Structure wordsTheory
signature wordsTheory =
sig
type thm = Thm.thm
(* Definitions *)
val BIT_SET_curried_def : thm
val BIT_SET_tupled_primitive_def : thm
val INT_MAX_def : thm
val INT_MIN_def : thm
val UINT_MAX_def : thm
val add_with_carry_def : thm
val bit_count_def : thm
val bit_count_upto_def : thm
val bit_field_insert_def : thm
val concat_word_list_def : thm
val dimword_def : thm
val l2w_def : thm
val n2w_def : thm
val n2w_itself_primitive_def : thm
val nzcv_def : thm
val reduce_and_def : thm
val reduce_nand_def : thm
val reduce_nor_def : thm
val reduce_or_def : thm
val reduce_xnor_def : thm
val reduce_xor_def : thm
val s2w_def : thm
val saturate_add_def : thm
val saturate_mul_def : thm
val saturate_n2w_def : thm
val saturate_sub_def : thm
val saturate_w2w_def : thm
val sw2sw_def : thm
val w2l_def : thm
val w2n_def : thm
val w2s_def : thm
val w2w_def : thm
val word_1comp_def : thm
val word_2comp_def : thm
val word_H_def : thm
val word_L2_def : thm
val word_L_def : thm
val word_T_def : thm
val word_abs_def : thm
val word_add_def : thm
val word_and_def : thm
val word_asr_bv_def : thm
val word_asr_def : thm
val word_bit_def : thm
val word_bits_def : thm
val word_compare_def : thm
val word_concat_def : thm
val word_div_def : thm
val word_extract_def : thm
val word_from_bin_list_def : thm
val word_from_bin_string_def : thm
val word_from_dec_list_def : thm
val word_from_dec_string_def : thm
val word_from_hex_list_def : thm
val word_from_hex_string_def : thm
val word_from_oct_list_def : thm
val word_from_oct_string_def : thm
val word_ge_def : thm
val word_gt_def : thm
val word_hi_def : thm
val word_hs_def : thm
val word_join_def : thm
val word_le_def : thm
val word_len_def : thm
val word_lo_def : thm
val word_log2_def : thm
val word_ls_def : thm
val word_lsb_def : thm
val word_lsl_bv_def : thm
val word_lsl_def : thm
val word_lsr_bv_def : thm
val word_lsr_def : thm
val word_lt_def : thm
val word_max_def : thm
val word_min_def : thm
val word_mod_def : thm
val word_modify_def : thm
val word_msb_def : thm
val word_mul_def : thm
val word_nand_def : thm
val word_nor_def : thm
val word_or_def : thm
val word_reduce_def : thm
val word_replicate_def : thm
val word_reverse_def : thm
val word_rol_bv_def : thm
val word_rol_def : thm
val word_ror_bv_def : thm
val word_ror_def : thm
val word_rrx_def : thm
val word_sdiv_def : thm
val word_sign_extend_def : thm
val word_signed_bits_def : thm
val word_slice_def : thm
val word_smax_def : thm
val word_smin_def : thm
val word_smod_def : thm
val word_srem_def : thm
val word_sub_def : thm
val word_to_bin_list_def : thm
val word_to_bin_string_def : thm
val word_to_dec_list_def : thm
val word_to_dec_string_def : thm
val word_to_hex_list_def : thm
val word_to_hex_string_def : thm
val word_to_oct_list_def : thm
val word_to_oct_string_def : thm
val word_xnor_def : thm
val word_xor_def : thm
(* Theorems *)
val ADD_WITH_CARRY_SUB : thm
val ASR_ADD : thm
val ASR_LIMIT : thm
val ASR_UINT_MAX : thm
val BITS_ZEROL_DIMINDEX : thm
val BIT_SET : thm
val BIT_SET_def : thm
val BIT_SET_ind : thm
val BIT_UPDATE : thm
val BOUND_ORDER : thm
val CONCAT_EXTRACT : thm
val DIMINDEX_GT_0 : thm
val EXISTS_HB : thm
val EXTEND_EXTRACT : thm
val EXTRACT_ALL_BITS : thm
val EXTRACT_CONCAT : thm
val EXTRACT_JOIN : thm
val EXTRACT_JOIN_ADD : thm
val EXTRACT_JOIN_ADD_LSL : thm
val EXTRACT_JOIN_LSL : thm
val FCP_T_F : thm
val FST_ADD_WITH_CARRY : thm
val INT_MAX_LT_DIMWORD : thm
val INT_MIN_1 : thm
val INT_MIN_10 : thm
val INT_MIN_11 : thm
val INT_MIN_12 : thm
val INT_MIN_128 : thm
val INT_MIN_16 : thm
val INT_MIN_2 : thm
val INT_MIN_20 : thm
val INT_MIN_24 : thm
val INT_MIN_28 : thm
val INT_MIN_3 : thm
val INT_MIN_30 : thm
val INT_MIN_32 : thm
val INT_MIN_4 : thm
val INT_MIN_48 : thm
val INT_MIN_5 : thm
val INT_MIN_6 : thm
val INT_MIN_64 : thm
val INT_MIN_7 : thm
val INT_MIN_8 : thm
val INT_MIN_9 : thm
val INT_MIN_96 : thm
val INT_MIN_LT_DIMWORD : thm
val INT_MIN_SUM : thm
val LEAST_BIT_LT : thm
val LOG2_w2n : thm
val LOG2_w2n_lt : thm
val LSL_ADD : thm
val LSL_BITWISE : thm
val LSL_LIMIT : thm
val LSL_ONE : thm
val LSL_UINT_MAX : thm
val LSR_ADD : thm
val LSR_BITWISE : thm
val LSR_LESS : thm
val LSR_LIMIT : thm
val MOD_2EXP_DIMINDEX : thm
val MOD_COMPLEMENT : thm
val MOD_DIMINDEX : thm
val NOT_0w : thm
val NOT_FINITE_IMP_dimword_2 : thm
val NOT_INT_MIN_ZERO : thm
val NOT_UINTMAXw : thm
val NUMERAL_LESS_THM : thm
val ONE_LT_dimword : thm
val ROL_ADD : thm
val ROL_BITWISE : thm
val ROL_MOD : thm
val ROR_ADD : thm
val ROR_BITWISE : thm
val ROR_CYCLE : thm
val ROR_MOD : thm
val ROR_ROL : thm
val ROR_UINT_MAX : thm
val SHIFT_1_SUB_1 : thm
val SHIFT_ZERO : thm
val SUC_WORD_PRED : thm
val TWO_COMP_NEG : thm
val TWO_COMP_POS : thm
val TWO_COMP_POS_NEG : thm
val WORD_0_LS : thm
val WORD_0_POS : thm
val WORD_2COMP_LSL : thm
val WORD_ADD_0 : thm
val WORD_ADD_ASSOC : thm
val WORD_ADD_BIT : thm
val WORD_ADD_BIT0 : thm
val WORD_ADD_COMM : thm
val WORD_ADD_EQ_SUB : thm
val WORD_ADD_INV_0_EQ : thm
val WORD_ADD_LEFT_LO : thm
val WORD_ADD_LEFT_LO2 : thm
val WORD_ADD_LEFT_LS : thm
val WORD_ADD_LEFT_LS2 : thm
val WORD_ADD_LID_UNIQ : thm
val WORD_ADD_LINV : thm
val WORD_ADD_LSL : thm
val WORD_ADD_OR : thm
val WORD_ADD_RID_UNIQ : thm
val WORD_ADD_RIGHT_LO : thm
val WORD_ADD_RIGHT_LO2 : thm
val WORD_ADD_RIGHT_LS : thm
val WORD_ADD_RIGHT_LS2 : thm
val WORD_ADD_RINV : thm
val WORD_ADD_SUB : thm
val WORD_ADD_SUB2 : thm
val WORD_ADD_SUB3 : thm
val WORD_ADD_SUB_ASSOC : thm
val WORD_ADD_SUB_SYM : thm
val WORD_ADD_XOR : thm
val WORD_ALL_BITS : thm
val WORD_AND_ABSORD : thm
val WORD_AND_ASSOC : thm
val WORD_AND_CLAUSES : thm
val WORD_AND_COMM : thm
val WORD_AND_COMP : thm
val WORD_AND_EXP_SUB1 : thm
val WORD_AND_IDEM : thm
val WORD_BITS_COMP_THM : thm
val WORD_BITS_EXTRACT : thm
val WORD_BITS_LSL : thm
val WORD_BITS_LSR : thm
val WORD_BITS_LT : thm
val WORD_BITS_MIN_HIGH : thm
val WORD_BITS_OVER_BITWISE : thm
val WORD_BITS_SLICE_THM : thm
val WORD_BITS_ZERO : thm
val WORD_BITS_ZERO2 : thm
val WORD_BITS_ZERO3 : thm
val WORD_BIT_BITS : thm
val WORD_DE_MORGAN_THM : thm
val WORD_DIVISION : thm
val WORD_DIV_LSR : thm
val WORD_EQ : thm
val WORD_EQ_ADD_LCANCEL : thm
val WORD_EQ_ADD_RCANCEL : thm
val WORD_EQ_NEG : thm
val WORD_EQ_SUB_LADD : thm
val WORD_EQ_SUB_RADD : thm
val WORD_EQ_SUB_ZERO : thm
val WORD_EXTRACT_BITS_COMP : thm
val WORD_EXTRACT_COMP_THM : thm
val WORD_EXTRACT_ID : thm
val WORD_EXTRACT_LSL : thm
val WORD_EXTRACT_LSL2 : thm
val WORD_EXTRACT_LT : thm
val WORD_EXTRACT_MIN_HIGH : thm
val WORD_EXTRACT_OVER_ADD : thm
val WORD_EXTRACT_OVER_ADD2 : thm
val WORD_EXTRACT_OVER_BITWISE : thm
val WORD_EXTRACT_OVER_MUL : thm
val WORD_EXTRACT_OVER_MUL2 : thm
val WORD_EXTRACT_ZERO : thm
val WORD_EXTRACT_ZERO2 : thm
val WORD_EXTRACT_ZERO3 : thm
val WORD_FINITE : thm
val WORD_GE : thm
val WORD_GREATER : thm
val WORD_GREATER_EQ : thm
val WORD_GREATER_OR_EQ : thm
val WORD_GT : thm
val WORD_HI : thm
val WORD_HIGHER : thm
val WORD_HIGHER_EQ : thm
val WORD_HIGHER_OR_EQ : thm
val WORD_HS : thm
val WORD_H_POS : thm
val WORD_H_WORD_L : thm
val WORD_INDUCT : thm
val WORD_LCANCEL_SUB : thm
val WORD_LE : thm
val WORD_LEFT_ADD_DISTRIB : thm
val WORD_LEFT_AND_OVER_OR : thm
val WORD_LEFT_AND_OVER_XOR : thm
val WORD_LEFT_OR_OVER_AND : thm
val WORD_LEFT_SUB_DISTRIB : thm
val WORD_LESS_0_word_T : thm
val WORD_LESS_ANTISYM : thm
val WORD_LESS_CASES : thm
val WORD_LESS_CASES_IMP : thm
val WORD_LESS_EQUAL_ANTISYM : thm
val WORD_LESS_EQ_ANTISYM : thm
val WORD_LESS_EQ_CASES : thm
val WORD_LESS_EQ_H : thm
val WORD_LESS_EQ_LESS_TRANS : thm
val WORD_LESS_EQ_REFL : thm
val WORD_LESS_EQ_TRANS : thm
val WORD_LESS_IMP_LESS_OR_EQ : thm
val WORD_LESS_LESS_CASES : thm
val WORD_LESS_LESS_EQ_TRANS : thm
val WORD_LESS_NEG_LEFT : thm
val WORD_LESS_NEG_RIGHT : thm
val WORD_LESS_NOT_EQ : thm
val WORD_LESS_OR_EQ : thm
val WORD_LESS_REFL : thm
val WORD_LESS_TRANS : thm
val WORD_LE_EQ_LS : thm
val WORD_LE_LS : thm
val WORD_LITERAL_ADD : thm
val WORD_LITERAL_AND : thm
val WORD_LITERAL_MULT : thm
val WORD_LITERAL_OR : thm
val WORD_LITERAL_XOR : thm
val WORD_LO : thm
val WORD_LOWER_ANTISYM : thm
val WORD_LOWER_CASES : thm
val WORD_LOWER_CASES_IMP : thm
val WORD_LOWER_EQUAL_ANTISYM : thm
val WORD_LOWER_EQ_ANTISYM : thm
val WORD_LOWER_EQ_CASES : thm
val WORD_LOWER_EQ_LOWER_TRANS : thm
val WORD_LOWER_EQ_REFL : thm
val WORD_LOWER_EQ_TRANS : thm
val WORD_LOWER_IMP_LOWER_OR_EQ : thm
val WORD_LOWER_LOWER_CASES : thm
val WORD_LOWER_LOWER_EQ_TRANS : thm
val WORD_LOWER_NOT_EQ : thm
val WORD_LOWER_OR_EQ : thm
val WORD_LOWER_REFL : thm
val WORD_LOWER_TRANS : thm
val WORD_LO_word_0 : thm
val WORD_LO_word_T : thm
val WORD_LS : thm
val WORD_LS_T : thm
val WORD_LS_word_0 : thm
val WORD_LS_word_T : thm
val WORD_LT : thm
val WORD_LT_EQ_LO : thm
val WORD_LT_LO : thm
val WORD_LT_SUB_UPPER : thm
val WORD_L_LESS_EQ : thm
val WORD_L_LESS_H : thm
val WORD_L_NEG : thm
val WORD_L_PLUS_H : thm
val WORD_MODIFY_BIT : thm
val WORD_MOD_1 : thm
val WORD_MOD_POW2 : thm
val WORD_MSB_1COMP : thm
val WORD_MSB_INT_MIN_LS : thm
val WORD_MULT_ASSOC : thm
val WORD_MULT_CLAUSES : thm
val WORD_MULT_COMM : thm
val WORD_MULT_SUC : thm
val WORD_MUL_LSL : thm
val WORD_NAND_NOT_AND : thm
val WORD_NEG : thm
val WORD_NEG_0 : thm
val WORD_NEG_1 : thm
val WORD_NEG_1_T : thm
val WORD_NEG_ADD : thm
val WORD_NEG_EQ : thm
val WORD_NEG_EQ_0 : thm
val WORD_NEG_L : thm
val WORD_NEG_LMUL : thm
val WORD_NEG_MUL : thm
val WORD_NEG_NEG : thm
val WORD_NEG_RMUL : thm
val WORD_NEG_SUB : thm
val WORD_NEG_T : thm
val WORD_NOR_NOT_OR : thm
val WORD_NOT : thm
val WORD_NOT_0 : thm
val WORD_NOT_GREATER : thm
val WORD_NOT_HIGHER : thm
val WORD_NOT_LESS : thm
val WORD_NOT_LESS_EQ : thm
val WORD_NOT_LESS_EQUAL : thm
val WORD_NOT_LOWER : thm
val WORD_NOT_LOWER_EQ : thm
val WORD_NOT_LOWER_EQUAL : thm
val WORD_NOT_NOT : thm
val WORD_NOT_T : thm
val WORD_NOT_XOR : thm
val WORD_OR_ABSORB : thm
val WORD_OR_ASSOC : thm
val WORD_OR_CLAUSES : thm
val WORD_OR_COMM : thm
val WORD_OR_COMP : thm
val WORD_OR_IDEM : thm
val WORD_PRED_THM : thm
val WORD_RCANCEL_SUB : thm
val WORD_RIGHT_ADD_DISTRIB : thm
val WORD_RIGHT_AND_OVER_OR : thm
val WORD_RIGHT_AND_OVER_XOR : thm
val WORD_RIGHT_OR_OVER_AND : thm
val WORD_RIGHT_SUB_DISTRIB : thm
val WORD_SET_INDUCT : thm
val WORD_SLICE_BITS_THM : thm
val WORD_SLICE_COMP_THM : thm
val WORD_SLICE_OVER_BITWISE : thm
val WORD_SLICE_THM : thm
val WORD_SLICE_ZERO : thm
val WORD_SLICE_ZERO2 : thm
val WORD_SUB : thm
val WORD_SUB_ADD : thm
val WORD_SUB_ADD2 : thm
val WORD_SUB_INTRO : thm
val WORD_SUB_LE : thm
val WORD_SUB_LNEG : thm
val WORD_SUB_LT : thm
val WORD_SUB_LZERO : thm
val WORD_SUB_NEG : thm
val WORD_SUB_PLUS : thm
val WORD_SUB_REFL : thm
val WORD_SUB_RNEG : thm
val WORD_SUB_RZERO : thm
val WORD_SUB_SUB : thm
val WORD_SUB_SUB2 : thm
val WORD_SUB_SUB3 : thm
val WORD_SUB_TRIANGLE : thm
val WORD_SUM_ZERO : thm
val WORD_XNOR_NOT_XOR : thm
val WORD_XOR : thm
val WORD_XOR_ASSOC : thm
val WORD_XOR_CLAUSES : thm
val WORD_XOR_COMM : thm
val WORD_XOR_COMP : thm
val WORD_ZERO_LE : thm
val WORD_w2w_EXTRACT : thm
val WORD_w2w_OVER_ADD : thm
val WORD_w2w_OVER_BITWISE : thm
val WORD_w2w_OVER_MUL : thm
val ZERO_LE_INT_MAX : thm
val ZERO_LO_INT_MIN : thm
val ZERO_LT_INT_MAX : thm
val ZERO_LT_INT_MIN : thm
val ZERO_LT_UINT_MAX : thm
val ZERO_LT_dimword : thm
val ZERO_SHIFT : thm
val bit_count_is_zero : thm
val bit_count_upto_0 : thm
val bit_count_upto_SUC : thm
val bit_count_upto_is_zero : thm
val bit_field_insert : thm
val dimindex_1 : thm
val dimindex_10 : thm
val dimindex_11 : thm
val dimindex_12 : thm
val dimindex_128 : thm
val dimindex_16 : thm
val dimindex_1_cases : thm
val dimindex_2 : thm
val dimindex_20 : thm
val dimindex_24 : thm
val dimindex_28 : thm
val dimindex_3 : thm
val dimindex_30 : thm
val dimindex_32 : thm
val dimindex_4 : thm
val dimindex_48 : thm
val dimindex_5 : thm
val dimindex_6 : thm
val dimindex_64 : thm
val dimindex_7 : thm
val dimindex_8 : thm
val dimindex_9 : thm
val dimindex_96 : thm
val dimindex_dimword_iso : thm
val dimindex_dimword_le_iso : thm
val dimindex_dimword_lt_iso : thm
val dimindex_int_max_iso : thm
val dimindex_int_max_le_iso : thm
val dimindex_int_max_lt_iso : thm
val dimindex_int_min_iso : thm
val dimindex_int_min_le_iso : thm
val dimindex_int_min_lt_iso : thm
val dimindex_lt_dimword : thm
val dimindex_uint_max_iso : thm
val dimindex_uint_max_le_iso : thm
val dimindex_uint_max_lt_iso : thm
val dimword_1 : thm
val dimword_10 : thm
val dimword_11 : thm
val dimword_12 : thm
val dimword_128 : thm
val dimword_16 : thm
val dimword_2 : thm
val dimword_20 : thm
val dimword_24 : thm
val dimword_28 : thm
val dimword_3 : thm
val dimword_30 : thm
val dimword_32 : thm
val dimword_4 : thm
val dimword_48 : thm
val dimword_5 : thm
val dimword_6 : thm
val dimword_64 : thm
val dimword_7 : thm
val dimword_8 : thm
val dimword_9 : thm
val dimword_96 : thm
val dimword_IS_TWICE_INT_MIN : thm
val dimword_sub_int_min : thm
val fcp_n2w : thm
val finite_1 : thm
val finite_10 : thm
val finite_11 : thm
val finite_12 : thm
val finite_128 : thm
val finite_16 : thm
val finite_2 : thm
val finite_20 : thm
val finite_24 : thm
val finite_28 : thm
val finite_3 : thm
val finite_30 : thm
val finite_32 : thm
val finite_4 : thm
val finite_48 : thm
val finite_5 : thm
val finite_6 : thm
val finite_64 : thm
val finite_7 : thm
val finite_8 : thm
val finite_9 : thm
val finite_96 : thm
val foldl_reduce_and : thm
val foldl_reduce_nand : thm
val foldl_reduce_nor : thm
val foldl_reduce_or : thm
val foldl_reduce_xnor : thm
val foldl_reduce_xor : thm
val l2w_w2l : thm
val lsr_1_word_T : thm
val mod_dimindex : thm
val n2w_11 : thm
val n2w_BITS : thm
val n2w_DIV : thm
val n2w_SUC : thm
val n2w_dimword : thm
val n2w_itself_def : thm
val n2w_itself_ind : thm
val n2w_mod : thm
val n2w_sub : thm
val n2w_sub_eq_0 : thm
val n2w_w2n : thm
val ranged_word_nchotomy : thm
val reduce_and : thm
val reduce_or : thm
val s2w_w2s : thm
val saturate_add : thm
val saturate_mul : thm
val saturate_sub : thm
val saturate_w2w : thm
val saturate_w2w_n2w : thm
val sw2sw : thm
val sw2sw_0 : thm
val sw2sw_id : thm
val sw2sw_sw2sw : thm
val sw2sw_w2w : thm
val sw2sw_w2w_add : thm
val sw2sw_word_T : thm
val w2l_l2w : thm
val w2n_11 : thm
val w2n_11_lift : thm
val w2n_add : thm
val w2n_eq_0 : thm
val w2n_lsr : thm
val w2n_lt : thm
val w2n_minus1 : thm
val w2n_n2w : thm
val w2n_w2w : thm
val w2n_w2w_le : thm
val w2s_s2w : thm
val w2w : thm
val w2w_0 : thm
val w2w_LSL : thm
val w2w_eq_n2w : thm
val w2w_id : thm
val w2w_lt : thm
val w2w_n2w : thm
val w2w_w2w : thm
val word_0 : thm
val word_0_n2w : thm
val word_1_n2w : thm
val word_1comp_n2w : thm
val word_2comp_dimindex_1 : thm
val word_2comp_n2w : thm
val word_H : thm
val word_L : thm
val word_L2 : thm
val word_L2_MULT : thm
val word_L_MULT : thm
val word_L_MULT_NEG : thm
val word_T : thm
val word_T_not_zero : thm
val word_abs : thm
val word_abs_diff : thm
val word_abs_neg : thm
val word_abs_word_abs : thm
val word_add_n2w : thm
val word_and_n2w : thm
val word_asr : thm
val word_asr_n2w : thm
val word_bin_list : thm
val word_bin_string : thm
val word_bit : thm
val word_bit_0 : thm
val word_bit_0_word_T : thm
val word_bit_n2w : thm
val word_bits_n2w : thm
val word_bits_w2w : thm
val word_concat_0 : thm
val word_concat_0_0 : thm
val word_concat_0_eq : thm
val word_concat_word_T : thm
val word_dec_list : thm
val word_dec_string : thm
val word_div_1 : thm
val word_eq_0 : thm
val word_eq_n2w : thm
val word_extract_eq_n2w : thm
val word_extract_mask : thm
val word_extract_n2w : thm
val word_extract_w2w : thm
val word_ge_n2w : thm
val word_gt_n2w : thm
val word_hex_list : thm
val word_hex_string : thm
val word_hi_n2w : thm
val word_hs_n2w : thm
val word_index : thm
val word_index_n2w : thm
val word_join_0 : thm
val word_join_index : thm
val word_join_word_T : thm
val word_le_n2w : thm
val word_lo_n2w : thm
val word_log2_1 : thm
val word_log2_n2w : thm
val word_ls_n2w : thm
val word_lsb : thm
val word_lsb_n2w : thm
val word_lsb_word_T : thm
val word_lsl_n2w : thm
val word_lsr_n2w : thm
val word_lt_n2w : thm
val word_modify_n2w : thm
val word_msb : thm
val word_msb_n2w : thm
val word_msb_n2w_numeric : thm
val word_msb_neg : thm
val word_msb_word_T : thm
val word_mul_n2w : thm
val word_nand_n2w : thm
val word_nchotomy : thm
val word_nor_n2w : thm
val word_oct_list : thm
val word_oct_string : thm
val word_or_n2w : thm
val word_reduce_n2w : thm
val word_replicate_concat_word_list : thm
val word_reverse_0 : thm
val word_reverse_n2w : thm
val word_reverse_thm : thm
val word_reverse_word_T : thm
val word_ror : thm
val word_ror_n2w : thm
val word_rrx_0 : thm
val word_rrx_n2w : thm
val word_rrx_word_T : thm
val word_shift_bv : thm
val word_sign_extend_bits : thm
val word_signed_bits_n2w : thm
val word_slice_n2w : thm
val word_sub_w2n : thm
val word_xnor_n2w : thm
val word_xor_n2w : thm
val words_grammars : type_grammar.grammar * term_grammar.grammar
(*
[ASCIInumbers] Parent theory of "words"
[fcp] Parent theory of "words"
[numeral_bit] Parent theory of "words"
[sum_num] Parent theory of "words"
[BIT_SET_curried_def] Definition
|- ∀x x1. BIT_SET x x1 = BIT_SET_tupled (x,x1)
[BIT_SET_tupled_primitive_def] Definition
|- BIT_SET_tupled =
WFREC
(@R.
WF R ∧ (∀i n. n ≠ 0 ∧ ODD n ⇒ R (SUC i,n DIV 2) (i,n)) ∧
∀i n. n ≠ 0 ∧ ¬ODD n ⇒ R (SUC i,n DIV 2) (i,n))
(λBIT_SET_tupled a.
case a of
(i,n) =>
I
(if n = 0 then ∅
else if ODD n then
i INSERT BIT_SET_tupled (SUC i,n DIV 2)
else BIT_SET_tupled (SUC i,n DIV 2)))
[INT_MAX_def] Definition
|- INT_MAX (:α) = INT_MIN (:α) − 1
[INT_MIN_def] Definition
|- INT_MIN (:α) = 2 ** (dimindex (:α) − 1)
[UINT_MAX_def] Definition
|- UINT_MAX (:α) = dimword (:α) − 1
[add_with_carry_def] Definition
|- ∀x y carry_in.
add_with_carry (x,y,carry_in) =
(let unsigned_sum = w2n x + w2n y + if carry_in then 1 else 0 in
let result = n2w unsigned_sum in
let carry_out = w2n result ≠ unsigned_sum and
overflow =
(word_msb x ⇔ word_msb y) ∧
(word_msb x ⇎ word_msb result)
in
(result,carry_out,overflow))
[bit_count_def] Definition
|- ∀w. bit_count w = bit_count_upto (dimindex (:α)) w
[bit_count_upto_def] Definition
|- ∀n w. bit_count_upto n w = SUM n (λi. if w ' i then 1 else 0)
[bit_field_insert_def] Definition
|- ∀h l a.
bit_field_insert h l a =
word_modify (λi. COND (l ≤ i ∧ i ≤ h) (a ' (i − l)))
[concat_word_list_def] Definition
|- (concat_word_list [] = 0w) ∧
∀h t.
concat_word_list (h::t) =
w2w h ‖ concat_word_list t ≪ dimindex (:α)
[dimword_def] Definition
|- dimword (:α) = 2 ** dimindex (:α)
[l2w_def] Definition
|- ∀b l. l2w b l = n2w (l2n b l)
[n2w_def] Definition
|- ∀n. n2w n = FCP i. BIT i n
[n2w_itself_primitive_def] Definition
|- n2w_itself =
WFREC (@R. WF R) (λn2w_itself a. case a of (n,v1) => I (n2w n))
[nzcv_def] Definition
|- ∀a b.
nzcv a b =
(let q = w2n a + w2n (-b) in
let r = n2w q
in
(word_msb r,r = 0w,BIT (dimindex (:α)) q ∨ (b = 0w),
(word_msb a ⇎ word_msb b) ∧ (word_msb r ⇎ word_msb a)))
[reduce_and_def] Definition
|- reduce_and = word_reduce $/\
[reduce_nand_def] Definition
|- reduce_nand = word_reduce (λa b. ¬(a ∧ b))
[reduce_nor_def] Definition
|- reduce_nor = word_reduce (λa b. ¬(a ∨ b))
[reduce_or_def] Definition
|- reduce_or = word_reduce $\/
[reduce_xnor_def] Definition
|- reduce_xnor = word_reduce $<=>
[reduce_xor_def] Definition
|- reduce_xor = word_reduce (λx y. x ⇎ y)
[s2w_def] Definition
|- ∀b f s. s2w b f s = n2w (s2n b f s)
[saturate_add_def] Definition
|- ∀a b. saturate_add a b = saturate_n2w (w2n a + w2n b)
[saturate_mul_def] Definition
|- ∀a b. saturate_mul a b = saturate_n2w (w2n a * w2n b)
[saturate_n2w_def] Definition
|- ∀n. saturate_n2w n = if dimword (:α) ≤ n then UINT_MAXw else n2w n
[saturate_sub_def] Definition
|- ∀a b. saturate_sub a b = n2w (w2n a − w2n b)
[saturate_w2w_def] Definition
|- ∀w. saturate_w2w w = saturate_n2w (w2n w)
[sw2sw_def] Definition
|- ∀w.
sw2sw w =
n2w (SIGN_EXTEND (dimindex (:α)) (dimindex (:β)) (w2n w))
[w2l_def] Definition
|- ∀b w. w2l b w = n2l b (w2n w)
[w2n_def] Definition
|- ∀w. w2n w = SUM (dimindex (:α)) (λi. SBIT (w ' i) i)
[w2s_def] Definition
|- ∀b f w. w2s b f w = n2s b f (w2n w)
[w2w_def] Definition
|- ∀w. w2w w = n2w (w2n w)
[word_1comp_def] Definition
|- ∀w. ¬w = FCP i. ¬w ' i
[word_2comp_def] Definition
|- ∀w. -w = n2w (dimword (:α) − w2n w)
[word_H_def] Definition
|- INT_MAXw = n2w (INT_MAX (:α))
[word_L2_def] Definition
|- INT_MINw2 = INT_MINw * INT_MINw
[word_L_def] Definition
|- INT_MINw = n2w (INT_MIN (:α))
[word_T_def] Definition
|- UINT_MAXw = n2w (UINT_MAX (:α))
[word_abs_def] Definition
|- ∀w. word_abs w = if w < 0w then -w else w
[word_add_def] Definition
|- ∀v w. v + w = n2w (w2n v + w2n w)
[word_and_def] Definition
|- ∀v w. v && w = FCP i. v ' i ∧ w ' i
[word_asr_bv_def] Definition
|- ∀w n. w >>~ n = w ≫ w2n n
[word_asr_def] Definition
|- ∀w n.
w ≫ n =
FCP i. if dimindex (:α) ≤ i + n then word_msb w else w ' (i + n)
[word_bit_def] Definition
|- ∀b w. word_bit b w ⇔ b ≤ dimindex (:α) − 1 ∧ w ' b
[word_bits_def] Definition
|- ∀h l.
h -- l =
(λw. FCP i. i + l ≤ MIN h (dimindex (:α) − 1) ∧ w ' (i + l))
[word_compare_def] Definition
|- ∀a b. word_compare a b = if a = b then 1w else 0w
[word_concat_def] Definition
|- ∀v w. v @@ w = w2w (word_join v w)
[word_div_def] Definition
|- ∀v w. v // w = n2w (w2n v DIV w2n w)
[word_extract_def] Definition
|- ∀h l. h >< l = w2w o (h -- l)
[word_from_bin_list_def] Definition
|- word_from_bin_list = l2w 2
[word_from_bin_string_def] Definition
|- word_from_bin_string = s2w 2 UNHEX
[word_from_dec_list_def] Definition
|- word_from_dec_list = l2w 10
[word_from_dec_string_def] Definition
|- word_from_dec_string = s2w 10 UNHEX
[word_from_hex_list_def] Definition
|- word_from_hex_list = l2w 16
[word_from_hex_string_def] Definition
|- word_from_hex_string = s2w 16 UNHEX
[word_from_oct_list_def] Definition
|- word_from_oct_list = l2w 8
[word_from_oct_string_def] Definition
|- word_from_oct_string = s2w 8 UNHEX
[word_ge_def] Definition
|- ∀a b. a ≥ b ⇔ (let (n,z,c,v) = nzcv a b in n ⇔ v)
[word_gt_def] Definition
|- ∀a b. a > b ⇔ (let (n,z,c,v) = nzcv a b in ¬z ∧ (n ⇔ v))
[word_hi_def] Definition
|- ∀a b. a >₊ b ⇔ (let (n,z,c,v) = nzcv a b in c ∧ ¬z)
[word_hs_def] Definition
|- ∀a b. a ≥₊ b ⇔ (let (n,z,c,v) = nzcv a b in c)
[word_join_def] Definition
|- ∀v w.
word_join v w =
(let cv = w2w v and cw = w2w w in cv ≪ dimindex (:β) ‖ cw)
[word_le_def] Definition
|- ∀a b. a ≤ b ⇔ (let (n,z,c,v) = nzcv a b in z ∨ (n ⇎ v))
[word_len_def] Definition
|- ∀w. word_len w = dimindex (:α)
[word_lo_def] Definition
|- ∀a b. a <₊ b ⇔ (let (n,z,c,v) = nzcv a b in ¬c)
[word_log2_def] Definition
|- ∀w. word_log2 w = n2w (LOG2 (w2n w))
[word_ls_def] Definition
|- ∀a b. a ≤₊ b ⇔ (let (n,z,c,v) = nzcv a b in ¬c ∨ z)
[word_lsb_def] Definition
|- ∀w. word_lsb w ⇔ w ' 0
[word_lsl_bv_def] Definition
|- ∀w n. w <<~ n = w ≪ w2n n
[word_lsl_def] Definition
|- ∀w n. w ≪ n = FCP i. i < dimindex (:α) ∧ n ≤ i ∧ w ' (i − n)
[word_lsr_bv_def] Definition
|- ∀w n. w >>>~ n = w ⋙ w2n n
[word_lsr_def] Definition
|- ∀w n. w ⋙ n = FCP i. i + n < dimindex (:α) ∧ w ' (i + n)
[word_lt_def] Definition
|- ∀a b. a < b ⇔ (let (n,z,c,v) = nzcv a b in n ⇎ v)
[word_max_def] Definition
|- ∀a b. word_max a b = if a <₊ b then b else a
[word_min_def] Definition
|- ∀a b. word_min a b = if a <₊ b then a else b
[word_mod_def] Definition
|- ∀v w. word_mod v w = n2w (w2n v MOD w2n w)
[word_modify_def] Definition
|- ∀f w. word_modify f w = FCP i. f i (w ' i)
[word_msb_def] Definition
|- ∀w. word_msb w ⇔ w ' (dimindex (:α) − 1)
[word_mul_def] Definition
|- ∀v w. v * w = n2w (w2n v * w2n w)
[word_nand_def] Definition
|- ∀v w. v ~&& w = FCP i. ¬(v ' i ∧ w ' i)
[word_nor_def] Definition
|- ∀v w. v ~|| w = FCP i. ¬(v ' i ∨ w ' i)
[word_or_def] Definition
|- ∀v w. v ‖ w = FCP i. v ' i ∨ w ' i
[word_reduce_def] Definition
|- ∀f w.
word_reduce f w =
$FCP
(K
(let l =
GENLIST (λi. w ' (dimindex (:α) − 1 − i))
(dimindex (:α))
in
FOLDL f (HD l) (TL l)))
[word_replicate_def] Definition
|- ∀n w.
word_replicate n w =
FCP i. i < n * dimindex (:α) ∧ w ' (i MOD dimindex (:α))
[word_reverse_def] Definition
|- ∀w. word_reverse w = FCP i. w ' (dimindex (:α) − 1 − i)
[word_rol_bv_def] Definition
|- ∀w n. w #<<~ n = w ⇆ w2n n
[word_rol_def] Definition
|- ∀w n. w ⇆ n = w ⇄ (dimindex (:α) − n MOD dimindex (:α))
[word_ror_bv_def] Definition
|- ∀w n. w #>>~ n = w ⇄ w2n n
[word_ror_def] Definition
|- ∀w n. w ⇄ n = FCP i. w ' ((i + n) MOD dimindex (:α))
[word_rrx_def] Definition
|- ∀c w.
word_rrx (c,w) =
(word_lsb w,
FCP i. if i = dimindex (:α) − 1 then c else (w ⋙ 1) ' i)
[word_sdiv_def] Definition
|- ∀a b.
a / b =
if word_msb a then if word_msb b then -a // -b else -(-a // b)
else if word_msb b then -(a // -b)
else a // b
[word_sign_extend_def] Definition
|- ∀n w.
word_sign_extend n w =
n2w (SIGN_EXTEND n (dimindex (:α)) (w2n w))
[word_signed_bits_def] Definition
|- ∀h l.
h --- l =
(λw.
FCP i.
l ≤ MIN h (dimindex (:α) − 1) ∧
w ' (MIN (i + l) (MIN h (dimindex (:α) − 1))))
[word_slice_def] Definition
|- ∀h l.
h '' l =
(λw. FCP i. l ≤ i ∧ i ≤ MIN h (dimindex (:α) − 1) ∧ w ' i)
[word_smax_def] Definition
|- ∀a b. word_smax a b = if a < b then b else a
[word_smin_def] Definition
|- ∀a b. word_smin a b = if a < b then a else b
[word_smod_def] Definition
|- ∀s t.
word_smod s t =
(let abs_s = if word_msb s then -s else s and
abs_t = if word_msb t then -t else t
in
let u = word_mod abs_s abs_t
in
if u = 0w then u
else if word_msb s then if word_msb t then -u else -u + t
else if word_msb t then u + t
else u)
[word_srem_def] Definition
|- ∀a b.
word_srem a b =
if word_msb a then
if word_msb b then -word_mod (-a) (-b) else -word_mod (-a) b
else if word_msb b then word_mod a (-b)
else word_mod a b
[word_sub_def] Definition
|- ∀v w. v − w = v + -w
[word_to_bin_list_def] Definition
|- word_to_bin_list = w2l 2
[word_to_bin_string_def] Definition
|- word_to_bin_string = w2s 2 HEX
[word_to_dec_list_def] Definition
|- word_to_dec_list = w2l 10
[word_to_dec_string_def] Definition
|- word_to_dec_string = w2s 10 HEX
[word_to_hex_list_def] Definition
|- word_to_hex_list = w2l 16
[word_to_hex_string_def] Definition
|- word_to_hex_string = w2s 16 HEX
[word_to_oct_list_def] Definition
|- word_to_oct_list = w2l 8
[word_to_oct_string_def] Definition
|- word_to_oct_string = w2s 8 HEX
[word_xnor_def] Definition
|- ∀v w. v ~?? w = FCP i. v ' i ⇔ w ' i
[word_xor_def] Definition
|- ∀v w. v ⊕ w = FCP i. v ' i ⇎ w ' i
[ADD_WITH_CARRY_SUB] Theorem
|- ∀x y.
add_with_carry (x,¬y,T) =
(x − y,y ≤₊ x,
(word_msb x ⇎ word_msb y) ∧ (word_msb (x − y) ⇎ word_msb x))
[ASR_ADD] Theorem
|- ∀w m n. w ≫ m ≫ n = w ≫ (m + n)
[ASR_LIMIT] Theorem
|- ∀w n.
dimindex (:α) ≤ n ⇒
(w ≫ n = if word_msb w then UINT_MAXw else 0w)
[ASR_UINT_MAX] Theorem
|- ∀n. UINT_MAXw ≫ n = UINT_MAXw
[BITS_ZEROL_DIMINDEX] Theorem
|- ∀n. n < dimword (:α) ⇒ (BITS (dimindex (:α) − 1) 0 n = n)
[BIT_SET] Theorem
|- ∀i n. BIT i n ⇔ i ∈ BIT_SET 0 n
[BIT_SET_def] Theorem
|- ∀n i.
BIT_SET i n =
if n = 0 then ∅
else if ODD n then i INSERT BIT_SET (SUC i) (n DIV 2)
else BIT_SET (SUC i) (n DIV 2)
[BIT_SET_ind] Theorem
|- ∀P.
(∀i n.
(n ≠ 0 ∧ ODD n ⇒ P (SUC i) (n DIV 2)) ∧
(n ≠ 0 ∧ ¬ODD n ⇒ P (SUC i) (n DIV 2)) ⇒
P i n) ⇒
∀v v1. P v v1
[BIT_UPDATE] Theorem
|- ∀n x. n :+ x = word_modify (λi b. if i = n then x else b)
[BOUND_ORDER] Theorem
|- INT_MAX (:α) < INT_MIN (:α) ∧ INT_MIN (:α) ≤ UINT_MAX (:α) ∧
UINT_MAX (:α) < dimword (:α)
[CONCAT_EXTRACT] Theorem
|- ∀h m l w.
(h − m = dimindex (:β)) ∧ (m + 1 − l = dimindex (:γ)) ∧
(h + 1 − l = dimindex (:δ)) ∧ dimindex (:β + γ) ≠ 1 ⇒
((h >< m + 1) w @@ (m >< l) w = (h >< l) w)
[DIMINDEX_GT_0] Theorem
|- 0 < dimindex (:α)
[EXISTS_HB] Theorem
|- ∃m. dimindex (:α) = SUC m
[EXTEND_EXTRACT] Theorem
|- ∀h l w.
(dimindex (:γ) = h + 1 − l) ⇒ ((h >< l) w = w2w ((h >< l) w))
[EXTRACT_ALL_BITS] Theorem
|- ∀h w. dimindex (:α) − 1 ≤ h ⇒ ((h >< 0) w = w2w w)
[EXTRACT_CONCAT] Theorem
|- ∀v w.
FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) ∧
dimindex (:α) + dimindex (:β) ≤ dimindex (:γ) ⇒
((dimindex (:β) − 1 >< 0) (v @@ w) = w) ∧
((dimindex (:α) + dimindex (:β) − 1 >< dimindex (:β)) (v @@ w) =
v)
[EXTRACT_JOIN] Theorem
|- ∀h m m' l s w.
l ≤ m ∧ m' ≤ h ∧ (m' = m + 1) ∧ (s = m' − l) ⇒
((h >< m') w ≪ s ‖ (m >< l) w =
(MIN h (MIN (dimindex (:β) + l − 1) (dimindex (:α) − 1)) >< l)
w)
[EXTRACT_JOIN_ADD] Theorem
|- ∀h m m' l s w.
l ≤ m ∧ m' ≤ h ∧ (m' = m + 1) ∧ (s = m' − l) ⇒
((h >< m') w ≪ s + (m >< l) w =
(MIN h (MIN (dimindex (:β) + l − 1) (dimindex (:α) − 1)) >< l)
w)
[EXTRACT_JOIN_ADD_LSL] Theorem
|- ∀h m m' l s n w.
l ≤ m ∧ m' ≤ h ∧ (m' = m + 1) ∧ (s = m' − l + n) ⇒
((h >< m') w ≪ s + (m >< l) w ≪ n =
(MIN h (MIN (dimindex (:β) + l − 1) (dimindex (:α) − 1)) >< l)
w ≪ n)
[EXTRACT_JOIN_LSL] Theorem
|- ∀h m m' l s n w.
l ≤ m ∧ m' ≤ h ∧ (m' = m + 1) ∧ (s = m' − l + n) ⇒
((h >< m') w ≪ s ‖ (m >< l) w ≪ n =
(MIN h (MIN (dimindex (:β) + l − 1) (dimindex (:α) − 1)) >< l)
w ≪ n)
[FCP_T_F] Theorem
|- ($FCP (K T) = UINT_MAXw) ∧ ($FCP (K F) = 0w)
[FST_ADD_WITH_CARRY] Theorem
|- ((∀a b. FST (add_with_carry (a,b,F)) = a + b) ∧
(∀a b. FST (add_with_carry (a,¬b,T)) = a − b) ∧
∀a b. FST (add_with_carry (¬a,b,T)) = b − a) ∧
(∀n a. FST (add_with_carry (a,n2w n,T)) = a − ¬n2w n) ∧
∀n b. FST (add_with_carry (n2w n,b,T)) = b − ¬n2w n
[INT_MAX_LT_DIMWORD] Theorem
|- INT_MAX (:α) < dimword (:α)
[INT_MIN_1] Theorem
|- INT_MIN (:unit) = 1
[INT_MIN_10] Theorem
|- INT_MIN (:10) = 512
[INT_MIN_11] Theorem
|- INT_MIN (:11) = 1024
[INT_MIN_12] Theorem
|- INT_MIN (:12) = 2048
[INT_MIN_128] Theorem
|- INT_MIN (:128) = 170141183460469231731687303715884105728
[INT_MIN_16] Theorem
|- INT_MIN (:16) = 32768
[INT_MIN_2] Theorem
|- INT_MIN (:2) = 2
[INT_MIN_20] Theorem
|- INT_MIN (:20) = 524288
[INT_MIN_24] Theorem
|- INT_MIN (:24) = 8388608
[INT_MIN_28] Theorem
|- INT_MIN (:28) = 134217728
[INT_MIN_3] Theorem
|- INT_MIN (:3) = 4
[INT_MIN_30] Theorem
|- INT_MIN (:30) = 536870912
[INT_MIN_32] Theorem
|- INT_MIN (:32) = 2147483648
[INT_MIN_4] Theorem
|- INT_MIN (:4) = 8
[INT_MIN_48] Theorem
|- INT_MIN (:48) = 140737488355328
[INT_MIN_5] Theorem
|- INT_MIN (:5) = 16
[INT_MIN_6] Theorem
|- INT_MIN (:6) = 32
[INT_MIN_64] Theorem
|- INT_MIN (:64) = 9223372036854775808
[INT_MIN_7] Theorem
|- INT_MIN (:7) = 64
[INT_MIN_8] Theorem
|- INT_MIN (:8) = 128
[INT_MIN_9] Theorem
|- INT_MIN (:9) = 256
[INT_MIN_96] Theorem
|- INT_MIN (:96) = 39614081257132168796771975168
[INT_MIN_LT_DIMWORD] Theorem
|- INT_MIN (:α) < dimword (:α)
[INT_MIN_SUM] Theorem
|- INT_MIN (:α + β) =
if FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) then dimword (:α) * INT_MIN (:β)
else INT_MIN (:α + β)
[LEAST_BIT_LT] Theorem
|- ∀w. w ≠ 0w ⇒ (LEAST i. w ' i) < dimindex (:α)
[LOG2_w2n] Theorem
|- ∀w.
w ≠ 0w ⇒
(LOG2 (w2n w) =
dimindex (:α) − 1 − LEAST i. w ' (dimindex (:α) − 1 − i))
[LOG2_w2n_lt] Theorem
|- ∀w. w ≠ 0w ⇒ LOG2 (w2n w) < dimindex (:α)
[LSL_ADD] Theorem
|- ∀w m n. w ≪ m ≪ n = w ≪ (m + n)
[LSL_BITWISE] Theorem
|- (∀n v w. w ≪ n && v ≪ n = (w && v) ≪ n) ∧
(∀n v w. w ≪ n ‖ v ≪ n = (w ‖ v) ≪ n) ∧
∀n v w. w ≪ n ⊕ v ≪ n = (w ⊕ v) ≪ n
[LSL_LIMIT] Theorem
|- ∀w n. dimindex (:α) ≤ n ⇒ (w ≪ n = 0w)
[LSL_ONE] Theorem
|- ∀w. w ≪ 1 = w + w
[LSL_UINT_MAX] Theorem
|- ∀n. UINT_MAXw ≪ n = n2w (dimword (:α) − 2 ** n)
[LSR_ADD] Theorem
|- ∀w m n. w ⋙ m ⋙ n = w ⋙ (m + n)
[LSR_BITWISE] Theorem
|- (∀n v w. w ⋙ n && v ⋙ n = (w && v) ⋙ n) ∧
(∀n v w. w ⋙ n ‖ v ⋙ n = (w ‖ v) ⋙ n) ∧
∀n v w. w ⋙ n ⊕ v ⋙ n = (w ⊕ v) ⋙ n
[LSR_LESS] Theorem
|- ∀m y. y ≠ 0w ∧ 0 < m ⇒ w2n (y ⋙ m) < w2n y
[LSR_LIMIT] Theorem
|- ∀w n. dimindex (:α) ≤ n ⇒ (w ⋙ n = 0w)
[MOD_2EXP_DIMINDEX] Theorem
|- ∀n. n MOD dimword (:α) = MOD_2EXP (dimindex (:α)) n
[MOD_COMPLEMENT] Theorem
|- ∀n q a.
0 < n ∧ 0 < q ∧ a < q * n ⇒
((q * n − a) MOD n = (n − a MOD n) MOD n)
[MOD_DIMINDEX] Theorem
|- ∀n. n MOD dimword (:α) = BITS (dimindex (:α) − 1) 0 n
[NOT_0w] Theorem
|- ∀w. w ≠ 0w ⇒ ∃i. i < dimindex (:α) ∧ w ' i
[NOT_FINITE_IMP_dimword_2] Theorem
|- INFINITE 𝕌(:α) ⇒ (dimword (:α) = 2)
[NOT_INT_MIN_ZERO] Theorem
|- INT_MINw ≠ 0w
[NOT_UINTMAXw] Theorem
|- ∀w. w ≠ UINT_MAXw ⇒ ∃i. i < dimindex (:α) ∧ ¬w ' i
[NUMERAL_LESS_THM] Theorem
|- (∀m n.
m < NUMERAL (BIT1 n) ⇔
(m = NUMERAL (BIT1 n) − 1) ∨ m < NUMERAL (BIT1 n) − 1) ∧
∀m n.
m < NUMERAL (BIT2 n) ⇔
(m = NUMERAL (BIT1 n)) ∨ m < NUMERAL (BIT1 n)
[ONE_LT_dimword] Theorem
|- 1 < dimword (:α)
[ROL_ADD] Theorem
|- ∀w m n. w ⇆ m ⇆ n = w ⇆ (m + n)
[ROL_BITWISE] Theorem
|- (∀n v w. w ⇆ n && v ⇆ n = (w && v) ⇆ n) ∧
(∀n v w. w ⇆ n ‖ v ⇆ n = (w ‖ v) ⇆ n) ∧
∀n v w. w ⇆ n ⊕ v ⇆ n = (w ⊕ v) ⇆ n
[ROL_MOD] Theorem
|- ∀w n. w ⇆ (n MOD dimindex (:α)) = w ⇆ n
[ROR_ADD] Theorem
|- ∀w m n. w ⇄ m ⇄ n = w ⇄ (m + n)
[ROR_BITWISE] Theorem
|- (∀n v w. w ⇄ n && v ⇄ n = (w && v) ⇄ n) ∧
(∀n v w. w ⇄ n ‖ v ⇄ n = (w ‖ v) ⇄ n) ∧
∀n v w. w ⇄ n ⊕ v ⇄ n = (w ⊕ v) ⇄ n
[ROR_CYCLE] Theorem
|- ∀w n. w ⇄ (n * dimindex (:α)) = w
[ROR_MOD] Theorem
|- ∀w n. w ⇄ (n MOD dimindex (:α)) = w ⇄ n
[ROR_ROL] Theorem
|- ∀w n. (w ⇄ n ⇆ n = w) ∧ (w ⇆ n ⇄ n = w)
[ROR_UINT_MAX] Theorem
|- ∀n. UINT_MAXw ⇄ n = UINT_MAXw
[SHIFT_1_SUB_1] Theorem
|- ∀i n. i < dimindex (:α) ⇒ ((1w ≪ n − 1w) ' i ⇔ i < n)
[SHIFT_ZERO] Theorem
|- (∀a. a ≪ 0 = a) ∧ (∀a. a ≫ 0 = a) ∧ (∀a. a ⋙ 0 = a) ∧
(∀a. a ⇆ 0 = a) ∧ ∀a. a ⇄ 0 = a
[SUC_WORD_PRED] Theorem
|- ∀x. x ≠ 0w ⇒ (SUC (w2n (x − 1w)) = w2n x)
[TWO_COMP_NEG] Theorem
|- ∀a.
word_msb a ⇒
if (dimindex (:α) − 1 = 0) ∨ (a = INT_MINw) then word_msb (-a)
else ¬word_msb (-a)
[TWO_COMP_POS] Theorem
|- ∀a. ¬word_msb a ⇒ (a = 0w) ∨ word_msb (-a)
[TWO_COMP_POS_NEG] Theorem
|- ∀a.
¬((dimindex (:α) − 1 = 0) ∨ (a = 0w) ∨ (a = INT_MINw)) ⇒
(¬word_msb a ⇔ word_msb (-a))
[WORD_0_LS] Theorem
|- ∀w. 0w ≤₊ w
[WORD_0_POS] Theorem
|- ¬word_msb 0w
[WORD_2COMP_LSL] Theorem
|- ∀n a. -a ≪ n = -(a ≪ n)
[WORD_ADD_0] Theorem
|- (∀w. w + 0w = w) ∧ ∀w. 0w + w = w
[WORD_ADD_ASSOC] Theorem
|- ∀v w x. v + (w + x) = v + w + x
[WORD_ADD_BIT] Theorem
|- ∀n a b.
n < dimindex (:α) ⇒
((a + b) ' n ⇔
if n = 0 then a ' 0 ⇎ b ' 0
else if ((n − 1 -- 0) a + (n − 1 -- 0) b) ' n then
a ' n ⇔ b ' n
else a ' n ⇎ b ' n)
[WORD_ADD_BIT0] Theorem
|- ∀a b. (a + b) ' 0 ⇔ (a ' 0 ⇎ b ' 0)
[WORD_ADD_COMM] Theorem
|- ∀v w. v + w = w + v
[WORD_ADD_EQ_SUB] Theorem
|- ∀v w x. (v + w = x) ⇔ (v = x − w)
[WORD_ADD_INV_0_EQ] Theorem
|- ∀v w. (v + w = v) ⇔ (w = 0w)
[WORD_ADD_LEFT_LO] Theorem
|- ∀b c a.
a + b <₊ c ⇔
if b ≤₊ c then
(let x = n2w (w2n c − w2n b) in a <₊ x ∨ b ≠ 0w ∧ -c + x ≤₊ a)
else -b ≤₊ a ∧ a <₊ -b + c
[WORD_ADD_LEFT_LO2] Theorem
|- ∀c a. c + a <₊ a ⇔ a ≠ 0w ∧ (c ≠ 0w ∧ -c <₊ a ∨ (a = -c))
[WORD_ADD_LEFT_LS] Theorem
|- ∀b c a.
a + b ≤₊ c ⇔
if b ≤₊ c then
(let x = n2w (w2n c − w2n b) in a ≤₊ x ∨ b ≠ 0w ∧ -c + x ≤₊ a)
else -b ≤₊ a ∧ a ≤₊ -b + c
[WORD_ADD_LEFT_LS2] Theorem
|- ∀c a. c + a ≤₊ a ⇔ (c = 0w) ∨ a ≠ 0w ∧ (-c <₊ a ∨ (a = -c))
[WORD_ADD_LID_UNIQ] Theorem
|- ∀v w. (v + w = w) ⇔ (v = 0w)
[WORD_ADD_LINV] Theorem
|- ∀w. -w + w = 0w
[WORD_ADD_LSL] Theorem
|- ∀n a b. (a + b) ≪ n = a ≪ n + b ≪ n
[WORD_ADD_OR] Theorem
|- ∀a b. (a && b = 0w) ⇒ (a + b = a ‖ b)
[WORD_ADD_RID_UNIQ] Theorem
|- ∀v w. (v + w = v) ⇔ (w = 0w)
[WORD_ADD_RIGHT_LO] Theorem
|- ∀c a b.
a <₊ b + c ⇔
if c ≤₊ a then
(let x = n2w (w2n a − w2n c)
in
x <₊ b ∧ ((c = 0w) ∨ b <₊ -a + x))
else b <₊ -c ∨ -c + a <₊ b
[WORD_ADD_RIGHT_LO2] Theorem
|- ∀c a. a <₊ c + a ⇔ c ≠ 0w ∧ ((a = 0w) ∨ a <₊ -c)
[WORD_ADD_RIGHT_LS] Theorem
|- ∀c a b.
a ≤₊ b + c ⇔
if c ≤₊ a then
(let x = n2w (w2n a − w2n c)
in
x ≤₊ b ∧ ((c = 0w) ∨ b <₊ -a + x))
else b <₊ -c ∨ -c + a ≤₊ b
[WORD_ADD_RIGHT_LS2] Theorem
|- ∀c a. a ≤₊ c + a ⇔ (a = 0w) ∨ (c = 0w) ∨ a <₊ -c
[WORD_ADD_RINV] Theorem
|- ∀w. w + -w = 0w
[WORD_ADD_SUB] Theorem
|- ∀v w. v + w − w = v
[WORD_ADD_SUB2] Theorem
|- ∀v w. w + v − w = v
[WORD_ADD_SUB3] Theorem
|- ∀v x. v − (v + x) = -x
[WORD_ADD_SUB_ASSOC] Theorem
|- ∀v w x. v + w − x = v + (w − x)
[WORD_ADD_SUB_SYM] Theorem
|- ∀v w x. v + w − x = v − x + w
[WORD_ADD_XOR] Theorem
|- ∀a b. (a && b = 0w) ⇒ (a + b = a ⊕ b)
[WORD_ALL_BITS] Theorem
|- ∀w h. dimindex (:α) − 1 ≤ h ⇒ ((h -- 0) w = w)
[WORD_AND_ABSORD] Theorem
|- ∀a b. a ‖ a && b = a
[WORD_AND_ASSOC] Theorem
|- ∀a b c. (a && b) && c = a && b && c
[WORD_AND_CLAUSES] Theorem
|- ∀a.
(UINT_MAXw && a = a) ∧ (a && UINT_MAXw = a) ∧ (0w && a = 0w) ∧
(a && 0w = 0w) ∧ (a && a = a)
[WORD_AND_COMM] Theorem
|- ∀a b. a && b = b && a
[WORD_AND_COMP] Theorem
|- ∀a. a && ¬a = 0w
[WORD_AND_EXP_SUB1] Theorem
|- ∀m n. n2w n && n2w (2 ** m − 1) = n2w (n MOD 2 ** m)
[WORD_AND_IDEM] Theorem
|- ∀a. a && a = a
[WORD_BITS_COMP_THM] Theorem
|- ∀h1 l1 h2 l2 w.
(h2 -- l2) ((h1 -- l1) w) = (MIN h1 (h2 + l1) -- l2 + l1) w
[WORD_BITS_EXTRACT] Theorem
|- ∀h l w. (h -- l) w = (h >< l) w
[WORD_BITS_LSL] Theorem
|- ∀h l n w.
h < dimindex (:α) ⇒
((h -- l) (w ≪ n) =
if n ≤ h then (h − n -- l − n) w ≪ (n − l) else 0w)
[WORD_BITS_LSR] Theorem
|- ∀h l w n. (h -- l) w ⋙ n = (h -- l + n) w
[WORD_BITS_LT] Theorem
|- ∀h l w. w2n ((h -- l) w) < 2 ** (SUC h − l)
[WORD_BITS_MIN_HIGH] Theorem
|- ∀w h l.
dimindex (:α) − 1 < h ⇒
((h -- l) w = (dimindex (:α) − 1 -- l) w)
[WORD_BITS_OVER_BITWISE] Theorem
|- (∀h l v w. (h -- l) v && (h -- l) w = (h -- l) (v && w)) ∧
(∀h l v w. (h -- l) v ‖ (h -- l) w = (h -- l) (v ‖ w)) ∧
∀h l v w. (h -- l) v ⊕ (h -- l) w = (h -- l) (v ⊕ w)
[WORD_BITS_SLICE_THM] Theorem
|- ∀h l w. (h -- l) ((h '' l) w) = (h -- l) w
[WORD_BITS_ZERO] Theorem
|- ∀h l w. h < l ⇒ ((h -- l) w = 0w)
[WORD_BITS_ZERO2] Theorem
|- ∀h l. (h -- l) 0w = 0w
[WORD_BITS_ZERO3] Theorem
|- ∀h l w. dimindex (:α) ≤ l ⇒ ((h -- l) w = 0w)
[WORD_BIT_BITS] Theorem
|- ∀b w. word_bit b w ⇔ ((b -- b) w = 1w)
[WORD_DE_MORGAN_THM] Theorem
|- ∀a b. (¬(a && b) = ¬a ‖ ¬b) ∧ (¬(a ‖ b) = ¬a && ¬b)
[WORD_DIVISION] Theorem
|- ∀w.
w ≠ 0w ⇒ ∀v. (v = v // w * w + word_mod v w) ∧ word_mod v w <₊ w
[WORD_DIV_LSR] Theorem
|- ∀m n. n < dimindex (:α) ⇒ (m ⋙ n = m // n2w (2 ** n))
[WORD_EQ] Theorem
|- ∀v w.
(∀x. x < dimindex (:α) ⇒ (word_bit x v ⇔ word_bit x w)) ⇔
(v = w)
[WORD_EQ_ADD_LCANCEL] Theorem
|- ∀v w x. (v + w = v + x) ⇔ (w = x)
[WORD_EQ_ADD_RCANCEL] Theorem
|- ∀v w x. (v + w = x + w) ⇔ (v = x)
[WORD_EQ_NEG] Theorem
|- ∀v w. (-v = -w) ⇔ (v = w)
[WORD_EQ_SUB_LADD] Theorem
|- ∀v w x. (v = w − x) ⇔ (v + x = w)
[WORD_EQ_SUB_RADD] Theorem
|- ∀v w x. (v − w = x) ⇔ (v = x + w)
[WORD_EQ_SUB_ZERO] Theorem
|- ∀w v. (v − w = 0w) ⇔ (v = w)
[WORD_EXTRACT_BITS_COMP] Theorem
|- ∀n l k j h. (j >< k) ((h -- l) n) = (MIN h (j + l) >< k + l) n
[WORD_EXTRACT_COMP_THM] Theorem
|- ∀w h l m n.
(h >< l) ((m >< n) w) =
(MIN m
(MIN (h + n)
(MIN (dimindex (:γ) − 1) (dimindex (:β) + n − 1))) ><
l + n) w
[WORD_EXTRACT_ID] Theorem
|- ∀w h. w2n w < 2 ** SUC h ⇒ ((h >< 0) w = w)
[WORD_EXTRACT_LSL] Theorem
|- ∀h l n w.
h < dimindex (:α) ⇒
((h >< l) (w ≪ n) =
if n ≤ h then (h − n >< l − n) w ≪ (n − l) else 0w)
[WORD_EXTRACT_LSL2] Theorem
|- ∀h l n w.
dimindex (:β) + l ≤ h + n ⇒
((h >< l) w ≪ n = (dimindex (:β) + l − (n + 1) >< l) w ≪ n)
[WORD_EXTRACT_LT] Theorem
|- ∀h l w. w2n ((h >< l) w) < 2 ** (SUC h − l)
[WORD_EXTRACT_MIN_HIGH] Theorem
|- (∀h l w.
dimindex (:α) ≤ dimindex (:β) + l ∧ dimindex (:α) ≤ h ⇒
((h >< l) w = (dimindex (:α) − 1 >< l) w)) ∧
∀h l w.
dimindex (:β) + l < dimindex (:α) ∧ dimindex (:β) + l ≤ h ⇒
((h >< l) w = (dimindex (:β) + l − 1 >< l) w)
[WORD_EXTRACT_OVER_ADD] Theorem
|- ∀a b h.
dimindex (:β) − 1 ≤ h ∧ dimindex (:β) ≤ dimindex (:α) ⇒
((h >< 0) (a + b) = (h >< 0) a + (h >< 0) b)
[WORD_EXTRACT_OVER_ADD2] Theorem
|- ∀a b h.
h < dimindex (:α) ⇒
((h >< 0) ((h >< 0) a + (h >< 0) b) = (h >< 0) (a + b))
[WORD_EXTRACT_OVER_BITWISE] Theorem
|- (∀h l v w. (h >< l) v && (h >< l) w = (h >< l) (v && w)) ∧
(∀h l v w. (h >< l) v ‖ (h >< l) w = (h >< l) (v ‖ w)) ∧
∀h l v w. (h >< l) v ⊕ (h >< l) w = (h >< l) (v ⊕ w)
[WORD_EXTRACT_OVER_MUL] Theorem
|- ∀a b h.
dimindex (:β) − 1 ≤ h ∧ dimindex (:β) ≤ dimindex (:α) ⇒
((h >< 0) (a * b) = (h >< 0) a * (h >< 0) b)
[WORD_EXTRACT_OVER_MUL2] Theorem
|- ∀a b h.
h < dimindex (:α) ⇒
((h >< 0) ((h >< 0) a * (h >< 0) b) = (h >< 0) (a * b))
[WORD_EXTRACT_ZERO] Theorem
|- ∀h l w. h < l ⇒ ((h >< l) w = 0w)
[WORD_EXTRACT_ZERO2] Theorem
|- ∀h l. (h >< l) 0w = 0w
[WORD_EXTRACT_ZERO3] Theorem
|- ∀h l w. dimindex (:α) ≤ l ⇒ ((h >< l) w = 0w)
[WORD_FINITE] Theorem
|- ∀s. FINITE s
[WORD_GE] Theorem
|- ∀a b.
a ≥ b ⇔
(word_msb b ⇔ word_msb a) ∧ w2n a ≥ w2n b ∨
word_msb b ∧ ¬word_msb a
[WORD_GREATER] Theorem
|- ∀a b. a > b ⇔ b < a
[WORD_GREATER_EQ] Theorem
|- ∀a b. a ≥ b ⇔ b ≤ a
[WORD_GREATER_OR_EQ] Theorem
|- ∀a b. a ≥ b ⇔ a > b ∨ (a = b)
[WORD_GT] Theorem
|- ∀a b.
a > b ⇔
(word_msb b ⇔ word_msb a) ∧ w2n a > w2n b ∨
word_msb b ∧ ¬word_msb a
[WORD_HI] Theorem
|- ∀a b. a >₊ b ⇔ w2n a > w2n b
[WORD_HIGHER] Theorem
|- ∀a b. a >₊ b ⇔ b <₊ a
[WORD_HIGHER_EQ] Theorem
|- ∀a b. a ≥₊ b ⇔ b ≤₊ a
[WORD_HIGHER_OR_EQ] Theorem
|- ∀a b. a ≥₊ b ⇔ a >₊ b ∨ (a = b)
[WORD_HS] Theorem
|- ∀a b. a ≥₊ b ⇔ w2n a ≥ w2n b
[WORD_H_POS] Theorem
|- ¬word_msb INT_MAXw
[WORD_H_WORD_L] Theorem
|- INT_MAXw = INT_MINw − 1w
[WORD_INDUCT] Theorem
|- ∀P.
P 0w ∧
(∀n. SUC n < dimword (:α) ⇒ P (n2w n) ⇒ P (n2w (SUC n))) ⇒
∀x. P x
[WORD_LCANCEL_SUB] Theorem
|- ∀v w x. (v − w = x − w) ⇔ (v = x)
[WORD_LE] Theorem
|- ∀a b.
a ≤ b ⇔
(word_msb a ⇔ word_msb b) ∧ w2n a ≤ w2n b ∨
word_msb a ∧ ¬word_msb b
[WORD_LEFT_ADD_DISTRIB] Theorem
|- ∀v w x. v * (w + x) = v * w + v * x
[WORD_LEFT_AND_OVER_OR] Theorem
|- ∀a b c. a && (b ‖ c) = a && b ‖ a && c
[WORD_LEFT_AND_OVER_XOR] Theorem
|- ∀a b c. a && (b ⊕ c) = a && b ⊕ a && c
[WORD_LEFT_OR_OVER_AND] Theorem
|- ∀a b c. a ‖ b && c = (a ‖ b) && (a ‖ c)
[WORD_LEFT_SUB_DISTRIB] Theorem
|- ∀v w x. v * (w − x) = v * w − v * x
[WORD_LESS_0_word_T] Theorem
|- ¬(0w < -1w) ∧ ¬(0w ≤ -1w) ∧ -1w < 0w ∧ -1w ≤ 0w
[WORD_LESS_ANTISYM] Theorem
|- ∀a b. ¬(a < b ∧ b < a)
[WORD_LESS_CASES] Theorem
|- ∀a b. a < b ∨ b ≤ a
[WORD_LESS_CASES_IMP] Theorem
|- ∀a b. ¬(a < b) ∧ a ≠ b ⇒ b < a
[WORD_LESS_EQUAL_ANTISYM] Theorem
|- ∀a b. a ≤ b ∧ b ≤ a ⇒ (a = b)
[WORD_LESS_EQ_ANTISYM] Theorem
|- ∀a b. ¬(a < b ∧ b ≤ a)
[WORD_LESS_EQ_CASES] Theorem
|- ∀a b. a ≤ b ∨ b ≤ a
[WORD_LESS_EQ_H] Theorem
|- ∀a. a ≤ INT_MAXw
[WORD_LESS_EQ_LESS_TRANS] Theorem
|- ∀a b c. a ≤ b ∧ b < c ⇒ a < c
[WORD_LESS_EQ_REFL] Theorem
|- ∀a. a ≤ a
[WORD_LESS_EQ_TRANS] Theorem
|- ∀a b c. a ≤ b ∧ b ≤ c ⇒ a ≤ c
[WORD_LESS_IMP_LESS_OR_EQ] Theorem
|- ∀a b. a < b ⇒ a ≤ b
[WORD_LESS_LESS_CASES] Theorem
|- ∀a b. (a = b) ∨ a < b ∨ b < a
[WORD_LESS_LESS_EQ_TRANS] Theorem
|- ∀a b c. a < b ∧ b ≤ c ⇒ a < c
[WORD_LESS_NEG_LEFT] Theorem
|- ∀a b. -a <₊ b ⇔ b ≠ 0w ∧ ((a = 0w) ∨ -b <₊ a)
[WORD_LESS_NEG_RIGHT] Theorem
|- ∀a b. a <₊ -b ⇔ b ≠ 0w ∧ ((a = 0w) ∨ b <₊ -a)
[WORD_LESS_NOT_EQ] Theorem
|- ∀a b. a < b ⇒ a ≠ b
[WORD_LESS_OR_EQ] Theorem
|- ∀a b. a ≤ b ⇔ a < b ∨ (a = b)
[WORD_LESS_REFL] Theorem
|- ∀a. ¬(a < a)
[WORD_LESS_TRANS] Theorem
|- ∀a b c. a < b ∧ b < c ⇒ a < c
[WORD_LE_EQ_LS] Theorem
|- ∀x y. 0w ≤ x ∧ 0w ≤ y ⇒ (x ≤ y ⇔ x ≤₊ y)
[WORD_LE_LS] Theorem
|- ∀a b.
a ≤ b ⇔
INT_MINw ≤₊ a ∧ (b <₊ INT_MINw ∨ a ≤₊ b) ∨
a <₊ INT_MINw ∧ b <₊ INT_MINw ∧ a ≤₊ b
[WORD_LITERAL_ADD] Theorem
|- (∀m n. -n2w m + -n2w n = -n2w (m + n)) ∧
∀m n. n2w m + -n2w n = if n ≤ m then n2w (m − n) else -n2w (n − m)
[WORD_LITERAL_AND] Theorem
|- (∀n m.
n2w n && n2w m =
n2w (BITWISE (SUC (MIN (LOG2 n) (LOG2 m))) $/\ n m)) ∧
(∀n m.
n2w n && ¬n2w m =
n2w (BITWISE (SUC (LOG2 n)) (λa b. a ∧ ¬b) n m)) ∧
(∀n m.
¬n2w m && n2w n =
n2w (BITWISE (SUC (LOG2 n)) (λa b. a ∧ ¬b) n m)) ∧
∀n m.
¬n2w n && ¬n2w m =
¬n2w (BITWISE (SUC (MAX (LOG2 n) (LOG2 m))) $\/ n m)
[WORD_LITERAL_MULT] Theorem
|- (∀m n. n2w m * -n2w n = -n2w (m * n)) ∧
∀m n. -n2w m * -n2w n = n2w (m * n)
[WORD_LITERAL_OR] Theorem
|- (∀n m.
n2w n ‖ n2w m =
n2w (BITWISE (SUC (MAX (LOG2 n) (LOG2 m))) $\/ n m)) ∧
(∀n m.
n2w n ‖ ¬n2w m =
¬n2w (BITWISE (SUC (LOG2 m)) (λa b. a ∧ ¬b) m n)) ∧
(∀n m.
¬n2w m ‖ n2w n =
¬n2w (BITWISE (SUC (LOG2 m)) (λa b. a ∧ ¬b) m n)) ∧
∀n m.
¬n2w n ‖ ¬n2w m =
¬n2w (BITWISE (SUC (MIN (LOG2 n) (LOG2 m))) $/\ n m)
[WORD_LITERAL_XOR] Theorem
|- ∀n m.
n2w n ⊕ n2w m =
n2w (BITWISE (SUC (MAX (LOG2 n) (LOG2 m))) (λx y. x ⇎ y) n m)
[WORD_LO] Theorem
|- ∀a b. a <₊ b ⇔ w2n a < w2n b
[WORD_LOWER_ANTISYM] Theorem
|- ∀a b. ¬(a <₊ b ∧ b <₊ a)
[WORD_LOWER_CASES] Theorem
|- ∀a b. a <₊ b ∨ b ≤₊ a
[WORD_LOWER_CASES_IMP] Theorem
|- ∀a b. ¬(a <₊ b) ∧ a ≠ b ⇒ b <₊ a
[WORD_LOWER_EQUAL_ANTISYM] Theorem
|- ∀a b. a ≤₊ b ∧ b ≤₊ a ⇒ (a = b)
[WORD_LOWER_EQ_ANTISYM] Theorem
|- ∀a b. ¬(a <₊ b ∧ b ≤₊ a)
[WORD_LOWER_EQ_CASES] Theorem
|- ∀a b. a ≤₊ b ∨ b ≤₊ a
[WORD_LOWER_EQ_LOWER_TRANS] Theorem
|- ∀a b c. a ≤₊ b ∧ b <₊ c ⇒ a <₊ c
[WORD_LOWER_EQ_REFL] Theorem
|- ∀a. a ≤₊ a
[WORD_LOWER_EQ_TRANS] Theorem
|- ∀a b c. a ≤₊ b ∧ b ≤₊ c ⇒ a ≤₊ c
[WORD_LOWER_IMP_LOWER_OR_EQ] Theorem
|- ∀a b. a <₊ b ⇒ a ≤₊ b
[WORD_LOWER_LOWER_CASES] Theorem
|- ∀a b. (a = b) ∨ a <₊ b ∨ b <₊ a
[WORD_LOWER_LOWER_EQ_TRANS] Theorem
|- ∀a b c. a <₊ b ∧ b ≤₊ c ⇒ a <₊ c
[WORD_LOWER_NOT_EQ] Theorem
|- ∀a b. a <₊ b ⇒ a ≠ b
[WORD_LOWER_OR_EQ] Theorem
|- ∀a b. a ≤₊ b ⇔ a <₊ b ∨ (a = b)
[WORD_LOWER_REFL] Theorem
|- ∀a. ¬(a <₊ a)
[WORD_LOWER_TRANS] Theorem
|- ∀a b c. a <₊ b ∧ b <₊ c ⇒ a <₊ c
[WORD_LO_word_0] Theorem
|- (∀n. 0w <₊ n ⇔ n ≠ 0w) ∧ ∀n. ¬(n <₊ 0w)
[WORD_LO_word_T] Theorem
|- (∀n. ¬(-1w <₊ n)) ∧ ∀n. n <₊ -1w ⇔ n ≠ -1w
[WORD_LS] Theorem
|- ∀a b. a ≤₊ b ⇔ w2n a ≤ w2n b
[WORD_LS_T] Theorem
|- ∀w. w ≤₊ UINT_MAXw
[WORD_LS_word_0] Theorem
|- ∀n. n ≤₊ 0w ⇔ (n = 0w)
[WORD_LS_word_T] Theorem
|- (∀n. -1w ≤₊ n ⇔ (n = -1w)) ∧ ∀n. n ≤₊ -1w
[WORD_LT] Theorem
|- ∀a b.
a < b ⇔
(word_msb a ⇔ word_msb b) ∧ w2n a < w2n b ∨
word_msb a ∧ ¬word_msb b
[WORD_LT_EQ_LO] Theorem
|- ∀x y. 0w ≤ x ∧ 0w ≤ y ⇒ (x < y ⇔ x <₊ y)
[WORD_LT_LO] Theorem
|- ∀a b.
a < b ⇔
INT_MINw ≤₊ a ∧ (b <₊ INT_MINw ∨ a <₊ b) ∨
a <₊ INT_MINw ∧ b <₊ INT_MINw ∧ a <₊ b
[WORD_LT_SUB_UPPER] Theorem
|- ∀x y. 0w < y ∧ y < x ⇒ x − y < x
[WORD_L_LESS_EQ] Theorem
|- ∀a. INT_MINw ≤ a
[WORD_L_LESS_H] Theorem
|- INT_MINw < INT_MAXw
[WORD_L_NEG] Theorem
|- word_msb INT_MINw
[WORD_L_PLUS_H] Theorem
|- INT_MINw + INT_MAXw = UINT_MAXw
[WORD_MODIFY_BIT] Theorem
|- ∀f w i. i < dimindex (:α) ⇒ (word_modify f w ' i ⇔ f i (w ' i))
[WORD_MOD_1] Theorem
|- ∀m. word_mod m 1w = 0w
[WORD_MOD_POW2] Theorem
|- ∀m v.
v < dimindex (:α) − 1 ⇒
(word_mod m (n2w (2 ** SUC v)) = (v -- 0) m)
[WORD_MSB_1COMP] Theorem
|- ∀w. word_msb (¬w) ⇔ ¬word_msb w
[WORD_MSB_INT_MIN_LS] Theorem
|- ∀a. word_msb a ⇔ INT_MINw ≤₊ a
[WORD_MULT_ASSOC] Theorem
|- ∀v w x. v * (w * x) = v * w * x
[WORD_MULT_CLAUSES] Theorem
|- ∀v w.
(0w * v = 0w) ∧ (v * 0w = 0w) ∧ (1w * v = v) ∧ (v * 1w = v) ∧
((v + 1w) * w = v * w + w) ∧ (v * (w + 1w) = v + v * w)
[WORD_MULT_COMM] Theorem
|- ∀v w. v * w = w * v
[WORD_MULT_SUC] Theorem
|- ∀v n. v * n2w (n + 1) = v * n2w n + v
[WORD_MUL_LSL] Theorem
|- ∀a n. a ≪ n = n2w (2 ** n) * a
[WORD_NAND_NOT_AND] Theorem
|- ∀a b. a ~&& b = ¬(a && b)
[WORD_NEG] Theorem
|- ∀w. -w = ¬w + 1w
[WORD_NEG_0] Theorem
|- -0w = 0w
[WORD_NEG_1] Theorem
|- -1w = UINT_MAXw
[WORD_NEG_1_T] Theorem
|- ∀i. i < dimindex (:α) ⇒ (-1w) ' i
[WORD_NEG_ADD] Theorem
|- ∀v w. -(v + w) = -v + -w
[WORD_NEG_EQ] Theorem
|- ∀w v. (-v = w) ⇔ (v = -w)
[WORD_NEG_EQ_0] Theorem
|- (-v = 0w) ⇔ (v = 0w)
[WORD_NEG_L] Theorem
|- -INT_MINw = INT_MINw
[WORD_NEG_LMUL] Theorem
|- ∀v w. -(v * w) = -v * w
[WORD_NEG_MUL] Theorem
|- ∀w. -w = -1w * w
[WORD_NEG_NEG] Theorem
|- ∀w. - -w = w
[WORD_NEG_RMUL] Theorem
|- ∀v w. -(v * w) = v * -w
[WORD_NEG_SUB] Theorem
|- ∀w v. -(v − w) = w − v
[WORD_NEG_T] Theorem
|- -UINT_MAXw = 1w
[WORD_NOR_NOT_OR] Theorem
|- ∀a b. a ~|| b = ¬(a ‖ b)
[WORD_NOT] Theorem
|- ∀w. ¬w = -w − 1w
[WORD_NOT_0] Theorem
|- ¬0w = UINT_MAXw
[WORD_NOT_GREATER] Theorem
|- ∀a b. ¬(a > b) ⇔ a ≤ b
[WORD_NOT_HIGHER] Theorem
|- ∀a b. ¬(a >₊ b) ⇔ a ≤₊ b
[WORD_NOT_LESS] Theorem
|- ∀a b. ¬(a < b) ⇔ b ≤ a
[WORD_NOT_LESS_EQ] Theorem
|- ∀a b. (a = b) ⇒ ¬(a < b)
[WORD_NOT_LESS_EQUAL] Theorem
|- ∀a b. ¬(a ≤ b) ⇔ b < a
[WORD_NOT_LOWER] Theorem
|- ∀a b. ¬(a <₊ b) ⇔ b ≤₊ a
[WORD_NOT_LOWER_EQ] Theorem
|- ∀a b. (a = b) ⇒ ¬(a <₊ b)
[WORD_NOT_LOWER_EQUAL] Theorem
|- ∀a b. ¬(a ≤₊ b) ⇔ b <₊ a
[WORD_NOT_NOT] Theorem
|- ∀a. ¬ ¬a = a
[WORD_NOT_T] Theorem
|- ¬UINT_MAXw = 0w
[WORD_NOT_XOR] Theorem
|- ∀a b.
(¬a ⊕ ¬b = a ⊕ b) ∧ (a ⊕ ¬b = ¬(a ⊕ b)) ∧ (¬a ⊕ b = ¬(a ⊕ b))
[WORD_OR_ABSORB] Theorem
|- ∀a b. a && (a ‖ b) = a
[WORD_OR_ASSOC] Theorem
|- ∀a b c. (a ‖ b) ‖ c = a ‖ b ‖ c
[WORD_OR_CLAUSES] Theorem
|- ∀a.
(UINT_MAXw ‖ a = UINT_MAXw) ∧ (a ‖ UINT_MAXw = UINT_MAXw) ∧
(0w ‖ a = a) ∧ (a ‖ 0w = a) ∧ (a ‖ a = a)
[WORD_OR_COMM] Theorem
|- ∀a b. a ‖ b = b ‖ a
[WORD_OR_COMP] Theorem
|- ∀a. a ‖ ¬a = UINT_MAXw
[WORD_OR_IDEM] Theorem
|- ∀a. a ‖ a = a
[WORD_PRED_THM] Theorem
|- ∀m. m ≠ 0w ⇒ w2n (m − 1w) < w2n m
[WORD_RCANCEL_SUB] Theorem
|- ∀v w x. (v − w = v − x) ⇔ (w = x)
[WORD_RIGHT_ADD_DISTRIB] Theorem
|- ∀v w x. (v + w) * x = v * x + w * x
[WORD_RIGHT_AND_OVER_OR] Theorem
|- ∀a b c. (a ‖ b) && c = a && c ‖ b && c
[WORD_RIGHT_AND_OVER_XOR] Theorem
|- ∀a b c. (a ⊕ b) && c = a && c ⊕ b && c
[WORD_RIGHT_OR_OVER_AND] Theorem
|- ∀a b c. a && b ‖ c = (a ‖ c) && (b ‖ c)
[WORD_RIGHT_SUB_DISTRIB] Theorem
|- ∀v w x. (w − x) * v = w * v − x * v
[WORD_SET_INDUCT] Theorem
|- ∀P. P ∅ ∧ (∀s. P s ⇒ ∀e. e ∉ s ⇒ P (e INSERT s)) ⇒ ∀s. P s
[WORD_SLICE_BITS_THM] Theorem
|- ∀h w. (h '' 0) w = (h -- 0) w
[WORD_SLICE_COMP_THM] Theorem
|- ∀h m' m l w.
l ≤ m ∧ (m' = m + 1) ∧ m < h ⇒
((h '' m') w ‖ (m '' l) w = (h '' l) w)
[WORD_SLICE_OVER_BITWISE] Theorem
|- (∀h l v w. (h '' l) v && (h '' l) w = (h '' l) (v && w)) ∧
(∀h l v w. (h '' l) v ‖ (h '' l) w = (h '' l) (v ‖ w)) ∧
∀h l v w. (h '' l) v ⊕ (h '' l) w = (h '' l) (v ⊕ w)
[WORD_SLICE_THM] Theorem
|- ∀h l w. (h '' l) w = (h -- l) w ≪ l
[WORD_SLICE_ZERO] Theorem
|- ∀h l w. h < l ⇒ ((h '' l) w = 0w)
[WORD_SLICE_ZERO2] Theorem
|- ∀l h. (h '' l) 0w = 0w
[WORD_SUB] Theorem
|- ∀v w. -w + v = v − w
[WORD_SUB_ADD] Theorem
|- ∀v w. v − w + w = v
[WORD_SUB_ADD2] Theorem
|- ∀v w. v + (w − v) = w
[WORD_SUB_INTRO] Theorem
|- (∀x y. -y + x = x − y) ∧ (∀x y. x + -y = x − y) ∧
(∀x y z. -x * y + z = z − x * y) ∧
(∀x y z. z + -x * y = z − x * y) ∧ (∀x. -1w * x = -x) ∧
(∀x y z. z − -x * y = z + x * y) ∧
∀x y z. -x * y − z = -(x * y + z)
[WORD_SUB_LE] Theorem
|- ∀x y. 0w ≤ y ∧ y ≤ x ⇒ 0w ≤ x − y ∧ x − y ≤ x
[WORD_SUB_LNEG] Theorem
|- ∀v w. -v − w = -(v + w)
[WORD_SUB_LT] Theorem
|- ∀x y. 0w < y ∧ y < x ⇒ 0w < x − y ∧ x − y < x
[WORD_SUB_LZERO] Theorem
|- ∀w. 0w − w = -w
[WORD_SUB_NEG] Theorem
|- ∀v w. -v − -w = w − v
[WORD_SUB_PLUS] Theorem
|- ∀v w x. v − (w + x) = v − w − x
[WORD_SUB_REFL] Theorem
|- ∀w. w − w = 0w
[WORD_SUB_RNEG] Theorem
|- ∀v w. v − -w = v + w
[WORD_SUB_RZERO] Theorem
|- ∀w. w − 0w = w
[WORD_SUB_SUB] Theorem
|- ∀v w x. v − (w − x) = v + x − w
[WORD_SUB_SUB2] Theorem
|- ∀v w. v − (v − w) = w
[WORD_SUB_SUB3] Theorem
|- ∀w v. v − w − v = -w
[WORD_SUB_TRIANGLE] Theorem
|- ∀v w x. v − w + (w − x) = v − x
[WORD_SUM_ZERO] Theorem
|- ∀a b. (a + b = 0w) ⇔ (a = -b)
[WORD_XNOR_NOT_XOR] Theorem
|- ∀a b. a ~?? b = ¬(a ⊕ b)
[WORD_XOR] Theorem
|- ∀a b. a ⊕ b = a && ¬b ‖ b && ¬a
[WORD_XOR_ASSOC] Theorem
|- ∀a b c. (a ⊕ b) ⊕ c = a ⊕ b ⊕ c
[WORD_XOR_CLAUSES] Theorem
|- ∀a.
(UINT_MAXw ⊕ a = ¬a) ∧ (a ⊕ UINT_MAXw = ¬a) ∧ (0w ⊕ a = a) ∧
(a ⊕ 0w = a) ∧ (a ⊕ a = 0w)
[WORD_XOR_COMM] Theorem
|- ∀a b. a ⊕ b = b ⊕ a
[WORD_XOR_COMP] Theorem
|- ∀a. a ⊕ ¬a = UINT_MAXw
[WORD_ZERO_LE] Theorem
|- ∀w. 0w ≤ w ⇔ w2n w < INT_MIN (:α)
[WORD_w2w_EXTRACT] Theorem
|- ∀w. w2w w = (dimindex (:α) − 1 >< 0) w
[WORD_w2w_OVER_ADD] Theorem
|- ∀a b. w2w (a + b) = (dimindex (:α) − 1 -- 0) (w2w a + w2w b)
[WORD_w2w_OVER_BITWISE] Theorem
|- (∀v w. w2w v && w2w w = w2w (v && w)) ∧
(∀v w. w2w v ‖ w2w w = w2w (v ‖ w)) ∧
∀v w. w2w v ⊕ w2w w = w2w (v ⊕ w)
[WORD_w2w_OVER_MUL] Theorem
|- ∀a b. w2w (a * b) = (dimindex (:α) − 1 -- 0) (w2w a * w2w b)
[ZERO_LE_INT_MAX] Theorem
|- 0 ≤ INT_MAX (:α)
[ZERO_LO_INT_MIN] Theorem
|- 0w <₊ INT_MINw
[ZERO_LT_INT_MAX] Theorem
|- 1 < dimindex (:α) ⇒ 0 < INT_MAX (:α)
[ZERO_LT_INT_MIN] Theorem
|- 0 < INT_MIN (:α)
[ZERO_LT_UINT_MAX] Theorem
|- 0 < UINT_MAX (:α)
[ZERO_LT_dimword] Theorem
|- 0 < dimword (:α)
[ZERO_SHIFT] Theorem
|- (∀n. 0w ≪ n = 0w) ∧ (∀n. 0w ≫ n = 0w) ∧ (∀n. 0w ⋙ n = 0w) ∧
(∀n. 0w ⇆ n = 0w) ∧ ∀n. 0w ⇄ n = 0w
[bit_count_is_zero] Theorem
|- ∀w. (bit_count w = 0) ⇔ (w = 0w)
[bit_count_upto_0] Theorem
|- ∀w. bit_count_upto 0 w = 0
[bit_count_upto_SUC] Theorem
|- ∀w n.
bit_count_upto (SUC n) w =
(if w ' n then 1 else 0) + bit_count_upto n w
[bit_count_upto_is_zero] Theorem
|- ∀n w. (bit_count_upto n w = 0) ⇔ ∀i. i < n ⇒ ¬w ' i
[bit_field_insert] Theorem
|- ∀h l a b.
h < l + dimindex (:α) ⇒
(bit_field_insert h l a b =
(let mask = (h '' l) (-1w) in w2w a ≪ l && mask ‖ b && ¬mask))
[dimindex_1] Theorem
|- dimindex (:unit) = 1
[dimindex_10] Theorem
|- dimindex (:10) = 10
[dimindex_11] Theorem
|- dimindex (:11) = 11
[dimindex_12] Theorem
|- dimindex (:12) = 12
[dimindex_128] Theorem
|- dimindex (:128) = 128
[dimindex_16] Theorem
|- dimindex (:16) = 16
[dimindex_1_cases] Theorem
|- ∀a. (dimindex (:α) = 1) ⇒ (a = 0w) ∨ (a = 1w)
[dimindex_2] Theorem
|- dimindex (:2) = 2
[dimindex_20] Theorem
|- dimindex (:20) = 20
[dimindex_24] Theorem
|- dimindex (:24) = 24
[dimindex_28] Theorem
|- dimindex (:28) = 28
[dimindex_3] Theorem
|- dimindex (:3) = 3
[dimindex_30] Theorem
|- dimindex (:30) = 30
[dimindex_32] Theorem
|- dimindex (:32) = 32
[dimindex_4] Theorem
|- dimindex (:4) = 4
[dimindex_48] Theorem
|- dimindex (:48) = 48
[dimindex_5] Theorem
|- dimindex (:5) = 5
[dimindex_6] Theorem
|- dimindex (:6) = 6
[dimindex_64] Theorem
|- dimindex (:64) = 64
[dimindex_7] Theorem
|- dimindex (:7) = 7
[dimindex_8] Theorem
|- dimindex (:8) = 8
[dimindex_9] Theorem
|- dimindex (:9) = 9
[dimindex_96] Theorem
|- dimindex (:96) = 96
[dimindex_dimword_iso] Theorem
|- (dimindex (:α) = dimindex (:β)) ⇔ (dimword (:α) = dimword (:β))
[dimindex_dimword_le_iso] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇔ dimword (:α) ≤ dimword (:β)
[dimindex_dimword_lt_iso] Theorem
|- dimindex (:α) < dimindex (:β) ⇔ dimword (:α) < dimword (:β)
[dimindex_int_max_iso] Theorem
|- (dimindex (:α) = dimindex (:β)) ⇔ (INT_MAX (:α) = INT_MAX (:β))
[dimindex_int_max_le_iso] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇔ INT_MAX (:α) ≤ INT_MAX (:β)
[dimindex_int_max_lt_iso] Theorem
|- dimindex (:α) < dimindex (:β) ⇔ INT_MAX (:α) < INT_MAX (:β)
[dimindex_int_min_iso] Theorem
|- (dimindex (:α) = dimindex (:β)) ⇔ (INT_MIN (:α) = INT_MIN (:β))
[dimindex_int_min_le_iso] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇔ INT_MIN (:α) ≤ INT_MIN (:β)
[dimindex_int_min_lt_iso] Theorem
|- dimindex (:α) < dimindex (:β) ⇔ INT_MIN (:α) < INT_MIN (:β)
[dimindex_lt_dimword] Theorem
|- dimindex (:α) < dimword (:α)
[dimindex_uint_max_iso] Theorem
|- (dimindex (:α) = dimindex (:β)) ⇔ (UINT_MAX (:α) = UINT_MAX (:β))
[dimindex_uint_max_le_iso] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇔ UINT_MAX (:α) ≤ UINT_MAX (:β)
[dimindex_uint_max_lt_iso] Theorem
|- dimindex (:α) < dimindex (:β) ⇔ UINT_MAX (:α) < UINT_MAX (:β)
[dimword_1] Theorem
|- dimword (:unit) = 2
[dimword_10] Theorem
|- dimword (:10) = 1024
[dimword_11] Theorem
|- dimword (:11) = 2048
[dimword_12] Theorem
|- dimword (:12) = 4096
[dimword_128] Theorem
|- dimword (:128) = 340282366920938463463374607431768211456
[dimword_16] Theorem
|- dimword (:16) = 65536
[dimword_2] Theorem
|- dimword (:2) = 4
[dimword_20] Theorem
|- dimword (:20) = 1048576
[dimword_24] Theorem
|- dimword (:24) = 16777216
[dimword_28] Theorem
|- dimword (:28) = 268435456
[dimword_3] Theorem
|- dimword (:3) = 8
[dimword_30] Theorem
|- dimword (:30) = 1073741824
[dimword_32] Theorem
|- dimword (:32) = 4294967296
[dimword_4] Theorem
|- dimword (:4) = 16
[dimword_48] Theorem
|- dimword (:48) = 281474976710656
[dimword_5] Theorem
|- dimword (:5) = 32
[dimword_6] Theorem
|- dimword (:6) = 64
[dimword_64] Theorem
|- dimword (:64) = 18446744073709551616
[dimword_7] Theorem
|- dimword (:7) = 128
[dimword_8] Theorem
|- dimword (:8) = 256
[dimword_9] Theorem
|- dimword (:9) = 512
[dimword_96] Theorem
|- dimword (:96) = 79228162514264337593543950336
[dimword_IS_TWICE_INT_MIN] Theorem
|- dimword (:α) = 2 * INT_MIN (:α)
[dimword_sub_int_min] Theorem
|- dimword (:α) − INT_MIN (:α) = INT_MIN (:α)
[fcp_n2w] Theorem
|- ∀f. $FCP f = word_modify (λi b. f i) 0w
[finite_1] Theorem
|- FINITE 𝕌(:unit)
[finite_10] Theorem
|- FINITE 𝕌(:10)
[finite_11] Theorem
|- FINITE 𝕌(:11)
[finite_12] Theorem
|- FINITE 𝕌(:12)
[finite_128] Theorem
|- FINITE 𝕌(:128)
[finite_16] Theorem
|- FINITE 𝕌(:16)
[finite_2] Theorem
|- FINITE 𝕌(:2)
[finite_20] Theorem
|- FINITE 𝕌(:20)
[finite_24] Theorem
|- FINITE 𝕌(:24)
[finite_28] Theorem
|- FINITE 𝕌(:28)
[finite_3] Theorem
|- FINITE 𝕌(:3)
[finite_30] Theorem
|- FINITE 𝕌(:30)
[finite_32] Theorem
|- FINITE 𝕌(:32)
[finite_4] Theorem
|- FINITE 𝕌(:4)
[finite_48] Theorem
|- FINITE 𝕌(:48)
[finite_5] Theorem
|- FINITE 𝕌(:5)
[finite_6] Theorem
|- FINITE 𝕌(:6)
[finite_64] Theorem
|- FINITE 𝕌(:64)
[finite_7] Theorem
|- FINITE 𝕌(:7)
[finite_8] Theorem
|- FINITE 𝕌(:8)
[finite_9] Theorem
|- FINITE 𝕌(:9)
[finite_96] Theorem
|- FINITE 𝕌(:96)
[foldl_reduce_and] Theorem
|- ∀w.
reduce_and w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $&& (HD l) (TL l))
[foldl_reduce_nand] Theorem
|- ∀w.
reduce_nand w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $~&& (HD l) (TL l))
[foldl_reduce_nor] Theorem
|- ∀w.
reduce_nor w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $~|| (HD l) (TL l))
[foldl_reduce_or] Theorem
|- ∀w.
reduce_or w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $|| (HD l) (TL l))
[foldl_reduce_xnor] Theorem
|- ∀w.
reduce_xnor w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $~?? (HD l) (TL l))
[foldl_reduce_xor] Theorem
|- ∀w.
reduce_xor w =
(let l =
GENLIST
(λi. (let n = dimindex (:α) − 1 − i in (n >< n) w))
(dimindex (:α))
in
FOLDL $?? (HD l) (TL l))
[l2w_w2l] Theorem
|- ∀b w. 1 < b ⇒ (l2w b (w2l b w) = w)
[lsr_1_word_T] Theorem
|- -1w ⋙ 1 = INT_MAXw
[mod_dimindex] Theorem
|- ∀n. n MOD dimindex (:α) < dimword (:α)
[n2w_11] Theorem
|- ∀m n. (n2w m = n2w n) ⇔ (m MOD dimword (:α) = n MOD dimword (:α))
[n2w_BITS] Theorem
|- ∀h l n. h < dimindex (:α) ⇒ (n2w (BITS h l n) = (h -- l) (n2w n))
[n2w_DIV] Theorem
|- ∀a n. a < dimword (:α) ⇒ (n2w (a DIV 2 ** n) = n2w a ⋙ n)
[n2w_SUC] Theorem
|- ∀n. n2w (SUC n) = n2w n + 1w
[n2w_dimword] Theorem
|- n2w (dimword (:α)) = 0w
[n2w_itself_def] Theorem
|- n2w_itself (n,(:α)) = n2w n
[n2w_itself_ind] Theorem
|- ∀P. (∀n. P (n,(:α))) ⇒ ∀v v1. P (v,v1)
[n2w_mod] Theorem
|- ∀n. n2w (n MOD dimword (:α)) = n2w n
[n2w_sub] Theorem
|- ∀a b. b ≤ a ⇒ (n2w (a − b) = n2w a − n2w b)
[n2w_sub_eq_0] Theorem
|- ∀a b. a ≤ b ⇒ (n2w (a − b) = 0w)
[n2w_w2n] Theorem
|- ∀w. n2w (w2n w) = w
[ranged_word_nchotomy] Theorem
|- ∀w. ∃n. (w = n2w n) ∧ n < dimword (:α)
[reduce_and] Theorem
|- ∀w. reduce_and w = if w = UINT_MAXw then 1w else 0w
[reduce_or] Theorem
|- ∀w. reduce_or w = if w = 0w then 0w else 1w
[s2w_w2s] Theorem
|- ∀c2n n2c b w.
1 < b ∧ (∀x. x < b ⇒ (c2n (n2c x) = x)) ⇒
(s2w b c2n (w2s b n2c w) = w)
[saturate_add] Theorem
|- ∀a b.
saturate_add a b =
if UINT_MAXw − a ≤₊ b then UINT_MAXw else a + b
[saturate_mul] Theorem
|- ∀a b.
saturate_mul a b =
if FINITE 𝕌(:α) ∧ w2w UINT_MAXw ≤₊ w2w a * w2w b then UINT_MAXw
else a * b
[saturate_sub] Theorem
|- ∀a b. saturate_sub a b = if a ≤₊ b then 0w else a − b
[saturate_w2w] Theorem
|- ∀w.
saturate_w2w w =
if dimindex (:β) ≤ dimindex (:α) ∧ w2w UINT_MAXw ≤₊ w then
UINT_MAXw
else w2w w
[saturate_w2w_n2w] Theorem
|- ∀n.
saturate_w2w (n2w n) =
(let m = n MOD dimword (:α)
in
if dimindex (:β) ≤ dimindex (:α) ∧ dimword (:β) ≤ m then
UINT_MAXw
else n2w m)
[sw2sw] Theorem
|- ∀w i.
i < dimindex (:β) ⇒
(sw2sw w ' i ⇔
if i < dimindex (:α) ∨ dimindex (:β) < dimindex (:α) then w ' i
else word_msb w)
[sw2sw_0] Theorem
|- sw2sw 0w = 0w
[sw2sw_id] Theorem
|- ∀w. sw2sw w = w
[sw2sw_sw2sw] Theorem
|- ∀w.
sw2sw (sw2sw w) =
if
dimindex (:β) < dimindex (:α) ∧ dimindex (:β) < dimindex (:γ)
then
sw2sw (w2w w)
else sw2sw w
[sw2sw_w2w] Theorem
|- ∀w.
sw2sw w =
(if word_msb w then -1w ≪ dimindex (:α) else 0w) ‖ w2w w
[sw2sw_w2w_add] Theorem
|- ∀w.
sw2sw w =
(if word_msb w then -1w ≪ dimindex (:α) else 0w) + w2w w
[sw2sw_word_T] Theorem
|- sw2sw (-1w) = -1w
[w2l_l2w] Theorem
|- ∀b l. w2l b (l2w b l) = n2l b (l2n b l MOD dimword (:α))
[w2n_11] Theorem
|- ∀v w. (w2n v = w2n w) ⇔ (v = w)
[w2n_11_lift] Theorem
|- ∀a b.
dimindex (:α) ≤ dimindex (:γ) ∧ dimindex (:β) ≤ dimindex (:γ) ⇒
((w2n a = w2n b) ⇔ (w2w a = w2w b))
[w2n_add] Theorem
|- ∀a b. ¬word_msb a ∧ ¬word_msb b ⇒ (w2n (a + b) = w2n a + w2n b)
[w2n_eq_0] Theorem
|- ∀w. (w2n w = 0) ⇔ (w = 0w)
[w2n_lsr] Theorem
|- ∀w m. w2n (w ⋙ m) = w2n w DIV 2 ** m
[w2n_lt] Theorem
|- ∀w. w2n w < dimword (:α)
[w2n_minus1] Theorem
|- w2n (-1w) = dimword (:α) − 1
[w2n_n2w] Theorem
|- ∀n. w2n (n2w n) = n MOD dimword (:α)
[w2n_w2w] Theorem
|- ∀w.
w2n (w2w w) =
if dimindex (:α) ≤ dimindex (:β) then w2n w
else w2n ((dimindex (:β) − 1 -- 0) w)
[w2n_w2w_le] Theorem
|- ∀w. w2n (w2w w) ≤ w2n w
[w2s_s2w] Theorem
|- ∀b c2n n2c s.
w2s b n2c (s2w b c2n s) =
n2s b n2c (s2n b c2n s MOD dimword (:α))
[w2w] Theorem
|- ∀w i. i < dimindex (:β) ⇒ (w2w w ' i ⇔ i < dimindex (:α) ∧ w ' i)
[w2w_0] Theorem
|- w2w 0w = 0w
[w2w_LSL] Theorem
|- ∀w n.
w2w (w ≪ n) =
if n < dimindex (:α) then
w2w ((dimindex (:α) − 1 − n -- 0) w) ≪ n
else 0w
[w2w_eq_n2w] Theorem
|- ∀x y.
dimindex (:α) ≤ dimindex (:β) ∧ y < dimword (:α) ⇒
((w2w x = n2w y) ⇔ (x = n2w y))
[w2w_id] Theorem
|- ∀w. w2w w = w
[w2w_lt] Theorem
|- ∀w. w2n (w2w w) < dimword (:α)
[w2w_n2w] Theorem
|- ∀n.
w2w (n2w n) =
if dimindex (:β) ≤ dimindex (:α) then n2w n
else n2w (BITS (dimindex (:α) − 1) 0 n)
[w2w_w2w] Theorem
|- ∀w. w2w (w2w w) = w2w ((dimindex (:β) − 1 -- 0) w)
[word_0] Theorem
|- ∀i. i < dimindex (:α) ⇒ ¬0w ' i
[word_0_n2w] Theorem
|- w2n 0w = 0
[word_1_n2w] Theorem
|- w2n 1w = 1
[word_1comp_n2w] Theorem
|- ∀n. ¬n2w n = n2w (dimword (:α) − 1 − n MOD dimword (:α))
[word_2comp_dimindex_1] Theorem
|- ∀w. (dimindex (:α) = 1) ⇒ (-w = w)
[word_2comp_n2w] Theorem
|- ∀n. -n2w n = n2w (dimword (:α) − n MOD dimword (:α))
[word_H] Theorem
|- ∀n. n < dimindex (:α) ⇒ (INT_MAXw ' n ⇔ n < dimindex (:α) − 1)
[word_L] Theorem
|- ∀n. n < dimindex (:α) ⇒ (INT_MINw ' n ⇔ (n = dimindex (:α) − 1))
[word_L2] Theorem
|- INT_MINw2 = if 1 < dimindex (:α) then 0w else INT_MINw
[word_L2_MULT] Theorem
|- (INT_MINw2 * INT_MINw2 = INT_MINw2) ∧
(INT_MINw * INT_MINw2 = INT_MINw2) ∧
(∀n. n2w n * INT_MINw2 = if EVEN n then 0w else INT_MINw2) ∧
∀n. -n2w n * INT_MINw2 = if EVEN n then 0w else INT_MINw2
[word_L_MULT] Theorem
|- ∀n. n2w n * INT_MINw = if EVEN n then 0w else INT_MINw
[word_L_MULT_NEG] Theorem
|- ∀n. -n2w n * INT_MINw = if EVEN n then 0w else INT_MINw
[word_T] Theorem
|- ∀i. i < dimindex (:α) ⇒ UINT_MAXw ' i
[word_T_not_zero] Theorem
|- -1w ≠ 0w
[word_abs] Theorem
|- ∀w.
word_abs w = FCP i. ¬word_msb w ∧ w ' i ∨ word_msb w ∧ (-w) ' i
[word_abs_diff] Theorem
|- ∀a b. word_abs (a − b) = word_abs (b − a)
[word_abs_neg] Theorem
|- ∀w. word_abs (-w) = word_abs w
[word_abs_word_abs] Theorem
|- ∀w. word_abs (word_abs w) = word_abs w
[word_add_n2w] Theorem
|- ∀m n. n2w m + n2w n = n2w (m + n)
[word_and_n2w] Theorem
|- ∀n m. n2w n && n2w m = n2w (BITWISE (dimindex (:α)) $/\ n m)
[word_asr] Theorem
|- ∀w n.
w ≫ n =
if word_msb w then
(dimindex (:α) − 1 '' dimindex (:α) − n) UINT_MAXw ‖ w ⋙ n
else w ⋙ n
[word_asr_n2w] Theorem
|- ∀n w.
w ≫ n =
if word_msb w then
n2w
(dimword (:α) −
2 ** (dimindex (:α) − MIN n (dimindex (:α)))) ‖ w ⋙ n
else w ⋙ n
[word_bin_list] Theorem
|- word_from_bin_list o word_to_bin_list = I
[word_bin_string] Theorem
|- word_from_bin_string o word_to_bin_string = I
[word_bit] Theorem
|- ∀w b. b < dimindex (:α) ⇒ (w ' b ⇔ word_bit b w)
[word_bit_0] Theorem
|- ∀h. ¬word_bit h 0w
[word_bit_0_word_T] Theorem
|- word_bit 0 (-1w)
[word_bit_n2w] Theorem
|- ∀b n. word_bit b (n2w n) ⇔ b ≤ dimindex (:α) − 1 ∧ BIT b n
[word_bits_n2w] Theorem
|- ∀h l n.
(h -- l) (n2w n) = n2w (BITS (MIN h (dimindex (:α) − 1)) l n)
[word_bits_w2w] Theorem
|- ∀w h l.
(h -- l) (w2w w) = w2w ((MIN h (dimindex (:β) − 1) -- l) w)
[word_concat_0] Theorem
|- ∀x. FINITE 𝕌(:α) ∧ x < dimword (:β) ⇒ (0w @@ n2w x = n2w x)
[word_concat_0_0] Theorem
|- 0w @@ 0w = 0w
[word_concat_0_eq] Theorem
|- ∀x y.
FINITE 𝕌(:α) ∧ dimindex (:β) ≤ dimindex (:γ) ∧
y < dimword (:β) ⇒
((0w @@ x = n2w y) ⇔ (x = n2w y))
[word_concat_word_T] Theorem
|- -1w @@ -1w = w2w (-1w)
[word_dec_list] Theorem
|- word_from_dec_list o word_to_dec_list = I
[word_dec_string] Theorem
|- word_from_dec_string o word_to_dec_string = I
[word_div_1] Theorem
|- ∀v. v // 1w = v
[word_eq_0] Theorem
|- ∀w. (w = 0w) ⇔ ∀i. i < dimindex (:α) ⇒ ¬w ' i
[word_eq_n2w] Theorem
|- (∀m n. (n2w m = n2w n) ⇔ MOD_2EXP_EQ (dimindex (:α)) m n) ∧
(∀n. (n2w n = -1w) ⇔ MOD_2EXP_MAX (dimindex (:α)) n) ∧
∀n. (-1w = n2w n) ⇔ MOD_2EXP_MAX (dimindex (:α)) n
[word_extract_eq_n2w] Theorem
|- ∀x h y.
dimindex (:α) ≤ dimindex (:β) ∧ dimindex (:α) − 1 ≤ h ∧
y < dimword (:α) ⇒
(((h >< 0) x = n2w y) ⇔ (x = n2w y))
[word_extract_mask] Theorem
|- ∀h l a.
(h >< l) a = if l ≤ h then a ⋙ l && 2w ≪ (h − l) − 1w else 0w
[word_extract_n2w] Theorem
|- (h >< l) (n2w n) =
if dimindex (:β) ≤ dimindex (:α) then
n2w (BITS (MIN h (dimindex (:α) − 1)) l n)
else
n2w
(BITS
(MIN (MIN h (dimindex (:α) − 1)) (dimindex (:α) − 1 + l)) l
n)
[word_extract_w2w] Theorem
|- ∀w h l.
dimindex (:α) ≤ dimindex (:β) ⇒ ((h >< l) (w2w w) = (h >< l) w)
[word_ge_n2w] Theorem
|- ∀a b.
n2w a ≥ n2w b ⇔
(let sa = BIT (dimindex (:α) − 1) a and
sb = BIT (dimindex (:α) − 1) b
in
(sa ⇔ sb) ∧ a MOD dimword (:α) ≥ b MOD dimword (:α) ∨
¬sa ∧ sb)
[word_gt_n2w] Theorem
|- ∀a b.
n2w a > n2w b ⇔
(let sa = BIT (dimindex (:α) − 1) a and
sb = BIT (dimindex (:α) − 1) b
in
(sa ⇔ sb) ∧ a MOD dimword (:α) > b MOD dimword (:α) ∨
¬sa ∧ sb)
[word_hex_list] Theorem
|- word_from_hex_list o word_to_hex_list = I
[word_hex_string] Theorem
|- word_from_hex_string o word_to_hex_string = I
[word_hi_n2w] Theorem
|- ∀a b. n2w a >₊ n2w b ⇔ a MOD dimword (:α) > b MOD dimword (:α)
[word_hs_n2w] Theorem
|- ∀a b. n2w a ≥₊ n2w b ⇔ a MOD dimword (:α) ≥ b MOD dimword (:α)
[word_index] Theorem
|- ∀n i. i < dimindex (:α) ⇒ (n2w n ' i ⇔ BIT i n)
[word_index_n2w] Theorem
|- ∀n i.
n2w n ' i ⇔
if i < dimindex (:α) then BIT i n
else FAIL $' index too large (n2w n) i
[word_join_0] Theorem
|- ∀a. word_join 0w a = w2w a
[word_join_index] Theorem
|- ∀i a b.
FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) ∧ i < dimindex (:α + β) ⇒
(word_join a b ' i ⇔
if i < dimindex (:β) then b ' i else a ' (i − dimindex (:β)))
[word_join_word_T] Theorem
|- word_join (-1w) (-1w) = -1w
[word_le_n2w] Theorem
|- ∀a b.
n2w a ≤ n2w b ⇔
(let sa = BIT (dimindex (:α) − 1) a and
sb = BIT (dimindex (:α) − 1) b
in
(sa ⇔ sb) ∧ a MOD dimword (:α) ≤ b MOD dimword (:α) ∨
sa ∧ ¬sb)
[word_lo_n2w] Theorem
|- ∀a b. n2w a <₊ n2w b ⇔ a MOD dimword (:α) < b MOD dimword (:α)
[word_log2_1] Theorem
|- word_log2 1w = 0w
[word_log2_n2w] Theorem
|- ∀n. word_log2 (n2w n) = n2w (LOG2 (n MOD dimword (:α)))
[word_ls_n2w] Theorem
|- ∀a b. n2w a ≤₊ n2w b ⇔ a MOD dimword (:α) ≤ b MOD dimword (:α)
[word_lsb] Theorem
|- word_lsb = word_bit 0
[word_lsb_n2w] Theorem
|- ∀n. word_lsb (n2w n) ⇔ ODD n
[word_lsb_word_T] Theorem
|- word_lsb (-1w)
[word_lsl_n2w] Theorem
|- ∀n m.
n2w m ≪ n =
if dimindex (:α) − 1 < n then 0w else n2w (m * 2 ** n)
[word_lsr_n2w] Theorem
|- ∀w n. w ⋙ n = (dimindex (:α) − 1 -- n) w
[word_lt_n2w] Theorem
|- ∀a b.
n2w a < n2w b ⇔
(let sa = BIT (dimindex (:α) − 1) a and
sb = BIT (dimindex (:α) − 1) b
in
(sa ⇔ sb) ∧ a MOD dimword (:α) < b MOD dimword (:α) ∨
sa ∧ ¬sb)
[word_modify_n2w] Theorem
|- ∀f n. word_modify f (n2w n) = n2w (BIT_MODIFY (dimindex (:α)) f n)
[word_msb] Theorem
|- word_msb = word_bit (dimindex (:α) − 1)
[word_msb_n2w] Theorem
|- ∀n. word_msb (n2w n) ⇔ BIT (dimindex (:α) − 1) n
[word_msb_n2w_numeric] Theorem
|- word_msb (n2w n) ⇔ INT_MIN (:α) ≤ n MOD dimword (:α)
[word_msb_neg] Theorem
|- ∀w. word_msb w ⇔ w < 0w
[word_msb_word_T] Theorem
|- word_msb (-1w)
[word_mul_n2w] Theorem
|- ∀m n. n2w m * n2w n = n2w (m * n)
[word_nand_n2w] Theorem
|- ∀n m.
n2w n ~&& n2w m =
n2w (BITWISE (dimindex (:α)) (λx y. ¬(x ∧ y)) n m)
[word_nchotomy] Theorem
|- ∀w. ∃n. w = n2w n
[word_nor_n2w] Theorem
|- ∀n m.
n2w n ~|| n2w m =
n2w (BITWISE (dimindex (:α)) (λx y. ¬(x ∨ y)) n m)
[word_oct_list] Theorem
|- word_from_oct_list o word_to_oct_list = I
[word_oct_string] Theorem
|- word_from_oct_string o word_to_oct_string = I
[word_or_n2w] Theorem
|- ∀n m. n2w n ‖ n2w m = n2w (BITWISE (dimindex (:α)) $\/ n m)
[word_reduce_n2w] Theorem
|- ∀n f.
word_reduce f (n2w n) =
$FCP
(K
(let l = BOOLIFY (dimindex (:α)) n []
in
FOLDL f (HD l) (TL l)))
[word_replicate_concat_word_list] Theorem
|- ∀n w. word_replicate n w = concat_word_list (GENLIST (K w) n)
[word_reverse_0] Theorem
|- word_reverse 0w = 0w
[word_reverse_n2w] Theorem
|- ∀n. word_reverse (n2w n) = n2w (BIT_REVERSE (dimindex (:α)) n)
[word_reverse_thm] Theorem
|- ∀w v n.
(word_reverse (word_reverse w) = w) ∧
(word_reverse (w ≪ n) = word_reverse w ⋙ n) ∧
(word_reverse (w ⋙ n) = word_reverse w ≪ n) ∧
(word_reverse (w ‖ v) = word_reverse w ‖ word_reverse v) ∧
(word_reverse (w && v) = word_reverse w && word_reverse v) ∧
(word_reverse (w ⊕ v) = word_reverse w ⊕ word_reverse v) ∧
(word_reverse (¬w) = ¬word_reverse w) ∧ (word_reverse 0w = 0w) ∧
(word_reverse (-1w) = -1w) ∧
((word_reverse w = 0w) ⇔ (w = 0w)) ∧
((word_reverse w = -1w) ⇔ (w = -1w))
[word_reverse_word_T] Theorem
|- word_reverse (-1w) = -1w
[word_ror] Theorem
|- ∀w n.
w ⇄ n =
(let x = n MOD dimindex (:α)
in
(dimindex (:α) − 1 -- x) w ‖
(x − 1 -- 0) w ≪ (dimindex (:α) − x))
[word_ror_n2w] Theorem
|- ∀n a.
n2w a ⇄ n =
(let x = n MOD dimindex (:α)
in
n2w
(BITS (dimindex (:α) − 1) x a +
BITS (x − 1) 0 a * 2 ** (dimindex (:α) − x)))
[word_rrx_0] Theorem
|- (word_rrx (F,0w) = (F,0w)) ∧ (word_rrx (T,0w) = (F,INT_MINw))
[word_rrx_n2w] Theorem
|- ∀c a.
word_rrx (c,n2w a) =
(ODD a,
n2w
(BITS (dimindex (:α) − 1) 1 a + SBIT c (dimindex (:α) − 1)))
[word_rrx_word_T] Theorem
|- (word_rrx (F,-1w) = (T,INT_MAXw)) ∧ (word_rrx (T,-1w) = (T,-1w))
[word_shift_bv] Theorem
|- (∀w n. n < dimword (:α) ⇒ (w ≪ n = w <<~ n2w n)) ∧
(∀w n. n < dimword (:α) ⇒ (w ≫ n = w >>~ n2w n)) ∧
(∀w n. n < dimword (:α) ⇒ (w ⋙ n = w >>>~ n2w n)) ∧
(∀w n. w ⇄ n = w #>>~ n2w (n MOD dimindex (:α))) ∧
∀w n. w ⇆ n = w #<<~ n2w (n MOD dimindex (:α))
[word_sign_extend_bits] Theorem
|- ∀h l w.
(h --- l) w =
word_sign_extend (MIN (SUC h) (dimindex (:α)) − l) ((h -- l) w)
[word_signed_bits_n2w] Theorem
|- ∀h l n.
(h --- l) (n2w n) =
n2w
(SIGN_EXTEND (MIN (SUC h) (dimindex (:α)) − l) (dimindex (:α))
(BITS (MIN h (dimindex (:α) − 1)) l n))
[word_slice_n2w] Theorem
|- ∀h l n.
(h '' l) (n2w n) = n2w (SLICE (MIN h (dimindex (:α) − 1)) l n)
[word_sub_w2n] Theorem
|- ∀x y. y ≤₊ x ⇒ (w2n (x − y) = w2n x − w2n y)
[word_xnor_n2w] Theorem
|- ∀n m. n2w n ~?? n2w m = n2w (BITWISE (dimindex (:α)) $<=> n m)
[word_xor_n2w] Theorem
|- ∀n m.
n2w n ⊕ n2w m = n2w (BITWISE (dimindex (:α)) (λx y. x ⇎ y) n m)
*)
end
HOL 4, Kananaskis-10