From 569513587ac168683c40cef03c1e3a74579e6e44 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Sun, 5 Feb 2023 15:15:42 +0100
Subject: Rename primitives.lean to Primitives.lean

---
 backends/lean/Primitives.lean | 373 ++++++++++++++++++++++++++++++++++++++++++
 backends/lean/primitives.lean | 373 ------------------------------------------
 2 files changed, 373 insertions(+), 373 deletions(-)
 create mode 100644 backends/lean/Primitives.lean
 delete mode 100644 backends/lean/primitives.lean

(limited to 'backends')

diff --git a/backends/lean/Primitives.lean b/backends/lean/Primitives.lean
new file mode 100644
index 00000000..79958d94
--- /dev/null
+++ b/backends/lean/Primitives.lean
@@ -0,0 +1,373 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+-- TODO: use syntactic conventions and capitalize error, result, etc.
+
+inductive error where
+   | assertionFailure: error
+   | integerOverflow: error
+   | arrayOutOfBounds: error
+   | maximumSizeExceeded: error
+   | panic: error
+deriving Repr, BEq
+
+open error
+
+inductive result (α : Type u) where
+  | ret (v: α): result α
+  | fail (e: error): result α 
+deriving Repr, BEq
+
+open result
+
+/- HELPERS -/
+
+-- TODO: is there automated syntax for these discriminators?
+def is_ret {α: Type} (r: result α): Bool :=
+  match r with
+  | result.ret _ => true
+  | result.fail _ => false
+
+def massert (b:Bool) : result Unit :=
+  if b then .ret () else fail assertionFailure
+
+def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+  match x with
+  | result.fail _ => by contradiction
+  | result.ret x => x
+
+/- DO-DSL SUPPORT -/
+
+def bind (x: result α) (f: α -> result β) : result β :=
+  match x with
+  | ret v  => f v 
+  | fail v => fail v
+
+-- Allows using result in do-blocks
+instance : Bind result where
+  bind := bind
+
+-- Allows using return x in do-blocks
+instance : Pure result where
+  pure := fun x => ret x
+
+/- CUSTOM-DSL SUPPORT -/
+
+-- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- because we may need a hypothesis for equational reasoning in the scope. We
+-- rely on subtype, and a custom let-binding operator, in effect recreating our
+-- own variant of the do-dsl
+
+def result.attach : (o : result α) → result { x : α // o = ret x }
+  | .ret x => .ret ⟨x, rfl⟩
+  | .fail e   => .fail e
+
+macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+
+-- Silly example of the kind of reasoning that this notation enables
+#eval do
+  let h: y <-- .ret (0: Nat)
+  let _: y = 0 := by cases h; decide
+  let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
+  .ret r
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- NOTE: we reuse the USize type from prelude.lean, because at least we know
+-- it's defined in an idiomatic style that is going to make proofs easy (and
+-- indeed, several proofs here are much shortened compared to Aymeric's earlier
+-- attempt.) This is not stricto sensu the *correct* thing to do, because one
+-- can query at run-time the value of USize, which we do *not* want to do (we
+-- don't know what target we'll run on!), but when the day comes, we'll just
+-- define our own USize.
+-- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
+-- not something we want to define (I think), so we use our own monadic sub (but
+-- is it in line with the Rust behavior?)
+
+-- TODO: I am somewhat under the impression that subtraction is defined as a
+-- total function over nats...? the hypothesis in the if condition is not used
+-- in the then-branch which confuses me quite a bit
+
+-- TODO: add a refinement for the result (just like vec_push_back below) that
+-- explains that the toNat of the result (in the case of success) is the sub of
+-- the toNat of the arguments (i.e. intrinsic specification)
+-- ... do we want intrinsic specifications for the builtins? that might require
+-- some careful type annotations in the monadic notation for clients, but may
+-- give us more "for free"
+
+-- Note from Chris Bailey: "If there's more than one salient property of your
+-- definition then the subtyping strategy might get messy, and the property part
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." Try to settle this with a Lean expert on what is the most
+-- productive way to go about this?
+
+-- One needs to perform a little bit of reasoning in order to successfully
+-- inject constants into USize, so we provide a general-purpose macro
+
+syntax "intlit" : tactic
+
+macro_rules
+  | `(tactic| intlit) => `(tactic|
+    match USize.size, usize_size_eq with
+    | _, Or.inl rfl => decide
+    | _, Or.inr rfl => decide)
+
+-- This is how the macro is expected to be used
+#eval USize.ofNatCore 0 (by intlit)
+
+-- Also works for other integer types (at the expense of a needless disjunction)
+#eval UInt32.ofNatCore 0 (by intlit)
+
+-- Further thoughts: look at what has been done here:
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
+-- and
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
+-- which both contain a fair amount of reasoning already!
+def USize.checked_sub (n: USize) (m: USize): result USize :=
+  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
+  if n >= m then
+    let n' := USize.toNat n
+    let m' := USize.toNat n
+    let r := USize.ofNatCore (n' - m') (by
+      have h: n' - m' <= n' := by
+        apply Nat.sub_le_of_le_add
+        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
+      apply Nat.lt_of_le_of_lt h
+      apply n.val.isLt
+    )
+    return r
+  else
+    fail integerOverflow
+
+def USize.checked_add (n: USize) (m: USize): result USize :=
+  if h: n.val.val + m.val.val <= 4294967295 then
+    .ret ⟨ n.val.val + m.val.val, by
+      have h': 4294967295 < USize.size := by intlit
+      apply Nat.lt_of_le_of_lt h h'
+    ⟩
+  else if h: n.val + m.val < USize.size then
+    .ret ⟨ n.val + m.val, h ⟩
+  else
+    .fail integerOverflow
+
+def USize.checked_rem (n: USize) (m: USize): result USize :=
+  if h: m > 0 then
+    .ret ⟨ n.val % m.val, by
+      have h1: ↑m.val < USize.size := m.val.isLt
+      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
+      apply Nat.lt_trans h2 h1
+    ⟩
+  else
+    .fail integerOverflow
+
+def USize.checked_mul (n: USize) (m: USize): result USize :=
+    if h: n.val.val * m.val.val <= 4294967295 then
+    .ret ⟨ n.val.val * m.val.val, by
+      have h': 4294967295 < USize.size := by intlit
+      apply Nat.lt_of_le_of_lt h h'
+    ⟩
+  else if h: n.val * m.val < USize.size then
+    .ret ⟨ n.val * m.val, h ⟩
+  else
+    .fail integerOverflow
+
+def USize.checked_div (n: USize) (m: USize): result USize :=
+  if m > 0 then
+    .ret ⟨ n.val / m.val, by
+      have h1: ↑n.val < USize.size := n.val.isLt
+      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
+      apply Nat.lt_of_le_of_lt h2 h1
+    ⟩
+  else
+    .fail integerOverflow
+
+class MachineInteger (t: Type) where
+  size: Nat
+  val: t -> Fin size
+  ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+set_option hygiene false in
+run_cmd
+  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+  Lean.Elab.Command.elabCommand (← `(
+    namespace $typeName
+    instance: MachineInteger $typeName where
+      size := size
+      val := val
+      ofNatCore := ofNatCore
+    end $typeName
+  ))
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+  if h: MachineInteger.val x < MachineInteger.size dst then
+    .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
+  else
+    .fail integerOverflow
+
+
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-------------
+-- VECTORS --
+-------------
+
+-- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
+-- rather than maximum values (usize_max).
+def vec (α : Type u) := { l : List α // List.length l < USize.size }
+
+def vec_new (α : Type u): vec α := ⟨ [], by {
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => simp
+  | _, Or.inr rfl => simp
+  } ⟩
+
+#check vec_new
+
+def vec_len (α : Type u) (v : vec α) : USize :=
+  let ⟨ v, l ⟩ := v
+  USize.ofNatCore (List.length v) l
+
+#eval vec_len Nat (vec_new Nat)
+ 
+def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+
+-- NOTE: old version trying to use a subtype notation, but probably better to
+-- leave result elimination to auxiliary lemmas with suitable preconditions
+-- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
+-- make the proof work in that case? Probably need to import tactics from
+-- mathlib to deal with inequalities... would love to see an example.
+def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+  match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
+  :=
+  if h : List.length v.val + 1 < USize.size then
+    ⟨ return ⟨List.concat v.val x,
+      by
+        rw [List.length_concat]
+        assumption
+     ⟩, by simp ⟩
+  else
+    ⟨ fail maximumSizeExceeded, by simp ⟩
+
+#eval do
+  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
+  -- fields val and property. However, Lean's elaborator can automatically
+  -- select the `val` field if the context provides a type annotation. We
+  -- annotate `x`, which relieves us of having to write `.val` on the right-hand
+  -- side of the monadic let.
+  let v := vec_new Nat
+  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  -- TODO: strengthen post-condition above and do a demo to show that we can
+  -- safely eliminate the `fail` case
+  return (vec_len Nat x)
+
+def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+  :=
+  if h : List.length v.val + 1 <= 4294967295 then
+    return ⟨ List.concat v.val x,
+      by
+        rw [List.length_concat]
+        have h': 4294967295 < USize.size := by intlit
+        apply Nat.lt_of_le_of_lt h h'
+    ⟩
+  else if h: List.length v.val + 1 < USize.size then
+    return ⟨List.concat v.val x,
+      by
+        rw [List.length_concat]
+        assumption
+     ⟩
+  else
+    fail maximumSizeExceeded
+
+def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+  if i.val < List.length v.val then
+    .ret ()
+  else
+    .fail arrayOutOfBounds
+
+def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+  if i.val < List.length v.val then
+    .ret ⟨ List.set v.val i.val x, by
+      have h: List.length v.val < USize.size := v.property
+      rewrite [ List.length_set v.val i.val x ]
+      assumption
+    ⟩
+  else
+    .fail arrayOutOfBounds
+
+def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+  if h: i.val < List.length v.val then
+    .ret (List.get v.val ⟨i.val, h⟩)
+  else
+    .fail arrayOutOfBounds
+
+def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+  if i.val < List.length v.val then
+    .ret ()
+  else
+    .fail arrayOutOfBounds
+
+def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+  if h: i.val < List.length v.val then
+    .ret (List.get v.val ⟨i.val, h⟩)
+  else
+    .fail arrayOutOfBounds
+
+def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+  if i.val < List.length v.val then
+    .ret ⟨ List.set v.val i.val x, by
+      have h: List.length v.val < USize.size := v.property
+      rewrite [ List.length_set v.val i.val x ]
+      assumption
+    ⟩
+  else
+    .fail arrayOutOfBounds
+
+----------
+-- MISC --
+----------
+
+def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
+  x
+
+def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
+  y
+
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  logInfo "Reducing and asserting: "
+  logInfo _stx[1]
+  runTermElabM (fun _ => do
+    let e ← Term.elabTerm _stx[1] none
+    logInfo (Expr.dbgToString e)
+    -- How to evaluate the term and compare the result to true?
+    pure ())
+  -- logInfo (Expr.dbgToString (``true))
+  -- throwError "TODO: assert"
+
+#eval 2 == 2
+#assert (2 == 2)
diff --git a/backends/lean/primitives.lean b/backends/lean/primitives.lean
deleted file mode 100644
index 79958d94..00000000
--- a/backends/lean/primitives.lean
+++ /dev/null
@@ -1,373 +0,0 @@
-import Lean
-import Lean.Meta.Tactic.Simp
-import Init.Data.List.Basic
-import Mathlib.Tactic.RunCmd
-
--------------
--- PRELUDE --
--------------
-
--- Results & monadic combinators
-
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
-deriving Repr, BEq
-
-open error
-
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
-deriving Repr, BEq
-
-open result
-
-/- HELPERS -/
-
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
-  match r with
-  | result.ret _ => true
-  | result.fail _ => false
-
-def massert (b:Bool) : result Unit :=
-  if b then .ret () else fail assertionFailure
-
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
-  match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
-
-/- DO-DSL SUPPORT -/
-
-def bind (x: result α) (f: α -> result β) : result β :=
-  match x with
-  | ret v  => f v 
-  | fail v => fail v
-
--- Allows using result in do-blocks
-instance : Bind result where
-  bind := bind
-
--- Allows using return x in do-blocks
-instance : Pure result where
-  pure := fun x => ret x
-
-/- CUSTOM-DSL SUPPORT -/
-
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
--- because we may need a hypothesis for equational reasoning in the scope. We
--- rely on subtype, and a custom let-binding operator, in effect recreating our
--- own variant of the do-dsl
-
-def result.attach : (o : result α) → result { x : α // o = ret x }
-  | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
-
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
-
--- Silly example of the kind of reasoning that this notation enables
-#eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
-  let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
-  .ret r
-
-----------------------
--- MACHINE INTEGERS --
-----------------------
-
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
-
--- Note from Chris Bailey: "If there's more than one salient property of your
--- definition then the subtyping strategy might get messy, and the property part
--- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
-
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
-
-syntax "intlit" : tactic
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- Also works for other integer types (at the expense of a needless disjunction)
-#eval UInt32.ofNatCore 0 (by intlit)
-
--- Further thoughts: look at what has been done here:
--- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
--- and
--- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
--- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
-    .ret ⟨ n.val + m.val, h ⟩
-  else
-    .fail integerOverflow
-
-def USize.checked_rem (n: USize) (m: USize): result USize :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
-
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
-    .ret ⟨ n.val * m.val, h ⟩
-  else
-    .fail integerOverflow
-
-def USize.checked_div (n: USize) (m: USize): result USize :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  else
-    .fail integerOverflow
-
-class MachineInteger (t: Type) where
-  size: Nat
-  val: t -> Fin size
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
-  if h: MachineInteger.val x < MachineInteger.size dst then
-    .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
-  else
-    .fail integerOverflow
-
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
--------------
--- VECTORS --
--------------
-
--- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
--- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
-
-def vec_new (α : Type u): vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
-
-#check vec_new
-
-def vec_len (α : Type u) (v : vec α) : USize :=
-  let ⟨ v, l ⟩ := v
-  USize.ofNatCore (List.length v) l
-
-#eval vec_len Nat (vec_new Nat)
- 
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
-
--- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
--- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
--- make the proof work in that case? Probably need to import tactics from
--- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
-  match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
-  :=
-  if h : List.length v.val + 1 < USize.size then
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- annotate `x`, which relieves us of having to write `.val` on the right-hand
-  -- side of the monadic let.
-  let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
-  -- TODO: strengthen post-condition above and do a demo to show that we can
-  -- safely eliminate the `fail` case
-  return (vec_len Nat x)
-
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
-  :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
-  else
-    fail maximumSizeExceeded
-
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
-  if i.val < List.length v.val then
-    .ret ()
-  else
-    .fail arrayOutOfBounds
-
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
-  if i.val < List.length v.val then
-    .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
-      rewrite [ List.length_set v.val i.val x ]
-      assumption
-    ⟩
-  else
-    .fail arrayOutOfBounds
-
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
-  if h: i.val < List.length v.val then
-    .ret (List.get v.val ⟨i.val, h⟩)
-  else
-    .fail arrayOutOfBounds
-
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
-  if i.val < List.length v.val then
-    .ret ()
-  else
-    .fail arrayOutOfBounds
-
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
-  if h: i.val < List.length v.val then
-    .ret (List.get v.val ⟨i.val, h⟩)
-  else
-    .fail arrayOutOfBounds
-
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
-  if i.val < List.length v.val then
-    .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
-      rewrite [ List.length_set v.val i.val x ]
-      assumption
-    ⟩
-  else
-    .fail arrayOutOfBounds
-
-----------
--- MISC --
-----------
-
-def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
-  x
-
-def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
-  y
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
-  runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
-    pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
-
-#eval 2 == 2
-#assert (2 == 2)
-- 
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