author | huffman |
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parent 45847 | b4254b2e2b4a |
child 45956 | ae70b6830f15 |
permissions | -rw-r--r-- |
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(* |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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Definitions and basic theorems for bit-wise logical operations |
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for integers expressed using Pls, Min, BIT, |
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and converting them to and from lists of bools. |
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*) |
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header {* Bitwise Operations on Binary Integers *} |
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theory Bit_Int |
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imports Bit_Representation Bit_Operations |
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begin |
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subsection {* Recursion combinators for bitstrings *} |
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function bin_rec :: "'a \<Rightarrow> 'a \<Rightarrow> (int \<Rightarrow> bit \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> int \<Rightarrow> 'a" where |
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"bin_rec f1 f2 f3 bin = (if bin = 0 then f1 |
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else if bin = - 1 then f2 |
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else f3 (bin_rest bin) (bin_last bin) (bin_rec f1 f2 f3 (bin_rest bin)))" |
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by pat_completeness auto |
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termination by (relation "measure (nat o abs o snd o snd o snd)") |
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(simp_all add: bin_last_def bin_rest_def) |
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declare bin_rec.simps [simp del] |
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lemma bin_rec_PM: |
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"f = bin_rec f1 f2 f3 ==> f Int.Pls = f1 & f Int.Min = f2" |
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by (unfold Pls_def Min_def) (simp add: bin_rec.simps) |
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lemma bin_rec_Pls: "bin_rec f1 f2 f3 Int.Pls = f1" |
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by (unfold Pls_def Min_def) (simp add: bin_rec.simps) |
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lemma bin_rec_Min: "bin_rec f1 f2 f3 Int.Min = f2" |
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by (unfold Pls_def Min_def) (simp add: bin_rec.simps) |
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lemma bin_rec_Bit0: |
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"f3 Int.Pls (0::bit) f1 = f1 \<Longrightarrow> |
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bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w (0::bit) (bin_rec f1 f2 f3 w)" |
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by (unfold Pls_def Min_def Bit0_def Bit1_def) (simp add: bin_rec.simps bin_last_def bin_rest_def) |
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lemma bin_rec_Bit1: |
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"f3 Int.Min (1::bit) f2 = f2 \<Longrightarrow> |
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bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w (1::bit) (bin_rec f1 f2 f3 w)" |
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apply (cases "w = Int.Min") |
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apply (simp add: bin_rec_Min) |
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apply (cases "w = Int.Pls") |
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apply (simp add: bin_rec_Pls number_of_is_id Pls_def [symmetric] bin_rec.simps) |
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apply (subst bin_rec.simps) |
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apply auto unfolding Pls_def Min_def Bit0_def Bit1_def number_of_is_id apply auto |
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done |
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lemma bin_rec_Bit: |
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"f = bin_rec f1 f2 f3 ==> f3 Int.Pls (0::bit) f1 = f1 ==> |
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f3 Int.Min (1::bit) f2 = f2 ==> f (w BIT b) = f3 w b (f w)" |
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by (cases b, simp add: bin_rec_Bit0 BIT_simps, simp add: bin_rec_Bit1 BIT_simps) |
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lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min |
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bin_rec_Bit0 bin_rec_Bit1 |
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subsection {* Logical operations *} |
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text "bit-wise logical operations on the int type" |
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instantiation int :: bit |
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begin |
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definition |
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int_not_def: "bitNOT = bin_rec (- 1) 0 |
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(\<lambda>w b s. s BIT (NOT b))" |
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definition |
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int_and_def: "bitAND = bin_rec (\<lambda>x. 0) (\<lambda>y. y) |
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(\<lambda>w b s y. s (bin_rest y) BIT (b AND bin_last y))" |
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definition |
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int_or_def: "bitOR = bin_rec (\<lambda>x. x) (\<lambda>y. - 1) |
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(\<lambda>w b s y. s (bin_rest y) BIT (b OR bin_last y))" |
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definition |
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int_xor_def: "bitXOR = bin_rec (\<lambda>x. x) bitNOT |
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(\<lambda>w b s y. s (bin_rest y) BIT (b XOR bin_last y))" |
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instance .. |
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end |
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subsubsection {* Basic simplification rules *} |
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lemma int_not_simps [simp]: |
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"NOT Int.Pls = Int.Min" |
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"NOT Int.Min = Int.Pls" |
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"NOT (Int.Bit0 w) = Int.Bit1 (NOT w)" |
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"NOT (Int.Bit1 w) = Int.Bit0 (NOT w)" |
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"NOT (w BIT b) = (NOT w) BIT (NOT b)" |
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unfolding int_not_def Pls_def [symmetric] Min_def [symmetric] |
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by (simp_all add: bin_rec_simps BIT_simps) |
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lemma int_xor_Pls [simp]: |
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"Int.Pls XOR x = x" |
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unfolding int_xor_def Pls_def [symmetric] Min_def [symmetric] by (simp add: bin_rec_PM) |
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lemma int_xor_Min [simp]: |
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"Int.Min XOR x = NOT x" |
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unfolding int_xor_def Pls_def [symmetric] Min_def [symmetric] by (simp add: bin_rec_PM) |
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lemma int_xor_Bits [simp]: |
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"(x BIT b) XOR (y BIT c) = (x XOR y) BIT (b XOR c)" |
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apply (unfold int_xor_def Pls_def [symmetric] Min_def [symmetric]) |
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apply (rule bin_rec_simps (1) [THEN fun_cong, THEN trans]) |
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apply (rule ext, simp) |
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prefer 2 |
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apply simp |
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apply (rule ext) |
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apply (simp add: int_not_simps [symmetric]) |
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done |
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lemma int_xor_Bits2 [simp]: |
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"(Int.Bit0 x) XOR (Int.Bit0 y) = Int.Bit0 (x XOR y)" |
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"(Int.Bit0 x) XOR (Int.Bit1 y) = Int.Bit1 (x XOR y)" |
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"(Int.Bit1 x) XOR (Int.Bit0 y) = Int.Bit1 (x XOR y)" |
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"(Int.Bit1 x) XOR (Int.Bit1 y) = Int.Bit0 (x XOR y)" |
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unfolding BIT_simps [symmetric] int_xor_Bits by simp_all |
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lemma int_or_Pls [simp]: |
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"Int.Pls OR x = x" |
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by (unfold int_or_def) (simp add: bin_rec_PM) |
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lemma int_or_Min [simp]: |
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"Int.Min OR x = Int.Min" |
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by (unfold int_or_def Pls_def [symmetric] Min_def [symmetric]) (simp add: bin_rec_PM) |
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lemma int_or_Bits [simp]: |
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"(x BIT b) OR (y BIT c) = (x OR y) BIT (b OR c)" |
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unfolding int_or_def Pls_def [symmetric] Min_def [symmetric] |
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by (simp add: bin_rec_simps BIT_simps) |
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lemma int_or_Bits2 [simp]: |
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"(Int.Bit0 x) OR (Int.Bit0 y) = Int.Bit0 (x OR y)" |
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"(Int.Bit0 x) OR (Int.Bit1 y) = Int.Bit1 (x OR y)" |
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"(Int.Bit1 x) OR (Int.Bit0 y) = Int.Bit1 (x OR y)" |
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"(Int.Bit1 x) OR (Int.Bit1 y) = Int.Bit1 (x OR y)" |
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unfolding BIT_simps [symmetric] int_or_Bits by simp_all |
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lemma int_and_Pls [simp]: |
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"Int.Pls AND x = Int.Pls" |
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unfolding int_and_def Pls_def [symmetric] by (simp add: bin_rec_PM) |
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lemma int_and_Min [simp]: |
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"Int.Min AND x = x" |
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unfolding int_and_def by (simp add: bin_rec_PM) |
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lemma int_and_Bits [simp]: |
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"(x BIT b) AND (y BIT c) = (x AND y) BIT (b AND c)" |
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unfolding int_and_def Pls_def [symmetric] Min_def [symmetric] |
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by (simp add: bin_rec_simps BIT_simps) |
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lemma int_and_Bits2 [simp]: |
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"(Int.Bit0 x) AND (Int.Bit0 y) = Int.Bit0 (x AND y)" |
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"(Int.Bit0 x) AND (Int.Bit1 y) = Int.Bit0 (x AND y)" |
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"(Int.Bit1 x) AND (Int.Bit0 y) = Int.Bit0 (x AND y)" |
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"(Int.Bit1 x) AND (Int.Bit1 y) = Int.Bit1 (x AND y)" |
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unfolding BIT_simps [symmetric] int_and_Bits by simp_all |
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subsubsection {* Binary destructors *} |
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lemma bin_rest_NOT [simp]: "bin_rest (NOT x) = NOT (bin_rest x)" |
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by (cases x rule: bin_exhaust, simp) |
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lemma bin_last_NOT [simp]: "bin_last (NOT x) = NOT (bin_last x)" |
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by (cases x rule: bin_exhaust, simp) |
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lemma bin_rest_AND [simp]: "bin_rest (x AND y) = bin_rest x AND bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bin_last_AND [simp]: "bin_last (x AND y) = bin_last x AND bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bin_rest_OR [simp]: "bin_rest (x OR y) = bin_rest x OR bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bin_last_OR [simp]: "bin_last (x OR y) = bin_last x OR bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bin_rest_XOR [simp]: "bin_rest (x XOR y) = bin_rest x XOR bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bin_last_XOR [simp]: "bin_last (x XOR y) = bin_last x XOR bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0" |
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by (induct b, simp_all) |
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lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1" |
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by (induct a, simp_all) |
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lemma bin_nth_ops: |
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"!!x y. bin_nth (x AND y) n = (bin_nth x n & bin_nth y n)" |
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201 |
"!!x y. bin_nth (x OR y) n = (bin_nth x n | bin_nth y n)" |
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202 |
"!!x y. bin_nth (x XOR y) n = (bin_nth x n ~= bin_nth y n)" |
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203 |
"!!x. bin_nth (NOT x) n = (~ bin_nth x n)" |
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204 |
by (induct n) auto |
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205 |
|
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206 |
subsubsection {* Derived properties *} |
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207 |
|
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208 |
lemma int_xor_extra_simps [simp]: |
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209 |
"w XOR Int.Pls = w" |
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210 |
"w XOR Int.Min = NOT w" |
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211 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
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212 |
|
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213 |
lemma int_or_extra_simps [simp]: |
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214 |
"w OR Int.Pls = w" |
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"w OR Int.Min = Int.Min" |
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216 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 217 |
|
37667 | 218 |
lemma int_and_extra_simps [simp]: |
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219 |
"w AND Int.Pls = Int.Pls" |
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220 |
"w AND Int.Min = w" |
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221 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 222 |
|
223 |
(* commutativity of the above *) |
|
224 |
lemma bin_ops_comm: |
|
225 |
shows |
|
24353 | 226 |
int_and_comm: "!!y::int. x AND y = y AND x" and |
227 |
int_or_comm: "!!y::int. x OR y = y OR x" and |
|
228 |
int_xor_comm: "!!y::int. x XOR y = y XOR x" |
|
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229 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 230 |
|
231 |
lemma bin_ops_same [simp]: |
|
24353 | 232 |
"(x::int) AND x = x" |
233 |
"(x::int) OR x = x" |
|
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234 |
"(x::int) XOR x = Int.Pls" |
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235 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 236 |
|
24353 | 237 |
lemma int_not_not [simp]: "NOT (NOT (x::int)) = x" |
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238 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 239 |
|
240 |
lemmas bin_log_esimps = |
|
241 |
int_and_extra_simps int_or_extra_simps int_xor_extra_simps |
|
242 |
int_and_Pls int_and_Min int_or_Pls int_or_Min int_xor_Pls int_xor_Min |
|
243 |
||
244 |
(* basic properties of logical (bit-wise) operations *) |
|
245 |
||
246 |
lemma bbw_ao_absorb: |
|
24353 | 247 |
"!!y::int. x AND (y OR x) = x & x OR (y AND x) = x" |
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248 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 249 |
|
250 |
lemma bbw_ao_absorbs_other: |
|
24353 | 251 |
"x AND (x OR y) = x \<and> (y AND x) OR x = (x::int)" |
252 |
"(y OR x) AND x = x \<and> x OR (x AND y) = (x::int)" |
|
253 |
"(x OR y) AND x = x \<and> (x AND y) OR x = (x::int)" |
|
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254 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24353 | 255 |
|
24333 | 256 |
lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other |
257 |
||
258 |
lemma int_xor_not: |
|
24353 | 259 |
"!!y::int. (NOT x) XOR y = NOT (x XOR y) & |
260 |
x XOR (NOT y) = NOT (x XOR y)" |
|
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261 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 262 |
|
263 |
lemma int_and_assoc: |
|
24353 | 264 |
"(x AND y) AND (z::int) = x AND (y AND z)" |
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265 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 266 |
|
267 |
lemma int_or_assoc: |
|
24353 | 268 |
"(x OR y) OR (z::int) = x OR (y OR z)" |
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269 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 270 |
|
271 |
lemma int_xor_assoc: |
|
24353 | 272 |
"(x XOR y) XOR (z::int) = x XOR (y XOR z)" |
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273 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 274 |
|
275 |
lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc |
|
276 |
||
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277 |
(* BH: Why are these declared as simp rules??? *) |
24333 | 278 |
lemma bbw_lcs [simp]: |
24353 | 279 |
"(y::int) AND (x AND z) = x AND (y AND z)" |
280 |
"(y::int) OR (x OR z) = x OR (y OR z)" |
|
281 |
"(y::int) XOR (x XOR z) = x XOR (y XOR z)" |
|
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282 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 283 |
|
284 |
lemma bbw_not_dist: |
|
24353 | 285 |
"!!y::int. NOT (x OR y) = (NOT x) AND (NOT y)" |
286 |
"!!y::int. NOT (x AND y) = (NOT x) OR (NOT y)" |
|
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287 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 288 |
|
289 |
lemma bbw_oa_dist: |
|
24353 | 290 |
"!!y z::int. (x AND y) OR z = |
291 |
(x OR z) AND (y OR z)" |
|
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292 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 293 |
|
294 |
lemma bbw_ao_dist: |
|
24353 | 295 |
"!!y z::int. (x OR y) AND z = |
296 |
(x AND z) OR (y AND z)" |
|
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297 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 298 |
|
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299 |
(* |
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300 |
Why were these declared simp??? |
24333 | 301 |
declare bin_ops_comm [simp] bbw_assocs [simp] |
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302 |
*) |
24333 | 303 |
|
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304 |
subsubsection {* Interactions with arithmetic *} |
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305 |
|
24333 | 306 |
lemma plus_and_or [rule_format]: |
24353 | 307 |
"ALL y::int. (x AND y) + (x OR y) = x + y" |
24333 | 308 |
apply (induct x rule: bin_induct) |
309 |
apply clarsimp |
|
310 |
apply clarsimp |
|
311 |
apply clarsimp |
|
312 |
apply (case_tac y rule: bin_exhaust) |
|
313 |
apply clarsimp |
|
314 |
apply (unfold Bit_def) |
|
315 |
apply clarsimp |
|
316 |
apply (erule_tac x = "x" in allE) |
|
37667 | 317 |
apply (simp add: bitval_def split: bit.split) |
24333 | 318 |
done |
319 |
||
320 |
lemma le_int_or: |
|
37667 | 321 |
"bin_sign (y::int) = Int.Pls ==> x <= x OR y" |
322 |
apply (induct y arbitrary: x rule: bin_induct) |
|
24333 | 323 |
apply clarsimp |
324 |
apply clarsimp |
|
325 |
apply (case_tac x rule: bin_exhaust) |
|
326 |
apply (case_tac b) |
|
327 |
apply (case_tac [!] bit) |
|
45847 | 328 |
apply (auto simp: less_eq_int_code BIT_simps) |
24333 | 329 |
done |
330 |
||
331 |
lemmas int_and_le = |
|
45475 | 332 |
xtr3 [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or] |
24333 | 333 |
|
24364 | 334 |
(* interaction between bit-wise and arithmetic *) |
335 |
(* good example of bin_induction *) |
|
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336 |
lemma bin_add_not: "x + NOT x = Int.Min" |
24364 | 337 |
apply (induct x rule: bin_induct) |
338 |
apply clarsimp |
|
339 |
apply clarsimp |
|
45847 | 340 |
apply (case_tac bit, auto simp: BIT_simps) |
24364 | 341 |
done |
342 |
||
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343 |
subsubsection {* Truncating results of bit-wise operations *} |
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344 |
|
24364 | 345 |
lemma bin_trunc_ao: |
346 |
"!!x y. (bintrunc n x) AND (bintrunc n y) = bintrunc n (x AND y)" |
|
347 |
"!!x y. (bintrunc n x) OR (bintrunc n y) = bintrunc n (x OR y)" |
|
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348 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 349 |
|
350 |
lemma bin_trunc_xor: |
|
351 |
"!!x y. bintrunc n (bintrunc n x XOR bintrunc n y) = |
|
352 |
bintrunc n (x XOR y)" |
|
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353 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 354 |
|
355 |
lemma bin_trunc_not: |
|
356 |
"!!x. bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)" |
|
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|
357 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 358 |
|
359 |
(* want theorems of the form of bin_trunc_xor *) |
|
360 |
lemma bintr_bintr_i: |
|
361 |
"x = bintrunc n y ==> bintrunc n x = bintrunc n y" |
|
362 |
by auto |
|
363 |
||
364 |
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i] |
|
365 |
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i] |
|
366 |
||
367 |
subsection {* Setting and clearing bits *} |
|
368 |
||
26558 | 369 |
primrec |
24364 | 370 |
bin_sc :: "nat => bit => int => int" |
26558 | 371 |
where |
372 |
Z: "bin_sc 0 b w = bin_rest w BIT b" |
|
373 |
| Suc: "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w" |
|
24364 | 374 |
|
24333 | 375 |
(** nth bit, set/clear **) |
376 |
||
377 |
lemma bin_nth_sc [simp]: |
|
45955 | 378 |
"bin_nth (bin_sc n b w) n = (b = 1)" |
379 |
by (induct n arbitrary: w) auto |
|
24333 | 380 |
|
381 |
lemma bin_sc_sc_same [simp]: |
|
45955 | 382 |
"bin_sc n c (bin_sc n b w) = bin_sc n c w" |
383 |
by (induct n arbitrary: w) auto |
|
24333 | 384 |
|
385 |
lemma bin_sc_sc_diff: |
|
45955 | 386 |
"m ~= n ==> |
24333 | 387 |
bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)" |
45955 | 388 |
apply (induct n arbitrary: w m) |
24333 | 389 |
apply (case_tac [!] m) |
390 |
apply auto |
|
391 |
done |
|
392 |
||
393 |
lemma bin_nth_sc_gen: |
|
45955 | 394 |
"bin_nth (bin_sc n b w) m = (if m = n then b = 1 else bin_nth w m)" |
395 |
by (induct n arbitrary: w m) (case_tac [!] m, auto) |
|
24333 | 396 |
|
397 |
lemma bin_sc_nth [simp]: |
|
45955 | 398 |
"(bin_sc n (If (bin_nth w n) 1 0) w) = w" |
399 |
by (induct n arbitrary: w) auto |
|
24333 | 400 |
|
401 |
lemma bin_sign_sc [simp]: |
|
45955 | 402 |
"bin_sign (bin_sc n b w) = bin_sign w" |
403 |
by (induct n arbitrary: w) auto |
|
24333 | 404 |
|
405 |
lemma bin_sc_bintr [simp]: |
|
45955 | 406 |
"bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)" |
407 |
apply (induct n arbitrary: w m) |
|
24333 | 408 |
apply (case_tac [!] w rule: bin_exhaust) |
409 |
apply (case_tac [!] m, auto) |
|
410 |
done |
|
411 |
||
412 |
lemma bin_clr_le: |
|
45955 | 413 |
"bin_sc n 0 w <= w" |
414 |
apply (induct n arbitrary: w) |
|
24333 | 415 |
apply (case_tac [!] w rule: bin_exhaust) |
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416 |
apply (auto simp del: BIT_simps) |
24333 | 417 |
apply (unfold Bit_def) |
37667 | 418 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 419 |
done |
420 |
||
421 |
lemma bin_set_ge: |
|
45955 | 422 |
"bin_sc n 1 w >= w" |
423 |
apply (induct n arbitrary: w) |
|
24333 | 424 |
apply (case_tac [!] w rule: bin_exhaust) |
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425 |
apply (auto simp del: BIT_simps) |
24333 | 426 |
apply (unfold Bit_def) |
37667 | 427 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 428 |
done |
429 |
||
430 |
lemma bintr_bin_clr_le: |
|
45955 | 431 |
"bintrunc n (bin_sc m 0 w) <= bintrunc n w" |
432 |
apply (induct n arbitrary: w m) |
|
24333 | 433 |
apply simp |
434 |
apply (case_tac w rule: bin_exhaust) |
|
435 |
apply (case_tac m) |
|
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|
436 |
apply (auto simp del: BIT_simps) |
24333 | 437 |
apply (unfold Bit_def) |
37667 | 438 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 439 |
done |
440 |
||
441 |
lemma bintr_bin_set_ge: |
|
45955 | 442 |
"bintrunc n (bin_sc m 1 w) >= bintrunc n w" |
443 |
apply (induct n arbitrary: w m) |
|
24333 | 444 |
apply simp |
445 |
apply (case_tac w rule: bin_exhaust) |
|
446 |
apply (case_tac m) |
|
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|
447 |
apply (auto simp del: BIT_simps) |
24333 | 448 |
apply (unfold Bit_def) |
37667 | 449 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 450 |
done |
451 |
||
37654 | 452 |
lemma bin_sc_FP [simp]: "bin_sc n 0 Int.Pls = Int.Pls" |
45847 | 453 |
by (induct n) (auto simp: BIT_simps) |
24333 | 454 |
|
37654 | 455 |
lemma bin_sc_TM [simp]: "bin_sc n 1 Int.Min = Int.Min" |
45847 | 456 |
by (induct n) (auto simp: BIT_simps) |
24333 | 457 |
|
458 |
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP |
|
459 |
||
460 |
lemma bin_sc_minus: |
|
461 |
"0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w" |
|
462 |
by auto |
|
463 |
||
464 |
lemmas bin_sc_Suc_minus = |
|
45604 | 465 |
trans [OF bin_sc_minus [symmetric] bin_sc.Suc] |
24333 | 466 |
|
467 |
lemmas bin_sc_Suc_pred [simp] = |
|
45604 | 468 |
bin_sc_Suc_minus [of "number_of bin", simplified nobm1] for bin |
24333 | 469 |
|
24465 | 470 |
|
24364 | 471 |
subsection {* Splitting and concatenation *} |
24333 | 472 |
|
26558 | 473 |
definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int" where |
474 |
"bin_rcat n = foldl (%u v. bin_cat u n v) Int.Pls" |
|
475 |
||
37667 | 476 |
lemma [code]: |
477 |
"bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0" |
|
478 |
by (simp add: bin_rcat_def Pls_def) |
|
479 |
||
28042 | 480 |
fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 481 |
"bin_rsplit_aux n m c bs = |
24364 | 482 |
(if m = 0 | n = 0 then bs else |
483 |
let (a, b) = bin_split n c |
|
26558 | 484 |
in bin_rsplit_aux n (m - n) a (b # bs))" |
24364 | 485 |
|
26558 | 486 |
definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
487 |
"bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []" |
|
488 |
||
28042 | 489 |
fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 490 |
"bin_rsplitl_aux n m c bs = |
24364 | 491 |
(if m = 0 | n = 0 then bs else |
492 |
let (a, b) = bin_split (min m n) c |
|
26558 | 493 |
in bin_rsplitl_aux n (m - n) a (b # bs))" |
24364 | 494 |
|
26558 | 495 |
definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
496 |
"bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []" |
|
497 |
||
24364 | 498 |
declare bin_rsplit_aux.simps [simp del] |
499 |
declare bin_rsplitl_aux.simps [simp del] |
|
500 |
||
501 |
lemma bin_sign_cat: |
|
45955 | 502 |
"bin_sign (bin_cat x n y) = bin_sign x" |
503 |
by (induct n arbitrary: y) auto |
|
24364 | 504 |
|
505 |
lemma bin_cat_Suc_Bit: |
|
506 |
"bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b" |
|
507 |
by auto |
|
508 |
||
509 |
lemma bin_nth_cat: |
|
45955 | 510 |
"bin_nth (bin_cat x k y) n = |
24364 | 511 |
(if n < k then bin_nth y n else bin_nth x (n - k))" |
45955 | 512 |
apply (induct k arbitrary: n y) |
24364 | 513 |
apply clarsimp |
514 |
apply (case_tac n, auto) |
|
24333 | 515 |
done |
516 |
||
24364 | 517 |
lemma bin_nth_split: |
45955 | 518 |
"bin_split n c = (a, b) ==> |
24364 | 519 |
(ALL k. bin_nth a k = bin_nth c (n + k)) & |
520 |
(ALL k. bin_nth b k = (k < n & bin_nth c k))" |
|
45955 | 521 |
apply (induct n arbitrary: b c) |
24364 | 522 |
apply clarsimp |
523 |
apply (clarsimp simp: Let_def split: ls_splits) |
|
524 |
apply (case_tac k) |
|
525 |
apply auto |
|
526 |
done |
|
527 |
||
528 |
lemma bin_cat_assoc: |
|
45955 | 529 |
"bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)" |
530 |
by (induct n arbitrary: z) auto |
|
24364 | 531 |
|
45955 | 532 |
lemma bin_cat_assoc_sym: |
533 |
"bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z" |
|
534 |
apply (induct n arbitrary: z m, clarsimp) |
|
24364 | 535 |
apply (case_tac m, auto) |
24333 | 536 |
done |
537 |
||
45955 | 538 |
lemma bin_cat_Pls [simp]: "bin_cat Int.Pls n w = bintrunc n w" |
539 |
by (induct n arbitrary: w) auto |
|
24364 | 540 |
|
541 |
lemma bintr_cat1: |
|
45955 | 542 |
"bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b" |
543 |
by (induct n arbitrary: b) auto |
|
24364 | 544 |
|
545 |
lemma bintr_cat: "bintrunc m (bin_cat a n b) = |
|
546 |
bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)" |
|
547 |
by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr) |
|
548 |
||
549 |
lemma bintr_cat_same [simp]: |
|
550 |
"bintrunc n (bin_cat a n b) = bintrunc n b" |
|
551 |
by (auto simp add : bintr_cat) |
|
552 |
||
553 |
lemma cat_bintr [simp]: |
|
45955 | 554 |
"bin_cat a n (bintrunc n b) = bin_cat a n b" |
555 |
by (induct n arbitrary: b) auto |
|
24364 | 556 |
|
557 |
lemma split_bintrunc: |
|
45955 | 558 |
"bin_split n c = (a, b) ==> b = bintrunc n c" |
559 |
by (induct n arbitrary: b c) (auto simp: Let_def split: ls_splits) |
|
24364 | 560 |
|
561 |
lemma bin_cat_split: |
|
45955 | 562 |
"bin_split n w = (u, v) ==> w = bin_cat u n v" |
563 |
by (induct n arbitrary: v w) (auto simp: Let_def split: ls_splits) |
|
24364 | 564 |
|
565 |
lemma bin_split_cat: |
|
45955 | 566 |
"bin_split n (bin_cat v n w) = (v, bintrunc n w)" |
567 |
by (induct n arbitrary: w) auto |
|
24364 | 568 |
|
569 |
lemma bin_split_Pls [simp]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
25762
diff
changeset
|
570 |
"bin_split n Int.Pls = (Int.Pls, Int.Pls)" |
45847 | 571 |
by (induct n) (auto simp: Let_def BIT_simps split: ls_splits) |
24364 | 572 |
|
573 |
lemma bin_split_Min [simp]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
25762
diff
changeset
|
574 |
"bin_split n Int.Min = (Int.Min, bintrunc n Int.Min)" |
24364 | 575 |
by (induct n) (auto simp: Let_def split: ls_splits) |
576 |
||
577 |
lemma bin_split_trunc: |
|
45955 | 578 |
"bin_split (min m n) c = (a, b) ==> |
24364 | 579 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)" |
45955 | 580 |
apply (induct n arbitrary: m b c, clarsimp) |
24364 | 581 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
582 |
apply (case_tac m) |
|
45847 | 583 |
apply (auto simp: Let_def BIT_simps split: ls_splits) |
24333 | 584 |
done |
585 |
||
24364 | 586 |
lemma bin_split_trunc1: |
45955 | 587 |
"bin_split n c = (a, b) ==> |
24364 | 588 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)" |
45955 | 589 |
apply (induct n arbitrary: m b c, clarsimp) |
24364 | 590 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
591 |
apply (case_tac m) |
|
45847 | 592 |
apply (auto simp: Let_def BIT_simps split: ls_splits) |
24364 | 593 |
done |
24333 | 594 |
|
24364 | 595 |
lemma bin_cat_num: |
45955 | 596 |
"bin_cat a n b = a * 2 ^ n + bintrunc n b" |
597 |
apply (induct n arbitrary: b, clarsimp) |
|
24364 | 598 |
apply (simp add: Bit_def cong: number_of_False_cong) |
599 |
done |
|
600 |
||
601 |
lemma bin_split_num: |
|
45955 | 602 |
"bin_split n b = (b div 2 ^ n, b mod 2 ^ n)" |
603 |
apply (induct n arbitrary: b, simp add: Pls_def) |
|
45529
0e1037d4e049
remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents:
45475
diff
changeset
|
604 |
apply (simp add: bin_rest_def zdiv_zmult2_eq) |
24364 | 605 |
apply (case_tac b rule: bin_exhaust) |
606 |
apply simp |
|
37667 | 607 |
apply (simp add: Bit_def mod_mult_mult1 p1mod22k bitval_def |
45955 | 608 |
split: bit.split) |
609 |
done |
|
24364 | 610 |
|
611 |
subsection {* Miscellaneous lemmas *} |
|
24333 | 612 |
|
613 |
lemma nth_2p_bin: |
|
45955 | 614 |
"bin_nth (2 ^ n) m = (m = n)" |
615 |
apply (induct n arbitrary: m) |
|
24333 | 616 |
apply clarsimp |
617 |
apply safe |
|
618 |
apply (case_tac m) |
|
619 |
apply (auto simp: trans [OF numeral_1_eq_1 [symmetric] number_of_eq]) |
|
620 |
apply (case_tac m) |
|
621 |
apply (auto simp: Bit_B0_2t [symmetric]) |
|
622 |
done |
|
623 |
||
624 |
(* for use when simplifying with bin_nth_Bit *) |
|
625 |
||
626 |
lemma ex_eq_or: |
|
627 |
"(EX m. n = Suc m & (m = k | P m)) = (n = Suc k | (EX m. n = Suc m & P m))" |
|
628 |
by auto |
|
629 |
||
630 |
end |
|
631 |