(* ID: $Id: BinGeneral.thy,v 1.19 2008/05/07 08:59:42 berghofe Exp $ Author: Jeremy Dawson, NICTA contains basic definition to do with integers expressed using Pls, Min, BIT and important resulting theorems, in particular, bin_rec and related work *) header {* Basic Definitions for Binary Integers *} theory BinGeneral imports Num_Lemmas begin subsection {* Further properties of numerals *} datatype bit = B0 | B1 definition Bit :: "int => bit => int" (infixl "BIT" 90) where "k BIT b = (case b of B0 => 0 | B1 => 1) + k + k" lemma BIT_B0_eq_Bit0 [simp]: "w BIT B0 = Int.Bit0 w" unfolding Bit_def Bit0_def by simp lemma BIT_B1_eq_Bit1 [simp]: "w BIT B1 = Int.Bit1 w" unfolding Bit_def Bit1_def by simp lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1 hide (open) const B0 B1 lemma Min_ne_Pls [iff]: "Int.Min ~= Int.Pls" unfolding Min_def Pls_def by auto lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric] lemmas PlsMin_defs [intro!] = Pls_def Min_def Pls_def [symmetric] Min_def [symmetric] lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI] lemma number_of_False_cong: "False ==> number_of x = number_of y" by (rule FalseE) (** ways in which type Bin resembles a datatype **) lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c" apply (unfold Bit_def) apply (simp (no_asm_use) split: bit.split_asm) apply simp_all apply (drule_tac f=even in arg_cong, clarsimp)+ done lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard] lemma BIT_eq_iff [simp]: "(u BIT b = v BIT c) = (u = v ∧ b = c)" by (rule iffI) auto lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]] lemma less_Bits: "(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)" unfolding Bit_def by (auto split: bit.split) lemma le_Bits: "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))" unfolding Bit_def by (auto split: bit.split) lemma no_no [simp]: "number_of (number_of i) = i" unfolding number_of_eq by simp lemma Bit_B0: "k BIT bit.B0 = k + k" by (unfold Bit_def) simp lemma Bit_B1: "k BIT bit.B1 = k + k + 1" by (unfold Bit_def) simp lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k" by (rule trans, rule Bit_B0) simp lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1" by (rule trans, rule Bit_B1) simp lemma B_mod_2': "X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0" apply (simp (no_asm) only: Bit_B0 Bit_B1) apply (simp add: z1pmod2) done lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1" unfolding numeral_simps number_of_is_id by (simp add: z1pmod2) lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0" unfolding numeral_simps number_of_is_id by simp lemma neB1E [elim!]: assumes ne: "y ≠ bit.B1" assumes y: "y = bit.B0 ==> 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" apply (unfold Bit_def) apply (cases "even bin") apply (clarsimp simp: even_equiv_def) apply (auto simp: odd_equiv_def split: bit.split) done lemma bin_exhaust: assumes Q: "!!x b. bin = x BIT b ==> Q" shows "Q" apply (insert bin_ex_rl [of bin]) apply (erule exE)+ apply (rule Q) apply force done subsection {* Destructors for binary integers *} definition bin_rl :: "int => int × bit" where [code func del]: "bin_rl w = (THE (r, l). w = r BIT l)" lemma bin_rl_char: "(bin_rl w = (r, l)) = (r BIT l = w)" apply (unfold bin_rl_def) apply safe apply (cases w rule: bin_exhaust) apply auto done definition bin_rest_def [code func del]: "bin_rest w = fst (bin_rl w)" definition bin_last_def [code func del] : "bin_last w = snd (bin_rl w)" primrec bin_nth where Z: "bin_nth w 0 = (bin_last w = bit.B1)" | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n" lemma bin_rl: "bin_rl w = (bin_rest w, bin_last w)" unfolding bin_rest_def bin_last_def by auto lemma bin_rl_simps [simp]: "bin_rl Int.Pls = (Int.Pls, bit.B0)" "bin_rl Int.Min = (Int.Min, bit.B1)" "bin_rl (Int.Bit0 r) = (r, bit.B0)" "bin_rl (Int.Bit1 r) = (r, bit.B1)" "bin_rl (r BIT b) = (r, b)" unfolding bin_rl_char by simp_all declare bin_rl_simps(1-4) [code func] lemmas bin_rl_simp [simp] = iffD1 [OF bin_rl_char bin_rl] lemma bin_abs_lem: "bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls --> nat (abs w) < nat (abs bin)" apply (clarsimp simp add: bin_rl_char) apply (unfold Pls_def Min_def Bit_def) apply (cases b) apply (clarsimp, arith) apply (clarsimp, arith) done lemma bin_induct: assumes PPls: "P Int.Pls" and PMin: "P Int.Min" and PBit: "!!bin bit. P bin ==> P (bin BIT bit)" shows "P bin" apply (rule_tac P=P and a=bin and f1="nat o abs" in wf_measure [THEN wf_induct]) apply (simp add: measure_def inv_image_def) apply (case_tac x rule: bin_exhaust) apply (frule bin_abs_lem) apply (auto simp add : PPls PMin PBit) done lemma numeral_induct: assumes Pls: "P Int.Pls" assumes Min: "P Int.Min" assumes Bit0: "!!w. [|P w; w ≠ Int.Pls|] ==> P (Int.Bit0 w)" assumes Bit1: "!!w. [|P w; w ≠ Int.Min|] ==> P (Int.Bit1 w)" shows "P x" apply (induct x rule: bin_induct) apply (rule Pls) apply (rule Min) apply (case_tac bit) apply (case_tac "bin = Int.Pls") apply simp apply (simp add: Bit0) apply (case_tac "bin = Int.Min") apply simp apply (simp add: Bit1) done lemma bin_rest_simps [simp]: "bin_rest Int.Pls = Int.Pls" "bin_rest Int.Min = Int.Min" "bin_rest (Int.Bit0 w) = w" "bin_rest (Int.Bit1 w) = w" "bin_rest (w BIT b) = w" unfolding bin_rest_def by auto declare bin_rest_simps(1-4) [code func] lemma bin_last_simps [simp]: "bin_last Int.Pls = bit.B0" "bin_last Int.Min = bit.B1" "bin_last (Int.Bit0 w) = bit.B0" "bin_last (Int.Bit1 w) = bit.B1" "bin_last (w BIT b) = b" unfolding bin_last_def by auto declare bin_last_simps(1-4) [code func] lemma bin_r_l_extras [simp]: "bin_last 0 = bit.B0" "bin_last (- 1) = bit.B1" "bin_last -1 = bit.B1" "bin_last 1 = bit.B1" "bin_rest 1 = 0" "bin_rest 0 = 0" "bin_rest (- 1) = - 1" "bin_rest -1 = -1" apply (unfold number_of_Min) apply (unfold Pls_def [symmetric] Min_def [symmetric]) apply (unfold numeral_1_eq_1 [symmetric]) apply (auto simp: number_of_eq) done lemma bin_last_mod: "bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)" apply (case_tac w rule: bin_exhaust) apply (case_tac b) apply auto done lemma bin_rest_div: "bin_rest w = w div 2" apply (case_tac w rule: bin_exhaust) apply (rule trans) apply clarsimp apply (rule refl) apply (drule trans) apply (rule Bit_def) apply (simp add: z1pdiv2 split: bit.split) done lemma Bit_div2 [simp]: "(w BIT b) div 2 = w" unfolding bin_rest_div [symmetric] by auto lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w" using Bit_div2 [where b=bit.B0] by simp lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w" using Bit_div2 [where b=bit.B1] by simp lemma bin_nth_lem [rule_format]: "ALL y. bin_nth x = bin_nth y --> x = y" apply (induct x rule: bin_induct) apply safe apply (erule rev_mp) apply (induct_tac y rule: bin_induct) apply (safe del: subset_antisym) apply (drule_tac x=0 in fun_cong, force) apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force) apply (drule_tac x=0 in fun_cong, force) apply (erule rev_mp) apply (induct_tac y rule: bin_induct) apply (safe del: subset_antisym) apply (drule_tac x=0 in fun_cong, force) apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force) apply (drule_tac x=0 in fun_cong, force) apply (case_tac y rule: bin_exhaust) apply clarify apply (erule allE) apply (erule impE) prefer 2 apply (erule BIT_eqI) apply (drule_tac x=0 in fun_cong, force) apply (rule ext) apply (drule_tac x="Suc ?x" in fun_cong, force) done lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)" by (auto elim: bin_nth_lem) lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1], standard] lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n" by (induct n) auto lemma bin_nth_Min [simp]: "bin_nth Int.Min n" by (induct n) auto lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = bit.B1)" by auto lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n" by auto lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)" by (cases n) auto lemma bin_nth_minus_Bit0 [simp]: "0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)" using bin_nth_minus [where b=bit.B0] by simp lemma bin_nth_minus_Bit1 [simp]: "0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)" using bin_nth_minus [where b=bit.B1] by simp lemmas bin_nth_0 = bin_nth.simps(1) lemmas bin_nth_Suc = bin_nth.simps(2) lemmas bin_nth_simps = bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus bin_nth_minus_Bit0 bin_nth_minus_Bit1 subsection {* Recursion combinator for binary integers *} lemma brlem: "(bin = Int.Min) = (- bin + Int.pred 0 = 0)" unfolding Min_def pred_def by arith function bin_rec :: "'a => 'a => (int => bit => 'a => 'a) => int => 'a" where "bin_rec f1 f2 f3 bin = (if bin = Int.Pls then f1 else if bin = Int.Min then f2 else case bin_rl bin of (w, b) => f3 w b (bin_rec f1 f2 f3 w))" by pat_completeness auto termination apply (relation "measure (nat o abs o snd o snd o snd)") apply simp apply (simp add: Pls_def brlem) apply (clarsimp simp: bin_rl_char pred_def) apply (frule thin_rl [THEN refl [THEN bin_abs_lem [rule_format]]]) apply (unfold Pls_def Min_def number_of_eq) prefer 2 apply (erule asm_rl) apply auto done declare bin_rec.simps [simp del] lemma bin_rec_PM: "f = bin_rec f1 f2 f3 ==> f Int.Pls = f1 & f Int.Min = f2" by (auto simp add: bin_rec.simps) lemma bin_rec_Pls: "bin_rec f1 f2 f3 Int.Pls = f1" by (simp add: bin_rec.simps) lemma bin_rec_Min: "bin_rec f1 f2 f3 Int.Min = f2" by (simp add: bin_rec.simps) lemma bin_rec_Bit0: "f3 Int.Pls bit.B0 f1 = f1 ==> bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w bit.B0 (bin_rec f1 f2 f3 w)" apply (simp add: bin_rec_Pls bin_rec.simps [of _ _ _ "Int.Bit0 w"]) unfolding Pls_def Min_def Bit0_def apply auto apply presburger apply (simp add: bin_rec.simps) done lemma bin_rec_Bit1: "f3 Int.Min bit.B1 f2 = f2 ==> bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w bit.B1 (bin_rec f1 f2 f3 w)" apply (simp add: bin_rec.simps [of _ _ _ "Int.Bit1 w"]) unfolding Pls_def Min_def Bit1_def apply auto apply (cases w) apply auto apply (simp add: bin_rec.simps) unfolding Min_def Pls_def number_of_is_id apply auto unfolding Bit0_def apply presburger done lemma bin_rec_Bit: "f = bin_rec f1 f2 f3 ==> f3 Int.Pls bit.B0 f1 = f1 ==> f3 Int.Min bit.B1 f2 = f2 ==> f (w BIT b) = f3 w b (f w)" by (cases b, simp add: bin_rec_Bit0, simp add: bin_rec_Bit1) lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min bin_rec_Bit0 bin_rec_Bit1 subsection {* Truncating binary integers *} definition bin_sign_def [code func del] : "bin_sign = bin_rec Int.Pls Int.Min (%w b s. s)" lemma bin_sign_simps [simp]: "bin_sign Int.Pls = Int.Pls" "bin_sign Int.Min = Int.Min" "bin_sign (Int.Bit0 w) = bin_sign w" "bin_sign (Int.Bit1 w) = bin_sign w" "bin_sign (w BIT b) = bin_sign w" unfolding bin_sign_def by (auto simp: bin_rec_simps) declare bin_sign_simps(1-4) [code func] lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = (bin_sign w)" by (cases w rule: bin_exhaust) auto consts bintrunc :: "nat => int => int" primrec Z : "bintrunc 0 bin = Int.Pls" Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)" consts sbintrunc :: "nat => int => int" primrec Z : "sbintrunc 0 bin = (case bin_last bin of bit.B1 => Int.Min | bit.B0 => Int.Pls)" Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)" lemma sign_bintr: "!!w. bin_sign (bintrunc n w) = Int.Pls" by (induct n) auto lemma bintrunc_mod2p: "!!w. bintrunc n w = (w mod 2 ^ n :: int)" apply (induct n, clarsimp) apply (simp add: bin_last_mod bin_rest_div Bit_def zmod_zmult2_eq cong: number_of_False_cong) done lemma sbintrunc_mod2p: "!!w. sbintrunc n w = ((w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n :: int)" apply (induct n) apply clarsimp apply (subst zmod_zadd_left_eq) apply (simp add: bin_last_mod) apply (simp add: number_of_eq) apply clarsimp apply (simp add: bin_last_mod bin_rest_div Bit_def cong: number_of_False_cong) apply (clarsimp simp: zmod_zmult_zmult1 [symmetric] zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]]) apply (rule trans [symmetric, OF _ emep1]) apply auto apply (auto simp: even_def) done subsection "Simplifications for (s)bintrunc" lemma bit_bool: "(b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))" by (cases b') auto lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric] lemma bin_sign_lem: "!!bin. (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n" apply (induct n) apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+ done lemma nth_bintr: "!!w m. bin_nth (bintrunc m w) n = (n < m & bin_nth w n)" apply (induct n) apply (case_tac m, auto)[1] apply (case_tac m, auto)[1] done lemma nth_sbintr: "!!w m. bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)" apply (induct n) apply (case_tac m, simp_all split: bit.splits)[1] apply (case_tac m, simp_all split: bit.splits)[1] done lemma bin_nth_Bit: "bin_nth (w BIT b) n = (n = 0 & b = bit.B1 | (EX m. n = Suc m & bin_nth w m))" by (cases n) auto lemma bin_nth_Bit0: "bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)" using bin_nth_Bit [where b=bit.B0] by simp lemma bin_nth_Bit1: "bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))" using bin_nth_Bit [where b=bit.B1] by simp lemma bintrunc_bintrunc_l: "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)" by (rule bin_eqI) (auto simp add : nth_bintr) lemma sbintrunc_sbintrunc_l: "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)" by (rule bin_eqI) (auto simp: nth_sbintr min_def) lemma bintrunc_bintrunc_ge: "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)" by (rule bin_eqI) (auto simp: nth_bintr) lemma bintrunc_bintrunc_min [simp]: "bintrunc m (bintrunc n w) = bintrunc (min m n) w" apply (unfold min_def) apply (rule bin_eqI) apply (auto simp: nth_bintr) done lemma sbintrunc_sbintrunc_min [simp]: "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w" apply (unfold min_def) apply (rule bin_eqI) apply (auto simp: nth_sbintr) done lemmas bintrunc_Pls = bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard] lemmas bintrunc_Min [simp] = bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard] lemmas bintrunc_BIT [simp] = bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard] lemma bintrunc_Bit0 [simp]: "bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)" using bintrunc_BIT [where b=bit.B0] by simp lemma bintrunc_Bit1 [simp]: "bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)" using bintrunc_BIT [where b=bit.B1] by simp lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT bintrunc_Bit0 bintrunc_Bit1 lemmas sbintrunc_Suc_Pls = sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard] lemmas sbintrunc_Suc_Min = sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard] lemmas sbintrunc_Suc_BIT [simp] = sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard] lemma sbintrunc_Suc_Bit0 [simp]: "sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)" using sbintrunc_Suc_BIT [where b=bit.B0] by simp lemma sbintrunc_Suc_Bit1 [simp]: "sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)" using sbintrunc_Suc_BIT [where b=bit.B1] by simp lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1 lemmas sbintrunc_Pls = sbintrunc.Z [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps bit.simps, standard] lemmas sbintrunc_Min = sbintrunc.Z [where bin="Int.Min", simplified bin_last_simps bin_rest_simps bit.simps, standard] lemmas sbintrunc_0_BIT_B0 [simp] = sbintrunc.Z [where bin="w BIT bit.B0", simplified bin_last_simps bin_rest_simps bit.simps, standard] lemmas sbintrunc_0_BIT_B1 [simp] = sbintrunc.Z [where bin="w BIT bit.B1", simplified bin_last_simps bin_rest_simps bit.simps, standard] lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls" using sbintrunc_0_BIT_B0 by simp lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min" using sbintrunc_0_BIT_B1 by simp lemmas sbintrunc_0_simps = sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1 sbintrunc_0_Bit0 sbintrunc_0_Bit1 lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs lemma bintrunc_minus: "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w" by auto lemma sbintrunc_minus: "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w" by auto lemmas bintrunc_minus_simps = bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans], standard] lemmas sbintrunc_minus_simps = sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans], standard] lemma bintrunc_n_Pls [simp]: "bintrunc n Int.Pls = Int.Pls" by (induct n) auto lemma sbintrunc_n_PM [simp]: "sbintrunc n Int.Pls = Int.Pls" "sbintrunc n Int.Min = Int.Min" by (induct n) auto lemmas thobini1 = arg_cong [where f = "%w. w BIT b", standard] lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1] lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1] lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard] lemmas bintrunc_Pls_minus_I = bmsts(1) lemmas bintrunc_Min_minus_I = bmsts(2) lemmas bintrunc_BIT_minus_I = bmsts(3) lemma bintrunc_0_Min: "bintrunc 0 Int.Min = Int.Pls" by auto lemma bintrunc_0_BIT: "bintrunc 0 (w BIT b) = Int.Pls" by auto lemma bintrunc_Suc_lem: "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y" by auto lemmas bintrunc_Suc_Ialts = bintrunc_Min_I [THEN bintrunc_Suc_lem, standard] bintrunc_BIT_I [THEN bintrunc_Suc_lem, standard] lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1] lemmas sbintrunc_Suc_Is = sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans], standard] lemmas sbintrunc_Suc_minus_Is = sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard] lemma sbintrunc_Suc_lem: "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y" by auto lemmas sbintrunc_Suc_Ialts = sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem, standard] lemma sbintrunc_bintrunc_lt: "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w" by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr) lemma bintrunc_sbintrunc_le: "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w" apply (rule bin_eqI) apply (auto simp: nth_sbintr nth_bintr) apply (subgoal_tac "x=n", safe, arith+)[1] apply (subgoal_tac "x=n", safe, arith+)[1] done lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le] lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt] lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l] lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l] lemma bintrunc_sbintrunc' [simp]: "0 < n ==> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w" by (cases n) (auto simp del: bintrunc.Suc) lemma sbintrunc_bintrunc' [simp]: "0 < n ==> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w" by (cases n) (auto simp del: bintrunc.Suc) lemma bin_sbin_eq_iff: "bintrunc (Suc n) x = bintrunc (Suc n) y <-> sbintrunc n x = sbintrunc n y" apply (rule iffI) apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc]) apply simp apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc]) apply simp done lemma bin_sbin_eq_iff': "0 < n ==> bintrunc n x = bintrunc n y <-> sbintrunc (n - 1) x = sbintrunc (n - 1) y" by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc) lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def] lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def] lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l] lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l] (* although bintrunc_minus_simps, if added to default simpset, tends to get applied where it's not wanted in developing the theories, we get a version for when the word length is given literally *) lemmas nat_non0_gr = trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl, standard] lemmas bintrunc_pred_simps [simp] = bintrunc_minus_simps [of "number_of bin", simplified nobm1, standard] lemmas sbintrunc_pred_simps [simp] = sbintrunc_minus_simps [of "number_of bin", simplified nobm1, standard] lemma no_bintr_alt: "number_of (bintrunc n w) = w mod 2 ^ n" by (simp add: number_of_eq bintrunc_mod2p) lemma no_bintr_alt1: "bintrunc n = (%w. w mod 2 ^ n :: int)" by (rule ext) (rule bintrunc_mod2p) lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}" apply (unfold no_bintr_alt1) apply (auto simp add: image_iff) apply (rule exI) apply (auto intro: int_mod_lem [THEN iffD1, symmetric]) done lemma no_bintr: "number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)" by (simp add : bintrunc_mod2p number_of_eq) lemma no_sbintr_alt2: "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)" by (rule ext) (simp add : sbintrunc_mod2p) lemma no_sbintr: "number_of (sbintrunc n w) = ((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)" by (simp add : no_sbintr_alt2 number_of_eq) lemma range_sbintrunc: "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}" apply (unfold no_sbintr_alt2) apply (auto simp add: image_iff eq_diff_eq) apply (rule exI) apply (auto intro: int_mod_lem [THEN iffD1, symmetric]) done lemma sb_inc_lem: "(a::int) + 2^k < 0 ==> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)" apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p]) apply (rule TrueI) done lemma sb_inc_lem': "(a::int) < - (2^k) ==> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)" by (rule iffD1 [OF less_diff_eq, THEN sb_inc_lem, simplified OrderedGroup.diff_0]) lemma sbintrunc_inc: "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x" unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp lemma sb_dec_lem: "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a" by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p, OF _ TrueI, simplified]) lemma sb_dec_lem': "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a" by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified]) lemma sbintrunc_dec: "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x" unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp lemmas zmod_uminus' = zmod_uminus [where b="c", standard] lemmas zpower_zmod' = zpower_zmod [where m="c" and y="k", standard] lemmas brdmod1s' [symmetric] = zmod_zadd_left_eq zmod_zadd_right_eq zmod_zsub_left_eq zmod_zsub_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev lemmas brdmods' [symmetric] = zpower_zmod' [symmetric] trans [OF zmod_zadd_left_eq zmod_zadd_right_eq] trans [OF zmod_zsub_left_eq zmod_zsub_right_eq] trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev] zmod_uminus' [symmetric] zmod_zadd_left_eq [where b = "1"] zmod_zsub_left_eq [where b = "1"] lemmas bintr_arith1s = brdmod1s' [where c="2^n", folded pred_def succ_def bintrunc_mod2p, standard] lemmas bintr_ariths = brdmods' [where c="2^n", folded pred_def succ_def bintrunc_mod2p, standard] lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p, standard] lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)" by (simp add : no_bintr m2pths) lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)" by (simp add : no_bintr m2pths) lemma bintr_Min: "number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1" by (simp add : no_bintr m1mod2k) lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)" by (simp add : no_sbintr m2pths) lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)" by (simp add : no_sbintr m2pths) lemma bintrunc_Suc: "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin" by (case_tac bin rule: bin_exhaust) auto lemma sign_Pls_ge_0: "(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))" by (induct bin rule: numeral_induct) auto lemma sign_Min_lt_0: "(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))" by (induct bin rule: numeral_induct) auto lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]] lemma bin_rest_trunc: "!!bin. (bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)" by (induct n) auto lemma bin_rest_power_trunc [rule_format] : "(bin_rest ^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^ k) bin)" by (induct k) (auto simp: bin_rest_trunc) lemma bin_rest_trunc_i: "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)" by auto lemma bin_rest_strunc: "!!bin. bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)" by (induct n) auto lemma bintrunc_rest [simp]: "!!bin. bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)" apply (induct n, simp) apply (case_tac bin rule: bin_exhaust) apply (auto simp: bintrunc_bintrunc_l) done lemma sbintrunc_rest [simp]: "!!bin. sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)" apply (induct n, simp) apply (case_tac bin rule: bin_exhaust) apply (auto simp: bintrunc_bintrunc_l split: bit.splits) done lemma bintrunc_rest': "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n" by (rule ext) auto lemma sbintrunc_rest' : "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n" by (rule ext) auto lemma rco_lem: "f o g o f = g o f ==> f o (g o f) ^ n = g ^ n o f" apply (rule ext) apply (induct_tac n) apply (simp_all (no_asm)) apply (drule fun_cong) apply (unfold o_def) apply (erule trans) apply simp done lemma rco_alt: "(f o g) ^ n o f = f o (g o f) ^ n" apply (rule ext) apply (induct n) apply (simp_all add: o_def) done lemmas rco_bintr = bintrunc_rest' [THEN rco_lem [THEN fun_cong], unfolded o_def] lemmas rco_sbintr = sbintrunc_rest' [THEN rco_lem [THEN fun_cong], unfolded o_def] subsection {* Splitting and concatenation *} primrec bin_split :: "nat => int => int × int" where Z: "bin_split 0 w = (w, Int.Pls)" | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))" primrec bin_cat :: "int => nat => int => int" where Z: "bin_cat w 0 v = w" | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v" subsection {* Miscellaneous lemmas *} lemmas funpow_minus_simp = trans [OF gen_minus [where f = "power f"] funpow_Suc, standard] lemmas funpow_pred_simp [simp] = funpow_minus_simp [of "number_of bin", simplified nobm1, standard] lemmas replicate_minus_simp = trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc, standard] lemmas replicate_pred_simp [simp] = replicate_minus_simp [of "number_of bin", simplified nobm1, standard] lemmas power_Suc_no [simp] = power_Suc [of "number_of a", standard] lemmas power_minus_simp = trans [OF gen_minus [where f = "power f"] power_Suc, standard] lemmas power_pred_simp = power_minus_simp [of "number_of bin", simplified nobm1, standard] lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f", standard] lemma list_exhaust_size_gt0: assumes y: "!!a list. y = a # list ==> P" shows "0 < length y ==> P" apply (cases y, simp) apply (rule y) apply fastsimp done lemma list_exhaust_size_eq0: assumes y: "y = [] ==> P" shows "length y = 0 ==> P" apply (cases y) apply (rule y, simp) apply simp done lemma size_Cons_lem_eq: "y = xa # list ==> size y = Suc k ==> size list = k" by auto lemma size_Cons_lem_eq_bin: "y = xa # list ==> size y = number_of (Int.succ k) ==> size list = number_of k" by (auto simp: pred_def succ_def split add : split_if_asm) lemmas ls_splits = prod.split split_split prod.split_asm split_split_asm split_if_asm lemma not_B1_is_B0: "y ≠ bit.B1 ==> y = bit.B0" by (cases y) auto lemma B1_ass_B0: assumes y: "y = bit.B0 ==> y = bit.B1" shows "y = bit.B1" apply (rule classical) apply (drule not_B1_is_B0) apply (erule y) done -- "simplifications for specific word lengths" lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc' lemmas s2n_ths = n2s_ths [symmetric] end
lemma BIT_B0_eq_Bit0:
w BIT B0 = Int.Bit0 w
lemma BIT_B1_eq_Bit1:
w BIT B1 = Int.Bit1 w
lemma BIT_simps:
w BIT B0 = Int.Bit0 w
w BIT B1 = Int.Bit1 w
lemma Min_ne_Pls:
Int.Min ≠ Int.Pls
lemma Pls_ne_Min:
Int.Pls ≠ Int.Min
lemma PlsMin_defs:
Int.Pls = 0
Int.Min = - 1
0 = Int.Pls
- 1 = Int.Min
lemma PlsMin_simps:
Int.Pls = 0 == True
Int.Min = - 1 == True
0 = Int.Pls == True
- 1 = Int.Min == True
lemma number_of_False_cong:
False ==> number_of x = number_of y
lemma BIT_eq:
u BIT b = v BIT c ==> u = v ∧ b = c
lemma BIT_eqE:
[| u BIT b = v BIT c; [| u = v; b = c |] ==> R |] ==> R
lemma BIT_eq_iff:
(u BIT b = v BIT c) = (u = v ∧ b = c)
lemma BIT_eqI:
[| u1 = v1; b1 = c1 |] ==> u1 BIT b1 = v1 BIT c1
lemma less_Bits:
(v BIT b < w BIT c) = (v < w ∨ v ≤ w ∧ b = bit.B0 ∧ c = bit.B1)
lemma le_Bits:
(v BIT b ≤ w BIT c) = (v < w ∨ v ≤ w ∧ (b ≠ bit.B1 ∨ c ≠ bit.B0))
lemma no_no:
number_of (number_of i) = i
lemma Bit_B0:
k BIT bit.B0 = k + k
lemma Bit_B1:
k BIT bit.B1 = k + k + 1
lemma Bit_B0_2t:
k BIT bit.B0 = 2 * k
lemma Bit_B1_2t:
k BIT bit.B1 = 2 * k + 1
lemma B_mod_2':
X = 2 ==> w BIT bit.B1 mod X = 1 ∧ w BIT bit.B0 mod X = 0
lemma B1_mod_2:
Int.Bit1 w mod 2 = 1
lemma B0_mod_2:
Int.Bit0 w mod 2 = 0
lemma neB1E:
[| y ≠ bit.B1; y = bit.B0 ==> P |] ==> P
lemma bin_ex_rl:
∃w b. w BIT b = bin
lemma bin_exhaust:
(!!x b. bin = x BIT b ==> Q) ==> Q
lemma bin_rl_char:
(bin_rl w = (r, l)) = (r BIT l = w)
lemma bin_rl:
bin_rl w = (bin_rest w, bin_last w)
lemma bin_rl_simps:
bin_rl Int.Pls = (Int.Pls, bit.B0)
bin_rl Int.Min = (Int.Min, bit.B1)
bin_rl (Int.Bit0 r) = (r, bit.B0)
bin_rl (Int.Bit1 r) = (r, bit.B1)
bin_rl (r BIT b) = (r, b)
lemma bin_rl_simp:
bin_rest w1 BIT bin_last w1 = w1
lemma bin_abs_lem:
bin = w BIT b ==> bin ≠ Int.Min --> bin ≠ Int.Pls --> nat ¦w¦ < nat ¦bin¦
lemma bin_induct:
[| P Int.Pls; P Int.Min; !!bin bit. P bin ==> P (bin BIT bit) |] ==> P bin
lemma numeral_induct:
[| P Int.Pls; P Int.Min; !!w. [| P w; w ≠ Int.Pls |] ==> P (Int.Bit0 w);
!!w. [| P w; w ≠ Int.Min |] ==> P (Int.Bit1 w) |]
==> P x
lemma bin_rest_simps:
bin_rest Int.Pls = Int.Pls
bin_rest Int.Min = Int.Min
bin_rest (Int.Bit0 w) = w
bin_rest (Int.Bit1 w) = w
bin_rest (w BIT b) = w
lemma bin_last_simps:
bin_last Int.Pls = bit.B0
bin_last Int.Min = bit.B1
bin_last (Int.Bit0 w) = bit.B0
bin_last (Int.Bit1 w) = bit.B1
bin_last (w BIT b) = b
lemma bin_r_l_extras:
bin_last 0 = bit.B0
bin_last (- 1) = bit.B1
bin_last -1 = bit.B1
bin_last 1 = bit.B1
bin_rest 1 = 0
bin_rest 0 = 0
bin_rest (- 1) = - 1
bin_rest -1 = -1
lemma bin_last_mod:
bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)
lemma bin_rest_div:
bin_rest w = w div 2
lemma Bit_div2:
w BIT b div 2 = w
lemma Bit0_div2:
Int.Bit0 w div 2 = w
lemma Bit1_div2:
Int.Bit1 w div 2 = w
lemma bin_nth_lem:
bin_nth x = bin_nth y ==> x = y
lemma bin_nth_eq_iff:
(bin_nth x = bin_nth y) = (x = y)
lemma bin_eqI:
(!!x. bin_nth x x = bin_nth y x) ==> x = y
lemma bin_nth_Pls:
¬ bin_nth Int.Pls n
lemma bin_nth_Min:
bin_nth Int.Min n
lemma bin_nth_0_BIT:
bin_nth (w BIT b) 0 = (b = bit.B1)
lemma bin_nth_Suc_BIT:
bin_nth (w BIT b) (Suc n) = bin_nth w n
lemma bin_nth_minus:
0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)
lemma bin_nth_minus_Bit0:
0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)
lemma bin_nth_minus_Bit1:
0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)
lemma bin_nth_0:
bin_nth w 0 = (bin_last w = bit.B1)
lemma bin_nth_Suc:
bin_nth w (Suc n) = bin_nth (bin_rest w) n
lemma bin_nth_simps:
bin_nth w 0 = (bin_last w = bit.B1)
bin_nth w (Suc n) = bin_nth (bin_rest w) n
¬ bin_nth Int.Pls n
bin_nth Int.Min n
0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)
0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)
0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)
lemma brlem:
(bin = Int.Min) = (- bin + Int.pred 0 = 0)
lemma bin_rec_PM:
f = bin_rec f1.0 f2.0 f3.0 ==> f Int.Pls = f1.0 ∧ f Int.Min = f2.0
lemma bin_rec_Pls:
bin_rec f1.0 f2.0 f3.0 Int.Pls = f1.0
lemma bin_rec_Min:
bin_rec f1.0 f2.0 f3.0 Int.Min = f2.0
lemma bin_rec_Bit0:
f3.0 Int.Pls bit.B0 f1.0 = f1.0
==> bin_rec f1.0 f2.0 f3.0 (Int.Bit0 w) =
f3.0 w bit.B0 (bin_rec f1.0 f2.0 f3.0 w)
lemma bin_rec_Bit1:
f3.0 Int.Min bit.B1 f2.0 = f2.0
==> bin_rec f1.0 f2.0 f3.0 (Int.Bit1 w) =
f3.0 w bit.B1 (bin_rec f1.0 f2.0 f3.0 w)
lemma bin_rec_Bit:
[| f = bin_rec f1.0 f2.0 f3.0; f3.0 Int.Pls bit.B0 f1.0 = f1.0;
f3.0 Int.Min bit.B1 f2.0 = f2.0 |]
==> f (w BIT b) = f3.0 w b (f w)
lemma bin_rec_simps:
[| f3.0 Int.Pls bit.B0 f1.0 = f1.0; f3.0 Int.Min bit.B1 f2.0 = f2.0 |]
==> bin_rec f1.0 f2.0 f3.0 (w BIT b) = f3.0 w b (bin_rec f1.0 f2.0 f3.0 w)
bin_rec f1.0 f2.0 f3.0 Int.Pls = f1.0
bin_rec f1.0 f2.0 f3.0 Int.Min = f2.0
f3.0 Int.Pls bit.B0 f1.0 = f1.0
==> bin_rec f1.0 f2.0 f3.0 (Int.Bit0 w) =
f3.0 w bit.B0 (bin_rec f1.0 f2.0 f3.0 w)
f3.0 Int.Min bit.B1 f2.0 = f2.0
==> bin_rec f1.0 f2.0 f3.0 (Int.Bit1 w) =
f3.0 w bit.B1 (bin_rec f1.0 f2.0 f3.0 w)
lemma bin_sign_simps:
bin_sign Int.Pls = Int.Pls
bin_sign Int.Min = Int.Min
bin_sign (Int.Bit0 w) = bin_sign w
bin_sign (Int.Bit1 w) = bin_sign w
bin_sign (w BIT b) = bin_sign w
lemma bin_sign_rest:
bin_sign (bin_rest w) = bin_sign w
lemma sign_bintr:
bin_sign (bintrunc n w) = Int.Pls
lemma bintrunc_mod2p:
bintrunc n w = w mod 2 ^ n
lemma sbintrunc_mod2p:
sbintrunc n w = (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n
lemma bit_bool:
(b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))
lemma bit_bool1:
(if s = bit.B1 then bit.B1 else bit.B0) = s
lemma bin_sign_lem:
(bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n
lemma nth_bintr:
bin_nth (bintrunc m w) n = (n < m ∧ bin_nth w n)
lemma nth_sbintr:
bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)
lemma bin_nth_Bit:
bin_nth (w BIT b) n = (n = 0 ∧ b = bit.B1 ∨ (∃m. n = Suc m ∧ bin_nth w m))
lemma bin_nth_Bit0:
bin_nth (Int.Bit0 w) n = (∃m. n = Suc m ∧ bin_nth w m)
lemma bin_nth_Bit1:
bin_nth (Int.Bit1 w) n = (n = 0 ∨ (∃m. n = Suc m ∧ bin_nth w m))
lemma bintrunc_bintrunc_l:
n ≤ m ==> bintrunc m (bintrunc n w) = bintrunc n w
lemma sbintrunc_sbintrunc_l:
n ≤ m ==> sbintrunc m (sbintrunc n w) = sbintrunc n w
lemma bintrunc_bintrunc_ge:
n ≤ m ==> bintrunc n (bintrunc m w) = bintrunc n w
lemma bintrunc_bintrunc_min:
bintrunc m (bintrunc n w) = bintrunc (min m n) w
lemma sbintrunc_sbintrunc_min:
sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w
lemma bintrunc_Pls:
bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0
lemma bintrunc_Min:
bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1
lemma bintrunc_BIT:
bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b
lemma bintrunc_Bit0:
bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)
lemma bintrunc_Bit1:
bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)
lemma bintrunc_Sucs:
bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0
bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1
bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b
bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)
bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)
lemma sbintrunc_Suc_Pls:
sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0
lemma sbintrunc_Suc_Min:
sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1
lemma sbintrunc_Suc_BIT:
sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b
lemma sbintrunc_Suc_Bit0:
sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)
lemma sbintrunc_Suc_Bit1:
sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)
lemma sbintrunc_Sucs:
sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0
sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1
sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b
sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)
sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)
lemma sbintrunc_Pls:
sbintrunc 0 Int.Pls = Int.Pls
lemma sbintrunc_Min:
sbintrunc 0 Int.Min = Int.Min
lemma sbintrunc_0_BIT_B0:
sbintrunc 0 (w BIT bit.B0) = Int.Pls
lemma sbintrunc_0_BIT_B1:
sbintrunc 0 (w BIT bit.B1) = Int.Min
lemma sbintrunc_0_Bit0:
sbintrunc 0 (Int.Bit0 w) = Int.Pls
lemma sbintrunc_0_Bit1:
sbintrunc 0 (Int.Bit1 w) = Int.Min
lemma sbintrunc_0_simps:
sbintrunc 0 Int.Pls = Int.Pls
sbintrunc 0 Int.Min = Int.Min
sbintrunc 0 (w BIT bit.B0) = Int.Pls
sbintrunc 0 (w BIT bit.B1) = Int.Min
sbintrunc 0 (Int.Bit0 w) = Int.Pls
sbintrunc 0 (Int.Bit1 w) = Int.Min
lemma bintrunc_simps:
bintrunc 0 bin = Int.Pls
bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0
bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1
bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b
bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)
bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)
lemma sbintrunc_simps:
sbintrunc 0 Int.Pls = Int.Pls
sbintrunc 0 Int.Min = Int.Min
sbintrunc 0 (w BIT bit.B0) = Int.Pls
sbintrunc 0 (w BIT bit.B1) = Int.Min
sbintrunc 0 (Int.Bit0 w) = Int.Pls
sbintrunc 0 (Int.Bit1 w) = Int.Min
sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0
sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1
sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b
sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)
sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)
lemma bintrunc_minus:
0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w
lemma sbintrunc_minus:
0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w
lemma bintrunc_minus_simps:
0 < n ==> bintrunc n Int.Pls = bintrunc (n - 1) Int.Pls BIT bit.B0
0 < n ==> bintrunc n Int.Min = bintrunc (n - 1) Int.Min BIT bit.B1
0 < n ==> bintrunc n (w BIT b) = bintrunc (n - 1) w BIT b
0 < n ==> bintrunc n (Int.Bit0 w) = Int.Bit0 (bintrunc (n - 1) w)
0 < n ==> bintrunc n (Int.Bit1 w) = Int.Bit1 (bintrunc (n - 1) w)
lemma sbintrunc_minus_simps:
0 < n ==> sbintrunc n Int.Pls = sbintrunc (n - 1) Int.Pls BIT bit.B0
0 < n ==> sbintrunc n Int.Min = sbintrunc (n - 1) Int.Min BIT bit.B1
0 < n ==> sbintrunc n (w BIT b) = sbintrunc (n - 1) w BIT b
0 < n ==> sbintrunc n (Int.Bit0 w) = Int.Bit0 (sbintrunc (n - 1) w)
0 < n ==> sbintrunc n (Int.Bit1 w) = Int.Bit1 (sbintrunc (n - 1) w)
lemma bintrunc_n_Pls:
bintrunc n Int.Pls = Int.Pls
lemma sbintrunc_n_PM:
sbintrunc n Int.Pls = Int.Pls
sbintrunc n Int.Min = Int.Min
lemma thobini1:
x = y ==> x BIT b = y BIT b
lemma bintrunc_BIT_I:
bintrunc n2 w2 = y1 ==> bintrunc (Suc n2) (w2 BIT b1) = y1 BIT b1
lemma bintrunc_Min_I:
bintrunc n2 Int.Min = y1 ==> bintrunc (Suc n2) Int.Min = y1 BIT bit.B1
lemma bmsts:
[| 0 < n; bintrunc (n - 1) Int.Pls = y |] ==> bintrunc n Int.Pls = y BIT bit.B0
[| 0 < n; bintrunc (n - 1) Int.Min = y |] ==> bintrunc n Int.Min = y BIT bit.B1
[| 0 < n; bintrunc (n - 1) w = y |] ==> bintrunc n (w BIT b) = y BIT b
lemma bintrunc_Pls_minus_I:
[| 0 < n; bintrunc (n - 1) Int.Pls = y |] ==> bintrunc n Int.Pls = y BIT bit.B0
lemma bintrunc_Min_minus_I:
[| 0 < n; bintrunc (n - 1) Int.Min = y |] ==> bintrunc n Int.Min = y BIT bit.B1
lemma bintrunc_BIT_minus_I:
[| 0 < n; bintrunc (n - 1) w = y |] ==> bintrunc n (w BIT b) = y BIT b
lemma bintrunc_0_Min:
bintrunc 0 Int.Min = Int.Pls
lemma bintrunc_0_BIT:
bintrunc 0 (w BIT b) = Int.Pls
lemma bintrunc_Suc_lem:
[| bintrunc (Suc n) x = y; m = Suc n |] ==> bintrunc m x = y
lemma bintrunc_Suc_Ialts:
[| bintrunc n Int.Min = y; m = Suc n |] ==> bintrunc m Int.Min = y BIT bit.B1
[| bintrunc n w = y; m = Suc n |] ==> bintrunc m (w BIT b) = y BIT b
lemma sbintrunc_BIT_I:
sbintrunc n2 w2 = y1 ==> sbintrunc (Suc n2) (w2 BIT b1) = y1 BIT b1
lemma sbintrunc_Suc_Is:
sbintrunc n Int.Pls = y ==> sbintrunc (Suc n) Int.Pls = y BIT bit.B0
sbintrunc n Int.Min = y ==> sbintrunc (Suc n) Int.Min = y BIT bit.B1
sbintrunc n w = y ==> sbintrunc (Suc n) (w BIT b) = y BIT b
lemma sbintrunc_Suc_minus_Is:
[| 0 < n; sbintrunc (n - 1) Int.Pls = y |]
==> sbintrunc n Int.Pls = y BIT bit.B0
[| 0 < n; sbintrunc (n - 1) Int.Min = y |]
==> sbintrunc n Int.Min = y BIT bit.B1
[| 0 < n; sbintrunc (n - 1) w = y |] ==> sbintrunc n (w BIT b) = y BIT b
lemma sbintrunc_Suc_lem:
[| sbintrunc (Suc n) x = y; m = Suc n |] ==> sbintrunc m x = y
lemma sbintrunc_Suc_Ialts:
[| sbintrunc n Int.Pls = y; m = Suc n |] ==> sbintrunc m Int.Pls = y BIT bit.B0
[| sbintrunc n Int.Min = y; m = Suc n |] ==> sbintrunc m Int.Min = y BIT bit.B1
[| sbintrunc n w = y; m = Suc n |] ==> sbintrunc m (w BIT b) = y BIT b
lemma sbintrunc_bintrunc_lt:
n < m ==> sbintrunc n (bintrunc m w) = sbintrunc n w
lemma bintrunc_sbintrunc_le:
m ≤ Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w
lemma bintrunc_sbintrunc:
bintrunc (Suc n) (sbintrunc n w) = bintrunc (Suc n) w
lemma sbintrunc_bintrunc:
sbintrunc n (bintrunc (Suc n) w) = sbintrunc n w
lemma bintrunc_bintrunc:
bintrunc m (bintrunc m w) = bintrunc m w
lemma sbintrunc_sbintrunc:
sbintrunc m (sbintrunc m w) = sbintrunc m w
lemma bintrunc_sbintrunc':
0 < n ==> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w
lemma sbintrunc_bintrunc':
0 < n ==> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w
lemma bin_sbin_eq_iff:
(bintrunc (Suc n) x = bintrunc (Suc n) y) = (sbintrunc n x = sbintrunc n y)
lemma bin_sbin_eq_iff':
0 < n
==> (bintrunc n x = bintrunc n y) = (sbintrunc (n - 1) x = sbintrunc (n - 1) y)
lemma bintrunc_sbintruncS0:
0 < n ==> bintrunc n (sbintrunc (n - Suc 0) w) = bintrunc n w
lemma sbintrunc_bintruncS0:
0 < n ==> sbintrunc (n - Suc 0) (bintrunc n w) = sbintrunc (n - Suc 0) w
lemma bintrunc_bintrunc_l':
bintrunc (n + m1) (bintrunc n w) = bintrunc n w
lemma sbintrunc_sbintrunc_l':
sbintrunc (n + m1) (sbintrunc n w) = sbintrunc n w
lemma nat_non0_gr:
(¬ iszero z) = (z ≠ (0::'a))
lemma bintrunc_pred_simps:
0 < number_of bin
==> bintrunc (number_of bin) Int.Pls =
bintrunc (number_of (Int.pred bin)) Int.Pls BIT bit.B0
0 < number_of bin
==> bintrunc (number_of bin) Int.Min =
bintrunc (number_of (Int.pred bin)) Int.Min BIT bit.B1
0 < number_of bin
==> bintrunc (number_of bin) (w BIT b) =
bintrunc (number_of (Int.pred bin)) w BIT b
0 < number_of bin
==> bintrunc (number_of bin) (Int.Bit0 w) =
Int.Bit0 (bintrunc (number_of (Int.pred bin)) w)
0 < number_of bin
==> bintrunc (number_of bin) (Int.Bit1 w) =
Int.Bit1 (bintrunc (number_of (Int.pred bin)) w)
lemma sbintrunc_pred_simps:
0 < number_of bin
==> sbintrunc (number_of bin) Int.Pls =
sbintrunc (number_of (Int.pred bin)) Int.Pls BIT bit.B0
0 < number_of bin
==> sbintrunc (number_of bin) Int.Min =
sbintrunc (number_of (Int.pred bin)) Int.Min BIT bit.B1
0 < number_of bin
==> sbintrunc (number_of bin) (w BIT b) =
sbintrunc (number_of (Int.pred bin)) w BIT b
0 < number_of bin
==> sbintrunc (number_of bin) (Int.Bit0 w) =
Int.Bit0 (sbintrunc (number_of (Int.pred bin)) w)
0 < number_of bin
==> sbintrunc (number_of bin) (Int.Bit1 w) =
Int.Bit1 (sbintrunc (number_of (Int.pred bin)) w)
lemma no_bintr_alt:
number_of (bintrunc n w) = w mod 2 ^ n
lemma no_bintr_alt1:
bintrunc n = (λw. w mod 2 ^ n)
lemma range_bintrunc:
range (bintrunc n) = {i. 0 ≤ i ∧ i < 2 ^ n}
lemma no_bintr:
number_of (bintrunc n w) = number_of w mod 2 ^ n
lemma no_sbintr_alt2:
sbintrunc n = (λw. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n)
lemma no_sbintr:
number_of (sbintrunc n w) = (number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n
lemma range_sbintrunc:
range (sbintrunc n) = {i. - (2 ^ n) ≤ i ∧ i < 2 ^ n}
lemma sb_inc_lem:
a + 2 ^ k < 0 ==> a + 2 ^ k + 2 ^ Suc k ≤ (a + 2 ^ k) mod 2 ^ Suc k
lemma sb_inc_lem':
a < - (2 ^ k) ==> a + 2 ^ k + 2 ^ Suc k ≤ (a + 2 ^ k) mod 2 ^ Suc k
lemma sbintrunc_inc:
x < - (2 ^ n) ==> x + 2 ^ Suc n ≤ sbintrunc n x
lemma sb_dec_lem:
0 ≤ - (2 ^ k) + a ==> (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a
lemma sb_dec_lem':
2 ^ k ≤ a ==> (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a
lemma sbintrunc_dec:
2 ^ n ≤ x ==> sbintrunc n x ≤ x - 2 ^ Suc n
lemma zmod_uminus':
- (a mod c) mod c = - a mod c
lemma zpower_zmod':
(x mod c) ^ k mod c = x ^ k mod c
lemma brdmod1s':
(a mod c + b) mod c = (a + b) mod c
(a + b mod c) mod c = (a + b) mod c
(a mod c - b) mod c = (a - b) mod c
(a - b mod c) mod c = (a - b) mod c
a * (b mod c) mod c = a * b mod c
b mod c * a mod c = b * a mod c
lemma brdmods':
(x mod c) ^ k mod c = x ^ k mod c
(a mod c + b mod c) mod c = (a + b) mod c
(a mod c - b mod c) mod c = (a - b) mod c
b mod c * (ba mod c) mod c = b * ba mod c
- (a mod c) mod c = - a mod c
(a mod c + 1) mod c = (a + 1) mod c
(a mod c - 1) mod c = (a - 1) mod c
lemma bintr_arith1s:
bintrunc n (bintrunc n a + b) = bintrunc n (a + b)
bintrunc n (a + bintrunc n b) = bintrunc n (a + b)
bintrunc n (bintrunc n a - b) = bintrunc n (a - b)
bintrunc n (a - bintrunc n b) = bintrunc n (a - b)
bintrunc n (a * bintrunc n b) = bintrunc n (a * b)
bintrunc n (bintrunc n b * a) = bintrunc n (b * a)
lemma bintr_ariths:
bintrunc n (bintrunc n x ^ k) = bintrunc n (x ^ k)
bintrunc n (bintrunc n a + bintrunc n b) = bintrunc n (a + b)
bintrunc n (bintrunc n a - bintrunc n b) = bintrunc n (a - b)
bintrunc n (bintrunc n b * bintrunc n ba) = bintrunc n (b * ba)
bintrunc n (- bintrunc n a) = bintrunc n (- a)
bintrunc n (Int.succ (bintrunc n a)) = bintrunc n (Int.succ a)
bintrunc n (Int.pred (bintrunc n a)) = bintrunc n (Int.pred a)
lemma m2pths:
0 < b ==> 0 ≤ a mod b
a mod 2 ^ n < 2 ^ n
lemma bintr_ge0:
0 ≤ number_of (bintrunc n w)
lemma bintr_lt2p:
number_of (bintrunc n w) < 2 ^ n
lemma bintr_Min:
number_of (bintrunc n Int.Min) = 2 ^ n - 1
lemma sbintr_ge:
- (2 ^ n) ≤ number_of (sbintrunc n w)
lemma sbintr_lt:
number_of (sbintrunc n w) < 2 ^ n
lemma bintrunc_Suc:
bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin
lemma sign_Pls_ge_0:
(bin_sign bin = Int.Pls) = (0 ≤ number_of bin)
lemma sign_Min_lt_0:
(bin_sign bin = Int.Min) = (number_of bin < 0)
lemma sign_Min_neg:
(bin_sign bin2 = Int.Min) = neg (number_of bin2)
lemma bin_rest_trunc:
bin_rest (bintrunc n bin) = bintrunc (n - 1) (bin_rest bin)
lemma bin_rest_power_trunc:
(bin_rest ^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^ k) bin)
lemma bin_rest_trunc_i:
bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)
lemma bin_rest_strunc:
bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)
lemma bintrunc_rest:
bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)
lemma sbintrunc_rest:
sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)
lemma bintrunc_rest':
bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n
lemma sbintrunc_rest':
sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n
lemma rco_lem:
f o g o f = g o f ==> f o (g o f) ^ n = g ^ n o f
lemma rco_alt:
(f o g) ^ n o f = f o (g o f) ^ n
lemma rco_bintr:
bintrunc n2 (((λx. bin_rest (bintrunc n2 x)) ^ n1) x) =
(bin_rest ^ n1) (bintrunc n2 x)
lemma rco_sbintr:
sbintrunc n2 (((λx. bin_rest (sbintrunc n2 x)) ^ n1) x) =
(bin_rest ^ n1) (sbintrunc n2 x)
lemma funpow_minus_simp:
0 < n ==> f ^ n = f o f ^ (n - 1)
lemma funpow_pred_simp:
0 < number_of bin ==> f ^ number_of bin = f o f ^ number_of (Int.pred bin)
lemma replicate_minus_simp:
0 < n ==> replicate n x = x # replicate (n - 1) x
lemma replicate_pred_simp:
0 < number_of bin
==> replicate (number_of bin) x = x # replicate (number_of (Int.pred bin)) x
lemma power_Suc_no:
number_of a ^ Suc n = number_of a * number_of a ^ n
lemma power_minus_simp:
0 < n ==> f ^ n = f * f ^ (n - 1)
lemma power_pred_simp:
0 < number_of bin ==> f ^ number_of bin = f * f ^ number_of (Int.pred bin)
lemma power_pred_simp_no:
0 < number_of bin
==> number_of f ^ number_of bin =
number_of f * number_of f ^ number_of (Int.pred bin)
lemma list_exhaust_size_gt0:
[| !!a list. y = a # list ==> P; 0 < length y |] ==> P
lemma list_exhaust_size_eq0:
[| y = [] ==> P; length y = 0 |] ==> P
lemma size_Cons_lem_eq:
[| y = xa # list; length y = Suc k |] ==> length list = k
lemma size_Cons_lem_eq_bin:
[| y = xa # list; length y = number_of (Int.succ k) |]
==> length list = number_of k
lemma ls_splits:
P (prod_case f1.0 x) = (∀a b. x = (a, b) --> P (f1.0 a b))
P (split f1.0 x) = (∀a b. x = (a, b) --> P (f1.0 a b))
P (prod_case f1.0 x) = (¬ (∃a b. x = (a, b) ∧ ¬ P (f1.0 a b)))
P (split f1.0 x) = (¬ (∃a b. x = (a, b) ∧ ¬ P (f1.0 a b)))
P (if Q then x else y) = (¬ (Q ∧ ¬ P x ∨ ¬ Q ∧ ¬ P y))
lemma not_B1_is_B0:
y ≠ bit.B1 ==> y = bit.B0
lemma B1_ass_B0:
(y = bit.B0 ==> y = bit.B1) ==> y = bit.B1
lemma n2s_ths:
2 + n1 == Suc (Suc n1)
n1 + 2 == Suc (Suc n1)
lemma s2n_ths:
Suc (Suc n) == 2 + n
Suc (Suc n) == n + 2