From 3e8060b5501ec83940a4309389a68898df26ebd0 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 17 Jul 2023 23:37:31 +0200
Subject: Reorganize the Lean backend

---
 backends/lean/Base/Primitives/Scalar.lean | 507 ++++++++++++++++++++++++++++++
 1 file changed, 507 insertions(+)
 create mode 100644 backends/lean/Base/Primitives/Scalar.lean

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
new file mode 100644
index 00000000..241dfa07
--- /dev/null
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -0,0 +1,507 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+
+namespace Primitives
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- We redefine our machine integers types.
+
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits`
+-- using the simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at
+-- least officially, 16-bit microcontrollers, so this seems like a fine design decision
+-- for now.)
+
+-- 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." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
+
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
+
+open Result Error
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+
+-- The "structured" bounds
+def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1))
+def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+def I8.smin    : Int := - (HPow.hPow 2 7)
+def I8.smax    : Int := HPow.hPow 2 7 - 1
+def I16.smin   : Int := - (HPow.hPow 2 15)
+def I16.smax   : Int := HPow.hPow 2 15 - 1
+def I32.smin   : Int := -(HPow.hPow 2 31)
+def I32.smax   : Int := HPow.hPow 2 31 - 1
+def I64.smin   : Int := -(HPow.hPow 2 63)
+def I64.smax   : Int := HPow.hPow 2 63 - 1
+def I128.smin  : Int := -(HPow.hPow 2 127)
+def I128.smax  : Int := HPow.hPow 2 127 - 1
+def Usize.smin : Int := 0
+def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1
+def U8.smin    : Int := 0
+def U8.smax    : Int := HPow.hPow 2 8 - 1
+def U16.smin   : Int := 0
+def U16.smax   : Int := HPow.hPow 2 16 - 1
+def U32.smin   : Int := 0
+def U32.smax   : Int := HPow.hPow 2 32 - 1
+def U64.smin   : Int := 0
+def U64.smax   : Int := HPow.hPow 2 64 - 1
+def U128.smin  : Int := 0
+def U128.smax  : Int := HPow.hPow 2 128 - 1
+
+-- The "normalized" bounds, that we use in practice
+def I8.min    := -128
+def I8.max   := 127
+def I16.min  := -32768
+def I16.max  := 32767
+def I32.min  := -2147483648
+def I32.max  := 2147483647
+def I64.min  := -9223372036854775808
+def I64.max  := 9223372036854775807
+def I128.min := -170141183460469231731687303715884105728
+def I128.max := 170141183460469231731687303715884105727
+@[simp] def U8.min   := 0
+def U8.max   := 255
+@[simp] def U16.min  := 0
+def U16.max  := 65535
+@[simp] def U32.min  := 0
+def U32.max  := 4294967295
+@[simp] def U64.min  := 0
+def U64.max  := 18446744073709551615
+@[simp] def U128.min := 0
+def U128.max := 340282366920938463463374607431768211455
+@[simp] def Usize.min := 0
+
+def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
+  ⟨ Isize.smin, by
+    simp [Isize.smin]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } :=
+  ⟨ Isize.smax, by
+    simp [Isize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } :=
+  ⟨ Usize.smax, by
+    simp [Usize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.min := Isize.refined_min.val
+def Isize.max := Isize.refined_max.val
+def Usize.max := Usize.refined_max.val
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.smin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smin
+  | .I8    => I8.smin
+  | .I16   => I16.smin
+  | .I32   => I32.smin
+  | .I64   => I64.smin
+  | .I128  => I128.smin
+  | .Usize => Usize.smin
+  | .U8    => U8.smin
+  | .U16   => U16.smin
+  | .U32   => U32.smin
+  | .U64   => U64.smin
+  | .U128  => U128.smin
+
+def Scalar.smax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smax
+  | .I8    => I8.smax
+  | .I16   => I16.smax
+  | .I32   => I32.smax
+  | .I64   => I64.smax
+  | .I128  => I128.smax
+  | .Usize => Usize.smax
+  | .U8    => U8.smax
+  | .U16   => U16.smax
+  | .U32   => U32.smax
+  | .U64   => U64.smax
+  | .U128  => U128.smax
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by
+  cases ty <;> rfl
+
+def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by
+  cases ty <;> rfl
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Scalar.min .I32
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Scalar.max .I32
+  | .Usize => Scalar.max .U32
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+  have h := Isize.refined_min.property
+  cases h <;> simp [*, Isize.min]
+
+theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+  . have h := Isize.refined_max.property
+    cases h <;> simp [*, Isize.max]
+  . have h := Usize.refined_max.property
+    cases h <;> simp [*, Usize.max]
+
+theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
+  have := Scalar.cMin_bound ty
+  linarith
+
+theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
+  have := Scalar.cMax_bound ty
+  linarith
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty ≤ val
+  hmax : val ≤ Scalar.max ty
+deriving Repr
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
+  :=
+  λ h => by
+  apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty ≤ x) (hmax : x ≤ Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+-- Tactic to prove that integers are in bounds
+-- TODO: use this: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam
+syntax "intlit" : tactic
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices; decide)
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by intlit) : Scalar ty :=
+  -- Remark: we initially wrote:
+  --  let ⟨ hmin, hmax ⟩ := h
+  --  Scalar.ofIntCore x hmin hmax
+  -- We updated to the line below because a similar pattern in `Scalar.tryMk`
+  -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
+  -- TODO: investigate
+  Scalar.ofIntCore x h.left h.right
+
+@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
+  (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
+
+theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
+  simp at *
+  have ⟨ hmin, hmax ⟩ := h
+  have hbmin := Scalar.cMin_bound ty
+  have hbmax := Scalar.cMax_bound ty
+  cases hmin <;> cases hmax <;> apply And.intro <;> linarith
+
+-- 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 Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  if h:Scalar.check_bounds ty x then
+    -- If we do:
+    -- ```
+    -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
+    -- Scalar.ofIntCore x hmin hmax
+    -- ```
+    -- then normalization blocks (for instance, some proofs which use reflexivity fail).
+    -- However, the version below doesn't block reduction (TODO: investigate):
+    return Scalar.ofInt x (Scalar.check_bounds_prop h)
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Our custom remainder operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_rem (x y : Int) : Int :=
+  if 0 ≤ x then |x| % |y|
+  else - (|x| % |y|)
+
+-- Our custom division operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_div (x y : Int) : Int :=
+  if 0 ≤ x && 0 ≤ y then |x| / |y|
+  else if 0 ≤ x && y < 0 then - (|x| / |y|)
+  else if x < 0 && 0 ≤ y then - (|x| / |y|)
+  else |x| / |y|
+
+-- Checking that the remainder operation is correct
+#assert scalar_rem 1 2 = 1
+#assert scalar_rem (-1) 2 = -1
+#assert scalar_rem 1 (-2) = 1
+#assert scalar_rem (-1) (-2) = -1
+#assert scalar_rem 7 3 = (1:Int)
+#assert scalar_rem (-7) 3 = -1
+#assert scalar_rem 7 (-3) = 1
+#assert scalar_rem (-7) (-3) = -1
+
+-- Checking that the division operation is correct
+#assert scalar_div 3 2 = 1
+#assert scalar_div (-3) 2 = -1
+#assert scalar_div 3 (-2) = -1
+#assert scalar_div (-3) (-2) = 1
+#assert scalar_div 7 3 = 2
+#assert scalar_div (-7) 3 = -2
+#assert scalar_div 7 (-3) = -2
+#assert scalar_div (-7) (-3) = 2
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+/- Remark: we can't write the following instance because of restrictions about
+   the type class parameters (`ty` doesn't appear in the return type, which is
+   forbidden):
+
+   ```
+   instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
+   ```
+ -/
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- -- We now define a type class that subsumes the various machine integer types, so
+-- -- as to write a concise definition for scalar_cast, rather than exhaustively
+-- -- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- -- and fails if a cast operation would involve a truncation or modulo.
+
+-- 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
+--   ))
+
+-- -- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- -- Lean to infer `src`.
+
+-- 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
+
+end Primitives
-- 
cgit v1.2.3


From 2fa3cb8ee04dd7ff4184e3e1000fdc025abc50a4 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 17 Jul 2023 23:37:48 +0200
Subject: Start proving theorems for primitive definitions

---
 backends/lean/Base/Primitives/Scalar.lean | 1 +
 1 file changed, 1 insertion(+)

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index 241dfa07..3f88caa2 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -2,6 +2,7 @@ import Lean
 import Lean.Meta.Tactic.Simp
 import Mathlib.Tactic.Linarith
 import Base.Primitives.Base
+import Base.Diverge.Base
 
 namespace Primitives
 
-- 
cgit v1.2.3


From 3df0b36891975935c3d8035f56389ee6bbcbf251 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Wed, 19 Jul 2023 18:13:31 +0200
Subject: Add arithmetic spec lemmas

---
 backends/lean/Base/Primitives/Scalar.lean | 167 ++++++++++++++++++++++++++++--
 1 file changed, 161 insertions(+), 6 deletions(-)

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index 3f88caa2..aaa4027f 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -3,6 +3,8 @@ import Lean.Meta.Tactic.Simp
 import Mathlib.Tactic.Linarith
 import Base.Primitives.Base
 import Base.Diverge.Base
+import Base.Progress.Base
+import Base.Arith.Int
 
 namespace Primitives
 
@@ -122,6 +124,22 @@ inductive ScalarTy :=
 | U64
 | U128
 
+def ScalarTy.isSigned (ty : ScalarTy) : Bool :=
+  match ty with
+  | Isize
+  | I8
+  | I16
+  | I32
+  | I64
+  | I128 => true
+  | Usize
+  | U8
+  | U16
+  | U32
+  | U64
+  | U128 => false
+
+
 def Scalar.smin (ty : ScalarTy) : Int :=
   match ty with
   | .Isize => Isize.smin
@@ -289,23 +307,30 @@ def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
 
 def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
 
-def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
-  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
-
 -- Our custom remainder operation, which satisfies the semantics of Rust
 -- TODO: is there a better way?
 def scalar_rem (x y : Int) : Int :=
-  if 0 ≤ x then |x| % |y|
+  if 0 ≤ x then x % y
   else - (|x| % |y|)
 
+@[simp]
+def scalar_rem_nonneg {x y : Int} (hx : 0 ≤ x) : scalar_rem x y = x % y := by
+  intros
+  simp [*, scalar_rem]
+
 -- Our custom division operation, which satisfies the semantics of Rust
 -- TODO: is there a better way?
 def scalar_div (x y : Int) : Int :=
-  if 0 ≤ x && 0 ≤ y then |x| / |y|
+  if 0 ≤ x && 0 ≤ y then x / y
   else if 0 ≤ x && y < 0 then - (|x| / |y|)
   else if x < 0 && 0 ≤ y then - (|x| / |y|)
   else |x| / |y|
 
+@[simp]
+def scalar_div_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : scalar_div x y = x / y := by
+  intros
+  simp [*, scalar_div]
+
 -- Checking that the remainder operation is correct
 #assert scalar_rem 1 2 = 1
 #assert scalar_rem (-1) 2 = -1
@@ -326,8 +351,11 @@ def scalar_div (x y : Int) : Int :=
 #assert scalar_div 7 (-3) = -2
 #assert scalar_div (-7) (-3) = 2
 
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (scalar_div x.val y.val) else fail divisionByZero
+
 def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
-  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+  if y.val != 0 then Scalar.tryMk ty (scalar_rem x.val y.val) else fail divisionByZero
 
 def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
   Scalar.tryMk ty (x.val + y.val)
@@ -410,6 +438,133 @@ instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
 instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
   hMod x y := Scalar.rem x y
 
+-- TODO: make progress work at a more fine grained level (see `Scalar.add_unsigned_spec`)
+@[cpspec]
+theorem Scalar.add_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val + y.val)
+  (hmax : x.val + y.val ≤ Scalar.max ty) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  simp [HAdd.hAdd, add, Add.add]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmax : x.val + y.val ≤ Scalar.max ty) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  have hmin : Scalar.min ty ≤ x.val + y.val := by
+    have hx := x.hmin
+    have hy := y.hmin
+    cases ty <;> simp [min] at * <;> linarith
+  apply add_spec <;> assumption
+
+-- TODO: make it finer grained
+@[cpspec]
+theorem Scalar.sub_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val - y.val)
+  (hmax : x.val - y.val ≤ Scalar.max ty) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  simp [HSub.hSub, sub, Sub.sub]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  have : x.val - y.val ≤ Scalar.max ty := by
+    have hx := x.hmin
+    have hxm := x.hmax
+    have hy := y.hmin
+    cases ty <;> simp [min, max] at * <;> linarith
+  intros
+  apply sub_spec <;> assumption
+
+-- TODO: make it finer grained
+@[cpspec]
+theorem Scalar.mul_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val * y.val)
+  (hmax : x.val * y.val ≤ Scalar.max ty) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  simp [HMul.hMul, mul, Mul.mul]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmax : x.val * y.val ≤ Scalar.max ty) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  have : Scalar.min ty ≤ x.val * y.val := by
+    have hx := x.hmin
+    have hy := y.hmin
+    cases ty <;> simp at * <;> apply mul_nonneg hx hy
+  apply mul_spec <;> assumption
+
+-- TODO: make it finer grained
+@[cpspec]
+theorem Scalar.div_spec {ty} {x y : Scalar ty}
+  (hnz : y.val ≠ 0)
+  (hmin : Scalar.min ty ≤ scalar_div x.val y.val)
+  (hmax : scalar_div x.val y.val ≤ Scalar.max ty) :
+  ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := by
+  simp [HDiv.hDiv, div, Div.div]
+  simp [tryMk, *]
+  simp [pure]
+  rfl
+
+theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+  (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+  have hx := x.hmin
+  have hy := y.hmin
+  simp [h] at hx hy
+  have hmin : 0 ≤ x.val / y.val := Int.ediv_nonneg hx hy
+  have hmax : x.val / y.val ≤ Scalar.max ty := by
+    have := Int.ediv_le_self y.val hx
+    have := x.hmax
+    linarith
+  have hs := @div_spec ty x y hnz
+  simp [*] at hs
+  apply hs
+
+-- TODO: make it finer grained
+@[cpspec]
+theorem Scalar.rem_spec {ty} {x y : Scalar ty}
+  (hnz : y.val ≠ 0)
+  (hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
+  (hmax : scalar_rem x.val y.val ≤ Scalar.max ty) :
+  ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
+  simp [HMod.hMod, rem]
+  simp [tryMk, *]
+  simp [pure]
+  rfl
+
+theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+  (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
+  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+  have hx := x.hmin
+  have hy := y.hmin
+  simp [h] at hx hy
+  have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz
+  have hmax : x.val % y.val ≤ Scalar.max ty := by
+    have h := @Int.ediv_emod_unique x.val y.val (x.val % y.val) (x.val / y.val)
+    simp at h
+    have : 0 < y.val := by int_tac
+    simp [*] at h
+    have := y.hmax
+    linarith
+  have hs := @rem_spec ty x y hnz
+  simp [*] at hs
+  simp [*]
+
 -- ofIntCore
 -- TODO: typeclass?
 def Isize.ofIntCore := @Scalar.ofIntCore .Isize
-- 
cgit v1.2.3


From d87e35e1a53b2252cc2c8c554216115773fd9678 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 20 Jul 2023 11:38:55 +0200
Subject: Add fine-grained lemmas for the arithmetic operations

---
 backends/lean/Base/Primitives/Scalar.lean | 137 ++++++++++++++++++++++++++++--
 1 file changed, 130 insertions(+), 7 deletions(-)

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index aaa4027f..1e9b51c2 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -438,7 +438,7 @@ instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
 instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
   hMod x y := Scalar.rem x y
 
--- TODO: make progress work at a more fine grained level (see `Scalar.add_unsigned_spec`)
+-- Generic theorem - shouldn't be used much
 @[cpspec]
 theorem Scalar.add_spec {ty} {x y : Scalar ty}
   (hmin : Scalar.min ty ≤ x.val + y.val)
@@ -460,7 +460,32 @@ theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
     cases ty <;> simp [min] at * <;> linarith
   apply add_spec <;> assumption
 
--- TODO: make it finer grained
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
 @[cpspec]
 theorem Scalar.sub_spec {ty} {x y : Scalar ty}
   (hmin : Scalar.min ty ≤ x.val - y.val)
@@ -484,8 +509,32 @@ theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
   intros
   apply sub_spec <;> assumption
 
--- TODO: make it finer grained
-@[cpspec]
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+-- Generic theorem - shouldn't be used much
 theorem Scalar.mul_spec {ty} {x y : Scalar ty}
   (hmin : Scalar.min ty ≤ x.val * y.val)
   (hmax : x.val * y.val ≤ Scalar.max ty) :
@@ -506,7 +555,32 @@ theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
     cases ty <;> simp at * <;> apply mul_nonneg hx hy
   apply mul_spec <;> assumption
 
--- TODO: make it finer grained
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
 @[cpspec]
 theorem Scalar.div_spec {ty} {x y : Scalar ty}
   (hnz : y.val ≠ 0)
@@ -534,7 +608,32 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   simp [*] at hs
   apply hs
 
--- TODO: make it finer grained
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
 @[cpspec]
 theorem Scalar.rem_spec {ty} {x y : Scalar ty}
   (hnz : y.val ≠ 0)
@@ -548,7 +647,7 @@ theorem Scalar.rem_spec {ty} {x y : Scalar ty}
 
 theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
   (hnz : y.val ≠ 0) :
-  ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
   have h : Scalar.min ty = 0 := by cases ty <;> simp at *
   have hx := x.hmin
   have hy := y.hmin
@@ -565,6 +664,30 @@ theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   simp [*] at hs
   simp [*]
 
+@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
 -- ofIntCore
 -- TODO: typeclass?
 def Isize.ofIntCore := @Scalar.ofIntCore .Isize
-- 
cgit v1.2.3


From 1854c631a6a7a3f8d45ad18e05547f9d3782c3ee Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Tue, 25 Jul 2023 16:26:08 +0200
Subject: Make progress on the hashmap properties

---
 backends/lean/Base/Primitives/Scalar.lean | 48 +++++++++++++++++--------------
 1 file changed, 27 insertions(+), 21 deletions(-)

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index 1e9b51c2..3beb7527 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -66,27 +66,33 @@ def U128.smin  : Int := 0
 def U128.smax  : Int := HPow.hPow 2 128 - 1
 
 -- The "normalized" bounds, that we use in practice
-def I8.min    := -128
-def I8.max   := 127
-def I16.min  := -32768
-def I16.max  := 32767
-def I32.min  := -2147483648
-def I32.max  := 2147483647
-def I64.min  := -9223372036854775808
-def I64.max  := 9223372036854775807
-def I128.min := -170141183460469231731687303715884105728
-def I128.max := 170141183460469231731687303715884105727
-@[simp] def U8.min   := 0
-def U8.max   := 255
-@[simp] def U16.min  := 0
-def U16.max  := 65535
-@[simp] def U32.min  := 0
-def U32.max  := 4294967295
-@[simp] def U64.min  := 0
-def U64.max  := 18446744073709551615
-@[simp] def U128.min := 0
-def U128.max := 340282366920938463463374607431768211455
-@[simp] def Usize.min := 0
+def I8.min    : Int   := -128
+def I8.max    : Int   := 127
+def I16.min   : Int  := -32768
+def I16.max   : Int  := 32767
+def I32.min   : Int  := -2147483648
+def I32.max   : Int  := 2147483647
+def I64.min   : Int  := -9223372036854775808
+def I64.max   : Int  := 9223372036854775807
+def I128.min  : Int := -170141183460469231731687303715884105728
+def I128.max  : Int := 170141183460469231731687303715884105727
+@[simp]
+def U8.min    : Int   := 0
+def U8.max    : Int   := 255
+@[simp]
+def U16.min   : Int  := 0
+def U16.max   : Int  := 65535
+@[simp]
+def U32.min   : Int  := 0
+def U32.max   : Int  := 4294967295
+@[simp]
+def U64.min   : Int  := 0
+def U64.max   : Int  := 18446744073709551615
+@[simp]
+def U128.min  : Int := 0
+def U128.max  : Int := 340282366920938463463374607431768211455
+@[simp]
+def Usize.min : Int := 0
 
 def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
   ⟨ Isize.smin, by
-- 
cgit v1.2.3


From 9e8fccbe4b667fc341b6544030f85af05fe89307 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Tue, 25 Jul 2023 20:12:48 +0200
Subject: Make progress on the proofs of the hashmap

---
 backends/lean/Base/Primitives/Scalar.lean | 47 ++++++++++++++++++++++++++++---
 1 file changed, 43 insertions(+), 4 deletions(-)

(limited to 'backends/lean/Base/Primitives/Scalar.lean')

diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index 3beb7527..2e5be8bf 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -660,10 +660,8 @@ theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
   simp [h] at hx hy
   have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz
   have hmax : x.val % y.val ≤ Scalar.max ty := by
-    have h := @Int.ediv_emod_unique x.val y.val (x.val % y.val) (x.val / y.val)
-    simp at h
-    have : 0 < y.val := by int_tac
-    simp [*] at h
+    have h : 0 < y.val := by int_tac
+    have h := Int.emod_lt_of_pos x.val h
     have := y.hmax
     linarith
   have hs := @rem_spec ty x y hnz
@@ -724,6 +722,47 @@ def U32.ofInt   := @Scalar.ofInt .U32
 def U64.ofInt   := @Scalar.ofInt .U64
 def U128.ofInt  := @Scalar.ofInt .U128
 
+-- TODO: factor those lemmas out
+@[simp] theorem Scalar.ofInt_val_eq {ty} (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : (Scalar.ofInt x h).val = x := by
+  simp [Scalar.ofInt, Scalar.ofIntCore]
+
+@[simp] theorem Isize.ofInt_val_eq (h : Scalar.min ScalarTy.Isize ≤ x ∧ x ≤ Scalar.max ScalarTy.Isize) : (Isize.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I8.ofInt_val_eq (h : Scalar.min ScalarTy.I8 ≤ x ∧ x ≤ Scalar.max ScalarTy.I8) : (I8.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I16.ofInt_val_eq (h : Scalar.min ScalarTy.I16 ≤ x ∧ x ≤ Scalar.max ScalarTy.I16) : (I16.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I32.ofInt_val_eq (h : Scalar.min ScalarTy.I32 ≤ x ∧ x ≤ Scalar.max ScalarTy.I32) : (I32.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I64.ofInt_val_eq (h : Scalar.min ScalarTy.I64 ≤ x ∧ x ≤ Scalar.max ScalarTy.I64) : (I64.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I128.ofInt_val_eq (h : Scalar.min ScalarTy.I128 ≤ x ∧ x ≤ Scalar.max ScalarTy.I128) : (I128.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem Usize.ofInt_val_eq (h : Scalar.min ScalarTy.Usize ≤ x ∧ x ≤ Scalar.max ScalarTy.Usize) : (Usize.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U8.ofInt_val_eq (h : Scalar.min ScalarTy.U8 ≤ x ∧ x ≤ Scalar.max ScalarTy.U8) : (U8.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U16.ofInt_val_eq (h : Scalar.min ScalarTy.U16 ≤ x ∧ x ≤ Scalar.max ScalarTy.U16) : (U16.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U32.ofInt_val_eq (h : Scalar.min ScalarTy.U32 ≤ x ∧ x ≤ Scalar.max ScalarTy.U32) : (U32.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U64.ofInt_val_eq (h : Scalar.min ScalarTy.U64 ≤ x ∧ x ≤ Scalar.max ScalarTy.U64) : (U64.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U128.ofInt_val_eq (h : Scalar.min ScalarTy.U128 ≤ x ∧ x ≤ Scalar.max ScalarTy.U128) : (U128.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+
 -- Comparisons
 instance {ty} : LT (Scalar ty) where
   lt a b := LT.lt a.val b.val
-- 
cgit v1.2.3