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Diffstat (limited to 'backends/hol4/primitivesScript.sml')
-rw-r--r-- | backends/hol4/primitivesScript.sml | 1270 |
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 () |