From 0cee49de70bec6d3ec2221b64a532d19ad71e5e0 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 29 Jun 2023 14:51:53 +0200
Subject: Generalize a bit FixI and add an example

---
 backends/lean/Base/Diverge/Base.lean | 260 ++++++++++++++++++++---------------
 1 file changed, 151 insertions(+), 109 deletions(-)

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 2e60f6e8..630c0bf6 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -57,7 +57,7 @@ deriving Repr, BEq
 
 open Result
 
-def bind (x: Result α) (f: α -> Result β) : Result β :=
+def bind {α : Type u} {β : Type v} (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
@@ -84,7 +84,7 @@ instance : Pure Result where
 @[simp] theorem bind_tc_div (f : α → Result β) :
   (do let y ← div; f y) = div := by simp [Bind.bind, bind]
 
-def div? {α: Type} (r: Result α): Bool :=
+def div? {α: Type u} (r: Result α): Bool :=
   match r with
   | div => true
   | ret _ | fail _ => false
@@ -96,8 +96,8 @@ namespace Fix
   open Primitives
   open Result
 
-  variable {a : Type} {b : a → Type}
-  variable {c d : Type}
+  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 -/
 
@@ -334,7 +334,8 @@ namespace Fix
     (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 => do let y ← g k; h y k) := by
+    @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
@@ -494,49 +495,49 @@ namespace FixI
   open Primitives Fix
 
   -- The index type
-  variable {id : Type}
+  variable {id : Type u}
 
   -- The input/output types
-  variable {a b : id → Type}
+  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) → a i → Result (b i)) : Prop :=
+  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) → a i → Result (b i)) :
-    (x: (i:id) × a i) → Result (b x.fst) :=
+  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)) :
-    (i:id) → a i → Result (b i) :=
+  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) → a i → Result (b i)) → (i:id) → a i → Result (b i)) :
-    ((x: (i:id) × a i) → Result (b x.fst)) → (x: (i:id) × a i) → Result (b x.fst) :=
+  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: (i:id) × a i) → Result (b x.fst)) :
-    ((i:id) → a i → Result (b i)) → (i:id) → a i → Result (b i) :=
+  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) → a i → Result (b i)) → Result c) :
-    ((x: (i:id) × a i) → Result (b x.fst)) → Result c :=
+  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) → a i → Result (b i)) → (i:id) → a i → Result (b i))
-    (e : ((i:id) → a i → Result (b i)) → Result c) : Prop :=
+  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) → a i → Result (b i)) → (i:id) → a i → Result (b i)) : Prop :=
+  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)
 
   noncomputable def fix
-    (f : ((i:id) → a i → Result (b i)) → (i:id) → a i → Result (b i)) :
-    (i:id) → a i → Result (b i) :=
+    (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) → a i → Result (b i)) → (i:id) → a i → Result (b i)}}
+    {{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
@@ -553,57 +554,43 @@ namespace FixI
   /- Some utilities to define the mutually recursive functions -/
 
   -- TODO: use more
-  @[simp] def kk_ty (id : Type) (a b : id → Type) := (i:id) → a i → Result (b i)
-  @[simp] def k_ty (id : Type) (a b : id → Type) := kk_ty id a b → kk_ty id a b
+  @[simp] def 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)
+  @[simp] def 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
+
+  def in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
+  @[simp] def 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) (a b : id → Type) :
-    List (Type u) → List (Type u) → Type (u + 1) :=
-    | Nil : Funs id a b [] []
-    | Cons {ity oty : Type u} {itys otys : List (Type u)}
-           (f : kk_ty id a b → ity → Result oty) (tl : Funs id a b itys otys) :
-           Funs id a b (ity :: itys) (oty :: otys)
-
-  theorem Funs.length_eq {itys otys : List (Type)} (fl : Funs id a b itys otys) :
-    otys.length = itys.length :=
-    match fl with
-    | .Nil => by simp
-    | .Cons f tl =>
-      have h:= Funs.length_eq tl
-      by simp [h]
-
-  def fin_cast {n m : Nat} (h : m = n) (i : Fin n) : Fin m :=
-    ⟨ i.val, by have h1:= i.isLt; simp_all ⟩
-
-  @[simp] def Funs.cast_fin {itys otys : List (Type)}
-    (fl : Funs id a b itys otys) (i : Fin itys.length) : Fin otys.length :=
-    fin_cast (fl.length_eq) i
-
-  def get_fun {itys otys : List (Type)} (fl : Funs id a b itys otys) :
-    (i : Fin itys.length) → kk_ty id a b → itys.get i → Result (otys.get (fl.cast_fin i)) :=
+  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 itys1 otys1 f tl =>
-      λ i =>
-      if h: i.val = 0 then
-        Eq.mp (by cases i; simp_all [List.get]) f
-      else
-        let j := i.val - 1
-        have Hj: j < itys1.length := by
-          have Hi := i.isLt
-          simp at Hi
-          revert Hi
-          cases Heq: i.val <;> simp_all
+    | @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 itys1.length := ⟨ j, Hj ⟩
-        Eq.mp
-          (by
-            cases Heq: i; rename_i val isLt;
-            cases Heq': j; rename_i val' isLt;
-            cases val <;> simp_all [List.get, fin_cast])
-          (get_fun tl j)
+        let j: Fin tys1.length := ⟨ j, jLt ⟩
+        Eq.mp (by simp) (get_fun tl j)
 
   -- TODO: move
   theorem add_one_le_iff_le_ne (n m : Nat) (h1 : m ≤ n) (h2 : m ≠ n) : m + 1 ≤ n := by
@@ -683,19 +670,19 @@ namespace FixI
 
   /- Automating the proofs -/
   @[simp] theorem is_valid_p_same
-    (k : ((i:id) → a i → Result (b i)) → (i:id) → a i → Result (b i)) (x : Result c) :
+    (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) → a i → Result (b i)) → (i:id) → a i → Result (b i)) (i : id) (x : a i) :
+     (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_bind
-    {{k : ((i:id) → a i → Result (b i)) → (i:id) → a i → Result (b i)}}
-    {{g : ((i:id) → a i → Result (b i)) → Result c}}
-    {{h : c → ((i:id) → a i → Result (b i)) → Result d}}
+    {{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
@@ -705,7 +692,7 @@ namespace FixI
 
   def Funs.is_valid_p
     (k : k_ty id a b)
-    (fl : Funs id a b itys otys) :
+    (fl : Funs id a b tys) :
     Prop :=
     match fl with
     | .Nil => True
@@ -713,31 +700,29 @@ namespace FixI
 
   def Funs.is_valid_p_is_valid_p_aux
     {k : k_ty id a b}
-    {itys otys : List Type}
-    (Heq : List.length otys = List.length itys)
-    (fl : Funs id a b itys otys) (Hvalid : is_valid_p k fl) :
-    ∀ (i : Fin itys.length) (x : itys.get i), FixI.is_valid_p k (fun k => get_fun fl i k x) := by
+    {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 // itys.length = n } := ⟨ itys.length, by rfl ⟩
-    revert itys otys Heq fl Hvalid
+    have ⟨ n, Hn ⟩ : { n : Nat // tys.length = n } := ⟨ tys.length, by rfl ⟩
+    revert tys fl Hvalid
     induction n
     --
     case zero =>
-      intro itys otys Heq fl Hvalid Hlen;
-      have Heq: itys = [] := by cases itys <;> simp_all
-      have Heq: otys = [] := by cases otys <;> simp_all
+      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 itys otys Heq fl Hvalid Hlen i x;
-      cases fl <;> simp at Hlen i x Heq Hvalid
-      rename_i ity oty itys otys f fl
+      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 itys otys (by simp[*]) fl Hvf1 (by simp [*])
+      have Hn := @Hn tys fl Hvf1 (by simp [*])
       -- Case disjunction on i
       match i with
       | ⟨ 0, _ ⟩ =>
@@ -747,19 +732,20 @@ namespace FixI
       | ⟨ .succ j, HiLt ⟩ =>
         simp_arith at HiLt
         simp at x
-        let j : Fin (List.length itys) := ⟨ j, by simp_arith [HiLt] ⟩
+        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
-    (itys otys : List (Type)) (Heq: otys.length = itys.length := by decide)
-    (k : k_ty (Fin (List.length itys)) (List.get itys) fun i => List.get otys (fin_cast Heq i))
-    (fl : Funs (Fin itys.length) itys.get (λ i => otys.get (fin_cast Heq i)) itys otys) :
+    (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 itys.length) (x : itys.get i), FixI.is_valid_p k (fun k => get_fun fl i k x)
+    ∀ (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 [*]
+    apply is_valid_p_is_valid_p_aux; simp [*]
 
 end FixI
 
@@ -960,27 +946,21 @@ namespace Ex4
   /- Mutually recursive functions - 2nd encoding -/
   open Primitives FixI
 
-  attribute [local simp] List.get
-
   /- We make the input type and output types dependent on a parameter -/
-  @[simp] def input_ty (i : Fin 2) : Type :=
-    [Int, Int].get i
-
-  @[simp] def output_ty (i : Fin 2) : Type :=
-    [Bool, Bool].get i
-
-  /- The continuation -/
-  variable (k : (i : Fin 2) → input_ty i → Result (output_ty i))
+  @[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) → input_ty i → Result (output_ty i)) (i : Int) : Result Bool :=
+  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 (i : Int) : Result Bool :=
+  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
@@ -988,18 +968,19 @@ namespace Ex4
       .ret b
 
   @[simp] def bodies :
-    Funs (Fin 2) input_ty output_ty [Int, Int] [Bool, Bool] :=
+    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) → input_ty i → Result (output_ty i)) (i: Fin 2) :
-    input_ty i → Result (output_ty i) := get_fun bodies i k
+  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 [Int, Int] [Bool, Bool])
+    apply (Funs.is_valid_p_is_valid_p tys)
     simp [Funs.is_valid_p]
     (repeat (apply And.intro)) <;> intro x <;> simp at x <;>
     simp only [is_even_body, is_odd_body]
@@ -1106,3 +1087,64 @@ namespace Ex5
     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 list_nth_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; simp at x
+    simp only [list_nth_body]
+    -- Prove the validity of the individual bodies
+    intro k x
+    simp [list_nth_body]
+    split <;> simp
+    split <;> simp
+
+  noncomputable
+  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 (@list_nth_body_is_valid a)
+    simp [list_nth]
+    conv => lhs; rw [Heq]
+
+end Ex6
-- 
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