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+Require Import Lia.
+Require Coq.Strings.Ascii.
+Require Coq.Strings.String.
+Require Import Coq.Program.Equality.
+Require Import Coq.ZArith.ZArith.
+Require Import Coq.ZArith.Znat.
+Require Import List.
+Import ListNotations.
+
+Module Primitives.
+
+ (* TODO: use more *)
+Declare Scope Primitives_scope.
+
+(*** Result *)
+
+Inductive error :=
+ | Failure
+ | OutOfFuel.
+
+Inductive result A :=
+ | Return : A -> result A
+ | Fail_ : error -> result A.
+
+Arguments Return {_} a.
+Arguments Fail_ {_}.
+
+Definition bind {A B} (m: result A) (f: A -> result B) : result B :=
+ match m with
+ | Fail_ e => Fail_ e
+ | Return x => f x
+ end.
+
+Definition return_ {A: Type} (x: A) : result A := Return x.
+Definition fail_ {A: Type} (e: error) : result A := Fail_ e.
+
+Notation "x <- c1 ; c2" := (bind c1 (fun x => c2))
+ (at level 61, c1 at next level, right associativity).
+
+(** Monadic assert *)
+Definition massert (b: bool) : result unit :=
+ if b then Return tt else Fail_ Failure.
+
+(** Normalize and unwrap a successful result (used for globals) *)
+Definition eval_result_refl {A} {x} (a: result A) (p: a = Return x) : A :=
+ match a as r return (r = Return x -> A) with
+ | Return a' => fun _ => a'
+ | Fail_ e => fun p' =>
+ False_rect _ (eq_ind (Fail_ e)
+ (fun e : result A =>
+ match e with
+ | Return _ => False
+ | Fail_ e => True
+ end)
+ I (Return x) p')
+ end p.
+
+Notation "x %global" := (eval_result_refl x eq_refl) (at level 40).
+Notation "x %return" := (eval_result_refl x eq_refl) (at level 40).
+
+(* Sanity check *)
+Check (if true then Return (1 + 2) else Fail_ Failure)%global = 3.
+
+(*** Misc *)
+
+
+Definition string := Coq.Strings.String.string.
+Definition char := Coq.Strings.Ascii.ascii.
+Definition char_of_byte := Coq.Strings.Ascii.ascii_of_byte.
+
+Definition mem_replace_fwd (a : Type) (x : a) (y : a) : a := x .
+Definition mem_replace_back (a : Type) (x : a) (y : a) : a := y .
+
+(*** Scalars *)
+
+Definition i8_min : Z := -128%Z.
+Definition i8_max : Z := 127%Z.
+Definition i16_min : Z := -32768%Z.
+Definition i16_max : Z := 32767%Z.
+Definition i32_min : Z := -2147483648%Z.
+Definition i32_max : Z := 2147483647%Z.
+Definition i64_min : Z := -9223372036854775808%Z.
+Definition i64_max : Z := 9223372036854775807%Z.
+Definition i128_min : Z := -170141183460469231731687303715884105728%Z.
+Definition i128_max : Z := 170141183460469231731687303715884105727%Z.
+Definition u8_min : Z := 0%Z.
+Definition u8_max : Z := 255%Z.
+Definition u16_min : Z := 0%Z.
+Definition u16_max : Z := 65535%Z.
+Definition u32_min : Z := 0%Z.
+Definition u32_max : Z := 4294967295%Z.
+Definition u64_min : Z := 0%Z.
+Definition u64_max : Z := 18446744073709551615%Z.
+Definition u128_min : Z := 0%Z.
+Definition u128_max : Z := 340282366920938463463374607431768211455%Z.
+
+(** The bounds of [isize] and [usize] vary with the architecture. *)
+Axiom isize_min : Z.
+Axiom isize_max : Z.
+Definition usize_min : Z := 0%Z.
+Axiom usize_max : Z.
+
+Open Scope Z_scope.
+
+(** We provide those lemmas to reason about the bounds of [isize] and [usize] *)
+Axiom isize_min_bound : isize_min <= i32_min.
+Axiom isize_max_bound : i32_max <= isize_max.
+Axiom usize_max_bound : u32_max <= usize_max.
+
+Inductive scalar_ty :=
+ | Isize
+ | I8
+ | I16
+ | I32
+ | I64
+ | I128
+ | Usize
+ | U8
+ | U16
+ | U32
+ | U64
+ | U128
+.
+
+Definition scalar_min (ty: scalar_ty) : Z :=
+ 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
+end.
+
+Definition scalar_max (ty: scalar_ty) : Z :=
+ 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
+end.
+
+(** We use the following conservative bounds to make sure we can compute bound
+ checks in most situations *)
+Definition scalar_min_cons (ty: scalar_ty) : Z :=
+ match ty with
+ | Isize => i32_min
+ | Usize => u32_min
+ | _ => scalar_min ty
+end.
+
+Definition scalar_max_cons (ty: scalar_ty) : Z :=
+ match ty with
+ | Isize => i32_max
+ | Usize => u32_max
+ | _ => scalar_max ty
+end.
+
+Lemma scalar_min_cons_valid : forall ty, scalar_min ty <= scalar_min_cons ty .
+Proof.
+ destruct ty; unfold scalar_min_cons, scalar_min; try lia.
+ - pose isize_min_bound; lia.
+ - apply Z.le_refl.
+Qed.
+
+Lemma scalar_max_cons_valid : forall ty, scalar_max ty >= scalar_max_cons ty .
+Proof.
+ destruct ty; unfold scalar_max_cons, scalar_max; try lia.
+ - pose isize_max_bound; lia.
+ - pose usize_max_bound. lia.
+Qed.
+
+Definition scalar (ty: scalar_ty) : Type :=
+ { x: Z | scalar_min ty <= x <= scalar_max ty }.
+
+Definition to_Z {ty} (x: scalar ty) : Z := proj1_sig x.
+
+(** Bounds checks: we start by using the conservative bounds, to make sure we
+ can compute in most situations, then we use the real bounds (for [isize]
+ and [usize]). *)
+Definition scalar_ge_min (ty: scalar_ty) (x: Z) : bool :=
+ Z.leb (scalar_min_cons ty) x || Z.leb (scalar_min ty) x.
+
+Definition scalar_le_max (ty: scalar_ty) (x: Z) : bool :=
+ Z.leb x (scalar_max_cons ty) || Z.leb x (scalar_max ty).
+
+Lemma scalar_ge_min_valid (ty: scalar_ty) (x: Z) :
+ scalar_ge_min ty x = true -> scalar_min ty <= x .
+Proof.
+ unfold scalar_ge_min.
+ pose (scalar_min_cons_valid ty).
+ lia.
+Qed.
+
+Lemma scalar_le_max_valid (ty: scalar_ty) (x: Z) :
+ scalar_le_max ty x = true -> x <= scalar_max ty .
+Proof.
+ unfold scalar_le_max.
+ pose (scalar_max_cons_valid ty).
+ lia.
+Qed.
+
+Definition scalar_in_bounds (ty: scalar_ty) (x: Z) : bool :=
+ scalar_ge_min ty x && scalar_le_max ty x .
+
+Lemma scalar_in_bounds_valid (ty: scalar_ty) (x: Z) :
+ scalar_in_bounds ty x = true -> scalar_min ty <= x <= scalar_max ty .
+Proof.
+ unfold scalar_in_bounds.
+ intros H.
+ destruct (scalar_ge_min ty x) eqn:Hmin.
+ - destruct (scalar_le_max ty x) eqn:Hmax.
+ + pose (scalar_ge_min_valid ty x Hmin).
+ pose (scalar_le_max_valid ty x Hmax).
+ lia.
+ + inversion H.
+ - inversion H.
+Qed.
+
+Import Sumbool.
+
+Definition mk_scalar (ty: scalar_ty) (x: Z) : result (scalar ty) :=
+ match sumbool_of_bool (scalar_in_bounds ty x) with
+ | left H => Return (exist _ x (scalar_in_bounds_valid _ _ H))
+ | right _ => Fail_ Failure
+ end.
+
+Definition scalar_add {ty} (x y: scalar ty) : result (scalar ty) := mk_scalar ty (to_Z x + to_Z y).
+
+Definition scalar_sub {ty} (x y: scalar ty) : result (scalar ty) := mk_scalar ty (to_Z x - to_Z y).
+
+Definition scalar_mul {ty} (x y: scalar ty) : result (scalar ty) := mk_scalar ty (to_Z x * to_Z y).
+
+Definition scalar_div {ty} (x y: scalar ty) : result (scalar ty) :=
+ if to_Z y =? 0 then Fail_ Failure else
+ mk_scalar ty (to_Z x / to_Z y).
+
+Definition scalar_rem {ty} (x y: scalar ty) : result (scalar ty) := mk_scalar ty (Z.rem (to_Z x) (to_Z y)).
+
+Definition scalar_neg {ty} (x: scalar ty) : result (scalar ty) := mk_scalar ty (-(to_Z x)).
+
+(** Cast an integer from a [src_ty] to a [tgt_ty] *)
+(* TODO: check the semantics of casts in Rust *)
+Definition scalar_cast (src_ty tgt_ty : scalar_ty) (x : scalar src_ty) : result (scalar tgt_ty) :=
+ mk_scalar tgt_ty (to_Z x).
+
+(** Comparisons *)
+Print Z.leb .
+
+Definition scalar_leb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ Z.leb (to_Z x) (to_Z y) .
+
+Definition scalar_ltb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ Z.ltb (to_Z x) (to_Z y) .
+
+Definition scalar_geb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ Z.geb (to_Z x) (to_Z y) .
+
+Definition scalar_gtb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ Z.gtb (to_Z x) (to_Z y) .
+
+Definition scalar_eqb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ Z.eqb (to_Z x) (to_Z y) .
+
+Definition scalar_neqb {ty : scalar_ty} (x : scalar ty) (y : scalar ty) : bool :=
+ negb (Z.eqb (to_Z x) (to_Z y)) .
+
+
+(** The scalar types *)
+Definition isize := scalar Isize.
+Definition i8 := scalar I8.
+Definition i16 := scalar I16.
+Definition i32 := scalar I32.
+Definition i64 := scalar I64.
+Definition i128 := scalar I128.
+Definition usize := scalar Usize.
+Definition u8 := scalar U8.
+Definition u16 := scalar U16.
+Definition u32 := scalar U32.
+Definition u64 := scalar U64.
+Definition u128 := scalar U128.
+
+(** Negaion *)
+Definition isize_neg := @scalar_neg Isize.
+Definition i8_neg := @scalar_neg I8.
+Definition i16_neg := @scalar_neg I16.
+Definition i32_neg := @scalar_neg I32.
+Definition i64_neg := @scalar_neg I64.
+Definition i128_neg := @scalar_neg I128.
+
+(** Division *)
+Definition isize_div := @scalar_div Isize.
+Definition i8_div := @scalar_div I8.
+Definition i16_div := @scalar_div I16.
+Definition i32_div := @scalar_div I32.
+Definition i64_div := @scalar_div I64.
+Definition i128_div := @scalar_div I128.
+Definition usize_div := @scalar_div Usize.
+Definition u8_div := @scalar_div U8.
+Definition u16_div := @scalar_div U16.
+Definition u32_div := @scalar_div U32.
+Definition u64_div := @scalar_div U64.
+Definition u128_div := @scalar_div U128.
+
+(** Remainder *)
+Definition isize_rem := @scalar_rem Isize.
+Definition i8_rem := @scalar_rem I8.
+Definition i16_rem := @scalar_rem I16.
+Definition i32_rem := @scalar_rem I32.
+Definition i64_rem := @scalar_rem I64.
+Definition i128_rem := @scalar_rem I128.
+Definition usize_rem := @scalar_rem Usize.
+Definition u8_rem := @scalar_rem U8.
+Definition u16_rem := @scalar_rem U16.
+Definition u32_rem := @scalar_rem U32.
+Definition u64_rem := @scalar_rem U64.
+Definition u128_rem := @scalar_rem U128.
+
+(** Addition *)
+Definition isize_add := @scalar_add Isize.
+Definition i8_add := @scalar_add I8.
+Definition i16_add := @scalar_add I16.
+Definition i32_add := @scalar_add I32.
+Definition i64_add := @scalar_add I64.
+Definition i128_add := @scalar_add I128.
+Definition usize_add := @scalar_add Usize.
+Definition u8_add := @scalar_add U8.
+Definition u16_add := @scalar_add U16.
+Definition u32_add := @scalar_add U32.
+Definition u64_add := @scalar_add U64.
+Definition u128_add := @scalar_add U128.
+
+(** Substraction *)
+Definition isize_sub := @scalar_sub Isize.
+Definition i8_sub := @scalar_sub I8.
+Definition i16_sub := @scalar_sub I16.
+Definition i32_sub := @scalar_sub I32.
+Definition i64_sub := @scalar_sub I64.
+Definition i128_sub := @scalar_sub I128.
+Definition usize_sub := @scalar_sub Usize.
+Definition u8_sub := @scalar_sub U8.
+Definition u16_sub := @scalar_sub U16.
+Definition u32_sub := @scalar_sub U32.
+Definition u64_sub := @scalar_sub U64.
+Definition u128_sub := @scalar_sub U128.
+
+(** Multiplication *)
+Definition isize_mul := @scalar_mul Isize.
+Definition i8_mul := @scalar_mul I8.
+Definition i16_mul := @scalar_mul I16.
+Definition i32_mul := @scalar_mul I32.
+Definition i64_mul := @scalar_mul I64.
+Definition i128_mul := @scalar_mul I128.
+Definition usize_mul := @scalar_mul Usize.
+Definition u8_mul := @scalar_mul U8.
+Definition u16_mul := @scalar_mul U16.
+Definition u32_mul := @scalar_mul U32.
+Definition u64_mul := @scalar_mul U64.
+Definition u128_mul := @scalar_mul U128.
+
+(** Small utility *)
+Definition usize_to_nat (x: usize) : nat := Z.to_nat (to_Z x).
+
+(** Notations *)
+Notation "x %isize" := ((mk_scalar Isize x)%return) (at level 9).
+Notation "x %i8" := ((mk_scalar I8 x)%return) (at level 9).
+Notation "x %i16" := ((mk_scalar I16 x)%return) (at level 9).
+Notation "x %i32" := ((mk_scalar I32 x)%return) (at level 9).
+Notation "x %i64" := ((mk_scalar I64 x)%return) (at level 9).
+Notation "x %i128" := ((mk_scalar I128 x)%return) (at level 9).
+Notation "x %usize" := ((mk_scalar Usize x)%return) (at level 9).
+Notation "x %u8" := ((mk_scalar U8 x)%return) (at level 9).
+Notation "x %u16" := ((mk_scalar U16 x)%return) (at level 9).
+Notation "x %u32" := ((mk_scalar U32 x)%return) (at level 9).
+Notation "x %u64" := ((mk_scalar U64 x)%return) (at level 9).
+Notation "x %u128" := ((mk_scalar U128 x)%return) (at level 9).
+
+Notation "x s= y" := (scalar_eqb x y) (at level 80) : Primitives_scope.
+Notation "x s<> y" := (scalar_neqb x y) (at level 80) : Primitives_scope.
+Notation "x s<= y" := (scalar_leb x y) (at level 80) : Primitives_scope.
+Notation "x s< y" := (scalar_ltb x y) (at level 80) : Primitives_scope.
+Notation "x s>= y" := (scalar_geb x y) (at level 80) : Primitives_scope.
+Notation "x s> y" := (scalar_gtb x y) (at level 80) : Primitives_scope.
+
+(*** Vectors *)
+
+Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+
+Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
+
+Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+
+Lemma le_0_usize_max : 0 <= usize_max.
+Proof.
+ pose (H := usize_max_bound).
+ unfold u32_max in H.
+ lia.
+Qed.
+
+Definition vec_new (T: Type) : vec T := (exist _ [] le_0_usize_max).
+
+Lemma vec_len_in_usize {T} (v: vec T) : usize_min <= vec_length v <= usize_max.
+Proof.
+ unfold vec_length, usize_min.
+ split.
+ - lia.
+ - apply (proj2_sig v).
+Qed.
+
+Definition vec_len (T: Type) (v: vec T) : usize :=
+ exist _ (vec_length v) (vec_len_in_usize v).
+
+Fixpoint list_update {A} (l: list A) (n: nat) (a: A)
+ : list A :=
+ match l with
+ | [] => []
+ | x :: t => match n with
+ | 0%nat => a :: t
+ | S m => x :: (list_update t m a)
+end end.
+
+Definition vec_bind {A B} (v: vec A) (f: list A -> result (list B)) : result (vec B) :=
+ l <- f (vec_to_list v) ;
+ match sumbool_of_bool (scalar_le_max Usize (Z.of_nat (length l))) with
+ | left H => Return (exist _ l (scalar_le_max_valid _ _ H))
+ | right _ => Fail_ Failure
+ end.
+
+(* The **forward** function shouldn't be used *)
+Definition vec_push_fwd (T: Type) (v: vec T) (x: T) : unit := tt.
+
+Definition vec_push_back (T: Type) (v: vec T) (x: T) : result (vec T) :=
+ vec_bind v (fun l => Return (l ++ [x])).
+
+(* The **forward** function shouldn't be used *)
+Definition vec_insert_fwd (T: Type) (v: vec T) (i: usize) (x: T) : result unit :=
+ if to_Z i <? vec_length v then Return tt else Fail_ Failure.
+
+Definition vec_insert_back (T: Type) (v: vec T) (i: usize) (x: T) : result (vec T) :=
+ vec_bind v (fun l =>
+ if to_Z i <? Z.of_nat (length l)
+ then Return (list_update l (usize_to_nat i) x)
+ else Fail_ Failure).
+
+(* The **backward** function shouldn't be used *)
+Definition vec_index_fwd (T: Type) (v: vec T) (i: usize) : result T :=
+ match nth_error (vec_to_list v) (usize_to_nat i) with
+ | Some n => Return n
+ | None => Fail_ Failure
+ end.
+
+Definition vec_index_back (T: Type) (v: vec T) (i: usize) (x: T) : result unit :=
+ if to_Z i <? vec_length v then Return tt else Fail_ Failure.
+
+(* The **backward** function shouldn't be used *)
+Definition vec_index_mut_fwd (T: Type) (v: vec T) (i: usize) : result T :=
+ match nth_error (vec_to_list v) (usize_to_nat i) with
+ | Some n => Return n
+ | None => Fail_ Failure
+ end.
+
+Definition vec_index_mut_back (T: Type) (v: vec T) (i: usize) (x: T) : result (vec T) :=
+ vec_bind v (fun l =>
+ if to_Z i <? Z.of_nat (length l)
+ then Return (list_update l (usize_to_nat i) x)
+ else Fail_ Failure).
+
+End Primitives.