diff options
Diffstat (limited to 'backends/hol4/primitivesTheory.sig')
-rw-r--r-- | backends/hol4/primitivesTheory.sig | 804 |
1 files changed, 401 insertions, 403 deletions
diff --git a/backends/hol4/primitivesTheory.sig b/backends/hol4/primitivesTheory.sig index cc2f3115..66adac64 100644 --- a/backends/hol4/primitivesTheory.sig +++ b/backends/hol4/primitivesTheory.sig @@ -3,8 +3,6 @@ sig type thm = Thm.thm (* Axioms *) - val MK_VEC_SPEC : thm - val VEC_TO_LIST_BOUNDS : thm val i128_to_int_bounds : thm val i16_to_int_bounds : thm val i32_to_int_bounds : thm @@ -44,6 +42,7 @@ sig val u8_to_int_bounds : thm val usize_bounds : thm val usize_to_int_bounds : thm + val vec_to_list_num_bounds : thm (* Definitions *) val bind_def : thm @@ -156,70 +155,6 @@ sig val vec_push_back_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 INT_OF_NUM : thm - val INT_OF_NUM_NEQ_INJ : 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 USIZE_TO_INT_INJ : thm - val USIZE_TO_INT_NEQ_INJ : thm - val VEC_NEW_SPEC : thm - val VEC_TO_LIST_INT_BOUNDS : thm val datatype_error : thm val datatype_result : thm val error2num_11 : thm @@ -233,9 +168,39 @@ sig val error_case_eq : thm val error_induction : thm val error_nchotomy : thm + val i128_add_eq : thm + val i128_div_eq : thm + val i128_mul_eq : thm + val i128_rem_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 index_update_diff : thm val index_update_same : thm - val int_induction : thm + val int_of_num_neq_inj : thm + val isize_add_eq : thm + val isize_div_eq : thm + val isize_mul_eq : thm + val isize_rem_eq : thm + val isize_sub_eq : thm val num2error_11 : thm val num2error_ONTO : thm val num2error_error2num : thm @@ -247,9 +212,41 @@ sig val result_distinct : thm val result_induction : thm val result_nchotomy : 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 update_ind : thm val update_len : thm val update_spec : 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 usize_to_int_inj : thm + val usize_to_int_neq_inj : thm val vec_insert_back_spec : thm val vec_len_spec : thm val vec_to_list_int_bounds : thm @@ -265,207 +262,201 @@ sig [oracles: ] [axioms: mk_vec_spec] [] ⊢ ∀l. len l ≤ usize_max ⇒ ∃v. mk_vec l = Return v ∧ vec_to_list v = l - [VEC_TO_LIST_BOUNDS] Axiom + [vec_to_list_num_bounds] Axiom - [oracles: ] [axioms: VEC_TO_LIST_BOUNDS] [] + [oracles: ] [axioms: vec_to_list_num_bounds] [] ⊢ ∀v. (let l = LENGTH (vec_to_list v) in 0 ≤ l ∧ l ≤ Num usize_max) - [isize_bounds] Axiom - - [oracles: ] [axioms: isize_bounds] [] - ⊢ isize_min ≤ i16_min ∧ isize_max ≥ i16_max - - [usize_bounds] Axiom + [int_to_usize_usize_to_int] Axiom - [oracles: ] [axioms: usize_bounds] [] ⊢ usize_max ≥ u16_max + [oracles: ] [axioms: int_to_usize_usize_to_int] [] + ⊢ ∀i. int_to_usize (usize_to_int i) = i - [isize_to_int_bounds] Axiom + [int_to_u128_u128_to_int] Axiom - [oracles: ] [axioms: isize_to_int_bounds] [] - ⊢ ∀n. isize_min ≤ isize_to_int n ∧ isize_to_int n ≤ isize_max + [oracles: ] [axioms: int_to_u128_u128_to_int] [] + ⊢ ∀i. int_to_u128 (u128_to_int i) = i - [i8_to_int_bounds] Axiom + [int_to_u64_u64_to_int] Axiom - [oracles: ] [axioms: i8_to_int_bounds] [] - ⊢ ∀n. i8_min ≤ i8_to_int n ∧ i8_to_int n ≤ i8_max + [oracles: ] [axioms: int_to_u64_u64_to_int] [] + ⊢ ∀i. int_to_u64 (u64_to_int i) = i - [i16_to_int_bounds] Axiom + [int_to_u32_u32_to_int] Axiom - [oracles: ] [axioms: i16_to_int_bounds] [] - ⊢ ∀n. i16_min ≤ i16_to_int n ∧ i16_to_int n ≤ i16_max + [oracles: ] [axioms: int_to_u32_u32_to_int] [] + ⊢ ∀i. int_to_u32 (u32_to_int i) = i - [i32_to_int_bounds] Axiom + [int_to_u16_u16_to_int] Axiom - [oracles: ] [axioms: i32_to_int_bounds] [] - ⊢ ∀n. i32_min ≤ i32_to_int n ∧ i32_to_int n ≤ i32_max + [oracles: ] [axioms: int_to_u16_u16_to_int] [] + ⊢ ∀i. int_to_u16 (u16_to_int i) = i - [i64_to_int_bounds] Axiom + [int_to_u8_u8_to_int] Axiom - [oracles: ] [axioms: i64_to_int_bounds] [] - ⊢ ∀n. i64_min ≤ i64_to_int n ∧ i64_to_int n ≤ i64_max + [oracles: ] [axioms: int_to_u8_u8_to_int] [] + ⊢ ∀i. int_to_u8 (u8_to_int i) = i - [i128_to_int_bounds] Axiom + [int_to_isize_isize_to_int] Axiom - [oracles: ] [axioms: i128_to_int_bounds] [] - ⊢ ∀n. i128_min ≤ i128_to_int n ∧ i128_to_int n ≤ i128_max + [oracles: ] [axioms: int_to_isize_isize_to_int] [] + ⊢ ∀i. int_to_isize (isize_to_int i) = i - [usize_to_int_bounds] Axiom + [int_to_i128_i128_to_int] Axiom - [oracles: ] [axioms: usize_to_int_bounds] [] - ⊢ ∀n. 0 ≤ usize_to_int n ∧ usize_to_int n ≤ usize_max + [oracles: ] [axioms: int_to_i128_i128_to_int] [] + ⊢ ∀i. int_to_i128 (i128_to_int i) = i - [u8_to_int_bounds] Axiom + [int_to_i64_i64_to_int] Axiom - [oracles: ] [axioms: u8_to_int_bounds] [] - ⊢ ∀n. 0 ≤ u8_to_int n ∧ u8_to_int n ≤ u8_max + [oracles: ] [axioms: int_to_i64_i64_to_int] [] + ⊢ ∀i. int_to_i64 (i64_to_int i) = i - [u16_to_int_bounds] Axiom + [int_to_i32_i32_to_int] Axiom - [oracles: ] [axioms: u16_to_int_bounds] [] - ⊢ ∀n. 0 ≤ u16_to_int n ∧ u16_to_int n ≤ u16_max + [oracles: ] [axioms: int_to_i32_i32_to_int] [] + ⊢ ∀i. int_to_i32 (i32_to_int i) = i - [u32_to_int_bounds] Axiom + [int_to_i16_i16_to_int] Axiom - [oracles: ] [axioms: u32_to_int_bounds] [] - ⊢ ∀n. 0 ≤ u32_to_int n ∧ u32_to_int n ≤ u32_max + [oracles: ] [axioms: int_to_i16_i16_to_int] [] + ⊢ ∀i. int_to_i16 (i16_to_int i) = i - [u64_to_int_bounds] Axiom + [int_to_i8_i8_to_int] Axiom - [oracles: ] [axioms: u64_to_int_bounds] [] - ⊢ ∀n. 0 ≤ u64_to_int n ∧ u64_to_int n ≤ u64_max + [oracles: ] [axioms: int_to_i8_i8_to_int] [] + ⊢ ∀i. int_to_i8 (i8_to_int i) = i - [u128_to_int_bounds] Axiom + [int_to_u128_id] Axiom - [oracles: ] [axioms: u128_to_int_bounds] [] - ⊢ ∀n. 0 ≤ u128_to_int n ∧ u128_to_int n ≤ u128_max + [oracles: ] [axioms: int_to_u128_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u128_max ⇒ u128_to_int (int_to_u128 n) = n - [int_to_isize_id] Axiom + [int_to_u64_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 + [oracles: ] [axioms: int_to_u64_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u64_max ⇒ u64_to_int (int_to_u64 n) = n - [int_to_usize_id] Axiom + [int_to_u32_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 + [oracles: ] [axioms: int_to_u32_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u32_max ⇒ u32_to_int (int_to_u32 n) = n - [int_to_i8_id] Axiom + [int_to_u16_id] Axiom - [oracles: ] [axioms: int_to_i8_id] [] - ⊢ ∀n. i8_min ≤ n ∧ n ≤ i8_max ⇒ i8_to_int (int_to_i8 n) = n + [oracles: ] [axioms: int_to_u16_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u16_max ⇒ u16_to_int (int_to_u16 n) = n - [int_to_i16_id] Axiom + [int_to_u8_id] Axiom - [oracles: ] [axioms: int_to_i16_id] [] - ⊢ ∀n. i16_min ≤ n ∧ n ≤ i16_max ⇒ i16_to_int (int_to_i16 n) = n + [oracles: ] [axioms: int_to_u8_id] [] + ⊢ ∀n. 0 ≤ n ∧ n ≤ u8_max ⇒ u8_to_int (int_to_u8 n) = n - [int_to_i32_id] Axiom + [int_to_i128_id] Axiom - [oracles: ] [axioms: int_to_i32_id] [] - ⊢ ∀n. i32_min ≤ n ∧ n ≤ i32_max ⇒ i32_to_int (int_to_i32 n) = n + [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_i128_id] Axiom + [int_to_i32_id] Axiom - [oracles: ] [axioms: int_to_i128_id] [] - ⊢ ∀n. i128_min ≤ n ∧ n ≤ i128_max ⇒ i128_to_int (int_to_i128 n) = n + [oracles: ] [axioms: int_to_i32_id] [] + ⊢ ∀n. i32_min ≤ n ∧ n ≤ i32_max ⇒ i32_to_int (int_to_i32 n) = n - [int_to_u8_id] Axiom + [int_to_i16_id] Axiom - [oracles: ] [axioms: int_to_u8_id] [] - ⊢ ∀n. 0 ≤ n ∧ n ≤ u8_max ⇒ u8_to_int (int_to_u8 n) = n + [oracles: ] [axioms: int_to_i16_id] [] + ⊢ ∀n. i16_min ≤ n ∧ n ≤ i16_max ⇒ i16_to_int (int_to_i16 n) = n - [int_to_u16_id] Axiom + [int_to_i8_id] Axiom - [oracles: ] [axioms: int_to_u16_id] [] - ⊢ ∀n. 0 ≤ n ∧ n ≤ u16_max ⇒ u16_to_int (int_to_u16 n) = n + [oracles: ] [axioms: int_to_i8_id] [] + ⊢ ∀n. i8_min ≤ n ∧ n ≤ i8_max ⇒ i8_to_int (int_to_i8 n) = n - [int_to_u32_id] Axiom + [int_to_usize_id] Axiom - [oracles: ] [axioms: int_to_u32_id] [] - ⊢ ∀n. 0 ≤ n ∧ n ≤ u32_max ⇒ u32_to_int (int_to_u32 n) = n + [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_u64_id] Axiom + [int_to_isize_id] Axiom - [oracles: ] [axioms: int_to_u64_id] [] - ⊢ ∀n. 0 ≤ n ∧ n ≤ u64_max ⇒ u64_to_int (int_to_u64 n) = n + [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 - [int_to_u128_id] Axiom + [u128_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u128_id] [] - ⊢ ∀n. 0 ≤ n ∧ n ≤ u128_max ⇒ u128_to_int (int_to_u128 n) = n + [oracles: ] [axioms: u128_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u128_to_int n ∧ u128_to_int n ≤ u128_max - [int_to_i8_i8_to_int] Axiom + [u64_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_i8_i8_to_int] [] - ⊢ ∀i. int_to_i8 (i8_to_int i) = i + [oracles: ] [axioms: u64_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u64_to_int n ∧ u64_to_int n ≤ u64_max - [int_to_i16_i16_to_int] Axiom + [u32_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_i16_i16_to_int] [] - ⊢ ∀i. int_to_i16 (i16_to_int i) = i + [oracles: ] [axioms: u32_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u32_to_int n ∧ u32_to_int n ≤ u32_max - [int_to_i32_i32_to_int] Axiom + [u16_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_i32_i32_to_int] [] - ⊢ ∀i. int_to_i32 (i32_to_int i) = i + [oracles: ] [axioms: u16_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u16_to_int n ∧ u16_to_int n ≤ u16_max - [int_to_i64_i64_to_int] Axiom + [u8_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_i64_i64_to_int] [] - ⊢ ∀i. int_to_i64 (i64_to_int i) = i + [oracles: ] [axioms: u8_to_int_bounds] [] + ⊢ ∀n. 0 ≤ u8_to_int n ∧ u8_to_int n ≤ u8_max - [int_to_i128_i128_to_int] Axiom + [usize_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_i128_i128_to_int] [] - ⊢ ∀i. int_to_i128 (i128_to_int i) = i + [oracles: ] [axioms: usize_to_int_bounds] [] + ⊢ ∀n. 0 ≤ usize_to_int n ∧ usize_to_int n ≤ usize_max - [int_to_isize_isize_to_int] Axiom + [i128_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_isize_isize_to_int] [] - ⊢ ∀i. int_to_isize (isize_to_int i) = i + [oracles: ] [axioms: i128_to_int_bounds] [] + ⊢ ∀n. i128_min ≤ i128_to_int n ∧ i128_to_int n ≤ i128_max - [int_to_u8_u8_to_int] Axiom + [i64_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u8_u8_to_int] [] - ⊢ ∀i. int_to_u8 (u8_to_int i) = i + [oracles: ] [axioms: i64_to_int_bounds] [] + ⊢ ∀n. i64_min ≤ i64_to_int n ∧ i64_to_int n ≤ i64_max - [int_to_u16_u16_to_int] Axiom + [i32_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u16_u16_to_int] [] - ⊢ ∀i. int_to_u16 (u16_to_int i) = i + [oracles: ] [axioms: i32_to_int_bounds] [] + ⊢ ∀n. i32_min ≤ i32_to_int n ∧ i32_to_int n ≤ i32_max - [int_to_u32_u32_to_int] Axiom + [i16_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u32_u32_to_int] [] - ⊢ ∀i. int_to_u32 (u32_to_int i) = i + [oracles: ] [axioms: i16_to_int_bounds] [] + ⊢ ∀n. i16_min ≤ i16_to_int n ∧ i16_to_int n ≤ i16_max - [int_to_u64_u64_to_int] Axiom + [i8_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u64_u64_to_int] [] - ⊢ ∀i. int_to_u64 (u64_to_int i) = i + [oracles: ] [axioms: i8_to_int_bounds] [] + ⊢ ∀n. i8_min ≤ i8_to_int n ∧ i8_to_int n ≤ i8_max - [int_to_u128_u128_to_int] Axiom + [isize_to_int_bounds] Axiom - [oracles: ] [axioms: int_to_u128_u128_to_int] [] - ⊢ ∀i. int_to_u128 (u128_to_int i) = i + [oracles: ] [axioms: isize_to_int_bounds] [] + ⊢ ∀n. isize_min ≤ isize_to_int n ∧ isize_to_int n ≤ isize_max - [int_to_usize_usize_to_int] Axiom + [usize_bounds] Axiom - [oracles: ] [axioms: int_to_usize_usize_to_int] [] - ⊢ ∀i. int_to_usize (usize_to_int i) = i + [oracles: ] [axioms: usize_bounds] [] ⊢ usize_max ≥ u16_max - [MK_VEC_SPEC] Axiom + [isize_bounds] Axiom - [oracles: ] [axioms: MK_VEC_SPEC] [] - ⊢ ∀l. &LENGTH l ≤ usize_max ⇒ - ∃v. mk_vec l = Return v ∧ vec_to_list v = l + [oracles: ] [axioms: isize_bounds] [] + ⊢ isize_min ≤ i16_min ∧ isize_max ≥ i16_max [bind_def] Definition @@ -1035,7 +1026,61 @@ sig ⊢ ∀v x. vec_push_back v x = mk_vec (vec_to_list v ⧺ [x]) - [I128_ADD_EQ] Theorem + [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 + + [i128_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] @@ -1045,7 +1090,7 @@ sig ∃z. i128_add x y = Return z ∧ i128_to_int z = i128_to_int x + i128_to_int y - [I128_DIV_EQ] Theorem + [i128_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] @@ -1056,7 +1101,7 @@ sig ∃z. i128_div x y = Return z ∧ i128_to_int z = i128_to_int x / i128_to_int y - [I128_MUL_EQ] Theorem + [i128_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] @@ -1066,7 +1111,19 @@ sig ∃z. i128_mul x y = Return z ∧ i128_to_int z = i128_to_int x * i128_to_int y - [I128_SUB_EQ] Theorem + [i128_rem_eq] Theorem + + [oracles: DISK_THM] + [axioms: isize_bounds, int_to_i128_id, i128_to_int_bounds, + usize_bounds] [] + ⊢ ∀x y. + i128_to_int y ≠ 0 ⇒ + i128_min ≤ int_rem (i128_to_int x) (i128_to_int y) ⇒ + int_rem (i128_to_int x) (i128_to_int y) ≤ i128_max ⇒ + ∃z. i128_rem x y = Return z ∧ + i128_to_int z = int_rem (i128_to_int x) (i128_to_int y) + + [i128_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i128_id, usize_bounds] [] @@ -1076,7 +1133,7 @@ sig ∃z. i128_sub x y = Return z ∧ i128_to_int z = i128_to_int x − i128_to_int y - [I16_ADD_EQ] Theorem + [i16_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] @@ -1086,7 +1143,7 @@ sig ∃z. i16_add x y = Return z ∧ i16_to_int z = i16_to_int x + i16_to_int y - [I16_DIV_EQ] Theorem + [i16_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] @@ -1097,7 +1154,7 @@ sig ∃z. i16_div x y = Return z ∧ i16_to_int z = i16_to_int x / i16_to_int y - [I16_MUL_EQ] Theorem + [i16_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] @@ -1107,7 +1164,7 @@ sig ∃z. i16_mul x y = Return z ∧ i16_to_int z = i16_to_int x * i16_to_int y - [I16_REM_EQ] Theorem + [i16_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i16_id, i16_to_int_bounds, usize_bounds] @@ -1119,7 +1176,7 @@ sig ∃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 + [i16_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i16_id, usize_bounds] [] @@ -1129,7 +1186,7 @@ sig ∃z. i16_sub x y = Return z ∧ i16_to_int z = i16_to_int x − i16_to_int y - [I32_ADD_EQ] Theorem + [i32_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] @@ -1139,7 +1196,7 @@ sig ∃z. i32_add x y = Return z ∧ i32_to_int z = i32_to_int x + i32_to_int y - [I32_DIV_EQ] Theorem + [i32_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] @@ -1150,7 +1207,7 @@ sig ∃z. i32_div x y = Return z ∧ i32_to_int z = i32_to_int x / i32_to_int y - [I32_MUL_EQ] Theorem + [i32_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] @@ -1160,7 +1217,7 @@ sig ∃z. i32_mul x y = Return z ∧ i32_to_int z = i32_to_int x * i32_to_int y - [I32_REM_EQ] Theorem + [i32_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i32_id, i32_to_int_bounds, usize_bounds] @@ -1172,7 +1229,7 @@ sig ∃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 + [i32_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i32_id, usize_bounds] [] @@ -1182,7 +1239,7 @@ sig ∃z. i32_sub x y = Return z ∧ i32_to_int z = i32_to_int x − i32_to_int y - [I64_ADD_EQ] Theorem + [i64_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] @@ -1192,7 +1249,7 @@ sig ∃z. i64_add x y = Return z ∧ i64_to_int z = i64_to_int x + i64_to_int y - [I64_DIV_EQ] Theorem + [i64_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] @@ -1203,7 +1260,7 @@ sig ∃z. i64_div x y = Return z ∧ i64_to_int z = i64_to_int x / i64_to_int y - [I64_MUL_EQ] Theorem + [i64_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] @@ -1213,7 +1270,7 @@ sig ∃z. i64_mul x y = Return z ∧ i64_to_int z = i64_to_int x * i64_to_int y - [I64_REM_EQ] Theorem + [i64_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i64_id, i64_to_int_bounds, usize_bounds] @@ -1225,7 +1282,7 @@ sig ∃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 + [i64_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i64_id, usize_bounds] [] @@ -1235,7 +1292,7 @@ sig ∃z. i64_sub x y = Return z ∧ i64_to_int z = i64_to_int x − i64_to_int y - [I8_ADD_EQ] Theorem + [i8_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] @@ -1245,7 +1302,7 @@ sig ∃z. i8_add x y = Return z ∧ i8_to_int z = i8_to_int x + i8_to_int y - [I8_DIV_EQ] Theorem + [i8_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] @@ -1256,7 +1313,7 @@ sig ∃z. i8_div x y = Return z ∧ i8_to_int z = i8_to_int x / i8_to_int y - [I8_MUL_EQ] Theorem + [i8_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] @@ -1266,7 +1323,7 @@ sig ∃z. i8_mul x y = Return z ∧ i8_to_int z = i8_to_int x * i8_to_int y - [I8_REM_EQ] Theorem + [i8_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i8_id, i8_to_int_bounds, usize_bounds] @@ -1278,7 +1335,7 @@ sig ∃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 + [i8_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_i8_id, usize_bounds] [] @@ -1288,15 +1345,24 @@ sig ∃z. i8_sub x y = Return z ∧ i8_to_int z = i8_to_int x − i8_to_int y - [INT_OF_NUM] Theorem + [index_update_diff] Theorem - ⊢ ∀i. 0 ≤ i ⇒ &Num i = i + ⊢ ∀ls i j y. 0 ≤ i ⇒ i < len ls ⇒ index i (update ls i y) = y - [INT_OF_NUM_NEQ_INJ] Theorem + [index_update_same] Theorem + + ⊢ ∀ls i j y. + 0 ≤ i ⇒ + i < len ls ⇒ + j < len ls ⇒ + j ≠ i ⇒ + index j (update ls i y) = index j ls + + [int_of_num_neq_inj] Theorem ⊢ ∀n m. &n ≠ &m ⇒ n ≠ m - [ISIZE_ADD_EQ] Theorem + [isize_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, @@ -1309,7 +1375,7 @@ sig ∃z. isize_add x y = Return z ∧ isize_to_int z = isize_to_int x + isize_to_int y - [ISIZE_DIV_EQ] Theorem + [isize_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, @@ -1323,7 +1389,7 @@ sig ∃z. isize_div x y = Return z ∧ isize_to_int z = isize_to_int x / isize_to_int y - [ISIZE_MUL_EQ] Theorem + [isize_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_isize_id, isize_to_int_bounds, @@ -1336,7 +1402,21 @@ sig ∃z. isize_mul x y = Return z ∧ isize_to_int z = isize_to_int x * isize_to_int y - [ISIZE_SUB_EQ] Theorem + [isize_rem_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 ≤ int_rem (isize_to_int x) (isize_to_int y) ∨ + isize_min ≤ int_rem (isize_to_int x) (isize_to_int y) ⇒ + int_rem (isize_to_int x) (isize_to_int y) ≤ i16_max ∨ + int_rem (isize_to_int x) (isize_to_int y) ≤ isize_max ⇒ + ∃z. isize_rem x y = Return z ∧ + isize_to_int z = int_rem (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, @@ -1349,7 +1429,60 @@ sig ∃z. isize_sub x y = Return z ∧ isize_to_int z = isize_to_int x − isize_to_int y - [U128_ADD_EQ] Theorem + [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 + + [u128_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, @@ -1359,7 +1492,7 @@ sig ∃z. u128_add x y = Return z ∧ u128_to_int z = u128_to_int x + u128_to_int y - [U128_DIV_EQ] Theorem + [u128_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, @@ -1369,7 +1502,7 @@ sig ∃z. u128_div x y = Return z ∧ u128_to_int z = u128_to_int x / u128_to_int y - [U128_MUL_EQ] Theorem + [u128_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, @@ -1379,7 +1512,7 @@ sig ∃z. u128_mul x y = Return z ∧ u128_to_int z = u128_to_int x * u128_to_int y - [U128_REM_EQ] Theorem + [u128_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, @@ -1389,7 +1522,7 @@ sig ∃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 + [u128_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u128_id, u128_to_int_bounds, @@ -1399,7 +1532,7 @@ sig ∃z. u128_sub x y = Return z ∧ u128_to_int z = u128_to_int x − u128_to_int y - [U16_ADD_EQ] Theorem + [u16_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] @@ -1409,7 +1542,7 @@ sig ∃z. u16_add x y = Return z ∧ u16_to_int z = u16_to_int x + u16_to_int y - [U16_DIV_EQ] Theorem + [u16_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] @@ -1419,7 +1552,7 @@ sig ∃z. u16_div x y = Return z ∧ u16_to_int z = u16_to_int x / u16_to_int y - [U16_MUL_EQ] Theorem + [u16_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] @@ -1429,7 +1562,7 @@ sig ∃z. u16_mul x y = Return z ∧ u16_to_int z = u16_to_int x * u16_to_int y - [U16_REM_EQ] Theorem + [u16_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] @@ -1439,7 +1572,7 @@ sig ∃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 + [u16_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u16_id, u16_to_int_bounds, usize_bounds] @@ -1449,7 +1582,7 @@ sig ∃z. u16_sub x y = Return z ∧ u16_to_int z = u16_to_int x − u16_to_int y - [U32_ADD_EQ] Theorem + [u32_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] @@ -1459,7 +1592,7 @@ sig ∃z. u32_add x y = Return z ∧ u32_to_int z = u32_to_int x + u32_to_int y - [U32_DIV_EQ] Theorem + [u32_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] @@ -1469,7 +1602,7 @@ sig ∃z. u32_div x y = Return z ∧ u32_to_int z = u32_to_int x / u32_to_int y - [U32_MUL_EQ] Theorem + [u32_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] @@ -1479,7 +1612,7 @@ sig ∃z. u32_mul x y = Return z ∧ u32_to_int z = u32_to_int x * u32_to_int y - [U32_REM_EQ] Theorem + [u32_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] @@ -1489,7 +1622,7 @@ sig ∃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 + [u32_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u32_id, u32_to_int_bounds, usize_bounds] @@ -1499,7 +1632,7 @@ sig ∃z. u32_sub x y = Return z ∧ u32_to_int z = u32_to_int x − u32_to_int y - [U64_ADD_EQ] Theorem + [u64_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] @@ -1509,7 +1642,7 @@ sig ∃z. u64_add x y = Return z ∧ u64_to_int z = u64_to_int x + u64_to_int y - [U64_DIV_EQ] Theorem + [u64_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] @@ -1519,7 +1652,7 @@ sig ∃z. u64_div x y = Return z ∧ u64_to_int z = u64_to_int x / u64_to_int y - [U64_MUL_EQ] Theorem + [u64_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] @@ -1529,7 +1662,7 @@ sig ∃z. u64_mul x y = Return z ∧ u64_to_int z = u64_to_int x * u64_to_int y - [U64_REM_EQ] Theorem + [u64_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] @@ -1539,7 +1672,7 @@ sig ∃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 + [u64_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u64_id, u64_to_int_bounds, usize_bounds] @@ -1549,7 +1682,7 @@ sig ∃z. u64_sub x y = Return z ∧ u64_to_int z = u64_to_int x − u64_to_int y - [U8_ADD_EQ] Theorem + [u8_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] @@ -1559,7 +1692,7 @@ sig ∃z. u8_add x y = Return z ∧ u8_to_int z = u8_to_int x + u8_to_int y - [U8_DIV_EQ] Theorem + [u8_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] @@ -1569,7 +1702,7 @@ sig ∃z. u8_div x y = Return z ∧ u8_to_int z = u8_to_int x / u8_to_int y - [U8_MUL_EQ] Theorem + [u8_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] @@ -1579,7 +1712,7 @@ sig ∃z. u8_mul x y = Return z ∧ u8_to_int z = u8_to_int x * u8_to_int y - [U8_REM_EQ] Theorem + [u8_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] @@ -1589,7 +1722,7 @@ sig ∃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 + [u8_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_u8_id, u8_to_int_bounds, usize_bounds] @@ -1599,7 +1732,25 @@ sig ∃z. u8_sub x y = Return z ∧ u8_to_int z = u8_to_int x − u8_to_int y - [USIZE_ADD_EQ] Theorem + [update_ind] Theorem + + ⊢ ∀P. (∀i y. P [] i y) ∧ (∀v0 ls y. P (v0::ls) 0 y) ∧ + (∀x ls i y. P ls i y ⇒ P (x::ls) (SUC i) y) ⇒ + ∀v v1 v2. P v v1 v2 + + [update_len] Theorem + + ⊢ ∀ls i y. len (update ls i y) = len ls + + [update_spec] Theorem + + ⊢ ∀ls i y. + 0 ≤ i ⇒ + i < len ls ⇒ + len (update ls i y) = len ls ∧ index i (update ls i y) = y ∧ + ∀j. j < len ls ⇒ j ≠ i ⇒ index j (update ls i y) = index j ls + + [usize_add_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, @@ -1610,7 +1761,7 @@ sig ∃z. usize_add x y = Return z ∧ usize_to_int z = usize_to_int x + usize_to_int y - [USIZE_DIV_EQ] Theorem + [usize_div_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, @@ -1620,7 +1771,7 @@ sig ∃z. usize_div x y = Return z ∧ usize_to_int z = usize_to_int x / usize_to_int y - [USIZE_MUL_EQ] Theorem + [usize_mul_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, @@ -1631,7 +1782,7 @@ sig ∃z. usize_mul x y = Return z ∧ usize_to_int z = usize_to_int x * usize_to_int y - [USIZE_REM_EQ] Theorem + [usize_rem_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, @@ -1641,7 +1792,7 @@ sig ∃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 + [usize_sub_eq] Theorem [oracles: DISK_THM] [axioms: isize_bounds, int_to_usize_id, usize_to_int_bounds, @@ -1651,169 +1802,16 @@ sig ∃z. usize_sub x y = Return z ∧ usize_to_int z = usize_to_int x − usize_to_int y - [USIZE_TO_INT_INJ] Theorem + [usize_to_int_inj] Theorem [oracles: DISK_THM] [axioms: int_to_usize_usize_to_int] [] ⊢ ∀i j. usize_to_int i = usize_to_int j ⇔ i = j - [USIZE_TO_INT_NEQ_INJ] Theorem + [usize_to_int_neq_inj] Theorem [oracles: DISK_THM] [axioms: int_to_usize_usize_to_int] [] ⊢ ∀i j. i ≠ j ⇒ usize_to_int i ≠ usize_to_int j - [VEC_NEW_SPEC] Theorem - - [oracles: DISK_THM] [axioms: usize_bounds, MK_VEC_SPEC] [] - ⊢ vec_to_list vec_new = [] - - [VEC_TO_LIST_INT_BOUNDS] Theorem - - [oracles: DISK_THM] [axioms: usize_bounds, VEC_TO_LIST_BOUNDS] [] - ⊢ ∀v. 0 ≤ &LENGTH (vec_to_list v) ∧ - &LENGTH (vec_to_list v) ≤ usize_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 - - [index_update_diff] Theorem - - ⊢ ∀ls i j y. 0 ≤ i ⇒ i < len ls ⇒ index i (update ls i y) = y - - [index_update_same] Theorem - - ⊢ ∀ls i j y. - 0 ≤ i ⇒ - i < len ls ⇒ - j < len ls ⇒ - j ≠ i ⇒ - index j (update ls i y) = index j ls - - [int_induction] Theorem - - ⊢ ∀P. P 0 ∧ (∀i. 0 ≤ i ∧ P i ⇒ P (i + 1)) ⇒ ∀i. 0 ≤ i ⇒ P i - - [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 - - [update_ind] Theorem - - ⊢ ∀P. (∀i y. P [] i y) ∧ (∀v0 ls y. P (v0::ls) 0 y) ∧ - (∀x ls i y. P ls i y ⇒ P (x::ls) (SUC i) y) ⇒ - ∀v v1 v2. P v v1 v2 - - [update_len] Theorem - - ⊢ ∀ls i y. len (update ls i y) = len ls - - [update_spec] Theorem - - ⊢ ∀ls i y. - 0 ≤ i ⇒ - i < len ls ⇒ - len (update ls i y) = len ls ∧ index i (update ls i y) = y ∧ - ∀j. j < len ls ⇒ j ≠ i ⇒ index j (update ls i y) = index j ls - [vec_insert_back_spec] Theorem [oracles: DISK_THM, cheat] |