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 result A := | Return : A -> result A | Fail_ : result A. Arguments Return {_} a. Arguments Fail_ {_}. Definition bind {A B} (m: result A) (f: A -> result B) : result B := match m with | Fail_ => Fail_ | Return x => f x end. Definition return_ {A: Type} (x: A) : result A := Return x . Definition fail_ {A: Type} : result A := Fail_ . 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_. (** 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_ => fun p' => False_rect _ (eq_ind Fail_ (fun e : result A => match e with | Return _ => False | Fail_ => 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_)%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_ 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_ 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_ 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 if to_Z i Return n | None => Fail_ end. Definition vec_index_back (T: Type) (v: vec T) (i: usize) (x: T) : result unit := if to_Z i Return n | None => Fail_ 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