--- a/src/HOL/Word/Bits_Bit.thy Mon Apr 03 21:17:47 2017 +0200
+++ b/src/HOL/Word/Bits_Bit.thy Mon Apr 03 23:12:16 2017 +0200
@@ -11,20 +11,24 @@
instantiation bit :: bit
begin
-primrec bitNOT_bit where
- "NOT 0 = (1::bit)"
+primrec bitNOT_bit
+ where
+ "NOT 0 = (1::bit)"
| "NOT 1 = (0::bit)"
-primrec bitAND_bit where
- "0 AND y = (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)"
+primrec bitOR_bit
+ where
+ "0 OR y = (y::bit)"
| "1 OR y = (1::bit)"
-primrec bitXOR_bit where
- "0 XOR y = (y::bit)"
+primrec bitXOR_bit
+ where
+ "0 XOR y = (y::bit)"
| "1 XOR y = (NOT y :: bit)"
instance ..
@@ -34,40 +38,48 @@
lemmas bit_simps =
bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
-lemma bit_extra_simps [simp]:
- "x AND 0 = (0::bit)"
- "x AND 1 = (x::bit)"
- "x OR 1 = (1::bit)"
- "x OR 0 = (x::bit)"
- "x XOR 1 = NOT (x::bit)"
- "x XOR 0 = (x::bit)"
+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::bit) AND y = y AND x"
- "(x::bit) OR y = y OR x"
- "(x::bit) XOR y = y XOR x"
+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::bit) AND x = x"
- "(x::bit) OR x = x"
- "(x::bit) XOR x = 0"
+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::bit)) = x"
+lemma bit_not_not [simp]: "NOT (NOT x) = x"
+ for x :: bit
by (cases x) auto
-lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)"
- by (induct b, simp_all)
+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::bit) XOR c = (b AND NOT c) OR (NOT b AND c)"
- 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::bit) = 1 \<longleftrightarrow> b = 0"
- 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::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
- by (induct a, 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
end