author | blanchet |
Tue, 05 Oct 2010 11:45:10 +0200 | |
changeset 39953 | aa54f347e5e2 |
parent 37667 | 41acc0fa6b6c |
child 44939 | 5930d35c976d |
permissions | -rw-r--r-- |
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(* |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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definition 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, simp add: bin_rec_Bit1) |
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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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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] by (simp_all add: bin_rec_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)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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unfolding BIT_simps [symmetric] int_xor_Bits by simp_all |
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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lemma int_xor_x_simps': |
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"w XOR (Int.Pls BIT 0) = w" |
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"w XOR (Int.Min BIT 1) = NOT w" |
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apply (induct w rule: bin_induct) |
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apply simp_all[4] |
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apply (unfold int_xor_Bits) |
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apply clarsimp+ |
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done |
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lemma int_xor_extra_simps [simp]: |
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"w XOR Int.Pls = w" |
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"w XOR Int.Min = NOT w" |
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using int_xor_x_simps' 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] by (simp add: bin_rec_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)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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parents:
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"(Int.Bit1 x) OR (Int.Bit0 y) = Int.Bit1 (x OR y)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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"(Int.Bit1 x) OR (Int.Bit1 y) = Int.Bit1 (x OR y)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
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changeset
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unfolding BIT_simps [symmetric] int_or_Bits by simp_all |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
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changeset
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lemma int_or_x_simps': |
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"w OR (Int.Pls BIT 0) = w" |
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"w OR (Int.Min BIT 1) = Int.Min" |
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apply (induct w rule: bin_induct) |
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apply simp_all[4] |
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apply (unfold int_or_Bits) |
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apply clarsimp+ |
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done |
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lemma int_or_extra_simps [simp]: |
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"w OR Int.Pls = w" |
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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"w OR Int.Min = Int.Min" |
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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using int_or_x_simps' 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 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] by (simp add: bin_rec_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)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
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"(Int.Bit0 x) AND (Int.Bit1 y) = Int.Bit0 (x AND y)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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"(Int.Bit1 x) AND (Int.Bit0 y) = Int.Bit0 (x AND y)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
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"(Int.Bit1 x) AND (Int.Bit1 y) = Int.Bit1 (x AND y)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
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unfolding BIT_simps [symmetric] int_and_Bits by simp_all |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
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lemma int_and_x_simps': |
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"w AND (Int.Pls BIT 0) = Int.Pls" |
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"w AND (Int.Min BIT 1) = w" |
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apply (induct w rule: bin_induct) |
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apply simp_all[4] |
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apply (unfold int_and_Bits) |
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apply clarsimp+ |
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done |
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lemma int_and_extra_simps [simp]: |
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parents:
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"w AND Int.Pls = Int.Pls" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
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changeset
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"w AND Int.Min = w" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
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changeset
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using int_and_x_simps' by simp_all |
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(* commutativity of the above *) |
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lemma bin_ops_comm: |
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shows |
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int_and_comm: "!!y::int. x AND y = y AND x" and |
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int_or_comm: "!!y::int. x OR y = y OR x" and |
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int_xor_comm: "!!y::int. x XOR y = y XOR x" |
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apply (induct x rule: bin_induct) |
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apply simp_all[6] |
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apply (case_tac y rule: bin_exhaust, simp add: bit_ops_comm)+ |
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done |
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lemma bin_ops_same [simp]: |
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"(x::int) AND x = x" |
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"(x::int) OR x = x" |
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"(x::int) XOR x = Int.Pls" |
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by (induct x rule: bin_induct) auto |
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lemma int_not_not [simp]: "NOT (NOT (x::int)) = x" |
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by (induct x rule: bin_induct) auto |
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lemmas bin_log_esimps = |
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int_and_extra_simps int_or_extra_simps int_xor_extra_simps |
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int_and_Pls int_and_Min int_or_Pls int_or_Min int_xor_Pls int_xor_Min |
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(* basic properties of logical (bit-wise) operations *) |
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lemma bbw_ao_absorb: |
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"!!y::int. x AND (y OR x) = x & x OR (y AND x) = x" |
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apply (induct x rule: bin_induct) |
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apply auto |
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apply (case_tac [!] y rule: bin_exhaust) |
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apply auto |
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apply (case_tac [!] bit) |
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apply auto |
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done |
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lemma bbw_ao_absorbs_other: |
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"x AND (x OR y) = x \<and> (y AND x) OR x = (x::int)" |
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"(y OR x) AND x = x \<and> x OR (x AND y) = (x::int)" |
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"(x OR y) AND x = x \<and> (x AND y) OR x = (x::int)" |
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apply (auto simp: bbw_ao_absorb int_or_comm) |
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apply (subst int_or_comm) |
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apply (simp add: bbw_ao_absorb) |
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apply (subst int_and_comm) |
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apply (subst int_or_comm) |
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apply (simp add: bbw_ao_absorb) |
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apply (subst int_and_comm) |
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apply (simp add: bbw_ao_absorb) |
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252 |
done |
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lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other |
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lemma int_xor_not: |
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"!!y::int. (NOT x) XOR y = NOT (x XOR y) & |
258 |
x XOR (NOT y) = NOT (x XOR y)" |
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apply (induct x rule: bin_induct) |
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apply auto |
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apply (case_tac y rule: bin_exhaust, auto, |
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case_tac b, auto)+ |
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done |
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lemma bbw_assocs': |
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"!!y z::int. (x AND y) AND z = x AND (y AND z) & |
267 |
(x OR y) OR z = x OR (y OR z) & |
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(x XOR y) XOR z = x XOR (y XOR z)" |
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apply (induct x rule: bin_induct) |
270 |
apply (auto simp: int_xor_not) |
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apply (case_tac [!] y rule: bin_exhaust) |
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apply (case_tac [!] z rule: bin_exhaust) |
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273 |
apply (case_tac [!] bit) |
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apply (case_tac [!] b) |
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26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
275 |
apply (auto simp del: BIT_simps) |
24333 | 276 |
done |
277 |
||
278 |
lemma int_and_assoc: |
|
24353 | 279 |
"(x AND y) AND (z::int) = x AND (y AND z)" |
24333 | 280 |
by (simp add: bbw_assocs') |
281 |
||
282 |
lemma int_or_assoc: |
|
24353 | 283 |
"(x OR y) OR (z::int) = x OR (y OR z)" |
24333 | 284 |
by (simp add: bbw_assocs') |
285 |
||
286 |
lemma int_xor_assoc: |
|
24353 | 287 |
"(x XOR y) XOR (z::int) = x XOR (y XOR z)" |
24333 | 288 |
by (simp add: bbw_assocs') |
289 |
||
290 |
lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc |
|
291 |
||
292 |
lemma bbw_lcs [simp]: |
|
24353 | 293 |
"(y::int) AND (x AND z) = x AND (y AND z)" |
294 |
"(y::int) OR (x OR z) = x OR (y OR z)" |
|
295 |
"(y::int) XOR (x XOR z) = x XOR (y XOR z)" |
|
24333 | 296 |
apply (auto simp: bbw_assocs [symmetric]) |
297 |
apply (auto simp: bin_ops_comm) |
|
298 |
done |
|
299 |
||
300 |
lemma bbw_not_dist: |
|
24353 | 301 |
"!!y::int. NOT (x OR y) = (NOT x) AND (NOT y)" |
302 |
"!!y::int. NOT (x AND y) = (NOT x) OR (NOT y)" |
|
24333 | 303 |
apply (induct x rule: bin_induct) |
304 |
apply auto |
|
305 |
apply (case_tac [!] y rule: bin_exhaust) |
|
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|
306 |
apply (case_tac [!] bit, auto simp del: BIT_simps) |
24333 | 307 |
done |
308 |
||
309 |
lemma bbw_oa_dist: |
|
24353 | 310 |
"!!y z::int. (x AND y) OR z = |
311 |
(x OR z) AND (y OR z)" |
|
24333 | 312 |
apply (induct x rule: bin_induct) |
313 |
apply auto |
|
314 |
apply (case_tac y rule: bin_exhaust) |
|
315 |
apply (case_tac z rule: bin_exhaust) |
|
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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changeset
|
316 |
apply (case_tac ba, auto simp del: BIT_simps) |
24333 | 317 |
done |
318 |
||
319 |
lemma bbw_ao_dist: |
|
24353 | 320 |
"!!y z::int. (x OR y) AND z = |
321 |
(x AND z) OR (y AND z)" |
|
24333 | 322 |
apply (induct x rule: bin_induct) |
323 |
apply auto |
|
324 |
apply (case_tac y rule: bin_exhaust) |
|
325 |
apply (case_tac z rule: bin_exhaust) |
|
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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|
326 |
apply (case_tac ba, auto simp del: BIT_simps) |
24333 | 327 |
done |
328 |
||
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|
329 |
(* |
3e29eafabe16
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|
330 |
Why were these declared simp??? |
24333 | 331 |
declare bin_ops_comm [simp] bbw_assocs [simp] |
24367
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changeset
|
332 |
*) |
24333 | 333 |
|
334 |
lemma plus_and_or [rule_format]: |
|
24353 | 335 |
"ALL y::int. (x AND y) + (x OR y) = x + y" |
24333 | 336 |
apply (induct x rule: bin_induct) |
337 |
apply clarsimp |
|
338 |
apply clarsimp |
|
339 |
apply clarsimp |
|
340 |
apply (case_tac y rule: bin_exhaust) |
|
341 |
apply clarsimp |
|
342 |
apply (unfold Bit_def) |
|
343 |
apply clarsimp |
|
344 |
apply (erule_tac x = "x" in allE) |
|
37667 | 345 |
apply (simp add: bitval_def split: bit.split) |
24333 | 346 |
done |
347 |
||
348 |
lemma le_int_or: |
|
37667 | 349 |
"bin_sign (y::int) = Int.Pls ==> x <= x OR y" |
350 |
apply (induct y arbitrary: x rule: bin_induct) |
|
24333 | 351 |
apply clarsimp |
352 |
apply clarsimp |
|
353 |
apply (case_tac x rule: bin_exhaust) |
|
354 |
apply (case_tac b) |
|
355 |
apply (case_tac [!] bit) |
|
26514 | 356 |
apply (auto simp: less_eq_int_code) |
24333 | 357 |
done |
358 |
||
359 |
lemmas int_and_le = |
|
360 |
xtr3 [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or] ; |
|
361 |
||
24364 | 362 |
lemma bin_nth_ops: |
363 |
"!!x y. bin_nth (x AND y) n = (bin_nth x n & bin_nth y n)" |
|
364 |
"!!x y. bin_nth (x OR y) n = (bin_nth x n | bin_nth y n)" |
|
365 |
"!!x y. bin_nth (x XOR y) n = (bin_nth x n ~= bin_nth y n)" |
|
366 |
"!!x. bin_nth (NOT x) n = (~ bin_nth x n)" |
|
367 |
apply (induct n) |
|
368 |
apply safe |
|
369 |
apply (case_tac [!] x rule: bin_exhaust) |
|
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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changeset
|
370 |
apply (simp_all del: BIT_simps) |
24364 | 371 |
apply (case_tac [!] y rule: bin_exhaust) |
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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changeset
|
372 |
apply (simp_all del: BIT_simps) |
24364 | 373 |
apply (auto dest: not_B1_is_B0 intro: B1_ass_B0) |
374 |
done |
|
375 |
||
376 |
(* interaction between bit-wise and arithmetic *) |
|
377 |
(* good example of bin_induction *) |
|
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changeset
|
378 |
lemma bin_add_not: "x + NOT x = Int.Min" |
24364 | 379 |
apply (induct x rule: bin_induct) |
380 |
apply clarsimp |
|
381 |
apply clarsimp |
|
382 |
apply (case_tac bit, auto) |
|
383 |
done |
|
384 |
||
385 |
(* truncating results of bit-wise operations *) |
|
386 |
lemma bin_trunc_ao: |
|
387 |
"!!x y. (bintrunc n x) AND (bintrunc n y) = bintrunc n (x AND y)" |
|
388 |
"!!x y. (bintrunc n x) OR (bintrunc n y) = bintrunc n (x OR y)" |
|
389 |
apply (induct n) |
|
390 |
apply auto |
|
391 |
apply (case_tac [!] x rule: bin_exhaust) |
|
392 |
apply (case_tac [!] y rule: bin_exhaust) |
|
393 |
apply auto |
|
394 |
done |
|
395 |
||
396 |
lemma bin_trunc_xor: |
|
397 |
"!!x y. bintrunc n (bintrunc n x XOR bintrunc n y) = |
|
398 |
bintrunc n (x XOR y)" |
|
399 |
apply (induct n) |
|
400 |
apply auto |
|
401 |
apply (case_tac [!] x rule: bin_exhaust) |
|
402 |
apply (case_tac [!] y rule: bin_exhaust) |
|
403 |
apply auto |
|
404 |
done |
|
405 |
||
406 |
lemma bin_trunc_not: |
|
407 |
"!!x. bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)" |
|
408 |
apply (induct n) |
|
409 |
apply auto |
|
410 |
apply (case_tac [!] x rule: bin_exhaust) |
|
411 |
apply auto |
|
412 |
done |
|
413 |
||
414 |
(* want theorems of the form of bin_trunc_xor *) |
|
415 |
lemma bintr_bintr_i: |
|
416 |
"x = bintrunc n y ==> bintrunc n x = bintrunc n y" |
|
417 |
by auto |
|
418 |
||
419 |
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i] |
|
420 |
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i] |
|
421 |
||
422 |
subsection {* Setting and clearing bits *} |
|
423 |
||
26558 | 424 |
primrec |
24364 | 425 |
bin_sc :: "nat => bit => int => int" |
26558 | 426 |
where |
427 |
Z: "bin_sc 0 b w = bin_rest w BIT b" |
|
428 |
| Suc: "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w" |
|
24364 | 429 |
|
24333 | 430 |
(** nth bit, set/clear **) |
431 |
||
432 |
lemma bin_nth_sc [simp]: |
|
37654 | 433 |
"!!w. bin_nth (bin_sc n b w) n = (b = 1)" |
24333 | 434 |
by (induct n) auto |
435 |
||
436 |
lemma bin_sc_sc_same [simp]: |
|
437 |
"!!w. bin_sc n c (bin_sc n b w) = bin_sc n c w" |
|
438 |
by (induct n) auto |
|
439 |
||
440 |
lemma bin_sc_sc_diff: |
|
441 |
"!!w m. m ~= n ==> |
|
442 |
bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)" |
|
443 |
apply (induct n) |
|
444 |
apply (case_tac [!] m) |
|
445 |
apply auto |
|
446 |
done |
|
447 |
||
448 |
lemma bin_nth_sc_gen: |
|
37654 | 449 |
"!!w m. bin_nth (bin_sc n b w) m = (if m = n then b = 1 else bin_nth w m)" |
24333 | 450 |
by (induct n) (case_tac [!] m, auto) |
451 |
||
452 |
lemma bin_sc_nth [simp]: |
|
37654 | 453 |
"!!w. (bin_sc n (If (bin_nth w n) 1 0) w) = w" |
24465 | 454 |
by (induct n) auto |
24333 | 455 |
|
456 |
lemma bin_sign_sc [simp]: |
|
457 |
"!!w. bin_sign (bin_sc n b w) = bin_sign w" |
|
458 |
by (induct n) auto |
|
459 |
||
460 |
lemma bin_sc_bintr [simp]: |
|
461 |
"!!w m. bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)" |
|
462 |
apply (induct n) |
|
463 |
apply (case_tac [!] w rule: bin_exhaust) |
|
464 |
apply (case_tac [!] m, auto) |
|
465 |
done |
|
466 |
||
467 |
lemma bin_clr_le: |
|
37654 | 468 |
"!!w. bin_sc n 0 w <= w" |
24333 | 469 |
apply (induct n) |
470 |
apply (case_tac [!] w rule: bin_exhaust) |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
471 |
apply (auto simp del: BIT_simps) |
24333 | 472 |
apply (unfold Bit_def) |
37667 | 473 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 474 |
done |
475 |
||
476 |
lemma bin_set_ge: |
|
37654 | 477 |
"!!w. bin_sc n 1 w >= w" |
24333 | 478 |
apply (induct n) |
479 |
apply (case_tac [!] w rule: bin_exhaust) |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
480 |
apply (auto simp del: BIT_simps) |
24333 | 481 |
apply (unfold Bit_def) |
37667 | 482 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 483 |
done |
484 |
||
485 |
lemma bintr_bin_clr_le: |
|
37654 | 486 |
"!!w m. bintrunc n (bin_sc m 0 w) <= bintrunc n w" |
24333 | 487 |
apply (induct n) |
488 |
apply simp |
|
489 |
apply (case_tac w rule: bin_exhaust) |
|
490 |
apply (case_tac m) |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
491 |
apply (auto simp del: BIT_simps) |
24333 | 492 |
apply (unfold Bit_def) |
37667 | 493 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 494 |
done |
495 |
||
496 |
lemma bintr_bin_set_ge: |
|
37654 | 497 |
"!!w m. bintrunc n (bin_sc m 1 w) >= bintrunc n w" |
24333 | 498 |
apply (induct n) |
499 |
apply simp |
|
500 |
apply (case_tac w rule: bin_exhaust) |
|
501 |
apply (case_tac m) |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
502 |
apply (auto simp del: BIT_simps) |
24333 | 503 |
apply (unfold Bit_def) |
37667 | 504 |
apply (simp_all add: bitval_def split: bit.split) |
24333 | 505 |
done |
506 |
||
37654 | 507 |
lemma bin_sc_FP [simp]: "bin_sc n 0 Int.Pls = Int.Pls" |
24333 | 508 |
by (induct n) auto |
509 |
||
37654 | 510 |
lemma bin_sc_TM [simp]: "bin_sc n 1 Int.Min = Int.Min" |
24333 | 511 |
by (induct n) auto |
512 |
||
513 |
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP |
|
514 |
||
515 |
lemma bin_sc_minus: |
|
516 |
"0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w" |
|
517 |
by auto |
|
518 |
||
519 |
lemmas bin_sc_Suc_minus = |
|
520 |
trans [OF bin_sc_minus [symmetric] bin_sc.Suc, standard] |
|
521 |
||
522 |
lemmas bin_sc_Suc_pred [simp] = |
|
523 |
bin_sc_Suc_minus [of "number_of bin", simplified nobm1, standard] |
|
524 |
||
24465 | 525 |
|
24364 | 526 |
subsection {* Splitting and concatenation *} |
24333 | 527 |
|
26558 | 528 |
definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int" where |
529 |
"bin_rcat n = foldl (%u v. bin_cat u n v) Int.Pls" |
|
530 |
||
37667 | 531 |
lemma [code]: |
532 |
"bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0" |
|
533 |
by (simp add: bin_rcat_def Pls_def) |
|
534 |
||
28042 | 535 |
fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 536 |
"bin_rsplit_aux n m c bs = |
24364 | 537 |
(if m = 0 | n = 0 then bs else |
538 |
let (a, b) = bin_split n c |
|
26558 | 539 |
in bin_rsplit_aux n (m - n) a (b # bs))" |
24364 | 540 |
|
26558 | 541 |
definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
542 |
"bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []" |
|
543 |
||
28042 | 544 |
fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 545 |
"bin_rsplitl_aux n m c bs = |
24364 | 546 |
(if m = 0 | n = 0 then bs else |
547 |
let (a, b) = bin_split (min m n) c |
|
26558 | 548 |
in bin_rsplitl_aux n (m - n) a (b # bs))" |
24364 | 549 |
|
26558 | 550 |
definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
551 |
"bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []" |
|
552 |
||
24364 | 553 |
declare bin_rsplit_aux.simps [simp del] |
554 |
declare bin_rsplitl_aux.simps [simp del] |
|
555 |
||
556 |
lemma bin_sign_cat: |
|
557 |
"!!y. bin_sign (bin_cat x n y) = bin_sign x" |
|
558 |
by (induct n) auto |
|
559 |
||
560 |
lemma bin_cat_Suc_Bit: |
|
561 |
"bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b" |
|
562 |
by auto |
|
563 |
||
564 |
lemma bin_nth_cat: |
|
565 |
"!!n y. bin_nth (bin_cat x k y) n = |
|
566 |
(if n < k then bin_nth y n else bin_nth x (n - k))" |
|
567 |
apply (induct k) |
|
568 |
apply clarsimp |
|
569 |
apply (case_tac n, auto) |
|
24333 | 570 |
done |
571 |
||
24364 | 572 |
lemma bin_nth_split: |
573 |
"!!b c. bin_split n c = (a, b) ==> |
|
574 |
(ALL k. bin_nth a k = bin_nth c (n + k)) & |
|
575 |
(ALL k. bin_nth b k = (k < n & bin_nth c k))" |
|
24333 | 576 |
apply (induct n) |
24364 | 577 |
apply clarsimp |
578 |
apply (clarsimp simp: Let_def split: ls_splits) |
|
579 |
apply (case_tac k) |
|
580 |
apply auto |
|
581 |
done |
|
582 |
||
583 |
lemma bin_cat_assoc: |
|
584 |
"!!z. bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)" |
|
585 |
by (induct n) auto |
|
586 |
||
587 |
lemma bin_cat_assoc_sym: "!!z m. |
|
588 |
bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z" |
|
589 |
apply (induct n, clarsimp) |
|
590 |
apply (case_tac m, auto) |
|
24333 | 591 |
done |
592 |
||
24364 | 593 |
lemma bin_cat_Pls [simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
25762
diff
changeset
|
594 |
"!!w. bin_cat Int.Pls n w = bintrunc n w" |
24364 | 595 |
by (induct n) auto |
596 |
||
597 |
lemma bintr_cat1: |
|
598 |
"!!b. bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b" |
|
599 |
by (induct n) auto |
|
600 |
||
601 |
lemma bintr_cat: "bintrunc m (bin_cat a n b) = |
|
602 |
bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)" |
|
603 |
by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr) |
|
604 |
||
605 |
lemma bintr_cat_same [simp]: |
|
606 |
"bintrunc n (bin_cat a n b) = bintrunc n b" |
|
607 |
by (auto simp add : bintr_cat) |
|
608 |
||
609 |
lemma cat_bintr [simp]: |
|
610 |
"!!b. bin_cat a n (bintrunc n b) = bin_cat a n b" |
|
611 |
by (induct n) auto |
|
612 |
||
613 |
lemma split_bintrunc: |
|
614 |
"!!b c. bin_split n c = (a, b) ==> b = bintrunc n c" |
|
615 |
by (induct n) (auto simp: Let_def split: ls_splits) |
|
616 |
||
617 |
lemma bin_cat_split: |
|
618 |
"!!v w. bin_split n w = (u, v) ==> w = bin_cat u n v" |
|
619 |
by (induct n) (auto simp: Let_def split: ls_splits) |
|
620 |
||
621 |
lemma bin_split_cat: |
|
622 |
"!!w. bin_split n (bin_cat v n w) = (v, bintrunc n w)" |
|
623 |
by (induct n) auto |
|
624 |
||
625 |
lemma bin_split_Pls [simp]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
25762
diff
changeset
|
626 |
"bin_split n Int.Pls = (Int.Pls, Int.Pls)" |
24364 | 627 |
by (induct n) (auto simp: Let_def split: ls_splits) |
628 |
||
629 |
lemma bin_split_Min [simp]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
25762
diff
changeset
|
630 |
"bin_split n Int.Min = (Int.Min, bintrunc n Int.Min)" |
24364 | 631 |
by (induct n) (auto simp: Let_def split: ls_splits) |
632 |
||
633 |
lemma bin_split_trunc: |
|
634 |
"!!m b c. bin_split (min m n) c = (a, b) ==> |
|
635 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)" |
|
636 |
apply (induct n, clarsimp) |
|
637 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
|
638 |
apply (case_tac m) |
|
639 |
apply (auto simp: Let_def split: ls_splits) |
|
24333 | 640 |
done |
641 |
||
24364 | 642 |
lemma bin_split_trunc1: |
643 |
"!!m b c. bin_split n c = (a, b) ==> |
|
644 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)" |
|
645 |
apply (induct n, clarsimp) |
|
646 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
|
647 |
apply (case_tac m) |
|
648 |
apply (auto simp: Let_def split: ls_splits) |
|
649 |
done |
|
24333 | 650 |
|
24364 | 651 |
lemma bin_cat_num: |
652 |
"!!b. bin_cat a n b = a * 2 ^ n + bintrunc n b" |
|
653 |
apply (induct n, clarsimp) |
|
654 |
apply (simp add: Bit_def cong: number_of_False_cong) |
|
655 |
done |
|
656 |
||
657 |
lemma bin_split_num: |
|
658 |
"!!b. bin_split n b = (b div 2 ^ n, b mod 2 ^ n)" |
|
659 |
apply (induct n, clarsimp) |
|
660 |
apply (simp add: bin_rest_div zdiv_zmult2_eq) |
|
661 |
apply (case_tac b rule: bin_exhaust) |
|
662 |
apply simp |
|
37667 | 663 |
apply (simp add: Bit_def mod_mult_mult1 p1mod22k bitval_def |
24364 | 664 |
split: bit.split |
665 |
cong: number_of_False_cong) |
|
666 |
done |
|
667 |
||
668 |
subsection {* Miscellaneous lemmas *} |
|
24333 | 669 |
|
670 |
lemma nth_2p_bin: |
|
671 |
"!!m. bin_nth (2 ^ n) m = (m = n)" |
|
672 |
apply (induct n) |
|
673 |
apply clarsimp |
|
674 |
apply safe |
|
675 |
apply (case_tac m) |
|
676 |
apply (auto simp: trans [OF numeral_1_eq_1 [symmetric] number_of_eq]) |
|
677 |
apply (case_tac m) |
|
678 |
apply (auto simp: Bit_B0_2t [symmetric]) |
|
679 |
done |
|
680 |
||
681 |
(* for use when simplifying with bin_nth_Bit *) |
|
682 |
||
683 |
lemma ex_eq_or: |
|
684 |
"(EX m. n = Suc m & (m = k | P m)) = (n = Suc k | (EX m. n = Suc m & P m))" |
|
685 |
by auto |
|
686 |
||
687 |
end |
|
688 |