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Diffstat (limited to 'backends/hol4/primitivesTheory.sig')
| -rw-r--r-- | backends/hol4/primitivesTheory.sig | 1612 |
1 files changed, 1612 insertions, 0 deletions
diff --git a/backends/hol4/primitivesTheory.sig b/backends/hol4/primitivesTheory.sig new file mode 100644 index 00000000..75098bfe --- /dev/null +++ b/backends/hol4/primitivesTheory.sig @@ -0,0 +1,1612 @@ +signature primitivesTheory = +sig + type thm = Thm.thm + + (* Axioms *) + val VEC_TO_LIST_BOUNDS : thm + val i128_to_int_bounds : thm + val i16_to_int_bounds : thm + val i32_to_int_bounds : thm + val i64_to_int_bounds : thm + val i8_to_int_bounds : thm + val int_to_i128_id : thm + val int_to_i16_id : thm + val int_to_i32_id : thm + val int_to_i64_id : thm + val int_to_i8_id : thm + val int_to_isize_id : thm + val int_to_u128_id : thm + val int_to_u16_id : thm + val int_to_u32_id : thm + val int_to_u64_id : thm + val int_to_u8_id : thm + val int_to_usize_id : thm + val isize_bounds : thm + val isize_to_int_bounds : thm + val u128_to_int_bounds : thm + val u16_to_int_bounds : thm + val u32_to_int_bounds : thm + val u64_to_int_bounds : thm + val u8_to_int_bounds : thm + val usize_bounds : thm + val usize_to_int_bounds : thm + + (* Definitions *) + val bind_def : thm + val error_BIJ : thm + val error_CASE : thm + val error_TY_DEF : thm + val error_size_def : thm + val i128_add_def : thm + val i128_div_def : thm + val i128_max_def : thm + val i128_min_def : thm + val i128_mul_def : thm + val i128_rem_def : thm + val i128_sub_def : thm + val i16_add_def : thm + val i16_div_def : thm + val i16_max_def : thm + val i16_min_def : thm + val i16_mul_def : thm + val i16_rem_def : thm + val i16_sub_def : thm + val i32_add_def : thm + val i32_div_def : thm + val i32_max_def : thm + val i32_min_def : thm + val i32_mul_def : thm + val i32_rem_def : thm + val i32_sub_def : thm + val i64_add_def : thm + val i64_div_def : thm + val i64_max_def : thm + val i64_min_def : thm + val i64_mul_def : thm + val i64_rem_def : thm + val i64_sub_def : thm + val i8_add_def : thm + val i8_div_def : thm + val i8_max_def : thm + val i8_min_def : thm + val i8_mul_def : thm + val i8_rem_def : thm + val i8_sub_def : thm + val int_rem_def : thm + val isize_add_def : thm + val isize_div_def : thm + val isize_mul_def : thm + val isize_rem_def : thm + val isize_sub_def : thm + val massert_def : thm + val mem_replace_back_def : thm + val mem_replace_fwd_def : thm + val mk_i128_def : thm + val mk_i16_def : thm + val mk_i32_def : thm + val mk_i64_def : thm + val mk_i8_def : thm + val mk_isize_def : thm + val mk_u128_def : thm + val mk_u16_def : thm + val mk_u32_def : thm + val mk_u64_def : thm + val mk_u8_def : thm + val mk_usize_def : thm + val result_TY_DEF : thm + val result_case_def : thm + val result_size_def : thm + val return_def : thm + val u128_add_def : thm + val u128_div_def : thm + val u128_max_def : thm + val u128_mul_def : thm + val u128_rem_def : thm + val u128_sub_def : thm + val u16_add_def : thm + val u16_div_def : thm + val u16_max_def : thm + val u16_mul_def : thm + val u16_rem_def : thm + val u16_sub_def : thm + val u32_add_def : thm + val u32_div_def : thm + val u32_max_def : thm + val u32_mul_def : thm + val u32_rem_def : thm + val u32_sub_def : thm + val u64_add_def : thm + val u64_div_def : thm + val u64_max_def : thm + val u64_mul_def : thm + val u64_rem_def : thm + val u64_sub_def : thm + val u8_add_def : thm + val u8_div_def : thm + val u8_max_def : thm + val u8_mul_def : thm + val u8_rem_def : thm + val u8_sub_def : thm + val usize_add_def : thm + val usize_div_def : thm + val usize_mul_def : thm + val usize_rem_def : thm + val usize_sub_def : thm + val vec_len_def : thm + + (* Theorems *) + val I128_ADD_EQ : thm + val I128_DIV_EQ : thm + val I128_MUL_EQ : thm + val I128_SUB_EQ : thm + val I16_ADD_EQ : thm + val I16_DIV_EQ : thm + val I16_MUL_EQ : thm + val I16_REM_EQ : thm + val I16_SUB_EQ : thm + val I32_ADD_EQ : thm + val I32_DIV_EQ : thm + val I32_MUL_EQ : thm + val I32_REM_EQ : thm + val I32_SUB_EQ : thm + val I64_ADD_EQ : thm + val I64_DIV_EQ : thm + val I64_MUL_EQ : thm + val I64_REM_EQ : thm + val I64_SUB_EQ : thm + val I8_ADD_EQ : thm + val I8_DIV_EQ : thm + val I8_MUL_EQ : thm + val I8_REM_EQ : thm + val I8_SUB_EQ : thm + val ISIZE_ADD_EQ : thm + val ISIZE_DIV_EQ : thm + val ISIZE_MUL_EQ : thm + val ISIZE_SUB_EQ : thm + val U128_ADD_EQ : thm + val U128_DIV_EQ : thm + val U128_MUL_EQ : thm + val U128_REM_EQ : thm + val U128_SUB_EQ : thm + val U16_ADD_EQ : thm + val U16_DIV_EQ : thm + val U16_MUL_EQ : thm + val U16_REM_EQ : thm + val U16_SUB_EQ : thm + val U32_ADD_EQ : thm + val U32_DIV_EQ : thm + val U32_MUL_EQ : thm + val U32_REM_EQ : thm + val U32_SUB_EQ : thm + val U64_ADD_EQ : thm + val U64_DIV_EQ : thm + val U64_MUL_EQ : thm + val U64_REM_EQ : thm + val U64_SUB_EQ : thm + val U8_ADD_EQ : thm + val U8_DIV_EQ : thm + val U8_MUL_EQ : thm + val U8_REM_EQ : thm + val U8_SUB_EQ : thm + val USIZE_ADD_EQ : thm + val USIZE_DIV_EQ : thm + val USIZE_MUL_EQ : thm + val USIZE_REM_EQ : thm + val USIZE_SUB_EQ : thm + val VEC_TO_LIST_INT_BOUNDS : thm + val datatype_error : thm + val datatype_result : thm + val error2num_11 : thm + val error2num_ONTO : thm + val error2num_num2error : thm + val error2num_thm : thm + val error_Axiom : thm + val error_EQ_error : thm + val error_case_cong : thm + val error_case_def : thm + val error_case_eq : thm + val error_induction : thm + val error_nchotomy : thm + val num2error_11 : thm + val num2error_ONTO : thm + val num2error_error2num : thm + val num2error_thm : thm + val result_11 : thm + val result_Axiom : thm + val result_case_cong : thm + val result_case_eq : thm + val result_distinct : thm + val result_induction : thm + val result_nchotomy : thm + + val primitives_grammars : type_grammar.grammar * term_grammar.grammar +(* + [primitivesArith] Parent theory of "primitives" + + [string] Parent theory of "primitives" + + [int_to_u128_id] Axiom + + [oracles: ] [axioms: int_to_u128_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u128_max ⇒ u128_to_int (int_to_u128 n) = n + + [int_to_u64_id] Axiom + + [oracles: ] [axioms: int_to_u64_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u64_max ⇒ u64_to_int (int_to_u64 n) = n + + [int_to_u32_id] Axiom + + [oracles: ] [axioms: int_to_u32_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u32_max ⇒ u32_to_int (int_to_u32 n) = n + + [int_to_u16_id] Axiom + + [oracles: ] [axioms: int_to_u16_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u16_max ⇒ u16_to_int (int_to_u16 n) = n + + [int_to_u8_id] Axiom + + [oracles: ] [axioms: int_to_u8_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u8_max ⇒ u8_to_int (int_to_u8 n) = n + + [int_to_i128_id] Axiom + + [oracles: ] [axioms: int_to_i128_id] [] + ⊢ ∀n. i128_min ≤ n ∧ n ≤ i128_max ⇒ i128_to_int (int_to_i128 n) = n + + [int_to_i64_id] Axiom + + [oracles: ] [axioms: int_to_i64_id] [] + ⊢ ∀n. i64_min ≤ n ∧ n ≤ i64_max ⇒ i64_to_int (int_to_i64 n) = n + + [int_to_i32_id] Axiom + + [oracles: ] [axioms: int_to_i32_id] [] + ⊢ ∀n. i32_min ≤ n ∧ n ≤ i32_max ⇒ i32_to_int (int_to_i32 n) = n + + [int_to_i16_id] Axiom + + [oracles: ] [axioms: int_to_i16_id] [] + ⊢ ∀n. i16_min ≤ n ∧ n ≤ i16_max ⇒ i16_to_int (int_to_i16 n) = n + + [int_to_i8_id] Axiom + + [oracles: ] [axioms: int_to_i8_id] [] + ⊢ ∀n. i8_min ≤ n ∧ n ≤ i8_max ⇒ i8_to_int (int_to_i8 n) = n + + [int_to_usize_id] Axiom + + [oracles: ] [axioms: int_to_usize_id] [] + ⊢ ∀n. 0 ≤ n ∧ (n ≤ u16_max ∨ n ≤ usize_max) ⇒ + usize_to_int (int_to_usize n) = n + + [int_to_isize_id] Axiom + + [oracles: ] [axioms: 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 + + [u128_to_int_bounds] Axiom + + [oracles: ] [axioms: u128_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u128_to_int n ∧ u128_to_int n ≤ u128_max + + [u64_to_int_bounds] Axiom + + [oracles: ] [axioms: u64_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u64_to_int n ∧ u64_to_int n ≤ u64_max + + [u32_to_int_bounds] Axiom + + [oracles: ] [axioms: u32_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u32_to_int n ∧ u32_to_int n ≤ u32_max + + [u16_to_int_bounds] Axiom + + [oracles: ] [axioms: u16_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u16_to_int n ∧ u16_to_int n ≤ u16_max + + [u8_to_int_bounds] Axiom + + [oracles: ] [axioms: u8_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u8_to_int n ∧ u8_to_int n ≤ u8_max + + [usize_to_int_bounds] Axiom + + [oracles: ] [axioms: usize_to_int_bounds] [] + ⊢ ∀n. 0 ≤ usize_to_int n ∧ usize_to_int n ≤ usize_max + + [i128_to_int_bounds] Axiom + + [oracles: ] [axioms: i128_to_int_bounds] [] + ⊢ ∀n. i128_min ≤ i128_to_int n ∧ i128_to_int n ≤ i128_max + + [i64_to_int_bounds] Axiom + + [oracles: ] [axioms: i64_to_int_bounds] [] + ⊢ ∀n. i64_min ≤ i64_to_int n ∧ i64_to_int n ≤ i64_max + + [i32_to_int_bounds] Axiom + + [oracles: ] [axioms: i32_to_int_bounds] [] + ⊢ ∀n. i32_min ≤ i32_to_int n ∧ i32_to_int n ≤ i32_max + + [i16_to_int_bounds] Axiom + + [oracles: ] [axioms: i16_to_int_bounds] [] + ⊢ ∀n. i16_min ≤ i16_to_int n ∧ i16_to_int n ≤ i16_max + + [i8_to_int_bounds] Axiom + + [oracles: ] [axioms: i8_to_int_bounds] [] + ⊢ ∀n. i8_min ≤ i8_to_int n ∧ i8_to_int n ≤ i8_max + + [isize_to_int_bounds] Axiom + + [oracles: ] [axioms: isize_to_int_bounds] [] + ⊢ ∀n. isize_min ≤ isize_to_int n ∧ isize_to_int n ≤ isize_max + + [usize_bounds] Axiom + + [oracles: ] [axioms: usize_bounds] [] ⊢ usize_max ≥ u16_max + + [isize_bounds] Axiom + + [oracles: ] [axioms: isize_bounds] [] + ⊢ isize_min ≤ i16_min ∧ isize_max ≥ i16_max + + [VEC_TO_LIST_BOUNDS] Axiom + + [oracles: ] [axioms: VEC_TO_LIST_BOUNDS] [] + ⊢ ∀v. (let l = LENGTH (vec_to_list v) in 0 ≤ l ∧ l ≤ 4294967295) + + [bind_def] Definition + + ⊢ ∀x f. + monad_bind x f = + case x of Return y => f y | Fail e => Fail e | Loop => Loop + + [error_BIJ] Definition + + ⊢ (∀a. num2error (error2num a) = a) ∧ + ∀r. (λn. n < 1) r ⇔ error2num (num2error r) = r + + [error_CASE] Definition + + ⊢ ∀x v0. (case x of Failure => v0) = (λm. v0) (error2num x) + + [error_TY_DEF] Definition + + ⊢ ∃rep. TYPE_DEFINITION (λn. n < 1) rep + + [error_size_def] Definition + + ⊢ ∀x. error_size x = 0 + + [i128_add_def] Definition + + ⊢ ∀x y. i128_add x y = mk_i128 (i128_to_int x + i128_to_int y) + + [i128_div_def] Definition + + ⊢ ∀x y. + i128_div x y = + if i128_to_int y = 0 then Fail Failure + else mk_i128 (i128_to_int x / i128_to_int y) + + [i128_max_def] Definition + + ⊢ i128_max = 170141183460469231731687303715884105727 + + [i128_min_def] Definition + + ⊢ i128_min = -170141183460469231731687303715884105728 + + [i128_mul_def] Definition + + ⊢ ∀x y. i128_mul x y = mk_i128 (i128_to_int x * i128_to_int y) + + [i128_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [i128_sub_def] Definition + + ⊢ ∀x y. i128_sub x y = mk_i128 (i128_to_int x − i128_to_int y) + + [i16_add_def] Definition + + ⊢ ∀x y. i16_add x y = mk_i16 (i16_to_int x + i16_to_int y) + + [i16_div_def] Definition + + ⊢ ∀x y. + i16_div x y = + if i16_to_int y = 0 then Fail Failure + else mk_i16 (i16_to_int x / i16_to_int y) + + [i16_max_def] Definition + + ⊢ i16_max = 32767 + + [i16_min_def] Definition + + ⊢ i16_min = -32768 + + [i16_mul_def] Definition + + ⊢ ∀x y. i16_mul x y = mk_i16 (i16_to_int x * i16_to_int y) + + [i16_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [i16_sub_def] Definition + + ⊢ ∀x y. i16_sub x y = mk_i16 (i16_to_int x − i16_to_int y) + + [i32_add_def] Definition + + ⊢ ∀x y. i32_add x y = mk_i32 (i32_to_int x + i32_to_int y) + + [i32_div_def] Definition + + ⊢ ∀x y. + i32_div x y = + if i32_to_int y = 0 then Fail Failure + else mk_i32 (i32_to_int x / i32_to_int y) + + [i32_max_def] Definition + + ⊢ i32_max = 2147483647 + + [i32_min_def] Definition + + ⊢ i32_min = -2147483648 + + [i32_mul_def] Definition + + ⊢ ∀x y. i32_mul x y = mk_i32 (i32_to_int x * i32_to_int y) + + [i32_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [i32_sub_def] Definition + + ⊢ ∀x y. i32_sub x y = mk_i32 (i32_to_int x − i32_to_int y) + + [i64_add_def] Definition + + ⊢ ∀x y. i64_add x y = mk_i64 (i64_to_int x + i64_to_int y) + + [i64_div_def] Definition + + ⊢ ∀x y. + i64_div x y = + if i64_to_int y = 0 then Fail Failure + else mk_i64 (i64_to_int x / i64_to_int y) + + [i64_max_def] Definition + + ⊢ i64_max = 9223372036854775807 + + [i64_min_def] Definition + + ⊢ i64_min = -9223372036854775808 + + [i64_mul_def] Definition + + ⊢ ∀x y. i64_mul x y = mk_i64 (i64_to_int x * i64_to_int y) + + [i64_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [i64_sub_def] Definition + + ⊢ ∀x y. i64_sub x y = mk_i64 (i64_to_int x − i64_to_int y) + + [i8_add_def] Definition + + ⊢ ∀x y. i8_add x y = mk_i8 (i8_to_int x + i8_to_int y) + + [i8_div_def] Definition + + ⊢ ∀x y. + i8_div x y = + if i8_to_int y = 0 then Fail Failure + else mk_i8 (i8_to_int x / i8_to_int y) + + [i8_max_def] Definition + + ⊢ i8_max = 127 + + [i8_min_def] Definition + + ⊢ i8_min = -128 + + [i8_mul_def] Definition + + ⊢ ∀x y. i8_mul x y = mk_i8 (i8_to_int x * i8_to_int y) + + [i8_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [i8_sub_def] Definition + + ⊢ ∀x y. i8_sub x y = mk_i8 (i8_to_int x − i8_to_int y) + + [int_rem_def] Definition + + ⊢ ∀x y. + int_rem x y = + if x ≥ 0 ∧ y ≥ 0 ∨ x < 0 ∧ y < 0 then x % y else -(x % y) + + [isize_add_def] Definition + + ⊢ ∀x y. isize_add x y = mk_isize (isize_to_int x + isize_to_int y) + + [isize_div_def] Definition + + ⊢ ∀x y. + isize_div x y = + if isize_to_int y = 0 then Fail Failure + else mk_isize (isize_to_int x / isize_to_int y) + + [isize_mul_def] Definition + + ⊢ ∀x y. isize_mul x y = mk_isize (isize_to_int x * isize_to_int y) + + [isize_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [isize_sub_def] Definition + + ⊢ ∀x y. isize_sub x y = mk_isize (isize_to_int x − isize_to_int y) + + [massert_def] Definition + + ⊢ ∀b. massert b = if b then Return () else Fail Failure + + [mem_replace_back_def] Definition + + ⊢ ∀x y. mem_replace_back x y = y + + [mem_replace_fwd_def] Definition + + ⊢ ∀x y. mem_replace_fwd x y = x + + [mk_i128_def] Definition + + ⊢ ∀n. mk_i128 n = + if i128_min ≤ n ∧ n ≤ i128_max then Return (int_to_i128 n) + else Fail Failure + + [mk_i16_def] Definition + + ⊢ ∀n. mk_i16 n = + if i16_min ≤ n ∧ n ≤ i16_max then Return (int_to_i16 n) + else Fail Failure + + [mk_i32_def] Definition + + ⊢ ∀n. mk_i32 n = + if i32_min ≤ n ∧ n ≤ i32_max then Return (int_to_i32 n) + else Fail Failure + + [mk_i64_def] Definition + + ⊢ ∀n. mk_i64 n = + if i64_min ≤ n ∧ n ≤ i64_max then Return (int_to_i64 n) + else Fail Failure + + [mk_i8_def] Definition + + ⊢ ∀n. mk_i8 n = + if i8_min ≤ n ∧ n ≤ i8_max then Return (int_to_i8 n) + else Fail Failure + + [mk_isize_def] Definition + + ⊢ ∀n. mk_isize n = + if isize_min ≤ n ∧ n ≤ isize_max then Return (int_to_isize n) + else Fail Failure + + [mk_u128_def] Definition + + ⊢ ∀n. mk_u128 n = + if 0 ≤ n ∧ n ≤ u128_max then Return (int_to_u128 n) + else Fail Failure + + [mk_u16_def] Definition + + ⊢ ∀n. mk_u16 n = + if 0 ≤ n ∧ n ≤ u16_max then Return (int_to_u16 n) + else Fail Failure + + [mk_u32_def] Definition + + ⊢ ∀n. mk_u32 n = + if 0 ≤ n ∧ n ≤ u32_max then Return (int_to_u32 n) + else Fail Failure + + [mk_u64_def] Definition + + ⊢ ∀n. mk_u64 n = + if 0 ≤ n ∧ n ≤ u64_max then Return (int_to_u64 n) + else Fail Failure + + [mk_u8_def] Definition + + ⊢ ∀n. mk_u8 n = + if 0 ≤ n ∧ n ≤ u8_max then Return (int_to_u8 n) + else Fail Failure + + [mk_usize_def] Definition + + ⊢ ∀n. mk_usize n = + if 0 ≤ n ∧ n ≤ usize_max then Return (int_to_usize n) + else Fail Failure + + [result_TY_DEF] Definition + + ⊢ ∃rep. + TYPE_DEFINITION + (λa0. + ∀ $var$('result'). + (∀a0. + (∃a. a0 = + (λa. + ind_type$CONSTR 0 (a,ARB) + (λn. ind_type$BOTTOM)) a) ∨ + (∃a. a0 = + (λa. + ind_type$CONSTR (SUC 0) (ARB,a) + (λn. ind_type$BOTTOM)) a) ∨ + a0 = + ind_type$CONSTR (SUC (SUC 0)) (ARB,ARB) + (λn. ind_type$BOTTOM) ⇒ + $var$('result') a0) ⇒ + $var$('result') a0) rep + + [result_case_def] Definition + + ⊢ (∀a f f1 v. result_CASE (Return a) f f1 v = f a) ∧ + (∀a f f1 v. result_CASE (Fail a) f f1 v = f1 a) ∧ + ∀f f1 v. result_CASE Loop f f1 v = v + + [result_size_def] Definition + + ⊢ (∀f a. result_size f (Return a) = 1 + f a) ∧ + (∀f a. result_size f (Fail a) = 1 + error_size a) ∧ + ∀f. result_size f Loop = 0 + + [return_def] Definition + + ⊢ ∀x. return x = Return x + + [u128_add_def] Definition + + ⊢ ∀x y. u128_add x y = mk_u128 (u128_to_int x + u128_to_int y) + + [u128_div_def] Definition + + ⊢ ∀x y. + u128_div x y = + if u128_to_int y = 0 then Fail Failure + else mk_u128 (u128_to_int x / u128_to_int y) + + [u128_max_def] Definition + + ⊢ u128_max = 340282366920938463463374607431768211455 + + [u128_mul_def] Definition + + ⊢ ∀x y. u128_mul x y = mk_u128 (u128_to_int x * u128_to_int y) + + [u128_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [u128_sub_def] Definition + + ⊢ ∀x y. u128_sub x y = mk_u128 (u128_to_int x − u128_to_int y) + + [u16_add_def] Definition + + ⊢ ∀x y. u16_add x y = mk_u16 (u16_to_int x + u16_to_int y) + + [u16_div_def] Definition + + ⊢ ∀x y. + u16_div x y = + if u16_to_int y = 0 then Fail Failure + else mk_u16 (u16_to_int x / u16_to_int y) + + [u16_max_def] Definition + + ⊢ u16_max = 65535 + + [u16_mul_def] Definition + + ⊢ ∀x y. u16_mul x y = mk_u16 (u16_to_int x * u16_to_int y) + + [u16_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [u16_sub_def] Definition + + ⊢ ∀x y. u16_sub x y = mk_u16 (u16_to_int x − u16_to_int y) + + [u32_add_def] Definition + + ⊢ ∀x y. u32_add x y = mk_u32 (u32_to_int x + u32_to_int y) + + [u32_div_def] Definition + + ⊢ ∀x y. + u32_div x y = + if u32_to_int y = 0 then Fail Failure + else mk_u32 (u32_to_int x / u32_to_int y) + + [u32_max_def] Definition + + ⊢ u32_max = 4294967295 + + [u32_mul_def] Definition + + ⊢ ∀x y. u32_mul x y = mk_u32 (u32_to_int x * u32_to_int y) + + [u32_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [u32_sub_def] Definition + + ⊢ ∀x y. u32_sub x y = mk_u32 (u32_to_int x − u32_to_int y) + + [u64_add_def] Definition + + ⊢ ∀x y. u64_add x y = mk_u64 (u64_to_int x + u64_to_int y) + + [u64_div_def] Definition + + ⊢ ∀x y. + u64_div x y = + if u64_to_int y = 0 then Fail Failure + else mk_u64 (u64_to_int x / u64_to_int y) + + [u64_max_def] Definition + + ⊢ u64_max = 18446744073709551615 + + [u64_mul_def] Definition + + ⊢ ∀x y. u64_mul x y = mk_u64 (u64_to_int x * u64_to_int y) + + [u64_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [u64_sub_def] Definition + + ⊢ ∀x y. u64_sub x y = mk_u64 (u64_to_int x − u64_to_int y) + + [u8_add_def] Definition + + ⊢ ∀x y. u8_add x y = mk_u8 (u8_to_int x + u8_to_int y) + + [u8_div_def] Definition + + ⊢ ∀x y. + u8_div x y = + if u8_to_int y = 0 then Fail Failure + else mk_u8 (u8_to_int x / u8_to_int y) + + [u8_max_def] Definition + + ⊢ u8_max = 255 + + [u8_mul_def] Definition + + ⊢ ∀x y. u8_mul x y = mk_u8 (u8_to_int x * u8_to_int y) + + [u8_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [u8_sub_def] Definition + + ⊢ ∀x y. u8_sub x y = mk_u8 (u8_to_int x − u8_to_int y) + + [usize_add_def] Definition + + ⊢ ∀x y. usize_add x y = mk_usize (usize_to_int x + usize_to_int y) + + [usize_div_def] Definition + + ⊢ ∀x y. + usize_div x y = + if usize_to_int y = 0 then Fail Failure + else mk_usize (usize_to_int x / usize_to_int y) + + [usize_mul_def] Definition + + ⊢ ∀x y. usize_mul x y = mk_usize (usize_to_int x * usize_to_int y) + + [usize_rem_def] Definition + + ⊢ ∀x y. + 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)) + + [usize_sub_def] Definition + + ⊢ ∀x y. usize_sub x y = mk_usize (usize_to_int x − usize_to_int y) + + [vec_len_def] Definition + + ⊢ ∀v. vec_len v = int_to_u32 (&LENGTH (vec_to_list v)) + + [I128_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] + ⊢ ∀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 + + [I128_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] + ⊢ ∀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 + + [I128_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] + ⊢ ∀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 + + [I128_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] + ⊢ ∀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 + + [I16_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] + ⊢ ∀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 + + [I16_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] + ⊢ ∀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 + + [I16_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] + ⊢ ∀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 + + [I16_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i16_id, i16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [I16_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] + ⊢ ∀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 + + [I32_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] + ⊢ ∀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 + + [I32_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] + ⊢ ∀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 + + [I32_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] + ⊢ ∀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 + + [I32_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i32_id, i32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [I32_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] + ⊢ ∀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 + + [I64_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] + ⊢ ∀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 + + [I64_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] + ⊢ ∀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 + + [I64_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] + ⊢ ∀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 + + [I64_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i64_id, i64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [I64_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] + ⊢ ∀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 + + [I8_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] + ⊢ ∀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 + + [I8_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] + ⊢ ∀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 + + [I8_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] + ⊢ ∀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 + + [I8_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i8_id, i8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [I8_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] + ⊢ ∀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 + + [ISIZE_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [ISIZE_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [ISIZE_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [ISIZE_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [U128_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [U128_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [U128_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [U128_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, + usize_bounds] [] + ⊢ ∀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) + + [U128_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [U16_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U16_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U16_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U16_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [U16_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U32_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U32_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U32_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U32_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [U32_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U64_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U64_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U64_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U64_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [U64_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U8_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U8_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U8_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [U8_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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) + + [U8_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] + [] + ⊢ ∀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 + + [USIZE_ADD_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [USIZE_DIV_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [USIZE_MUL_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [USIZE_REM_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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) + + [USIZE_SUB_EQ] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, + usize_bounds] [] + ⊢ ∀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 + + [VEC_TO_LIST_INT_BOUNDS] Theorem + + [oracles: DISK_THM] [axioms: VEC_TO_LIST_BOUNDS] [] + ⊢ ∀v. (let l = &LENGTH (vec_to_list v) in 0 ≤ l ∧ l ≤ u32_max) + + [datatype_error] Theorem + + ⊢ DATATYPE (error Failure) + + [datatype_result] Theorem + + ⊢ DATATYPE (result Return Fail Loop) + + [error2num_11] Theorem + + ⊢ ∀a a'. error2num a = error2num a' ⇔ a = a' + + [error2num_ONTO] Theorem + + ⊢ ∀r. r < 1 ⇔ ∃a. r = error2num a + + [error2num_num2error] Theorem + + ⊢ ∀r. r < 1 ⇔ error2num (num2error r) = r + + [error2num_thm] Theorem + + ⊢ error2num Failure = 0 + + [error_Axiom] Theorem + + ⊢ ∀x0. ∃f. f Failure = x0 + + [error_EQ_error] Theorem + + ⊢ ∀a a'. a = a' ⇔ error2num a = error2num a' + + [error_case_cong] Theorem + + ⊢ ∀M M' v0. + M = M' ∧ (M' = Failure ⇒ v0 = v0') ⇒ + (case M of Failure => v0) = case M' of Failure => v0' + + [error_case_def] Theorem + + ⊢ ∀v0. (case Failure of Failure => v0) = v0 + + [error_case_eq] Theorem + + ⊢ (case x of Failure => v0) = v ⇔ x = Failure ∧ v0 = v + + [error_induction] Theorem + + ⊢ ∀P. P Failure ⇒ ∀a. P a + + [error_nchotomy] Theorem + + ⊢ ∀a. a = Failure + + [num2error_11] Theorem + + ⊢ ∀r r'. r < 1 ⇒ r' < 1 ⇒ (num2error r = num2error r' ⇔ r = r') + + [num2error_ONTO] Theorem + + ⊢ ∀a. ∃r. a = num2error r ∧ r < 1 + + [num2error_error2num] Theorem + + ⊢ ∀a. num2error (error2num a) = a + + [num2error_thm] Theorem + + ⊢ num2error 0 = Failure + + [result_11] Theorem + + ⊢ (∀a a'. Return a = Return a' ⇔ a = a') ∧ + ∀a a'. Fail a = Fail a' ⇔ a = a' + + [result_Axiom] Theorem + + ⊢ ∀f0 f1 f2. ∃fn. + (∀a. fn (Return a) = f0 a) ∧ (∀a. fn (Fail a) = f1 a) ∧ + fn Loop = f2 + + [result_case_cong] Theorem + + ⊢ ∀M M' f f1 v. + M = M' ∧ (∀a. M' = Return a ⇒ f a = f' a) ∧ + (∀a. M' = Fail a ⇒ f1 a = f1' a) ∧ (M' = Loop ⇒ v = v') ⇒ + result_CASE M f f1 v = result_CASE M' f' f1' v' + + [result_case_eq] Theorem + + ⊢ result_CASE x f f1 v = v' ⇔ + (∃a. x = Return a ∧ f a = v') ∨ (∃e. x = Fail e ∧ f1 e = v') ∨ + x = Loop ∧ v = v' + + [result_distinct] Theorem + + ⊢ (∀a' a. Return a ≠ Fail a') ∧ (∀a. Return a ≠ Loop) ∧ + ∀a. Fail a ≠ Loop + + [result_induction] Theorem + + ⊢ ∀P. (∀a. P (Return a)) ∧ (∀e. P (Fail e)) ∧ P Loop ⇒ ∀r. P r + + [result_nchotomy] Theorem + + ⊢ ∀rr. (∃a. rr = Return a) ∨ (∃e. rr = Fail e) ∨ rr = Loop + + +*) +end |