(* Version of sail_values.lem that uses Lem's machine words library *) open import Pervasives_extra open import Machine_word open import Sail_impl_base type ii = integer type nn = natural val pow : integer -> integer -> integer let pow m n = m ** (natFromInteger n) let pow2 n = pow 2 n let add_int (l,r) = integerAdd l r let add_signed (l,r) = integerAdd l r let sub_int (l,r) = integerMinus l r let mult_int (l,r) = integerMult l r let quotient_int (l,r) = integerDiv l r let quotient_nat (l,r) = natDiv l r let power_int_nat (l,r) = integerPow l r let power_int_int (l, r) = integerPow l (natFromInteger r) let negate_int i = integerNegate i let min_int (l, r) = integerMin l r let max_int (l, r) = integerMax l r let add_real (l, r) = realAdd l r let sub_real (l, r) = realMinus l r let mult_real (l, r) = realMult l r let div_real (l, r) = realDiv l r let negate_real r = realNegate r let abs_real r = realAbs r let power_real (b, e) = realPowInteger b e let or_bool (l, r) = (l || r) let and_bool (l, r) = (l && r) let xor_bool (l, r) = xor l r let list_append (l, r) = l ++ r let list_length xs = integerFromNat (List.length xs) let list_take (n, xs) = List.take (natFromInteger n) xs let list_drop (n, xs) = List.drop (natFromInteger n) xs val repeat : forall 'a. list 'a -> integer -> list 'a let rec repeat xs n = if n <= 0 then [] else xs ++ repeat xs (n-1) let duplicate_to_list (bit, length) = repeat [bit] length let rec replace bs ((n : integer),b') = match bs with | [] -> [] | b :: bs -> if n = 0 then b' :: bs else b :: replace bs (n - 1,b') end let upper n = n (*** Bits *) type bitU = B0 | B1 | BU let showBitU = function | B0 -> "O" | B1 -> "I" | BU -> "U" end instance (Show bitU) let show = showBitU end class (BitU 'a) val to_bitU : 'a -> bitU val of_bitU : bitU -> 'a end instance (BitU bitU) let to_bitU b = b let of_bitU b = b end let bitU_to_bool = function | B0 -> false | B1 -> true | BU -> failwith "to_bool applied to BU" end let bool_to_bitU b = if b then B1 else B0 instance (BitU bool) let to_bitU = bool_to_bitU let of_bitU = bitU_to_bool end let cast_bit_bool = bitU_to_bool let bit_lifted_of_bitU = function | B0 -> Bitl_zero | B1 -> Bitl_one | BU -> Bitl_undef end let bitU_of_bit = function | Bitc_zero -> B0 | Bitc_one -> B1 end let bit_of_bitU = function | B0 -> Bitc_zero | B1 -> Bitc_one | BU -> failwith "bit_of_bitU: BU" end let bitU_of_bit_lifted = function | Bitl_zero -> B0 | Bitl_one -> B1 | Bitl_undef -> BU | Bitl_unknown -> failwith "bitU_of_bit_lifted Bitl_unknown" end let bitwise_not_bit = function | B1 -> B0 | B0 -> B1 | BU -> BU end (* let inline (~) = bitwise_not_bit *) val is_one : integer -> bitU let is_one i = if i = 1 then B1 else B0 let bitwise_binop_bit op = function | (BU,_) -> BU (*Do we want to do this or to respect | of I and & of B0 rules?*) | (_,BU) -> BU (*Do we want to do this or to respect | of I and & of B0 rules?*) | (x,y) -> bool_to_bitU (op (bitU_to_bool x) (bitU_to_bool y)) end val bitwise_and_bit : bitU * bitU -> bitU let bitwise_and_bit = bitwise_binop_bit (&&) val bitwise_or_bit : bitU * bitU -> bitU let bitwise_or_bit = bitwise_binop_bit (||) val bitwise_xor_bit : bitU * bitU -> bitU let bitwise_xor_bit = bitwise_binop_bit xor val (&.) : bitU -> bitU -> bitU let inline (&.) x y = bitwise_and_bit (x,y) val (|.) : bitU -> bitU -> bitU let inline (|.) x y = bitwise_or_bit (x,y) val (+.) : bitU -> bitU -> bitU let inline (+.) x y = bitwise_xor_bit (x,y) val to_bin_aux : natural -> list bitU let rec to_bin_aux x = if x = 0 then [] else (if x mod 2 = 1 then B1 else B0) :: to_bin_aux (x / 2) let to_bin n = List.reverse (to_bin_aux n) val of_bin : list bitU -> natural let of_bin bits = let (sum,_) = List.foldr (fun b (acc,exp) -> match b with | B1 -> (acc + naturalPow 2 exp, exp + 1) | B0 -> (acc, exp + 1) | BU -> failwith "of_bin: bitvector has undefined bits" end) (0,0) bits in sum val bitlist_to_integer : list bitU -> integer let bitlist_to_integer bs = integerFromNatural (of_bin bs) val pad_zero : list bitU -> integer -> list bitU let rec pad_zero bits n = if n <= 0 then bits else pad_zero (B0 :: bits) (n -1) let bitwise_not_bitlist = List.map bitwise_not_bit let rec add_one_bit_ignore_overflow_aux bits = match bits with | [] -> [] | B0 :: bits -> B1 :: bits | B1 :: bits -> B0 :: add_one_bit_ignore_overflow_aux bits | BU :: _ -> failwith "add_one_bit_ignore_overflow: undefined bit" end let add_one_bit_ignore_overflow bits = List.reverse (add_one_bit_ignore_overflow_aux (List.reverse bits)) let bits_of_nat ((len : integer),(n : natural)) = let bits = to_bin n in let len_bits = integerFromNat (List.length bits) in let longer = len - len_bits in if longer < 0 then drop (natFromInteger (abs (longer))) bits else pad_zero bits longer let bits_of_int ((len : integer),(n : integer)) = let bits = bits_of_nat (len, naturalFromInteger (abs n)) in if n > (0 : integer) then bits else (add_one_bit_ignore_overflow (bitwise_not_bitlist bits)) (*** Vectors *) (* element list * start * has increasing direction *) type vector 'a = Vector of list 'a * integer * bool let showVector (Vector elems start inc) = "Vector " ^ show elems ^ " " ^ show start ^ " " ^ show inc let get_dir (Vector _ _ ord) = ord let get_start (Vector _ s _) = s let get_elems (Vector elems _ _) = elems let length (Vector bs _ _) = integerFromNat (length bs) let vector_length = length instance forall 'a. Show 'a => (Show (vector 'a)) let show = showVector end let dir is_inc = if is_inc then D_increasing else D_decreasing let bool_of_dir = function | D_increasing -> true | D_decreasing -> false end (*** Vector operations *) val set_vector_start : forall 'a. (integer * vector 'a) -> vector 'a let set_vector_start (new_start, Vector bs _ is_inc) = Vector bs new_start is_inc let reset_vector_start v = set_vector_start (if (get_dir v) then 0 else (length v - 1), v) let set_vector_start_to_length v = set_vector_start (length v - 1, v) let vector_concat (Vector bs start is_inc, Vector bs' _ _) = Vector (bs ++ bs') start is_inc let inline (^^) = vector_concat val sublist : forall 'a. list 'a -> (nat * nat) -> list 'a let sublist xs (i,j) = let (toJ,_suffix) = List.splitAt (j+1) xs in let (_prefix,fromItoJ) = List.splitAt i toJ in fromItoJ val update_sublist : forall 'a. list 'a -> (nat * nat) -> list 'a -> list 'a let update_sublist xs (i,j) xs' = let (toJ,suffix) = List.splitAt (j+1) xs in let (prefix,_fromItoJ) = List.splitAt i toJ in prefix ++ xs' ++ suffix val slice_aux : forall 'a. bool -> integer -> list 'a -> integer -> integer -> list 'a let slice_aux is_inc start bs i j = let iN = natFromInteger i in let jN = natFromInteger j in let startN = natFromInteger start in sublist bs (if is_inc then (iN-startN,jN-startN) else (startN-iN,startN-jN)) val slice : forall 'a. vector 'a -> integer -> integer -> vector 'a let slice (Vector bs start is_inc) i j = Vector (slice_aux is_inc start bs i j) i is_inc let vector_subrange_inc (start, v, i, j) = slice v i j let vector_subrange_dec (start, v, i, j) = slice v i j (* this is for the vector slicing introduced in vector-concat patterns: i and j index into the "raw data", the list of bits. Therefore getting the bit list is easy, but the start index has to be transformed to match the old vector start and the direction. *) val slice_raw : forall 'a. vector 'a -> integer -> integer -> vector 'a let slice_raw (Vector bs start is_inc) i j = let iN = natFromInteger i in let jN = natFromInteger j in let bits = sublist bs (iN,jN) in let len = integerFromNat (List.length bits) in Vector bits (if is_inc then 0 else len - 1) is_inc val update_aux : forall 'a. bool -> integer -> list 'a -> integer -> integer -> list 'a -> list 'a let update_aux is_inc start bs i j bs' = let iN = natFromInteger i in let jN = natFromInteger j in let startN = natFromInteger start in update_sublist bs (if is_inc then (iN-startN,jN-startN) else (startN-iN,startN-jN)) bs' val update : forall 'a. vector 'a -> integer -> integer -> vector 'a -> vector 'a let update (Vector bs start is_inc) i j (Vector bs' _ _) = Vector (update_aux is_inc start bs i j bs') start is_inc let vector_update_subrange_inc (start, v, i, j, v') = update v i j v' let vector_update_subrange_dec (start, v, i, j, v') = update v i j v' val access_aux : forall 'a. bool -> integer -> list 'a -> integer -> 'a let access_aux is_inc start xs n = if is_inc then List_extra.nth xs (natFromInteger (n - start)) else List_extra.nth xs (natFromInteger (start - n)) val access : forall 'a. vector 'a -> integer -> 'a let access (Vector bs start is_inc) n = access_aux is_inc start bs n let vector_access_inc (start, v, i) = access v i let vector_access_dec (start, v, i) = access v i val update_pos : forall 'a. vector 'a -> integer -> 'a -> vector 'a let update_pos v n b = update v n n (Vector [b] 0 false) let vector_update_pos_inc (start, v, i, x) = update_pos v i x let vector_update_pos_dec (start, v, i, x) = update_pos v i x let extract_only_element (Vector elems _ _) = match elems with | [] -> failwith "extract_only_element called for empty vector" | [e] -> e | _ -> failwith "extract_only_element called for vector with more elements" end (*** Bitvectors *) (* element list * start * has increasing direction *) type bitvector 'a = mword 'a (* Bitvector of mword 'a * integer * bool *) declare isabelle target_sorts bitvector = `len` class (Bitvector 'a) val bits_of : 'a -> list bitU val of_bits : list bitU -> 'a val unsigned : 'a -> integer (* The first two parameters of the following specify indexing: indexing order and start index *) val get_bit : bool -> integer -> 'a -> integer -> bitU val set_bit : bool -> integer -> 'a -> integer -> bitU -> 'a val get_bits : bool -> integer -> 'a -> integer -> integer -> list bitU val set_bits : bool -> integer -> 'a -> integer -> integer -> list bitU -> 'a end instance forall 'a. BitU 'a => (Bitvector (list 'a)) let bits_of v = List.map to_bitU v let of_bits v = List.map of_bitU v let unsigned v = bitlist_to_integer (List.map to_bitU v) let get_bit is_inc start v n = to_bitU (access_aux is_inc start v n) let set_bit is_inc start v n b = update_aux is_inc start v n n [of_bitU b] let get_bits is_inc start v i j = List.map to_bitU (slice_aux is_inc start v i j) let set_bits is_inc start v i j v' = update_aux is_inc start v i j (List.map of_bitU v') end instance forall 'a. BitU 'a => (Bitvector (vector 'a)) let bits_of v = List.map to_bitU (get_elems v) let of_bits v = Vector (List.map of_bitU v) (integerFromNat (List.length v) - 1) false let unsigned v = unsigned (get_elems v) let get_bit is_inc start v n = to_bitU (access v n) let set_bit is_inc start v n b = update_pos v n (of_bitU b) let get_bits is_inc start v i j = List.map to_bitU (get_elems (slice v i j)) let set_bits is_inc start v i j v' = update v i j (Vector (List.map of_bitU v') (integerFromNat (List.length v') - 1) false) end instance forall 'a. Size 'a => (Bitvector (mword 'a)) let bits_of v = List.map to_bitU (bitlistFromWord v) let of_bits v = wordFromBitlist (List.map of_bitU v) let unsigned v = unsignedIntegerFromWord v let get_bit is_inc start v n = to_bitU (access_aux is_inc start (bitlistFromWord v) n) let set_bit is_inc start v n b = wordFromBitlist (update_aux is_inc start (bitlistFromWord v) n n [of_bitU b]) let get_bits is_inc start v i j = slice_aux is_inc start (List.map to_bitU (bitlistFromWord v)) i j let set_bits is_inc start v i j v' = wordFromBitlist (update_aux is_inc start (bitlistFromWord v) i j (List.map of_bitU v')) end (*let showBitvector (Bitvector elems start inc) = "Bitvector " ^ show elems ^ " " ^ show start ^ " " ^ show inc let bvget_dir (Bitvector _ _ ord) = ord let bvget_start (Bitvector _ s _) = s let bvget_elems (Bitvector elems _ _) = elems instance forall 'a. (Show (bitvector 'a)) let show = showBitvector end*) let bvec_to_vec is_inc start bs = let bits = List.map bool_to_bitU (bitlistFromWord bs) in Vector bits start is_inc let vec_to_bvec (Vector elems start is_inc) = (*let word =*) wordFromBitlist (List.map bitU_to_bool elems) (*in Bitvector word start is_inc*) (*** Vector operations *) (* Bytes and addresses *) val byte_chunks : forall 'a. nat -> list 'a -> list (list 'a) let rec byte_chunks n list = match (n,list) with | (0,_) -> [] | (n+1, a::b::c::d::e::f::g::h::rest) -> [a;b;c;d;e;f;g;h] :: byte_chunks n rest | _ -> failwith "byte_chunks not given enough bits" end val bitv_of_byte_lifteds : bool -> list Sail_impl_base.byte_lifted -> list bitU let bitv_of_byte_lifteds dir v = let bits = foldl (fun x (Byte_lifted y) -> x ++ (List.map bitU_of_bit_lifted y)) [] v in let len = integerFromNat (List.length bits) in bits (*Vector bits (if dir then 0 else len - 1) dir*) val bitv_of_bytes : bool -> list Sail_impl_base.byte -> list bitU let bitv_of_bytes dir v = let bits = foldl (fun x (Byte y) -> x ++ (List.map bitU_of_bit y)) [] v in let len = integerFromNat (List.length bits) in bits (*Vector bits (if dir then 0 else len - 1) dir*) val byte_lifteds_of_bitv : list bitU -> list byte_lifted let byte_lifteds_of_bitv bits = let bits = List.map bit_lifted_of_bitU bits in byte_lifteds_of_bit_lifteds bits val bytes_of_bitv : list bitU -> list byte let bytes_of_bitv bits = let bits = List.map bit_of_bitU bits in bytes_of_bits bits val bit_lifteds_of_bitUs : list bitU -> list bit_lifted let bit_lifteds_of_bitUs bits = List.map bit_lifted_of_bitU bits val bit_lifteds_of_bitv : list bitU -> list bit_lifted let bit_lifteds_of_bitv v = bit_lifteds_of_bitUs v val address_lifted_of_bitv : list bitU -> address_lifted let address_lifted_of_bitv v = let byte_lifteds = byte_lifteds_of_bitv v in let maybe_address_integer = match (maybe_all (List.map byte_of_byte_lifted byte_lifteds)) with | Just bs -> Just (integer_of_byte_list bs) | _ -> Nothing end in Address_lifted byte_lifteds maybe_address_integer val address_of_bitv : list bitU -> address let address_of_bitv v = let bytes = bytes_of_bitv v in address_of_byte_list bytes let rec reverse_endianness_bl bits = if List.length bits <= 8 then bits else list_append(reverse_endianness_bl(list_drop(8, bits)), list_take(8, bits)) val reverse_endianness : forall 'a. Bitvector 'a => 'a -> 'a let reverse_endianness v = of_bits (reverse_endianness_bl (bits_of v)) (*** Registers *) type register_field = string type register_field_index = string * (integer * integer) (* name, start and end *) type register = | Register of string * (* name *) integer * (* length *) integer * (* start index *) bool * (* is increasing *) list register_field_index | UndefinedRegister of integer (* length *) | RegisterPair of register * register type register_ref 'regstate 'a = <| reg_name : string; reg_start : integer; reg_is_inc : bool; read_from : 'regstate -> 'a; write_to : 'regstate -> 'a -> 'regstate |> type field_ref 'regtype 'a = <| field_name : string; field_start : integer; field_is_inc : bool; get_field : 'regtype -> 'a; set_field : 'regtype -> 'a -> 'regtype |> let name_of_reg = function | Register name _ _ _ _ -> name | UndefinedRegister _ -> failwith "name_of_reg UndefinedRegister" | RegisterPair _ _ -> failwith "name_of_reg RegisterPair" end let size_of_reg = function | Register _ size _ _ _ -> size | UndefinedRegister size -> size | RegisterPair _ _ -> failwith "size_of_reg RegisterPair" end let start_of_reg = function | Register _ _ start _ _ -> start | UndefinedRegister _ -> failwith "start_of_reg UndefinedRegister" | RegisterPair _ _ -> failwith "start_of_reg RegisterPair" end let is_inc_of_reg = function | Register _ _ _ is_inc _ -> is_inc | UndefinedRegister _ -> failwith "is_inc_of_reg UndefinedRegister" | RegisterPair _ _ -> failwith "in_inc_of_reg RegisterPair" end let dir_of_reg = function | Register _ _ _ is_inc _ -> dir is_inc | UndefinedRegister _ -> failwith "dir_of_reg UndefinedRegister" | RegisterPair _ _ -> failwith "dir_of_reg RegisterPair" end let size_of_reg_nat reg = natFromInteger (size_of_reg reg) let start_of_reg_nat reg = natFromInteger (start_of_reg reg) val register_field_indices_aux : register -> register_field -> maybe (integer * integer) let rec register_field_indices_aux register rfield = match register with | Register _ _ _ _ rfields -> List.lookup rfield rfields | RegisterPair r1 r2 -> let m_indices = register_field_indices_aux r1 rfield in if isJust m_indices then m_indices else register_field_indices_aux r2 rfield | UndefinedRegister _ -> Nothing end val register_field_indices : register -> register_field -> integer * integer let register_field_indices register rfield = match register_field_indices_aux register rfield with | Just indices -> indices | Nothing -> failwith "Invalid register/register-field combination" end let register_field_indices_nat reg regfield= let (i,j) = register_field_indices reg regfield in (natFromInteger i,natFromInteger j) let rec external_reg_value reg_name v = let (internal_start, external_start, direction) = match reg_name with | Reg _ start size dir -> (start, (if dir = D_increasing then start else (start - (size +1))), dir) | Reg_slice _ reg_start dir (slice_start, slice_end) -> ((if dir = D_increasing then slice_start else (reg_start - slice_start)), slice_start, dir) | Reg_field _ reg_start dir _ (slice_start, slice_end) -> ((if dir = D_increasing then slice_start else (reg_start - slice_start)), slice_start, dir) | Reg_f_slice _ reg_start dir _ _ (slice_start, slice_end) -> ((if dir = D_increasing then slice_start else (reg_start - slice_start)), slice_start, dir) end in let bits = bit_lifteds_of_bitv v in <| rv_bits = bits; rv_dir = direction; rv_start = external_start; rv_start_internal = internal_start |> val internal_reg_value : register_value -> list bitU let internal_reg_value v = List.map bitU_of_bit_lifted v.rv_bits (*(integerFromNat v.rv_start_internal) (v.rv_dir = D_increasing)*) let external_slice (d:direction) (start:nat) ((i,j):(nat*nat)) = match d with (*This is the case the thread/concurrecny model expects, so no change needed*) | D_increasing -> (i,j) | D_decreasing -> let slice_i = start - i in let slice_j = (i - j) + slice_i in (slice_i,slice_j) end let external_reg_whole reg = Reg (name_of_reg reg) (start_of_reg_nat reg) (size_of_reg_nat reg) (dir_of_reg reg) let external_reg_slice reg (i,j) = let start = start_of_reg_nat reg in let dir = dir_of_reg reg in Reg_slice (name_of_reg reg) start dir (external_slice dir start (i,j)) let external_reg_field_whole reg rfield = let (m,n) = register_field_indices_nat reg rfield in let start = start_of_reg_nat reg in let dir = dir_of_reg reg in Reg_field (name_of_reg reg) start dir rfield (external_slice dir start (m,n)) let external_reg_field_slice reg rfield (i,j) = let (m,n) = register_field_indices_nat reg rfield in let start = start_of_reg_nat reg in let dir = dir_of_reg reg in Reg_f_slice (name_of_reg reg) start dir rfield (external_slice dir start (m,n)) (external_slice dir start (i,j)) let external_mem_value v = byte_lifteds_of_bitv v $> List.reverse let internal_mem_value direction bytes = List.reverse bytes $> bitv_of_byte_lifteds direction val foreach_inc : forall 'vars. (integer * integer * integer) -> 'vars -> (integer -> 'vars -> 'vars) -> 'vars let rec foreach_inc (i,stop,by) vars body = if (by > 0 && i <= stop) || (by < 0 && stop <= i) then let vars = body i vars in foreach_inc (i + by,stop,by) vars body else vars val foreach_dec : forall 'vars. (integer * integer * integer) -> 'vars -> (integer -> 'vars -> 'vars) -> 'vars let rec foreach_dec (i,stop,by) vars body = if (by > 0 && i >= stop) || (by < 0 && stop >= i) then let vars = body i vars in foreach_dec (i - by,stop,by) vars body else vars let assert' b msg_opt = let msg = match msg_opt with | Just msg -> msg | Nothing -> "unspecified error" end in if b then () else failwith msg (* convert numbers unsafely to naturals *) class (ToNatural 'a) val toNatural : 'a -> natural end (* eta-expanded for Isabelle output, otherwise it breaks *) instance (ToNatural integer) let toNatural = (fun n -> naturalFromInteger n) end instance (ToNatural int) let toNatural = (fun n -> naturalFromInt n) end instance (ToNatural nat) let toNatural = (fun n -> naturalFromNat n) end instance (ToNatural natural) let toNatural = (fun n -> n) end let toNaturalFiveTup (n1,n2,n3,n4,n5) = (toNatural n1, toNatural n2, toNatural n3, toNatural n4, toNatural n5) (* Let the following types be generated by Sail per spec, using either bitlists or machine words as bitvector representation *) (*type regfp = | RFull of (string) | RSlice of (string * integer * integer) | RSliceBit of (string * integer) | RField of (string * string) type niafp = | NIAFP_successor | NIAFP_concrete_address of vector bitU | NIAFP_LR | NIAFP_CTR | NIAFP_register of regfp (* only for MIPS *) type diafp = | DIAFP_none | DIAFP_concrete of vector bitU | DIAFP_reg of regfp let regfp_to_reg (reg_info : string -> maybe string -> (nat * nat * direction * (nat * nat))) = function | RFull name -> let (start,length,direction,_) = reg_info name Nothing in Reg name start length direction | RSlice (name,i,j) -> let i = natFromInteger i in let j = natFromInteger j in let (start,length,direction,_) = reg_info name Nothing in let slice = external_slice direction start (i,j) in Reg_slice name start direction slice | RSliceBit (name,i) -> let i = natFromInteger i in let (start,length,direction,_) = reg_info name Nothing in let slice = external_slice direction start (i,i) in Reg_slice name start direction slice | RField (name,field_name) -> let (start,length,direction,span) = reg_info name (Just field_name) in let slice = external_slice direction start span in Reg_field name start direction field_name slice end let niafp_to_nia reginfo = function | NIAFP_successor -> NIA_successor | NIAFP_concrete_address v -> NIA_concrete_address (address_of_bitv (bits_of v)) | NIAFP_LR -> NIA_LR | NIAFP_CTR -> NIA_CTR | NIAFP_register r -> NIA_register (regfp_to_reg reginfo r) end let diafp_to_dia reginfo = function | DIAFP_none -> DIA_none | DIAFP_concrete v -> DIA_concrete_address (address_of_bitv (bits_of v)) | DIAFP_reg r -> DIA_register (regfp_to_reg reginfo r) end *)