(* Author: Jeremy Dawson and Gerwin Klein, NICTA contains arithmetic theorems for word, instantiations to arithmetic type classes and tactics for reducing word arithmetic to linear arithmetic on int or nat *) header {* Word Arithmetic *} theory WordArith imports WordDefinition begin lemma word_less_alt: "(a < b) = (uint a < uint b)" unfolding word_less_def word_le_def by (auto simp del: word_uint.Rep_inject simp: word_uint.Rep_inject [symmetric]) lemma signed_linorder: "class.linorder word_sle word_sless" proof qed (unfold word_sle_def word_sless_def, auto) interpretation signed: linorder "word_sle" "word_sless" by (rule signed_linorder) lemmas word_arith_wis = word_add_def word_mult_def word_minus_def word_succ_def word_pred_def word_0_wi word_1_wi lemma udvdI: "0 \ n ==> uint b = n * uint a ==> a udvd b" by (auto simp: udvd_def) lemmas word_div_no [simp] = word_div_def [of "number_of a" "number_of b", standard] lemmas word_mod_no [simp] = word_mod_def [of "number_of a" "number_of b", standard] lemmas word_less_no [simp] = word_less_def [of "number_of a" "number_of b", standard] lemmas word_le_no [simp] = word_le_def [of "number_of a" "number_of b", standard] lemmas word_sless_no [simp] = word_sless_def [of "number_of a" "number_of b", standard] lemmas word_sle_no [simp] = word_sle_def [of "number_of a" "number_of b", standard] (* following two are available in class number_ring, but convenient to have them here here; note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1 are in the default simpset, so to use the automatic simplifications for (eg) sint (number_of bin) on sint 1, must do (simp add: word_1_no del: numeral_1_eq_1) *) lemmas word_0_wi_Pls = word_0_wi [folded Pls_def] lemmas word_0_no = word_0_wi_Pls [folded word_no_wi] lemma int_one_bin: "(1 :: int) == (Int.Pls BIT bit.B1)" unfolding Pls_def Bit_def by auto lemma word_1_no: "(1 :: 'a :: len0 word) == number_of (Int.Pls BIT bit.B1)" unfolding word_1_wi word_number_of_def int_one_bin by auto lemma word_m1_wi: "-1 == word_of_int -1" by (rule word_number_of_alt) lemma word_m1_wi_Min: "-1 = word_of_int Int.Min" by (simp add: word_m1_wi number_of_eq) lemma word_0_bl: "of_bl [] = 0" unfolding word_0_wi of_bl_def by (simp add : Pls_def) lemma word_1_bl: "of_bl [True] = 1" unfolding word_1_wi of_bl_def by (simp add : bl_to_bin_def Bit_def Pls_def) lemma uint_0 [simp] : "(uint 0 = 0)" unfolding word_0_wi by (simp add: word_ubin.eq_norm Pls_def [symmetric]) lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0" by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def) lemma to_bl_0: "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" unfolding uint_bl by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric]) lemma uint_0_iff: "(uint x = 0) = (x = 0)" by (auto intro!: word_uint.Rep_eqD) lemma unat_0_iff: "(unat x = 0) = (x = 0)" unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) lemma unat_0 [simp]: "unat 0 = 0" unfolding unat_def by auto lemma size_0_same': "size w = 0 ==> w = (v :: 'a :: len0 word)" apply (unfold word_size) apply (rule box_equals) defer apply (rule word_uint.Rep_inverse)+ apply (rule word_ubin.norm_eq_iff [THEN iffD1]) apply simp done lemmas size_0_same = size_0_same' [folded word_size] lemmas unat_eq_0 = unat_0_iff lemmas unat_eq_zero = unat_0_iff lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" by (auto simp: unat_0_iff [symmetric]) lemma ucast_0 [simp] : "ucast 0 = 0" unfolding ucast_def by simp (simp add: word_0_wi) lemma sint_0 [simp] : "sint 0 = 0" unfolding sint_uint by (simp add: Pls_def [symmetric]) lemma scast_0 [simp] : "scast 0 = 0" apply (unfold scast_def) apply simp apply (simp add: word_0_wi) done lemma sint_n1 [simp] : "sint -1 = -1" apply (unfold word_m1_wi_Min) apply (simp add: word_sbin.eq_norm) apply (unfold Min_def number_of_eq) apply simp done lemma scast_n1 [simp] : "scast -1 = -1" apply (unfold scast_def sint_n1) apply (unfold word_number_of_alt) apply (rule refl) done lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1" unfolding word_1_wi by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1" by (unfold unat_def uint_1) auto lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1" unfolding ucast_def word_1_wi by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) (* abstraction preserves the operations (the definitions tell this for bins in range uint) *) lemmas arths = bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm, standard] lemma wi_homs: shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and wi_hom_neg: "- word_of_int a = word_of_int (- a)" and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (Int.succ a)" and wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)" by (auto simp: word_arith_wis arths) lemmas wi_hom_syms = wi_homs [symmetric] lemma word_sub_def: "a - b == a + - (b :: 'a :: len0 word)" unfolding word_sub_wi diff_def by (simp only : word_uint.Rep_inverse wi_hom_syms) lemmas word_diff_minus = word_sub_def [THEN meta_eq_to_obj_eq, standard] lemma word_of_int_sub_hom: "(word_of_int a) - word_of_int b = word_of_int (a - b)" unfolding word_sub_def diff_def by (simp only : wi_homs) lemmas new_word_of_int_homs = word_of_int_sub_hom wi_homs word_0_wi word_1_wi lemmas new_word_of_int_hom_syms = new_word_of_int_homs [symmetric, standard] lemmas word_of_int_hom_syms = new_word_of_int_hom_syms [unfolded succ_def pred_def] lemmas word_of_int_homs = new_word_of_int_homs [unfolded succ_def pred_def] lemmas word_of_int_add_hom = word_of_int_homs (2) lemmas word_of_int_mult_hom = word_of_int_homs (3) lemmas word_of_int_minus_hom = word_of_int_homs (4) lemmas word_of_int_succ_hom = word_of_int_homs (5) lemmas word_of_int_pred_hom = word_of_int_homs (6) lemmas word_of_int_0_hom = word_of_int_homs (7) lemmas word_of_int_1_hom = word_of_int_homs (8) (* now, to get the weaker results analogous to word_div/mod_def *) lemmas word_arith_alts = word_sub_wi [unfolded succ_def pred_def, standard] word_arith_wis [unfolded succ_def pred_def, standard] lemmas word_sub_alt = word_arith_alts (1) lemmas word_add_alt = word_arith_alts (2) lemmas word_mult_alt = word_arith_alts (3) lemmas word_minus_alt = word_arith_alts (4) lemmas word_succ_alt = word_arith_alts (5) lemmas word_pred_alt = word_arith_alts (6) lemmas word_0_alt = word_arith_alts (7) lemmas word_1_alt = word_arith_alts (8) subsection "Transferring goals from words to ints" lemma word_ths: shows word_succ_p1: "word_succ a = a + 1" and word_pred_m1: "word_pred a = a - 1" and word_pred_succ: "word_pred (word_succ a) = a" and word_succ_pred: "word_succ (word_pred a) = a" and word_mult_succ: "word_succ a * b = b + a * b" by (rule word_uint.Abs_cases [of b], rule word_uint.Abs_cases [of a], simp add: pred_def succ_def add_commute mult_commute ring_distribs new_word_of_int_homs)+ lemmas uint_cong = arg_cong [where f = uint] lemmas uint_word_ariths = word_arith_alts [THEN trans [OF uint_cong int_word_uint], standard] lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] (* similar expressions for sint (arith operations) *) lemmas sint_word_ariths = uint_word_arith_bintrs [THEN uint_sint [symmetric, THEN trans], unfolded uint_sint bintr_arith1s bintr_ariths len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep, standard] lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint], standard] lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint], standard] lemma word_pred_0_n1: "word_pred 0 = word_of_int -1" unfolding word_pred_def number_of_eq by (simp add : pred_def word_no_wi) lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min" by (simp add: word_pred_0_n1 number_of_eq) lemma word_m1_Min: "- 1 = word_of_int Int.Min" unfolding Min_def by (simp only: word_of_int_hom_syms) lemma succ_pred_no [simp]: "word_succ (number_of bin) = number_of (Int.succ bin) & word_pred (number_of bin) = number_of (Int.pred bin)" unfolding word_number_of_def by (simp add : new_word_of_int_homs) lemma word_sp_01 [simp] : "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0" by (unfold word_0_no word_1_no) auto (* alternative approach to lifting arithmetic equalities *) lemma word_of_int_Ex: "\y. x = word_of_int y" by (rule_tac x="uint x" in exI) simp lemma word_arith_eqs: fixes a :: "'a::len0 word" fixes b :: "'a::len0 word" shows word_add_0: "0 + a = a" and word_add_0_right: "a + 0 = a" and word_mult_1: "1 * a = a" and word_mult_1_right: "a * 1 = a" and word_add_commute: "a + b = b + a" and word_add_assoc: "a + b + c = a + (b + c)" and word_add_left_commute: "a + (b + c) = b + (a + c)" and word_mult_commute: "a * b = b * a" and word_mult_assoc: "a * b * c = a * (b * c)" and word_mult_left_commute: "a * (b * c) = b * (a * c)" and word_left_distrib: "(a + b) * c = a * c + b * c" and word_right_distrib: "a * (b + c) = a * b + a * c" and word_left_minus: "- a + a = 0" and word_diff_0_right: "a - 0 = a" and word_diff_self: "a - a = 0" using word_of_int_Ex [of a] word_of_int_Ex [of b] word_of_int_Ex [of c] by (auto simp: word_of_int_hom_syms [symmetric] zadd_0_right add_commute add_assoc add_left_commute mult_commute mult_assoc mult_left_commute left_distrib right_distrib) lemmas word_add_ac = word_add_commute word_add_assoc word_add_left_commute lemmas word_mult_ac = word_mult_commute word_mult_assoc word_mult_left_commute lemmas word_plus_ac0 = word_add_0 word_add_0_right word_add_ac lemmas word_times_ac1 = word_mult_1 word_mult_1_right word_mult_ac subsection "Order on fixed-length words" lemma word_order_trans: "x <= y ==> y <= z ==> x <= (z :: 'a :: len0 word)" unfolding word_le_def by auto lemma word_order_refl: "z <= (z :: 'a :: len0 word)" unfolding word_le_def by auto lemma word_order_antisym: "x <= y ==> y <= x ==> x = (y :: 'a :: len0 word)" unfolding word_le_def by (auto intro!: word_uint.Rep_eqD) lemma word_order_linear: "y <= x | x <= (y :: 'a :: len0 word)" unfolding word_le_def by auto lemma word_zero_le [simp] : "0 <= (y :: 'a :: len0 word)" unfolding word_le_def by auto instance word :: (len0) semigroup_add by intro_classes (simp add: word_add_assoc) instance word :: (len0) linorder by intro_classes (auto simp: word_less_def word_le_def) instance word :: (len0) ring by intro_classes (auto simp: word_arith_eqs word_diff_minus word_diff_self [unfolded word_diff_minus]) lemma word_m1_ge [simp] : "word_pred 0 >= y" unfolding word_le_def by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto lemmas word_n1_ge [simp] = word_m1_ge [simplified word_sp_01] lemmas word_not_simps [simp] = word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))" unfolding word_less_def by auto lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y", standard] lemma word_sless_alt: "(a =0. unat b = n * unat a)" apply (unfold udvd_def) apply safe apply (simp add: unat_def nat_mult_distrib) apply (simp add: uint_nat int_mult) apply (rule exI) apply safe prefer 2 apply (erule notE) apply (rule refl) apply force done lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y" unfolding dvd_def udvd_nat_alt by force lemmas unat_mono = word_less_nat_alt [THEN iffD1, standard] lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) ==> (0 :: 'a word) ~= 1"; unfolding word_arith_wis by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc) lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one] lemma no_no [simp] : "number_of (number_of b) = number_of b" by (simp add: number_of_eq) lemma unat_minus_one: "x ~= 0 ==> unat (x - 1) = unat x - 1" apply (unfold unat_def) apply (simp only: int_word_uint word_arith_alts rdmods) apply (subgoal_tac "uint x >= 1") prefer 2 apply (drule contrapos_nn) apply (erule word_uint.Rep_inverse' [symmetric]) apply (insert uint_ge_0 [of x])[1] apply arith apply (rule box_equals) apply (rule nat_diff_distrib) prefer 2 apply assumption apply simp apply (subst mod_pos_pos_trivial) apply arith apply (insert uint_lt2p [of x])[1] apply arith apply (rule refl) apply simp done lemma measure_unat: "p ~= 0 ==> unat (p - 1) < unat p" by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric]) lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0, standard] lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0, standard] lemma uint_sub_lt2p [simp]: "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 2 ^ len_of TYPE('a)" using uint_ge_0 [of y] uint_lt2p [of x] by arith subsection "Conditions for the addition (etc) of two words to overflow" lemma uint_add_lem: "(uint x + uint y < 2 ^ len_of TYPE('a)) = (uint (x + y :: 'a :: len0 word) = uint x + uint y)" by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) lemma uint_mult_lem: "(uint x * uint y < 2 ^ len_of TYPE('a)) = (uint (x * y :: 'a :: len0 word) = uint x * uint y)" by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) lemma uint_sub_lem: "(uint x >= uint y) = (uint (x - y) = uint x - uint y)" by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) lemma uint_add_le: "uint (x + y) <= uint x + uint y" unfolding uint_word_ariths by (auto simp: mod_add_if_z) lemma uint_sub_ge: "uint (x - y) >= uint x - uint y" unfolding uint_word_ariths by (auto simp: mod_sub_if_z) lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified, standard] lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified, standard] subsection {* Definition of uint\_arith *} lemma word_of_int_inverse: "word_of_int r = a ==> 0 <= r ==> r < 2 ^ len_of TYPE('a) ==> uint (a::'a::len0 word) = r" apply (erule word_uint.Abs_inverse' [rotated]) apply (simp add: uints_num) done lemma uint_split: fixes x::"'a::len0 word" shows "P (uint x) = (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)" apply (fold word_int_case_def) apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq' split: word_int_split) done lemma uint_split_asm: fixes x::"'a::len0 word" shows "P (uint x) = (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))" by (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq' split: uint_split) lemmas uint_splits = uint_split uint_split_asm lemmas uint_arith_simps = word_le_def word_less_alt word_uint.Rep_inject [symmetric] uint_sub_if' uint_plus_if' (* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *) lemma power_False_cong: "False ==> a ^ b = c ^ d" by auto (* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *) ML {* fun uint_arith_ss_of ss = ss addsimps @{thms uint_arith_simps} delsimps @{thms word_uint.Rep_inject} addsplits @{thms split_if_asm} addcongs @{thms power_False_cong} fun uint_arith_tacs ctxt = let fun arith_tac' n t = Arith_Data.verbose_arith_tac ctxt n t handle COOPER => Seq.empty; val cs = claset_of ctxt; val ss = simpset_of ctxt; in [ clarify_tac cs 1, full_simp_tac (uint_arith_ss_of ss) 1, ALLGOALS (full_simp_tac (HOL_ss addsplits @{thms uint_splits} addcongs @{thms power_False_cong})), rewrite_goals_tac @{thms word_size}, ALLGOALS (fn n => REPEAT (resolve_tac [allI, impI] n) THEN REPEAT (etac conjE n) THEN REPEAT (dtac @{thm word_of_int_inverse} n THEN atac n THEN atac n)), TRYALL arith_tac' ] end fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt)) *} method_setup uint_arith = {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *} "solving word arithmetic via integers and arith" subsection "More on overflows and monotonicity" lemma no_plus_overflow_uint_size: "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)" unfolding word_size by uint_arith lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size] lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)" by uint_arith lemma no_olen_add': fixes x :: "'a::len0 word" shows "(x \ y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))" by (simp add: word_add_ac add_ac no_olen_add) lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric], standard] lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem, standard] lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1, standard] lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem, standard] lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def] lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def] lemmas word_sub_le = word_sub_le_iff [THEN iffD2, standard] lemma word_less_sub1: "(x :: 'a :: len word) ~= 0 ==> (1 < x) = (0 < x - 1)" by uint_arith lemma word_le_sub1: "(x :: 'a :: len word) ~= 0 ==> (1 <= x) = (0 <= x - 1)" by uint_arith lemma sub_wrap_lt: "((x :: 'a :: len0 word) < x - z) = (x < z)" by uint_arith lemma sub_wrap: "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)" by uint_arith lemma plus_minus_not_NULL_ab: "(x :: 'a :: len0 word) <= ab - c ==> c <= ab ==> c ~= 0 ==> x + c ~= 0" by uint_arith lemma plus_minus_no_overflow_ab: "(x :: 'a :: len0 word) <= ab - c ==> c <= ab ==> x <= x + c" by uint_arith lemma le_minus': "(a :: 'a :: len0 word) + c <= b ==> a <= a + c ==> c <= b - a" by uint_arith lemma le_plus': "(a :: 'a :: len0 word) <= b ==> c <= b - a ==> a + c <= b" by uint_arith lemmas le_plus = le_plus' [rotated] lemmas le_minus = leD [THEN thin_rl, THEN le_minus', standard] lemma word_plus_mono_right: "(y :: 'a :: len0 word) <= z ==> x <= x + z ==> x + y <= x + z" by uint_arith lemma word_less_minus_cancel: "y - x < z - x ==> x <= z ==> (y :: 'a :: len0 word) < z" by uint_arith lemma word_less_minus_mono_left: "(y :: 'a :: len0 word) < z ==> x <= y ==> y - x < z - x" by uint_arith lemma word_less_minus_mono: "a < c ==> d < b ==> a - b < a ==> c - d < c ==> a - b < c - (d::'a::len word)" by uint_arith lemma word_le_minus_cancel: "y - x <= z - x ==> x <= z ==> (y :: 'a :: len0 word) <= z" by uint_arith lemma word_le_minus_mono_left: "(y :: 'a :: len0 word) <= z ==> x <= y ==> y - x <= z - x" by uint_arith lemma word_le_minus_mono: "a <= c ==> d <= b ==> a - b <= a ==> c - d <= c ==> a - b <= c - (d::'a::len word)" by uint_arith lemma plus_le_left_cancel_wrap: "(x :: 'a :: len0 word) + y' < x ==> x + y < x ==> (x + y' < x + y) = (y' < y)" by uint_arith lemma plus_le_left_cancel_nowrap: "(x :: 'a :: len0 word) <= x + y' ==> x <= x + y ==> (x + y' < x + y) = (y' < y)" by uint_arith lemma word_plus_mono_right2: "(a :: 'a :: len0 word) <= a + b ==> c <= b ==> a <= a + c" by uint_arith lemma word_less_add_right: "(x :: 'a :: len0 word) < y - z ==> z <= y ==> x + z < y" by uint_arith lemma word_less_sub_right: "(x :: 'a :: len0 word) < y + z ==> y <= x ==> x - y < z" by uint_arith lemma word_le_plus_either: "(x :: 'a :: len0 word) <= y | x <= z ==> y <= y + z ==> x <= y + z" by uint_arith lemma word_less_nowrapI: "(x :: 'a :: len0 word) < z - k ==> k <= z ==> 0 < k ==> x < x + k" by uint_arith lemma inc_le: "(i :: 'a :: len word) < m ==> i + 1 <= m" by uint_arith lemma inc_i: "(1 :: 'a :: len word) <= i ==> i < m ==> 1 <= (i + 1) & i + 1 <= m" by uint_arith lemma udvd_incr_lem: "up < uq ==> up = ua + n * uint K ==> uq = ua + n' * uint K ==> up + uint K <= uq" apply clarsimp apply (drule less_le_mult) apply safe done lemma udvd_incr': "p < q ==> uint p = ua + n * uint K ==> uint q = ua + n' * uint K ==> p + K <= q" apply (unfold word_less_alt word_le_def) apply (drule (2) udvd_incr_lem) apply (erule uint_add_le [THEN order_trans]) done lemma udvd_decr': "p < q ==> uint p = ua + n * uint K ==> uint q = ua + n' * uint K ==> p <= q - K" apply (unfold word_less_alt word_le_def) apply (drule (2) udvd_incr_lem) apply (drule le_diff_eq [THEN iffD2]) apply (erule order_trans) apply (rule uint_sub_ge) done lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, simplified] lemmas udvd_incr0 = udvd_incr' [where ua=0, simplified] lemmas udvd_decr0 = udvd_decr' [where ua=0, simplified] lemma udvd_minus_le': "xy < k ==> z udvd xy ==> z udvd k ==> xy <= k - z" apply (unfold udvd_def) apply clarify apply (erule (2) udvd_decr0) done ML {* Delsimprocs Numeral_Simprocs.cancel_factors *} lemma udvd_incr2_K: "p < a + s ==> a <= a + s ==> K udvd s ==> K udvd p - a ==> a <= p ==> 0 < K ==> p <= p + K & p + K <= a + s" apply (unfold udvd_def) apply clarify apply (simp add: uint_arith_simps split: split_if_asm) prefer 2 apply (insert uint_range' [of s])[1] apply arith apply (drule add_commute [THEN xtr1]) apply (simp add: diff_less_eq [symmetric]) apply (drule less_le_mult) apply arith apply simp done ML {* Addsimprocs Numeral_Simprocs.cancel_factors *} (* links with rbl operations *) lemma word_succ_rbl: "to_bl w = bl ==> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))" apply (unfold word_succ_def) apply clarify apply (simp add: to_bl_of_bin) apply (simp add: to_bl_def rbl_succ) done lemma word_pred_rbl: "to_bl w = bl ==> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))" apply (unfold word_pred_def) apply clarify apply (simp add: to_bl_of_bin) apply (simp add: to_bl_def rbl_pred) done lemma word_add_rbl: "to_bl v = vbl ==> to_bl w = wbl ==> to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))" apply (unfold word_add_def) apply clarify apply (simp add: to_bl_of_bin) apply (simp add: to_bl_def rbl_add) done lemma word_mult_rbl: "to_bl v = vbl ==> to_bl w = wbl ==> to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))" apply (unfold word_mult_def) apply clarify apply (simp add: to_bl_of_bin) apply (simp add: to_bl_def rbl_mult) done lemma rtb_rbl_ariths: "rev (to_bl w) = ys \ rev (to_bl (word_succ w)) = rbl_succ ys" "rev (to_bl w) = ys \ rev (to_bl (word_pred w)) = rbl_pred ys" "[| rev (to_bl v) = ys; rev (to_bl w) = xs |] ==> rev (to_bl (v * w)) = rbl_mult ys xs" "[| rev (to_bl v) = ys; rev (to_bl w) = xs |] ==> rev (to_bl (v + w)) = rbl_add ys xs" by (auto simp: rev_swap [symmetric] word_succ_rbl word_pred_rbl word_mult_rbl word_add_rbl) subsection "Arithmetic type class instantiations" instance word :: (len0) comm_monoid_add .. instance word :: (len0) comm_monoid_mult apply (intro_classes) apply (simp add: word_mult_commute) apply (simp add: word_mult_1) done instance word :: (len0) comm_semiring by (intro_classes) (simp add : word_left_distrib) instance word :: (len0) ab_group_add .. instance word :: (len0) comm_ring .. instance word :: (len) comm_semiring_1 by (intro_classes) (simp add: lenw1_zero_neq_one) instance word :: (len) comm_ring_1 .. instance word :: (len0) comm_semiring_0 .. instance word :: (len0) order .. (* note that iszero_def is only for class comm_semiring_1_cancel, which requires word length >= 1, ie 'a :: len word *) lemma zero_bintrunc: "iszero (number_of x :: 'a :: len word) = (bintrunc (len_of TYPE('a)) x = Int.Pls)" apply (unfold iszero_def word_0_wi word_no_wi) apply (rule word_ubin.norm_eq_iff [symmetric, THEN trans]) apply (simp add : Pls_def [symmetric]) done lemmas word_le_0_iff [simp] = word_zero_le [THEN leD, THEN linorder_antisym_conv1] lemma word_of_nat: "of_nat n = word_of_int (int n)" by (induct n) (auto simp add : word_of_int_hom_syms) lemma word_of_int: "of_int = word_of_int" apply (rule ext) apply (unfold of_int_def) apply (rule contentsI) apply safe apply (simp_all add: word_of_nat word_of_int_homs) defer apply (rule Rep_Integ_ne [THEN nonemptyE]) apply (rule bexI) prefer 2 apply assumption apply (auto simp add: RI_eq_diff) done lemma word_of_int_nat: "0 <= x ==> word_of_int x = of_nat (nat x)" by (simp add: of_nat_nat word_of_int) lemma word_number_of_eq: "number_of w = (of_int w :: 'a :: len word)" unfolding word_number_of_def word_of_int by auto instance word :: (len) number_ring by (intro_classes) (simp add : word_number_of_eq) lemma iszero_word_no [simp] : "iszero (number_of bin :: 'a :: len word) = iszero (number_of (bintrunc (len_of TYPE('a)) bin) :: int)" apply (simp add: zero_bintrunc number_of_is_id) apply (unfold iszero_def Pls_def) apply (rule refl) done subsection "Word and nat" lemma td_ext_unat': "n = len_of TYPE ('a :: len) ==> td_ext (unat :: 'a word => nat) of_nat (unats n) (%i. i mod 2 ^ n)" apply (unfold td_ext_def' unat_def word_of_nat unats_uints) apply (auto intro!: imageI simp add : word_of_int_hom_syms) apply (erule word_uint.Abs_inverse [THEN arg_cong]) apply (simp add: int_word_uint nat_mod_distrib nat_power_eq) done lemmas td_ext_unat = refl [THEN td_ext_unat'] lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm, standard] interpretation word_unat: td_ext "unat::'a::len word => nat" of_nat "unats (len_of TYPE('a::len))" "%i. i mod 2 ^ len_of TYPE('a::len)" by (rule td_ext_unat) lemmas td_unat = word_unat.td_thm lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq] lemma unat_le: "y <= unat (z :: 'a :: len word) ==> y : unats (len_of TYPE ('a))" apply (unfold unats_def) apply clarsimp apply (rule xtrans, rule unat_lt2p, assumption) done lemma word_nchotomy: "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)" apply (rule allI) apply (rule word_unat.Abs_cases) apply (unfold unats_def) apply auto done lemma of_nat_eq: fixes w :: "'a::len word" shows "(of_nat n = w) = (\q. n = unat w + q * 2 ^ len_of TYPE('a))" apply (rule trans) apply (rule word_unat.inverse_norm) apply (rule iffI) apply (rule mod_eqD) apply simp apply clarsimp done lemma of_nat_eq_size: "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)" unfolding word_size by (rule of_nat_eq) lemma of_nat_0: "(of_nat m = (0::'a::len word)) = (\q. m = q * 2 ^ len_of TYPE('a))" by (simp add: of_nat_eq) lemmas of_nat_2p = mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]] lemma of_nat_gt_0: "of_nat k ~= 0 ==> 0 < k" by (cases k) auto lemma of_nat_neq_0: "0 < k ==> k < 2 ^ len_of TYPE ('a :: len) ==> of_nat k ~= (0 :: 'a word)" by (clarsimp simp add : of_nat_0) lemma Abs_fnat_hom_add: "of_nat a + of_nat b = of_nat (a + b)" by simp lemma Abs_fnat_hom_mult: "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)" by (simp add: word_of_nat word_of_int_mult_hom zmult_int) lemma Abs_fnat_hom_Suc: "word_succ (of_nat a) = of_nat (Suc a)" by (simp add: word_of_nat word_of_int_succ_hom add_ac) lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0" by (simp add: word_of_nat word_0_wi) lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)" by (simp add: word_of_nat word_1_wi) lemmas Abs_fnat_homs = Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc Abs_fnat_hom_0 Abs_fnat_hom_1 lemma word_arith_nat_add: "a + b = of_nat (unat a + unat b)" by simp lemma word_arith_nat_mult: "a * b = of_nat (unat a * unat b)" by (simp add: Abs_fnat_hom_mult [symmetric]) lemma word_arith_nat_Suc: "word_succ a = of_nat (Suc (unat a))" by (subst Abs_fnat_hom_Suc [symmetric]) simp lemma word_arith_nat_div: "a div b = of_nat (unat a div unat b)" by (simp add: word_div_def word_of_nat zdiv_int uint_nat) lemma word_arith_nat_mod: "a mod b = of_nat (unat a mod unat b)" by (simp add: word_mod_def word_of_nat zmod_int uint_nat) lemmas word_arith_nat_defs = word_arith_nat_add word_arith_nat_mult word_arith_nat_Suc Abs_fnat_hom_0 Abs_fnat_hom_1 word_arith_nat_div word_arith_nat_mod lemmas unat_cong = arg_cong [where f = "unat"] lemmas unat_word_ariths = word_arith_nat_defs [THEN trans [OF unat_cong unat_of_nat], standard] lemmas word_sub_less_iff = word_sub_le_iff [simplified linorder_not_less [symmetric], simplified] lemma unat_add_lem: "(unat x + unat y < 2 ^ len_of TYPE('a)) = (unat (x + y :: 'a :: len word) = unat x + unat y)" unfolding unat_word_ariths by (auto intro!: trans [OF _ nat_mod_lem]) lemma unat_mult_lem: "(unat x * unat y < 2 ^ len_of TYPE('a)) = (unat (x * y :: 'a :: len word) = unat x * unat y)" unfolding unat_word_ariths by (auto intro!: trans [OF _ nat_mod_lem]) lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified, standard] lemma le_no_overflow: "x <= b ==> a <= a + b ==> x <= a + (b :: 'a :: len0 word)" apply (erule order_trans) apply (erule olen_add_eqv [THEN iffD1]) done lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def, standard] lemma unat_sub_if_size: "unat (x - y) = (if unat y <= unat x then unat x - unat y else unat x + 2 ^ size x - unat y)" apply (unfold word_size) apply (simp add: un_ui_le) apply (auto simp add: unat_def uint_sub_if') apply (rule nat_diff_distrib) prefer 3 apply (simp add: algebra_simps) apply (rule nat_diff_distrib [THEN trans]) prefer 3 apply (subst nat_add_distrib) prefer 3 apply (simp add: nat_power_eq) apply auto apply uint_arith done lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size] lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y" apply (simp add : unat_word_ariths) apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) apply (rule div_le_dividend) done lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y" apply (clarsimp simp add : unat_word_ariths) apply (cases "unat y") prefer 2 apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) apply (rule mod_le_divisor) apply auto done lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y" unfolding uint_nat by (simp add : unat_div zdiv_int) lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y" unfolding uint_nat by (simp add : unat_mod zmod_int) subsection {* Definition of unat\_arith tactic *} lemma unat_split: fixes x::"'a::len word" shows "P (unat x) = (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)" by (auto simp: unat_of_nat) lemma unat_split_asm: fixes x::"'a::len word" shows "P (unat x) = (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))" by (auto simp: unat_of_nat) lemmas of_nat_inverse = word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified] lemmas unat_splits = unat_split unat_split_asm lemmas unat_arith_simps = word_le_nat_alt word_less_nat_alt word_unat.Rep_inject [symmetric] unat_sub_if' unat_plus_if' unat_div unat_mod (* unat_arith_tac: tactic to reduce word arithmetic to nat, try to solve via arith *) ML {* fun unat_arith_ss_of ss = ss addsimps @{thms unat_arith_simps} delsimps @{thms word_unat.Rep_inject} addsplits @{thms split_if_asm} addcongs @{thms power_False_cong} fun unat_arith_tacs ctxt = let fun arith_tac' n t = Arith_Data.verbose_arith_tac ctxt n t handle COOPER => Seq.empty; val cs = claset_of ctxt; val ss = simpset_of ctxt; in [ clarify_tac cs 1, full_simp_tac (unat_arith_ss_of ss) 1, ALLGOALS (full_simp_tac (HOL_ss addsplits @{thms unat_splits} addcongs @{thms power_False_cong})), rewrite_goals_tac @{thms word_size}, ALLGOALS (fn n => REPEAT (resolve_tac [allI, impI] n) THEN REPEAT (etac conjE n) THEN REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)), TRYALL arith_tac' ] end fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt)) *} method_setup unat_arith = {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *} "solving word arithmetic via natural numbers and arith" lemma no_plus_overflow_unat_size: "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" unfolding word_size by unat_arith lemma unat_sub: "b <= a ==> unat (a - b) = unat a - unat (b :: 'a :: len word)" by unat_arith lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size] lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem, standard] lemma word_div_mult: "(0 :: 'a :: len word) < y ==> unat x * unat y < 2 ^ len_of TYPE('a) ==> x * y div y = x" apply unat_arith apply clarsimp apply (subst unat_mult_lem [THEN iffD1]) apply auto done lemma div_lt': "(i :: 'a :: len word) <= k div x ==> unat i * unat x < 2 ^ len_of TYPE('a)" apply unat_arith apply clarsimp apply (drule mult_le_mono1) apply (erule order_le_less_trans) apply (rule xtr7 [OF unat_lt2p div_mult_le]) done lemmas div_lt'' = order_less_imp_le [THEN div_lt'] lemma div_lt_mult: "(i :: 'a :: len word) < k div x ==> 0 < x ==> i * x < k" apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]]) apply (simp add: unat_arith_simps) apply (drule (1) mult_less_mono1) apply (erule order_less_le_trans) apply (rule div_mult_le) done lemma div_le_mult: "(i :: 'a :: len word) <= k div x ==> 0 < x ==> i * x <= k" apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]]) apply (simp add: unat_arith_simps) apply (drule mult_le_mono1) apply (erule order_trans) apply (rule div_mult_le) done lemma div_lt_uint': "(i :: 'a :: len word) <= k div x ==> uint i * uint x < 2 ^ len_of TYPE('a)" apply (unfold uint_nat) apply (drule div_lt') apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] nat_power_eq) done lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint'] lemma word_le_exists': "(x :: 'a :: len0 word) <= y ==> (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))" apply (rule exI) apply (rule conjI) apply (rule zadd_diff_inverse) apply uint_arith done lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab] lemmas plus_minus_no_overflow = order_less_imp_le [THEN plus_minus_no_overflow_ab] lemmas mcs = word_less_minus_cancel word_less_minus_mono_left word_le_minus_cancel word_le_minus_mono_left lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel, standard] lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel, standard] lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel, standard] lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse] lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1] lemma thd1: "a div b * b \ (a::nat)" using gt_or_eq_0 [of b] apply (rule disjE) apply (erule xtr4 [OF thd mult_commute]) apply clarsimp done lemmas uno_simps [THEN le_unat_uoi, standard] = mod_le_divisor div_le_dividend thd1 lemma word_mod_div_equality: "(n div b) * b + (n mod b) = (n :: 'a :: len word)" apply (unfold word_less_nat_alt word_arith_nat_defs) apply (cut_tac y="unat b" in gt_or_eq_0) apply (erule disjE) apply (simp add: mod_div_equality uno_simps) apply simp done lemma word_div_mult_le: "a div b * b <= (a::'a::len word)" apply (unfold word_le_nat_alt word_arith_nat_defs) apply (cut_tac y="unat b" in gt_or_eq_0) apply (erule disjE) apply (simp add: div_mult_le uno_simps) apply simp done lemma word_mod_less_divisor: "0 < n ==> m mod n < (n :: 'a :: len word)" apply (simp only: word_less_nat_alt word_arith_nat_defs) apply (clarsimp simp add : uno_simps) done lemma word_of_int_power_hom: "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)" by (induct n) (simp_all add : word_of_int_hom_syms power_Suc) lemma word_arith_power_alt: "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)" by (simp add : word_of_int_power_hom [symmetric]) lemma of_bl_length_less: "length x = k ==> k < len_of TYPE('a) ==> (of_bl x :: 'a :: len word) < 2 ^ k" apply (unfold of_bl_no [unfolded word_number_of_def] word_less_alt word_number_of_alt) apply safe apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm del: word_of_int_bin) apply (simp add: mod_pos_pos_trivial) apply (subst mod_pos_pos_trivial) apply (rule bl_to_bin_ge0) apply (rule order_less_trans) apply (rule bl_to_bin_lt2p) apply simp apply (rule bl_to_bin_lt2p) done subsection "Cardinality, finiteness of set of words" lemmas card_lessThan' = card_lessThan [unfolded lessThan_def] lemmas card_eq = word_unat.Abs_inj_on [THEN card_image, unfolded word_unat.image, unfolded unats_def, standard] lemmas card_word = trans [OF card_eq card_lessThan', standard] lemma finite_word_UNIV: "finite (UNIV :: 'a :: len word set)" apply (rule contrapos_np) prefer 2 apply (erule card_infinite) apply (simp add: card_word) done lemma card_word_size: "card (UNIV :: 'a :: len word set) = (2 ^ size (x :: 'a word))" unfolding word_size by (rule card_word) end