path: root/backends/lean/Base/Diverge/Base.lean

import Lean
import Lean.Meta.Tactic.Simp
import Init.Data.List.Basic
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
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
  open Result

  variable {a : Type u} {b : a → Type v}
  variable {c d : Type w} -- TODO: why do we have to make them both : Type w?

  /-! # The least fixed point definition and its properties -/

  def least_p (p : Nat → Prop) (n : Nat) : Prop := p n ∧ (∀ m, m < n → ¬ p m)
  noncomputable def least (p : Nat → Prop) : Nat :=
    Classical.epsilon (least_p p)

  -- Auxiliary theorem for [least_spec]: if there exists an `n` satisfying `p`,
  -- there there exists a least `m` satisfying `p`.
  theorem least_spec_aux (p : Nat → Prop) : ∀ (n : Nat), (hn : p n) → ∃ m, least_p p m := by
    apply Nat.strongRec'
    intros n hi hn
    -- Case disjunction on: is n the smallest n satisfying p?
    match Classical.em (∀ m, m < n → ¬ p m) with
    | .inl hlt =>
      -- Yes: trivial
      exists n
    | .inr hlt =>
      simp at *
      let ⟨ m, ⟨ hmlt, hm ⟩ ⟩ := hlt
      have hi := hi m hmlt hm
      apply hi

  -- The specification of [least]: either `p` is never satisfied, or it is satisfied
  -- by `least p` and no `n < least p` satisfies `p`.
  theorem least_spec (p : Nat → Prop) : (∀ n, ¬ p n) ∨ (p (least p) ∧ ∀ n, n < least p → ¬ p n) := by
    -- Case disjunction on the existence of an `n` which satisfies `p`
    match Classical.em (∀ n, ¬ p n) with
    | .inl h =>
      -- There doesn't exist: trivial
      apply (Or.inl h)
    | .inr h =>
      -- There exists: we simply use `least_spec_aux` in combination with the property
      -- of the epsilon operator
      simp at *
      let ⟨ n, hn ⟩ := h
      apply Or.inr
      have hl := least_spec_aux p n hn
      have he := Classical.epsilon_spec hl
      apply he

  /-! # The fixed point definitions -/

  def fix_fuel (n : Nat) (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
    Result (b x) :=
    match n with
    | 0 => .div
    | n + 1 =>
      f (fix_fuel n f) x

  @[simp] def fix_fuel_pred (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (x : a) (n : Nat) :=
    not (div? (fix_fuel n f x))

  def fix_fuel_P (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (x : a) (n : Nat) : Prop :=
    fix_fuel_pred f x n

  partial
  def fixImpl (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
    f (fixImpl f) x

  -- The fact that `fix` is implemented by `fixImpl` allows us to not mark the
  -- functions defined with the fixed-point as noncomputable. One big advantage
  -- is that it allows us to evaluate those functions, for instance with #eval.
  @[implemented_by fixImpl]
  def fix (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
    fix_fuel (least (fix_fuel_P f x)) f x

  /-! # The validity property -/

  -- Monotonicity relation over results
  -- TODO: generalize (we should parameterize the definition by a relation over `a`)
  def result_rel {a : Type u} (x1 x2 : Result a) : Prop :=
    match x1 with
    | div => True
    | fail _ => x2 = x1
    | ret _ => x2 = x1 -- TODO: generalize

  -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
  def karrow_rel (k1 k2 : (x:a) → Result (b x)) : Prop :=
    ∀ x, result_rel (k1 x) (k2 x)

  -- Monotonicity property for function bodies
  def is_mono (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
    ∀ {{k1 k2}}, karrow_rel k1 k2 → karrow_rel (f k1) (f k2)

  -- "Continuity" property.
  -- We need this, and this looks a lot like continuity. Also see this paper:
  -- https://inria.hal.science/file/index/docid/216187/filename/tarski.pdf
  -- We define our "continuity" criteria so that it gives us what we need to
  -- prove the fixed-point equation, and we can also easily manipulate it.
  def is_cont (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
    ∀ x, (Hdiv : ∀ n, fix_fuel (.succ n) f x = div) → f (fix f) x = div

  /-! # The proof of the fixed-point equation -/
  theorem fix_fuel_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}
    (Hmono : is_mono f) :
    ∀ {{n m}}, n ≤ m → karrow_rel (fix_fuel n f) (fix_fuel m f) := by
    intros n
    induction n
    case zero => simp [karrow_rel, fix_fuel, result_rel]
    case succ n1 Hi =>
      intros m Hle x
      simp [result_rel]
      match m with
      | 0 =>
        exfalso
        zify at *
        linarith
      | Nat.succ m1 =>
        simp_arith at Hle
        simp [fix_fuel]
        have Hi := Hi Hle
        have Hmono := Hmono Hi x
        simp [result_rel] at Hmono
        apply Hmono

  @[simp] theorem neg_fix_fuel_P
    {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} {x : a} {n : Nat} :
    ¬ fix_fuel_P f x n ↔ (fix_fuel n f x = div) := by
    simp [fix_fuel_P, div?]
    cases fix_fuel n f x <;> simp  

  theorem fix_fuel_fix_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
    ∀ n, karrow_rel (fix_fuel n f) (fix f) := by
    intros n x
    simp [result_rel]
    have Hl := least_spec (fix_fuel_P f x)
    simp at Hl
    match Hl with
    | .inl Hl => simp [*]
    | .inr ⟨ Hl, Hn ⟩ =>
      match Classical.em (fix_fuel n f x = div) with
      | .inl Hd =>
        simp [*]
      | .inr Hd =>
        have Hineq : least (fix_fuel_P f x) ≤ n := by
          -- Proof by contradiction
          cases Classical.em (least (fix_fuel_P f x) ≤ n) <;> simp [*]
          simp at *
          rename_i Hineq
          have Hn := Hn n Hineq
          contradiction
        have Hfix : ¬ (fix f x = div) := by
          simp [fix]
          -- By property of the least upper bound
          revert Hd Hl
          -- TODO: there is no conversion to select the head of a function!
          conv => lhs; apply congr_fun; apply congr_fun; apply congr_fun; simp [fix_fuel_P, div?]
          cases fix_fuel (least (fix_fuel_P f x)) f x <;> simp
        have Hmono := fix_fuel_mono Hmono Hineq x
        simp [result_rel] at Hmono
        simp [fix] at *
        cases Heq: fix_fuel (least (fix_fuel_P f x)) f x <;>
        cases Heq':fix_fuel n f x <;>
        simp_all

  theorem fix_fuel_P_least {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
    ∀ {{x n}}, fix_fuel_P f x n → fix_fuel_P f x (least (fix_fuel_P f x)) := by
    intros x n Hf
    have Hfmono := fix_fuel_fix_mono Hmono n x
    -- TODO: there is no conversion to select the head of a function!
    conv => apply congr_fun; simp [fix_fuel_P]
    simp [fix_fuel_P] at Hf
    revert Hf Hfmono
    simp [div?, result_rel, fix]
    cases fix_fuel n f x <;> simp_all

  -- Prove the fixed point equation in the case there exists some fuel for which
  -- the execution terminates
  theorem fix_fixed_eq_terminates (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (Hmono : is_mono f)
    (x : a) (n : Nat) (He : fix_fuel_P f x n) :
    fix f x = f (fix f) x := by
    have Hl := fix_fuel_P_least Hmono He
    -- TODO: better control of simplification
    conv at Hl =>
      apply congr_fun
      simp [fix_fuel_P]
    -- The least upper bound is > 0
    have ⟨ n, Hsucc ⟩ : ∃ n, least (fix_fuel_P f x) = Nat.succ n := by
      revert Hl
      simp [div?]
      cases least (fix_fuel_P f x) <;> simp [fix_fuel]
    simp [Hsucc] at Hl
    revert Hl
    simp [*, div?, fix, fix_fuel]
    -- Use the monotonicity
    have Hfixmono := fix_fuel_fix_mono Hmono n
    have Hvm := Hmono Hfixmono x
    -- Use functional extensionality
    simp [result_rel, fix] at Hvm
    revert Hvm
    split <;> simp [*] <;> intros <;> simp [*]

  theorem fix_fixed_eq_forall {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
    (Hmono : is_mono f) (Hcont : is_cont f) :
    ∀ x, fix f x = f (fix f) x := by
    intros x
    -- Case disjunction: is there a fuel such that the execution successfully execute?
    match Classical.em (∃ n, fix_fuel_P f x n) with
    | .inr He =>
      -- No fuel: the fixed point evaluates to `div`
      --simp [fix] at *
      simp at *
      conv => lhs; simp [fix]
      have Hel := He (Nat.succ (least (fix_fuel_P f x))); simp [*, fix_fuel] at *; clear Hel
      -- Use the "continuity" of `f`
      have He : ∀ n, fix_fuel (.succ n) f x = div := by intros; simp [*]
      have Hcont := Hcont x He
      simp [Hcont]
    | .inl ⟨ n, He ⟩ => apply fix_fixed_eq_terminates f Hmono x n He

  -- The final fixed point equation
  theorem fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
    (Hmono : is_mono f) (Hcont : is_cont f) :
    fix f = f (fix f) := by
    have Heq := fix_fixed_eq_forall Hmono Hcont
    have Heq1 : fix f = (λ x => fix f x) := by simp
    rw [Heq1]
    conv => lhs; ext; simp [Heq]

  /-! # Making the proofs of validity manageable (and automatable) -/

  -- Monotonicity property for expressions
  def is_mono_p (e : ((x:a) → Result (b x)) → Result c) : Prop :=
    ∀ {{k1 k2}}, karrow_rel k1 k2 → result_rel (e k1) (e k2)

  theorem is_mono_p_same (x : Result c) :
    @is_mono_p a b c (λ _ => x) := by
    simp [is_mono_p, karrow_rel, result_rel]
    split <;> simp

  theorem is_mono_p_rec (x : a) :
    @is_mono_p a b (b x) (λ f => f x) := by
    simp_all [is_mono_p, karrow_rel, result_rel]

  -- The important lemma about `is_mono_p`
  theorem is_mono_p_bind
    (g : ((x:a) → Result (b x)) → Result c)
    (h : c → ((x:a) → Result (b x)) → Result d) :
    is_mono_p g →
    (∀ y, is_mono_p (h y)) →
    @is_mono_p a b d (λ k => @Bind.bind Result _ c d (g k) (fun y => h y k)) := by
--    @is_mono_p a b d (λ k => do let (y : c) ← g k; h y k) := by
    intro hg hh
    simp [is_mono_p]
    intro fg fh Hrgh
    simp [karrow_rel, result_rel]
    have hg := hg Hrgh; simp [result_rel] at hg
    cases heq0: g fg <;> simp_all
    rename_i _ y
    have hh := hh y Hrgh; simp [result_rel] at hh
    simp_all

  -- Continuity property for expressions - note that we take the continuation
  -- as parameter
  def is_cont_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (e : ((x:a) → Result (b x)) → Result c) : Prop :=
    (Hc : ∀ n, e (fix_fuel n k) = .div) →
    e (fix k) = .div

  theorem is_cont_p_same (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (x : Result c) :
    is_cont_p k (λ _ => x) := by
    simp [is_cont_p]

  theorem is_cont_p_rec (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
    is_cont_p f (λ f => f x) := by
    simp_all [is_cont_p, fix]

  -- The important lemma about `is_cont_p`
  theorem is_cont_p_bind
    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (Hkmono : is_mono k)
    (g : ((x:a) → Result (b x)) → Result c)
    (h : c → ((x:a) → Result (b x)) → Result d) :
    is_mono_p g →
    is_cont_p k g →
    (∀ y, is_mono_p (h y)) →
    (∀ y, is_cont_p k (h y)) →
    is_cont_p k (λ k => do let y ← g k; h y k) := by
    intro Hgmono Hgcont Hhmono Hhcont
    simp [is_cont_p]
    intro Hdiv
    -- Case on `g (fix... k)`: is there an n s.t. it terminates?
    cases Classical.em (∀ n, g (fix_fuel n k) = .div) <;> rename_i Hn
    . -- Case 1: g diverges
      have Hgcont := Hgcont Hn
      simp_all
    . -- Case 2: g doesn't diverge
      simp at Hn
      let ⟨ n, Hn ⟩ := Hn
      have Hdivn := Hdiv n
      have Hffmono := fix_fuel_fix_mono Hkmono n
      have Hgeq := Hgmono Hffmono
      simp [result_rel] at Hgeq
      cases Heq: g (fix_fuel n k) <;> rename_i y <;> simp_all
      -- Remains the .ret case
      -- Use Hdiv to prove that: ∀ n, h y (fix_fuel n f) = div
      -- We do this in two steps: first we prove it for m ≥ n
      have Hhdiv: ∀ m, h y (fix_fuel m k) = .div := by
        have Hhdiv : ∀ m, n ≤ m → h y (fix_fuel m k) = .div := by
          -- We use the fact that `g (fix_fuel n f) = .div`, combined with Hdiv
          intro m Hle
          have Hdivm := Hdiv m
          -- Monotonicity of g
          have Hffmono := fix_fuel_mono Hkmono Hle
          have Hgmono := Hgmono Hffmono
          -- We need to clear Hdiv because otherwise simp_all rewrites Hdivm with Hdiv
          clear Hdiv
          simp_all [result_rel]
        intro m
        -- TODO: we shouldn't need the excluded middle here because it is decidable
        cases Classical.em (n ≤ m) <;> rename_i Hl
        . apply Hhdiv; assumption
        . simp at Hl
          -- Make a case disjunction on `h y (fix_fuel m k)`: if it is not equal
          -- to div, use the monotonicity of `h y`
          have Hle : m ≤ n := by linarith
          have Hffmono := fix_fuel_mono Hkmono Hle
          have Hmono := Hhmono y Hffmono
          simp [result_rel] at Hmono
          cases Heq: h y (fix_fuel m k) <;> simp_all
      -- We can now use the continuity hypothesis for h
      apply Hhcont; assumption

  -- The validity property for an expression
  def is_valid_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (e : ((x:a) → Result (b x)) → Result c) : Prop :=
    is_mono_p e ∧
    (is_mono k → is_cont_p k e)

  @[simp] theorem is_valid_p_same
    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : Result c) :
    is_valid_p k (λ _ => x) := by
    simp [is_valid_p, is_mono_p_same, is_cont_p_same]

  @[simp] theorem is_valid_p_rec
    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
    is_valid_p k (λ k => k x) := by
    simp_all [is_valid_p, is_mono_p_rec, is_cont_p_rec]

  theorem is_valid_p_ite
    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (cond : Prop) [h : Decidable cond]
    {e1 e2 : ((x:a) → Result (b x)) → Result c}
    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
    is_valid_p k (ite cond e1 e2) := by
    split <;> assumption

  theorem is_valid_p_dite
    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
    (cond : Prop) [h : Decidable cond]
    {e1 : cond → ((x:a) → Result (b x)) → Result c}
    {e2 : Not cond → ((x:a) → Result (b x)) → Result c}
    (he1: ∀ x, is_valid_p k (e1 x)) (he2 : ∀ x, is_valid_p k (e2 x)) :
    is_valid_p k (dite cond e1 e2) := by
    split <;> simp [*]

  -- Lean is good at unification: we can write a very general version
  -- (in particular, it will manage to figure out `g` and `h` when we
  -- apply the lemma)
  theorem is_valid_p_bind
    {{k : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
    {{g : ((x:a) → Result (b x)) → Result c}}
    {{h : c → ((x:a) → Result (b x)) → 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
    let ⟨ Hgmono, Hgcont ⟩ := Hgvalid
    simp [is_valid_p, forall_and] at Hhvalid
    have ⟨ Hhmono, Hhcont ⟩ := Hhvalid
    simp [← imp_forall_iff] at Hhcont
    simp [is_valid_p]; constructor
    . -- Monotonicity
      apply is_mono_p_bind <;> assumption
    . -- Continuity
      intro Hkmono
      have Hgcont := Hgcont Hkmono
      have Hhcont := Hhcont Hkmono
      apply is_cont_p_bind <;> assumption

  def is_valid (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
    ∀ k x, is_valid_p k (λ k => f k x)

  theorem is_valid_p_imp_is_valid {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
    (Hvalid : is_valid f) :
    is_mono f ∧ is_cont f := by
    have Hmono : is_mono f := by
      intro f h Hr x
      have Hmono := Hvalid (λ _ _ => .div) x
      have Hmono := Hmono.left
      apply Hmono; assumption
    have Hcont : is_cont f := by
      intro x Hdiv
      have Hcont := (Hvalid f x).right Hmono
      simp [is_cont_p] at Hcont
      apply Hcont
      intro n
      have Hdiv := Hdiv n
      simp [fix_fuel] at Hdiv
      simp [*]
    simp [*]

  theorem is_valid_fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
    (Hvalid : is_valid f) :
    fix f = f (fix f) := by
    have ⟨ Hmono, Hcont ⟩ := is_valid_p_imp_is_valid Hvalid
    exact fix_fixed_eq Hmono Hcont

end Fix

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

  -- The index type
  variable {id : Type u}

  -- The input/output types
  variable {a : id → Type v} {b : (i:id) → a i → Type w}

  -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
  def karrow_rel (k1 k2 : (i:id) → (x:a i) → Result (b i x)) : Prop :=
    ∀ i x, result_rel (k1 i x) (k2 i x)

  def kk_to_gen (k : (i:id) → (x:a i) → Result (b i x)) :
    (x: (i:id) × a i) → Result (b x.fst x.snd) :=
    λ ⟨ i, x ⟩ => k i x

  def kk_of_gen (k : (x: (i:id) × a i) → Result (b x.fst x.snd)) :
    (i:id) → (x:a i) → Result (b i x) :=
    λ i x => k ⟨ i, x ⟩

  def k_to_gen (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
    ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd) :=
    λ kk => kk_to_gen (k (kk_of_gen kk))

  def k_of_gen (k : ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd)) :
    ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x) :=
    λ kk => kk_of_gen (k (kk_to_gen kk))

  def e_to_gen (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) :
    ((x: (i:id) × a i) → Result (b x.fst x.snd)) → Result c :=
    λ k => e (kk_of_gen k)

  def is_valid_p (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
    (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) : Prop :=
    Fix.is_valid_p (k_to_gen k) (e_to_gen e)

  def is_valid (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) : Prop :=
    ∀ k i x, is_valid_p k (λ k => f k i x)

  def fix
    (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
    (i:id) → (x:a i) → Result (b i x) :=
    kk_of_gen (Fix.fix (k_to_gen f))

  theorem is_valid_fix_fixed_eq
    {{f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
    (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, x ⟩ := x
      have Hvalid := Hvalid (k_of_gen k) i 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) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
    (i:id) → (x:a i) → Result (b i x)
  abbrev k_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
    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)
  abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
    in_out_ty :=
    Sigma.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) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :
    List in_out_ty.{v, w} → Type (max (u + 1) (max (v + 1) (w + 1))) :=
    | Nil : Funs id a b []
    | Cons {ity : Type v} {oty : ity → Type w} {tys : List in_out_ty}
           (f : kk_ty id a b → (x:ity) → Result (oty x)) (tl : Funs id a b tys) :
           Funs id a b (⟨ ity, oty ⟩ :: tys)

  def get_fun {tys : List in_out_ty} (fl : Funs id a b tys) :
    (i : Fin tys.length) → kk_ty id a b → (x : (tys.get i).fst) →
         Result ((tys.get i).snd x) :=
    match fl with
    | .Nil => λ i => by have h:= i.isLt; simp at h
    | @Funs.Cons id a b 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) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (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) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (i : id) (x : a i) :
    is_valid_p k (λ k => k i 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) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
    (cond : Prop) [h : Decidable cond]
    {e1 e2 : ((i:id) → (x:a i) → Result (b i x)) → 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) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
    (cond : Prop) [h : Decidable cond]
    {e1 : ((i:id) → (x:a i) → Result (b i x)) → cond → Result c}
    {e2 : ((i:id) → (x:a i) → Result (b i x)) → 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) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
    {{g : ((i:id) → (x:a i) → Result (b i x)) → Result c}}
    {{h : c → ((i:id) → (x:a i) → Result (b i x)) → 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 a b)
    (fl : Funs id a b tys) :
    Prop :=
    match fl with
    | .Nil => True
    | .Cons f fl => (∀ x, FixI.is_valid_p k (λ k => f k x)) ∧ fl.is_valid_p k

  theorem Funs.is_valid_p_Nil (k : k_ty id 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 a b}
    {tys : List in_out_ty}
    (fl : Funs id a b tys) (Hvalid : is_valid_p k fl) :
    ∀ (i : Fin tys.length) (x : (tys.get i).fst), FixI.is_valid_p k (fun k => get_fun fl i k 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 x => (List.get tys i).snd x))
    (fl : Funs (Fin tys.length) (λ i => (tys.get i).fst) (λ i x => (tys.get i).snd x) tys) :
    fl.is_valid_p k  →
    ∀ (i : Fin tys.length) (x : (tys.get i).fst),
      FixI.is_valid_p k (fun k => get_fun fl i k x)
    := by
    intro Hvalid
    apply is_valid_p_is_valid_p_aux; simp [*]

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

  variable {a : Type} (k : (List a × Int) → Result a)

  def list_nth_body (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 (tl, i - 1)

  theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
    intro k x
    simp [list_nth_body]
    split <;> try simp
    split <;> try simp

  def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (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 (@list_nth_body_is_valid a)
    simp [list_nth]
    conv => lhs; rw [Heq]

end Ex1

namespace Ex2
  /- Same as Ex1, but we make the body of nth non tail-rec (this is mostly
     to see what happens when there are let-bindings) -/
  open Primitives Fix

  variable {a : Type} (k : (List a × Int) → Result a)

  def list_nth_body (x : (List a × Int)) : Result a :=
    let (ls, i) := x
    match ls with
    | [] => .fail .panic
    | hd :: tl =>
      if i = 0 then .ret hd
      else
        do
          let y ← k (tl, i - 1)
          .ret y

  theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
    intro k x
    simp [list_nth_body]
    split <;> try simp
    split <;> try simp
    apply is_valid_p_bind <;> intros <;> simp_all

  def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (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
           do
             let y ← list_nth tl (i - 1)
             .ret y)
    := by
    have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
    simp [list_nth]
    conv => lhs; rw [Heq]

end Ex2

namespace Ex3
  /- Mutually recursive functions - first encoding (see Ex4 for a better encoding) -/
  open Primitives Fix

  /- Because we have mutually recursive functions, we use a sum for the inputs
     and the output types:
     - inputs: the sum allows to select the function to call in the recursive
       calls (and the functions may not have the same input types)
     - outputs: this case is degenerate because `even` and `odd` have the same
       return type `Bool`, but generally speaking we need a sum type because
       the functions in the mutually recursive group may have different
       return types.
   -/
  variable (k : (Int ⊕ Int) → Result (Bool ⊕ Bool))

  def is_even_is_odd_body (x : (Int ⊕ Int)) : Result (Bool ⊕ Bool) :=
    match x with
    | .inl i =>
      -- Body of `is_even`
      if i = 0
      then .ret (.inl true) -- We use .inl because this is `is_even`
      else
        do
          let b ←
            do
              -- Call `odd`: we need to wrap the input value in `.inr`, then
              -- extract the output value
              let r ← k (.inr (i- 1))
              match r with
              | .inl _ => .fail .panic -- Invalid output
              | .inr b => .ret b
          -- Wrap the return value
          .ret (.inl b)
    | .inr i =>
      -- Body of `is_odd`
      if i = 0
      then .ret (.inr false) -- We use .inr because this is `is_odd`
      else
        do
          let b ←
            do
              -- Call `is_even`: we need to wrap the input value in .inr, then
              -- extract the output value
              let r ← k (.inl (i- 1))
              match r with
              | .inl b => .ret b
              | .inr _ => .fail .panic -- Invalid output
          -- Wrap the return value
          .ret (.inr b)

  theorem is_even_is_odd_body_is_valid:
    ∀ k x, is_valid_p k (λ k => is_even_is_odd_body k x) := by
    intro k x
    simp [is_even_is_odd_body]
    split <;> (try simp) <;> split <;> try simp
    apply is_valid_p_bind; simp
    intros; split <;> simp
    apply is_valid_p_bind; simp
    intros; split <;> simp

  def is_even (i : Int): Result Bool :=
    do
      let r ← fix is_even_is_odd_body (.inl i)
      match r with
      | .inl b => .ret b
      | .inr _ => .fail .panic

  def is_odd (i : Int): Result Bool :=
    do
      let r ← fix is_even_is_odd_body (.inr i)
      match r with
      | .inl _ => .fail .panic
      | .inr b => .ret b

  -- The unfolding equation for `is_even` - diverges if `i < 0`
  theorem is_even_eq (i : Int) :
    is_even i = (if i = 0 then .ret true else is_odd (i - 1))
    := by
    have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
    simp [is_even, is_odd]
    conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
    -- Very annoying: we need to swap the matches
    -- Doing this with rewriting lemmas is hard generally speaking
    -- (especially as we may have to generate lemmas for user-defined
    -- inductives on the fly).
    -- The simplest is to repeatedly split then simplify (we identify
    -- the outer match or monadic let-binding, and split on its scrutinee)
    split <;> try simp
    cases H0 : fix is_even_is_odd_body (Sum.inr (i - 1)) <;> simp
    rename_i v
    split <;> simp

  -- The unfolding equation for `is_odd` - diverges if `i < 0`
  theorem is_odd_eq (i : Int) :
    is_odd i = (if i = 0 then .ret false else is_even (i - 1))
    := by
    have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
    simp [is_even, is_odd]
    conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
    -- Same remark as for `even`
    split <;> try simp
    cases H0 : fix is_even_is_odd_body (Sum.inl (i - 1)) <;> simp
    rename_i v
    split <;> simp

end Ex3

namespace Ex4
  /- Mutually recursive functions - 2nd encoding -/
  open Primitives FixI

  /- We make the input type and output types dependent on a parameter -/
  @[simp] def tys : List in_out_ty := [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)]
  @[simp] def input_ty (i : Fin 2) : Type := (tys.get i).fst
  @[simp] def output_ty (i : Fin 2) (x : input_ty i) : Type :=
    (tys.get i).snd x

  /- The bodies are more natural -/
  def is_even_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
    if i = 0
      then .ret true
    else do
      let b ← k 1 (i - 1)
      .ret b

  def is_odd_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
    if i = 0
      then .ret false
    else do
      let b ← k 0 (i - 1)
      .ret b

  @[simp] def bodies :
    Funs (Fin 2) input_ty output_ty
      [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)] :=
    Funs.Cons (is_even_body) (Funs.Cons (is_odd_body) Funs.Nil)

  def body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 2) :
    (x : input_ty i) → Result (output_ty i x) := 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 [is_even_body, is_odd_body]
    -- Prove the validity of the individual bodies
    . split <;> simp
      apply is_valid_p_bind <;> simp
    . split <;> simp
      apply is_valid_p_bind <;> simp

  theorem body_fix_eq : fix body = body (fix body) :=
    is_valid_fix_fixed_eq body_is_valid

  def is_even (i : Int) : Result Bool := fix body 0 i
  def is_odd (i : Int) : Result Bool := fix body 1 i

  theorem is_even_eq (i : Int) : is_even i =
    (if i = 0
       then .ret true
     else do
       let b ← is_odd (i - 1)
       .ret b) := by
    simp [is_even, is_odd];
    conv => lhs; rw [body_fix_eq]

  theorem is_odd_eq (i : Int) : is_odd i =
    (if i = 0
       then .ret false
     else do
       let b ← is_even (i - 1)
       .ret b) := by
    simp [is_even, is_odd];
    conv => lhs; rw [body_fix_eq]
end Ex4

namespace Ex5
  /- Higher-order example -/
  open Primitives Fix

  variable {a b : Type}

  /- An auxiliary function, which doesn't require the fixed-point -/
  def map (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 theorem for `map`, generic in `f` -/
  theorem map_is_valid
    {{f : (a → Result b) → a → Result c}}
    (Hfvalid : ∀ k x, is_valid_p k (λ k => f k x))
    (k : (a → Result b) → a → Result b)
    (ls : List a) :
    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

  /- An example which uses map -/
  inductive Tree (a : Type) :=
  | leaf (x : a)
  | node (tl : List (Tree a))

  def id_body (k : Tree a → Result (Tree a)) (t : Tree a) : Result (Tree a) :=
    match t with
    | .leaf x => .ret (.leaf x)
    | .node tl =>
      do
        let tl ← map k tl
        .ret (.node tl)

  theorem id_body_is_valid :
    ∀ k x, is_valid_p k (λ k => @id_body a k x) := by
    intro k 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
    apply map_is_valid;
    -- Remark: if we don't do the intro, then the last step is expensive:
    -- "typeclass inference of Nonempty took 119ms"
    intro k x
    simp only [is_valid_p_same, is_valid_p_rec]

  def id (t : Tree a) := fix id_body t

  -- The unfolding equation
  theorem id_eq (t : Tree a) :
    (id t =
       match t with
       | .leaf x => .ret (.leaf x)
       | .node tl =>
       do
         let tl ← map id tl
         .ret (.node tl))
    := by
    have Heq := is_valid_fix_fixed_eq (@id_body_is_valid a)
    simp [id]
    conv => lhs; rw [Heq]; simp; rw [id_body]

end Ex5

namespace Ex6
  /- `list_nth` again, but this time we use FixI -/
  open Primitives FixI

  @[simp] def tys.{u} : List in_out_ty :=
    [mk_in_out_ty ((a:Type u) × (List a × Int)) (λ ⟨ a, _ ⟩ => a)]

  @[simp] def input_ty (i : Fin 1) := (tys.get i).fst
  @[simp] def output_ty (i : Fin 1) (x : input_ty i) :=
    (tys.get i).snd x

  def list_nth_body.{u} (k : (i:Fin 1) → (x:input_ty i) → Result (output_ty i x))
    (x : (a : Type u) × List a × Int) : Result x.fst :=
    let ⟨ a, 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) input_ty output_ty tys :=
    Funs.Cons list_nth_body Funs.Nil

  def body (k : (i : Fin 1) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 1) :
    (x : input_ty i) → Result (output_ty i x) := 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) input_ty output_ty)
    (x : (a : Type u) × List a × Int) : is_valid_p k (fun k => list_nth_body k x) :=
    let ⟨ a, 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 input
    have Heqix := congr_fun Heqi { fst := a, snd := (ls, i) }
    -- Done
    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