path: root/tests/coq/betree/Primitives.v

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 core_mem_replace (a : Type) (x : a) (y : a) : a := x .
Definition core_mem_replace_back (a : Type) (x : a) (y : a) : a := y .

Record mut_raw_ptr (T : Type) := { mut_raw_ptr_v : T }.
Record const_raw_ptr (T : Type) := { const_raw_ptr_v : T }.

(*** 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 *)
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.

(** Constants *)
Definition core_u8_max    := u8_max %u32.
Definition core_u16_max   := u16_max %u32.
Definition core_u32_max   := u32_max %u32.
Definition core_u64_max   := u64_max %u64.
Definition core_u128_max  := u64_max %u128.
Axiom core_usize_max : usize. (** TODO *)
Definition core_i8_max    := i8_max %i32.
Definition core_i16_max   := i16_max %i32.
Definition core_i32_max   := i32_max %i32.
Definition core_i64_max   := i64_max %i64.
Definition core_i128_max  := i64_max %i128.
Axiom core_isize_max : isize. (** TODO *)

(*** core::ops *)

(* Trait declaration: [core::ops::index::Index] *)
Record core_ops_index_Index (Self Idx : Type) := mk_core_ops_index_Index {
  core_ops_index_Index_Output : Type;
  core_ops_index_Index_index : Self -> Idx -> result core_ops_index_Index_Output;
}.
Arguments mk_core_ops_index_Index {_ _}.
Arguments core_ops_index_Index_Output {_ _}.
Arguments core_ops_index_Index_index {_ _}.

(* Trait declaration: [core::ops::index::IndexMut] *)
Record core_ops_index_IndexMut (Self Idx : Type) := mk_core_ops_index_IndexMut {
  core_ops_index_IndexMut_indexInst : core_ops_index_Index Self Idx;
  core_ops_index_IndexMut_index_mut : Self -> Idx -> result core_ops_index_IndexMut_indexInst.(core_ops_index_Index_Output);
  core_ops_index_IndexMut_index_mut_back : Self -> Idx -> core_ops_index_IndexMut_indexInst.(core_ops_index_Index_Output) -> result Self;
}.
Arguments mk_core_ops_index_IndexMut {_ _}.
Arguments core_ops_index_IndexMut_indexInst {_ _}.
Arguments core_ops_index_IndexMut_index_mut {_ _}.
Arguments core_ops_index_IndexMut_index_mut_back {_ _}.

(* Trait declaration [core::ops::deref::Deref] *)
Record core_ops_deref_Deref (Self : Type) := mk_core_ops_deref_Deref {
  core_ops_deref_Deref_target : Type;
  core_ops_deref_Deref_deref : Self -> result core_ops_deref_Deref_target;
}.
Arguments mk_core_ops_deref_Deref {_}.
Arguments core_ops_deref_Deref_target {_}.
Arguments core_ops_deref_Deref_deref {_}.

(* Trait declaration [core::ops::deref::DerefMut] *)
Record core_ops_deref_DerefMut (Self : Type) := mk_core_ops_deref_DerefMut {
  core_ops_deref_DerefMut_derefInst : core_ops_deref_Deref Self;
  core_ops_deref_DerefMut_deref_mut : Self -> result core_ops_deref_DerefMut_derefInst.(core_ops_deref_Deref_target);
  core_ops_deref_DerefMut_deref_mut_back : Self -> core_ops_deref_DerefMut_derefInst.(core_ops_deref_Deref_target) -> result Self;
}.
Arguments mk_core_ops_deref_DerefMut {_}.
Arguments core_ops_deref_DerefMut_derefInst {_}.
Arguments core_ops_deref_DerefMut_deref_mut {_}.
Arguments core_ops_deref_DerefMut_deref_mut_back {_}.

Record core_ops_range_Range (T : Type) := mk_core_ops_range_Range {
  core_ops_range_Range_start : T;
  core_ops_range_Range_end_ : T;
}.
Arguments mk_core_ops_range_Range {_}.
Arguments core_ops_range_Range_start {_}.
Arguments core_ops_range_Range_end_ {_}.

(*** [alloc] *)

Definition alloc_boxed_Box_deref (T : Type) (x : T) : result T := Return x.
Definition alloc_boxed_Box_deref_mut (T : Type) (x : T) : result T := Return x.
Definition alloc_boxed_Box_deref_mut_back (T : Type) (_ : T) (x : T) : result T := Return x.

(* Trait instance *)
Definition alloc_boxed_Box_coreOpsDerefInst (Self : Type) : core_ops_deref_Deref Self := {|
  core_ops_deref_Deref_target := Self;
  core_ops_deref_Deref_deref := alloc_boxed_Box_deref Self;
|}.

(* Trait instance *)
Definition alloc_boxed_Box_coreOpsDerefMutInst (Self : Type) : core_ops_deref_DerefMut Self := {|
  core_ops_deref_DerefMut_derefInst := alloc_boxed_Box_coreOpsDerefInst Self;
  core_ops_deref_DerefMut_deref_mut := alloc_boxed_Box_deref_mut Self;
  core_ops_deref_DerefMut_deref_mut_back := alloc_boxed_Box_deref_mut_back Self;
|}.


(*** Arrays *)
Definition array T (n : usize) := { l: list T | Z.of_nat (length l) = to_Z n}.

Lemma le_0_usize_max : 0 <= usize_max.
Proof.
  pose (H := usize_max_bound).
  unfold u32_max in H.
  lia.
Qed.

Lemma eqb_imp_eq (x y : Z) : Z.eqb x y = true -> x = y.
Proof.
  lia.
Qed.

(* TODO: finish the definitions *)
Axiom mk_array : forall (T : Type) (n : usize) (l : list T), array T n.

(* For initialization *)
Axiom array_repeat : forall (T : Type) (n : usize) (x : T), array T n.

Axiom array_index_usize : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
Axiom array_update_usize : forall (T : Type) (n : usize) (x : array T n) (i : usize) (nx : T), result (array T n).

(*** Slice *)
Definition slice T := { l: list T | Z.of_nat (length l) <= usize_max}.

Axiom slice_len : forall (T : Type) (s : slice T), usize.
Axiom slice_index_usize : forall (T : Type) (x : slice T) (i : usize), result T.
Axiom slice_update_usize : forall (T : Type) (x : slice T) (i : usize) (nx : T), result (slice T).

(*** Subslices *)

Axiom array_to_slice : forall (T : Type) (n : usize) (x : array T n), result (slice T).
Axiom array_from_slice : forall (T : Type) (n : usize) (x : array T n) (s : slice T), result (array T n).

Axiom array_subslice: forall (T : Type) (n : usize) (x : array T n) (r : core_ops_range_Range usize), result (slice T).
Axiom array_update_subslice: forall (T : Type) (n : usize) (x : array T n) (r : core_ops_range_Range usize) (ns : slice T), result (array T n).

Axiom slice_subslice: forall (T : Type) (x : slice T) (r : core_ops_range_Range usize), result (slice T).
Axiom slice_update_subslice: forall (T : Type) (x : slice T) (r : core_ops_range_Range usize) (ns : slice T), result (slice T).

(*** Vectors *)

Definition alloc_vec_Vec T := { l: list T | Z.of_nat (length l) <= usize_max }.

Definition alloc_vec_Vec_to_list {T: Type} (v: alloc_vec_Vec T) : list T := proj1_sig v.

Definition alloc_vec_Vec_length {T: Type} (v: alloc_vec_Vec T) : Z := Z.of_nat (length (alloc_vec_Vec_to_list v)).

Definition alloc_vec_Vec_new (T: Type) : alloc_vec_Vec T := (exist _ [] le_0_usize_max).

Lemma alloc_vec_Vec_len_in_usize {T} (v: alloc_vec_Vec T) : usize_min <= alloc_vec_Vec_length v <= usize_max.
Proof.
  unfold alloc_vec_Vec_length, usize_min.
  split.
  - lia.
  - apply (proj2_sig v).
Qed.

Definition alloc_vec_Vec_len (T: Type) (v: alloc_vec_Vec T) : usize :=
  exist _ (alloc_vec_Vec_length v) (alloc_vec_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 alloc_vec_Vec_bind {A B} (v: alloc_vec_Vec A) (f: list A -> result (list B)) : result (alloc_vec_Vec B) :=
  l <- f (alloc_vec_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 alloc_vec_Vec_push_fwd (T: Type) (v: alloc_vec_Vec T) (x: T) : unit := tt.

Definition alloc_vec_Vec_push (T: Type) (v: alloc_vec_Vec T) (x: T) : result (alloc_vec_Vec T) :=
  alloc_vec_Vec_bind v (fun l => Return (l ++ [x])).

(* The **forward** function shouldn't be used *)
Definition alloc_vec_Vec_insert_fwd (T: Type) (v: alloc_vec_Vec T) (i: usize) (x: T) : result unit :=
  if to_Z i <? alloc_vec_Vec_length v then Return tt else Fail_ Failure.

Definition alloc_vec_Vec_insert (T: Type) (v: alloc_vec_Vec T) (i: usize) (x: T) : result (alloc_vec_Vec T) :=
  alloc_vec_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).

(* Helper *)
Axiom alloc_vec_Vec_index_usize : forall {T : Type} (v : alloc_vec_Vec T) (i : usize) (x : T), result T.

(* Helper *)
Axiom alloc_vec_Vec_update_usize : forall {T : Type} (v : alloc_vec_Vec T) (i : usize) (x : T), result (alloc_vec_Vec T).

(* Trait declaration: [core::slice::index::private_slice_index::Sealed] *)
Definition core_slice_index_private_slice_index_Sealed (self : Type) := unit.

(* Trait declaration: [core::slice::index::SliceIndex] *)
Record core_slice_index_SliceIndex (Self T : Type) := mk_core_slice_index_SliceIndex {
  core_slice_index_SliceIndex_sealedInst : core_slice_index_private_slice_index_Sealed Self;
  core_slice_index_SliceIndex_Output : Type;
  core_slice_index_SliceIndex_get : Self -> T -> result (option core_slice_index_SliceIndex_Output);
  core_slice_index_SliceIndex_get_mut : Self -> T -> result (option core_slice_index_SliceIndex_Output);
  core_slice_index_SliceIndex_get_mut_back : Self -> T -> option core_slice_index_SliceIndex_Output -> result T;
  core_slice_index_SliceIndex_get_unchecked : Self -> const_raw_ptr T -> result (const_raw_ptr core_slice_index_SliceIndex_Output);
  core_slice_index_SliceIndex_get_unchecked_mut : Self -> mut_raw_ptr T -> result (mut_raw_ptr core_slice_index_SliceIndex_Output);
  core_slice_index_SliceIndex_index : Self -> T -> result core_slice_index_SliceIndex_Output;
  core_slice_index_SliceIndex_index_mut : Self -> T -> result core_slice_index_SliceIndex_Output;
  core_slice_index_SliceIndex_index_mut_back : Self -> T -> core_slice_index_SliceIndex_Output -> result T;
}.
Arguments mk_core_slice_index_SliceIndex {_ _}.
Arguments core_slice_index_SliceIndex_sealedInst {_ _}.
Arguments core_slice_index_SliceIndex_Output {_ _}.
Arguments core_slice_index_SliceIndex_get {_ _}.
Arguments core_slice_index_SliceIndex_get_mut {_ _}.
Arguments core_slice_index_SliceIndex_get_mut_back {_ _}.
Arguments core_slice_index_SliceIndex_get_unchecked {_ _}.
Arguments core_slice_index_SliceIndex_get_unchecked_mut {_ _}.
Arguments core_slice_index_SliceIndex_index {_ _}.
Arguments core_slice_index_SliceIndex_index_mut {_ _}.
Arguments core_slice_index_SliceIndex_index_mut_back {_ _}.

(* [core::slice::index::[T]::index]: forward function *)
Definition core_slice_index_Slice_index
  (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T))
  (s : slice T) (i : Idx) : result inst.(core_slice_index_SliceIndex_Output) :=
  x <- inst.(core_slice_index_SliceIndex_get) i s;
  match x with
  | None => Fail_ Failure
  | Some x => Return x
  end.

(* [core::slice::index::Range:::get]: forward function *)
Axiom core_slice_index_Range_get : forall (T : Type) (i : core_ops_range_Range usize) (s : slice T), result (option (slice T)).

(* [core::slice::index::Range::get_mut]: forward function *)
Axiom core_slice_index_Range_get_mut :
  forall (T : Type), core_ops_range_Range usize -> slice T -> result (option (slice T)).

(* [core::slice::index::Range::get_mut]: backward function 0 *)
Axiom core_slice_index_Range_get_mut_back :
  forall (T : Type), core_ops_range_Range usize -> slice T -> option (slice T) -> result (slice T).

(* [core::slice::index::Range::get_unchecked]: forward function *)
Definition core_slice_index_Range_get_unchecked
  (T : Type) :
  core_ops_range_Range usize -> const_raw_ptr (slice T) -> result (const_raw_ptr (slice T)) :=
  (* Don't know what the model should be - for now we always fail to make
     sure code which uses it fails *)
  fun _ _ => Fail_ Failure.

(* [core::slice::index::Range::get_unchecked_mut]: forward function *)
Definition core_slice_index_Range_get_unchecked_mut
  (T : Type) :
  core_ops_range_Range usize -> mut_raw_ptr (slice T) -> result (mut_raw_ptr (slice T)) :=
  (* Don't know what the model should be - for now we always fail to make
    sure code which uses it fails *)
  fun _ _ => Fail_ Failure.

(* [core::slice::index::Range::index]: forward function *)
Axiom core_slice_index_Range_index :
  forall (T : Type), core_ops_range_Range usize -> slice T -> result (slice T).

(* [core::slice::index::Range::index_mut]: forward function *)
Axiom core_slice_index_Range_index_mut :
  forall (T : Type), core_ops_range_Range usize -> slice T -> result (slice T).

(* [core::slice::index::Range::index_mut]: backward function 0 *)
Axiom core_slice_index_Range_index_mut_back :
  forall (T : Type), core_ops_range_Range usize -> slice T -> slice T -> result (slice T).

(* [core::slice::index::[T]::index_mut]: forward function *)
Axiom core_slice_index_Slice_index_mut :
  forall (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T)),
  slice T -> Idx -> result inst.(core_slice_index_SliceIndex_Output).

(* [core::slice::index::[T]::index_mut]: backward function 0 *)
Axiom core_slice_index_Slice_index_mut_back :
  forall (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T)),
  slice T -> Idx -> inst.(core_slice_index_SliceIndex_Output) -> result (slice T).

(* [core::array::[T; N]::index]: forward function *)
Axiom core_array_Array_index :
  forall (T Idx : Type) (N : usize) (inst : core_ops_index_Index (slice T) Idx)
  (a : array T N) (i : Idx), result inst.(core_ops_index_Index_Output).

(* [core::array::[T; N]::index_mut]: forward function *)
Axiom core_array_Array_index_mut :
  forall (T Idx : Type) (N : usize) (inst : core_ops_index_IndexMut (slice T) Idx)
  (a : array T N) (i : Idx), result inst.(core_ops_index_IndexMut_indexInst).(core_ops_index_Index_Output).

(* [core::array::[T; N]::index_mut]: backward function 0 *)
Axiom core_array_Array_index_mut_back :
  forall (T Idx : Type) (N : usize) (inst : core_ops_index_IndexMut (slice T) Idx)
  (a : array T N) (i : Idx) (x : inst.(core_ops_index_IndexMut_indexInst).(core_ops_index_Index_Output)), result (array T N).

(* Trait implementation: [core::slice::index::[T]] *)
Definition core_slice_index_Slice_coreopsindexIndexInst (T Idx : Type)
  (inst : core_slice_index_SliceIndex Idx (slice T)) :
  core_ops_index_Index (slice T) Idx := {|
  core_ops_index_Index_Output := inst.(core_slice_index_SliceIndex_Output);
  core_ops_index_Index_index := core_slice_index_Slice_index T Idx inst;
|}.

(* Trait implementation: [core::slice::index::private_slice_index::Range] *)
Definition core_slice_index_private_slice_index_Range_coresliceindexprivate_slice_indexSealedInst
  : core_slice_index_private_slice_index_Sealed (core_ops_range_Range usize) := tt.

(* Trait implementation: [core::slice::index::Range] *)
Definition core_slice_index_Range_coresliceindexSliceIndexInst (T : Type) :
  core_slice_index_SliceIndex (core_ops_range_Range usize) (slice T) := {|
  core_slice_index_SliceIndex_sealedInst := core_slice_index_private_slice_index_Range_coresliceindexprivate_slice_indexSealedInst;
  core_slice_index_SliceIndex_Output := slice T;
  core_slice_index_SliceIndex_get := core_slice_index_Range_get T;
  core_slice_index_SliceIndex_get_mut := core_slice_index_Range_get_mut T;
  core_slice_index_SliceIndex_get_mut_back := core_slice_index_Range_get_mut_back T;
  core_slice_index_SliceIndex_get_unchecked := core_slice_index_Range_get_unchecked T;
  core_slice_index_SliceIndex_get_unchecked_mut := core_slice_index_Range_get_unchecked_mut T;
  core_slice_index_SliceIndex_index := core_slice_index_Range_index T;
  core_slice_index_SliceIndex_index_mut := core_slice_index_Range_index_mut T;
  core_slice_index_SliceIndex_index_mut_back := core_slice_index_Range_index_mut_back T;
|}.

(* Trait implementation: [core::slice::index::[T]] *)
Definition core_slice_index_Slice_coreopsindexIndexMutInst (T Idx : Type)
  (inst : core_slice_index_SliceIndex Idx (slice T)) :
  core_ops_index_IndexMut (slice T) Idx := {|
  core_ops_index_IndexMut_indexInst := core_slice_index_Slice_coreopsindexIndexInst T Idx inst;
  core_ops_index_IndexMut_index_mut := core_slice_index_Slice_index_mut T Idx inst;
  core_ops_index_IndexMut_index_mut_back := core_slice_index_Slice_index_mut_back T Idx inst;
|}.

(* Trait implementation: [core::array::[T; N]] *)
Definition core_array_Array_coreopsindexIndexInst (T Idx : Type) (N : usize)
  (inst : core_ops_index_Index (slice T) Idx) :
  core_ops_index_Index (array T N) Idx := {|
  core_ops_index_Index_Output := inst.(core_ops_index_Index_Output);
  core_ops_index_Index_index := core_array_Array_index T Idx N inst;
|}.

(* Trait implementation: [core::array::[T; N]] *)
Definition core_array_Array_coreopsindexIndexMutInst (T Idx : Type) (N : usize)
  (inst : core_ops_index_IndexMut (slice T) Idx) :
  core_ops_index_IndexMut (array T N) Idx := {|
  core_ops_index_IndexMut_indexInst := core_array_Array_coreopsindexIndexInst T Idx N inst.(core_ops_index_IndexMut_indexInst);
  core_ops_index_IndexMut_index_mut := core_array_Array_index_mut T Idx N inst;
  core_ops_index_IndexMut_index_mut_back := core_array_Array_index_mut_back T Idx N inst;
|}.

(* [core::slice::index::usize::get]: forward function *)
Axiom core_slice_index_usize_get : forall (T : Type), usize -> slice T -> result (option T).

(* [core::slice::index::usize::get_mut]: forward function *)
Axiom core_slice_index_usize_get_mut : forall (T : Type), usize -> slice T -> result (option T).

(* [core::slice::index::usize::get_mut]: backward function 0 *)
Axiom core_slice_index_usize_get_mut_back :
  forall (T : Type), usize -> slice T -> option T -> result (slice T).

(* [core::slice::index::usize::get_unchecked]: forward function *)
Axiom core_slice_index_usize_get_unchecked :
  forall (T : Type), usize -> const_raw_ptr (slice T) -> result (const_raw_ptr T).

(* [core::slice::index::usize::get_unchecked_mut]: forward function *)
Axiom core_slice_index_usize_get_unchecked_mut :
  forall (T : Type), usize -> mut_raw_ptr (slice T) -> result (mut_raw_ptr T).

(* [core::slice::index::usize::index]: forward function *)
Axiom core_slice_index_usize_index : forall (T : Type), usize -> slice T -> result T.

(* [core::slice::index::usize::index_mut]: forward function *)
Axiom core_slice_index_usize_index_mut : forall (T : Type), usize -> slice T -> result T.

(* [core::slice::index::usize::index_mut]: backward function 0 *)
Axiom core_slice_index_usize_index_mut_back :
  forall (T : Type), usize -> slice T -> T -> result (slice T).

(* Trait implementation: [core::slice::index::private_slice_index::usize] *)
Definition core_slice_index_private_slice_index_usize_coresliceindexprivate_slice_indexSealedInst
  : core_slice_index_private_slice_index_Sealed usize := tt.

(* Trait implementation: [core::slice::index::usize] *)
Definition core_slice_index_usize_coresliceindexSliceIndexInst (T : Type) :
  core_slice_index_SliceIndex usize (slice T) := {|
  core_slice_index_SliceIndex_sealedInst := core_slice_index_private_slice_index_usize_coresliceindexprivate_slice_indexSealedInst;
  core_slice_index_SliceIndex_Output := T;
  core_slice_index_SliceIndex_get := core_slice_index_usize_get T;
  core_slice_index_SliceIndex_get_mut := core_slice_index_usize_get_mut T;
  core_slice_index_SliceIndex_get_mut_back := core_slice_index_usize_get_mut_back T;
  core_slice_index_SliceIndex_get_unchecked := core_slice_index_usize_get_unchecked T;
  core_slice_index_SliceIndex_get_unchecked_mut := core_slice_index_usize_get_unchecked_mut T;
  core_slice_index_SliceIndex_index := core_slice_index_usize_index T;
  core_slice_index_SliceIndex_index_mut := core_slice_index_usize_index_mut T;
  core_slice_index_SliceIndex_index_mut_back := core_slice_index_usize_index_mut_back T;
|}.

(* [alloc::vec::Vec::index]: forward function *)
Axiom alloc_vec_Vec_index : forall (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T))
  (Self : alloc_vec_Vec T) (i : Idx), result inst.(core_slice_index_SliceIndex_Output).

(* [alloc::vec::Vec::index_mut]: forward function *)
Axiom alloc_vec_Vec_index_mut : forall (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T))
  (Self : alloc_vec_Vec T) (i : Idx), result inst.(core_slice_index_SliceIndex_Output).

(* [alloc::vec::Vec::index_mut]: backward function 0 *)
Axiom alloc_vec_Vec_index_mut_back :
  forall (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T))
  (Self : alloc_vec_Vec T) (i : Idx) (x : inst.(core_slice_index_SliceIndex_Output)), result (alloc_vec_Vec T).

(* Trait implementation: [alloc::vec::Vec] *)
Definition alloc_vec_Vec_coreopsindexIndexInst (T Idx : Type)
  (inst : core_slice_index_SliceIndex Idx (slice T)) :
  core_ops_index_Index (alloc_vec_Vec T) Idx := {|
  core_ops_index_Index_Output := inst.(core_slice_index_SliceIndex_Output);
  core_ops_index_Index_index := alloc_vec_Vec_index T Idx inst;
|}.

(* Trait implementation: [alloc::vec::Vec] *)
Definition alloc_vec_Vec_coreopsindexIndexMutInst (T Idx : Type)
  (inst : core_slice_index_SliceIndex Idx (slice T)) :
  core_ops_index_IndexMut (alloc_vec_Vec T) Idx := {|
  core_ops_index_IndexMut_indexInst := alloc_vec_Vec_coreopsindexIndexInst T Idx inst;
  core_ops_index_IndexMut_index_mut := alloc_vec_Vec_index_mut T Idx inst;
  core_ops_index_IndexMut_index_mut_back := alloc_vec_Vec_index_mut_back T Idx inst;
|}.

(*** Theorems *)

Axiom alloc_vec_Vec_index_mut_back_eq : forall {a : Type} (v : alloc_vec_Vec a) (i : usize) (x : a),
  alloc_vec_Vec_index_mut_back a usize (core_slice_index_usize_coresliceindexSliceIndexInst a) v i x =
    alloc_vec_Vec_update_usize v i x.

End Primitives.