diff -r bf86b2002c96 -r d6cf9a5b9be9 src/HOL/Word/Bit_Representation.thy --- a/src/HOL/Word/Bit_Representation.thy Mon Dec 23 09:21:38 2013 +0100 +++ b/src/HOL/Word/Bit_Representation.thy Mon Dec 23 14:24:20 2013 +0100 @@ -5,38 +5,34 @@ header {* Integers as implict bit strings *} theory Bit_Representation -imports "~~/src/HOL/Library/Bit" Misc_Numeric +imports Misc_Numeric begin subsection {* Constructors and destructors for binary integers *} -definition bitval :: "bit \ 'a\zero_neq_one" where - "bitval = bit_case 0 1" - -lemma bitval_simps [simp]: - "bitval 0 = 0" - "bitval 1 = 1" - by (simp_all add: bitval_def) - -definition Bit :: "int \ bit \ int" (infixl "BIT" 90) where - "k BIT b = bitval b + k + k" +definition Bit :: "int \ bool \ int" (infixl "BIT" 90) where + "k BIT b = (if b then 1 else 0) + k + k" lemma Bit_B0: - "k BIT (0::bit) = k + k" + "k BIT False = k + k" by (unfold Bit_def) simp lemma Bit_B1: - "k BIT (1::bit) = k + k + 1" + "k BIT True = k + k + 1" by (unfold Bit_def) simp -lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k" +lemma Bit_B0_2t: "k BIT False = 2 * k" by (rule trans, rule Bit_B0) simp -lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1" +lemma Bit_B1_2t: "k BIT True = 2 * k + 1" by (rule trans, rule Bit_B1) simp -definition bin_last :: "int \ bit" where - "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))" +definition bin_last :: "int \ bool" where + "bin_last w \ w mod 2 = 1" + +lemma bin_last_odd: + "bin_last = odd" + by (rule ext) (simp add: bin_last_def even_def) definition bin_rest :: "int \ int" where "bin_rest w = w div 2" @@ -56,48 +52,55 @@ by (cases b, simp_all add: z1pmod2) lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \ u = v \ b = c" - by (metis bin_rest_BIT bin_last_BIT) + apply (auto simp add: Bit_def) + apply arith + apply arith + done lemma BIT_bin_simps [simp]: - "numeral k BIT 0 = numeral (Num.Bit0 k)" - "numeral k BIT 1 = numeral (Num.Bit1 k)" - "(- numeral k) BIT 0 = - numeral (Num.Bit0 k)" - "(- numeral k) BIT 1 = - numeral (Num.BitM k)" + "numeral k BIT False = numeral (Num.Bit0 k)" + "numeral k BIT True = numeral (Num.Bit1 k)" + "(- numeral k) BIT False = - numeral (Num.Bit0 k)" + "(- numeral k) BIT True = - numeral (Num.BitM k)" unfolding numeral.simps numeral_BitM - unfolding Bit_def bitval_simps + unfolding Bit_def by (simp_all del: arith_simps add_numeral_special diff_numeral_special) lemma BIT_special_simps [simp]: - shows "0 BIT 0 = 0" and "0 BIT 1 = 1" - and "1 BIT 0 = 2" and "1 BIT 1 = 3" - and "(- 1) BIT 0 = - 2" and "(- 1) BIT 1 = - 1" + shows "0 BIT False = 0" and "0 BIT True = 1" + and "1 BIT False = 2" and "1 BIT True = 3" + and "(- 1) BIT False = - 2" and "(- 1) BIT True = - 1" unfolding Bit_def by simp_all -lemma Bit_eq_0_iff: "w BIT b = 0 \ w = 0 \ b = 0" - by (subst BIT_eq_iff [symmetric], simp) +lemma Bit_eq_0_iff: "w BIT b = 0 \ w = 0 \ \ b" + apply (auto simp add: Bit_def) + apply arith + done -lemma Bit_eq_m1_iff: "w BIT b = - 1 \ w = - 1 \ b = 1" - by (cases b) (auto simp add: Bit_def, arith) +lemma Bit_eq_m1_iff: "w BIT b = -1 \ w = -1 \ b" + apply (auto simp add: Bit_def) + apply arith + done lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w" by (induct w, simp_all) lemma expand_BIT: - "numeral (Num.Bit0 w) = numeral w BIT 0" - "numeral (Num.Bit1 w) = numeral w BIT 1" - "- numeral (Num.Bit0 w) = - numeral w BIT 0" - "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT 1" + "numeral (Num.Bit0 w) = numeral w BIT False" + "numeral (Num.Bit1 w) = numeral w BIT True" + "- numeral (Num.Bit0 w) = (- numeral w) BIT False" + "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True" unfolding add_One by (simp_all add: BitM_inc) lemma bin_last_numeral_simps [simp]: - "bin_last 0 = 0" - "bin_last 1 = 1" - "bin_last -1 = 1" - "bin_last Numeral1 = 1" - "bin_last (numeral (Num.Bit0 w)) = 0" - "bin_last (numeral (Num.Bit1 w)) = 1" - "bin_last (- numeral (Num.Bit0 w)) = 0" - "bin_last (- numeral (Num.Bit1 w)) = 1" + "\ bin_last 0" + "bin_last 1" + "bin_last -1" + "bin_last Numeral1" + "\ bin_last (numeral (Num.Bit0 w))" + "bin_last (numeral (Num.Bit1 w))" + "\ bin_last (- numeral (Num.Bit0 w))" + "bin_last (- numeral (Num.Bit1 w))" unfolding expand_BIT bin_last_BIT by (simp_all add: bin_last_def zmod_zminus1_eq_if) lemma bin_rest_numeral_simps [simp]: @@ -112,51 +115,42 @@ unfolding expand_BIT bin_rest_BIT by (simp_all add: bin_rest_def zdiv_zminus1_eq_if) lemma less_Bits: - "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))" - unfolding Bit_def by (auto simp add: bitval_def split: bit.split) + "v BIT b < w BIT c \ v < w \ v \ w \ \ b \ c" + unfolding Bit_def by auto lemma le_Bits: - "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))" - unfolding Bit_def by (auto simp add: bitval_def split: bit.split) + "v BIT b \ w BIT c \ v < w \ v \ w \ (\ b \ c)" + unfolding Bit_def by auto lemma pred_BIT_simps [simp]: - "x BIT 0 - 1 = (x - 1) BIT 1" - "x BIT 1 - 1 = x BIT 0" + "x BIT False - 1 = (x - 1) BIT True" + "x BIT True - 1 = x BIT False" by (simp_all add: Bit_B0_2t Bit_B1_2t) lemma succ_BIT_simps [simp]: - "x BIT 0 + 1 = x BIT 1" - "x BIT 1 + 1 = (x + 1) BIT 0" + "x BIT False + 1 = x BIT True" + "x BIT True + 1 = (x + 1) BIT False" by (simp_all add: Bit_B0_2t Bit_B1_2t) lemma add_BIT_simps [simp]: - "x BIT 0 + y BIT 0 = (x + y) BIT 0" - "x BIT 0 + y BIT 1 = (x + y) BIT 1" - "x BIT 1 + y BIT 0 = (x + y) BIT 1" - "x BIT 1 + y BIT 1 = (x + y + 1) BIT 0" + "x BIT False + y BIT False = (x + y) BIT False" + "x BIT False + y BIT True = (x + y) BIT True" + "x BIT True + y BIT False = (x + y) BIT True" + "x BIT True + y BIT True = (x + y + 1) BIT False" by (simp_all add: Bit_B0_2t Bit_B1_2t) lemma mult_BIT_simps [simp]: - "x BIT 0 * y = (x * y) BIT 0" - "x * y BIT 0 = (x * y) BIT 0" - "x BIT 1 * y = (x * y) BIT 0 + y" + "x BIT False * y = (x * y) BIT False" + "x * y BIT False = (x * y) BIT False" + "x BIT True * y = (x * y) BIT False + y" by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps) lemma B_mod_2': - "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0" + "X = 2 ==> (w BIT True) mod X = 1 & (w BIT False) mod X = 0" apply (simp (no_asm) only: Bit_B0 Bit_B1) apply (simp add: z1pmod2) done -lemma neB1E [elim!]: - assumes ne: "y \ (1::bit)" - assumes y: "y = (0::bit) \ P" - shows "P" - apply (rule y) - apply (cases y rule: bit.exhaust, simp) - apply (simp add: ne) - done - lemma bin_ex_rl: "EX w b. w BIT b = bin" by (metis bin_rl_simp) @@ -170,8 +164,10 @@ done primrec bin_nth where - Z: "bin_nth w 0 = (bin_last w = (1::bit))" - | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n" + Z: "bin_nth w 0 \ bin_last w" + | Suc: "bin_nth w (Suc n) \ bin_nth (bin_rest w) n" + +find_theorems "bin_rest _ = _" lemma bin_abs_lem: "bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 --> @@ -248,7 +244,7 @@ lemma bin_nth_minus1 [simp]: "bin_nth -1 n" by (induct n) auto -lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))" +lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 \ b" by auto lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n" @@ -285,8 +281,8 @@ "bin_sign (numeral k) = 0" "bin_sign (- numeral k) = -1" "bin_sign (w BIT b) = bin_sign w" - unfolding bin_sign_def Bit_def bitval_def - by (simp_all split: bit.split) + unfolding bin_sign_def Bit_def + by simp_all lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = bin_sign w" @@ -297,7 +293,7 @@ | Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)" primrec sbintrunc :: "nat => int => int" where - Z : "sbintrunc 0 bin = (case bin_last bin of (1::bit) \ -1 | (0::bit) \ 0)" + Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)" | Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)" lemma sign_bintr: "bin_sign (bintrunc n w) = 0" @@ -313,7 +309,8 @@ apply simp apply (subst mod_add_left_eq) apply (simp add: bin_last_def) - apply (simp add: bin_last_def bin_rest_def Bit_def) + apply arith + apply (simp add: bin_last_def bin_rest_def Bit_def mod_2_neq_1_eq_eq_0) apply (clarsimp simp: mod_mult_mult1 [symmetric] zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]]) apply (rule trans [symmetric, OF _ emep1]) @@ -334,13 +331,13 @@ lemma bintrunc_Suc_numeral: "bintrunc (Suc n) 1 = 1" - "bintrunc (Suc n) -1 = bintrunc n -1 BIT 1" - "bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT 0" - "bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT 1" + "bintrunc (Suc n) -1 = bintrunc n -1 BIT True" + "bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT False" + "bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT True" "bintrunc (Suc n) (- numeral (Num.Bit0 w)) = - bintrunc n (- numeral w) BIT 0" + bintrunc n (- numeral w) BIT False" "bintrunc (Suc n) (- numeral (Num.Bit1 w)) = - bintrunc n (- numeral (w + Num.One)) BIT 1" + bintrunc n (- numeral (w + Num.One)) BIT True" by simp_all lemma sbintrunc_0_numeral [simp]: @@ -354,21 +351,15 @@ lemma sbintrunc_Suc_numeral: "sbintrunc (Suc n) 1 = 1" "sbintrunc (Suc n) (numeral (Num.Bit0 w)) = - sbintrunc n (numeral w) BIT 0" + sbintrunc n (numeral w) BIT False" "sbintrunc (Suc n) (numeral (Num.Bit1 w)) = - sbintrunc n (numeral w) BIT 1" + sbintrunc n (numeral w) BIT True" "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) = - sbintrunc n (- numeral w) BIT 0" + sbintrunc n (- numeral w) BIT False" "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) = - sbintrunc n (- numeral (w + Num.One)) BIT 1" + sbintrunc n (- numeral (w + Num.One)) BIT True" by simp_all -lemma bit_bool: - "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))" - by (cases b') auto - -lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric] - lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n" apply (induct n arbitrary: bin) apply (case_tac bin rule: bin_exhaust, case_tac b, auto) @@ -384,23 +375,25 @@ "bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)" apply (induct n arbitrary: w m) - apply (case_tac m, simp_all split: bit.splits)[1] - apply (case_tac m, simp_all split: bit.splits)[1] + apply (case_tac m) + apply simp_all + apply (case_tac m) + apply simp_all done lemma bin_nth_Bit: - "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))" + "bin_nth (w BIT b) n = (n = 0 & b | (EX m. n = Suc m & bin_nth w m))" by (cases n) auto lemma bin_nth_Bit0: "bin_nth (numeral (Num.Bit0 w)) n \ (\m. n = Suc m \ bin_nth (numeral w) m)" - using bin_nth_Bit [where w="numeral w" and b="(0::bit)"] by simp + using bin_nth_Bit [where w="numeral w" and b="False"] by simp lemma bin_nth_Bit1: "bin_nth (numeral (Num.Bit1 w)) n \ n = 0 \ (\m. n = Suc m \ bin_nth (numeral w) m)" - using bin_nth_Bit [where w="numeral w" and b="(1::bit)"] by simp + using bin_nth_Bit [where w="numeral w" and b="True"] by simp lemma bintrunc_bintrunc_l: "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)" @@ -452,19 +445,19 @@ lemmas sbintrunc_Pls = sbintrunc.Z [where bin="0", - simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] + simplified bin_last_numeral_simps bin_rest_numeral_simps] lemmas sbintrunc_Min = sbintrunc.Z [where bin="-1", - simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] + simplified bin_last_numeral_simps bin_rest_numeral_simps] lemmas sbintrunc_0_BIT_B0 [simp] = - sbintrunc.Z [where bin="w BIT (0::bit)", - simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] for w + sbintrunc.Z [where bin="w BIT False", + simplified bin_last_numeral_simps bin_rest_numeral_simps] for w lemmas sbintrunc_0_BIT_B1 [simp] = - sbintrunc.Z [where bin="w BIT (1::bit)", - simplified bin_last_BIT bin_rest_numeral_simps bit.simps] for w + sbintrunc.Z [where bin="w BIT True", + simplified bin_last_BIT bin_rest_numeral_simps] for w lemmas sbintrunc_0_simps = sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1 @@ -583,25 +576,25 @@ lemma bintrunc_numeral_simps [simp]: "bintrunc (numeral k) (numeral (Num.Bit0 w)) = - bintrunc (pred_numeral k) (numeral w) BIT 0" + bintrunc (pred_numeral k) (numeral w) BIT False" "bintrunc (numeral k) (numeral (Num.Bit1 w)) = - bintrunc (pred_numeral k) (numeral w) BIT 1" + bintrunc (pred_numeral k) (numeral w) BIT True" "bintrunc (numeral k) (- numeral (Num.Bit0 w)) = - bintrunc (pred_numeral k) (- numeral w) BIT 0" + bintrunc (pred_numeral k) (- numeral w) BIT False" "bintrunc (numeral k) (- numeral (Num.Bit1 w)) = - bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1" + bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True" "bintrunc (numeral k) 1 = 1" by (simp_all add: bintrunc_numeral) lemma sbintrunc_numeral_simps [simp]: "sbintrunc (numeral k) (numeral (Num.Bit0 w)) = - sbintrunc (pred_numeral k) (numeral w) BIT 0" + sbintrunc (pred_numeral k) (numeral w) BIT False" "sbintrunc (numeral k) (numeral (Num.Bit1 w)) = - sbintrunc (pred_numeral k) (numeral w) BIT 1" + sbintrunc (pred_numeral k) (numeral w) BIT True" "sbintrunc (numeral k) (- numeral (Num.Bit0 w)) = - sbintrunc (pred_numeral k) (- numeral w) BIT 0" + sbintrunc (pred_numeral k) (- numeral w) BIT False" "sbintrunc (numeral k) (- numeral (Num.Bit1 w)) = - sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1" + sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True" "sbintrunc (numeral k) 1 = 1" by (simp_all add: sbintrunc_numeral) @@ -728,7 +721,7 @@ "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)" apply (induct n arbitrary: bin, simp) apply (case_tac bin rule: bin_exhaust) - apply (auto simp: bintrunc_bintrunc_l split: bit.splits) + apply (auto simp: bintrunc_bintrunc_l split: bool.splits) done lemma bintrunc_rest':