Organize a bit the HOL files

1 files changed, 1270 insertions, 0 deletions

diff --git a/backends/hol4/primitivesScript.sml b/backends/hol4/primitivesScript.sml
new file mode 100644
index 00000000..56a876d4
--- /dev/null
+++ b/backends/hol4/primitivesScript.sml
@@ -0,0 +1,1270 @@
+open HolKernel boolLib bossLib Parse
+open boolTheory arithmeticTheory integerTheory intLib listTheory stringTheory
+
+open primitivesArithTheory
+
+val primitives_theory_name = "primitives"
+val _ = new_theory primitives_theory_name
+
+(*** Result *)
+Datatype:
+  error = Failure
+End
+
+Datatype:
+  result = Return 'a | Fail error | Loop
+End
+
+Type M = ``: 'a result``
+
+val bind_def = Define `
+  (bind : 'a M -> ('a -> 'b M) -> 'b M) x f =
+    case x of
+      Return y => f y
+    | Fail e => Fail e
+    | Loop => Loop`;
+
+val bind_name = fst (dest_const “bind”)
+
+val return_def = Define `
+  (return : 'a -> 'a M) x =
+    Return x`;
+
+val massert_def = Define ‘massert b = if b then Return () else Fail Failure’
+
+Overload monad_bind = ``bind``
+Overload monad_unitbind = ``\x y. bind x (\z. y)``
+Overload monad_ignore_bind = ``\x y. bind x (\z. y)``
+
+(* Allow the use of monadic syntax *)
+val _ = monadsyntax.enable_monadsyntax ()
+
+(*** Misc *)
+Type char = “:char”
+Type string = “:string”
+
+val mem_replace_fwd_def = Define ‘mem_replace_fwd (x : 'a) (y :'a) : 'a = x’
+val mem_replace_back_def = Define ‘mem_replace_back (x : 'a) (y :'a) : 'a = y’
+
+(*** Scalars *)
+(* Remark: most of the following code was partially generated *)
+
+(* The bounds for the isize/usize types are opaque, because they vary with
+   the architecture *)
+val _ = new_constant ("isize_min", “:int”)
+val _ = new_constant ("isize_max", “:int”)
+val _ = new_constant ("usize_max", “:int”)
+
+val _ = new_type ("usize", 0)
+val _ = new_type ("u8", 0)
+val _ = new_type ("u16", 0)
+val _ = new_type ("u32", 0)
+val _ = new_type ("u64", 0)
+val _ = new_type ("u128", 0)
+val _ = new_type ("isize", 0)
+val _ = new_type ("i8", 0)
+val _ = new_type ("i16", 0)
+val _ = new_type ("i32", 0)
+val _ = new_type ("i64", 0)
+val _ = new_type ("i128", 0)
+
+val _ = new_constant ("isize_to_int", “:isize -> int”)
+val _ = new_constant ("i8_to_int",    “:i8 -> int”)
+val _ = new_constant ("i16_to_int",   “:i16 -> int”)
+val _ = new_constant ("i32_to_int",   “:i32 -> int”)
+val _ = new_constant ("i64_to_int",   “:i64 -> int”)
+val _ = new_constant ("i128_to_int",  “:i128 -> int”)
+val _ = new_constant ("usize_to_int", “:usize -> int”)
+val _ = new_constant ("u8_to_int",    “:u8 -> int”)
+val _ = new_constant ("u16_to_int",   “:u16 -> int”)
+val _ = new_constant ("u32_to_int",   “:u32 -> int”)
+val _ = new_constant ("u64_to_int",   “:u64 -> int”)
+val _ = new_constant ("u128_to_int",  “:u128 -> int”)
+
+val _ = new_constant ("int_to_isize", “:int -> isize”)
+val _ = new_constant ("int_to_i8", “:int -> i8”)
+val _ = new_constant ("int_to_i16", “:int -> i16”)
+val _ = new_constant ("int_to_i32", “:int -> i32”)
+val _ = new_constant ("int_to_i64", “:int -> i64”)
+val _ = new_constant ("int_to_i128", “:int -> i128”)
+val _ = new_constant ("int_to_usize", “:int -> usize”)
+val _ = new_constant ("int_to_u8", “:int -> u8”)
+val _ = new_constant ("int_to_u16", “:int -> u16”)
+val _ = new_constant ("int_to_u32", “:int -> u32”)
+val _ = new_constant ("int_to_u64", “:int -> u64”)
+val _ = new_constant ("int_to_u128", “:int -> u128”)
+
+(* The bounds *)
+val i8_min_def   = Define ‘i8_min   = (-128:int)’
+val i8_max_def   = Define ‘i8_max   = (127:int)’
+val i16_min_def  = Define ‘i16_min  = (-32768:int)’
+val i16_max_def  = Define ‘i16_max  = (32767:int)’
+val i32_min_def  = Define ‘i32_min  = (-2147483648:int)’
+val i32_max_def  = Define ‘i32_max  = (2147483647:int)’
+val i64_min_def  = Define ‘i64_min  = (-9223372036854775808:int)’
+val i64_max_def  = Define ‘i64_max  = (9223372036854775807:int)’
+val i128_min_def = Define ‘i128_min = (-170141183460469231731687303715884105728:int)’
+val i128_max_def = Define ‘i128_max = (170141183460469231731687303715884105727:int)’
+val u8_max_def   = Define ‘u8_max   = (255:int)’
+val u16_max_def  = Define ‘u16_max  = (65535:int)’
+val u32_max_def  = Define ‘u32_max  = (4294967295:int)’
+val u64_max_def  = Define ‘u64_max  = (18446744073709551615:int)’
+val u128_max_def = Define ‘u128_max = (340282366920938463463374607431768211455:int)’
+
+val all_bound_defs = [
+  i8_min_def,   i8_max_def,
+  i16_min_def,  i16_max_def,
+  i32_min_def,  i32_max_def,
+  i64_min_def,  i64_max_def,
+  i128_min_def, i128_max_def,
+  u8_max_def,
+  u16_max_def,
+  u32_max_def,
+  u64_max_def,
+  u128_max_def
+]
+
+(* The following bounds are valid for all architectures *)
+val isize_bounds = new_axiom ("isize_bounds", “isize_min <= i16_min /\ isize_max >= i16_max”)
+val usize_bounds = new_axiom ("usize_bounds", “usize_max >= u16_max”)
+
+(* Conversion bounds *)
+val isize_to_int_bounds = new_axiom ("isize_to_int_bounds",
+  “!n. isize_min <= isize_to_int n /\ isize_to_int n <= isize_max”)
+
+val i8_to_int_bounds = new_axiom ("i8_to_int_bounds",
+  “!n. i8_min <= i8_to_int n /\ i8_to_int n <= i8_max”)
+
+val i16_to_int_bounds = new_axiom ("i16_to_int_bounds",
+  “!n. i16_min <= i16_to_int n /\ i16_to_int n <= i16_max”)
+
+val i32_to_int_bounds = new_axiom ("i32_to_int_bounds",
+  “!n. i32_min <= i32_to_int n /\ i32_to_int n <= i32_max”)
+
+val i64_to_int_bounds = new_axiom ("i64_to_int_bounds",
+  “!n. i64_min <= i64_to_int n /\ i64_to_int n <= i64_max”)
+
+val i128_to_int_bounds = new_axiom ("i128_to_int_bounds",
+  “!n. i128_min <= i128_to_int n /\ i128_to_int n <= i128_max”)
+
+val usize_to_int_bounds = new_axiom ("usize_to_int_bounds",
+  “!n. 0 <= usize_to_int n /\ usize_to_int n <= usize_max”)
+
+val u8_to_int_bounds = new_axiom ("u8_to_int_bounds",
+  “!n. 0 <= u8_to_int n /\ u8_to_int n <= u8_max”)
+
+val u16_to_int_bounds = new_axiom ("u16_to_int_bounds",
+  “!n. 0 <= u16_to_int n /\ u16_to_int n <= u16_max”)
+
+val u32_to_int_bounds = new_axiom ("u32_to_int_bounds",
+  “!n. 0 <= u32_to_int n /\ u32_to_int n <= u32_max”)
+
+val u64_to_int_bounds = new_axiom ("u64_to_int_bounds",
+  “!n. 0 <= u64_to_int n /\ u64_to_int n <= u64_max”)
+
+val u128_to_int_bounds = new_axiom ("u128_to_int_bounds",
+  “!n. 0 <= u128_to_int n /\ u128_to_int n <= u128_max”)
+
+val all_to_int_bounds_lemmas = [
+  isize_to_int_bounds,
+  i8_to_int_bounds,
+  i16_to_int_bounds,
+  i32_to_int_bounds,
+  i64_to_int_bounds,
+  i128_to_int_bounds,
+  usize_to_int_bounds,
+  u8_to_int_bounds,
+  u16_to_int_bounds,
+  u32_to_int_bounds,
+  u64_to_int_bounds,
+  u128_to_int_bounds
+]
+
+(* Conversion to and from int.
+
+   Note that for isize and usize, we write the lemmas in such a way that the
+   proofs are easily automatable for constants.
+ *)
+val int_to_isize_id =
+  new_axiom ("int_to_isize_id",
+    “!n. (i16_min <= n \/ isize_min <= n) /\ (n <= i16_max \/ n <= isize_max) ==>
+         isize_to_int (int_to_isize n) = n”)
+
+val int_to_usize_id =
+  new_axiom ("int_to_usize_id",
+    “!n. 0 <= n /\ (n <= u16_max \/ n <= usize_max) ==> usize_to_int (int_to_usize n) = n”)
+
+val int_to_i8_id =
+  new_axiom ("int_to_i8_id",
+    “!n. i8_min <= n /\ n <= i8_max ==> i8_to_int (int_to_i8 n) = n”)
+
+val int_to_i16_id =
+  new_axiom ("int_to_i16_id",
+    “!n. i16_min <= n /\ n <= i16_max ==> i16_to_int (int_to_i16 n) = n”)
+
+val int_to_i32_id =
+  new_axiom ("int_to_i32_id",
+    “!n. i32_min <= n /\ n <= i32_max ==> i32_to_int (int_to_i32 n) = n”)
+
+val int_to_i64_id =
+  new_axiom ("int_to_i64_id",
+    “!n. i64_min <= n /\ n <= i64_max ==> i64_to_int (int_to_i64 n) = n”)
+
+val int_to_i128_id =
+  new_axiom ("int_to_i128_id",
+    “!n. i128_min <= n /\ n <= i128_max ==> i128_to_int (int_to_i128 n) = n”)
+
+val int_to_u8_id =
+  new_axiom ("int_to_u8_id",
+    “!n. 0 <= n /\ n <= u8_max ==> u8_to_int (int_to_u8 n) = n”)
+
+val int_to_u16_id =
+  new_axiom ("int_to_u16_id",
+    “!n. 0 <= n /\ n <= u16_max ==> u16_to_int (int_to_u16 n) = n”)
+
+val int_to_u32_id =
+  new_axiom ("int_to_u32_id",
+    “!n. 0 <= n /\ n <= u32_max ==> u32_to_int (int_to_u32 n) = n”)
+
+val int_to_u64_id =
+  new_axiom ("int_to_u64_id",
+    “!n. 0 <= n /\ n <= u64_max ==> u64_to_int (int_to_u64 n) = n”)
+
+val int_to_u128_id =
+  new_axiom ("int_to_u128_id",
+    “!n. 0 <= n /\ n <= u128_max ==> u128_to_int (int_to_u128 n) = n”)
+
+val all_conversion_id_lemmas = [
+  int_to_isize_id,
+  int_to_i8_id,
+  int_to_i16_id,
+  int_to_i32_id,
+  int_to_i64_id,
+  int_to_i128_id,
+  int_to_usize_id,
+  int_to_u8_id,
+  int_to_u16_id,
+  int_to_u32_id,
+  int_to_u64_id,
+  int_to_u128_id
+]
+
+(** Utilities to define the arithmetic operations *)
+val mk_isize_def = Define
+  ‘mk_isize n =
+    if isize_min <= n /\ n <= isize_max then Return (int_to_isize n)
+    else Fail Failure’
+
+val mk_i8_def = Define
+  ‘mk_i8 n =
+    if i8_min <= n /\ n <= i8_max then Return (int_to_i8 n)
+    else Fail Failure’
+
+val mk_i16_def = Define
+  ‘mk_i16 n =
+    if i16_min <= n /\ n <= i16_max then Return (int_to_i16 n)
+    else Fail Failure’
+
+val mk_i32_def = Define
+  ‘mk_i32 n =
+    if i32_min <= n /\ n <= i32_max then Return (int_to_i32 n)
+    else Fail Failure’
+
+val mk_i64_def = Define
+  ‘mk_i64 n =
+    if i64_min <= n /\ n <= i64_max then Return (int_to_i64 n)
+    else Fail Failure’
+
+val mk_i128_def = Define
+  ‘mk_i128 n =
+    if i128_min <= n /\ n <= i128_max then Return (int_to_i128 n)
+    else Fail Failure’
+
+val mk_usize_def = Define
+  ‘mk_usize n =
+    if 0 <= n /\ n <= usize_max then Return (int_to_usize n)
+    else Fail Failure’
+
+val mk_u8_def = Define
+  ‘mk_u8 n =
+    if 0 <= n /\ n <= u8_max then Return (int_to_u8 n)
+    else Fail Failure’
+
+val mk_u16_def = Define
+  ‘mk_u16 n =
+    if 0 <= n /\ n <= u16_max then Return (int_to_u16 n)
+    else Fail Failure’
+
+val mk_u32_def = Define
+  ‘mk_u32 n =
+    if 0 <= n /\ n <= u32_max then Return (int_to_u32 n)
+    else Fail Failure’
+
+val mk_u64_def = Define
+  ‘mk_u64 n =
+    if 0 <= n /\ n <= u64_max then Return (int_to_u64 n)
+    else Fail Failure’
+
+val mk_u128_def = Define
+  ‘mk_u128 n =
+    if 0 <= n /\ n <= u128_max then Return (int_to_u128 n)
+    else Fail Failure’
+
+val all_mk_int_defs = [
+  mk_isize_def,
+  mk_i8_def,
+  mk_i16_def,
+  mk_i32_def,
+  mk_i64_def,
+  mk_i128_def,
+  mk_usize_def,
+  mk_u8_def,
+  mk_u16_def,
+  mk_u32_def,
+  mk_u64_def,
+  mk_u128_def
+]
+
+
+val isize_add_def = Define ‘isize_add x y = mk_isize ((isize_to_int x) + (isize_to_int y))’
+val i8_add_def    = Define ‘i8_add x y    = mk_i8 ((i8_to_int x) + (i8_to_int y))’
+val i16_add_def   = Define ‘i16_add x y   = mk_i16 ((i16_to_int x) + (i16_to_int y))’
+val i32_add_def   = Define ‘i32_add x y   = mk_i32 ((i32_to_int x) + (i32_to_int y))’
+val i64_add_def   = Define ‘i64_add x y   = mk_i64 ((i64_to_int x) + (i64_to_int y))’
+val i128_add_def  = Define ‘i128_add x y  = mk_i128 ((i128_to_int x) + (i128_to_int y))’
+val usize_add_def = Define ‘usize_add x y = mk_usize ((usize_to_int x) + (usize_to_int y))’
+val u8_add_def    = Define ‘u8_add x y    = mk_u8 ((u8_to_int x) + (u8_to_int y))’
+val u16_add_def   = Define ‘u16_add x y   = mk_u16 ((u16_to_int x) + (u16_to_int y))’
+val u32_add_def   = Define ‘u32_add x y   = mk_u32 ((u32_to_int x) + (u32_to_int y))’
+val u64_add_def   = Define ‘u64_add x y   = mk_u64 ((u64_to_int x) + (u64_to_int y))’
+val u128_add_def  = Define ‘u128_add x y  = mk_u128 ((u128_to_int x) + (u128_to_int y))’
+
+val all_add_defs = [
+  isize_add_def,
+  i8_add_def,
+  i16_add_def,
+  i32_add_def,
+  i64_add_def,
+  i128_add_def,
+  usize_add_def,
+  u8_add_def,
+  u16_add_def,
+  u32_add_def,
+  u64_add_def,
+  u128_add_def
+]
+
+val isize_sub_def = Define ‘isize_sub x y = mk_isize ((isize_to_int x) - (isize_to_int y))’
+val i8_sub_def    = Define ‘i8_sub x y    = mk_i8 ((i8_to_int x) - (i8_to_int y))’
+val i16_sub_def   = Define ‘i16_sub x y   = mk_i16 ((i16_to_int x) - (i16_to_int y))’
+val i32_sub_def   = Define ‘i32_sub x y   = mk_i32 ((i32_to_int x) - (i32_to_int y))’
+val i64_sub_def   = Define ‘i64_sub x y   = mk_i64 ((i64_to_int x) - (i64_to_int y))’
+val i128_sub_def  = Define ‘i128_sub x y  = mk_i128 ((i128_to_int x) - (i128_to_int y))’
+val usize_sub_def = Define ‘usize_sub x y = mk_usize ((usize_to_int x) - (usize_to_int y))’
+val u8_sub_def    = Define ‘u8_sub x y    = mk_u8 ((u8_to_int x) - (u8_to_int y))’
+val u16_sub_def   = Define ‘u16_sub x y   = mk_u16 ((u16_to_int x) - (u16_to_int y))’
+val u32_sub_def   = Define ‘u32_sub x y   = mk_u32 ((u32_to_int x) - (u32_to_int y))’
+val u64_sub_def   = Define ‘u64_sub x y   = mk_u64 ((u64_to_int x) - (u64_to_int y))’
+val u128_sub_def  = Define ‘u128_sub x y  = mk_u128 ((u128_to_int x) - (u128_to_int y))’
+
+val all_sub_defs = [
+  isize_sub_def,
+  i8_sub_def,
+  i16_sub_def,
+  i32_sub_def,
+  i64_sub_def,
+  i128_sub_def,
+  usize_sub_def,
+  u8_sub_def,
+  u16_sub_def,
+  u32_sub_def,
+  u64_sub_def,
+  u128_sub_def
+]
+
+val isize_mul_def = Define ‘isize_mul x y = mk_isize ((isize_to_int x) * (isize_to_int y))’
+val i8_mul_def    = Define ‘i8_mul x y    = mk_i8 ((i8_to_int x) * (i8_to_int y))’
+val i16_mul_def   = Define ‘i16_mul x y   = mk_i16 ((i16_to_int x) * (i16_to_int y))’
+val i32_mul_def   = Define ‘i32_mul x y   = mk_i32 ((i32_to_int x) * (i32_to_int y))’
+val i64_mul_def   = Define ‘i64_mul x y   = mk_i64 ((i64_to_int x) * (i64_to_int y))’
+val i128_mul_def  = Define ‘i128_mul x y  = mk_i128 ((i128_to_int x) * (i128_to_int y))’
+val usize_mul_def = Define ‘usize_mul x y = mk_usize ((usize_to_int x) * (usize_to_int y))’
+val u8_mul_def    = Define ‘u8_mul x y    = mk_u8 ((u8_to_int x) * (u8_to_int y))’
+val u16_mul_def   = Define ‘u16_mul x y   = mk_u16 ((u16_to_int x) * (u16_to_int y))’
+val u32_mul_def   = Define ‘u32_mul x y   = mk_u32 ((u32_to_int x) * (u32_to_int y))’
+val u64_mul_def   = Define ‘u64_mul x y   = mk_u64 ((u64_to_int x) * (u64_to_int y))’
+val u128_mul_def  = Define ‘u128_mul x y  = mk_u128 ((u128_to_int x) * (u128_to_int y))’
+
+val all_mul_defs = [
+  isize_mul_def,
+  i8_mul_def,
+  i16_mul_def,
+  i32_mul_def,
+  i64_mul_def,
+  i128_mul_def,
+  usize_mul_def,
+  u8_mul_def,
+  u16_mul_def,
+  u32_mul_def,
+  u64_mul_def,
+  u128_mul_def
+]
+
+val isize_div_def = Define ‘isize_div x y =
+  if isize_to_int y = 0 then Fail Failure else mk_isize ((isize_to_int x) / (isize_to_int y))’
+val i8_div_def = Define ‘i8_div x y =
+  if i8_to_int y = 0 then Fail Failure else mk_i8 ((i8_to_int x) / (i8_to_int y))’
+val i16_div_def = Define ‘i16_div x y =
+  if i16_to_int y = 0 then Fail Failure else mk_i16 ((i16_to_int x) / (i16_to_int y))’
+val i32_div_def = Define ‘i32_div x y =
+  if i32_to_int y = 0 then Fail Failure else mk_i32 ((i32_to_int x) / (i32_to_int y))’
+val i64_div_def = Define ‘i64_div x y =
+  if i64_to_int y = 0 then Fail Failure else mk_i64 ((i64_to_int x) / (i64_to_int y))’
+val i128_div_def = Define ‘i128_div x y =
+  if i128_to_int y = 0 then Fail Failure else mk_i128 ((i128_to_int x) / (i128_to_int y))’
+val usize_div_def = Define ‘usize_div x y =
+  if usize_to_int y = 0 then Fail Failure else mk_usize ((usize_to_int x) / (usize_to_int y))’
+val u8_div_def = Define ‘u8_div x y =
+  if u8_to_int y = 0 then Fail Failure else mk_u8 ((u8_to_int x) / (u8_to_int y))’
+val u16_div_def = Define ‘u16_div x y =
+  if u16_to_int y = 0 then Fail Failure else mk_u16 ((u16_to_int x) / (u16_to_int y))’
+val u32_div_def = Define ‘u32_div x y =
+  if u32_to_int y = 0 then Fail Failure else mk_u32 ((u32_to_int x) / (u32_to_int y))’
+val u64_div_def = Define ‘u64_div x y =
+  if u64_to_int y = 0 then Fail Failure else mk_u64 ((u64_to_int x) / (u64_to_int y))’
+val u128_div_def = Define ‘u128_div x y =
+  if u128_to_int y = 0 then Fail Failure else mk_u128 ((u128_to_int x) / (u128_to_int y))’
+
+val all_div_defs = [
+  isize_div_def,
+  i8_div_def,
+  i16_div_def,
+  i32_div_def,
+  i64_div_def,
+  i128_div_def,
+  usize_div_def,
+  u8_div_def,
+  u16_div_def,
+  u32_div_def,
+  u64_div_def,
+  u128_div_def
+]
+
+(* The remainder operation is not a modulo.
+
+   In Rust, the remainder has the sign of the dividend.
+   In HOL4, it has the sign of the divisor.
+ *)
+val int_rem_def = Define ‘int_rem (x : int) (y : int) : int =
+  if (x >= 0 /\ y >= 0) \/ (x < 0 /\ y < 0) then x % y else -(x % y)’
+
+(* Checking consistency with Rust *)
+val _ = prove(“int_rem 1 2 = 1”, EVAL_TAC)
+val _ = prove(“int_rem (-1) 2 = -1”, EVAL_TAC)
+val _ = prove(“int_rem 1 (-2) = 1”, EVAL_TAC)
+val _ = prove(“int_rem (-1) (-2) = -1”, EVAL_TAC)
+
+val isize_rem_def = Define ‘isize_rem x y =
+  if isize_to_int y = 0 then Fail Failure else mk_isize (int_rem (isize_to_int x) (isize_to_int y))’
+val i8_rem_def = Define ‘i8_rem x y =
+  if i8_to_int y = 0 then Fail Failure else mk_i8 (int_rem (i8_to_int x) (i8_to_int y))’
+val i16_rem_def = Define ‘i16_rem x y =
+  if i16_to_int y = 0 then Fail Failure else mk_i16 (int_rem (i16_to_int x) (i16_to_int y))’
+val i32_rem_def = Define ‘i32_rem x y =
+  if i32_to_int y = 0 then Fail Failure else mk_i32 (int_rem (i32_to_int x) (i32_to_int y))’
+val i64_rem_def = Define ‘i64_rem x y =
+  if i64_to_int y = 0 then Fail Failure else mk_i64 (int_rem (i64_to_int x) (i64_to_int y))’
+val i128_rem_def = Define ‘i128_rem x y =
+  if i128_to_int y = 0 then Fail Failure else mk_i128 (int_rem (i128_to_int x) (i128_to_int y))’
+val usize_rem_def = Define ‘usize_rem x y =
+  if usize_to_int y = 0 then Fail Failure else mk_usize (int_rem (usize_to_int x) (usize_to_int y))’
+val u8_rem_def = Define ‘u8_rem x y =
+  if u8_to_int y = 0 then Fail Failure else mk_u8 (int_rem (u8_to_int x) (u8_to_int y))’
+val u16_rem_def = Define ‘u16_rem x y =
+  if u16_to_int y = 0 then Fail Failure else mk_u16 (int_rem (u16_to_int x) (u16_to_int y))’
+val u32_rem_def = Define ‘u32_rem x y =
+  if u32_to_int y = 0 then Fail Failure else mk_u32 (int_rem (u32_to_int x) (u32_to_int y))’
+val u64_rem_def = Define ‘u64_rem x y =
+  if u64_to_int y = 0 then Fail Failure else mk_u64 (int_rem (u64_to_int x) (u64_to_int y))’
+val u128_rem_def = Define ‘u128_rem x y =
+  if u128_to_int y = 0 then Fail Failure else mk_u128 (int_rem (u128_to_int x) (u128_to_int y))’
+
+val all_rem_defs = [
+  isize_rem_def,
+  i8_rem_def,
+  i16_rem_def,
+  i32_rem_def,
+  i64_rem_def,
+  i128_rem_def,
+  usize_rem_def,
+  u8_rem_def,
+  u16_rem_def,
+  u32_rem_def,
+  u64_rem_def,
+  u128_rem_def
+]
+
+(*
+val (asms,g) = top_goal ()
+*)
+
+fun prove_arith_op_eq (asms, g) =
+  let
+    val (_, t) = (dest_exists o snd o strip_imp o snd o strip_forall) g;
+    val (x_to_int, y_to_int) =
+      case (snd o strip_comb o rhs o snd o dest_conj) t of
+        [x, y] => (x,y)
+      | _ => failwith "Unexpected"
+    val x = (snd o dest_comb) x_to_int;
+    val y = (snd o dest_comb) y_to_int;
+    fun inst_first_lem arg lems =
+      MAP_FIRST (fn th => (ASSUME_TAC (SPEC arg th) handle HOL_ERR _ => FAIL_TAC "")) lems;
+  in
+    (rpt gen_tac >>
+     rpt DISCH_TAC >>
+     ASSUME_TAC usize_bounds >> (* Only useful for usize of course *)
+     ASSUME_TAC isize_bounds >> (* Only useful for isize of course *)
+     rw (int_rem_def :: List.concat [all_rem_defs, all_add_defs, all_sub_defs, all_mul_defs, all_div_defs, all_mk_int_defs, all_to_int_bounds_lemmas, all_conversion_id_lemmas]) >>
+     fs (int_rem_def :: List.concat [all_rem_defs, all_add_defs, all_sub_defs, all_mul_defs, all_div_defs, all_mk_int_defs, all_to_int_bounds_lemmas, all_conversion_id_lemmas]) >>
+     inst_first_lem x all_to_int_bounds_lemmas >>
+     inst_first_lem y all_to_int_bounds_lemmas >>
+     gs [NOT_LE_EQ_GT, NOT_LT_EQ_GE, NOT_GE_EQ_LT, NOT_GT_EQ_LE, GE_EQ_LE, GT_EQ_LT] >>
+     TRY COOPER_TAC >>
+     FIRST [
+       (* For division *)
+       qspecl_then [‘^x_to_int’, ‘^y_to_int’] ASSUME_TAC POS_DIV_POS_IS_POS >>
+       qspecl_then [‘^x_to_int’, ‘^y_to_int’] ASSUME_TAC POS_DIV_POS_LE_INIT >>
+       COOPER_TAC,
+       (* For remainder *)
+       dep_rewrite.DEP_PURE_ONCE_REWRITE_TAC all_conversion_id_lemmas >> fs [] >>
+       qspecl_then [‘^x_to_int’, ‘^y_to_int’] ASSUME_TAC POS_MOD_POS_IS_POS >>
+       qspecl_then [‘^x_to_int’, ‘^y_to_int’] ASSUME_TAC POS_MOD_POS_LE_INIT >>
+       COOPER_TAC,
+       dep_rewrite.DEP_PURE_ONCE_REWRITE_TAC all_conversion_id_lemmas >> fs []
+     ]) (asms, g)
+  end
+
+Theorem U8_ADD_EQ:
+  !x y.
+    u8_to_int x + u8_to_int y <= u8_max ==>
+    ?z. u8_add x y = Return z /\ u8_to_int z = u8_to_int x + u8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_ADD_EQ:
+  !(x y).
+    u16_to_int x + u16_to_int y <= u16_max ==>
+    ?(z). u16_add x y = Return z /\ u16_to_int z = u16_to_int x + u16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_ADD_EQ:
+  !x y.
+    u32_to_int x + u32_to_int y <= u32_max ==>
+    ?z. u32_add x y = Return z /\ u32_to_int z = u32_to_int x + u32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_ADD_EQ:
+  !x y.
+    u64_to_int x + u64_to_int y <= u64_max ==>
+    ?z. u64_add x y = Return z /\ u64_to_int z = u64_to_int x + u64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_ADD_EQ:
+  !x y.
+    u128_to_int x + u128_to_int y <= u128_max ==>
+    ?z. u128_add x y = Return z /\ u128_to_int z = u128_to_int x + u128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_ADD_EQ:
+  !x y.
+    (usize_to_int x + usize_to_int y <= u16_max) \/ (usize_to_int x + usize_to_int y <= usize_max) ==>
+    ?z. usize_add x y = Return z /\ usize_to_int z = usize_to_int x + usize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_ADD_EQ:
+  !x y.
+    i8_min <= i8_to_int x + i8_to_int y ==>
+    i8_to_int x + i8_to_int y <= i8_max ==>
+    ?z. i8_add x y = Return z /\ i8_to_int z = i8_to_int x + i8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_ADD_EQ:
+  !x y.
+    i16_min <= i16_to_int x + i16_to_int y ==>
+    i16_to_int x + i16_to_int y <= i16_max ==>
+    ?z. i16_add x y = Return z /\ i16_to_int z = i16_to_int x + i16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_ADD_EQ:
+  !x y.
+    i32_min <= i32_to_int x + i32_to_int y ==>
+    i32_to_int x + i32_to_int y <= i32_max ==>
+    ?z. i32_add x y = Return z /\ i32_to_int z = i32_to_int x + i32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_ADD_EQ:
+  !x y.
+    i64_min <= i64_to_int x + i64_to_int y ==>
+    i64_to_int x + i64_to_int y <= i64_max ==>
+    ?z. i64_add x y = Return z /\ i64_to_int z = i64_to_int x + i64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I128_ADD_EQ:
+  !x y.
+    i128_min <= i128_to_int x + i128_to_int y ==>
+    i128_to_int x + i128_to_int y <= i128_max ==>
+    ?z. i128_add x y = Return z /\ i128_to_int z = i128_to_int x + i128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem ISIZE_ADD_EQ:
+  !x y.
+    (i16_min <= isize_to_int x + isize_to_int y \/ isize_min <= isize_to_int x + isize_to_int y) ==>
+    (isize_to_int x + isize_to_int y <= i16_max \/ isize_to_int x + isize_to_int y <= isize_max) ==>
+    ?z. isize_add x y = Return z /\ isize_to_int z = isize_to_int x + isize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+val all_add_eqs = [
+  ISIZE_ADD_EQ,
+  I8_ADD_EQ,
+  I16_ADD_EQ,
+  I32_ADD_EQ,
+  I64_ADD_EQ,
+  I128_ADD_EQ,
+  USIZE_ADD_EQ,
+  U8_ADD_EQ,
+  U16_ADD_EQ,
+  U32_ADD_EQ,
+  U64_ADD_EQ,
+  U128_ADD_EQ
+]
+      
+Theorem U8_SUB_EQ:
+  !x y.
+    0 <= u8_to_int x - u8_to_int y ==>
+    ?z. u8_sub x y = Return z /\ u8_to_int z = u8_to_int x - u8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_SUB_EQ:
+  !x y.
+    0 <= u16_to_int x - u16_to_int y ==>
+    ?z. u16_sub x y = Return z /\ u16_to_int z = u16_to_int x - u16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_SUB_EQ:
+  !x y.
+    0 <= u32_to_int x - u32_to_int y ==>
+    ?z. u32_sub x y = Return z /\ u32_to_int z = u32_to_int x - u32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_SUB_EQ:
+  !x y.
+    0 <= u64_to_int x - u64_to_int y ==>
+    ?z. u64_sub x y = Return z /\ u64_to_int z = u64_to_int x - u64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_SUB_EQ:
+  !x y.
+    0 <= u128_to_int x - u128_to_int y ==>
+    ?z. u128_sub x y = Return z /\ u128_to_int z = u128_to_int x - u128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_SUB_EQ:
+  !x y.
+    0 <= usize_to_int x - usize_to_int y ==>
+    ?z. usize_sub x y = Return z /\ usize_to_int z = usize_to_int x - usize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_SUB_EQ:
+  !x y.
+    i8_min <= i8_to_int x - i8_to_int y ==>
+    i8_to_int x - i8_to_int y <= i8_max ==>
+    ?z. i8_sub x y = Return z /\ i8_to_int z = i8_to_int x - i8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_SUB_EQ:
+  !x y.
+    i16_min <= i16_to_int x - i16_to_int y ==>
+    i16_to_int x - i16_to_int y <= i16_max ==>
+    ?z. i16_sub x y = Return z /\ i16_to_int z = i16_to_int x - i16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_SUB_EQ:
+  !x y.
+    i32_min <= i32_to_int x - i32_to_int y ==>
+    i32_to_int x - i32_to_int y <= i32_max ==>
+    ?z. i32_sub x y = Return z /\ i32_to_int z = i32_to_int x - i32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_SUB_EQ:
+  !x y.
+    i64_min <= i64_to_int x - i64_to_int y ==>
+    i64_to_int x - i64_to_int y <= i64_max ==>
+    ?z. i64_sub x y = Return z /\ i64_to_int z = i64_to_int x - i64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I128_SUB_EQ:
+  !x y.
+    i128_min <= i128_to_int x - i128_to_int y ==>
+    i128_to_int x - i128_to_int y <= i128_max ==>
+    ?z. i128_sub x y = Return z /\ i128_to_int z = i128_to_int x - i128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem ISIZE_SUB_EQ:
+  !x y.
+    (i16_min <= isize_to_int x - isize_to_int y \/ isize_min <= isize_to_int x - isize_to_int y) ==>
+    (isize_to_int x - isize_to_int y <= i16_max \/ isize_to_int x - isize_to_int y <= isize_max) ==>
+    ?z. isize_sub x y = Return z /\ isize_to_int z = isize_to_int x - isize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+val all_sub_eqs = [
+  ISIZE_SUB_EQ,
+  I8_SUB_EQ,
+  I16_SUB_EQ,
+  I32_SUB_EQ,
+  I64_SUB_EQ,
+  I128_SUB_EQ,
+  USIZE_SUB_EQ,
+  U8_SUB_EQ,
+  U16_SUB_EQ,
+  U32_SUB_EQ,
+  U64_SUB_EQ,
+  U128_SUB_EQ
+]
+
+Theorem U8_MUL_EQ:
+  !x y.
+    u8_to_int x * u8_to_int y <= u8_max ==>
+    ?z. u8_mul x y = Return z /\ u8_to_int z = u8_to_int x * u8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_MUL_EQ:
+  !x y.
+    u16_to_int x * u16_to_int y <= u16_max ==>
+    ?z. u16_mul x y = Return z /\ u16_to_int z = u16_to_int x * u16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_MUL_EQ:
+  !x y.
+    u32_to_int x * u32_to_int y <= u32_max ==>
+    ?z. u32_mul x y = Return z /\ u32_to_int z = u32_to_int x * u32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_MUL_EQ:
+  !x y.
+    u64_to_int x * u64_to_int y <= u64_max ==>
+    ?z. u64_mul x y = Return z /\ u64_to_int z = u64_to_int x * u64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_MUL_EQ:
+  !x y.
+    u128_to_int x * u128_to_int y <= u128_max ==>
+    ?z. u128_mul x y = Return z /\ u128_to_int z = u128_to_int x * u128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_MUL_EQ:
+  !x y.
+    (usize_to_int x * usize_to_int y <= u16_max) \/ (usize_to_int x * usize_to_int y <= usize_max) ==>
+    ?z. usize_mul x y = Return z /\ usize_to_int z = usize_to_int x * usize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_MUL_EQ:
+  !x y.
+    i8_min <= i8_to_int x * i8_to_int y ==>
+    i8_to_int x * i8_to_int y <= i8_max ==>
+    ?z. i8_mul x y = Return z /\ i8_to_int z = i8_to_int x * i8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_MUL_EQ:
+  !x y.
+    i16_min <= i16_to_int x * i16_to_int y ==>
+    i16_to_int x * i16_to_int y <= i16_max ==>
+    ?z. i16_mul x y = Return z /\ i16_to_int z = i16_to_int x * i16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_MUL_EQ:
+  !x y.
+    i32_min <= i32_to_int x * i32_to_int y ==>
+    i32_to_int x * i32_to_int y <= i32_max ==>
+    ?z. i32_mul x y = Return z /\ i32_to_int z = i32_to_int x * i32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_MUL_EQ:
+  !x y.
+    i64_min <= i64_to_int x * i64_to_int y ==>
+    i64_to_int x * i64_to_int y <= i64_max ==>
+    ?z. i64_mul x y = Return z /\ i64_to_int z = i64_to_int x * i64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I128_MUL_EQ:
+  !x y.
+    i128_min <= i128_to_int x * i128_to_int y ==>
+    i128_to_int x * i128_to_int y <= i128_max ==>
+    ?z. i128_mul x y = Return z /\ i128_to_int z = i128_to_int x * i128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem ISIZE_MUL_EQ:
+  !x y.
+    (i16_min <= isize_to_int x * isize_to_int y \/ isize_min <= isize_to_int x * isize_to_int y) ==>
+    (isize_to_int x * isize_to_int y <= i16_max \/ isize_to_int x * isize_to_int y <= isize_max) ==>
+    ?z. isize_mul x y = Return z /\ isize_to_int z = isize_to_int x * isize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+val all_mul_eqs = [
+  ISIZE_MUL_EQ,
+  I8_MUL_EQ,
+  I16_MUL_EQ,
+  I32_MUL_EQ,
+  I64_MUL_EQ,
+  I128_MUL_EQ,
+  USIZE_MUL_EQ,
+  U8_MUL_EQ,
+  U16_MUL_EQ,
+  U32_MUL_EQ,
+  U64_MUL_EQ,
+  U128_MUL_EQ
+]
+
+Theorem U8_DIV_EQ:
+  !x y.
+    u8_to_int y <> 0 ==>
+    ?z. u8_div x y = Return z /\ u8_to_int z = u8_to_int x / u8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_DIV_EQ:
+  !x y.
+    u16_to_int y <> 0 ==>
+    ?z. u16_div x y = Return z /\ u16_to_int z = u16_to_int x / u16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_DIV_EQ:
+  !x y.
+    u32_to_int y <> 0 ==>
+    ?z. u32_div x y = Return z /\ u32_to_int z = u32_to_int x / u32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_DIV_EQ:
+  !x y.
+    u64_to_int y <> 0 ==>
+    ?z. u64_div x y = Return z /\ u64_to_int z = u64_to_int x / u64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_DIV_EQ:
+  !x y.
+    u128_to_int y <> 0 ==>
+    ?z. u128_div x y = Return z /\ u128_to_int z = u128_to_int x / u128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_DIV_EQ:
+  !x y.
+    usize_to_int y <> 0 ==>
+    ?z. usize_div x y = Return z /\ usize_to_int z = usize_to_int x / usize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_DIV_EQ:
+  !x y.
+    i8_to_int y <> 0 ==>
+    i8_min <= i8_to_int x / i8_to_int y ==>
+    i8_to_int x / i8_to_int y <= i8_max ==>
+    ?z. i8_div x y = Return z /\ i8_to_int z = i8_to_int x / i8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_DIV_EQ:
+  !x y.
+    i16_to_int y <> 0 ==>
+    i16_min <= i16_to_int x / i16_to_int y ==>
+    i16_to_int x / i16_to_int y <= i16_max ==>
+    ?z. i16_div x y = Return z /\ i16_to_int z = i16_to_int x / i16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_DIV_EQ:
+  !x y.
+    i32_to_int y <> 0 ==>
+    i32_min <= i32_to_int x / i32_to_int y ==>
+    i32_to_int x / i32_to_int y <= i32_max ==>
+    ?z. i32_div x y = Return z /\ i32_to_int z = i32_to_int x / i32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_DIV_EQ:
+  !x y.
+    i64_to_int y <> 0 ==>
+    i64_min <= i64_to_int x / i64_to_int y ==>
+    i64_to_int x / i64_to_int y <= i64_max ==>
+    ?z. i64_div x y = Return z /\ i64_to_int z = i64_to_int x / i64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I128_DIV_EQ:
+  !x y.
+    i128_to_int y <> 0 ==>
+    i128_min <= i128_to_int x / i128_to_int y ==>
+    i128_to_int x / i128_to_int y <= i128_max ==>
+    ?z. i128_div x y = Return z /\ i128_to_int z = i128_to_int x / i128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem ISIZE_DIV_EQ:
+  !x y.
+    isize_to_int y <> 0 ==>
+    (i16_min <= isize_to_int x / isize_to_int y \/ isize_min <= isize_to_int x / isize_to_int y) ==>
+    (isize_to_int x / isize_to_int y <= i16_max \/ isize_to_int x / isize_to_int y <= isize_max) ==>
+    ?z. isize_div x y = Return z /\ isize_to_int z = isize_to_int x / isize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+val all_div_eqs = [
+  ISIZE_DIV_EQ,
+  I8_DIV_EQ,
+  I16_DIV_EQ,
+  I32_DIV_EQ,
+  I64_DIV_EQ,
+  I128_DIV_EQ,
+  USIZE_DIV_EQ,
+  U8_DIV_EQ,
+  U16_DIV_EQ,
+  U32_DIV_EQ,
+  U64_DIV_EQ,
+  U128_DIV_EQ
+]
+
+Theorem U8_REM_EQ:
+  !x y.
+    u8_to_int y <> 0 ==>
+    ?z. u8_rem x y = Return z /\ u8_to_int z = int_rem (u8_to_int x) (u8_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_REM_EQ:
+  !x y.
+    u16_to_int y <> 0 ==>
+    ?z. u16_rem x y = Return z /\ u16_to_int z = int_rem (u16_to_int x) (u16_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_REM_EQ:
+  !x y.
+    u32_to_int y <> 0 ==>
+    ?z. u32_rem x y = Return z /\ u32_to_int z = int_rem (u32_to_int x) (u32_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_REM_EQ:
+  !x y.
+    u64_to_int y <> 0 ==>
+    ?z. u64_rem x y = Return z /\ u64_to_int z = int_rem (u64_to_int x) (u64_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_REM_EQ:
+  !x y.
+    u128_to_int y <> 0 ==>
+    ?z. u128_rem x y = Return z /\ u128_to_int z = int_rem (u128_to_int x) (u128_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_REM_EQ:
+  !x y.
+    usize_to_int y <> 0 ==>
+    ?z. usize_rem x y = Return z /\ usize_to_int z = int_rem (usize_to_int x) (usize_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_REM_EQ:
+  !x y.
+    i8_to_int y <> 0 ==>
+    i8_min <= int_rem (i8_to_int x) (i8_to_int y) ==>
+    int_rem (i8_to_int x) (i8_to_int y) <= i8_max ==>
+    ?z. i8_rem x y = Return z /\ i8_to_int z = int_rem (i8_to_int x) (i8_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_REM_EQ:
+  !x y.
+    i16_to_int y <> 0 ==>
+    i16_min <= int_rem (i16_to_int x) (i16_to_int y) ==>
+    int_rem (i16_to_int x) (i16_to_int y) <= i16_max ==>
+    ?z. i16_rem x y = Return z /\ i16_to_int z = int_rem (i16_to_int x) (i16_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_REM_EQ:
+  !x y.
+    i32_to_int y <> 0 ==>
+    i32_min <= int_rem (i32_to_int x) (i32_to_int y) ==>
+    int_rem (i32_to_int x) (i32_to_int y) <= i32_max ==>
+    ?z. i32_rem x y = Return z /\ i32_to_int z = int_rem (i32_to_int x) (i32_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_REM_EQ:
+  !x y.
+    i64_to_int y <> 0 ==>
+    i64_min <= int_rem (i64_to_int x) (i64_to_int y) ==>
+    int_rem (i64_to_int x) (i64_to_int y) <= i64_max ==>
+    ?z. i64_rem x y = Return z /\ i64_to_int z = int_rem (i64_to_int x) (i64_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_REM_EQ:
+  !x y.
+    i8_to_int y <> 0 ==>
+    i8_min <= int_rem (i8_to_int x) (i8_to_int y) ==>
+    int_rem (i8_to_int x) (i8_to_int y) <= i8_max ==>
+    ?z. i8_rem x y = Return z /\ i8_to_int z = int_rem (i8_to_int x) (i8_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_REM_EQ:
+  !x y.
+    i8_to_int y <> 0 ==>
+    i8_min <= int_rem (i8_to_int x) (i8_to_int y) ==>
+    int_rem (i8_to_int x) (i8_to_int y) <= i8_max ==>
+    ?z. i8_rem x y = Return z /\ i8_to_int z = int_rem (i8_to_int x) (i8_to_int y)
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U16_DIV_EQ:
+  !x y.
+    u16_to_int y <> 0 ==>
+    ?z. u16_div x y = Return z /\ u16_to_int z = u16_to_int x / u16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U32_DIV_EQ:
+  !x y.
+    u32_to_int y <> 0 ==>
+    ?z. u32_div x y = Return z /\ u32_to_int z = u32_to_int x / u32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U64_DIV_EQ:
+  !x y.
+    u64_to_int y <> 0 ==>
+    ?z. u64_div x y = Return z /\ u64_to_int z = u64_to_int x / u64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem U128_DIV_EQ:
+  !x y.
+    u128_to_int y <> 0 ==>
+    ?z. u128_div x y = Return z /\ u128_to_int z = u128_to_int x / u128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem USIZE_DIV_EQ:
+  !x y.
+    usize_to_int y <> 0 ==>
+    ?z. usize_div x y = Return z /\ usize_to_int z = usize_to_int x / usize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I8_DIV_EQ:
+  !x y.
+    i8_to_int y <> 0 ==>
+    i8_min <= i8_to_int x / i8_to_int y ==>
+    i8_to_int x / i8_to_int y <= i8_max ==>
+    ?z. i8_div x y = Return z /\ i8_to_int z = i8_to_int x / i8_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I16_DIV_EQ:
+  !x y.
+    i16_to_int y <> 0 ==>
+    i16_min <= i16_to_int x / i16_to_int y ==>
+    i16_to_int x / i16_to_int y <= i16_max ==>
+    ?z. i16_div x y = Return z /\ i16_to_int z = i16_to_int x / i16_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I32_DIV_EQ:
+  !x y.
+    i32_to_int y <> 0 ==>
+    i32_min <= i32_to_int x / i32_to_int y ==>
+    i32_to_int x / i32_to_int y <= i32_max ==>
+    ?z. i32_div x y = Return z /\ i32_to_int z = i32_to_int x / i32_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I64_DIV_EQ:
+  !x y.
+    i64_to_int y <> 0 ==>
+    i64_min <= i64_to_int x / i64_to_int y ==>
+    i64_to_int x / i64_to_int y <= i64_max ==>
+    ?z. i64_div x y = Return z /\ i64_to_int z = i64_to_int x / i64_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem I128_DIV_EQ:
+  !x y.
+    i128_to_int y <> 0 ==>
+    i128_min <= i128_to_int x / i128_to_int y ==>
+    i128_to_int x / i128_to_int y <= i128_max ==>
+    ?z. i128_div x y = Return z /\ i128_to_int z = i128_to_int x / i128_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+Theorem ISIZE_DIV_EQ:
+  !x y.
+    isize_to_int y <> 0 ==>
+    (i16_min <= isize_to_int x / isize_to_int y \/ isize_min <= isize_to_int x / isize_to_int y) ==>
+    (isize_to_int x / isize_to_int y <= i16_max \/ isize_to_int x / isize_to_int y <= isize_max) ==>
+    ?z. isize_div x y = Return z /\ isize_to_int z = isize_to_int x / isize_to_int y
+Proof
+  prove_arith_op_eq
+QED
+
+val all_div_eqs = [
+  ISIZE_DIV_EQ,
+  I8_DIV_EQ,
+  I16_DIV_EQ,
+  I32_DIV_EQ,
+  I64_DIV_EQ,
+  I128_DIV_EQ,
+  USIZE_DIV_EQ,
+  U8_DIV_EQ,
+  U16_DIV_EQ,
+  U32_DIV_EQ,
+  U64_DIV_EQ,
+  U128_DIV_EQ
+]
+
+(***
+ * Vectors
+ *)
+
+val _ = new_type ("vec", 1)
+val _ = new_constant ("vec_to_list", “:'a vec -> 'a list”)
+
+val VEC_TO_LIST_NUM_BOUNDS =
+  new_axiom (
+    "VEC_TO_LIST_BOUNDS",
+    “!v. let l = LENGTH (vec_to_list v) in
+     (0:num) <= l /\ l <= (4294967295:num)”)
+
+Theorem VEC_TO_LIST_INT_BOUNDS:
+  !v. let l = int_of_num (LENGTH (vec_to_list v)) in
+     0 <= l /\ l <= u32_max
+Proof
+  gen_tac >>
+  rw [u32_max_def] >>
+  assume_tac VEC_TO_LIST_NUM_BOUNDS >>
+  fs[]
+QED
+
+val VEC_LEN_DEF = Define ‘vec_len v = int_to_u32 (int_of_num (LENGTH (vec_to_list v)))’
+
+
+val _ = export_theory ()