Start working on a better encoding of mut rec defs for Diverge

1 files changed, 100 insertions, 2 deletions

diff --git a/backends/lean/Base/Diverge.lean b/backends/lean/Base/Diverge.lean
index 1ff34516..a5cf3459 100644
--- a/backends/lean/Base/Diverge.lean
+++ b/backends/lean/Base/Diverge.lean
@@ -550,7 +550,7 @@ namespace Ex2
 end Ex2
 
 namespace Ex3
-  /- Mutually recursive functions -/
+  /- 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
@@ -671,6 +671,104 @@ namespace Ex3
 end Ex3
 
 namespace Ex4
+  /- Mutually recursive functions - 2nd encoding -/
+  open Primitives Fix
+
+  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))
+
+  /- The bodies are more natural -/
+  def is_even_body (k : (i : Fin 2) → input_ty i → Result (output_ty i)) (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 :=
+    if i = 0
+      then .ret false
+    else do
+      let b ← k 0 (i - 1)
+      .ret b
+
+  inductive Funs : List (Type 0) → List (Type 0) → Type 1 :=
+    | Nil : Funs [] []
+    | Cons {ity oty : Type 0} {itys otys : List (Type 0)} (f : ity → Result oty) (tl : Funs itys otys) : Funs (ity :: itys) (oty :: otys)
+
+  theorem Funs.length_eq {itys otys : List (Type 0)} (fl : Funs itys otys) : itys.length = otys.length :=
+    match fl with
+    | .Nil => by simp
+    | .Cons f tl =>
+      have h:= Funs.length_eq tl
+      by simp [h]
+
+  @[simp] def Funs.cast_fin {itys otys : List (Type 0)} (fl : Funs itys otys) (i : Fin itys.length) : Fin otys.length :=
+    ⟨ i.val, by have h:= fl.length_eq; have h1:= i.isLt; simp_all ⟩
+
+  @[simp] def bodies (k : (i : Fin 2) → input_ty i → Result (output_ty i)) : Funs [Int, Int] [Bool, Bool] :=
+    Funs.Cons (is_even_body k) (Funs.Cons (is_odd_body k) Funs.Nil)
+
+  @[simp] def get_fun {itys otys : List (Type 0)} (fl : Funs itys otys) :
+    (i : Fin itys.length) → itys.get i → Result (otys.get (fl.cast_fin i)) :=
+    match fl with
+    | .Nil => λ i => by have h:= i.isLt; simp at h
+    | @Funs.Cons 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
+          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]) (get_fun tl j)
+
+  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 k) i
+
+  def fix_ {n : Nat} {ity oty : Fin n → Type 0} (f : ((i:Fin n) → ity i → Result (oty i)) → (i:Fin n) → ity i → Result (oty i)) :
+    (i:Fin n) → ity i → Result (oty i) :=
+    sorry
+
+  theorem body_fix_eq : fix_ body = body (fix_ body) := sorry
+
+  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
 
@@ -736,6 +834,6 @@ namespace Ex4
     simp [id]
     conv => lhs; rw [Heq]; simp; rw [id_body]
 
-end Ex4
+end Ex5
 
 end Diverge