We first compute the list of pairs: (input type × output type)
-  let inOutTys : Array (Expr × Expr) ←
-      preDefs.mapM (fun preDef => do
-        withRef preDef.ref do -- is the withRef useful?
-        -- Check the universe parameters - TODO: I'm not sure what the best thing
-        -- to do is. In practice, all the type parameters should be in Type 0, so
-        -- we shouldn't have universe issues.
-        if preDef.levelParams ≠ grLvlParams then
-          throwError "Non-uniform polymorphism in the universes"
-        forallTelescope preDef.type (fun in_tys out_ty => do
-          let in_ty ← liftM (mkSigmasType in_tys.toList)
-          -- Retrieve the type in the "Result"
-          let out_ty ← getResultTy out_ty
-          let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
-          pure (in_ty, out_ty)
-        )
-      )
-  trace[Diverge.def] "inOutTys: {inOutTys}"
-  -- Turn the list of input/output type pairs into an expresion
-  let inOutTysExpr ← inOutTys.mapM (λ (x, y) => mkInOutTy x y)
-  let inOutTysExpr ← mkListLit (← inferType (inOutTysExpr.get! 0)) inOutTysExpr.toList
-
-  -- From the list of pairs of input/output types, actually compute the
-  -- type of the continuation `k`.
-  -- We first introduce the index `i : Fin n` where `n` is the number of
-  -- functions in the group.
+  /- We first compute the tuples: (type parameters × input type × output type)
+     - type parameters: this is a sigma type
+     - input type: λ params_type => product type
+     - output type: λ params_type => output type
+     For instance, on the function:
+     `list_nth (α : Type) (ls : List α) (i : Int) : Result α`:
+     we generate:
+     `(Type, λ α => List α × i, λ α => Result α)`
+   -/
+  let paramInOutTys : Array TypeInfo ←
+    preDefs.mapM (fun preDef => do
+      -- Check the universe parameters - TODO: I'm not sure what the best thing
+      -- to do is. In practice, all the type parameters should be in Type 0, so
+      -- we shouldn't have universe issues.
+      if preDef.levelParams ≠ grLvlParams then
+        throwError "Non-uniform polymorphism in the universes"
+      forallTelescope preDef.type (fun in_tys out_ty => do
+        let total_num_args := in_tys.size
+        let (params, in_tys) ← splitInputArgs in_tys out_ty
+        trace[Diverge.def] "Decomposed arguments: {preDef.declName}: {params}, {in_tys}, {out_ty}"
+        let num_params := params.size
+        let params_ty ← mkSigmasType params.data
+        let in_ty ← mkSigmasMatchOrUnit params.data (← mkProdsType in_tys.data)
+        -- Retrieve the type in the "Result"
+        let out_ty ← getResultTy out_ty
+        let out_ty ← mkSigmasMatchOrUnit params.data out_ty
+        trace[Diverge.def] "inOutTy: {preDef.declName}: {params_ty}, {in_tys}, {out_ty}"
+        pure ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩))
+  trace[Diverge.def] "paramInOutTys: {paramInOutTys}"
+  -- Turn the list of input types/input args/output type tuples into expressions
+  let paramInOutTysExpr ← liftM (paramInOutTys.mapM mkInOutTyFromTypeInfo)
+  let paramInOutTysExpr ← mkListLit (← inferType (paramInOutTysExpr.get! 0)) paramInOutTysExpr.toList
+  trace[Diverge.def] "paramInOutTys: {paramInOutTys}"
+
+  /- From the list of pairs of input/output types, actually compute the
+     type of the continuation `k`.
+     We first introduce the index `i : Fin n` where `n` is the number of
+     functions in the group.
+   -/
   let i_var_ty := mkFin preDefs.size
   withLocalDeclD (mkAnonymous "i" 0) i_var_ty fun i_var => do
-  let in_out_ty ← mkAppM ``List.get #[inOutTysExpr, i_var]
-  trace[Diverge.def] "in_out_ty := {in_out_ty} : {← inferType in_out_ty}"
-  -- Add an auxiliary definition for `in_out_ty`
-  let in_out_ty ← do
-    let value ← mkLambdaFVars #[i_var] in_out_ty
-    let name := grName.append "in_out_ty"
+  let param_in_out_ty ← mkAppM ``List.get #[paramInOutTysExpr, i_var]
+  trace[Diverge.def] "param_in_out_ty := {param_in_out_ty} : {← inferType param_in_out_ty}"
+  -- Add an auxiliary definition for `param_in_out_ty` (this is a potentially big term)
+  let param_in_out_ty ← do
+    let value ← mkLambdaFVars #[i_var] param_in_out_ty
+    let name := grName.append "param_in_out_ty"
     let levelParams := grLvlParams
     let decl := Declaration.defnDecl {
       name := name
@@ -905,19 +1289,28 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
     }
     addDecl decl
     -- Return the constant
-    let in_out_ty := Lean.mkConst name (levelParams.map .param)
-    mkAppM' in_out_ty #[i_var]
-  trace[Diverge.def] "in_out_ty (after decl) := {in_out_ty} : {← inferType in_out_ty}"
-  let in_ty ← mkAppM ``Sigma.fst #[in_out_ty]
+    let param_in_out_ty := Lean.mkConst name (levelParams.map .param)
+    mkAppM' param_in_out_ty #[i_var]
+  trace[Diverge.def] "param_in_out_ty (after decl) := {param_in_out_ty} : {← inferType param_in_out_ty}"
+  -- Decompose between: param_ty, in_ty, out_ty
+  let param_ty ← mkAppM ``Sigma.fst #[param_in_out_ty]
+  let in_out_ty ← mkAppM ``Sigma.snd #[param_in_out_ty]
+  let in_ty ← mkAppM ``Prod.fst #[in_out_ty]
+  let out_ty ← mkAppM ``Prod.snd #[in_out_ty]
+  trace[Diverge.def] "param_ty: {param_ty}"
+  trace[Diverge.def] "in_ty: {in_ty}"
+  trace[Diverge.def] "out_ty: {out_ty}"
+  withLocalDeclD (mkAnonymous "t" 1) param_ty fun param => do
+  let in_ty ← mkAppM' in_ty #[param]
+  let out_ty ← mkAppM' out_ty #[param]
   trace[Diverge.def] "in_ty: {in_ty}"
-  withLocalDeclD (mkAnonymous "x" 1) in_ty fun input => do
-  let out_ty ← mkAppM' (← mkAppM ``Sigma.snd #[in_out_ty]) #[input]
   trace[Diverge.def] "out_ty: {out_ty}"
 
   -- Introduce the continuation `k`
-  let in_ty ← mkLambdaFVars #[i_var] in_ty
-  let out_ty ← mkLambdaFVars #[i_var, input] out_ty
-  let kk_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty]
+  let param_ty ← mkLambdaFVars #[i_var] param_ty
+  let in_ty ← mkLambdaFVars #[i_var, param] in_ty
+  let out_ty ← mkLambdaFVars #[i_var, param] out_ty
+  let kk_var_ty ← mkAppM ``FixII.kk_ty #[i_var_ty, param_ty, in_ty, out_ty]
   trace[Diverge.def] "kk_var_ty: {kk_var_ty}"
   withLocalDeclD (mkAnonymous "kk" 2) kk_var_ty fun kk_var => do
   trace[Diverge.def] "kk_var: {kk_var}"
@@ -925,29 +1318,30 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   -- Replace the recursive calls in all the function bodies by calls to the
   -- continuation `k` and and generate for those bodies declarations
   trace[Diverge.def] "# Generating the unary bodies"
-  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var inOutTys preDefs
+  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var paramInOutTys preDefs
   trace[Diverge.def] "Unary bodies (after decl): {bodies}"
+
   -- Generate the mutually recursive body
   trace[Diverge.def] "# Generating  the mut rec body"
-  let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var in_ty out_ty inOutTys.toList bodies
+  let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var param_ty in_ty out_ty paramInOutTys bodies
   trace[Diverge.def] "mut rec body (after decl): {mutRecBody}"
 
   -- Prove that the mut rec body satisfies the validity criteria required by
   -- our fixed-point
-  let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
+  let k_var_ty ← mkAppM ``FixII.k_ty #[i_var_ty, param_ty, in_ty, out_ty]
   withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do
   trace[Diverge.def] "# Proving that the mut rec body is valid"
-  let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
+  let isValidThm ← proveMutRecIsValid grName grLvlParams paramInOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
 
   -- Generate the final definitions
   trace[Diverge.def] "# Generating the final definitions"
-  let decls ← mkDeclareFixDefs mutRecBody inOutTys preDefs
+  let decls ← mkDeclareFixDefs mutRecBody paramInOutTys preDefs
 
   -- Prove the unfolding theorems
   trace[Diverge.def] "# Proving the unfolding theorems"
-  proveUnfoldingThms isValidThm inOutTys preDefs decls
+  proveUnfoldingThms isValidThm paramInOutTys preDefs decls
 
-  -- Generating code -- TODO
+  -- Generating code
   addAndCompilePartialRec preDefs
 
 -- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
@@ -1068,15 +1462,23 @@ elab_rules : command
       Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
 
 namespace Tests
+
   /- Some examples of partial functions -/
 
-  divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+  --set_option trace.Diverge.def true
+  --set_option trace.Diverge.def.genBody true
+  --set_option trace.Diverge.def.valid true
+  --set_option trace.Diverge.def.genBody.visit true
+
+  divergent def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
     match ls with
     | [] => .fail .panic
     | x :: ls =>
       if i = 0 then return x
       else return (← list_nth ls (i - 1))
 
+  --set_option trace.Diverge false
+
   #check list_nth.unfold
 
   example {a: Type} (ls : List a) :
@@ -1087,17 +1489,33 @@ namespace Tests
     . intro i hpos h; simp at h; linarith
     . rename_i hd tl ih
       intro i hpos h
-      -- We can directly use `rw [list_nth]`!
+      -- We can directly use `rw [list_nth]`
       rw [list_nth]; simp
       split <;> try simp [*]
       . tauto
-      . -- TODO: we shouldn't have to do that
+      . -- We don't have to do this if we use scalar_tac
         have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
         simp at h
         have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
         simp [ih]
         tauto
 
+  -- Return a continuation
+  divergent def list_nth_with_back {a: Type} (ls : List a) (i : Int) :
+    Result (a × (a → Result (List a))) :=
+    match ls with
+    | [] => .fail .panic
+    | x :: ls =>
+      if i = 0 then return (x, (λ ret => return (ret :: ls)))
+      else do
+        let (x, back) ← list_nth_with_back ls (i - 1)
+        return (x,
+          (λ ret => do
+           let ls ← back ret
+           return (x :: ls)))
+
+  #check list_nth_with_back.unfold
+
   mutual
     divergent def is_even (i : Int) : Result Bool :=
       if i = 0 then return true else return (← is_odd (i - 1))
@@ -1121,7 +1539,6 @@ namespace Tests
   #check bar.unfold
 
   -- Testing dependent branching and let-bindings
-  -- TODO: why the linter warning?
   divergent def isNonZero (i : Int) : Result Bool :=
     if _h:i = 0 then return false
     else
@@ -1157,6 +1574,82 @@ namespace Tests
 
   #check test1.unfold
 
+  /- Tests with higher-order functions -/
+  section HigherOrder
+    open Ex8
+
+    inductive Tree (a : Type u) :=
+    | leaf (x : a)
+    | node (tl : List (Tree a))
+
+    divergent def id {a : Type u} (t : Tree a) : Result (Tree a) :=
+      match t with
+      | .leaf x => .ret (.leaf x)
+      | .node tl =>
+        do
+          let tl ← map id tl
+          .ret (.node tl)
+
+    #check id.unfold
+
+    divergent def id1 {a : Type u} (t : Tree a) : Result (Tree a) :=
+      match t with
+      | .leaf x => .ret (.leaf x)
+      | .node tl =>
+        do
+          let tl ← map (fun x => id1 x) tl
+          .ret (.node tl)
+
+    #check id1.unfold
+
+    divergent def id2 {a : Type u} (t : Tree a) : Result (Tree a) :=
+      match t with
+      | .leaf x => .ret (.leaf x)
+      | .node tl =>
+        do
+          let tl ← map (fun x => do let _ ← id2 x; id2 x) tl
+          .ret (.node tl)
+
+    #check id2.unfold
+
+    divergent def incr (t : Tree Nat) : Result (Tree Nat) :=
+      match t with
+      | .leaf x => .ret (.leaf (x + 1))
+      | .node tl =>
+        do
+          let tl ← map incr tl
+          .ret (.node tl)
+
+    -- We handle this by inlining the let-binding
+    divergent def id3 (t : Tree Nat) : Result (Tree Nat) :=
+      match t with
+      | .leaf x => .ret (.leaf (x + 1))
+      | .node tl =>
+        do
+          let f := id3
+          let tl ← map f tl
+          .ret (.node tl)
+
+    #check id3.unfold
+
+    /-
+    -- This is not handled yet: we can only do it if we introduce "general"
+    -- relations for the input types and output types (result_rel should
+    -- be parameterized by something).
+    divergent def id4 (t : Tree Nat) : Result (Tree Nat) :=
+      match t with
+      | .leaf x => .ret (.leaf (x + 1))
+      | .node tl =>
+        do
+          let f ← .ret id4
+          let tl ← map f tl
+          .ret (.node tl)
+
+    #check id4.unfold
+    -/
+
+  end HigherOrder
+
 end Tests
 
 end Diverge
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index fedb1c74..0d33e9d2 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -1,15 +1,84 @@
 import Lean
+import Base.Utils
+import Base.Primitives.Base
+import Base.Extensions
 
 namespace Diverge
 
 open Lean Elab Term Meta
+open Utils Extensions
 
 -- We can't define and use trace classes in the same file
+initialize registerTraceClass `Diverge
 initialize registerTraceClass `Diverge.elab
 initialize registerTraceClass `Diverge.def
 initialize registerTraceClass `Diverge.def.sigmas
+initialize registerTraceClass `Diverge.def.prods
 initialize registerTraceClass `Diverge.def.genBody
+initialize registerTraceClass `Diverge.def.genBody.visit
 initialize registerTraceClass `Diverge.def.valid
 initialize registerTraceClass `Diverge.def.unfold
 
+-- For the attribute (for higher-order functions)
+initialize registerTraceClass `Diverge.attr
+
+-- Attribute
+
+-- divspec attribute
+structure DivSpecAttr where
+  attr : AttributeImpl
+  ext  : DiscrTreeExtension Name true
+  deriving Inhabited
+
+/- The persistent map from expressions to divspec theorems. -/
+initialize divspecAttr : DivSpecAttr ← do
+  let ext ← mkDiscrTreeExtention `divspecMap true
+  let attrImpl : AttributeImpl := {
+    name := `divspec
+    descr := "Marks theorems to use with the `divergent` encoding"
+    add := fun thName stx attrKind => do
+      Attribute.Builtin.ensureNoArgs stx
+      -- TODO: use the attribute kind
+      unless attrKind == AttributeKind.global do
+        throwError "invalid attribute divspec, must be global"
+      -- Lookup the theorem
+      let env ← getEnv
+      let thDecl := env.constants.find! thName
+      let fKey : Array (DiscrTree.Key true) ← MetaM.run' (do
+        /- The theorem should have the shape:
+           `∀ ..., is_valid_p k (λ k => ...)`
+
+           Dive into the ∀:
+         -/
+        let (_, _, fExpr) ← forallMetaTelescope thDecl.type.consumeMData
+        /- Dive into the argument of `is_valid_p`: -/
+        fExpr.consumeMData.withApp fun _ args => do
+        if args.size ≠ 7 then throwError "Invalid number of arguments to is_valid_p"
+        let fExpr := args.get! 6
+        /- Dive into the lambda: -/
+        let (_, _, fExpr) ← lambdaMetaTelescope fExpr.consumeMData
+        trace[Diverge] "Registering divspec theorem for {fExpr}"
+        -- Convert the function expression to a discrimination tree key
+        DiscrTree.mkPath fExpr)
+      let env := ext.addEntry env (fKey, thName)
+      setEnv env
+      trace[Diverge] "Saved the environment"
+      pure ()
+  }
+  registerBuiltinAttribute attrImpl
+  pure { attr := attrImpl, ext := ext }
+
+def DivSpecAttr.find? (s : DivSpecAttr) (e : Expr) : MetaM (Array Name) := do
+  (s.ext.getState (← getEnv)).getMatch e
+
+def DivSpecAttr.getState (s : DivSpecAttr) : MetaM (DiscrTree Name true) := do
+  pure (s.ext.getState (← getEnv))
+
+def showStoredDivSpec : MetaM Unit := do
+  let st ← divspecAttr.getState
+  -- TODO: how can we iterate over (at least) the values stored in the tree?
+  --let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+  let s := f!"{st}"
+  IO.println s
+
 end Diverge
diff --git a/backends/lean/Base/Extensions.lean b/backends/lean/Base/Extensions.lean
new file mode 100644
index 00000000..b34f41dc
--- /dev/null
+++ b/backends/lean/Base/Extensions.lean
@@ -0,0 +1,47 @@
+import Lean
+import Std.Lean.HashSet
+import Base.Utils
+import Base.Primitives.Base
+
+import Lean.Meta.DiscrTree
+import Lean.Meta.Tactic.Simp
+
+/-! Various state extensions used in the library -/
+namespace Extensions
+
+open Lean Elab Term Meta
+open Utils
+
+-- This is not used anymore but we keep it here.
+-- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
+def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) :
+  IO (MapDeclarationExtension α) :=
+  registerSimplePersistentEnvExtension {
+    name          := name,
+    addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
+    addEntryFn    := fun s n => s.insert n.1 n.2 ,
+    toArrayFn     := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
+  }
+
+/- Discrimination trees map expressions to values. When storing an expression
+   in a discrimination tree, the expression is first converted to an array
+   of `DiscrTree.Key`, which are the keys actually used by the discrimination
+   trees. The conversion operation is monadic, however, and extensions require
+   all the operations to be pure. For this reason, in the state extension, we
+   store the keys from *after* the transformation (i.e., the `DiscrTreeKey`
+   below). The transformation itself can be done elsewhere.
+ -/
+abbrev DiscrTreeKey (simpleReduce : Bool) := Array (DiscrTree.Key simpleReduce)
+
+abbrev DiscrTreeExtension (α : Type) (simpleReduce : Bool) :=
+  SimplePersistentEnvExtension (DiscrTreeKey simpleReduce × α) (DiscrTree α simpleReduce)
+
+def mkDiscrTreeExtention [Inhabited α] [BEq α] (name : Name := by exact decl_name%) (simpleReduce : Bool) :
+  IO (DiscrTreeExtension α simpleReduce) :=
+  registerSimplePersistentEnvExtension {
+    name          := name,
+    addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insertCore k v) s) DiscrTree.empty,
+    addEntryFn    := fun s n => s.insertCore n.1 n.2 ,
+  }
+
+end Extensions
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index f74fecd4..db522df2 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -528,7 +528,7 @@ instance {ty} : HAnd (Scalar ty) (Scalar ty) (Scalar ty) where
   hAnd x y := Scalar.and x y
 
 -- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
 theorem Scalar.add_spec {ty} {x y : Scalar ty}
   (hmin : Scalar.min ty ≤ x.val + y.val)
   (hmax : x.val + y.val ≤ Scalar.max ty) :
@@ -550,62 +550,62 @@ theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
   apply add_spec <;> assumption
 
 /- Fine-grained theorems -/
-@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
+@[pspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
+@[pspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
+@[pspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
+@[pspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
+@[pspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
+@[pspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
   apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem Isize.add_spec {x y : Isize}
+@[pspec] theorem Isize.add_spec {x y : Isize}
   (hmin : Isize.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ Isize.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
-@[cepspec] theorem I8.add_spec {x y : I8}
+@[pspec] theorem I8.add_spec {x y : I8}
   (hmin : I8.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I8.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
-@[cepspec] theorem I16.add_spec {x y : I16}
+@[pspec] theorem I16.add_spec {x y : I16}
   (hmin : I16.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I16.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
-@[cepspec] theorem I32.add_spec {x y : I32}
+@[pspec] theorem I32.add_spec {x y : I32}
   (hmin : I32.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I32.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
-@[cepspec] theorem I64.add_spec {x y : I64}
+@[pspec] theorem I64.add_spec {x y : I64}
   (hmin : I64.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I64.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
-@[cepspec] theorem I128.add_spec {x y : I128}
+@[pspec] theorem I128.add_spec {x y : I128}
   (hmin : I128.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I128.max) :
   ∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
   Scalar.add_spec hmin hmax
 
 -- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
 theorem Scalar.sub_spec {ty} {x y : Scalar ty}
   (hmin : Scalar.min ty ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ Scalar.max ty) :
@@ -629,56 +629,56 @@ theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
   apply sub_spec <;> assumption
 
 /- Fine-grained theorems -/
-@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
+@[pspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
+@[pspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
+@[pspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
+@[pspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
+@[pspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
+@[pspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
   apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
 
-@[cepspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ x.val - y.val)
+@[pspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ Isize.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
 
-@[cepspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ x.val - y.val)
+@[pspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ I8.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
 
-@[cepspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ x.val - y.val)
+@[pspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ I16.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
 
-@[cepspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ x.val - y.val)
+@[pspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ I32.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
 
-@[cepspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ x.val - y.val)
+@[pspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ I64.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
 
-@[cepspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ x.val - y.val)
+@[pspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ x.val - y.val)
   (hmax : x.val - y.val ≤ I128.max) :
   ∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
   Scalar.sub_spec hmin hmax
@@ -705,62 +705,62 @@ theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
   apply mul_spec <;> assumption
 
 /- Fine-grained theorems -/
-@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
+@[pspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
+@[pspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
+@[pspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
+@[pspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
+@[pspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
+@[pspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
   apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
 
-@[cepspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ x.val * y.val)
+@[pspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ Isize.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
-@[cepspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ x.val * y.val)
+@[pspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ I8.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
-@[cepspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ x.val * y.val)
+@[pspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ I16.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
-@[cepspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ x.val * y.val)
+@[pspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ I32.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
-@[cepspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ x.val * y.val)
+@[pspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ I64.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
-@[cepspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ x.val * y.val)
+@[pspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ I128.max) :
   ∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
   Scalar.mul_spec hmin hmax
 
 -- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
 theorem Scalar.div_spec {ty} {x y : Scalar ty}
   (hnz : y.val ≠ 0)
   (hmin : Scalar.min ty ≤ scalar_div x.val y.val)
@@ -788,66 +788,66 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   apply hs
 
 /- Fine-grained theorems -/
-@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+@[pspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [*]
 
-@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+@[pspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+@[pspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+@[pspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+@[pspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+@[pspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
   ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
   apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem Isize.div_spec (x : Isize) {y : Isize}
+@[pspec] theorem Isize.div_spec (x : Isize) {y : Isize}
   (hnz : y.val ≠ 0)
   (hmin : Isize.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ Isize.max):
   ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
   Scalar.div_spec hnz hmin hmax
 
-@[cepspec] theorem I8.div_spec (x : I8) {y : I8}
+@[pspec] theorem I8.div_spec (x : I8) {y : I8}
   (hnz : y.val ≠ 0)
   (hmin : I8.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ I8.max):
   ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
   Scalar.div_spec hnz hmin hmax
 
-@[cepspec] theorem I16.div_spec (x : I16) {y : I16}
+@[pspec] theorem I16.div_spec (x : I16) {y : I16}
   (hnz : y.val ≠ 0)
   (hmin : I16.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ I16.max):
   ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
   Scalar.div_spec hnz hmin hmax
 
-@[cepspec] theorem I32.div_spec (x : I32) {y : I32}
+@[pspec] theorem I32.div_spec (x : I32) {y : I32}
   (hnz : y.val ≠ 0)
   (hmin : I32.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ I32.max):
   ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
   Scalar.div_spec hnz hmin hmax
 
-@[cepspec] theorem I64.div_spec (x : I64) {y : I64}
+@[pspec] theorem I64.div_spec (x : I64) {y : I64}
   (hnz : y.val ≠ 0)
   (hmin : I64.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ I64.max):
   ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
   Scalar.div_spec hnz hmin hmax
 
-@[cepspec] theorem I128.div_spec (x : I128) {y : I128}
+@[pspec] theorem I128.div_spec (x : I128) {y : I128}
   (hnz : y.val ≠ 0)
   (hmin : I128.min ≤ scalar_div x.val y.val)
   (hmax : scalar_div x.val y.val ≤ I128.max):
@@ -855,7 +855,7 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   Scalar.div_spec hnz hmin hmax
 
 -- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
 theorem Scalar.rem_spec {ty} {x y : Scalar ty}
   (hnz : y.val ≠ 0)
   (hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
@@ -883,59 +883,59 @@ theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   simp [*] at hs
   simp [*]
 
-@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+@[pspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [*]
 
-@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+@[pspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+@[pspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+@[pspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+@[pspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+@[pspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
   ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
 
-@[cepspec] theorem I8.rem_spec (x : I8) {y : I8}
+@[pspec] theorem I8.rem_spec (x : I8) {y : I8}
   (hnz : y.val ≠ 0)
   (hmin : I8.min ≤ scalar_rem x.val y.val)
   (hmax : scalar_rem x.val y.val ≤ I8.max):
   ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
   Scalar.rem_spec hnz hmin hmax
 
-@[cepspec] theorem I16.rem_spec (x : I16) {y : I16}
+@[pspec] theorem I16.rem_spec (x : I16) {y : I16}
   (hnz : y.val ≠ 0)
   (hmin : I16.min ≤ scalar_rem x.val y.val)
   (hmax : scalar_rem x.val y.val ≤ I16.max):
   ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
   Scalar.rem_spec hnz hmin hmax
 
-@[cepspec] theorem I32.rem_spec (x : I32) {y : I32}
+@[pspec] theorem I32.rem_spec (x : I32) {y : I32}
   (hnz : y.val ≠ 0)
   (hmin : I32.min ≤ scalar_rem x.val y.val)
   (hmax : scalar_rem x.val y.val ≤ I32.max):
   ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
   Scalar.rem_spec hnz hmin hmax
 
-@[cepspec] theorem I64.rem_spec (x : I64) {y : I64}
+@[pspec] theorem I64.rem_spec (x : I64) {y : I64}
   (hnz : y.val ≠ 0)
   (hmin : I64.min ≤ scalar_rem x.val y.val)
   (hmax : scalar_rem x.val y.val ≤ I64.max):
   ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
   Scalar.rem_spec hnz hmin hmax
 
-@[cepspec] theorem I128.rem_spec (x : I128) {y : I128}
+@[pspec] theorem I128.rem_spec (x : I128) {y : I128}
   (hnz : y.val ≠ 0)
   (hmin : I128.min ≤ scalar_rem x.val y.val)
   (hmax : scalar_rem x.val y.val ≤ I128.max):
diff --git a/backends/lean/Base/Progress/Base.lean b/backends/lean/Base/Progress/Base.lean
index 76a92795..0ad16ab6 100644
--- a/backends/lean/Base/Progress/Base.lean
+++ b/backends/lean/Base/Progress/Base.lean
@@ -2,11 +2,12 @@ import Lean
 import Std.Lean.HashSet
 import Base.Utils
 import Base.Primitives.Base
+import Base.Extensions
 
 namespace Progress
 
 open Lean Elab Term Meta
-open Utils
+open Utils Extensions
 
 -- We can't define and use trace classes in the same file
 initialize registerTraceClass `Progress
@@ -15,17 +16,17 @@ initialize registerTraceClass `Progress
 
 structure PSpecDesc where
   -- The universally quantified variables
+  -- Can be fvars or mvars
   fvars : Array Expr
   -- The existentially quantified variables
   evars : Array Expr
+  -- The function applied to its arguments
+  fArgsExpr : Expr
   -- The function
-  fExpr : Expr
   fName : Name
   -- The function arguments
   fLevels : List Level
   args : Array Expr
-  -- The universally quantified variables which appear in the function arguments
-  argsFVars : Array FVarId
   -- The returned value
   ret : Expr
   -- The postcondition (if there is)
@@ -37,7 +38,7 @@ section Methods
   variable [MonadError m]
   variable {a : Type}
 
-  /- Analyze a pspec theorem to decompose its arguments.
+  /- Analyze a goal or a pspec theorem to decompose its arguments.
 
      PSpec theorems should be of the following shape:
      ```
@@ -56,12 +57,20 @@ section Methods
      TODO: generalize for when we do inductive proofs
   -/
   partial
-  def withPSpec [Inhabited (m a)] [Nonempty (m a)] (th : Expr) (k : PSpecDesc → m a)
-    (sanityChecks : Bool := false) :
+  def withPSpec [Inhabited (m a)] [Nonempty (m a)]
+    (isGoal : Bool) (th : Expr) (k : PSpecDesc → m a) :
     m a := do
     trace[Progress] "Proposition: {th}"
     -- Dive into the quantified variables and the assumptions
-    forallTelescope th.consumeMData fun fvars th => do
+    -- Note that if we analyze a pspec theorem to register it in a database (i.e.
+    -- a discrimination tree), we need to introduce *meta-variables* for the
+    -- quantified variables.
+    let telescope (k : Array Expr → Expr → m a) : m a :=
+      if isGoal then forallTelescope th.consumeMData (fun fvars th => k fvars th)
+      else do
+        let (fvars, _, th) ← forallMetaTelescope th.consumeMData
+        k fvars th
+    telescope fun fvars th => do
     trace[Progress] "Universally quantified arguments and assumptions: {fvars}"
     -- Dive into the existentials
     existsTelescope th.consumeMData fun evars th => do
@@ -78,7 +87,7 @@ section Methods
     -- destruct the application to get the function name
     mExpr.consumeMData.withApp fun mf margs => do
     trace[Progress] "After stripping the arguments of the monad expression:\n- mf: {mf}\n- margs: {margs}"
-    let (fExpr, f, args) ← do
+    let (fArgsExpr, f, args) ← do
       if mf.isConst ∧ mf.constName = ``Bind.bind then do
         -- Dive into the bind
         let fExpr := (margs.get! 4).consumeMData
@@ -86,29 +95,27 @@ section Methods
       else pure (mExpr, mf, margs)
     trace[Progress] "After stripping the arguments of the function call:\n- f: {f}\n- args: {args}"
     if ¬ f.isConst then throwError "Not a constant: {f}"
-    -- Compute the set of universally quantified variables which appear in the function arguments
-    let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
-    -- Sanity check
-    if sanityChecks then
-      -- All the variables which appear in the inputs given to the function are
-      -- universally quantified (in particular, they are not *existentially* quantified)
-      let fvarsSet : HashSet FVarId := HashSet.ofArray (fvars.map (fun x => x.fvarId!))
-      let filtArgsFVars := allArgsFVars.toArray.filter (fun fvar => ¬ fvarsSet.contains fvar)
-      if ¬ filtArgsFVars.isEmpty then
+    -- *Sanity check* (activated if we are analyzing a theorem to register it in a DB)
+    -- Check if some existentially quantified variables
+    let _ := do
+      -- Collect all the free variables in the arguments
+      let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
+      -- Check if they intersect the fvars we introduced for the existentially quantified variables
+      let evarsSet : HashSet FVarId := HashSet.ofArray (evars.map (fun (x : Expr) => x.fvarId!))
+      let filtArgsFVars := allArgsFVars.toArray.filter (fun var => evarsSet.contains var)
+      if filtArgsFVars.isEmpty then pure ()
+      else
         let filtArgsFVars := filtArgsFVars.map (fun fvarId => Expr.fvar fvarId)
         throwError "Some of the function inputs are not universally quantified: {filtArgsFVars}"
-    let argsFVars := fvars.map (fun x => x.fvarId!)
-    let argsFVars := argsFVars.filter (fun fvar => allArgsFVars.contains fvar)
     -- Return
-    trace[Progress] "Function: {f.constName!}";
+    trace[Progress] "Function with arguments: {fArgsExpr}";
     let thDesc := {
       fvars := fvars
       evars := evars
-      fExpr
+      fArgsExpr
       fName := f.constName!
       fLevels := f.constLevels!
       args := args
-      argsFVars
       ret := ret
       post := post
     }
@@ -116,117 +123,18 @@ section Methods
 
 end Methods
 
-def getPSpecFunName (th : Expr) : MetaM Name :=
-  withPSpec th (fun d => do pure d.fName) true
+def getPSpecFunArgsExpr (isGoal : Bool) (th : Expr) : MetaM Expr :=
+  withPSpec isGoal th (fun d => do pure d.fArgsExpr)
 
-def getPSpecClassFunNames (th : Expr) : MetaM (Name × Name) :=
-  withPSpec th (fun d => do
-    let arg0 := d.args.get! 0
-    arg0.withApp fun f _ => do
-    if ¬ f.isConst then throwError "Not a constant: {f}"
-    pure (d.fName, f.constName)
-    ) true
-
-def getPSpecClassFunNameArg (th : Expr) : MetaM (Name × Expr) :=
-  withPSpec th (fun d => do
-    let arg0 := d.args.get! 0
-    pure (d.fName, arg0)
-    ) true
-
--- "Regular" pspec attribute
+-- pspec attribute
 structure PSpecAttr where
   attr : AttributeImpl
-  ext  : MapDeclarationExtension Name
-  deriving Inhabited
-
-/- pspec attribute for type classes: we use the name of the type class to
-   lookup another map. We use the *first* argument of the type class to lookup
-   into this second map.
-
-   Example:
-   ========
-   We use type classes for addition. For instance, the addition between two
-   U32 is written (without syntactic sugar) as `HAdd.add (Scalar ty) x y`. As a consequence,
-   we store the theorem through the bindings: HAdd.add → Scalar → ...
-
-   SH: TODO: this (and `PSpecClassExprAttr`) is a bit ad-hoc. For now it works for the
-   specs of the scalar operations, which is what I really need, but I'm not sure it
-   applies well to other situations. A better way would probably to use type classes, but
-   I couldn't get them to work on those cases. It is worth retrying.
--/
-structure PSpecClassAttr where
-  attr : AttributeImpl
-  ext  : MapDeclarationExtension (NameMap Name)
-  deriving Inhabited
-
-/- Same as `PSpecClassAttr` but we use the full first argument (it works when it
-   is a constant). -/
-structure PSpecClassExprAttr where
-  attr : AttributeImpl
-  ext  : MapDeclarationExtension (HashMap Expr Name)
+  ext  : DiscrTreeExtension Name true
   deriving Inhabited
 
--- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
-def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) :
-  IO (MapDeclarationExtension α) :=
-  registerSimplePersistentEnvExtension {
-    name          := name,
-    addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
-    addEntryFn    := fun s n => s.insert n.1 n.2 ,
-    toArrayFn     := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
-  }
-
--- Declare an extension of maps of maps (using [RBMap]).
--- The important point is that we need to merge the bound values (which are maps).
-def mkMapMapDeclarationExtension [Inhabited β] (ord : α → α → Ordering)
-  (name : Name := by exact decl_name%) :
-  IO (MapDeclarationExtension (RBMap α β ord)) :=
-  registerSimplePersistentEnvExtension {
-    name          := name,
-    addImportedFn := fun a =>
-      a.foldl (fun s a => a.foldl (
-        -- We need to merge the maps
-        fun s (k0, k1_to_v) =>
-        match s.find? k0 with
-        | none =>
-          -- No binding: insert one
-          s.insert k0 k1_to_v
-        | some m =>
-          -- There is already a binding: merge
-          let m := RBMap.fold (fun m k v => m.insert k v) m k1_to_v
-          s.insert k0 m)
-          s) RBMap.empty,
-    addEntryFn    := fun s n => s.insert n.1 n.2 ,
-    toArrayFn     := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
-  }
-
--- Declare an extension of maps of maps (using [HashMap]).
--- The important point is that we need to merge the bound values (which are maps).
-def mkMapHashMapDeclarationExtension [BEq α] [Hashable α] [Inhabited β]
-  (name : Name := by exact decl_name%) :
-  IO (MapDeclarationExtension (HashMap α β)) :=
-  registerSimplePersistentEnvExtension {
-    name          := name,
-    addImportedFn := fun a =>
-      a.foldl (fun s a => a.foldl (
-        -- We need to merge the maps
-        fun s (k0, k1_to_v) =>
-        match s.find? k0 with
-        | none =>
-          -- No binding: insert one
-          s.insert k0 k1_to_v
-        | some m =>
-          -- There is already a binding: merge
-          let m := HashMap.fold (fun m k v => m.insert k v) m k1_to_v
-          s.insert k0 m)
-          s) RBMap.empty,
-    addEntryFn    := fun s n => s.insert n.1 n.2 ,
-    toArrayFn     := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
-  }
-
-/- The persistent map from function to pspec theorems. -/
+/- The persistent map from expressions to pspec theorems. -/
 initialize pspecAttr : PSpecAttr ← do
-  let ext ← mkMapDeclarationExtension `pspecMap
+  let ext ← mkDiscrTreeExtention `pspecMap true
   let attrImpl : AttributeImpl := {
     name := `pspec
     descr := "Marks theorems to use with the `progress` tactic"
@@ -238,130 +146,30 @@ initialize pspecAttr : PSpecAttr ← do
       -- Lookup the theorem
       let env ← getEnv
       let thDecl := env.constants.find! thName
-      let fName ← MetaM.run' (getPSpecFunName thDecl.type)
-      trace[Progress] "Registering spec theorem for {fName}"
-      let env := ext.addEntry env (fName, thName)
-      setEnv env
-      pure ()
-  }
-  registerBuiltinAttribute attrImpl
-  pure { attr := attrImpl, ext := ext }
-
-/- The persistent map from type classes to pspec theorems -/
-initialize pspecClassAttr : PSpecClassAttr ← do
-  let ext : MapDeclarationExtension (NameMap Name) ←
-    mkMapMapDeclarationExtension Name.quickCmp `pspecClassMap
-  let attrImpl : AttributeImpl  := {
-    name := `cpspec
-    descr := "Marks theorems to use for type classes with the `progress` tactic"
-    add := fun thName stx attrKind => do
-      Attribute.Builtin.ensureNoArgs stx
-      -- TODO: use the attribute kind
-      unless attrKind == AttributeKind.global do
-        throwError "invalid attribute 'cpspec', must be global"
-      -- Lookup the theorem
-      let env ← getEnv
-      let thDecl := env.constants.find! thName
-      let (fName, argName) ← MetaM.run' (getPSpecClassFunNames thDecl.type)
-      trace[Progress] "Registering class spec theorem for ({fName}, {argName})"
-      -- Update the entry if there is one, add an entry if there is none
-      let env :=
-        match (ext.getState (← getEnv)).find? fName with
-        | none =>
-          let m := RBMap.ofList [(argName, thName)]
-          ext.addEntry env (fName, m)
-        | some m =>
-          let m := m.insert argName thName
-          ext.addEntry env (fName, m)
+      let fKey ← MetaM.run' (do
+        let fExpr ← getPSpecFunArgsExpr false thDecl.type
+        trace[Progress] "Registering spec theorem for {fExpr}"
+        -- Convert the function expression to a discrimination tree key
+        DiscrTree.mkPath fExpr)
+      let env := ext.addEntry env (fKey, thName)
       setEnv env
+      trace[Progress] "Saved the environment"
       pure ()
   }
   registerBuiltinAttribute attrImpl
   pure { attr := attrImpl, ext := ext }
 
-/- The 2nd persistent map from type classes to pspec theorems -/
-initialize pspecClassExprAttr : PSpecClassExprAttr ← do
-  let ext : MapDeclarationExtension (HashMap Expr Name) ←
-    mkMapHashMapDeclarationExtension `pspecClassExprMap
-  let attrImpl : AttributeImpl  := {
-    name := `cepspec
-    descr := "Marks theorems to use for type classes with the `progress` tactic"
-    add := fun thName stx attrKind => do
-      Attribute.Builtin.ensureNoArgs stx
-      -- TODO: use the attribute kind
-      unless attrKind == AttributeKind.global do
-        throwError "invalid attribute 'cpspec', must be global"
-      -- Lookup the theorem
-      let env ← getEnv
-      let thDecl := env.constants.find! thName
-      let (fName, arg) ← MetaM.run' (getPSpecClassFunNameArg thDecl.type)
-      -- Sanity check: no variables appear in the argument
-      MetaM.run' do
-        let fvars ← getFVarIds arg
-        if ¬ fvars.isEmpty then throwError "The first argument ({arg}) contains variables"
-      -- We store two bindings:
-      -- - arg to theorem name
-      -- - reduced arg to theorem name
-      let rarg ← MetaM.run' (reduceAll arg)
-      trace[Progress] "Registering class spec theorem for ({fName}, {arg}) and ({fName}, {rarg})"
-      -- Update the entry if there is one, add an entry if there is none
-      let env :=
-        match (ext.getState (← getEnv)).find? fName with
-        | none =>
-          let m := HashMap.ofList [(arg, thName), (rarg, thName)]
-          ext.addEntry env (fName, m)
-        | some m =>
-          let m := m.insert arg thName
-          let m := m.insert rarg thName
-          ext.addEntry env (fName, m)
-      setEnv env
-      pure ()
-  }
-  registerBuiltinAttribute attrImpl
-  pure { attr := attrImpl, ext := ext }
-
-
-def PSpecAttr.find? (s : PSpecAttr) (name : Name) : MetaM (Option Name) := do
-  return (s.ext.getState (← getEnv)).find? name
-
-def PSpecClassAttr.find? (s : PSpecClassAttr) (className argName : Name) : MetaM (Option Name) := do
-  match (s.ext.getState (← getEnv)).find? className with
-  | none => return none
-  | some map => return map.find? argName
-
-def PSpecClassExprAttr.find? (s : PSpecClassExprAttr) (className : Name) (arg : Expr) : MetaM (Option Name) := do
-  match (s.ext.getState (← getEnv)).find? className with
-  | none => return none
-  | some map => return map.find? arg
-
-def PSpecAttr.getState (s : PSpecAttr) : MetaM (NameMap Name) := do
-  pure (s.ext.getState (← getEnv))
+def PSpecAttr.find? (s : PSpecAttr) (e : Expr) : MetaM (Array Name) := do
+  (s.ext.getState (← getEnv)).getMatch e
 
-def PSpecClassAttr.getState (s : PSpecClassAttr) : MetaM (NameMap (NameMap Name)) := do
-  pure (s.ext.getState (← getEnv))
-
-def PSpecClassExprAttr.getState (s : PSpecClassExprAttr) : MetaM (NameMap (HashMap Expr Name)) := do
+def PSpecAttr.getState (s : PSpecAttr) : MetaM (DiscrTree Name true) := do
   pure (s.ext.getState (← getEnv))
 
 def showStoredPSpec : MetaM Unit := do
   let st ← pspecAttr.getState
-  let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
-  IO.println s
-
-def showStoredPSpecClass : MetaM Unit := do
-  let st ← pspecClassAttr.getState
-  let s := st.toList.foldl (fun s (f, m) =>
-    let ms := m.toList.foldl (fun s (f, th) =>
-      f!"{s}\n  {f} → {th}") f!""
-    f!"{s}\n{f} → [{ms}]") f!""
-  IO.println s
-
-def showStoredPSpecExprClass : MetaM Unit := do
-  let st ← pspecClassExprAttr.getState
-  let s := st.toList.foldl (fun s (f, m) =>
-    let ms := m.toList.foldl (fun s (f, th) =>
-      f!"{s}\n  {f} → {th}") f!""
-    f!"{s}\n{f} → [{ms}]") f!""
+  -- TODO: how can we iterate over (at least) the values stored in the tree?
+  --let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+  let s := f!"{st}"
   IO.println s
 
 end Progress
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
index ba63f09d..a6a4e82a 100644
--- a/backends/lean/Base/Progress/Progress.lean
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -204,11 +204,11 @@ def getFirstArg (args : Array Expr) : Option Expr := do
   if args.size = 0 then none
   else some (args.get! 0)
 
-/- Helper: try to lookup a theorem and apply it, or continue with another tactic
-   if it fails -/
+/- Helper: try to lookup a theorem and apply it.
+   Return true if it succeeded. -/
 def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
   (asmTac : TacticM Unit) (fExpr : Expr)
-  (kind : String) (th : Option TheoremOrLocal) (x : TacticM Unit) : TacticM Unit := do
+  (kind : String) (th : Option TheoremOrLocal) : TacticM Bool := do
   let res ← do
     match th with
     | none =>
@@ -223,9 +223,9 @@ def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost :
           pure (some res)
         catch _ => none
   match res with
-  | some .Ok => return ()
+  | some .Ok => return true
   | some (.Error msg) => throwError msg
-  | none => x
+  | none => return false
 
 -- The array of ids are identifiers to use when introducing fresh variables
 def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrLocal)
@@ -236,11 +236,19 @@ def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrL
   let goalTy ← mgoal.getType
   trace[Progress] "goal: {goalTy}"
   -- Dive into the goal to lookup the theorem
-  let (fExpr, fName, args) ← do
-    withPSpec goalTy fun desc =>
-    -- TODO: check that no quantified variables in the arguments
-    pure (desc.fExpr, desc.fName, desc.args)
-  trace[Progress] "Function: {fName}"
+  -- Remark: if we don't isolate the call to `withPSpec` to immediately "close"
+  -- the terms immediately, we may end up with the error:
+  -- "(kernel) declaration has free variables"
+  -- I'm not sure I understand why.
+  -- TODO: we should also check that no quantified variable appears in fExpr.
+  -- If such variables appear, we should just fail because the goal doesn't
+  -- have the proper shape.
+  let fExpr ← do
+    let isGoal := true
+    withPSpec isGoal goalTy fun desc => do
+    let fExpr := desc.fArgsExpr
+    trace[Progress] "Expression to match: {fExpr}"
+    pure fExpr
   -- If the user provided a theorem/assumption: use it.
   -- Otherwise, lookup one.
   match withTh with
@@ -258,36 +266,24 @@ def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrL
       match res with
       | .Ok => return ()
       | .Error msg => throwError msg
-    -- It failed: try to lookup a theorem
-    -- TODO: use a list of theorems, and try them one by one?
-    trace[Progress] "No assumption succeeded: trying to lookup a theorem"
-    let pspec ← do
-      let thName ← pspecAttr.find? fName
-      pure (thName.map fun th => .Theorem th)
-    tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec do
-    -- It failed: try to lookup a *class* expr spec theorem (those are more
-    -- specific than class spec theorems)
-    trace[Progress] "Failed using a pspec theorem: trying to lookup a pspec class expr theorem"
-    let pspecClassExpr ← do
-      match getFirstArg args with
-      | none => pure none
-      | some arg => do
-        trace[Progress] "Using: f:{fName}, arg: {arg}"
-        let thName ← pspecClassExprAttr.find? fName arg
-        pure (thName.map fun th => .Theorem th)
-    tryLookupApply keep ids splitPost asmTac fExpr "pspec class expr theorem" pspecClassExpr do
-    -- It failed: try to lookup a *class* spec theorem
-    trace[Progress] "Failed using a pspec class expr theorem: trying to lookup a pspec class theorem"
-    let pspecClass ← do
-      match ← getFirstArgAppName args with
-      | none => pure none
-      | some argName => do
-        trace[Progress] "Using: f: {fName}, arg: {argName}"
-        let thName ← pspecClassAttr.find? fName argName
-        pure (thName.map fun th => .Theorem th)
-    tryLookupApply keep ids splitPost asmTac fExpr "pspec class theorem" pspecClass do
-    trace[Progress] "Failed using a pspec class theorem: trying to use a recursive assumption"
-    -- Try a recursive call - we try the assumptions of kind "auxDecl"
+    -- It failed: lookup the pspec theorems which match the expression
+    trace[Progress] "No assumption succeeded: trying to lookup a pspec theorem"
+    let pspecs : Array TheoremOrLocal ← do
+      let thNames ← pspecAttr.find? fExpr
+      -- TODO: because of reduction, there may be several valid theorems (for
+      -- instance for the scalars). We need to sort them from most specific to
+      -- least specific. For now, we assume the most specific theorems are at
+      -- the end.
+      let thNames := thNames.reverse
+      trace[Progress] "Looked up pspec theorems: {thNames}"
+      pure (thNames.map fun th => TheoremOrLocal.Theorem th)
+    -- Try the theorems one by one
+    for pspec in pspecs do
+      if ← tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec then return ()
+      else pure ()
+    -- It failed: try to use the recursive assumptions
+    trace[Progress] "Failed using a pspec theorem: trying to use a recursive assumption"
+    -- We try to apply the assumptions of kind "auxDecl"
     let ctx ← Lean.MonadLCtx.getLCtx
     let decls ← ctx.getAllDecls
     let decls := decls.filter (λ decl => match decl.kind with
@@ -381,8 +377,6 @@ namespace Test
 
   -- The following commands display the databases of theorems
   -- #eval showStoredPSpec
-  -- #eval showStoredPSpecClass
-  -- #eval showStoredPSpecExprClass
   open alloc.vec
 
   example {ty} {x y : Scalar ty}
@@ -402,6 +396,19 @@ namespace Test
   example {x y : U32}
     (hmax : x.val + y.val ≤ U32.max) :
     ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+    -- This spec theorem is suboptimal, but it is good to check that it works
+    progress with Scalar.add_spec as ⟨ z, h1 .. ⟩
+    simp [*, h1]
+ 
+  example {x y : U32}
+    (hmax : x.val + y.val ≤ U32.max) :
+    ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+    progress with U32.add_spec as ⟨ z, h1 .. ⟩
+    simp [*, h1]
+
+  example {x y : U32}
+    (hmax : x.val + y.val ≤ U32.max) :
+    ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
     progress keep _ as ⟨ z, h1 .. ⟩
     simp [*, h1]
 
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
index b917a789..b0032281 100644
--- a/backends/lean/Base/Utils.lean
+++ b/backends/lean/Base/Utils.lean
@@ -159,47 +159,96 @@ elab "print_ctx_decls" : tactic => do
   let decls ← ctx.getDecls
   printDecls decls
 
--- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- A map-reduce visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
 -- The continuation takes as parameters:
 -- - the current depth of the expression (useful for printing/debugging)
 -- - the expression to explore
-partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
-  let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
+partial def mapreduceVisit {a : Type} (k : Nat → a → Expr → MetaM (a × Expr))
+  (state : a) (e : Expr) : MetaM (a × Expr) := do
+  let mapreduceVisitBinders (state : a) (xs : Array Expr) (k2 : a → MetaM (a × Expr)) :
+    MetaM (a × Expr) := do
     let localInstances ← getLocalInstances
-    let mut lctx ← getLCtx
-    for x in xs do
-      let xFVarId := x.fvarId!
-      let localDecl ← xFVarId.getDecl
-      let type      ← mapVisit k localDecl.type
-      let localDecl := localDecl.setType type
-      let localDecl ← match localDecl.value? with
-         | some value => let value ← mapVisit k value; pure <| localDecl.setValue value
-         | none       => pure localDecl
-      lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
-    withLCtx lctx localInstances k2
+    -- Update the local declarations for the bindings in context `lctx`
+    let rec visit_xs (lctx : LocalContext) (state : a) (xs : List Expr) : MetaM (LocalContext × a) := do
+      match xs with
+      | [] => pure (lctx, state)
+      | x :: xs => do
+        let xFVarId := x.fvarId!
+        let localDecl ← xFVarId.getDecl
+        let (state, type) ← mapreduceVisit k state localDecl.type
+        let localDecl := localDecl.setType type
+        let (state, localDecl) ← match localDecl.value? with
+           | some value =>
+             let (state, value) ← mapreduceVisit k state value
+             pure (state, localDecl.setValue value)
+           | none => pure (state, localDecl)
+        let lctx := lctx.modifyLocalDecl xFVarId fun _ => localDecl
+        -- Recursive call
+        visit_xs lctx state xs
+    let (lctx, state) ← visit_xs (← getLCtx) state xs.toList
+    -- Call the continuation with the updated context
+    withLCtx lctx localInstances (k2 state)
   -- TODO: use a cache? (Lean.checkCache)
-  let rec visit (i : Nat) (e : Expr) : MetaM Expr := do
+  let rec visit (i : Nat) (state : a) (e : Expr) : MetaM (a × Expr) := do
     -- Explore
-    let e ← k i e
+    let (state, e) ← k i state e
     match e with
     | .bvar _
     | .fvar _
     | .mvar _
     | .sort _
     | .lit _
-    | .const _ _ => pure e
-    | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (visit (i + 1)))
+    | .const _ _ => pure (state, e)
+    | .app .. => do e.withApp fun f args => do
+      let (state, args) ← args.foldlM (fun (state, args) arg => do let (state, arg) ← visit (i + 1) state arg; pure (state, arg :: args)) (state, [])
+      let args := args.reverse
+      let (state, f) ← visit (i + 1) state f
+      let e' := mkAppN f (Array.mk args)
+      return (state, e')
     | .lam .. =>
       lambdaLetTelescope e fun xs b =>
-        mapVisitBinders xs do mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+        mapreduceVisitBinders state xs fun state => do
+        let (state, b) ← visit (i + 1) state b
+        let e' ← mkLambdaFVars xs b (usedLetOnly := false)
+        return (state, e')
     | .forallE .. => do
-      forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← visit (i + 1) b)
+      forallTelescope e fun xs b =>
+         mapreduceVisitBinders state xs fun state => do
+         let (state, b) ← visit (i + 1) state b
+         let e' ← mkForallFVars xs b
+         return (state, e')
     | .letE .. => do
-      lambdaLetTelescope e fun xs b => mapVisitBinders xs do
-        mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
-    | .mdata _ b => return e.updateMData! (← visit (i + 1) b)
-    | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
-  visit 0 e
+      lambdaLetTelescope e fun xs b =>
+        mapreduceVisitBinders state xs fun state => do
+        let (state, b) ← visit (i + 1) state b
+        let e' ← mkLambdaFVars xs b (usedLetOnly := false)
+        return (state, e')
+    | .mdata _ b => do
+      let (state, b) ← visit (i + 1) state b
+      return (state, e.updateMData! b)
+    | .proj _ _ b => do
+      let (state, b) ← visit (i + 1) state b
+      return (state, e.updateProj! b)
+  visit 0 state e
+
+-- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- The continuation takes as parameters:
+-- - the current depth of the expression (useful for printing/debugging)
+-- - the expression to explore
+partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+  let k' i (_ : Unit) e := do
+    let e ← k i e
+    pure ((), e)
+  let (_, e) ← mapreduceVisit k' () e
+  pure e
+
+-- A reduce visitor
+partial def reduceVisit {a : Type} (k : Nat → a → Expr → MetaM a) (s : a) (e : Expr) : MetaM a := do
+  let k' i (s : a) e := do
+    let s ← k i s e
+    pure (s, e)
+  let (s, _) ← mapreduceVisit k' s e
+  pure s
 
 -- Generate a fresh user name for an anonymous proposition to introduce in the
 -- assumptions
@@ -371,15 +420,22 @@ def splitConjTarget : TacticM Unit := do
 
 -- Destruct an equaliy and return the two sides
 def destEq (e : Expr) : MetaM (Expr × Expr) := do
-  e.withApp fun f args =>
+  e.consumeMData.withApp fun f args =>
   if f.isConstOf ``Eq ∧ args.size = 3 then pure (args.get! 1, args.get! 2)
   else throwError "Not an equality: {e}"
 
 -- Return the set of FVarIds in the expression
+-- TODO: this collects fvars introduced in the inner bindings
 partial def getFVarIds (e : Expr) (hs : HashSet FVarId := HashSet.empty) : MetaM (HashSet FVarId) := do
-  e.withApp fun body args => do
-  let hs := if body.isFVar then hs.insert body.fvarId! else hs
-  args.foldlM (fun hs arg => getFVarIds arg hs) hs
+  reduceVisit (fun _ (hs : HashSet FVarId) e =>
+    if e.isFVar then pure (hs.insert e.fvarId!) else pure hs)
+    hs e
+
+-- Return the set of MVarIds in the expression
+partial def getMVarIds (e : Expr) (hs : HashSet MVarId := HashSet.empty) : MetaM (HashSet MVarId) := do
+  reduceVisit (fun _ (hs : HashSet MVarId) e =>
+    if e.isMVar then pure (hs.insert e.mvarId!) else pure hs)
+    hs e
 
 -- Tactic to split on a disjunction.
 -- The expression `h` should be an fvar.