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 := | Ok : A -> result A | Fail_ : error -> result A. Arguments Ok {_} a. Arguments Fail_ {_}. Definition bind {A B} (m: result A) (f: A -> result B) : result B := match m with | Fail_ e => Fail_ e | Ok x => f x end. Definition return_ {A: Type} (x: A) : result A := Ok 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 Ok tt else Fail_ Failure. (** Normalize and unwrap a successful result (used for globals) *) Definition eval_result_refl {A} {x} (a: result A) (p: a = Ok x) : A := match a as r return (r = Ok x -> A) with | Ok a' => fun _ => a' | Fail_ e => fun p' => False_rect _ (eq_ind (Fail_ e) (fun e : result A => match e with | Ok _ => False | Fail_ e => True end) I (Ok 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 Ok (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 * a := (x, x) . 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 => Ok (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)). Axiom scalar_xor : forall ty, scalar ty -> scalar ty -> scalar ty. (* TODO *) Axiom scalar_or : forall ty, scalar ty -> scalar ty -> scalar ty. (* TODO *) Axiom scalar_and : forall ty, scalar ty -> scalar ty -> scalar ty. (* TODO *) Axiom scalar_shl : forall ty0 ty1, scalar ty0 -> scalar ty1 -> result (scalar ty0). (* TODO *) Axiom scalar_shr : forall ty0 ty1, scalar ty0 -> scalar ty1 -> result (scalar ty0). (* TODO *) (** 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). (* This can't fail, but for now we make all casts faillible (easier for the translation) *) Definition scalar_cast_bool (tgt_ty : scalar_ty) (x : bool) : result (scalar tgt_ty) := mk_scalar tgt_ty (if x then 1 else 0). (** 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. (** Xor *) Definition u8_xor := @scalar_xor U8. Definition u16_xor := @scalar_xor U16. Definition u32_xor := @scalar_xor U32. Definition u64_xor := @scalar_xor U64. Definition u128_xor := @scalar_xor U128. Definition usize_xor := @scalar_xor Usize. Definition i8_xor := @scalar_xor I8. Definition i16_xor := @scalar_xor I16. Definition i32_xor := @scalar_xor I32. Definition i64_xor := @scalar_xor I64. Definition i128_xor := @scalar_xor I128. Definition isize_xor := @scalar_xor Isize. (** Or *) Definition u8_or := @scalar_or U8. Definition u16_or := @scalar_or U16. Definition u32_or := @scalar_or U32. Definition u64_or := @scalar_or U64. Definition u128_or := @scalar_or U128. Definition usize_or := @scalar_or Usize. Definition i8_or := @scalar_or I8. Definition i16_or := @scalar_or I16. Definition i32_or := @scalar_or I32. Definition i64_or := @scalar_or I64. Definition i128_or := @scalar_or I128. Definition isize_or := @scalar_or Isize. (** And *) Definition u8_and := @scalar_and U8. Definition u16_and := @scalar_and U16. Definition u32_and := @scalar_and U32. Definition u64_and := @scalar_and U64. Definition u128_and := @scalar_and U128. Definition usize_and := @scalar_and Usize. Definition i8_and := @scalar_and I8. Definition i16_and := @scalar_and I16. Definition i32_and := @scalar_and I32. Definition i64_and := @scalar_and I64. Definition i128_and := @scalar_and I128. Definition isize_and := @scalar_and Isize. (** Shift left *) Definition u8_shl {ty} := @scalar_shl U8 ty. Definition u16_shl {ty} := @scalar_shl U16 ty. Definition u32_shl {ty} := @scalar_shl U32 ty. Definition u64_shl {ty} := @scalar_shl U64 ty. Definition u128_shl {ty} := @scalar_shl U128 ty. Definition usize_shl {ty} := @scalar_shl Usize ty. Definition i8_shl {ty} := @scalar_shl I8 ty. Definition i16_shl {ty} := @scalar_shl I16 ty. Definition i32_shl {ty} := @scalar_shl I32 ty. Definition i64_shl {ty} := @scalar_shl I64 ty. Definition i128_shl {ty} := @scalar_shl I128 ty. Definition isize_shl {ty} := @scalar_shl Isize ty. (** Shift right *) Definition u8_shr {ty} := @scalar_shr U8 ty. Definition u16_shr {ty} := @scalar_shr U16 ty. Definition u32_shr {ty} := @scalar_shr U32 ty. Definition u64_shr {ty} := @scalar_shr U64 ty. Definition u128_shr {ty} := @scalar_shr U128 ty. Definition usize_shr {ty} := @scalar_shr Usize ty. Definition i8_shr {ty} := @scalar_shr I8 ty. Definition i16_shr {ty} := @scalar_shr I16 ty. Definition i32_shr {ty} := @scalar_shr I32 ty. Definition i64_shr {ty} := @scalar_shr I64 ty. Definition i128_shr {ty} := @scalar_shr I128 ty. Definition isize_shr {ty} := @scalar_shr Isize ty. (** 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_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 {_ _}. (* 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_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 {_}. 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 := Ok x. Definition alloc_boxed_Box_deref_mut (T : Type) (x : T) : result (T * (T -> result T)) := Ok (x, fun x => Ok 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; |}. (*** 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). Definition array_index_mut_usize (T : Type) (n : usize) (a : array T n) (i : usize) : result (T * (T -> result (array T n))) := match array_index_usize T n a i with | Fail_ e => Fail_ e | Ok x => Ok (x, array_update_usize T n a i) end. (*** 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). Definition slice_index_mut_usize (T : Type) (s : slice T) (i : usize) : result (T * (T -> result (slice T))) := match slice_index_usize T s i with | Fail_ e => Fail_ e | Ok x => Ok (x, slice_update_usize T s i) end. (*** 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). Definition array_to_slice_mut (T : Type) (n : usize) (a : array T n) : result (slice T * (slice T -> result (array T n))) := match array_to_slice T n a with | Fail_ e => Fail_ e | Ok x => Ok (x, array_from_slice T n a) end. 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 => Ok (exist _ l (scalar_le_max_valid _ _ H)) | right _ => Fail_ Failure end. 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 => Ok (l ++ [x])). 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 result (alloc_vec_Vec T))) := match alloc_vec_Vec_index_usize v i with | Ok x => Ok (x, alloc_vec_Vec_update_usize v i) | Fail_ e => Fail_ e end. (* 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 * (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_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_unchecked {_ _}. Arguments core_slice_index_SliceIndex_get_unchecked_mut {_ _}. Arguments core_slice_index_SliceIndex_index {_ _}. Arguments core_slice_index_SliceIndex_index_mut {_ _}. (* [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 => Ok x end. (* [core::slice::index::Range:::get]: forward function *) Axiom core_slice_index_RangeUsize_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_RangeUsize_get_mut : forall (T : Type), core_ops_range_Range usize -> slice T -> result (option (slice T) * (option (slice T) -> result (slice T))). (* [core::slice::index::Range::get_unchecked]: forward function *) Definition core_slice_index_RangeUsize_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_RangeUsize_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_RangeUsize_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_RangeUsize_index_mut : forall (T : Type), core_ops_range_Range usize -> slice T -> result (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) * (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) * (inst.(core_ops_index_IndexMut_indexInst).(core_ops_index_Index_Output) -> result (array T N))). (* Trait implementation: [core::slice::index::private_slice_index::Range] *) Definition core_slice_index_private_slice_index_SealedRangeUsizeInst : core_slice_index_private_slice_index_Sealed (core_ops_range_Range usize) := tt. (* Trait implementation: [core::slice::index::Range] *) Definition core_slice_index_SliceIndexRangeUsizeSliceTInst (T : Type) : core_slice_index_SliceIndex (core_ops_range_Range usize) (slice T) := {| core_slice_index_SliceIndex_sealedInst := core_slice_index_private_slice_index_SealedRangeUsizeInst; core_slice_index_SliceIndex_Output := slice T; core_slice_index_SliceIndex_get := core_slice_index_RangeUsize_get T; core_slice_index_SliceIndex_get_mut := core_slice_index_RangeUsize_get_mut T; core_slice_index_SliceIndex_get_unchecked := core_slice_index_RangeUsize_get_unchecked T; core_slice_index_SliceIndex_get_unchecked_mut := core_slice_index_RangeUsize_get_unchecked_mut T; core_slice_index_SliceIndex_index := core_slice_index_RangeUsize_index T; core_slice_index_SliceIndex_index_mut := core_slice_index_RangeUsize_index_mut T; |}. (* Trait implementation: [core::slice::index::[T]] *) Definition core_ops_index_IndexSliceTIInst (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::[T]] *) Definition core_ops_index_IndexMutSliceTIInst (T Idx : Type) (inst : core_slice_index_SliceIndex Idx (slice T)) : core_ops_index_IndexMut (slice T) Idx := {| core_ops_index_IndexMut_indexInst := core_ops_index_IndexSliceTIInst T Idx inst; core_ops_index_IndexMut_index_mut := core_slice_index_Slice_index_mut T Idx inst; |}. (* Trait implementation: [core::array::[T; N]] *) Definition core_ops_index_IndexArrayInst (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_ops_index_IndexMutArrayInst (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_ops_index_IndexArrayInst 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::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 * (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 * (T -> result (slice T))). (* Trait implementation: [core::slice::index::private_slice_index::usize] *) Definition core_slice_index_private_slice_index_SealedUsizeInst : core_slice_index_private_slice_index_Sealed usize := tt. (* Trait implementation: [core::slice::index::usize] *) Definition core_slice_index_SliceIndexUsizeSliceTInst (T : Type) : core_slice_index_SliceIndex usize (slice T) := {| core_slice_index_SliceIndex_sealedInst := core_slice_index_private_slice_index_SealedUsizeInst; 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_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; |}. (* [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) * (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; |}. (*** Theorems *) Axiom alloc_vec_Vec_index_eq : forall {a : Type} (v : alloc_vec_Vec a) (i : usize) (x : a), alloc_vec_Vec_index a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i = alloc_vec_Vec_index_usize v i. Axiom alloc_vec_Vec_index_mut_eq : forall {a : Type} (v : alloc_vec_Vec a) (i : usize) (x : a), alloc_vec_Vec_index_mut a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i = alloc_vec_Vec_index_mut_usize v i. End Primitives.