diff options
| -rw-r--r-- | backends/hol4/Primitives.sml | 1021 |
1 files changed, 997 insertions, 24 deletions
diff --git a/backends/hol4/Primitives.sml b/backends/hol4/Primitives.sml index 4d77b84a..be384012 100644 --- a/backends/hol4/Primitives.sml +++ b/backends/hol4/Primitives.sml @@ -165,6 +165,21 @@ val u64_to_int_bounds = new_axiom ("u64_to_int_bounds", 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 @@ -295,6 +310,22 @@ val mk_u128_def = Define 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))’ @@ -308,6 +339,21 @@ val u32_add_def = Define ‘u32_add x y = mk_u32 ((u32_to_int x) + (u32_to_i 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))’ @@ -321,6 +367,21 @@ val u32_sub_def = Define ‘u32_sub x y = mk_u32 ((u32_to_int x) - (u32_to_i 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))’ @@ -334,6 +395,21 @@ val u32_mul_def = Define ‘u32_mul x y = mk_u32 ((u32_to_int x) * (u32_to_i 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 = @@ -359,27 +435,924 @@ val u64_div_def = Define ‘u64_div x 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 isize_mod_def = Define ‘isize_mod x y = - if isize_to_int y = 0 then Fail Failure else mk_isize ((isize_to_int x) % (isize_to_int y))’ -val i8_mod_def = Define ‘i8_mod x y = - if i8_to_int y = 0 then Fail Failure else mk_i8 ((i8_to_int x) % (i8_to_int y))’ -val i16_mod_def = Define ‘i16_mod x y = - if i16_to_int y = 0 then Fail Failure else mk_i16 ((i16_to_int x) % (i16_to_int y))’ -val i32_mod_def = Define ‘i32_mod x y = - if i32_to_int y = 0 then Fail Failure else mk_i32 ((i32_to_int x) % (i32_to_int y))’ -val i64_mod_def = Define ‘i64_mod x y = - if i64_to_int y = 0 then Fail Failure else mk_i64 ((i64_to_int x) % (i64_to_int y))’ -val i128_mod_def = Define ‘i128_mod x y = - if i128_to_int y = 0 then Fail Failure else mk_i128 ((i128_to_int x) % (i128_to_int y))’ -val usize_mod_def = Define ‘usize_mod x y = - if usize_to_int y = 0 then Fail Failure else mk_usize ((usize_to_int x) % (usize_to_int y))’ -val u8_mod_def = Define ‘u8_mod x y = - if u8_to_int y = 0 then Fail Failure else mk_u8 ((u8_to_int x) % (u8_to_int y))’ -val u16_mod_def = Define ‘u16_mod x y = - if u16_to_int y = 0 then Fail Failure else mk_u16 ((u16_to_int x) % (u16_to_int y))’ -val u32_mod_def = Define ‘u32_mod x y = - if u32_to_int y = 0 then Fail Failure else mk_u32 ((u32_to_int x) % (u32_to_int y))’ -val u64_mod_def = Define ‘u64_mod x y = - if u64_to_int y = 0 then Fail Failure else mk_u64 ((u64_to_int x) % (u64_to_int y))’ -val u128_mod_def = Define ‘u128_mod 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 +] + +(* Ignore a theorem. + + To be used in conjunction with {!pop_assum} for instance. + *) +fun IGNORE_TAC (_ : thm) : tactic = ALL_TAC + +val POP_IGNORE_TAC = POP_ASSUM IGNORE_TAC + +(* TODO: we need a better library of lemmas about arithmetic *) + +(* TODO: add those as rewriting tactics by default *) +val NOT_LE_EQ_GT = store_thm("NOT_LE_EQ_GT", “!(x y: int). ~(x <= y) <=> x > y”, COOPER_TAC) +val NOT_LT_EQ_GE = store_thm("NOT_LT_EQ_GE", “!(x y: int). ~(x < y) <=> x >= y”, COOPER_TAC) +val NOT_GE_EQ_LT = store_thm("NOT_GE_EQ_LT", “!(x y: int). ~(x >= y) <=> x < y”, COOPER_TAC) +val NOT_GT_EQ_LE = store_thm("NOT_GT_EQ_LE", “!(x y: int). ~(x > y) <=> x <= y”, COOPER_TAC) + +Theorem POS_MUL_POS_IS_POS: + !(x y : int). 0 <= x ==> 0 <= y ==> 0 <= x * y +Proof + rpt strip_tac >> + sg ‘0 <= &(Num x) * &(Num y)’ >- (rw [INT_MUL_CALCULATE] >> COOPER_TAC) >> + sg ‘&(Num x) = x’ >- (irule EQ_SYM >> rw [INT_OF_NUM] >> COOPER_TAC) >> + sg ‘&(Num y) = y’ >- (irule EQ_SYM >> rw [INT_OF_NUM] >> COOPER_TAC) >> + metis_tac[] +QED + +val GE_EQ_LE = store_thm("GE_EQ_LE", “!(x y : int). x >= y <=> y <= x”, COOPER_TAC) +val LE_EQ_GE = store_thm("LE_EQ_GE", “!(x y : int). x <= y <=> y >= x”, COOPER_TAC) +val GT_EQ_LT = store_thm("GT_EQ_LT", “!(x y : int). x > y <=> y < x”, COOPER_TAC) +val LT_EQ_GT = store_thm("LT_EQ_GT", “!(x y : int). x < y <=> y > x”, COOPER_TAC) + +Theorem POS_DIV_POS_IS_POS: + !(x y : int). 0 <= x ==> 0 < y ==> 0 <= x / y +Proof + rpt strip_tac >> + rw [LE_EQ_GE] >> + sg ‘y <> 0’ >- COOPER_TAC >> + qspecl_then [‘\x. x >= 0’, ‘x’, ‘y’] ASSUME_TAC INT_DIV_FORALL_P >> + fs [] >> POP_IGNORE_TAC >> rw [] >- COOPER_TAC >> + fs [NOT_LT_EQ_GE] >> + (* Proof by contradiction: assume k < 0 *) + spose_not_then ASSUME_TAC >> + fs [NOT_GE_EQ_LT] >> + sg ‘k * y = (k + 1) * y + - y’ >- (fs [INT_RDISTRIB] >> COOPER_TAC) >> + sg ‘0 <= (-(k + 1)) * y’ >- (irule POS_MUL_POS_IS_POS >> COOPER_TAC) >> + COOPER_TAC +QED + +Theorem POS_DIV_POS_LE: + !(x y d : int). 0 <= x ==> 0 <= y ==> 0 < d ==> x <= y ==> x / d <= y / d +Proof + rpt strip_tac >> + sg ‘d <> 0’ >- COOPER_TAC >> + qspecl_then [‘\k. k = x / d’, ‘x’, ‘d’] ASSUME_TAC INT_DIV_P >> + qspecl_then [‘\k. k = y / d’, ‘y’, ‘d’] ASSUME_TAC INT_DIV_P >> + rfs [NOT_LT_EQ_GE] >> TRY COOPER_TAC >> + sg ‘y = (x / d) * d + (r' + y - x)’ >- COOPER_TAC >> + qspecl_then [‘(x / d) * d’, ‘r' + y - x’, ‘d’] ASSUME_TAC INT_ADD_DIV >> + rfs [] >> + Cases_on ‘x = y’ >- fs [] >> + sg ‘r' + y ≠ x’ >- COOPER_TAC >> fs [] >> + sg ‘((x / d) * d) / d = x / d’ >- (irule INT_DIV_RMUL >> COOPER_TAC) >> + fs [] >> + sg ‘0 <= (r' + y − x) / d’ >- (irule POS_DIV_POS_IS_POS >> COOPER_TAC) >> + metis_tac [INT_LE_ADDR] +QED + +Theorem POS_DIV_POS_LE_INIT: + !(x y : int). 0 <= x ==> 0 < y ==> x / y <= x +Proof + rpt strip_tac >> + sg ‘y <> 0’ >- COOPER_TAC >> + qspecl_then [‘\k. k = x / y’, ‘x’, ‘y’] ASSUME_TAC INT_DIV_P >> + rfs [NOT_LT_EQ_GE] >- COOPER_TAC >> + sg ‘y = (y - 1) + 1’ >- rw [] >> + sg ‘x = x / y + ((x / y) * (y - 1) + r)’ >-( + qspecl_then [‘1’, ‘y-1’, ‘x / y’] ASSUME_TAC INT_LDISTRIB >> + rfs [] >> + COOPER_TAC + ) >> + sg ‘!a b c. 0 <= c ==> a = b + c ==> b <= a’ >- (COOPER_TAC) >> + pop_assum irule >> + exists_tac “x / y * (y − 1) + r” >> + sg ‘0 <= x / y’ >- (irule POS_DIV_POS_IS_POS >> COOPER_TAC) >> + sg ‘0 <= (x / y) * (y - 1)’ >- (irule POS_MUL_POS_IS_POS >> COOPER_TAC) >> + COOPER_TAC +QED + +Theorem POS_MOD_POS_IS_POS: + !(x y : int). 0 <= x ==> 0 < y ==> 0 <= x % y +Proof + rpt strip_tac >> + sg ‘y <> 0’ >- COOPER_TAC >> + imp_res_tac INT_DIVISION >> + first_x_assum (qspec_then ‘x’ assume_tac) >> + first_x_assum (qspec_then ‘x’ assume_tac) >> + sg ‘~(y < 0)’ >- COOPER_TAC >> fs [] +QED + +Theorem POS_MOD_POS_LE_INIT: + !(x y : int). 0 <= x ==> 0 < y ==> x % y <= x +Proof + rpt strip_tac >> + sg ‘y <> 0’ >- COOPER_TAC >> + imp_res_tac INT_DIVISION >> + first_x_assum (qspec_then ‘x’ assume_tac) >> + first_x_assum (qspec_then ‘x’ assume_tac) >> + sg ‘~(y < 0)’ >- COOPER_TAC >> fs [] >> + sg ‘0 <= x % y’ >- (irule POS_MOD_POS_IS_POS >> COOPER_TAC) >> + sg ‘0 <= x / y’ >- (irule POS_DIV_POS_IS_POS >> COOPER_TAC) >> + sg ‘0 <= (x / y) * y’ >- (irule POS_MUL_POS_IS_POS >> COOPER_TAC) >> + COOPER_TAC +QED + +(* +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 +] |