--- a/NEWS Fri Jun 14 08:34:27 2019 +0000
+++ b/NEWS Fri Jun 14 08:34:27 2019 +0000
@@ -14,6 +14,9 @@
* ASCII membership syntax concerning big operators for infimum
and supremum is gone. INCOMPATIBILITY.
+* Clear distinction between types for bits (False / True) and
+Z2 (0 / 1): theory HOL/Library/Bit.thy has been renamed accordingly.
+INCOMPATIBILITY.
New in Isabelle2019 (June 2019)
--- a/src/HOL/Library/Bit.thy Fri Jun 14 08:34:27 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,174 +0,0 @@
-(* Title: HOL/Library/Bit.thy
- Author: Brian Huffman
-*)
-
-section \<open>The Field of Integers mod 2\<close>
-
-theory Bit
-imports Main
-begin
-
-subsection \<open>Bits as a datatype\<close>
-
-typedef bit = "UNIV :: bool set"
- morphisms set Bit ..
-
-instantiation bit :: "{zero, one}"
-begin
-
-definition zero_bit_def: "0 = Bit False"
-
-definition one_bit_def: "1 = Bit True"
-
-instance ..
-
-end
-
-old_rep_datatype "0::bit" "1::bit"
-proof -
- fix P :: "bit \<Rightarrow> bool"
- fix x :: bit
- assume "P 0" and "P 1"
- then have "\<forall>b. P (Bit b)"
- unfolding zero_bit_def one_bit_def
- by (simp add: all_bool_eq)
- then show "P x"
- by (induct x) simp
-next
- show "(0::bit) \<noteq> (1::bit)"
- unfolding zero_bit_def one_bit_def
- by (simp add: Bit_inject)
-qed
-
-lemma Bit_set_eq [simp]: "Bit (set b) = b"
- by (fact set_inverse)
-
-lemma set_Bit_eq [simp]: "set (Bit P) = P"
- by (rule Bit_inverse) rule
-
-lemma bit_eq_iff: "x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)"
- by (auto simp add: set_inject)
-
-lemma Bit_inject [simp]: "Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)"
- by (auto simp add: Bit_inject)
-
-lemma set [iff]:
- "\<not> set 0"
- "set 1"
- by (simp_all add: zero_bit_def one_bit_def Bit_inverse)
-
-lemma [code]:
- "set 0 \<longleftrightarrow> False"
- "set 1 \<longleftrightarrow> True"
- by simp_all
-
-lemma set_iff: "set b \<longleftrightarrow> b = 1"
- by (cases b) simp_all
-
-lemma bit_eq_iff_set:
- "b = 0 \<longleftrightarrow> \<not> set b"
- "b = 1 \<longleftrightarrow> set b"
- by (simp_all add: bit_eq_iff)
-
-lemma Bit [simp, code]:
- "Bit False = 0"
- "Bit True = 1"
- by (simp_all add: zero_bit_def one_bit_def)
-
-lemma bit_not_0_iff [iff]: "x \<noteq> 0 \<longleftrightarrow> x = 1" for x :: bit
- by (simp add: bit_eq_iff)
-
-lemma bit_not_1_iff [iff]: "x \<noteq> 1 \<longleftrightarrow> x = 0" for x :: bit
- by (simp add: bit_eq_iff)
-
-lemma [code]:
- "HOL.equal 0 b \<longleftrightarrow> \<not> set b"
- "HOL.equal 1 b \<longleftrightarrow> set b"
- by (simp_all add: equal set_iff)
-
-
-subsection \<open>Type \<^typ>\<open>bit\<close> forms a field\<close>
-
-instantiation bit :: field
-begin
-
-definition plus_bit_def: "x + y = case_bit y (case_bit 1 0 y) x"
-
-definition times_bit_def: "x * y = case_bit 0 y x"
-
-definition uminus_bit_def [simp]: "- x = x" for x :: bit
-
-definition minus_bit_def [simp]: "x - y = x + y" for x y :: bit
-
-definition inverse_bit_def [simp]: "inverse x = x" for x :: bit
-
-definition divide_bit_def [simp]: "x div y = x * y" for x y :: bit
-
-lemmas field_bit_defs =
- plus_bit_def times_bit_def minus_bit_def uminus_bit_def
- divide_bit_def inverse_bit_def
-
-instance
- by standard (auto simp: field_bit_defs split: bit.split)
-
-end
-
-lemma bit_add_self: "x + x = 0" for x :: bit
- unfolding plus_bit_def by (simp split: bit.split)
-
-lemma bit_mult_eq_1_iff [simp]: "x * y = 1 \<longleftrightarrow> x = 1 \<and> y = 1" for x y :: bit
- unfolding times_bit_def by (simp split: bit.split)
-
-text \<open>Not sure whether the next two should be simp rules.\<close>
-
-lemma bit_add_eq_0_iff: "x + y = 0 \<longleftrightarrow> x = y" for x y :: bit
- unfolding plus_bit_def by (simp split: bit.split)
-
-lemma bit_add_eq_1_iff: "x + y = 1 \<longleftrightarrow> x \<noteq> y" for x y :: bit
- unfolding plus_bit_def by (simp split: bit.split)
-
-
-subsection \<open>Numerals at type \<^typ>\<open>bit\<close>\<close>
-
-text \<open>All numerals reduce to either 0 or 1.\<close>
-
-lemma bit_minus1 [simp]: "- 1 = (1 :: bit)"
- by (simp only: uminus_bit_def)
-
-lemma bit_neg_numeral [simp]: "(- numeral w :: bit) = numeral w"
- by (simp only: uminus_bit_def)
-
-lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)"
- by (simp only: numeral_Bit0 bit_add_self)
-
-lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)"
- by (simp only: numeral_Bit1 bit_add_self add_0_left)
-
-
-subsection \<open>Conversion from \<^typ>\<open>bit\<close>\<close>
-
-context zero_neq_one
-begin
-
-definition of_bit :: "bit \<Rightarrow> 'a"
- where "of_bit b = case_bit 0 1 b"
-
-lemma of_bit_eq [simp, code]:
- "of_bit 0 = 0"
- "of_bit 1 = 1"
- by (simp_all add: of_bit_def)
-
-lemma of_bit_eq_iff: "of_bit x = of_bit y \<longleftrightarrow> x = y"
- by (cases x) (cases y; simp)+
-
-end
-
-lemma (in semiring_1) of_nat_of_bit_eq: "of_nat (of_bit b) = of_bit b"
- by (cases b) simp_all
-
-lemma (in ring_1) of_int_of_bit_eq: "of_int (of_bit b) = of_bit b"
- by (cases b) simp_all
-
-hide_const (open) set
-
-end
--- a/src/HOL/Library/Library.thy Fri Jun 14 08:34:27 2019 +0000
+++ b/src/HOL/Library/Library.thy Fri Jun 14 08:34:27 2019 +0000
@@ -4,7 +4,6 @@
AList
Adhoc_Overloading
BigO
- Bit
BNF_Axiomatization
BNF_Corec
Boolean_Algebra
@@ -94,6 +93,7 @@
Type_Length
Uprod
While_Combinator
+ Z2
begin
end
(*>*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Z2.thy Fri Jun 14 08:34:27 2019 +0000
@@ -0,0 +1,180 @@
+(* Title: HOL/Library/Z2.thy
+ Author: Brian Huffman
+*)
+
+section \<open>The Field of Integers mod 2\<close>
+
+theory Z2
+imports Main
+begin
+
+text \<open>
+ Note that in most cases \<^typ>\<open>bool\<close> is appropriate hen a binary type is needed; the
+ type provided here, for historical reasons named \<guillemotright>bit\<guillemotleft>, is only needed if proper
+ field operations are required.
+\<close>
+
+subsection \<open>Bits as a datatype\<close>
+
+typedef bit = "UNIV :: bool set"
+ morphisms set Bit ..
+
+instantiation bit :: "{zero, one}"
+begin
+
+definition zero_bit_def: "0 = Bit False"
+
+definition one_bit_def: "1 = Bit True"
+
+instance ..
+
+end
+
+old_rep_datatype "0::bit" "1::bit"
+proof -
+ fix P :: "bit \<Rightarrow> bool"
+ fix x :: bit
+ assume "P 0" and "P 1"
+ then have "\<forall>b. P (Bit b)"
+ unfolding zero_bit_def one_bit_def
+ by (simp add: all_bool_eq)
+ then show "P x"
+ by (induct x) simp
+next
+ show "(0::bit) \<noteq> (1::bit)"
+ unfolding zero_bit_def one_bit_def
+ by (simp add: Bit_inject)
+qed
+
+lemma Bit_set_eq [simp]: "Bit (set b) = b"
+ by (fact set_inverse)
+
+lemma set_Bit_eq [simp]: "set (Bit P) = P"
+ by (rule Bit_inverse) rule
+
+lemma bit_eq_iff: "x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)"
+ by (auto simp add: set_inject)
+
+lemma Bit_inject [simp]: "Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)"
+ by (auto simp add: Bit_inject)
+
+lemma set [iff]:
+ "\<not> set 0"
+ "set 1"
+ by (simp_all add: zero_bit_def one_bit_def Bit_inverse)
+
+lemma [code]:
+ "set 0 \<longleftrightarrow> False"
+ "set 1 \<longleftrightarrow> True"
+ by simp_all
+
+lemma set_iff: "set b \<longleftrightarrow> b = 1"
+ by (cases b) simp_all
+
+lemma bit_eq_iff_set:
+ "b = 0 \<longleftrightarrow> \<not> set b"
+ "b = 1 \<longleftrightarrow> set b"
+ by (simp_all add: bit_eq_iff)
+
+lemma Bit [simp, code]:
+ "Bit False = 0"
+ "Bit True = 1"
+ by (simp_all add: zero_bit_def one_bit_def)
+
+lemma bit_not_0_iff [iff]: "x \<noteq> 0 \<longleftrightarrow> x = 1" for x :: bit
+ by (simp add: bit_eq_iff)
+
+lemma bit_not_1_iff [iff]: "x \<noteq> 1 \<longleftrightarrow> x = 0" for x :: bit
+ by (simp add: bit_eq_iff)
+
+lemma [code]:
+ "HOL.equal 0 b \<longleftrightarrow> \<not> set b"
+ "HOL.equal 1 b \<longleftrightarrow> set b"
+ by (simp_all add: equal set_iff)
+
+
+subsection \<open>Type \<^typ>\<open>bit\<close> forms a field\<close>
+
+instantiation bit :: field
+begin
+
+definition plus_bit_def: "x + y = case_bit y (case_bit 1 0 y) x"
+
+definition times_bit_def: "x * y = case_bit 0 y x"
+
+definition uminus_bit_def [simp]: "- x = x" for x :: bit
+
+definition minus_bit_def [simp]: "x - y = x + y" for x y :: bit
+
+definition inverse_bit_def [simp]: "inverse x = x" for x :: bit
+
+definition divide_bit_def [simp]: "x div y = x * y" for x y :: bit
+
+lemmas field_bit_defs =
+ plus_bit_def times_bit_def minus_bit_def uminus_bit_def
+ divide_bit_def inverse_bit_def
+
+instance
+ by standard (auto simp: field_bit_defs split: bit.split)
+
+end
+
+lemma bit_add_self: "x + x = 0" for x :: bit
+ unfolding plus_bit_def by (simp split: bit.split)
+
+lemma bit_mult_eq_1_iff [simp]: "x * y = 1 \<longleftrightarrow> x = 1 \<and> y = 1" for x y :: bit
+ unfolding times_bit_def by (simp split: bit.split)
+
+text \<open>Not sure whether the next two should be simp rules.\<close>
+
+lemma bit_add_eq_0_iff: "x + y = 0 \<longleftrightarrow> x = y" for x y :: bit
+ unfolding plus_bit_def by (simp split: bit.split)
+
+lemma bit_add_eq_1_iff: "x + y = 1 \<longleftrightarrow> x \<noteq> y" for x y :: bit
+ unfolding plus_bit_def by (simp split: bit.split)
+
+
+subsection \<open>Numerals at type \<^typ>\<open>bit\<close>\<close>
+
+text \<open>All numerals reduce to either 0 or 1.\<close>
+
+lemma bit_minus1 [simp]: "- 1 = (1 :: bit)"
+ by (simp only: uminus_bit_def)
+
+lemma bit_neg_numeral [simp]: "(- numeral w :: bit) = numeral w"
+ by (simp only: uminus_bit_def)
+
+lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)"
+ by (simp only: numeral_Bit0 bit_add_self)
+
+lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)"
+ by (simp only: numeral_Bit1 bit_add_self add_0_left)
+
+
+subsection \<open>Conversion from \<^typ>\<open>bit\<close>\<close>
+
+context zero_neq_one
+begin
+
+definition of_bit :: "bit \<Rightarrow> 'a"
+ where "of_bit b = case_bit 0 1 b"
+
+lemma of_bit_eq [simp, code]:
+ "of_bit 0 = 0"
+ "of_bit 1 = 1"
+ by (simp_all add: of_bit_def)
+
+lemma of_bit_eq_iff: "of_bit x = of_bit y \<longleftrightarrow> x = y"
+ by (cases x) (cases y; simp)+
+
+end
+
+lemma (in semiring_1) of_nat_of_bit_eq: "of_nat (of_bit b) = of_bit b"
+ by (cases b) simp_all
+
+lemma (in ring_1) of_int_of_bit_eq: "of_int (of_bit b) = of_bit b"
+ by (cases b) simp_all
+
+hide_const (open) set
+
+end
--- a/src/HOL/Word/Bits_Bit.thy Fri Jun 14 08:34:27 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,89 +0,0 @@
-(* Title: HOL/Word/Bits_Bit.thy
- Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
-*)
-
-section \<open>Bit operations in $\cal Z_2$\<close>
-
-theory Bits_Bit
- imports Bits "HOL-Library.Bit"
-begin
-
-instantiation bit :: bit_operations
-begin
-
-primrec bitNOT_bit
- where
- "NOT 0 = (1::bit)"
- | "NOT 1 = (0::bit)"
-
-primrec bitAND_bit
- where
- "0 AND y = (0::bit)"
- | "1 AND y = (y::bit)"
-
-primrec bitOR_bit
- where
- "0 OR y = (y::bit)"
- | "1 OR y = (1::bit)"
-
-primrec bitXOR_bit
- where
- "0 XOR y = (y::bit)"
- | "1 XOR y = (NOT y :: bit)"
-
-instance ..
-
-end
-
-lemmas bit_simps =
- bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
-
-lemma bit_extra_simps [simp]:
- "x AND 0 = 0"
- "x AND 1 = x"
- "x OR 1 = 1"
- "x OR 0 = x"
- "x XOR 1 = NOT x"
- "x XOR 0 = x"
- for x :: bit
- by (cases x; auto)+
-
-lemma bit_ops_comm:
- "x AND y = y AND x"
- "x OR y = y OR x"
- "x XOR y = y XOR x"
- for x :: bit
- by (cases y; auto)+
-
-lemma bit_ops_same [simp]:
- "x AND x = x"
- "x OR x = x"
- "x XOR x = 0"
- for x :: bit
- by (cases x; auto)+
-
-lemma bit_not_not [simp]: "NOT (NOT x) = x"
- for x :: bit
- by (cases x) auto
-
-lemma bit_or_def: "b OR c = NOT (NOT b AND NOT c)"
- for b c :: bit
- by (induct b) simp_all
-
-lemma bit_xor_def: "b XOR c = (b AND NOT c) OR (NOT b AND c)"
- for b c :: bit
- by (induct b) simp_all
-
-lemma bit_NOT_eq_1_iff [simp]: "NOT b = 1 \<longleftrightarrow> b = 0"
- for b :: bit
- by (induct b) simp_all
-
-lemma bit_AND_eq_1_iff [simp]: "a AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
- for a b :: bit
- by (induct a) simp_all
-
-lemma bit_nand_same [simp]: "x AND NOT x = 0"
- for x :: bit
- by (cases x) simp_all
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Word/Bits_Z2.thy Fri Jun 14 08:34:27 2019 +0000
@@ -0,0 +1,89 @@
+(* Title: HOL/Word/Bits_Z2.thy
+ Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
+*)
+
+section \<open>Bit operations in $\cal Z_2$\<close>
+
+theory Bits_Z2
+ imports Bits "HOL-Library.Z2"
+begin
+
+instantiation bit :: bit_operations
+begin
+
+primrec bitNOT_bit
+ where
+ "NOT 0 = (1::bit)"
+ | "NOT 1 = (0::bit)"
+
+primrec bitAND_bit
+ where
+ "0 AND y = (0::bit)"
+ | "1 AND y = (y::bit)"
+
+primrec bitOR_bit
+ where
+ "0 OR y = (y::bit)"
+ | "1 OR y = (1::bit)"
+
+primrec bitXOR_bit
+ where
+ "0 XOR y = (y::bit)"
+ | "1 XOR y = (NOT y :: bit)"
+
+instance ..
+
+end
+
+lemmas bit_simps =
+ bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
+
+lemma bit_extra_simps [simp]:
+ "x AND 0 = 0"
+ "x AND 1 = x"
+ "x OR 1 = 1"
+ "x OR 0 = x"
+ "x XOR 1 = NOT x"
+ "x XOR 0 = x"
+ for x :: bit
+ by (cases x; auto)+
+
+lemma bit_ops_comm:
+ "x AND y = y AND x"
+ "x OR y = y OR x"
+ "x XOR y = y XOR x"
+ for x :: bit
+ by (cases y; auto)+
+
+lemma bit_ops_same [simp]:
+ "x AND x = x"
+ "x OR x = x"
+ "x XOR x = 0"
+ for x :: bit
+ by (cases x; auto)+
+
+lemma bit_not_not [simp]: "NOT (NOT x) = x"
+ for x :: bit
+ by (cases x) auto
+
+lemma bit_or_def: "b OR c = NOT (NOT b AND NOT c)"
+ for b c :: bit
+ by (induct b) simp_all
+
+lemma bit_xor_def: "b XOR c = (b AND NOT c) OR (NOT b AND c)"
+ for b c :: bit
+ by (induct b) simp_all
+
+lemma bit_NOT_eq_1_iff [simp]: "NOT b = 1 \<longleftrightarrow> b = 0"
+ for b :: bit
+ by (induct b) simp_all
+
+lemma bit_AND_eq_1_iff [simp]: "a AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
+ for a b :: bit
+ by (induct a) simp_all
+
+lemma bit_nand_same [simp]: "x AND NOT x = 0"
+ for x :: bit
+ by (cases x) simp_all
+
+end
--- a/src/HOL/Word/Misc_Arithmetic.thy Fri Jun 14 08:34:27 2019 +0000
+++ b/src/HOL/Word/Misc_Arithmetic.thy Fri Jun 14 08:34:27 2019 +0000
@@ -3,7 +3,7 @@
section \<open>Miscellaneous lemmas, mostly for arithmetic\<close>
theory Misc_Arithmetic
- imports Misc_Auxiliary "HOL-Library.Bit"
+ imports Misc_Auxiliary "HOL-Library.Z2"
begin
lemma one_mod_exp_eq_one [simp]:
--- a/src/HOL/Word/Word.thy Fri Jun 14 08:34:27 2019 +0000
+++ b/src/HOL/Word/Word.thy Fri Jun 14 08:34:27 2019 +0000
@@ -9,7 +9,7 @@
"HOL-Library.Type_Length"
"HOL-Library.Boolean_Algebra"
Bits_Int
- Bits_Bit
+ Bits_Z2
Bit_Comprehension
Misc_Typedef
Misc_Arithmetic