# HG changeset patch
# User haftmann
# Date 1404497927 -7200
# Node ID cc97b347b301508a3b420c085d1c9a01f8268e7d
# Parent  de51a86fc9035bede32334f1806503687e6b1178
reduced name variants for assoc and commute on plus and mult

diff -r de51a86fc903 -r cc97b347b301 NEWS
--- a/NEWS	Fri Jul 04 20:07:08 2014 +0200
+++ b/NEWS	Fri Jul 04 20:18:47 2014 +0200
@@ -221,6 +221,22 @@
 * The symbol "\<newline>" may be used within char or string literals
 to represent (Char Nibble0 NibbleA), i.e. ASCII newline.
 
+* Reduced name variants for rules on associativity and commutativity:
+
+    add_assoc ~> add.assoc
+    add_commute ~> add.commute
+    add_left_commute ~> add.left_commute
+    mult_assoc ~> mult.assoc
+    mult_commute ~> mult.commute
+    mult_left_commute ~> mult.left_commute
+    nat_add_assoc ~> add.assoc
+    nat_add_commute ~> add.commute
+    nat_add_left_commute ~> add.left_commute
+    nat_mult_assoc ~> mult.assoc
+    nat_mult_commute ~> mult.commute
+    eq_assoc ~> iff_assoc
+    eq_left_commute ~> iff_left_commute
+
 * Qualified String.implode and String.explode.  INCOMPATIBILITY.
 
 * Simplifier: Enhanced solver of preconditions of rewrite rules can
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Codegen/Further.thy
--- a/src/Doc/Codegen/Further.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Codegen/Further.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -205,7 +205,7 @@
 interpretation %quote fun_power: power "\<lambda>n (f :: 'a \<Rightarrow> 'a). f ^^ n" where
   "power.powers (\<lambda>n f. f ^^ n) = funpows"
   by unfold_locales
-    (simp_all add: fun_eq_iff funpow_mult mult_commute funpows_def)
+    (simp_all add: fun_eq_iff funpow_mult mult.commute funpows_def)
 
 text {*
   \noindent This additional equation is trivially proved by the
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Tutorial/Advanced/simp2.thy
--- a/src/Doc/Tutorial/Advanced/simp2.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Tutorial/Advanced/simp2.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -130,7 +130,7 @@
 Each occurrence of an unknown is of the form
 $\Var{f}~x@1~\dots~x@n$, where the $x@i$ are distinct bound
 variables. Thus all ordinary rewrite rules, where all unknowns are
-of base type, for example @{thm add_assoc}, are acceptable: if an unknown is
+of base type, for example @{thm add.assoc}, are acceptable: if an unknown is
 of base type, it cannot have any arguments. Additionally, the rule
 @{text"(\<forall>x. ?P x \<and> ?Q x) = ((\<forall>x. ?P x) \<and> (\<forall>x. ?Q x))"} is also acceptable, in
 both directions: all arguments of the unknowns @{text"?P"} and
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Tutorial/Rules/Basic.thy
--- a/src/Doc/Tutorial/Rules/Basic.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Tutorial/Rules/Basic.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -64,11 +64,11 @@
 text {*
 the subst method
 
-@{thm[display] mult_commute}
-\rulename{mult_commute}
+@{thm[display] mult.commute}
+\rulename{mult.commute}
 
 this would fail:
-apply (simp add: mult_commute) 
+apply (simp add: mult.commute) 
 *}
 
 
@@ -76,7 +76,7 @@
 txt{*
 @{subgoals[display,indent=0,margin=65]}
 *}
-apply (subst mult_commute) 
+apply (subst mult.commute) 
 txt{*
 @{subgoals[display,indent=0,margin=65]}
 *}
@@ -84,7 +84,7 @@
 
 (*exercise involving THEN*)
 lemma "\<lbrakk>P x y z; Suc x < y\<rbrakk> \<Longrightarrow> f z = x*y"
-apply (rule mult_commute [THEN ssubst]) 
+apply (rule mult.commute [THEN ssubst]) 
 oops
 
 
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Tutorial/Types/Numbers.thy
--- a/src/Doc/Tutorial/Types/Numbers.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Tutorial/Types/Numbers.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -31,14 +31,14 @@
 @{thm[display] add_2_eq_Suc'[no_vars]}
 \rulename{add_2_eq_Suc'}
 
-@{thm[display] add_assoc[no_vars]}
-\rulename{add_assoc}
+@{thm[display] add.assoc[no_vars]}
+\rulename{add.assoc}
 
-@{thm[display] add_commute[no_vars]}
-\rulename{add_commute}
+@{thm[display] add.commute[no_vars]}
+\rulename{add.commute}
 
-@{thm[display] add_left_commute[no_vars]}
-\rulename{add_left_commute}
+@{thm[display] add.left_commute[no_vars]}
+\rulename{add.left_commute}
 
 these form add_ac; similarly there is mult_ac
 *}
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Tutorial/document/numerics.tex
--- a/src/Doc/Tutorial/document/numerics.tex	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Tutorial/document/numerics.tex	Fri Jul 04 20:18:47 2014 +0200
@@ -20,7 +20,7 @@
 infix symbols. Algebraic properties are organized using type classes
 around algebraic concepts such as rings and fields;
 a property such as the commutativity of addition is a single theorem 
-(\isa{add_commute}) that applies to all numeric types.
+(\isa{add.commute}) that applies to all numeric types.
 
 \index{linear arithmetic}%
 Many theorems involving numeric types can be proved automatically by
@@ -441,11 +441,11 @@
 the two expressions identical.
 \begin{isabelle}
 a\ +\ b\ +\ c\ =\ a\ +\ (b\ +\ c)
-\rulenamedx{add_assoc}\isanewline
+\rulenamedx{add.assoc}\isanewline
 a\ +\ b\ =\ b\ +\ a%
-\rulenamedx{add_commute}\isanewline
+\rulenamedx{add.commute}\isanewline
 a\ +\ (b\ +\ c)\ =\ b\ +\ (a\ +\ c)
-\rulename{add_left_commute}
+\rulename{add.left_commute}
 \end{isabelle}
 The name \isa{add_ac}\index{*add_ac (theorems)} 
 refers to the list of all three theorems; similarly
diff -r de51a86fc903 -r cc97b347b301 src/Doc/Tutorial/document/rules.tex
--- a/src/Doc/Tutorial/document/rules.tex	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/Doc/Tutorial/document/rules.tex	Fri Jul 04 20:18:47 2014 +0200
@@ -751,11 +751,11 @@
 Now we wish to apply a commutative law:
 \begin{isabelle}
 ?m\ *\ ?n\ =\ ?n\ *\ ?m%
-\rulename{mult_commute}
+\rulename{mult.commute}
 \end{isabelle}
 Isabelle rejects our first attempt:
 \begin{isabelle}
-apply (simp add: mult_commute)
+apply (simp add: mult.commute)
 \end{isabelle}
 The simplifier notices the danger of looping and refuses to apply the
 rule.%
@@ -764,9 +764,9 @@
 is already smaller than
 \isa{y\ *\ x}.}
 %
-The \isa{subst} method applies \isa{mult_commute} exactly once.  
+The \isa{subst} method applies \isa{mult.commute} exactly once.  
 \begin{isabelle}
-\isacommand{apply}\ (subst\ mult_commute)\isanewline
+\isacommand{apply}\ (subst\ mult.commute)\isanewline
 \ 1.\ \isasymlbrakk P\ x\ y\ z;\ Suc\ x\ <\ y\isasymrbrakk \
 \isasymLongrightarrow \ f\ z\ =\ y\ *\ x%
 \end{isabelle}
@@ -775,7 +775,7 @@
 \medskip
 This use of the \methdx{subst} method has the same effect as the command
 \begin{isabelle}
-\isacommand{apply}\ (rule\ mult_commute [THEN ssubst])
+\isacommand{apply}\ (rule\ mult.commute [THEN ssubst])
 \end{isabelle}
 The attribute \isa{THEN}, which combines two rules, is described in 
 {\S}\ref{sec:THEN} below. The \methdx{subst} method is more powerful than
@@ -1986,9 +1986,9 @@
 %  exercise worth including? E.g. find a difference between the two ways
 %  of substituting.
 %\begin{exercise}
-%In {\S}\ref{sec:subst} the method \isa{subst\ mult_commute} was applied.  How
+%In {\S}\ref{sec:subst} the method \isa{subst\ mult.commute} was applied.  How
 %can we achieve the same effect using \isa{THEN} with the rule \isa{ssubst}?
-%% answer  rule (mult_commute [THEN ssubst])
+%% answer  rule (mult.commute [THEN ssubst])
 %\end{exercise}
 
 \subsection{Modifying a Theorem using {\tt\slshape OF}}
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Algebra/Coset.thy
--- a/src/HOL/Algebra/Coset.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Algebra/Coset.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -815,7 +815,7 @@
      "\<lbrakk>finite(carrier G); subgroup H G\<rbrakk>
       \<Longrightarrow> card(rcosets H) * card(H) = order(G)"
 apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
-apply (subst mult_commute)
+apply (subst mult.commute)
 apply (rule card_partition)
    apply (simp add: rcosets_subset_PowG [THEN finite_subset])
   apply (simp add: rcosets_part_G)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Algebra/Exponent.thy
--- a/src/HOL/Algebra/Exponent.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Algebra/Exponent.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -80,7 +80,7 @@
 lemma div_combine:
   fixes p::nat
   shows "[| prime p; ~ (p ^ (Suc r) dvd n);  p^(a+r) dvd n*k |] ==> p ^ a dvd k"
-by (metis add_Suc nat_add_commute prime_power_dvd_cases)
+by (metis add_Suc add.commute prime_power_dvd_cases)
 
 (*Lemma for power_dvd_bound*)
 lemma Suc_le_power: "Suc 0 < p ==> Suc n <= p^n"
@@ -185,7 +185,7 @@
 text{*Main Combinatorial Argument*}
 
 lemma gcd_mult': fixes a::nat shows "gcd b (a * b) = b"
-by (simp add: mult_commute[of a b]) 
+by (simp add: mult.commute[of a b]) 
 
 lemma le_extend_mult: "[| c > 0; a <= b |] ==> a <= b * (c::nat)"
 apply (rule_tac P = "%x. x <= b * c" in subst)
@@ -201,7 +201,7 @@
 apply (drule less_imp_le [of a])
 apply (drule le_imp_power_dvd)
 apply (drule_tac b = "p ^ r" in dvd_trans, assumption)
-apply (metis diff_is_0_eq dvd_diffD1 gcd_dvd2_nat gcd_mult' gr0I le_extend_mult less_diff_conv nat_dvd_not_less nat_mult_commute not_add_less2 xt1(10))
+apply (metis diff_is_0_eq dvd_diffD1 gcd_dvd2_nat gcd_mult' gr0I le_extend_mult less_diff_conv nat_dvd_not_less mult.commute not_add_less2 xt1(10))
 done
 
 lemma p_fac_forw: "[| (m::nat) > 0; k>0; k < p^a; (p^r) dvd (p^a)* m - k |]  
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Algebra/Group.thy
--- a/src/HOL/Algebra/Group.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Algebra/Group.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -211,7 +211,7 @@
 
 lemma (in monoid) nat_pow_pow:
   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
-  by (induct m) (simp, simp add: nat_pow_mult add_commute)
+  by (induct m) (simp, simp add: nat_pow_mult add.commute)
 
 
 (* Jacobson defines submonoid here. *)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Algebra/IntRing.thy
--- a/src/HOL/Algebra/IntRing.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Algebra/IntRing.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -271,11 +271,11 @@
   then have "p dvd a \<or> p dvd b"
     by (metis prime prime_dvd_mult_eq_int)
   then show "(\<exists>x. a = x * int p) \<or> (\<exists>x. b = x * int p)"
-    by (metis dvd_def mult_commute)
+    by (metis dvd_def mult.commute)
 next
   assume "UNIV = {uu. EX x. uu = x * int p}"
   then obtain x where "1 = x * int p" by best
-  then have "\<bar>int p * x\<bar> = 1" by (simp add: mult_commute)
+  then have "\<bar>int p * x\<bar> = 1" by (simp add: mult.commute)
   then show False
     by (metis abs_of_nat int_1 of_nat_eq_iff abs_zmult_eq_1 one_not_prime_nat prime)
 qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Algebra/UnivPoly.thy
--- a/src/HOL/Algebra/UnivPoly.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Algebra/UnivPoly.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1353,7 +1353,7 @@
   case 0 from R show ?case by simp
 next
   case Suc with R show ?case
-    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
+    by (simp del: monom_mult add: monom_mult [THEN sym] add.commute)
 qed
 
 lemma (in ring_hom_cring) hom_pow [simp]:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Code_Numeral.thy
--- a/src/HOL/Code_Numeral.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Code_Numeral.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -494,7 +494,7 @@
     show "k = integer_of_int (int_of_integer k)" by simp
   qed
   moreover have "2 * (j div 2) = j - j mod 2"
-    by (simp add: zmult_div_cancel mult_commute)
+    by (simp add: zmult_div_cancel mult.commute)
   ultimately show ?thesis
     by (auto simp add: split_def Let_def mod_integer_def nat_of_integer_def not_le
       nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Complex.thy
--- a/src/HOL/Complex.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Complex.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -118,7 +118,7 @@
   show "scaleR (a + b) x = scaleR a x + scaleR b x"
     by (simp add: complex_eq_iff distrib_right)
   show "scaleR a (scaleR b x) = scaleR (a * b) x"
-    by (simp add: complex_eq_iff mult_assoc)
+    by (simp add: complex_eq_iff mult.assoc)
   show "scaleR 1 x = x"
     by (simp add: complex_eq_iff)
   show "scaleR a x * y = scaleR a (x * y)"
@@ -190,7 +190,7 @@
   by (simp add: divide_complex_def)
 
 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
-  by (simp add: mult_assoc [symmetric])
+  by (simp add: mult.assoc [symmetric])
 
 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
   by (simp add: complex_eq_iff)
@@ -298,14 +298,14 @@
   apply (rule abs_sqrt_wlog [where x="Re z"])
   apply (rule abs_sqrt_wlog [where x="Im z"])
   apply (rule power2_le_imp_le)
-  apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])
+  apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   done
 
 
 text {* Properties of complex signum. *}
 
 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
-  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
+  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
 
 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   by (simp add: complex_sgn_def divide_inverse)
@@ -703,7 +703,7 @@
 
 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
 apply (insert rcis_Ex [of z])
-apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
+apply (auto simp add: expi_def rcis_def mult.assoc [symmetric])
 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
 done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/Approximation.thy
--- a/src/HOL/Decision_Procs/Approximation.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/Approximation.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -247,7 +247,7 @@
         unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
       also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
         by (rule mult_strict_left_mono, auto intro: E_mod_pow)
-      also have "\<dots> = 2 powr (?E div 2 + 1)" unfolding add_commute[of _ 1] powr_add[symmetric]
+      also have "\<dots> = 2 powr (?E div 2 + 1)" unfolding add.commute[of _ 1] powr_add[symmetric]
         by simp
       finally show ?thesis by auto
     qed
@@ -386,8 +386,8 @@
 
   { have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
       using bounds(1) `0 \<le> real x`
-      unfolding power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
-      unfolding mult_commute[where 'a=real] mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
+      unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
+      unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
       by (auto intro!: mult_left_mono)
     also have "\<dots> \<le> arctan x" using arctan_bounds ..
     finally have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan x" . }
@@ -395,8 +395,8 @@
   { have "arctan x \<le> ?S (Suc n)" using arctan_bounds ..
     also have "\<dots> \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
       using bounds(2)[of "Suc n"] `0 \<le> real x`
-      unfolding power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
-      unfolding mult_commute[where 'a=real] mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
+      unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
+      unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
       by (auto intro!: mult_left_mono)
     finally have "arctan x \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
   ultimately show ?thesis by auto
@@ -842,8 +842,8 @@
   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
     OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
   show "?lb" and "?ub" using `0 \<le> real x`
-    unfolding power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
-    unfolding mult_commute[where 'a=real]
+    unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
+    unfolding mult.commute[where 'a=real]
     by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
 qed
 
@@ -1597,7 +1597,7 @@
   { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
     proof (rule mult_mono)
       show "0 \<le> x ^ Suc (Suc n)" by (auto simp add: `0 \<le> x`)
-      have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult_assoc[symmetric]
+      have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult.assoc[symmetric]
         by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto simp: `0 \<le> x`)
       thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
     qed auto }
@@ -1615,7 +1615,7 @@
 
   let "?s n" = "-1^n * (1 / real (1 + n)) * (real x)^(Suc n)"
 
-  have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 mult_assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult_commute[of "real x"] ev
+  have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult.commute[of "real x"] ev
     using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
     by (rule mult_right_mono)
@@ -1623,7 +1623,7 @@
   finally show "?lb \<le> ?ln" .
 
   have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
-  also have "\<dots> \<le> ?ub" unfolding power_Suc2 mult_assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult_commute[of "real x"] od
+  also have "\<dots> \<le> ?ub" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult.commute[of "real x"] od
     using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
     by (rule mult_right_mono)
@@ -2961,7 +2961,7 @@
                inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
         unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
         by (auto simp add: algebra_simps)
-          (simp only: mult_left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
+          (simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
       finally have "?T \<in> {l .. u}" .
     }
     thus ?thesis using DERIV by blast
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/Ferrack.thy
--- a/src/HOL/Decision_Procs/Ferrack.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/Ferrack.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1407,7 +1407,7 @@
   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
   from uset_l[OF lp] smU have mp: "real m > 0" by auto
   from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
@@ -1423,7 +1423,7 @@
   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
   from uset_l[OF lp] smU have mp: "real m > 0" by auto
   from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
@@ -1674,7 +1674,7 @@
       with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
         by auto
       let ?st = "Add (Mul m t) (Mul n s)"
-      from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult_commute)
+      from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult.commute)
       from tnb snb have st_nb: "numbound0 ?st" by simp
       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
@@ -1690,7 +1690,7 @@
     and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
     with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
     let ?st = "Add (Mul l t) (Mul k s)"
-    from np mp have mnp: "real (2*k*l) > 0" by (simp add: mult_commute)
+    from np mp have mnp: "real (2*k*l) > 0" by (simp add: mult.commute)
     from tnb snb have st_nb: "numbound0 ?st" by simp
     from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
   ultimately show "?E" by blast
@@ -1729,7 +1729,7 @@
          (set U \<times> set U)"using mnz nnz th  
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
     by (rule_tac x="(s,m)" in bexI,simp_all) 
-  (rule_tac x="(t,n)" in bexI,simp_all add: mult_commute)
+  (rule_tac x="(t,n)" in bexI,simp_all add: mult.commute)
 next
   fix t n s m
   assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
@@ -1778,7 +1778,7 @@
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
   from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
@@ -1806,7 +1806,7 @@
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
   from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/MIR.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -4514,7 +4514,7 @@
   then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<ge> ?N a s" by blast
   from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
   from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
@@ -4530,7 +4530,7 @@
   then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<le> ?N a s" by blast
   from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
   from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
@@ -4747,7 +4747,7 @@
       with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
         by auto
       let ?st = "Add (Mul m t) (Mul n s)"
-      from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult_commute)
+      from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult.commute)
       from tnb snb have st_nb: "numbound0 ?st" by simp
       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
@@ -4763,7 +4763,7 @@
     and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))"
     with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
     let ?st = "Add (Mul l t) (Mul k s)"
-    from np mp have mnp: "real (2*k*l) > 0" by (simp add: mult_commute)
+    from np mp have mnp: "real (2*k*l) > 0" by (simp add: mult.commute)
     from tnb snb have st_nb: "numbound0 ?st" by simp
     from \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
   ultimately show "?E" by blast
@@ -4883,7 +4883,7 @@
     from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def)
     let ?st = "Add (Mul m t) (Mul n s)"
     from tnb snb have stnb: "numbound0 ?st" by simp
-    from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult_commute)
+    from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult.commute)
     from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx
     have "\<exists> x. ?I x ?rq" by auto
     thus "?E" 
@@ -4915,7 +4915,7 @@
          (set U \<times> set U)"using mnz nnz th  
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
     by (rule_tac x="(s,m)" in bexI,simp_all) 
-  (rule_tac x="(t,n)" in bexI,simp_all add: mult_commute)
+  (rule_tac x="(t,n)" in bexI,simp_all add: mult.commute)
 next
   fix t n s m
   assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
@@ -4964,7 +4964,7 @@
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
   from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
@@ -4992,7 +4992,7 @@
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
   from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
--- a/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -428,7 +428,7 @@
   apply (simp add: Let_def split_def field_simps)
   apply (simp add: Let_def split_def field_simps)
   apply (simp add: Let_def split_def field_simps)
-  apply (simp add: Let_def split_def mult_assoc distrib_left[symmetric])
+  apply (simp add: Let_def split_def mult.assoc distrib_left[symmetric])
   apply (simp add: Let_def split_def field_simps)
   apply (simp add: Let_def split_def field_simps)
   done
@@ -1686,7 +1686,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x < - ?e"
         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
@@ -1701,7 +1701,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x > - ?e"
         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
@@ -1734,7 +1734,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x < - ?e"
         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
@@ -1749,7 +1749,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x > - ?e"
         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
@@ -1783,7 +1783,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x < - ?e"
         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0" by simp
       then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto
@@ -1797,7 +1797,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x > - ?e"
         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
@@ -1830,7 +1830,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x < - ?e"
         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
@@ -1846,7 +1846,7 @@
       assume xz: "x < -?e / ?c"
       then have "?c * x > - ?e"
         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
@@ -1899,7 +1899,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x > - ?e"
         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
@@ -1914,7 +1914,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x < - ?e"
         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0" by simp
       then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto
@@ -1944,7 +1944,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x > - ?e"
         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
@@ -1959,7 +1959,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x < - ?e"
         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
@@ -1992,7 +1992,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x > - ?e"
         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
@@ -2007,7 +2007,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x < - ?e"
         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
@@ -2040,7 +2040,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x > - ?e"
         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e > 0"
         by simp
       then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
@@ -2055,7 +2055,7 @@
       assume xz: "x > -?e / ?c"
       then have "?c * x < - ?e"
         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
-        by (simp add: mult_commute)
+        by (simp add: mult.commute)
       then have "?c * x + ?e < 0"
         by simp
       then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
@@ -2144,7 +2144,7 @@
       (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" by blast
   then have "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   then show ?thesis
     using csU by auto
 qed
@@ -2188,7 +2188,7 @@
     by blast
   then have "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   then show ?thesis
     using csU by auto
 qed
@@ -3811,7 +3811,7 @@
       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
       have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and>
           Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
-        by (simp add: add_divide_distrib diff_divide_distrib mult_minus2_left mult_commute)
+        by (simp add: add_divide_distrib diff_divide_distrib mult_minus2_left mult.commute)
     }
     moreover
     {
@@ -3828,7 +3828,7 @@
         using H(3,4) by (simp_all add: polymul_norm n2)
       from msubst2[OF lp nn, of x bs ] H(3,4,5)
       have "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
-        by (simp add: diff_divide_distrib add_divide_distrib mult_minus2_left mult_commute)
+        by (simp add: diff_divide_distrib add_divide_distrib mult_minus2_left mult.commute)
     }
     ultimately show ?thesis by blast
   qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/Polynomial_List.thy
--- a/src/HOL/Decision_Procs/Polynomial_List.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -121,7 +121,7 @@
   thus ?case by auto
 next
   case (Cons b bs a)
-  thus ?case by (cases a) (simp_all add: add_commute)
+  thus ?case by (cases a) (simp_all add: add.commute)
 qed
 
 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
@@ -192,13 +192,13 @@
   by simp
 
 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
-  by (simp add: poly_mult mult_assoc)
+  by (simp add: poly_mult mult.assoc)
 
 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
   by (induct p) auto
 
 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
-  by (induct n) (auto simp add: poly_mult mult_assoc)
+  by (induct n) (auto simp add: poly_mult mult.assoc)
 
 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
  @{term "p(x)"} *}
@@ -445,7 +445,7 @@
 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
   apply (simp add: fun_eq)
   apply (rule_tac x = "minus one a" in exI)
-  apply (simp add: add_commute [of a])
+  apply (simp add: add.commute [of a])
   done
 
 lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
@@ -517,7 +517,7 @@
 lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
   apply (simp add: divides_def, safe)
   apply (rule_tac x = "pmult qa qaa" in exI)
-  apply (auto simp add: poly_mult fun_eq mult_assoc)
+  apply (auto simp add: poly_mult fun_eq mult.assoc)
   done
 
 lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Decision_Procs/Rat_Pair.thy
--- a/src/HOL/Decision_Procs/Rat_Pair.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Decision_Procs/Rat_Pair.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -464,7 +464,7 @@
 qed
 
 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
-  by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)
+  by (simp add: Nmul_def split_def Let_def gcd_commute_int mult.commute)
 
 lemma Nmul_assoc:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Deriv.thy
--- a/src/HOL/Deriv.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Deriv.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -558,7 +558,7 @@
 lemma has_derivative_imp_has_field_derivative:
   "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
   unfolding has_field_derivative_def 
-  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult_commute)
+  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
 
 lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
   by (simp add: has_field_derivative_def)
@@ -587,7 +587,7 @@
   then obtain f' where *: "(f has_derivative f') (at x within s)"
     unfolding differentiable_def by auto
   then obtain c where "f' = (op * c)"
-    by (metis real_bounded_linear has_derivative_bounded_linear mult_commute fun_eq_iff)
+    by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
   with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
     unfolding has_field_derivative_def by auto
 qed (auto simp: differentiable_def has_field_derivative_def)
@@ -610,7 +610,7 @@
   done
 
 lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
-  by (simp add: fun_eq_iff mult_commute)
+  by (simp add: fun_eq_iff mult.commute)
 
 subsection {* Derivatives *}
 
@@ -732,7 +732,7 @@
   "(f has_field_derivative d) (at x within s) \<Longrightarrow>
   (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
   ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
-  by (drule (2) DERIV_divide) (simp add: mult_commute)
+  by (drule (2) DERIV_divide) (simp add: mult.commute)
 
 lemma DERIV_power_Suc:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
@@ -765,7 +765,7 @@
 lemma DERIV_chain:
   "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
   (f o g has_field_derivative Da * Db) (at x within s)"
-  by (drule (1) DERIV_chain', simp add: o_def mult_commute)
+  by (drule (1) DERIV_chain', simp add: o_def mult.commute)
 
 lemma DERIV_image_chain: 
   "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
@@ -779,7 +779,7 @@
       and "DERIV f x :> f'" 
       and "f x \<in> s"
     shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
-  by (metis (full_types) DERIV_chain' mult_commute assms)
+  by (metis (full_types) DERIV_chain' mult.commute assms)
 
 lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
   assumes "(\<And>x. DERIV g x :> g'(x))"
@@ -800,7 +800,7 @@
 apply (drule_tac k="- a" in LIM_offset)
 apply simp
 apply (drule_tac k="a" in LIM_offset)
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 done
 
 lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
@@ -1215,7 +1215,7 @@
   fixes f :: "real => real"
   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
 apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
-apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
+apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
 done
 
 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
@@ -1248,7 +1248,7 @@
   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
-  ultimately show ?thesis using neq by (force simp add: add_commute)
+  ultimately show ?thesis using neq by (force simp add: add.commute)
 qed
 
 (* A function with positive derivative is increasing. 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Divides.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -29,7 +29,7 @@
 text {* @{const div} and @{const mod} *}
 
 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
-  unfolding mult_commute [of b]
+  unfolding mult.commute [of b]
   by (rule mod_div_equality)
 
 lemma mod_div_equality': "a mod b + a div b * b = a"
@@ -51,7 +51,7 @@
 lemma div_mult_self2 [simp]:
   assumes "b \<noteq> 0"
   shows "(a + b * c) div b = c + a div b"
-  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
+  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
 
 lemma div_mult_self3 [simp]:
   assumes "b \<noteq> 0"
@@ -74,7 +74,7 @@
     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
       by (simp add: algebra_simps)
   finally have "a = a div b * b + (a + c * b) mod b"
-    by (simp add: add_commute [of a] add_assoc distrib_right)
+    by (simp add: add.commute [of a] add.assoc distrib_right)
   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
     by (simp add: mod_div_equality)
   then show ?thesis by simp
@@ -82,7 +82,7 @@
 
 lemma mod_mult_self2 [simp]: 
   "(a + b * c) mod b = a mod b"
-  by (simp add: mult_commute [of b])
+  by (simp add: mult.commute [of b])
 
 lemma mod_mult_self3 [simp]:
   "(c * b + a) mod b = a mod b"
@@ -123,16 +123,16 @@
 lemma div_add_self1 [simp]:
   assumes "b \<noteq> 0"
   shows "(b + a) div b = a div b + 1"
-  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
+  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
 
 lemma div_add_self2 [simp]:
   assumes "b \<noteq> 0"
   shows "(a + b) div b = a div b + 1"
-  using assms div_add_self1 [of b a] by (simp add: add_commute)
+  using assms div_add_self1 [of b a] by (simp add: add.commute)
 
 lemma mod_add_self1 [simp]:
   "(b + a) mod b = a mod b"
-  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
+  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
 
 lemma mod_add_self2 [simp]:
   "(a + b) mod b = a mod b"
@@ -153,7 +153,7 @@
 proof
   assume "b mod a = 0"
   with mod_div_equality [of b a] have "b div a * a = b" by simp
-  then have "b = a * (b div a)" unfolding mult_commute ..
+  then have "b = a * (b div a)" unfolding mult.commute ..
   then have "\<exists>c. b = a * c" ..
   then show "a dvd b" unfolding dvd_def .
 next
@@ -161,7 +161,7 @@
   then have "\<exists>c. b = a * c" unfolding dvd_def .
   then obtain c where "b = a * c" ..
   then have "b mod a = a * c mod a" by simp
-  then have "b mod a = c * a mod a" by (simp add: mult_commute)
+  then have "b mod a = c * a mod a" by (simp add: mult.commute)
   then show "b mod a = 0" by simp
 qed
 
@@ -197,12 +197,12 @@
 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
 
 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
-by (drule dvd_div_mult_self) (simp add: mult_commute)
+by (drule dvd_div_mult_self) (simp add: mult.commute)
 
 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
 apply (cases "a = 0")
  apply simp
-apply (auto simp: dvd_def mult_assoc)
+apply (auto simp: dvd_def mult.assoc)
 done
 
 lemma div_dvd_div[simp]:
@@ -211,8 +211,8 @@
  apply simp
 apply (unfold dvd_def)
 apply auto
- apply(blast intro:mult_assoc[symmetric])
-apply(fastforce simp add: mult_assoc)
+ apply(blast intro:mult.assoc[symmetric])
+apply(fastforce simp add: mult.assoc)
 done
 
 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
@@ -332,7 +332,7 @@
   apply (cases "y = 0", simp)
   apply (cases "z = 0", simp)
   apply (auto elim!: dvdE simp add: algebra_simps)
-  apply (subst mult_assoc [symmetric])
+  apply (subst mult.assoc [symmetric])
   apply (simp add: no_zero_divisors)
   done
 
@@ -342,12 +342,12 @@
 proof -
   from assms have "b div c * (a div 1) = b * a div (c * 1)"
     by (simp only: div_mult_div_if_dvd one_dvd)
-  then show ?thesis by (simp add: mult_commute)
+  then show ?thesis by (simp add: mult.commute)
 qed
    
 lemma div_mult_mult2 [simp]:
   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
-  by (drule div_mult_mult1) (simp add: mult_commute)
+  by (drule div_mult_mult1) (simp add: mult.commute)
 
 lemma div_mult_mult1_if [simp]:
   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
@@ -368,7 +368,7 @@
   
 lemma mod_mult_mult2:
   "(a * c) mod (b * c) = (a mod b) * c"
-  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
+  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
 
 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
   by (fact mod_mult_mult2 [symmetric])
@@ -405,7 +405,7 @@
 lemma dvd_div_div_eq_mult:
   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
-  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
+  using assms by (auto simp add: mult.commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
 
 end
 
@@ -1065,7 +1065,7 @@
 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
 
 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
-by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
+by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
 
 
 subsubsection {* Further Facts about Quotient and Remainder *}
@@ -1692,7 +1692,7 @@
   fix a b :: int
   show "a div b * b + a mod b = a"
     using divmod_int_rel_div_mod [of a b]
-    unfolding divmod_int_rel_def by (simp add: mult_commute)
+    unfolding divmod_int_rel_def by (simp add: mult.commute)
 next
   fix a b c :: int
   assume "b \<noteq> 0"
@@ -2022,7 +2022,7 @@
 apply (subgoal_tac "b*q = r' - r + b'*q'")
  prefer 2 apply simp
 apply (simp (no_asm_simp) add: distrib_left)
-apply (subst add_commute, rule add_less_le_mono, arith)
+apply (subst add.commute, rule add_less_le_mono, arith)
 apply (rule mult_right_mono, auto)
 done
 
@@ -2314,7 +2314,7 @@
   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
     (numeral v div (numeral w :: int))"
   unfolding numeral.simps
-  unfolding mult_2 [symmetric] add_commute [of _ 1]
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zdiv_mult_2, simp)
 
 lemma pos_zmod_mult_2:
@@ -2342,7 +2342,7 @@
   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
     2 * (numeral v mod numeral w) + (1::int)"
   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] add_commute [of _ 1]
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zmod_mult_2, simp)
 
 lemma zdiv_eq_0_iff:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Fact.thy
--- a/src/HOL/Fact.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Fact.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -230,7 +230,7 @@
   apply auto
   apply (induct k rule: int_ge_induct)
   apply auto
-  apply (subst add_assoc [symmetric])
+  apply (subst add.assoc [symmetric])
   apply (subst fact_plus_one_int)
   apply auto
   apply (erule order_trans)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Fields.thy
--- a/src/HOL/Fields.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Fields.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -51,7 +51,7 @@
     assume ab: "a * b = 0"
     hence "0 = inverse a * (a * b) * inverse b" by simp
     also have "\<dots> = (inverse a * a) * (b * inverse b)"
-      by (simp only: mult_assoc)
+      by (simp only: mult.assoc)
     also have "\<dots> = 1" using a b by simp
     finally show False by simp
   qed
@@ -81,7 +81,7 @@
 proof -
   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
-  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
+  ultimately show ?thesis by (simp add: mult.assoc [symmetric])
 qed
 
 lemma nonzero_inverse_minus_eq:
@@ -110,7 +110,7 @@
   shows "inverse (a * b) = inverse b * inverse a"
 proof -
   have "a * (b * inverse b) * inverse a = 1" using assms by simp
-  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
+  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
   thus ?thesis by (rule inverse_unique)
 qed
 
@@ -126,7 +126,7 @@
 proof
   assume neq: "b \<noteq> 0"
   {
-    hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc)
+    hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc)
     also assume "a / b = 1"
     finally show "a = b" by simp
   next
@@ -154,7 +154,7 @@
   by (simp add: divide_inverse)
 
 lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
-  by (simp add: divide_inverse mult_assoc)
+  by (simp add: divide_inverse mult.assoc)
 
 lemma minus_divide_left: "- (a / b) = (-a) / b"
   by (simp add: divide_inverse)
@@ -175,7 +175,7 @@
 proof -
   assume [simp]: "c \<noteq> 0"
   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
-  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
+  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
   finally show ?thesis .
 qed
 
@@ -183,7 +183,7 @@
 proof -
   assume [simp]: "c \<noteq> 0"
   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
-  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
+  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -194,10 +194,10 @@
   using nonzero_neg_divide_eq_eq[of b a c] by auto
 
 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
-  by (simp add: divide_inverse mult_assoc)
+  by (simp add: divide_inverse mult.assoc)
 
 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
-  by (drule sym) (simp add: divide_inverse mult_assoc)
+  by (drule sym) (simp add: divide_inverse mult.assoc)
 
 lemma add_divide_eq_iff [field_simps]:
   "z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
@@ -298,7 +298,7 @@
   fix a :: 'a
   assume "a \<noteq> 0"
   thus "inverse a * a = 1" by (rule field_inverse)
-  thus "a * inverse a = 1" by (simp only: mult_commute)
+  thus "a * inverse a = 1" by (simp only: mult.commute)
 next
   fix a b :: 'a
   show "a / b = a * inverse b" by (rule field_divide_inverse)
@@ -325,7 +325,7 @@
 
 lemma nonzero_mult_divide_mult_cancel_right [simp]:
   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
-by (simp add: mult_commute [of _ c])
+by (simp add: mult.commute [of _ c])
 
 lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
   by (simp add: divide_inverse mult_ac)
@@ -346,7 +346,7 @@
   also have "\<dots> = (x * z + y * w) / (y * z)"
     by (simp only: add_divide_distrib)
   finally show ?thesis
-    by (simp only: mult_commute)
+    by (simp only: mult.commute)
 qed
 
 text{*Special Cancellation Simprules for Division*}
@@ -406,7 +406,7 @@
 
 lemma inverse_divide [simp]:
   "inverse (a / b) = b / a"
-  by (simp add: divide_inverse mult_commute)
+  by (simp add: divide_inverse mult.commute)
 
 
 text {* Calculations with fractions *}
@@ -433,7 +433,7 @@
 
 lemma divide_divide_eq_left [simp]:
   "(a / b) / c = a / (b * c)"
-  by (simp add: divide_inverse mult_assoc)
+  by (simp add: divide_inverse mult.assoc)
 
 lemma divide_divide_times_eq:
   "(x / y) / (z / w) = (x * w) / (y * z)"
@@ -538,7 +538,7 @@
     by (simp add: apos invle less_imp_le mult_left_mono)
   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
     by (simp add: bpos less_imp_le mult_right_mono)
-  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
+  thus "b \<le> a"  by (simp add: mult.assoc apos bpos less_imp_not_eq2)
 qed
 
 lemma inverse_positive_imp_positive:
@@ -669,7 +669,7 @@
   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
   also have "... = (a*c \<le> b)"
-    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -679,7 +679,7 @@
   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
   also have "... = (b \<le> a*c)"
-    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -689,7 +689,7 @@
   hence "(a < b/c) = (a*c < (b/c)*c)"
     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
   also have "... = (a*c < b)"
-    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -699,7 +699,7 @@
   hence "(a < b/c) = ((b/c)*c < a*c)"
     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
   also have "... = (b < a*c)"
-    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -709,7 +709,7 @@
   hence "(b/c < a) = ((b/c)*c < a*c)"
     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
   also have "... = (b < a*c)"
-    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -719,7 +719,7 @@
   hence "(b/c < a) = (a*c < (b/c)*c)"
     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
   also have "... = (a*c < b)"
-    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -729,7 +729,7 @@
   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
   also have "... = (b \<le> a*c)"
-    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -739,7 +739,7 @@
   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
   also have "... = (a*c \<le> b)"
-    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
+    by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc) 
   finally show ?thesis .
 qed
 
@@ -1049,7 +1049,7 @@
 lemma divide_left_mono_neg: "a <= b 
     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
   apply (drule divide_left_mono [of _ _ "- c"])
-  apply (auto simp add: mult_commute)
+  apply (auto simp add: mult.commute)
 done
 
 lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/GCD.thy
--- a/src/HOL/GCD.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/GCD.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -410,7 +410,7 @@
   apply (rule_tac n = k in coprime_dvd_mult_nat)
   apply (simp add: gcd_assoc_nat)
   apply (simp add: gcd_commute_nat)
-  apply (simp_all add: mult_commute)
+  apply (simp_all add: mult.commute)
 done
 
 lemma gcd_mult_cancel_int:
@@ -432,9 +432,9 @@
   with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
   from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
   with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
-  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult_commute)
+  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult.commute)
   with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
-  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult_commute)
+  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult.commute)
   with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
   from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
   moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
@@ -456,7 +456,7 @@
 
 lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
   apply (subst (1 2) gcd_commute_nat)
-  apply (subst add_commute)
+  apply (subst add.commute)
   apply simp
 done
 
@@ -496,16 +496,16 @@
 done
 
 lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
-by (metis gcd_red_int mod_add_self1 add_commute)
+by (metis gcd_red_int mod_add_self1 add.commute)
 
 lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
-by (metis gcd_add1_int gcd_commute_int add_commute)
+by (metis gcd_add1_int gcd_commute_int add.commute)
 
 lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
 by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
 
 lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
-by (metis gcd_commute_int gcd_red_int mod_mult_self1 add_commute)
+by (metis gcd_commute_int gcd_red_int mod_mult_self1 add.commute)
 
 
 (* to do: differences, and all variations of addition rules
@@ -778,7 +778,7 @@
     by (auto simp:div_gcd_coprime_nat)
   hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
       ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
-    apply (subst (1 2) mult_commute)
+    apply (subst (1 2) mult.commute)
     apply (subst gcd_mult_distrib_nat [symmetric])
     apply simp
     done
@@ -820,10 +820,10 @@
     from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
     with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
     from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
-    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
     with z have th_1: "a' dvd b' * c" by auto
     from coprime_dvd_mult_nat[OF ab'(3)] th_1
-    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
+    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
     from ab' have "a = ?g*a'" by algebra
     with thb thc have ?thesis by blast }
   ultimately show ?thesis by blast
@@ -844,10 +844,10 @@
     with dc have th0: "a' dvd b*c"
       using dvd_trans[of a' a "b*c"] by simp
     from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
-    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
     with z have th_1: "a' dvd b' * c" by auto
     from coprime_dvd_mult_int[OF ab'(3)] th_1
-    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
+    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
     from ab' have "a = ?g*a'" by algebra
     with thb thc have ?thesis by blast }
   ultimately show ?thesis by blast
@@ -869,13 +869,13 @@
     from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
       by (simp add: ab'(1,2)[symmetric])
     hence "?g^n*a'^n dvd ?g^n *b'^n"
-      by (simp only: power_mult_distrib mult_commute)
+      by (simp only: power_mult_distrib mult.commute)
     with zn z n have th0:"a'^n dvd b'^n" by auto
     have "a' dvd a'^n" by (simp add: m)
     with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
-    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
     from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
-    have "a' dvd b'" by (subst (asm) mult_commute, blast)
+    have "a' dvd b'" by (subst (asm) mult.commute, blast)
     hence "a'*?g dvd b'*?g" by simp
     with ab'(1,2)  have ?thesis by simp }
   ultimately show ?thesis by blast
@@ -897,14 +897,14 @@
     from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
       by (simp add: ab'(1,2)[symmetric])
     hence "?g^n*a'^n dvd ?g^n *b'^n"
-      by (simp only: power_mult_distrib mult_commute)
+      by (simp only: power_mult_distrib mult.commute)
     with zn z n have th0:"a'^n dvd b'^n" by auto
     have "a' dvd a'^n" by (simp add: m)
     with th0 have "a' dvd b'^n"
       using dvd_trans[of a' "a'^n" "b'^n"] by simp
-    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
     from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
-    have "a' dvd b'" by (subst (asm) mult_commute, blast)
+    have "a' dvd b'" by (subst (asm) mult.commute, blast)
     hence "a'*?g dvd b'*?g" by simp
     with ab'(1,2)  have ?thesis by simp }
   ultimately show ?thesis by blast
@@ -922,7 +922,7 @@
 proof-
   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
     unfolding dvd_def by blast
-  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
   hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   from n' k show ?thesis unfolding dvd_def by auto
@@ -934,7 +934,7 @@
 proof-
   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
     unfolding dvd_def by blast
-  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
   hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   from n' k show ?thesis unfolding dvd_def by auto
@@ -1218,14 +1218,14 @@
           from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
             by simp
           hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
-            by (simp only: mult_assoc distrib_left)
+            by (simp only: mult.assoc distrib_left)
           hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
             by algebra
           hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
           hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
             by (simp only: diff_add_assoc[OF dble, of d, symmetric])
           hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
-            by (simp only: diff_mult_distrib2 add_commute mult_ac)
+            by (simp only: diff_mult_distrib2 add.commute mult_ac)
           hence ?thesis using H(1,2)
             apply -
             apply (rule exI[where x=d], simp)
@@ -1674,7 +1674,7 @@
 apply(auto simp add:inj_on_def)
 apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
 apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
-             dvd.neq_le_trans dvd_triv_right mult_commute)
+             dvd.neq_le_trans dvd_triv_right mult.commute)
 done
 
 text{* Nitpick: *}
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Groups.thy
--- a/src/HOL/Groups.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Groups.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -158,6 +158,8 @@
 
 end
 
+hide_fact add_assoc
+
 class ab_semigroup_add = semigroup_add +
   assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
 begin
@@ -165,13 +167,15 @@
 sublocale add!: abel_semigroup plus
   by default (fact add_commute)
 
-lemmas add_left_commute [algebra_simps, field_simps] = add.left_commute
+declare add.left_commute [algebra_simps, field_simps]
 
-theorems add_ac = add_assoc add_commute add_left_commute
+theorems add_ac = add.assoc add.commute add.left_commute
 
 end
 
-theorems add_ac = add_assoc add_commute add_left_commute
+hide_fact add_commute
+
+theorems add_ac = add.assoc add.commute add.left_commute
 
 class semigroup_mult = times +
   assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
@@ -182,6 +186,8 @@
 
 end
 
+hide_fact mult_assoc
+
 class ab_semigroup_mult = semigroup_mult +
   assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
 begin
@@ -189,13 +195,15 @@
 sublocale mult!: abel_semigroup times
   by default (fact mult_commute)
 
-lemmas mult_left_commute [algebra_simps, field_simps] = mult.left_commute
+declare mult.left_commute [algebra_simps, field_simps]
 
-theorems mult_ac = mult_assoc mult_commute mult_left_commute
+theorems mult_ac = mult.assoc mult.commute mult.left_commute
 
 end
 
-theorems mult_ac = mult_assoc mult_commute mult_left_commute
+hide_fact mult_commute
+
+theorems mult_ac = mult.assoc mult.commute mult.left_commute
 
 class monoid_add = zero + semigroup_add +
   assumes add_0_left: "0 + a = a"
@@ -325,7 +333,7 @@
 next
   fix a b c :: 'a
   assume "b + a = c + a"
-  then have "a + b = a + c" by (simp only: add_commute)
+  then have "a + b = a + c" by (simp only: add.commute)
   then show "b = c" by (rule add_imp_eq)
 qed
 
@@ -349,7 +357,7 @@
   assumes "a + b = 0" shows "- a = b"
 proof -
   have "- a = - a + (a + b)" using assms by simp
-  also have "\<dots> = b" by (simp add: add_assoc [symmetric])
+  also have "\<dots> = b" by (simp add: add.assoc [symmetric])
   finally show ?thesis .
 qed
 
@@ -381,36 +389,36 @@
   fix a b c :: 'a
   assume "a + b = a + c"
   then have "- a + a + b = - a + a + c"
-    unfolding add_assoc by simp
+    unfolding add.assoc by simp
   then show "b = c" by simp
 next
   fix a b c :: 'a
   assume "b + a = c + a"
   then have "b + a + - a = c + a  + - a" by simp
-  then show "b = c" unfolding add_assoc by simp
+  then show "b = c" unfolding add.assoc by simp
 qed
 
 lemma minus_add_cancel [simp]:
   "- a + (a + b) = b"
-  by (simp add: add_assoc [symmetric])
+  by (simp add: add.assoc [symmetric])
 
 lemma add_minus_cancel [simp]:
   "a + (- a + b) = b"
-  by (simp add: add_assoc [symmetric])
+  by (simp add: add.assoc [symmetric])
 
 lemma diff_add_cancel [simp]:
   "a - b + b = a"
-  by (simp only: diff_conv_add_uminus add_assoc) simp
+  by (simp only: diff_conv_add_uminus add.assoc) simp
 
 lemma add_diff_cancel [simp]:
   "a + b - b = a"
-  by (simp only: diff_conv_add_uminus add_assoc) simp
+  by (simp only: diff_conv_add_uminus add.assoc) simp
 
 lemma minus_add:
   "- (a + b) = - b + - a"
 proof -
   have "(a + b) + (- b + - a) = 0"
-    by (simp only: add_assoc add_minus_cancel) simp
+    by (simp only: add.assoc add_minus_cancel) simp
   then show "- (a + b) = - b + - a"
     by (rule minus_unique)
 qed
@@ -419,7 +427,7 @@
   "a - b = 0 \<longleftrightarrow> a = b"
 proof
   assume "a - b = 0"
-  have "a = (a - b) + b" by (simp add: add_assoc)
+  have "a = (a - b) + b" by (simp add: add.assoc)
   also have "\<dots> = b" using `a - b = 0` by simp
   finally show "a = b" .
 next
@@ -484,7 +492,7 @@
 next
   assume "a + b = 0"
   moreover have "a + (b + - b) = (a + b) + - b"
-    by (simp only: add_assoc)
+    by (simp only: add.assoc)
   ultimately show "a = - b" by simp
 qed
 
@@ -502,15 +510,15 @@
 
 lemma minus_diff_eq [simp]:
   "- (a - b) = b - a"
-  by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add_assoc minus_add_cancel) simp
+  by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add.assoc minus_add_cancel) simp
 
 lemma add_diff_eq [algebra_simps, field_simps]:
   "a + (b - c) = (a + b) - c"
-  by (simp only: diff_conv_add_uminus add_assoc)
+  by (simp only: diff_conv_add_uminus add.assoc)
 
 lemma diff_add_eq_diff_diff_swap:
   "a - (b + c) = a - c - b"
-  by (simp only: diff_conv_add_uminus add_assoc minus_add)
+  by (simp only: diff_conv_add_uminus add.assoc minus_add)
 
 lemma diff_eq_eq [algebra_simps, field_simps]:
   "a - b = c \<longleftrightarrow> a = c + b"
@@ -522,7 +530,7 @@
 
 lemma diff_diff_eq2 [algebra_simps, field_simps]:
   "a - (b - c) = (a + c) - b"
-  by (simp only: diff_conv_add_uminus add_assoc) simp
+  by (simp only: diff_conv_add_uminus add.assoc) simp
 
 lemma diff_eq_diff_eq:
   "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
@@ -543,13 +551,13 @@
   fix a b c :: 'a
   assume "a + b = a + c"
   then have "- a + a + b = - a + a + c"
-    by (simp only: add_assoc)
+    by (simp only: add.assoc)
   then show "b = c" by simp
 qed
 
 lemma uminus_add_conv_diff [simp]:
   "- a + b = b - a"
-  by (simp add: add_commute)
+  by (simp add: add.commute)
 
 lemma minus_add_distrib [simp]:
   "- (a + b) = - a + - b"
@@ -595,13 +603,13 @@
 
 lemma add_right_mono:
   "a \<le> b \<Longrightarrow> a + c \<le> b + c"
-by (simp add: add_commute [of _ c] add_left_mono)
+by (simp add: add.commute [of _ c] add_left_mono)
 
 text {* non-strict, in both arguments *}
 lemma add_mono:
   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
   apply (erule add_right_mono [THEN order_trans])
-  apply (simp add: add_commute add_left_mono)
+  apply (simp add: add.commute add_left_mono)
   done
 
 end
@@ -616,7 +624,7 @@
 
 lemma add_strict_right_mono:
   "a < b \<Longrightarrow> a + c < b + c"
-by (simp add: add_commute [of _ c] add_strict_left_mono)
+by (simp add: add.commute [of _ c] add_strict_left_mono)
 
 text{*Strict monotonicity in both arguments*}
 lemma add_strict_mono:
@@ -665,7 +673,7 @@
 lemma add_less_imp_less_right:
   "a + c < b + c \<Longrightarrow> a < b"
 apply (rule add_less_imp_less_left [of c])
-apply (simp add: add_commute)  
+apply (simp add: add.commute)  
 done
 
 lemma add_less_cancel_left [simp]:
@@ -682,7 +690,7 @@
 
 lemma add_le_cancel_right [simp]:
   "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
-  by (simp add: add_commute [of a c] add_commute [of b c])
+  by (simp add: add.commute [of a c] add.commute [of b c])
 
 lemma add_le_imp_le_right:
   "a + c \<le> b + c \<Longrightarrow> a \<le> b"
@@ -721,45 +729,45 @@
 
 lemma add_diff_assoc:
   "c + (b - a) = c + b - a"
-  using `a \<le> b` by (auto simp add: le_iff_add add_left_commute [of c])
+  using `a \<le> b` by (auto simp add: le_iff_add add.left_commute [of c])
 
 lemma add_diff_assoc2:
   "b - a + c = b + c - a"
-  using `a \<le> b` by (auto simp add: le_iff_add add_assoc)
+  using `a \<le> b` by (auto simp add: le_iff_add add.assoc)
 
 lemma diff_add_assoc:
   "c + b - a = c + (b - a)"
-  using `a \<le> b` by (simp add: add_commute add_diff_assoc)
+  using `a \<le> b` by (simp add: add.commute add_diff_assoc)
 
 lemma diff_add_assoc2:
   "b + c - a = b - a + c"
-  using `a \<le> b`by (simp add: add_commute add_diff_assoc)
+  using `a \<le> b`by (simp add: add.commute add_diff_assoc)
 
 lemma diff_diff_right:
   "c - (b - a) = c + a - b"
-  by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add_commute)
+  by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute)
 
 lemma diff_add:
   "b - a + a = b"
-  by (simp add: add_commute add_diff_inverse)
+  by (simp add: add.commute add_diff_inverse)
 
 lemma le_add_diff:
   "c \<le> b + c - a"
-  by (auto simp add: add_commute diff_add_assoc2 le_iff_add)
+  by (auto simp add: add.commute diff_add_assoc2 le_iff_add)
 
 lemma le_imp_diff_is_add:
   "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
-  by (auto simp add: add_commute add_diff_inverse)
+  by (auto simp add: add.commute add_diff_inverse)
 
 lemma le_diff_conv2:
   "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P
   then have "c + a \<le> b - a + a" by (rule add_right_mono)
-  then show ?Q by (simp add: add_diff_inverse add_commute)
+  then show ?Q by (simp add: add_diff_inverse add.commute)
 next
   assume ?Q
-  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add_commute)
+  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add.commute)
   then show ?P by simp
 qed
 
@@ -860,7 +868,7 @@
 
 lemma add_increasing2:
   "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
-  by (simp add: add_increasing add_commute [of a])
+  by (simp add: add_increasing add.commute [of a])
 
 lemma add_strict_increasing:
   "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
@@ -883,7 +891,7 @@
   fix a b c :: 'a
   assume "c + a \<le> c + b"
   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
-  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
+  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)
   thus "a \<le> b" by simp
 qed
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Groups_Big.thy
--- a/src/HOL/Groups_Big.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Groups_Big.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -958,7 +958,7 @@
   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
 proof-
   have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
-  also have "\<dots> = ?r" by (simp add: mult_commute)
+  also have "\<dots> = ?r" by (simp add: mult.commute)
   finally show ?thesis by auto
 qed
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/HOL.thy
--- a/src/HOL/HOL.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/HOL.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -977,10 +977,10 @@
   by blast
 
 lemma eq_ac:
-  shows eq_commute: "(a=b) = (b=a)"
-    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
-    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
-lemma neq_commute: "(a~=b) = (b~=a)" by iprover
+  shows eq_commute: "a = b \<longleftrightarrow> b = a"
+    and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
+    and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
+lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
 
 lemma conj_comms:
   shows conj_commute: "(P&Q) = (Q&P)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/HOLCF/FOCUS/Buffer_adm.thy
--- a/src/HOL/HOLCF/FOCUS/Buffer_adm.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/HOLCF/FOCUS/Buffer_adm.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -123,7 +123,7 @@
 apply (rule allI)
 apply (induct_tac "i")
 apply ( simp)
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 apply (intro strip)
 apply (subst BufAC_Cmt_F_def3)
 apply (drule_tac P="%x. x" in BufAC_Cmt_F_def3 [THEN subst])
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Hahn_Banach/Function_Norm.thy
--- a/src/HOL/Hahn_Banach/Function_Norm.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Hahn_Banach/Function_Norm.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -141,7 +141,7 @@
             then show "0 \<le> inverse \<parallel>x\<parallel>" by (rule order_less_imp_le)
           qed
           also have "\<dots> = c * (\<parallel>x\<parallel> * inverse \<parallel>x\<parallel>)"
-            by (rule Groups.mult_assoc)
+            by (rule Groups.mult.assoc)
           also
           from gt have "\<parallel>x\<parallel> \<noteq> 0" by simp
           then have "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by simp 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Hahn_Banach/Vector_Space.thy
--- a/src/HOL/Hahn_Banach/Vector_Space.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Hahn_Banach/Vector_Space.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -209,9 +209,9 @@
 proof
   from y have "y = 0 + y" by simp
   also from x y have "\<dots> = (- x + x) + y" by simp
-  also from x y have "\<dots> = - x + (x + y)" by (simp add: add_assoc)
+  also from x y have "\<dots> = - x + (x + y)" by (simp add: add.assoc)
   also assume "x + y = x + z"
-  also from x z have "- x + (x + z) = - x + x + z" by (simp add: add_assoc)
+  also from x z have "- x + (x + z) = - x + x + z" by (simp add: add.assoc)
   also from x z have "\<dots> = z" by simp
   finally show "y = z" .
 next
@@ -228,7 +228,7 @@
   by (simp only: add_assoc [symmetric])
 
 lemma mult_left_commute: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
-  by (simp add: mult_commute mult_assoc2)
+  by (simp add: mult.commute mult_assoc2)
 
 lemma mult_zero_uniq:
   assumes x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Hoare_Parallel/RG_Examples.thy
--- a/src/HOL/Hoare_Parallel/RG_Examples.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Hoare_Parallel/RG_Examples.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -52,7 +52,7 @@
    apply force
   apply(erule le_less_trans2) 
   apply(case_tac t,simp+)
-  apply (simp add:add_commute)
+  apply (simp add:add.commute)
   apply(simp add: add_le_mono)
 apply(rule Basic)
    apply simp
@@ -61,7 +61,7 @@
     apply simp
    apply(erule le_less_trans2)
    apply(case_tac t,simp+)
-   apply (simp add:add_commute)
+   apply (simp add:add.commute)
    apply(rule add_le_mono, auto)
 done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/IMP/ACom.thy
--- a/src/HOL/IMP/ACom.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/IMP/ACom.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -104,7 +104,7 @@
 apply(induction c arbitrary: f p)
 apply (auto simp: anno_def nth_append nth_Cons numeral_eq_Suc shift_def
             split: nat.split)
-  apply (metis add_Suc_right add_diff_inverse nat_add_commute)
+  apply (metis add_Suc_right add_diff_inverse add.commute)
  apply(rule_tac f=f in arg_cong)
  apply arith
 apply (metis less_Suc_eq)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/IMP/Abs_Int0.thy
--- a/src/HOL/IMP/Abs_Int0.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/IMP/Abs_Int0.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -317,7 +317,7 @@
 "m_s S X = (\<Sum> x \<in> X. m(S x))"
 
 lemma m_s_h: "finite X \<Longrightarrow> m_s S X \<le> h * card X"
-by(simp add: m_s_def) (metis nat_mult_commute of_nat_id setsum_bounded[OF h])
+by(simp add: m_s_def) (metis mult.commute of_nat_id setsum_bounded[OF h])
 
 fun m_o :: "'av st option \<Rightarrow> vname set \<Rightarrow> nat" ("m\<^sub>o") where
 "m_o (Some S) X = m_s S X" |
diff -r de51a86fc903 -r cc97b347b301 src/HOL/IMP/Abs_Int1.thy
--- a/src/HOL/IMP/Abs_Int1.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/IMP/Abs_Int1.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -109,7 +109,7 @@
 "m_s S X = (\<Sum> x \<in> X. m(fun S x))"
 
 lemma m_s_h: "finite X \<Longrightarrow> m_s S X \<le> h * card X"
-by(simp add: m_s_def) (metis nat_mult_commute of_nat_id setsum_bounded[OF h])
+by(simp add: m_s_def) (metis mult.commute of_nat_id setsum_bounded[OF h])
 
 definition m_o :: "'av st option \<Rightarrow> vname set \<Rightarrow> nat" ("m\<^sub>o") where
 "m_o opt X = (case opt of None \<Rightarrow> h * card X + 1 | Some S \<Rightarrow> m_s S X)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/IMP/Abs_Int_Den/Abs_Int_den1_ivl.thy
--- a/src/HOL/IMP/Abs_Int_Den/Abs_Int_den1_ivl.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/IMP/Abs_Int_Den/Abs_Int_den1_ivl.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -193,7 +193,7 @@
 proof
   case goal1 thus ?case
     by(auto simp add: filter_plus_ivl_def)
-      (metis rep_minus_ivl add_diff_cancel add_commute)+
+      (metis rep_minus_ivl add_diff_cancel add.commute)+
 next
   case goal2 thus ?case
     by(cases a1, cases a2,
diff -r de51a86fc903 -r cc97b347b301 src/HOL/IMP/Abs_Int_ITP/Abs_Int2_ivl_ITP.thy
--- a/src/HOL/IMP/Abs_Int_ITP/Abs_Int2_ivl_ITP.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/IMP/Abs_Int_ITP/Abs_Int2_ivl_ITP.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -208,7 +208,7 @@
 next
   case goal2 thus ?case
     by(auto simp add: filter_plus_ivl_def)
-      (metis gamma_minus_ivl add_diff_cancel add_commute)+
+      (metis gamma_minus_ivl add_diff_cancel add.commute)+
 next
   case goal3 thus ?case
     by(cases a1, cases a2,
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Int.thy
--- a/src/HOL/Int.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Int.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -567,7 +567,7 @@
   "(1 + z + z < 0) = (z < (0::int))"
 proof (cases z)
   case (nonneg n)
-  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
+  thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
                              le_imp_0_less [THEN order_less_imp_le])  
 next
   case (neg n)
@@ -585,7 +585,7 @@
   case (nonneg n)
   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
   thus ?thesis using  le_imp_0_less [OF le]
-    by (auto simp add: add_assoc) 
+    by (auto simp add: add.assoc) 
 next
   case (neg n)
   show ?thesis
@@ -594,7 +594,7 @@
     have "(0::int) < 1 + (int n + int n)"
       by (simp add: le_imp_0_less add_increasing) 
     also have "... = - (1 + z + z)" 
-      by (simp add: neg add_assoc [symmetric]) 
+      by (simp add: neg add.assoc [symmetric]) 
     also have "... = 0" by (simp add: eq) 
     finally have "0<0" ..
     thus False by blast
@@ -1079,7 +1079,7 @@
 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
 apply (rule iffI) 
  apply (frule pos_zmult_eq_1_iff_lemma)
- apply (simp add: mult_commute [of m]) 
+ apply (simp add: mult.commute [of m]) 
  apply (frule pos_zmult_eq_1_iff_lemma, auto) 
 done
 
@@ -1238,7 +1238,7 @@
 lemma zdvd_antisym_nonneg:
     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
   apply (simp add: dvd_def, auto)
-  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
+  apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
   done
 
 lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
@@ -1251,7 +1251,7 @@
   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
   from k k' have "a = a*k*k'" by simp
   with mult_cancel_left1[where c="a" and b="k*k'"]
-  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
+  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult.assoc)
   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
   thus ?thesis using k k' by auto
 qed
@@ -1301,7 +1301,7 @@
 proof-
   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
-    with h have False by (simp add: mult_assoc)}
+    with h have False by (simp add: mult.assoc)}
   hence "n = m * h" by blast
   thus ?thesis by simp
 qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Isar_Examples/Fibonacci.thy
--- a/src/HOL/Isar_Examples/Fibonacci.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Isar_Examples/Fibonacci.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -92,7 +92,7 @@
 proof -
   assume "0 < n"
   then have "gcd (n * k + m) n = gcd n (m mod n)"
-    by (simp add: gcd_non_0_nat add_commute)
+    by (simp add: gcd_non_0_nat add.commute)
   also from `0 < n` have "\<dots> = gcd m n"
     by (simp add: gcd_non_0_nat)
   finally show ?thesis .
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/BigO.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -489,7 +489,7 @@
   shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
   apply (simp add: bigo_def)
   apply (rule_tac x = "abs (inverse c)" in exI)
-  apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
+  apply (simp add: abs_mult [symmetric] mult.assoc [symmetric])
   done
 
 lemma bigo_const_mult4:
@@ -517,10 +517,10 @@
   apply (simp add: func_times)
   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
-  apply (simp add: mult_assoc [symmetric] abs_mult)
+  apply (simp add: mult.assoc [symmetric] abs_mult)
   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   apply (rule allI)
-  apply (subst mult_assoc)
+  apply (subst mult.assoc)
   apply (rule mult_left_mono)
   apply (erule spec)
   apply force
@@ -613,7 +613,7 @@
   apply (rule_tac x = c in exI)
   apply (rule allI)+
   apply (subst abs_mult)+
-  apply (subst mult_left_commute)
+  apply (subst mult.left_commute)
   apply (rule mult_left_mono)
   apply (erule spec)
   apply (rule abs_ge_zero)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Code_Target_Int.thy
--- a/src/HOL/Library/Code_Target_Int.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Code_Target_Int.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -103,7 +103,7 @@
   from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
   show ?thesis
     by (simp add: Let_def divmod_int_mod_div of_int_add [symmetric])
-      (simp add: * mult_commute)
+      (simp add: * mult.commute)
 qed
 
 declare of_int_code_if [code]
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Code_Target_Nat.thy
--- a/src/HOL/Library/Code_Target_Nat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Code_Target_Nat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -112,7 +112,7 @@
   from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
   show ?thesis
     by (simp add: Let_def divmod_nat_div_mod of_nat_add [symmetric])
-      (simp add: * mult_commute of_nat_mult add_commute)
+      (simp add: * mult.commute of_nat_mult add.commute)
 qed
 
 declare of_nat_code_if [code]
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Convex.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -210,7 +210,7 @@
     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
       by auto
     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
-      using s by (auto simp:add_commute)
+      using s by (auto simp:add.commute)
   }
   then show "convex s"
     unfolding convex_alt by auto
@@ -525,7 +525,7 @@
       using a_nonneg a1 asms by blast
     have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF `finite s` `i \<notin> s`] asms
-      by (auto simp only:add_commute)
+      by (auto simp only:add.commute)
     also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using i0 by auto
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
@@ -535,7 +535,7 @@
       by (auto simp: divide_inverse)
     also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
-      by (auto simp add:add_commute)
+      by (auto simp add:add.commute)
     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
       using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
         OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
@@ -768,7 +768,7 @@
     by auto
   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
     DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
-    unfolding inverse_eq_divide by (auto simp add: mult_assoc)
+    unfolding inverse_eq_divide by (auto simp add: mult.assoc)
   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
     using `b > 1` by (auto intro!:less_imp_le)
   from f''_ge0_imp_convex[OF pos_is_convex,
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Extended_Nat.thy
--- a/src/HOL/Library/Extended_Nat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Extended_Nat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -175,7 +175,7 @@
   by (simp_all add: eSuc_plus_1 add_ac)
 
 lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
-  by (simp only: add_commute[of m] iadd_Suc)
+  by (simp only: add.commute[of m] iadd_Suc)
 
 lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
   by (cases m, cases n, simp_all add: zero_enat_def)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Extended_Real.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -701,7 +701,7 @@
 lemma ereal_mult_strict_left_mono:
   "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
   using ereal_mult_strict_right_mono
-  by (simp add: mult_commute[of c])
+  by (simp add: mult.commute[of c])
 
 lemma ereal_mult_right_mono:
   fixes a b c :: ereal
@@ -717,7 +717,7 @@
   fixes a b c :: ereal
   shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
   using ereal_mult_right_mono
-  by (simp add: mult_commute[of c])
+  by (simp add: mult.commute[of c])
 
 lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
   by (simp add: one_ereal_def zero_ereal_def)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Float.thy
--- a/src/HOL/Library/Float.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Float.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -844,7 +844,7 @@
     hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
       by (auto simp: powr_minus)
     hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
-    hence "?S \<le> real m" unfolding mult_assoc by auto
+    hence "?S \<le> real m" unfolding mult.assoc by auto
     hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
     from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
     have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Formal_Power_Series.thy
--- a/src/HOL/Library/Formal_Power_Series.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -128,14 +128,14 @@
 proof
   fix a b c :: "'a fps"
   show "a + b + c = a + (b + c)"
-    by (simp add: fps_ext add_assoc)
+    by (simp add: fps_ext add.assoc)
 qed
 
 instance fps :: (ab_semigroup_add) ab_semigroup_add
 proof
   fix a b :: "'a fps"
   show "a + b = b + a"
-    by (simp add: fps_ext add_commute)
+    by (simp add: fps_ext add.commute)
 qed
 
 lemma fps_mult_assoc_lemma:
@@ -143,7 +143,7 @@
     and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
          (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
-  by (induct k) (simp_all add: Suc_diff_le setsum.distrib add_assoc)
+  by (induct k) (simp_all add: Suc_diff_le setsum.distrib add.assoc)
 
 instance fps :: (semiring_0) semigroup_mult
 proof
@@ -155,7 +155,7 @@
           (\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))"
       by (rule fps_mult_assoc_lemma)
     then show "((a * b) * c) $ n = (a * (b * c)) $ n"
-      by (simp add: fps_mult_nth setsum_right_distrib setsum_left_distrib mult_assoc)
+      by (simp add: fps_mult_nth setsum_right_distrib setsum_left_distrib mult.assoc)
   qed
 qed
 
@@ -174,7 +174,7 @@
     have "(\<Sum>i=0..n. a$i * b$(n - i)) = (\<Sum>i=0..n. a$(n - i) * b$i)"
       by (rule fps_mult_commute_lemma)
     then show "(a * b) $ n = (b * a) $ n"
-      by (simp add: fps_mult_nth mult_commute)
+      by (simp add: fps_mult_nth mult.commute)
   qed
 qed
 
@@ -397,7 +397,7 @@
 
 lemma X_mult_right_nth[simp]:
     "((f :: 'a::comm_semiring_1 fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))"
-  by (metis X_mult_nth mult_commute)
+  by (metis X_mult_nth mult.commute)
 
 lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then 1::'a::comm_ring_1 else 0)"
 proof (induct k)
@@ -419,14 +419,14 @@
     "(X^k * (f :: 'a::comm_ring_1 fps)) $n = (if n < k then 0 else f $ (n - k))"
   apply (induct k arbitrary: n)
   apply simp
-  unfolding power_Suc mult_assoc
+  unfolding power_Suc mult.assoc
   apply (case_tac n)
   apply auto
   done
 
 lemma X_power_mult_right_nth:
     "((f :: 'a::comm_ring_1 fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
-  by (metis X_power_mult_nth mult_commute)
+  by (metis X_power_mult_nth mult.commute)
 
 
 subsection{* Formal Power series form a metric space *}
@@ -666,7 +666,7 @@
   shows "inverse f * f = 1"
 proof -
   have c: "inverse f * f = f * inverse f"
-    by (simp add: mult_commute)
+    by (simp add: mult.commute)
   from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n"
     by (simp add: fps_inverse_def)
   from f0 have th0: "(inverse f * f) $ 0 = 1"
@@ -809,7 +809,7 @@
       setsum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
        by (rule setsum.reindex_bij_witness[where i="op - (n + 1)" and j="op - (n + 1)"]) auto
     have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n"
-      by (simp only: mult_commute)
+      by (simp only: mult.commute)
     also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
       by (simp add: fps_mult_nth setsum.distrib[symmetric])
     also have "\<dots> = setsum ?h {0..n+1}"
@@ -948,7 +948,7 @@
 
 lemma fps_nth_deriv_mult_const_right[simp]:
   "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
-  using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
+  using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult.commute)
 
 lemma fps_nth_deriv_setsum:
   "fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: 'a::comm_ring_1 fps)) S"
@@ -1026,7 +1026,7 @@
       {
         assume m0: "m \<noteq> 0"
         have "a ^k $ m = (a^l * a) $m"
-          by (simp add: k mult_commute)
+          by (simp add: k mult.commute)
         also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))"
           by (simp add: fps_mult_nth)
         also have "\<dots> = 0"
@@ -1114,7 +1114,7 @@
       unfolding power_mult_distrib[symmetric]
       apply (rule ssubst[where t = "a * inverse a" and s= 1])
       apply simp_all
-      apply (subst mult_commute)
+      apply (subst mult.commute)
       apply (rule inverse_mult_eq_1[OF a0])
       done
   }
@@ -1144,7 +1144,7 @@
   have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) = 0"
     unfolding power2_eq_square
     apply (simp add: field_simps)
-    apply (simp add: mult_assoc[symmetric])
+    apply (simp add: mult.assoc[symmetric])
     done
   then have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * (inverse a)\<^sup>2 =
       0 - fps_deriv a * (inverse a)\<^sup>2"
@@ -1194,7 +1194,7 @@
 lemma inverse_mult_eq_1':
   assumes f0: "f$0 \<noteq> (0::'a::field)"
   shows "f * inverse f= 1"
-  by (metis mult_commute inverse_mult_eq_1 f0)
+  by (metis mult.commute inverse_mult_eq_1 f0)
 
 lemma fps_divide_deriv:
   fixes a :: "'a::field fps"
@@ -1398,7 +1398,7 @@
     by simp
   have "a /?X = ?X *  Abs_fps (\<lambda>n::nat. setsum (op $ a) {0..n}) * inverse ?X"
     using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
-    by (simp add: fps_divide_def mult_assoc)
+    by (simp add: fps_divide_def mult.assoc)
   also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n::nat. setsum (op $ a) {0..n}) "
     by (simp add: mult_ac)
   finally show ?thesis
@@ -2289,14 +2289,14 @@
       done
     also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
       unfolding fps_deriv_nth
-      by (rule setsum.reindex_cong [of Suc]) (auto simp add: mult_assoc)
+      by (rule setsum.reindex_cong [of Suc]) (auto simp add: mult.assoc)
     finally have th0: "(fps_deriv (a oo b))$n =
       setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" .
 
     have "(((fps_deriv a) oo b) * (fps_deriv b))$n = setsum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
       unfolding fps_mult_nth by (simp add: mult_ac)
     also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
-      unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
+      unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult.assoc
       apply (rule setsum.cong)
       apply (rule refl)
       apply (rule setsum.mono_neutral_left)
@@ -2503,7 +2503,7 @@
     apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
     apply (rule setsum.cong)
     apply (rule refl)
-    apply (simp add: setsum.cartesian_product mult_assoc)
+    apply (simp add: setsum.cartesian_product mult.assoc)
     apply (rule setsum.mono_neutral_right[OF f])
     apply (simp add: subset_eq)
     apply presburger
@@ -2689,11 +2689,11 @@
 qed
 
 lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
-  by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc)
+  by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult.assoc)
 
 lemma fps_const_mult_apply_right:
   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
-  by (auto simp add: fps_const_mult_apply_left mult_commute)
+  by (auto simp add: fps_const_mult_apply_left mult.commute)
 
 lemma fps_compose_assoc:
   assumes c0: "c$0 = (0::'a::idom)"
@@ -2704,11 +2704,11 @@
     fix n
     have "?l$n = (setsum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n"
       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
-        setsum_right_distrib mult_assoc fps_setsum_nth)
+        setsum_right_distrib mult.assoc fps_setsum_nth)
     also have "\<dots> = ((setsum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
       by (simp add: fps_compose_setsum_distrib)
     also have "\<dots> = ?r$n"
-      apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
+      apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult.assoc)
       apply (rule setsum.cong)
       apply (rule refl)
       apply (rule setsum.mono_neutral_right)
@@ -3052,7 +3052,7 @@
     using a
     by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
   then have "fps_deriv ?l = fps_deriv ?r"
-    by (simp add: fps_deriv_L add_commute fps_divide_def divide_inverse)
+    by (simp add: fps_deriv_L add.commute fps_divide_def divide_inverse)
   then show ?thesis unfolding fps_deriv_eq_iff
     by (simp add: L_nth fps_inv_def)
 qed
@@ -3094,7 +3094,7 @@
   have x10: "?x1 $ 0 \<noteq> 0" by simp
   have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
   also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
-    apply (simp only: fps_divide_def  mult_assoc[symmetric] inverse_mult_eq_1[OF x10])
+    apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
     apply (simp add: field_simps)
     done
   finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
@@ -3119,7 +3119,7 @@
         case (Suc m)
         then show ?case unfolding th0
           apply (simp add: field_simps del: of_nat_Suc)
-          unfolding mult_assoc[symmetric] gbinomial_mult_1
+          unfolding mult.assoc[symmetric] gbinomial_mult_1
           apply (simp add: field_simps)
           done
       qed
@@ -3135,12 +3135,12 @@
   {
     assume h: ?rhs
     have th00: "\<And>x y. x * (a$0 * y) = a$0 * (x*y)"
-      by (simp add: mult_commute)
+      by (simp add: mult.commute)
     have "?l = ?r"
       apply (subst h)
       apply (subst (2) h)
       apply (clarsimp simp add: fps_eq_iff field_simps)
-      unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
+      unfolding mult.assoc[symmetric] th00 gbinomial_mult_1
       apply (simp add: field_simps gbinomial_mult_1)
       done
   }
@@ -3181,7 +3181,7 @@
   have th: "?r$0 \<noteq> 0" by simp
   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
     by (simp add: fps_inverse_deriv[OF th] fps_divide_def
-      power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg)
+      power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg)
   have eq: "inverse ?r $ 0 = 1"
     by (simp add: fps_inverse_def)
   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Fraction_Field.thy
--- a/src/HOL/Library/Fraction_Field.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Fraction_Field.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -252,7 +252,7 @@
   assume "q \<noteq> 0"
   then show "inverse q * q = 1"
     by (cases q rule: Fract_cases_nonzero)
-      (simp_all add: fract_expand eq_fract mult_commute)
+      (simp_all add: fract_expand eq_fract mult.commute)
 next
   fix q r :: "'a fract"
   show "q / r = q * inverse r" by (simp add: divide_fract_def)
@@ -398,7 +398,7 @@
     by (simp only: less_fract_def)
   show "q \<le> r \<or> r \<le> q"
     by (induct q, induct r)
-       (simp add: mult_commute, rule linorder_linear)
+       (simp add: mult.commute, rule linorder_linear)
 qed
 
 end
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Fundamental_Theorem_Algebra.thy
--- a/src/HOL/Library/Fundamental_Theorem_Algebra.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1006,9 +1006,9 @@
               apply (subst s)
               apply (subst r)
               apply (simp only: power_mult_distrib)
-              apply (subst mult_assoc [where b=s])
-              apply (subst mult_assoc [where a=u])
-              apply (subst mult_assoc [where b=u, symmetric])
+              apply (subst mult.assoc [where b=s])
+              apply (subst mult.assoc [where a=u])
+              apply (subst mult.assoc [where b=u, symmetric])
               apply (subst u [symmetric])
               apply (simp add: mult_ac power_add [symmetric])
               done
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Inner_Product.thy
--- a/src/HOL/Library/Inner_Product.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Inner_Product.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -136,7 +136,7 @@
     by (simp add: real_sqrt_mult_distrib)
   then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
     unfolding norm_eq_sqrt_inner
-    by (simp add: power2_eq_square mult_assoc)
+    by (simp add: power2_eq_square mult.assoc)
 qed
 
 end
@@ -209,11 +209,11 @@
 instance proof
   fix x y z r :: real
   show "inner x y = inner y x"
-    unfolding inner_real_def by (rule mult_commute)
+    unfolding inner_real_def by (rule mult.commute)
   show "inner (x + y) z = inner x z + inner y z"
     unfolding inner_real_def by (rule distrib_right)
   show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_real_def real_scaleR_def by (rule mult_assoc)
+    unfolding inner_real_def real_scaleR_def by (rule mult.assoc)
   show "0 \<le> inner x x"
     unfolding inner_real_def by simp
   show "inner x x = 0 \<longleftrightarrow> x = 0"
@@ -233,7 +233,7 @@
 instance proof
   fix x y z :: complex and r :: real
   show "inner x y = inner y x"
-    unfolding inner_complex_def by (simp add: mult_commute)
+    unfolding inner_complex_def by (simp add: mult.commute)
   show "inner (x + y) z = inner x z + inner y z"
     unfolding inner_complex_def by (simp add: distrib_right)
   show "inner (scaleR r x) y = r * inner x y"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Lattice_Algebras.thy
--- a/src/HOL/Library/Lattice_Algebras.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Lattice_Algebras.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -13,7 +13,7 @@
   apply (rule antisym)
   apply (simp_all add: le_infI)
   apply (rule add_le_imp_le_left [of "uminus a"])
-  apply (simp only: add_assoc [symmetric], simp add: diff_le_eq add.commute)
+  apply (simp only: add.assoc [symmetric], simp add: diff_le_eq add.commute)
   done
 
 lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
@@ -21,7 +21,7 @@
   have "c + inf a b = inf (c + a) (c + b)"
     by (simp add: add_inf_distrib_left)
   then show ?thesis
-    by (simp add: add_commute)
+    by (simp add: add.commute)
 qed
 
 end
@@ -32,10 +32,10 @@
 lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
   apply (rule antisym)
   apply (rule add_le_imp_le_left [of "uminus a"])
-  apply (simp only: add_assoc [symmetric], simp)
+  apply (simp only: add.assoc [symmetric], simp)
   apply (simp add: le_diff_eq add.commute)
   apply (rule le_supI)
-  apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
+  apply (rule add_le_imp_le_left [of "a"], simp only: add.assoc[symmetric], simp)+
   done
 
 lemma add_sup_distrib_right: "sup a b + c = sup (a + c) (b + c)"
@@ -43,7 +43,7 @@
   have "c + sup a b = sup (c+a) (c+b)"
     by (simp add: add_sup_distrib_left)
   then show ?thesis
-    by (simp add: add_commute)
+    by (simp add: add.commute)
 qed
 
 end
@@ -151,7 +151,7 @@
   then show ?r
     apply -
     apply (rule add_le_imp_le_right[of _ "uminus b" _])
-    apply (simp add: add_assoc)
+    apply (simp add: add.assoc)
     done
 next
   assume ?r
@@ -243,7 +243,7 @@
   then have "a + a + - a = - a"
     by simp
   then have "a + (a + - a) = - a"
-    by (simp only: add_assoc)
+    by (simp only: add.assoc)
   then have a: "- a = a"
     by simp
   show "a = 0"
@@ -309,7 +309,7 @@
 proof -
   from add_le_cancel_left [of "uminus a" "plus a a" zero]
   have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"
-    by (simp add: add_assoc[symmetric])
+    by (simp add: add.assoc[symmetric])
   then show ?thesis
     by simp
 qed
@@ -318,7 +318,7 @@
 proof -
   from add_le_cancel_left [of "uminus a" zero "plus a a"]
   have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
-    by (simp add: add_assoc[symmetric])
+    by (simp add: add.assoc[symmetric])
   then show ?thesis
     by simp
 qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/ListVector.thy
--- a/src/HOL/Library/ListVector.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/ListVector.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -94,7 +94,7 @@
 apply(simp)
 apply(case_tac zs)
  apply(simp)
-apply(simp add: add_assoc)
+apply(simp add: add.assoc)
 done
 
 subsection "Inner product"
@@ -146,7 +146,7 @@
 apply simp
 apply(case_tac ys)
 apply (simp)
-apply (simp add: distrib_left mult_assoc)
+apply (simp add: distrib_left mult.assoc)
 done
 
 end
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Multiset.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -225,8 +225,8 @@
 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
 proof
   assume ?rhs then show ?lhs
-  by (auto simp add: add_assoc add_commute [of "{#b#}"])
-    (drule sym, simp add: add_assoc [symmetric])
+  by (auto simp add: add.assoc add.commute [of "{#b#}"])
+    (drule sym, simp add: add.assoc [symmetric])
 next
   assume ?lhs
   show ?rhs
@@ -1494,9 +1494,9 @@
     case (Suc i')
     with Cons show ?thesis
       apply simp
-      apply (subst add_assoc)
-      apply (subst add_commute [of "{#v#}" "{#x#}"])
-      apply (subst add_assoc [symmetric])
+      apply (subst add.assoc)
+      apply (subst add.commute [of "{#v#}" "{#x#}"])
+      apply (subst add.assoc [symmetric])
       apply simp
       apply (rule mset_le_multiset_union_diff_commute)
       apply (simp add: mset_le_single nth_mem_multiset_of)
@@ -1597,7 +1597,7 @@
           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
           moreover from add have "M0 + K \<in> ?W" by simp
           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
-          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
+          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
         qed
         then show "N \<in> ?W" by (simp only: N)
       qed
@@ -1652,7 +1652,7 @@
  apply (rule_tac x = I in exI)
  apply (simp (no_asm))
  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
- apply (simp (no_asm_simp) add: add_assoc [symmetric])
+ apply (simp (no_asm_simp) add: add.assoc [symmetric])
  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="?S + ?T" in arg_cong)
  apply (simp add: diff_union_single_conv)
  apply (simp (no_asm_use) add: trans_def)
@@ -1695,7 +1695,7 @@
     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
  prefer 2
  apply force
-apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
+apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
 apply (erule trancl_trans)
 apply (rule r_into_trancl)
 apply (simp add: mult1_def set_of_def)
@@ -1760,7 +1760,7 @@
 apply auto
 apply (rule_tac x = a in exI)
 apply (rule_tac x = "C + M0" in exI)
-apply (simp add: add_assoc)
+apply (simp add: add.assoc)
 done
 
 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
@@ -1771,8 +1771,8 @@
 done
 
 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
-apply (subst add_commute [of B C])
-apply (subst add_commute [of D C])
+apply (subst add.commute [of B C])
+apply (subst add.commute [of D C])
 apply (erule union_less_mono2)
 done
 
@@ -1941,13 +1941,13 @@
 lemmas multi_count_eq = multiset_eq_iff [symmetric]
 
 lemma union_commute: "M + N = N + (M::'a multiset)"
-  by (fact add_commute)
+  by (fact add.commute)
 
 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
-  by (fact add_assoc)
+  by (fact add.assoc)
 
 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
-  by (fact add_left_commute)
+  by (fact add.left_commute)
 
 lemmas union_ac = union_assoc union_commute union_lcomm
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Nat_Bijection.thy
--- a/src/HOL/Library/Nat_Bijection.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Nat_Bijection.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -61,7 +61,7 @@
   "prod_decode (triangle k + m) = prod_decode_aux k m"
 apply (induct k arbitrary: m)
 apply (simp add: prod_decode_def)
-apply (simp only: triangle_Suc add_assoc)
+apply (simp only: triangle_Suc add.assoc)
 apply (subst prod_decode_aux.simps, simp)
 done
 
@@ -304,7 +304,7 @@
 apply (erule finite_induct, simp)
 apply (case_tac x)
 apply (simp add: even_nat_Suc_div_2 even_set_encode_iff vimage_Suc_insert_0)
-apply (simp add: finite_vimageI add_commute vimage_Suc_insert_Suc)
+apply (simp add: finite_vimageI add.commute vimage_Suc_insert_Suc)
 apply (simp add: set_encode_def finite_vimage_Suc_iff)
 done
 
@@ -334,7 +334,7 @@
   "n \<notin> set_decode z \<Longrightarrow> set_decode (2 ^ n + z) = insert n (set_decode z)"
  apply (induct n arbitrary: z, simp_all)
   apply (rule set_eqI, induct_tac x, simp, simp add: even_nat_Suc_div_2)
- apply (rule set_eqI, induct_tac x, simp, simp add: add_commute)
+ apply (rule set_eqI, induct_tac x, simp, simp add: add.commute)
 done
 
 lemma finite_set_decode [simp]: "finite (set_decode n)"
@@ -377,7 +377,7 @@
       by (metis finite_set_decode set_decode_inverse)
   thus ?thesis using assms
     apply auto
-    apply (simp add: set_encode_def nat_add_commute setsum.subset_diff)
+    apply (simp add: set_encode_def add.commute setsum.subset_diff)
     done
   qed
   thus ?thesis
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Polynomial.thy
--- a/src/HOL/Library/Polynomial.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Polynomial.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -624,9 +624,9 @@
 instance proof
   fix p q r :: "'a poly"
   show "(p + q) + r = p + (q + r)"
-    by (simp add: poly_eq_iff add_assoc)
+    by (simp add: poly_eq_iff add.assoc)
   show "p + q = q + p"
-    by (simp add: poly_eq_iff add_commute)
+    by (simp add: poly_eq_iff add.commute)
   show "0 + p = p"
     by (simp add: poly_eq_iff)
 qed
@@ -713,7 +713,7 @@
 lemma degree_add_eq_left:
   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
   using degree_add_eq_right [of q p]
-  by (simp add: add_commute)
+  by (simp add: add.commute)
 
 lemma degree_minus [simp]: "degree (- p) = degree p"
   unfolding degree_def by simp
@@ -824,7 +824,7 @@
   by (rule degree_le, simp add: coeff_eq_0)
 
 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
-  by (rule poly_eqI, simp add: mult_assoc)
+  by (rule poly_eqI, simp add: mult.assoc)
 
 lemma smult_0_right [simp]: "smult a 0 = 0"
   by (rule poly_eqI, simp)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Product_plus.thy
--- a/src/HOL/Library/Product_plus.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Product_plus.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -81,10 +81,10 @@
 subsection {* Class instances *}
 
 instance prod :: (semigroup_add, semigroup_add) semigroup_add
-  by default (simp add: prod_eq_iff add_assoc)
+  by default (simp add: prod_eq_iff add.assoc)
 
 instance prod :: (ab_semigroup_add, ab_semigroup_add) ab_semigroup_add
-  by default (simp add: prod_eq_iff add_commute)
+  by default (simp add: prod_eq_iff add.commute)
 
 instance prod :: (monoid_add, monoid_add) monoid_add
   by default (simp_all add: prod_eq_iff)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/RBT_Impl.thy
--- a/src/HOL/Library/RBT_Impl.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/RBT_Impl.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1416,7 +1416,7 @@
         moreover note feven[unfolded feven_def]
           (* FIXME: why does by(rule feven[unfolded feven_def]) not work? *)
         ultimately have "P (2 * (n div 2)) kvs" by -
-        thus ?thesis using True by (metis div_mod_equality' minus_nat.diff_0 nat_mult_commute)
+        thus ?thesis using True by (metis div_mod_equality' minus_nat.diff_0 mult.commute)
       next
         case False note ge0
         moreover from "1.prems" have n2: "n div 2 \<le> length kvs" by simp
@@ -1462,7 +1462,7 @@
         moreover note geven[unfolded geven_def]
         ultimately have "Q (2 * (n div 2)) kvs" by -
         thus ?thesis using True 
-          by(metis div_mod_equality' minus_nat.diff_0 nat_mult_commute)
+          by(metis div_mod_equality' minus_nat.diff_0 mult.commute)
       next
         case False note ge0
         moreover from "2.prems" have n2: "n div 2 \<le> length kvs" by simp
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Saturated.thy
--- a/src/HOL/Library/Saturated.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Saturated.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -61,7 +61,7 @@
   less_sat_def: "x < y \<longleftrightarrow> nat_of x < nat_of y"
 
 instance
-by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min.coboundedI1 nat_mult_commute)
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min.coboundedI1 mult.commute)
 
 end
 
@@ -117,14 +117,14 @@
       case True thus ?thesis by (simp add: sat_eq_iff)
     next
       case False with `a \<noteq> 0` show ?thesis
-        by (simp add: sat_eq_iff nat_mult_min_left nat_mult_min_right mult_assoc min.assoc min.absorb2)
+        by (simp add: sat_eq_iff nat_mult_min_left nat_mult_min_right mult.assoc min.assoc min.absorb2)
     qed
   qed
 next
   fix a :: "('a::len) sat"
   show "1 * a = a"
     apply (simp add: sat_eq_iff)
-    apply (metis One_nat_def len_gt_0 less_Suc0 less_zeroE linorder_not_less min.absorb_iff1 min_nat_of_len_of nat_mult_1_right nat_mult_commute)
+    apply (metis One_nat_def len_gt_0 less_Suc0 less_zeroE linorder_not_less min.absorb_iff1 min_nat_of_len_of nat_mult_1_right mult.commute)
     done
 next
   fix a b c :: "('a::len) sat"
@@ -143,7 +143,7 @@
 begin
 
 instance
-by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min.coboundedI1 nat_mult_commute)
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min.coboundedI1 mult.commute)
 
 end
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Library/Set_Algebras.thy
--- a/src/HOL/Library/Set_Algebras.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Library/Set_Algebras.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -108,7 +108,7 @@
   done
 
 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
-  by (auto simp add: elt_set_plus_def add_assoc)
+  by (auto simp add: elt_set_plus_def add.assoc)
 
 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)"
   apply (auto simp add: elt_set_plus_def set_plus_def)
@@ -216,7 +216,7 @@
 
 lemma set_times_rearrange2:
   "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
-  by (auto simp add: elt_set_times_def mult_assoc)
+  by (auto simp add: elt_set_times_def mult.assoc)
 
 lemma set_times_rearrange3:
   "((a::'a::semigroup_mult) *o B) * C = a *o (B * C)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Limits.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -702,7 +702,7 @@
       by (intro mult_mono' order_refl norm_g norm_ge_zero
                 mult_nonneg_nonneg K elim)
     also have "\<dots> = norm (f x) * (B * K)"
-      by (rule mult_assoc)
+      by (rule mult.assoc)
     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
   qed
   with f show ?thesis
@@ -716,7 +716,7 @@
   apply (rule add_left)
   apply (rule scaleR_right)
   apply (rule scaleR_left)
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   using bounded by fast
 
 lemma (in bounded_bilinear) Bfun_prod_Zfun:
@@ -1273,7 +1273,7 @@
 
 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
   unfolding tendsto_def eventually_sequentially
-  by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
+  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
 
 lemma Bseq_inverse_lemma:
   fixes x :: "'a::real_normed_div_algebra"
@@ -1530,7 +1530,7 @@
 lemma LIM_offset_zero:
   fixes a :: "'a::real_normed_vector"
   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
-by (drule_tac k="a" in LIM_offset, simp add: add_commute)
+by (drule_tac k="a" in LIM_offset, simp add: add.commute)
 
 lemma LIM_offset_zero_cancel:
   fixes a :: "'a::real_normed_vector"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/List.thy
--- a/src/HOL/List.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/List.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -3846,7 +3846,7 @@
 lemma append_replicate_commute:
   "replicate n x @ replicate k x = replicate k x @ replicate n x"
 apply (simp add: replicate_add [THEN sym])
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 done
 
 text{* Courtesy of Andreas Lochbihler: *}
@@ -4192,14 +4192,14 @@
 lemma sublist_shift_lemma:
      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
-by (induct xs rule: rev_induct) (simp_all add: add_commute)
+by (induct xs rule: rev_induct) (simp_all add: add.commute)
 
 lemma sublist_append:
      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
 apply (unfold sublist_def)
 apply (induct l' rule: rev_induct, simp)
 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 done
 
 lemma sublist_Cons:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Matrix_LP/Matrix.thy
--- a/src/HOL/Matrix_LP/Matrix.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Matrix_LP/Matrix.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1508,7 +1508,7 @@
   show "A + B + C = A + (B + C)"    
     apply (simp add: plus_matrix_def)
     apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
-    apply (simp_all add: add_assoc)
+    apply (simp_all add: add.assoc)
     done
   show "0 + A = A"
     apply (simp add: plus_matrix_def)
@@ -1528,7 +1528,7 @@
   show "A + B = B + A"
     apply (simp add: plus_matrix_def)
     apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
-    apply (simp_all add: add_commute)
+    apply (simp_all add: add.commute)
     done
   show "0 + A = A"
     apply (simp add: plus_matrix_def)
@@ -1689,7 +1689,7 @@
 lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
 apply (simp add: times_matrix_def)
 apply (subst transpose_mult_matrix)
-apply (simp_all add: mult_commute)
+apply (simp_all add: mult.commute)
 done
 
 lemma transpose_matrix_add: "transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B"
@@ -1731,7 +1731,7 @@
     apply (insert assms)
     apply (frule right_inverse_matrix_dim)
     by (simp add: right_inverse_matrix_def)
-  also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
+  also have "\<dots> = (Y * A) * X" by (simp add: mult.assoc)
   also have "\<dots> = X" 
     apply (insert assms)
     apply (frule left_inverse_matrix_dim)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Metis_Examples/Big_O.thy
--- a/src/HOL/Metis_Examples/Big_O.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Metis_Examples/Big_O.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -309,7 +309,7 @@
 apply (erule_tac x = x in allE)+
 apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs(f x)) * (ca * abs (g x))")
  apply (metis (no_types) abs_ge_zero abs_mult mult_mono')
-by (metis mult_assoc mult_left_commute abs_of_pos mult_left_commute abs_mult)
+by (metis mult.assoc mult.left_commute abs_of_pos mult.left_commute abs_mult)
 
 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
 by (metis bigo_mult bigo_refl set_times_mono3 subset_trans)
@@ -421,7 +421,7 @@
 
 lemma bigo_const_mult3: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
 apply (simp add: bigo_def)
-by (metis (no_types) abs_mult mult_assoc mult_1 order_refl left_inverse)
+by (metis (no_types) abs_mult mult.assoc mult_1 order_refl left_inverse)
 
 lemma bigo_const_mult4:
 "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> O(f) \<le> O(\<lambda>x. c * f x)"
@@ -444,7 +444,7 @@
   apply (rule_tac [2] ext)
    prefer 2
    apply simp
-  apply (simp add: mult_assoc [symmetric] abs_mult)
+  apply (simp add: mult.assoc [symmetric] abs_mult)
   (* couldn't get this proof without the step above *)
 proof -
   fix g :: "'b \<Rightarrow> 'a" and d :: 'a
@@ -530,7 +530,7 @@
  apply (rule abs_ge_zero)
 apply (unfold bigo_def)
 apply (auto simp add: abs_mult)
-by (metis abs_ge_zero mult_left_commute mult_left_mono)
+by (metis abs_ge_zero mult.left_commute mult_left_mono)
 
 lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
     (\<lambda>x. SUM y : A x. l x y * f(k x y)) =o
@@ -710,6 +710,6 @@
 lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs (h x)"
 apply (simp only: lesso_def bigo_alt_def)
 apply clarsimp
-by (metis add_commute diff_le_eq)
+by (metis add.commute diff_le_eq)
 
 end
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Metis_Examples/Binary_Tree.thy
--- a/src/HOL/Metis_Examples/Binary_Tree.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Metis_Examples/Binary_Tree.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -66,7 +66,7 @@
   qed
 next
   case (Br a t1 t2) thus ?case
-    by (metis n_leaves.simps(2) nat_add_commute reflect.simps(2))
+    by (metis n_leaves.simps(2) add.commute reflect.simps(2))
 qed
 
 lemma n_nodes_reflect: "n_nodes (reflect t) = n_nodes t"
@@ -74,7 +74,7 @@
   case Lf thus ?case by (metis reflect.simps(1))
 next
   case (Br a t1 t2) thus ?case
-    by (metis add_commute n_nodes.simps(2) reflect.simps(2))
+    by (metis add.commute n_nodes.simps(2) reflect.simps(2))
 qed
 
 lemma depth_reflect: "depth (reflect t) = depth t"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/MicroJava/Comp/CorrComp.thy
--- a/src/HOL/MicroJava/Comp/CorrComp.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/MicroJava/Comp/CorrComp.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -137,7 +137,7 @@
 apply (simp only: append_assoc)
 apply (erule thin_rl, erule thin_rl)
 apply (drule_tac x="pre @ instrs0" in spec)
-apply (simp add: add_assoc)
+apply (simp add: add.assoc)
 done
 
 lemma progression_refl: 
@@ -196,7 +196,7 @@
 apply simp
 apply (erule thin_rl, erule thin_rl)
 apply (drule_tac x="pre @ instr # instrs0" in spec)
-apply (simp add: add_assoc)
+apply (simp add: add.assoc)
 done
 
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -183,13 +183,13 @@
 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
 
 instance vec :: (semigroup_mult, finite) semigroup_mult
-  by default (vector mult_assoc)
+  by default (vector mult.assoc)
 
 instance vec :: (monoid_mult, finite) monoid_mult
   by default vector+
 
 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
-  by default (vector mult_commute)
+  by default (vector mult.commute)
 
 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
   by default vector
@@ -254,7 +254,7 @@
 instance vec :: (ring_char_0, finite) ring_char_0 ..
 
 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
-  by (vector mult_assoc)
+  by (vector mult.assoc)
 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
   by (vector field_simps)
 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
@@ -380,14 +380,14 @@
   done
 
 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
-  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
+  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)
   apply (subst setsum.commute)
   apply simp
   done
 
 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
-    setsum_right_distrib setsum_left_distrib mult_assoc)
+    setsum_right_distrib setsum_left_distrib mult.assoc)
   apply (subst setsum.commute)
   apply simp
   done
@@ -399,7 +399,7 @@
 
 lemma matrix_transpose_mul:
     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
-  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
+  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
 
 lemma matrix_eq:
   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
@@ -452,7 +452,7 @@
 
 lemma matrix_mult_vsum:
   "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
-  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
+  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
 
 lemma vector_componentwise:
   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
@@ -498,7 +498,7 @@
 lemma matrix_works:
   assumes lf: "linear f"
   shows "matrix f *v x = f (x::real ^ 'n)"
-  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
+  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
   apply clarify
   apply (rule linear_componentwise[OF lf, symmetric])
   done
@@ -518,7 +518,7 @@
 
 lemma matrix_vector_column:
   "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
-  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
+  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
 
 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   apply (rule adjoint_unique)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -3283,7 +3283,7 @@
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and `e > 0`
-      by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult_commute)
+      by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult.commute)
     finally have "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
@@ -5923,7 +5923,7 @@
         by (auto simp: inner_simps)
       then have "1 \<ge> inverse d * (x \<bullet> i - y \<bullet> i)" "1 \<ge> inverse d * (y \<bullet> i - x \<bullet> i)"
         apply (rule_tac[!] mult_left_le_imp_le[OF _ assms])
-        unfolding mult_assoc[symmetric]
+        unfolding mult.assoc[symmetric]
         using assms
         by (auto simp add: field_simps)
       then have "inverse d * (x \<bullet> i * 2) \<le> 2 + inverse d * (y \<bullet> i * 2)"
@@ -6493,7 +6493,7 @@
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and `e > 0`
-      by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult_commute)
+      by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult.commute)
     finally show "y \<in> s"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -127,7 +127,7 @@
   "(f has_field_derivative D) (at a within s) \<longleftrightarrow> ((\<lambda>z. (f z - f a) / (z - a)) ---> D) (at a within s)"
 proof -
   have 1: "\<And>w y. ~(w = a) ==> y / (w - a) - D = (y - (w - a)*D)/(w - a)"
-    by (metis divide_diff_eq_iff eq_iff_diff_eq_0 mult_commute)
+    by (metis divide_diff_eq_iff eq_iff_diff_eq_0 mult.commute)
   show ?thesis
     apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
     apply (simp add: LIM_zero_iff [where l = D, symmetric])
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Determinants.thy
--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -98,7 +98,7 @@
 lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
   apply (simp add: trace_def matrix_matrix_mult_def)
   apply (subst setsum.commute)
-  apply (simp add: mult_commute)
+  apply (simp add: mult.commute)
   done
 
 text {* Definition of determinant. *}
@@ -294,7 +294,7 @@
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes p: "p permutes (UNIV :: 'n::finite set)"
   shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
-  apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric])
+  apply (simp add: det_def setsum_right_distrib mult.assoc[symmetric])
   apply (subst sum_permutations_compose_right[OF p])
 proof (rule setsum.cong)
   let ?U = "UNIV :: 'n set"
@@ -318,7 +318,7 @@
     by blast
   show "of_int (sign (q \<circ> p)) * setprod (\<lambda>i. A$ p i$ (q \<circ> p) i) ?U =
     of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
-    by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
+    by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult)
 qed rule
 
 lemma det_permute_columns:
@@ -819,7 +819,7 @@
         unfolding permutation_permutes by auto
       have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
         "\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
-        unfolding mult_assoc[symmetric]
+        unfolding mult.assoc[symmetric]
         unfolding of_int_mult[symmetric]
         by (simp_all add: sign_idempotent)
       have ths: "?s q = ?s p * ?s (q \<circ> inv p)"
@@ -891,7 +891,7 @@
       by blast
     have *: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
       apply (drule_tac f="op + (- a)" in cong[OF refl])
-      apply (simp only: ab_left_minus add_assoc[symmetric])
+      apply (simp only: ab_left_minus add.assoc[symmetric])
       apply simp
       done
     from c ci
@@ -958,7 +958,7 @@
     apply (subst U)
     unfolding setsum.insert[OF fUk kUk]
     apply (subst th00)
-    unfolding add_assoc
+    unfolding add.assoc
     apply (subst det_row_add)
     unfolding thd0
     unfolding det_row_mul
@@ -1278,10 +1278,10 @@
   by (fact setprod_singleton_nat_seg)
 
 lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
-  by (simp add: eval_nat_numeral setprod_numseg mult_commute)
+  by (simp add: eval_nat_numeral setprod_numseg mult.commute)
 
 lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
-  by (simp add: eval_nat_numeral setprod_numseg mult_commute)
+  by (simp add: eval_nat_numeral setprod_numseg mult.commute)
 
 lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
   by (simp add: det_def sign_id)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
--- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -95,10 +95,10 @@
   unfolding uminus_vec_def by simp
 
 instance vec :: (semigroup_add, finite) semigroup_add
-  by default (simp add: vec_eq_iff add_assoc)
+  by default (simp add: vec_eq_iff add.assoc)
 
 instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
-  by default (simp add: vec_eq_iff add_commute)
+  by default (simp add: vec_eq_iff add.commute)
 
 instance vec :: (monoid_add, finite) monoid_add
   by default (simp_all add: vec_eq_iff)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Integration.thy
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -4926,7 +4926,7 @@
   apply (rule order_trans[OF mult_left_mono])
   apply (rule assms)
   apply (rule setsum_abs_ge_zero)
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   apply (rule mult_left_mono)
   apply (rule order_trans[of _ "setsum content p"])
   apply (rule eq_refl)
@@ -4961,7 +4961,7 @@
     apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
     defer
     unfolding setsum_left_distrib[symmetric]
-    apply (subst mult_commute)
+    apply (subst mult.commute)
     apply (rule mult_left_mono)
     apply (rule order_trans[of _ "setsum (content \<circ> snd) p"])
     apply (rule eq_refl)
@@ -6072,7 +6072,7 @@
       proof -
         case goal1
         then show ?case
-          apply (subst mult_commute, subst pos_le_divide_eq[symmetric])
+          apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
           using d(2)[rule_format,of "q i" i]
           using q[rule_format]
           apply (auto simp add: field_simps)
@@ -8349,7 +8349,7 @@
       then show "norm (f c) * norm (c - t) < e / 3"
         using False
         apply -
-        apply (subst mult_commute)
+        apply (subst mult.commute)
         apply (subst pos_less_divide_eq[symmetric])
         apply auto
         done
@@ -11081,7 +11081,7 @@
     show ?case
       apply (rule le_less_trans[OF norm_triangle_sub[of ig sg]])
       apply (subst real_sum_of_halves[of e,symmetric])
-      unfolding add_assoc[symmetric]
+      unfolding add.assoc[symmetric]
       apply (rule add_le_less_mono)
       defer
       apply (subst norm_minus_commute)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -425,7 +425,7 @@
       unfolding linear_setsum[OF lf]
       by (simp add: linear_cmul[OF lf])
     finally show "f x \<bullet> y = x \<bullet> ?w"
-      by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
   qed
   then show ?thesis
     unfolding adjoint_def choice_iff
@@ -848,7 +848,7 @@
         from h have "(x + y) \<in> span_induct_alt_help S"
           apply (induct rule: span_induct_alt_help.induct)
           apply simp
-          unfolding add_assoc
+          unfolding add.assoc
           apply (rule span_induct_alt_help_S)
           apply assumption
           apply simp
@@ -1083,7 +1083,7 @@
       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
         by (simp add: setsum.remove [OF fS xS] algebra_simps)
       also have "\<dots> = c*\<^sub>R x + y"
-        by (simp add: add_commute u)
+        by (simp add: add.commute u)
       finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
       then show ?thesis using th0 by blast
     next
@@ -1095,7 +1095,7 @@
         apply auto
         done
       show ?thesis using fS xS th0
-        by (simp add: th00 add_commute cong del: if_weak_cong)
+        by (simp add: th00 add.commute cong del: if_weak_cong)
     qed
     then show "?h (c*\<^sub>R x + y)"
       by fast
@@ -1472,7 +1472,7 @@
       from Basis_le_norm[OF i, of x]
       show "norm (?g i) \<le> norm (f i) * norm x"
         unfolding norm_scaleR
-        apply (subst mult_commute)
+        apply (subst mult.commute)
         apply (rule mult_mono)
         apply (auto simp add: field_simps)
         done
@@ -1494,7 +1494,7 @@
     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
       using `linear f` by (rule linear_bounded)
     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
-      by (simp add: mult_commute)
+      by (simp add: mult.commute)
   qed
 next
   assume "bounded_linear f"
@@ -1511,7 +1511,7 @@
     using lf unfolding linear_conv_bounded_linear
     by (rule bounded_linear.pos_bounded)
   then show ?thesis
-    by (simp only: mult_commute)
+    by (simp only: mult.commute)
 qed
 
 lemma bounded_linearI':
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Operator_Norm.thy
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -48,7 +48,7 @@
     using f.nonneg_bounded by auto
   then have "\<forall>x. norm (f x) / norm x \<le> b"
     by (clarify, case_tac "x = 0",
-      simp_all add: f.zero pos_divide_le_eq mult_commute)
+      simp_all add: f.zero pos_divide_le_eq mult.commute)
   then have "bdd_above (range (\<lambda>x. norm (f x) / norm x))"
     unfolding bdd_above_def by fast
   with UNIV_I show ?thesis
@@ -110,7 +110,7 @@
   also have "onorm f * norm (g x) \<le> onorm f * (onorm g * norm x)"
     by (rule mult_left_mono [OF onorm [OF g] onorm_pos_le [OF f]])
   finally show "norm ((f \<circ> g) x) \<le> onorm f * onorm g * norm x"
-    by (simp add: mult_assoc)
+    by (simp add: mult.assoc)
 qed
 
 lemma onorm_scaleR_lemma:
@@ -124,7 +124,7 @@
   have "\<bar>r\<bar> * norm (f x) \<le> \<bar>r\<bar> * (onorm f * norm x)"
     by (intro mult_left_mono onorm abs_ge_zero f)
   then show "norm (r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f * norm x"
-    by (simp only: norm_scaleR mult_assoc)
+    by (simp only: norm_scaleR mult.assoc)
 qed
 
 lemma onorm_scaleR:
@@ -142,7 +142,7 @@
     then have "onorm (\<lambda>x. inverse r *\<^sub>R r *\<^sub>R f x) \<le> \<bar>inverse r\<bar> * onorm (\<lambda>x. r *\<^sub>R f x)"
       by (rule onorm_scaleR_lemma)
     with `r \<noteq> 0` show "\<bar>r\<bar> * onorm f \<le> onorm (\<lambda>x. r *\<^sub>R f x)"
-      by (simp add: inverse_eq_divide pos_le_divide_eq mult_commute)
+      by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
   qed
 qed (simp add: onorm_zero)
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/PolyRoots.thy
--- a/src/HOL/Multivariate_Analysis/PolyRoots.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/PolyRoots.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -21,7 +21,7 @@
  shows   "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
 using setsum_gp_basic[of x n]
 apply (simp add: real_of_nat_def)
-by (metis eq_iff_diff_eq_0 mult_commute nonzero_eq_divide_eq)
+by (metis eq_iff_diff_eq_0 mult.commute nonzero_eq_divide_eq)
 
 lemma setsum_power_shift:
   fixes x :: "'a::{comm_ring,monoid_mult}"
@@ -41,7 +41,7 @@
   shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
 proof -
   have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
-    by (metis ab_semigroup_mult_class.mult_ac(1) assms mult_commute setsum_power_shift)
+    by (metis ab_semigroup_mult_class.mult_ac(1) assms mult.commute setsum_power_shift)
   also have "... =x^m * (1 - x^Suc(n-m))"
     by (metis ab_semigroup_mult_class.mult_ac(1) setsum_gp_basic)
   also have "... = x^m - x^Suc n"
@@ -58,7 +58,7 @@
                 else (x^m - x^Suc n) / (1 - x))"
 using setsum_gp_multiplied [of m n x] 
 apply (auto simp: real_of_nat_def)
-by (metis eq_iff_diff_eq_0 mult_commute nonzero_divide_eq_eq)
+by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
 
 lemma setsum_gp_offset:
   fixes x :: "'a::{comm_ring,division_ring_inverse_zero}"
@@ -133,7 +133,7 @@
 proof (induction n)
   case 0
   show ?case 
-    by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult_commute pos_divide_le_eq assms)
+    by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult.commute pos_divide_le_eq assms)
 next
   case (Suc n)
   then obtain M where M: "\<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n" ..
@@ -144,7 +144,7 @@
     then have norm1: "0 < norm z" "M \<le> norm z" "(e + norm (c (Suc n))) / e \<le> norm z"
       by auto
     then have norm2: "(e + norm (c (Suc n))) \<le> e * norm z"  "(norm z * norm z ^ n) > 0"
-      apply (metis assms less_divide_eq mult_commute not_le) 
+      apply (metis assms less_divide_eq mult.commute not_le) 
       using norm1 apply (metis mult_pos_pos zero_less_power)
       done
     have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n)) =
@@ -162,7 +162,7 @@
 qed
 
 lemma norm_lemma_xy: "\<lbrakk>abs b + 1 \<le> norm(y) - a; norm(x) \<le> a\<rbrakk> \<Longrightarrow> b \<le> norm(x + y)"
-  by (metis abs_add_one_not_less_self add_commute diff_le_eq dual_order.trans le_less_linear 
+  by (metis abs_add_one_not_less_self add.commute diff_le_eq dual_order.trans le_less_linear 
          norm_diff_ineq)
 
 lemma polyfun_extremal:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -2725,7 +2725,7 @@
 
 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
   unfolding bounded_def
-  by auto (metis add_commute add_le_cancel_right dist_commute dist_triangle_le)
+  by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
 
 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
   unfolding bounded_any_center [where a=0]
@@ -2859,7 +2859,7 @@
   then show ?thesis
     unfolding bounded_pos
     apply (rule_tac x="b*B" in exI)
-    using b B by (auto simp add: mult_commute)
+    using b B by (auto simp add: mult.commute)
 qed
 
 lemma bounded_scaling:
@@ -2901,7 +2901,7 @@
 
 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
   by (auto simp: bounded_def bdd_below_def dist_real_def)
-     (metis abs_le_D1 add_commute diff_le_eq)
+     (metis abs_le_D1 add.commute diff_le_eq)
 
 (* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)
 
@@ -6801,7 +6801,7 @@
     then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
       unfolding image_iff Bex_def mem_box
       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
-      apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_distrib inner_diff_left)
+      apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
       done
   }
   moreover
@@ -6811,7 +6811,7 @@
     then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
       unfolding image_iff Bex_def mem_box
       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
-      apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_distrib inner_diff_left)
+      apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
       done
   }
   ultimately show ?thesis using False by (auto simp: cbox_def)
@@ -7175,7 +7175,7 @@
       then show "norm (f b) / norm b * norm x \<le> norm (f x)"
         using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
         unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
-        by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
+        by (auto simp add: mult.commute pos_le_divide_eq pos_divide_le_eq)
     qed
   }
   ultimately show ?thesis by auto
@@ -7406,7 +7406,7 @@
     proof (cases "d = 0")
       case True
       have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
-        by (metis mult_zero_left mult_commute real_mult_le_cancel_iff1)
+        by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
       from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
         by (simp add: *)
       then show ?thesis using `e>0` by auto
@@ -7431,12 +7431,12 @@
         have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
           using cf_z2[of n "m - n"] and `m>n`
           unfolding pos_le_divide_eq[OF `1-c>0`]
-          by (auto simp add: mult_commute dist_commute)
+          by (auto simp add: mult.commute dist_commute)
         also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
           using mult_right_mono[OF * order_less_imp_le[OF **]]
-          unfolding mult_assoc by auto
+          unfolding mult.assoc by auto
         also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
-          using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
+          using mult_strict_right_mono[OF N **] unfolding mult.assoc by auto
         also have "\<dots> = e * (1 - c ^ (m - n))"
           using c and `d>0` and `1 - c > 0` by auto
         also have "\<dots> \<le> e" using c and `1 - c ^ (m - n) > 0` and `e>0`
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/CLim.thy
--- a/src/HOL/NSA/CLim.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/CLim.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -22,7 +22,7 @@
 lemma all_shift: "(\<forall>x::'a::comm_ring_1. P x) = (\<forall>x. P (x-a))";
 apply auto 
 apply (drule_tac x="x+a" in spec) 
-apply (simp add: add_assoc) 
+apply (simp add: add.assoc) 
 done
 
 lemma complex_add_minus_iff [simp]: "(x + - a = (0::complex)) = (x=a)"
@@ -142,7 +142,7 @@
 apply (drule_tac [2] DERIV_ident [THEN DERIV_mult])
 apply (auto simp add: distrib_right real_of_nat_Suc)
 apply (case_tac "n")
-apply (auto simp add: mult_ac add_commute)
+apply (auto simp add: mult_ac add.commute)
 done
 
 text{*Nonstandard version*}
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/Examples/NSPrimes.thy
--- a/src/HOL/NSA/Examples/NSPrimes.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/Examples/NSPrimes.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -274,7 +274,7 @@
 apply (subgoal_tac "k \<notin> hypnat_of_nat ` {p. prime p}")
 apply (force simp add: starprime_def, safe)
 apply (drule_tac x = x in bspec, auto)
-apply (metis add_commute hdvd_diff hdvd_one_eq_one hypnat_diff_add_inverse2 hypnat_one_not_prime)
+apply (metis add.commute hdvd_diff hdvd_one_eq_one hypnat_diff_add_inverse2 hypnat_one_not_prime)
 done
 
 end
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/HDeriv.thy
--- a/src/HOL/NSA/HDeriv.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/HDeriv.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -133,7 +133,7 @@
 apply (auto simp add: nsderiv_def)
 apply (rule ccontr, drule_tac x = h in bspec)
 apply (drule_tac [2] c = h in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mult_assoc)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mult.assoc)
 done
 
 text{*Differentiability implies continuity
@@ -143,15 +143,15 @@
 apply (drule approx_minus_iff [THEN iffD1])
 apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
 apply (drule_tac x = "xa - star_of x" in bspec)
- prefer 2 apply (simp add: add_assoc [symmetric])
-apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
+ prefer 2 apply (simp add: add.assoc [symmetric])
+apply (auto simp add: mem_infmal_iff [symmetric] add.commute)
 apply (drule_tac c = "xa - star_of x" in approx_mult1)
 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-            simp add: mult_assoc nonzero_mult_divide_cancel_right)
+            simp add: mult.assoc nonzero_mult_divide_cancel_right)
 apply (drule_tac x3=D in
            HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult,
              THEN mem_infmal_iff [THEN iffD1]])
-apply (auto simp add: mult_commute
+apply (auto simp add: mult.commute
             intro: approx_trans approx_minus_iff [THEN iffD2])
 done
 
@@ -189,7 +189,7 @@
       ==> x - y \<approx> 0"
 apply (simp add: nonzero_divide_eq_eq)
 apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
-            simp add: mult_assoc mem_infmal_iff [symmetric])
+            simp add: mult.assoc mem_infmal_iff [symmetric])
 apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
 done
 
@@ -206,14 +206,14 @@
 apply (drule_tac
      approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
 apply (auto intro!: approx_add_mono1
-            simp add: distrib_right distrib_left mult_commute add_assoc)
+            simp add: distrib_right distrib_left mult.commute add.assoc)
 apply (rule_tac b1 = "star_of Db * star_of (f x)"
-         in add_commute [THEN subst])
+         in add.commute [THEN subst])
 apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
                     Infinitesimal_add Infinitesimal_mult
                     Infinitesimal_star_of_mult
                     Infinitesimal_star_of_mult2
-          simp add: add_assoc [symmetric])
+          simp add: add.assoc [symmetric])
 done
 
 text{*Multiplying by a constant*}
@@ -309,7 +309,7 @@
 proof -
   assume z: "z \<noteq> 0"
   have "x * y = x * (inverse z * z) * y" by (simp add: z)
-  thus ?thesis by (simp add: mult_assoc)
+  thus ?thesis by (simp add: mult.assoc)
 qed
 
 text{*This proof uses both definitions of differentiability.*}
@@ -373,7 +373,7 @@
 subsubsection {* Equivalence of NS and Standard definitions *}
 
 lemma divideR_eq_divide: "x /\<^sub>R y = x / y"
-by (simp add: divide_inverse mult_commute)
+by (simp add: divide_inverse mult.commute)
 
 text{*Now equivalence between NSDERIV and DERIV*}
 lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
@@ -403,7 +403,7 @@
 lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
       \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
 by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
-                   mult_commute)
+                   mult.commute)
 
 lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
 by auto
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/HLim.thy
--- a/src/HOL/NSA/HLim.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/HLim.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -95,7 +95,7 @@
 
 lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
 apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
-apply (auto simp add: add_assoc)
+apply (auto simp add: add.assoc)
 done
 
 lemma NSLIM_const_not_eq:
@@ -243,10 +243,10 @@
 apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
 apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
 apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
- prefer 2 apply (simp add: add_commute)
+ prefer 2 apply (simp add: add.commute)
 apply (rule_tac x = x in star_cases)
 apply (rule_tac [2] x = x in star_cases)
-apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc star_n_zero_num)
+apply (auto simp add: starfun star_of_def star_n_minus star_n_add add.assoc star_n_zero_num)
 done
 
 lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/HTranscendental.thy
--- a/src/HOL/NSA/HTranscendental.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/HTranscendental.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -254,7 +254,7 @@
 apply (drule_tac x = x in bspec, auto)
 apply (drule_tac c = x in approx_mult1)
 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] 
-            simp add: mult_assoc)
+            simp add: mult.assoc)
 apply (rule approx_add_right_cancel [where d="-1"])
 apply (rule approx_sym [THEN [2] approx_trans2])
 apply (auto simp add: mem_infmal_iff)
@@ -424,7 +424,7 @@
 apply (drule bspec [where x = x], auto)
 apply (drule approx_mult1 [where c = x])
 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-           simp add: mult_assoc)
+           simp add: mult.assoc)
 done
 
 lemma HFinite_cos [simp]: "sumhr (0, whn, %n. cos_coeff n * x ^ n) \<in> HFinite"
@@ -448,7 +448,7 @@
 apply auto
 apply (drule approx_mult1 [where c = x])
 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-            simp add: mult_assoc)
+            simp add: mult.assoc)
 apply (rule approx_add_right_cancel [where d = "-1"])
 apply simp
 done
@@ -463,7 +463,7 @@
 apply (drule bspec [where x = x], auto)
 apply (drule approx_mult1 [where c = x])
 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-             simp add: mult_assoc)
+             simp add: mult.assoc)
 done
 
 lemma STAR_sin_cos_Infinitesimal_mult:
@@ -481,7 +481,7 @@
      "N \<in> HNatInfinite  
       ==> hypreal_of_real a =  
           hypreal_of_hypnat N * (inverse(hypreal_of_hypnat N) * hypreal_of_real a)"
-by (simp add: mult_assoc [symmetric] zero_less_HNatInfinite)
+by (simp add: mult.assoc [symmetric] zero_less_HNatInfinite)
 
 lemma STAR_sin_Infinitesimal_divide:
      "[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x @= 1"
@@ -535,7 +535,7 @@
       ==> hypreal_of_hypnat n *  
           ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n))  
           @= hypreal_of_real pi"
-apply (rule mult_commute [THEN subst])
+apply (rule mult.commute [THEN subst])
 apply (erule STAR_sin_pi_divide_HNatInfinite_approx_pi)
 done
 
@@ -564,7 +564,7 @@
 apply (auto simp add: starfun_mult [symmetric] starfunNat_real_of_nat divide_inverse)
 apply (rule_tac f1 = inverse in starfun_o2 [THEN subst])
 apply (auto dest: STAR_sin_pi_divide_HNatInfinite_approx_pi 
-            simp add: starfunNat_real_of_nat mult_commute divide_inverse)
+            simp add: starfunNat_real_of_nat mult.commute divide_inverse)
 done
 
 lemma NSLIMSEQ_cos_one: "(%n. cos (pi / real n))----NS> 1"
@@ -587,7 +587,7 @@
      "x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - x\<^sup>2"
 apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
 apply (auto simp add: Infinitesimal_approx_minus [symmetric] 
-            add_assoc [symmetric] numeral_2_eq_2)
+            add.assoc [symmetric] numeral_2_eq_2)
 done
 
 lemma STAR_cos_Infinitesimal_approx2:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/HyperNat.thy
--- a/src/HOL/NSA/HyperNat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/HyperNat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -124,7 +124,7 @@
 done
 
 lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"
-by (simp add: linorder_not_le [symmetric] add_commute [of x]) 
+by (simp add: linorder_not_le [symmetric] add.commute [of x]) 
 
 lemma hypnat_diff_split:
     "P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
@@ -164,9 +164,9 @@
 lemma of_nat_eq_add [rule_format]:
      "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
 apply (induct n) 
-apply (auto simp add: add_assoc) 
+apply (auto simp add: add.assoc) 
 apply (case_tac x) 
-apply (auto simp add: add_commute [of 1]) 
+apply (auto simp add: add.commute [of 1]) 
 done
 
 lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/NSA.thy
--- a/src/HOL/NSA/NSA.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/NSA.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -430,7 +430,7 @@
 apply (drule_tac x="r / \<bar>star_of a\<bar>" in bspec)
 apply (simp add: Reals_eq_Standard)
 apply simp
-apply (simp add: hnorm_scaleR pos_less_divide_eq mult_commute)
+apply (simp add: hnorm_scaleR pos_less_divide_eq mult.commute)
 done
 
 lemma Compl_HFinite: "- HFinite = HInfinite"
@@ -473,11 +473,11 @@
 lemma HInfinite_add_ge_zero:
      "[|(x::hypreal) \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (x + y): HInfinite"
 by (auto intro!: hypreal_add_zero_less_le_mono
-       simp add: abs_if add_commute add_nonneg_nonneg HInfinite_def)
+       simp add: abs_if add.commute add_nonneg_nonneg HInfinite_def)
 
 lemma HInfinite_add_ge_zero2:
      "[|(x::hypreal) \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (y + x): HInfinite"
-by (auto intro!: HInfinite_add_ge_zero simp add: add_commute)
+by (auto intro!: HInfinite_add_ge_zero simp add: add.commute)
 
 lemma HInfinite_add_gt_zero:
      "[|(x::hypreal) \<in> HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite"
@@ -621,13 +621,13 @@
 by (simp add: approx_def)
 
 lemma approx_minus_iff2: " (x @= y) = (-y + x @= 0)"
-by (simp add: approx_def add_commute)
+by (simp add: approx_def add.commute)
 
 lemma approx_refl [iff]: "x @= x"
 by (simp add: approx_def Infinitesimal_def)
 
 lemma hypreal_minus_distrib1: "-(y + -(x::'a::ab_group_add)) = x + -y"
-by (simp add: add_commute)
+by (simp add: add.commute)
 
 lemma approx_sym: "x @= y ==> y @= x"
 apply (simp add: approx_def)
@@ -688,7 +688,7 @@
 lemma approx_minus: "a @= b ==> -a @= -b"
 apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
 apply (drule approx_minus_iff [THEN iffD1])
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 done
 
 lemma approx_minus2: "-a @= -b ==> a @= b"
@@ -746,17 +746,17 @@
 lemma Infinitesimal_add_approx: "[| y \<in> Infinitesimal; x + y = z |] ==> x @= z"
 apply (rule bex_Infinitesimal_iff [THEN iffD1])
 apply (drule Infinitesimal_minus_iff [THEN iffD2])
-apply (auto simp add: add_assoc [symmetric])
+apply (auto simp add: add.assoc [symmetric])
 done
 
 lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal ==> x @= x + y"
 apply (rule bex_Infinitesimal_iff [THEN iffD1])
 apply (drule Infinitesimal_minus_iff [THEN iffD2])
-apply (auto simp add: add_assoc [symmetric])
+apply (auto simp add: add.assoc [symmetric])
 done
 
 lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal ==> x @= y + x"
-by (auto dest: Infinitesimal_add_approx_self simp add: add_commute)
+by (auto dest: Infinitesimal_add_approx_self simp add: add.commute)
 
 lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal ==> x @= x + -y"
 by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
@@ -770,7 +770,7 @@
      "[| y \<in> Infinitesimal; x @= z + y|] ==> x @= z"
 apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
 apply (erule approx_trans3 [THEN approx_sym])
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 apply (erule approx_sym)
 done
 
@@ -781,7 +781,7 @@
 
 lemma approx_add_right_cancel: "b + d @= c + d ==> b @= c"
 apply (rule approx_add_left_cancel)
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 done
 
 lemma approx_add_mono1: "b @= c ==> d + b @= d + c"
@@ -790,19 +790,19 @@
 done
 
 lemma approx_add_mono2: "b @= c ==> b + a @= c + a"
-by (simp add: add_commute approx_add_mono1)
+by (simp add: add.commute approx_add_mono1)
 
 lemma approx_add_left_iff [simp]: "(a + b @= a + c) = (b @= c)"
 by (fast elim: approx_add_left_cancel approx_add_mono1)
 
 lemma approx_add_right_iff [simp]: "(b + a @= c + a) = (b @= c)"
-by (simp add: add_commute)
+by (simp add: add.commute)
 
 lemma approx_HFinite: "[| x \<in> HFinite; x @= y |] ==> y \<in> HFinite"
 apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
 apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
 apply (drule HFinite_add)
-apply (auto simp add: add_assoc)
+apply (auto simp add: add.assoc)
 done
 
 lemma approx_star_of_HFinite: "x @= star_of D ==> x \<in> HFinite"
@@ -851,7 +851,7 @@
 lemma approx_SReal_mult_cancel_zero:
      "[| (a::hypreal) \<in> Reals; a \<noteq> 0; a*x @= 0 |] ==> x @= 0"
 apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
-apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
+apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
 done
 
 lemma approx_mult_SReal1: "[| (a::hypreal) \<in> Reals; x @= 0 |] ==> x*a @= 0"
@@ -867,7 +867,7 @@
 lemma approx_SReal_mult_cancel:
      "[| (a::hypreal) \<in> Reals; a \<noteq> 0; a* w @= a*z |] ==> w @= z"
 apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
-apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
+apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
 done
 
 lemma approx_SReal_mult_cancel_iff1 [simp]:
@@ -991,7 +991,7 @@
   shows "[| y \<noteq> 0;  y \<in> Infinitesimal;  x/y \<in> HFinite |]
          ==> x \<in> Infinitesimal"
 apply (drule Infinitesimal_HFinite_mult2, assumption)
-apply (simp add: divide_inverse mult_assoc)
+apply (simp add: divide_inverse mult.assoc)
 done
 
 lemma Infinitesimal_SReal_divide:
@@ -1161,7 +1161,7 @@
 
 lemma lemma_minus_le_zero: "t \<le> t + -r ==> r \<le> (0::hypreal)"
 apply (drule_tac c = "-t" in add_left_mono)
-apply (auto simp add: add_assoc [symmetric])
+apply (auto simp add: add.assoc [symmetric])
 done
 
 lemma lemma_st_part_le2:
@@ -1352,7 +1352,7 @@
 apply (drule HFinite_inverse2)+
 apply (drule approx_mult2, assumption, auto)
 apply (drule_tac c = "inverse x" in approx_mult1, assumption)
-apply (auto intro: approx_sym simp add: mult_assoc)
+apply (auto intro: approx_sym simp add: mult.assoc)
 done
 
 (*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
@@ -1372,7 +1372,7 @@
   shows
      "[| x \<in> HFinite - Infinitesimal;
          h \<in> Infinitesimal |] ==> inverse(h + x) @= inverse x"
-apply (rule add_commute [THEN subst])
+apply (rule add.commute [THEN subst])
 apply (blast intro: inverse_add_Infinitesimal_approx)
 done
 
@@ -1392,7 +1392,7 @@
 apply (auto intro: Infinitesimal_mult)
 apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
 apply (frule not_Infinitesimal_not_zero)
-apply (auto dest: Infinitesimal_HFinite_mult simp add: mult_assoc)
+apply (auto dest: Infinitesimal_HFinite_mult simp add: mult.assoc)
 done
 declare Infinitesimal_square_iff [symmetric, simp]
 
@@ -1414,7 +1414,7 @@
 apply safe
 apply (frule HFinite_inverse, assumption)
 apply (drule not_Infinitesimal_not_zero)
-apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
+apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
 done
 
 lemma approx_HFinite_mult_cancel_iff1:
@@ -1432,7 +1432,7 @@
 lemma HInfinite_HFinite_add:
      "[| x \<in> HInfinite; y \<in> HFinite |] ==> x + y \<in> HInfinite"
 apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
-apply (auto simp add: add_assoc HFinite_minus_iff)
+apply (auto simp add: add.assoc HFinite_minus_iff)
 done
 
 lemma HInfinite_ge_HInfinite:
@@ -1454,7 +1454,7 @@
 apply (frule HFinite_Infinitesimal_not_zero)
 apply (drule HFinite_not_Infinitesimal_inverse)
 apply (safe, drule HFinite_mult)
-apply (auto simp add: mult_assoc HFinite_HInfinite_iff)
+apply (auto simp add: mult.assoc HFinite_HInfinite_iff)
 done
 
 lemma HInfinite_HFinite_not_Infinitesimal_mult2:
@@ -1466,7 +1466,7 @@
 apply (drule HFinite_not_Infinitesimal_inverse)
 apply (safe, drule_tac x="inverse y" in HFinite_mult)
 apply assumption
-apply (auto simp add: mult_assoc [symmetric] HFinite_HInfinite_iff)
+apply (auto simp add: mult.assoc [symmetric] HFinite_HInfinite_iff)
 done
 
 lemma HInfinite_gt_SReal:
@@ -1634,7 +1634,7 @@
 lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:
      "[| x \<in> Infinitesimal;  abs(hypreal_of_real r) < hypreal_of_real y |]
       ==> abs (x + hypreal_of_real r) < hypreal_of_real y"
-apply (rule add_commute [THEN subst])
+apply (rule add.commute [THEN subst])
 apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
 done
 
@@ -1684,14 +1684,14 @@
 lemma Infinitesimal_square_cancel2 [simp]:
      "(x::hypreal)*x + y*y \<in> Infinitesimal ==> y*y \<in> Infinitesimal"
 apply (rule Infinitesimal_square_cancel)
-apply (rule add_commute [THEN subst])
+apply (rule add.commute [THEN subst])
 apply (simp (no_asm))
 done
 
 lemma HFinite_square_cancel2 [simp]:
   "(x::hypreal)*x + y*y \<in> HFinite ==> y*y \<in> HFinite"
 apply (rule HFinite_square_cancel)
-apply (rule add_commute [THEN subst])
+apply (rule add.commute [THEN subst])
 apply (simp (no_asm))
 done
 
@@ -2143,7 +2143,7 @@
 apply (subst pos_less_divide_eq, assumption)
 apply (subst pos_less_divide_eq)
  apply (simp add: real_of_nat_Suc_gt_zero)
-apply (simp add: mult_commute)
+apply (simp add: mult.commute)
 done
 
 lemma finite_inverse_real_of_posnat_gt_real:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/NSCA.thy
--- a/src/HOL/NSA/NSCA.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/NSCA.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -137,7 +137,7 @@
 lemma approx_SComplex_mult_cancel_zero:
      "[| a \<in> SComplex; a \<noteq> 0; a*x @= 0 |] ==> x @= 0"
 apply (drule Standard_inverse [THEN Standard_subset_HFinite [THEN subsetD]])
-apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
+apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
 done
 
 lemma approx_mult_SComplex1: "[| a \<in> SComplex; x @= 0 |] ==> x*a @= 0"
@@ -153,7 +153,7 @@
 lemma approx_SComplex_mult_cancel:
      "[| a \<in> SComplex; a \<noteq> 0; a* w @= a*z |] ==> w @= z"
 apply (drule Standard_inverse [THEN Standard_subset_HFinite [THEN subsetD]])
-apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
+apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
 done
 
 lemma approx_SComplex_mult_cancel_iff1 [simp]:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NSA/StarDef.thy
--- a/src/HOL/NSA/StarDef.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NSA/StarDef.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -766,16 +766,16 @@
 subsection {* Ordered group classes *}
 
 instance star :: (semigroup_add) semigroup_add
-by (intro_classes, transfer, rule add_assoc)
+by (intro_classes, transfer, rule add.assoc)
 
 instance star :: (ab_semigroup_add) ab_semigroup_add
-by (intro_classes, transfer, rule add_commute)
+by (intro_classes, transfer, rule add.commute)
 
 instance star :: (semigroup_mult) semigroup_mult
-by (intro_classes, transfer, rule mult_assoc)
+by (intro_classes, transfer, rule mult.assoc)
 
 instance star :: (ab_semigroup_mult) ab_semigroup_mult
-by (intro_classes, transfer, rule mult_commute)
+by (intro_classes, transfer, rule mult.commute)
 
 instance star :: (comm_monoid_add) comm_monoid_add
 by (intro_classes, transfer, rule comm_monoid_add_class.add_0)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Nat.thy
--- a/src/HOL/Nat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Nat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -244,10 +244,10 @@
   by (induct m) simp_all
 
 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
-  by (induct m) (simp_all add: add_left_commute)
+  by (induct m) (simp_all add: add.left_commute)
 
 lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
-  by (induct m) (simp_all add: add_assoc)
+  by (induct m) (simp_all add: add.assoc)
 
 instance proof
   fix n m q :: nat
@@ -263,20 +263,15 @@
 
 subsubsection {* Addition *}
 
-lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
-  by (rule add_assoc)
-
-lemma nat_add_commute: "m + n = n + (m::nat)"
-  by (rule add_commute)
+lemma nat_add_left_cancel:
+  fixes k m n :: nat
+  shows "k + m = k + n \<longleftrightarrow> m = n"
+  by (fact add_left_cancel)
 
-lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
-  by (rule add_left_commute)
-
-lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
-  by (rule add_left_cancel)
-
-lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
-  by (rule add_right_cancel)
+lemma nat_add_right_cancel:
+  fixes k m n :: nat
+  shows "m + k = n + k \<longleftrightarrow> m = n"
+  by (fact add_right_cancel)
 
 text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
 
@@ -315,31 +310,31 @@
 subsubsection {* Difference *}
 
 lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
-  by (induct m) simp_all
+  by (fact diff_cancel)
 
 lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
-  by (induct i j rule: diff_induct) simp_all
+  by (fact diff_diff_add)
 
 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
   by (simp add: diff_diff_left)
 
 lemma diff_commute: "(i::nat) - j - k = i - k - j"
-  by (simp add: diff_diff_left add_commute)
+  by (fact diff_right_commute)
 
 lemma diff_add_inverse: "(n + m) - n = (m::nat)"
-  by (induct n) simp_all
+  by (fact add_diff_cancel_left')
 
 lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
-  by (simp add: diff_add_inverse add_commute [of m n])
+  by (fact add_diff_cancel_right')
 
 lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
-  by (induct k) simp_all
+  by (fact comm_monoid_diff_class.add_diff_cancel_left)
 
 lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
-  by (simp add: diff_cancel add_commute)
+  by (fact add_diff_cancel_right)
 
 lemma diff_add_0: "n - (n + m) = (0::nat)"
-  by (induct n) simp_all
+  by (fact diff_add_zero)
 
 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
   unfolding One_nat_def by simp
@@ -350,20 +345,14 @@
 by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
 
 lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
-by (simp add: diff_mult_distrib mult_commute [of k])
+by (simp add: diff_mult_distrib mult.commute [of k])
   -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
 
 
 subsubsection {* Multiplication *}
 
-lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
-  by (rule mult_assoc)
-
-lemma nat_mult_commute: "m * n = n * (m::nat)"
-  by (rule mult_commute)
-
 lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
-  by (rule distrib_left)
+  by (fact distrib_left)
 
 lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
   by (induct m) auto
@@ -402,7 +391,7 @@
 qed
 
 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
-  by (simp add: mult_commute)
+  by (simp add: mult.commute)
 
 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
   by (subst mult_cancel1) simp
@@ -950,7 +939,7 @@
   next
     case (Suc k)
     have "0 + i < Suc k + i" by (rule add_less_mono1) simp
-    hence "i < Suc (i + k)" by (simp add: add_commute)
+    hence "i < Suc (i + k)" by (simp add: add.commute)
     from trans[OF this lessI Suc step]
     show ?case by simp
   qed
@@ -1036,7 +1025,7 @@
 by (insert add_right_mono [of 0 m n], simp)
 
 lemma le_add1: "n \<le> ((n + m)::nat)"
-by (simp add: add_commute, rule le_add2)
+by (simp add: add.commute, rule le_add2)
 
 lemma less_add_Suc1: "i < Suc (i + m)"
 by (rule le_less_trans, rule le_add1, rule lessI)
@@ -1071,7 +1060,7 @@
 done
 
 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
-by (simp add: add_commute)
+by (simp add: add.commute)
 
 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
 apply (rule order_trans [of _ "m+k"])
@@ -1079,7 +1068,7 @@
 done
 
 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
-apply (simp add: add_commute)
+apply (simp add: add.commute)
 apply (erule add_leD1)
 done
 
@@ -1103,7 +1092,7 @@
 by (simp add: add_diff_inverse linorder_not_less)
 
 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
-by (simp add: add_commute)
+by (simp add: add.commute)
 
 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
 by (induct m n rule: diff_induct) simp_all
@@ -1135,7 +1124,7 @@
 by (induct j k rule: diff_induct) simp_all
 
 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
-by (simp add: add_commute diff_add_assoc)
+by (simp add: add.commute diff_add_assoc)
 
 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
 by (auto simp add: diff_add_inverse2)
@@ -1233,7 +1222,7 @@
   done
 
 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
-by (simp add: mult_commute [of k])
+by (simp add: mult.commute [of k])
 
 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
 by (simp add: linorder_not_less [symmetric], auto)
@@ -1435,7 +1424,7 @@
     by (induct n) simp_all
   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
     by simp
-  with Suc show ?case by (simp add: add_commute)
+  with Suc show ?case by (simp add: add.commute)
 qed
 
 end
@@ -1693,13 +1682,13 @@
 by arith
 
 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
-by arith
+  by (fact le_diff_conv2) -- {* FIXME delete *}
 
 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
 by arith
 
 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
-by arith
+  by (fact le_add_diff) -- {* FIXME delete *}
 
 (*Replaces the previous diff_less and le_diff_less, which had the stronger
   second premise n\<le>m*)
@@ -1847,7 +1836,7 @@
 
 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   unfolding dvd_def
-  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
+  by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
 
 text {* @{term "op dvd"} is a partial order *}
 
@@ -1890,7 +1879,7 @@
   done
 
 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   apply (erule dvd_mult_cancel1)
   done
 
@@ -1940,7 +1929,7 @@
   fixes m n q :: nat
   assumes "m dvd q"
   shows "m dvd n + q \<longleftrightarrow> m dvd n"
-  using assms by (simp add: dvd_plus_eq_right add_commute [of n])
+  using assms by (simp add: dvd_plus_eq_right add.commute [of n])
 
 lemma less_eq_dvd_minus:
   fixes m n :: nat
@@ -1949,7 +1938,7 @@
 proof -
   from assms have "n = m + (n - m)" by simp
   then obtain q where "n = m + q" ..
-  then show ?thesis by (simp add: dvd_reduce add_commute [of m])
+  then show ?thesis by (simp add: dvd_reduce add.commute [of m])
 qed
 
 lemma dvd_minus_self:
@@ -1966,7 +1955,7 @@
     by (auto elim: dvd_plusE)
   also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
   also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
-  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add_commute)
+  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
   finally show ?thesis .
 qed
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Nominal/Examples/Standardization.thy
--- a/src/HOL/Nominal/Examples/Standardization.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Nominal/Examples/Standardization.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -213,7 +213,7 @@
     prefer 2
     apply (erule allE, erule impE, rule refl, erule spec)
     apply simp
-    apply (simp only: foldl_conv_fold add_commute fold_plus_listsum_rev)
+    apply (simp only: foldl_conv_fold add.commute fold_plus_listsum_rev)
     apply (fastforce simp add: listsum_map_remove1)
   apply clarify
   apply (subgoal_tac "\<exists>y::name. y \<sharp> (x, u, z)")
@@ -235,7 +235,7 @@
    prefer 2
    apply blast
   apply simp
-  apply (simp only: foldl_conv_fold add_commute fold_plus_listsum_rev)
+  apply (simp only: foldl_conv_fold add.commute fold_plus_listsum_rev)
   apply (fastforce simp add: listsum_map_remove1)
   done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/NthRoot.thy
--- a/src/HOL/NthRoot.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/NthRoot.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -222,7 +222,7 @@
            simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq)
 
 lemma real_root_commute: "root m (root n x) = root n (root m x)"
-  by (simp add: real_root_mult_exp [symmetric] mult_commute)
+  by (simp add: real_root_mult_exp [symmetric] mult.commute)
 
 text {* Monotonicity in first argument *}
 
@@ -432,7 +432,7 @@
   from n obtain m where m: "n = 2 * m"
     unfolding even_mult_two_ex ..
   from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
-    by (simp only: power_mult[symmetric] mult_commute)
+    by (simp only: power_mult[symmetric] mult.commute)
   then show ?thesis
     using m by simp
 qed
@@ -512,7 +512,7 @@
   proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
     show "inverse (sqrt x) / (sqrt x / x) = 1"
-      by (simp add: divide_inverse mult_assoc [symmetric] 
+      by (simp add: divide_inverse mult.assoc [symmetric] 
                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
   qed
 qed
@@ -603,7 +603,7 @@
   "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
 apply (rule power2_le_imp_le, simp)
 apply (simp add: power2_sum)
-apply (simp only: mult_assoc distrib_left [symmetric])
+apply (simp only: mult.assoc distrib_left [symmetric])
 apply (rule mult_left_mono)
 apply (rule power2_le_imp_le)
 apply (simp add: power2_sum power_mult_distrib)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Num.thy
--- a/src/HOL/Num.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Num.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -254,8 +254,8 @@
 lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
   apply (induct x)
   apply simp
-  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
-  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
+  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
+  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
   done
 
 lemma numeral_inc: "numeral (inc x) = numeral x + 1"
@@ -264,7 +264,7 @@
   have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
     by (simp only: one_plus_numeral_commute)
   with Bit1 show ?case
-    by (simp add: add_assoc)
+    by (simp add: add.assoc)
 qed simp_all
 
 declare numeral.simps [simp del]
@@ -350,7 +350,7 @@
 
 lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
   by (induct n rule: num_induct)
-     (simp_all only: numeral_One add_One add_inc numeral_inc add_assoc)
+     (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
 
 lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
   by (rule numeral_add [symmetric])
@@ -397,20 +397,20 @@
   apply simp
   apply (rule_tac a=x in add_left_imp_eq)
   apply (rule_tac a=x in add_right_imp_eq)
-  apply (simp add: add_assoc)
-  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
+  apply (simp add: add.assoc)
+  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
   apply (rule_tac a=x in add_left_imp_eq)
   apply (rule_tac a=x in add_right_imp_eq)
-  apply (simp add: add_assoc)
-  apply (simp add: add_assoc, simp add: add_assoc [symmetric])
+  apply (simp add: add.assoc)
+  apply (simp add: add.assoc, simp add: add.assoc [symmetric])
   done
 
 lemma is_num_add_left_commute:
   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
-  by (simp only: add_assoc [symmetric] is_num_add_commute)
+  by (simp only: add.assoc [symmetric] is_num_add_commute)
 
 lemmas is_num_normalize =
-  add_assoc is_num_add_commute is_num_add_left_commute
+  add.assoc is_num_add_commute is_num_add_left_commute
   is_num.intros is_num_numeral
   minus_add
 
@@ -1172,7 +1172,7 @@
   minus_zero left_minus right_minus
   mult_1_left mult_1_right
   mult_minus_left mult_minus_right
-  minus_add_distrib minus_minus mult_assoc
+  minus_add_distrib minus_minus mult.assoc
 
 lemmas of_nat_simps =
   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
@@ -1225,20 +1225,20 @@
 
 lemma add_numeral_left [simp]:
   "numeral v + (numeral w + z) = (numeral(v + w) + z)"
-  by (simp_all add: add_assoc [symmetric])
+  by (simp_all add: add.assoc [symmetric])
 
 lemma add_neg_numeral_left [simp]:
   "numeral v + (- numeral w + y) = (sub v w + y)"
   "- numeral v + (numeral w + y) = (sub w v + y)"
   "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
-  by (simp_all add: add_assoc [symmetric])
+  by (simp_all add: add.assoc [symmetric])
 
 lemma mult_numeral_left [simp]:
   "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
   "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
   "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
   "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
-  by (simp_all add: mult_assoc [symmetric])
+  by (simp_all add: mult.assoc [symmetric])
 
 hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Binomial.thy
--- a/src/HOL/Number_Theory/Binomial.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Binomial.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -33,7 +33,7 @@
 lemma choose_reduce_nat: 
   "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
     (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
-  by (metis Suc_diff_1 binomial.simps(2) nat_add_commute neq0_conv)
+  by (metis Suc_diff_1 binomial.simps(2) add.commute neq0_conv)
 
 lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
   by (induct n arbitrary: k) auto
@@ -356,7 +356,7 @@
   assumes kn: "k \<le> n"
   shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
   unfolding pochhammer_minus[OF kn, where b=b]
-  unfolding mult_assoc[symmetric]
+  unfolding mult.assoc[symmetric]
   unfolding power_add[symmetric]
   by simp
 
@@ -461,7 +461,7 @@
       apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
         of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
       unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
-      unfolding mult_assoc[symmetric]
+      unfolding mult.assoc[symmetric]
       unfolding setprod.distrib[symmetric]
       apply simp
       apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
@@ -494,7 +494,7 @@
 
 lemma gbinomial_mult_1':
     "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
-  by (simp add: mult_commute gbinomial_mult_1)
+  by (simp add: mult.commute gbinomial_mult_1)
 
 lemma gbinomial_Suc:
     "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
@@ -509,7 +509,7 @@
   "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
   using gbinomial_mult_fact[of k a]
-  by (subst mult_commute)
+  by (subst mult.commute)
 
 
 lemma gbinomial_Suc_Suc:
@@ -531,8 +531,8 @@
     unfolding gbinomial_mult_fact'
     apply (subst fact_Suc)
     unfolding of_nat_mult
-    apply (subst mult_commute)
-    unfolding mult_assoc
+    apply (subst mult.commute)
+    unfolding mult.assoc
     unfolding gbinomial_mult_fact
     apply (simp add: field_simps)
     done
@@ -597,7 +597,7 @@
     from assms have "n * i \<ge> i * k" by simp
     then have "n * k - n * i \<le> n * k - i * k" by arith
     then have "n * (k - i) \<le> (n - i) * k"
-      by (simp add: diff_mult_distrib2 nat_mult_commute)
+      by (simp add: diff_mult_distrib2 mult.commute)
     then have "of_nat n * of_nat (k - i) \<le> of_nat (n - i) * (of_nat k :: 'a)"
       unfolding of_nat_mult[symmetric] of_nat_le_iff .
     with assms show "of_nat n / of_nat k \<le> of_nat (n - i) / (of_nat (k - i) :: 'a)"
@@ -634,20 +634,20 @@
   also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
     apply (subst div_mult_div_if_dvd)
     apply (auto simp: fact_fact_dvd_fact)
-    apply (metis ab_semigroup_add_class.add_ac(1) add_commute fact_fact_dvd_fact)
+    apply (metis ab_semigroup_add_class.add_ac(1) add.commute fact_fact_dvd_fact)
     done
   also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
     apply (subst div_mult_div_if_dvd [symmetric])
     apply (auto simp: fact_fact_dvd_fact)
-    apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult_left_commute)
+    apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult.left_commute)
     done
   also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
     apply (subst div_mult_div_if_dvd)
     apply (auto simp: fact_fact_dvd_fact)
-    apply(metis mult_left_commute)
+    apply(metis mult.left_commute)
     done
   finally show ?thesis
-    by (simp add: binomial_altdef_nat mult_commute)
+    by (simp add: binomial_altdef_nat mult.commute)
 qed
 
 lemma choose_mult:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Cong.thy
--- a/src/HOL/Number_Theory/Cong.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Cong.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -275,11 +275,11 @@
 
 lemma cong_mult_lcancel_nat:
     "coprime k (m::nat) \<Longrightarrow> [k * a = k * b ] (mod m) = [a = b] (mod m)"
-  by (simp add: mult_commute cong_mult_rcancel_nat)
+  by (simp add: mult.commute cong_mult_rcancel_nat)
 
 lemma cong_mult_lcancel_int:
     "coprime k (m::int) \<Longrightarrow> [k * a = k * b] (mod m) = [a = b] (mod m)"
-  by (simp add: mult_commute cong_mult_rcancel_int)
+  by (simp add: mult.commute cong_mult_rcancel_int)
 
 (* was zcong_zgcd_zmult_zmod *)
 lemma coprime_cong_mult_int:
@@ -321,7 +321,7 @@
   proof (cases "b \<le> a")
     case True
     then show ?rhs using eqm
-      by (metis cong_altdef_nat dvd_def le_add_diff_inverse add_0_right mult_0 mult_commute)
+      by (metis cong_altdef_nat dvd_def le_add_diff_inverse add_0_right mult_0 mult.commute)
   next
     case False
     then show ?rhs using eqm 
@@ -333,7 +333,7 @@
 next
   assume ?rhs
   then show ?lhs
-    by (metis cong_nat_def mod_mult_self2 nat_mult_commute)
+    by (metis cong_nat_def mod_mult_self2 mult.commute)
 qed
 
 lemma cong_gcd_eq_int: "[(a::int) = b] (mod m) \<Longrightarrow> gcd a m = gcd b m"
@@ -467,18 +467,18 @@
   apply force
   apply (cases "a = 0", simp add: cong_0_1_nat)
   apply (rule iffI)
-  apply (metis cong_to_1_nat dvd_def monoid_mult_class.mult.right_neutral mult_commute mult_eq_if)
+  apply (metis cong_to_1_nat dvd_def monoid_mult_class.mult.right_neutral mult.commute mult_eq_if)
   apply (metis cong_add_lcancel_0_nat cong_mult_self_nat)
   done
 
 lemma cong_le_nat: "(y::nat) <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
-  by (metis cong_altdef_nat Nat.le_imp_diff_is_add dvd_def nat_mult_commute)
+  by (metis cong_altdef_nat Nat.le_imp_diff_is_add dvd_def mult.commute)
 
 lemma cong_solve_nat: "(a::nat) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
   apply (cases "n = 0")
   apply force
   apply (frule bezout_nat [of a n], auto)
-  by (metis cong_add_rcancel_0_nat cong_mult_self_nat mult_commute)
+  by (metis cong_add_rcancel_0_nat cong_mult_self_nat mult.commute)
 
 lemma cong_solve_int: "(a::int) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
   apply (cases "n = 0")
@@ -487,7 +487,7 @@
   apply (rule_tac x = "-1" in exI)
   apply auto
   apply (insert bezout_int [of a n], auto)
-  by (metis cong_iff_lin_int mult_commute)
+  by (metis cong_iff_lin_int mult.commute)
 
 lemma cong_solve_dvd_nat:
   assumes a: "(a::nat) \<noteq> 0" and b: "gcd a n dvd d"
@@ -603,9 +603,9 @@
   from cong_solve_coprime_nat [OF b] obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
     by auto
   have "[m1 * x1 = 0] (mod m1)"
-    by (subst mult_commute, rule cong_mult_self_nat)
+    by (subst mult.commute, rule cong_mult_self_nat)
   moreover have "[m2 * x2 = 0] (mod m2)"
-    by (subst mult_commute, rule cong_mult_self_nat)
+    by (subst mult.commute, rule cong_mult_self_nat)
   moreover note one two
   ultimately show ?thesis by blast
 qed
@@ -622,9 +622,9 @@
   from cong_solve_coprime_int [OF b] obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
     by auto
   have "[m1 * x1 = 0] (mod m1)"
-    by (subst mult_commute, rule cong_mult_self_int)
+    by (subst mult.commute, rule cong_mult_self_int)
   moreover have "[m2 * x2 = 0] (mod m2)"
-    by (subst mult_commute, rule cong_mult_self_int)
+    by (subst mult.commute, rule cong_mult_self_int)
   moreover note one two
   ultimately show ?thesis by blast
 qed
@@ -729,7 +729,7 @@
     apply (rule cong_trans_nat)
     prefer 2
     apply (rule `[y = u2] (mod m2)`)
-    apply (subst mult_commute)
+    apply (subst mult.commute)
     apply (rule cong_modulus_mult_nat)
     apply (rule cong_mod_nat)
     using nz apply auto
@@ -778,7 +778,7 @@
     by auto
   moreover have "[(PROD j : A - {i}. m j) * x = 0]
     (mod (PROD j : A - {i}. m j))"
-    by (subst mult_commute, rule cong_mult_self_nat)
+    by (subst mult.commute, rule cong_mult_self_nat)
   ultimately show "\<exists>a. [a = 1] (mod m i) \<and> [a = 0]
       (mod setprod m (A - {i}))"
     by blast
@@ -835,7 +835,7 @@
          [x = y] (mod (PROD i:A. m i))"
   apply (induct set: finite)
   apply auto
-  apply (metis coprime_cong_mult_nat nat_mult_commute setprod_coprime_nat)
+  apply (metis coprime_cong_mult_nat mult.commute setprod_coprime_nat)
   done
 
 lemma chinese_remainder_unique_nat:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Eratosthenes.thy
--- a/src/HOL/Number_Theory/Eratosthenes.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Eratosthenes.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -50,7 +50,7 @@
 
 lemma in_numbers_of_marks_eq:
   "m \<in> numbers_of_marks n bs \<longleftrightarrow> m \<in> {n..<n + length bs} \<and> bs ! (m - n)"
-  by (simp add: numbers_of_marks_def in_set_enumerate_eq image_iff add_commute)
+  by (simp add: numbers_of_marks_def in_set_enumerate_eq image_iff add.commute)
 
 lemma sorted_list_of_set_numbers_of_marks:
   "sorted_list_of_set (numbers_of_marks n bs) = map fst (filter snd (enumerate n bs))"
@@ -141,7 +141,7 @@
         by (simp add: div_mod_equality' [symmetric])
     }
     then have "w dvd v + w + r + (w - v mod w) \<longleftrightarrow> w dvd m + w + r + (w - m mod w)"
-      by (simp add: add_assoc add_left_commute [of m] add_left_commute [of v]
+      by (simp add: add.assoc add.left_commute [of m] add.left_commute [of v]
         dvd_plus_eq_left dvd_plus_eq_right)
     moreover from q have "Suc q = m + w + r" by (simp add: w_def)
     moreover from q have "Suc (Suc q) = v + w + r" by (simp add: v_def w_def)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Fib.thy
--- a/src/HOL/Number_Theory/Fib.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Fib.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -60,7 +60,7 @@
 lemma coprime_fib_Suc_nat: "coprime (fib (n::nat)) (fib (Suc n))"
   apply (induct n rule: fib.induct)
   apply auto
-  apply (metis gcd_add1_nat nat_add_commute)
+  apply (metis gcd_add1_nat add.commute)
   done
 
 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
@@ -68,7 +68,7 @@
   apply (cases m)
   apply (auto simp add: fib_add)
   apply (subst gcd_commute_nat)
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   apply (metis coprime_fib_Suc_nat gcd_add_mult_nat gcd_mult_cancel_nat gcd_nat.commute)
   done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Gauss.thy
--- a/src/HOL/Number_Theory/Gauss.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Gauss.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -260,11 +260,11 @@
   fix y::int and z::int
   assume "p - (y*a) mod p = (z*a) mod p"
   then have "[(y*a) mod p + (z*a) mod p = 0] (mod p)"
-    by (metis add_commute diff_eq_eq dvd_refl cong_int_def dvd_eq_mod_eq_0 mod_0)
+    by (metis add.commute diff_eq_eq dvd_refl cong_int_def dvd_eq_mod_eq_0 mod_0)
   moreover have "[y * a = (y*a) mod p] (mod p)"
     by (metis cong_int_def mod_mod_trivial)
   ultimately have "[a * (y + z) = 0] (mod p)"
-    by (metis cong_int_def mod_add_left_eq mod_add_right_eq mult_commute ring_class.ring_distribs(1))
+    by (metis cong_int_def mod_add_left_eq mod_add_right_eq mult.commute ring_class.ring_distribs(1))
   with p_prime a_nonzero p_a_relprime
   have a: "[y + z = 0] (mod p)"
     by (metis cong_prime_prod_zero_int)
@@ -319,19 +319,19 @@
 subsection {* Gauss' Lemma *}
 
 lemma aux: "setprod id A * -1 ^ card E * a ^ card A * -1 ^ card E = setprod id A * a ^ card A"
-by (metis (no_types) minus_minus mult_commute mult_left_commute power_minus power_one)
+by (metis (no_types) minus_minus mult.commute mult.left_commute power_minus power_one)
 
 theorem pre_gauss_lemma:
   "[a ^ nat((int p - 1) div 2) = (-1) ^ (card E)] (mod p)"
 proof -
   have "[setprod id A = setprod id F * setprod id D](mod p)"
-    by (auto simp add: prod_D_F_eq_prod_A mult_commute cong del:setprod.cong)
+    by (auto simp add: prod_D_F_eq_prod_A mult.commute cong del:setprod.cong)
   then have "[setprod id A = ((-1)^(card E) * setprod id E) * setprod id D] (mod p)"
     apply (rule cong_trans_int)
     apply (metis cong_scalar_int prod_F_zcong)
     done
   then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)"
-    by (metis C_prod_eq_D_times_E mult_commute mult_left_commute)
+    by (metis C_prod_eq_D_times_E mult.commute mult.left_commute)
   then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)"
     by (rule cong_trans_int) (metis C_B_zcong_prod cong_scalar2_int)
   then have "[setprod id A = ((-1)^(card E) *
@@ -349,7 +349,7 @@
   then have "[setprod id A = ((-1)^(card E) * a^(card A) *
       setprod id A)](mod p)"
     apply (rule cong_trans_int)
-    apply (simp add: cong_scalar2_int cong_scalar_int finite_A setprod_constant mult_assoc)
+    apply (simp add: cong_scalar2_int cong_scalar_int finite_A setprod_constant mult.assoc)
     done
   then have a: "[setprod id A * (-1)^(card E) =
       ((-1)^(card E) * a^(card A) * setprod id A * (-1)^(card E))](mod p)"
@@ -357,7 +357,7 @@
   then have "[setprod id A * (-1)^(card E) = setprod id A *
       (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
     apply (rule cong_trans_int)
-    apply (simp add: a mult_commute mult_left_commute)
+    apply (simp add: a mult.commute mult.left_commute)
     done
   then have "[setprod id A * (-1)^(card E) = setprod id A * a^(card A)](mod p)"
     apply (rule cong_trans_int)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Pocklington.thy
--- a/src/HOL/Number_Theory/Pocklington.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Pocklington.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -264,7 +264,7 @@
           mod_div_equality[of "(n - 1)" m, symmetric]
         by (simp add:power_add[symmetric] cong_nat_def th3 del: One_nat_def)
       have th1: "[a ^ ((n - 1) mod m) = 1] (mod n)"
-        by (metis cong_mult_rcancel_nat nat_mult_commute th2 yn)
+        by (metis cong_mult_rcancel_nat mult.commute th2 yn)
       from m(4)[rule_format, OF th0] nm1
         less_trans[OF mod_less_divisor[OF m(1), of "n - 1"] m(2)] th1
       have False by blast }
@@ -385,7 +385,7 @@
     from H have onz: "?o \<noteq> 0" by (simp add: ord_eq_0)
     hence op: "?o > 0" by simp
     from mod_div_equality[of d "ord n a"] lh
-    have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: cong_nat_def mult_commute)
+    have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: cong_nat_def mult.commute)
     hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
       by (simp add: cong_nat_def power_mult[symmetric] power_add[symmetric])
     hence th: "[a^?r = 1] (mod n)"
@@ -493,7 +493,7 @@
         from H[rule_format, of d] h d have "d = 1" by blast}
       moreover
       {assume h: "e\<^sup>2 \<le> n"
-        from e have "e dvd n" unfolding dvd_def by (simp add: mult_commute)
+        from e have "e dvd n" unfolding dvd_def by (simp add: mult.commute)
         with H[rule_format, of e] h have "e=1" by simp
         with e have "d = n" by simp}
       ultimately have "d=1 \<or> d=n"  by blast}
@@ -552,7 +552,7 @@
   hence "a ^ (r * q) + p * 0 = 1 + p * (l*k)" by (simp add: mult_ac)
   hence odq: "ord p (a^r) dvd q"
     unfolding ord_divides[symmetric] power_mult[symmetric]
-    by (metis an cong_dvd_modulus_nat mult_commute nqr pn) 
+    by (metis an cong_dvd_modulus_nat mult.commute nqr pn) 
   from odq[unfolded dvd_def] obtain d where d: "q = ord p (a^r) * d" by blast
   {assume d1: "d \<noteq> 1"
     obtain P where P: "prime P" "P dvd d"
@@ -564,9 +564,9 @@
       by (metis zero_not_prime_nat) 
     from P(2) obtain t where t: "d = P*t" unfolding dvd_def by blast
     from d s t P0  have s': "ord p (a^r) * t = s"
-      by (metis mult_commute mult_cancel1 nat_mult_assoc) 
+      by (metis mult.commute mult_cancel1 mult.assoc) 
     have "ord p (a^r) * t*r = r * ord p (a^r) * t"
-      by (metis mult_assoc mult_commute)
+      by (metis mult.assoc mult.commute)
     hence exps: "a^(ord p (a^r) * t*r) = ((a ^ r) ^ ord p (a^r)) ^ t"
       by (simp only: power_mult)
     then have th: "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)"
@@ -597,7 +597,7 @@
     by (metis p01(1)) 
   from p0 d have "p + q * 0 = 1 + q * d" by simp
   then show ?thesis
-    by (metis cong_iff_lin_nat mult_commute)
+    by (metis cong_iff_lin_nat mult.commute)
 qed
 
 theorem pocklington:
@@ -767,7 +767,7 @@
       by auto
     from div_mult1_eq[of r q p] p(2)
     have eq1: "r* (q div p) = (n - 1) div p"
-      unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult_commute)
+      unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult.commute)
     have ath: "\<And>a (b::nat). a <= b \<Longrightarrow> a \<noteq> 0 ==> 1 <= a \<and> 1 <= b" by arith
     {assume "a ^ ((n - 1) div p) mod n = 0"
       then obtain s where s: "a ^ ((n - 1) div p) = n*s"
@@ -779,7 +779,7 @@
       with eq0 have "a^ (n - 1) = (n*s)^p"
         by (simp add: power_mult[symmetric])
       hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
-      also have "\<dots> = 0" by (simp add: mult_assoc)
+      also have "\<dots> = 0" by (simp add: mult.assoc)
       finally have False by simp }
       then have th11: "a ^ ((n - 1) div p) mod n \<noteq> 0" by auto
     have th1: "[a ^ ((n - 1) div p) mod n = a ^ ((n - 1) div p)] (mod n)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Primes.thy
--- a/src/HOL/Number_Theory/Primes.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Primes.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -106,7 +106,7 @@
 lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
     EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
   unfolding prime_nat_def dvd_def apply auto
-  by (metis mult_commute linorder_neq_iff linorder_not_le mult_1
+  by (metis mult.commute linorder_neq_iff linorder_not_le mult_1
       n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
 
 lemma prime_dvd_power_nat: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p dvd x"
@@ -209,7 +209,7 @@
       from coprime_dvd_mult_nat[OF pnb pab] have ?thesis by blast }
     moreover
     { assume pb: "p dvd b"
-      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+      have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute)
       from coprime_common_divisor_nat [OF ab, of p] pb p have "\<not> p dvd a"
         by auto
       with p have "coprime a p"
@@ -323,7 +323,7 @@
   moreover 
   {assume px: "p dvd y"
     then obtain d where d: "y = p*d" unfolding dvd_def by blast
-    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
+    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult.commute)
     hence th: "d*x = p^k" using p0 by simp
     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
     with d have "y = p^Suc i" by simp 
@@ -355,7 +355,7 @@
   shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
 proof
   assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" 
-    unfolding dvd_def  apply (auto simp add: mult_commute) by blast
+    unfolding dvd_def  apply (auto simp add: mult.commute) by blast
   from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
   from e ij have "p^(i + j) = p^k" by (simp add: power_add)
   hence "i + j = k" using p prime_gt_1_nat power_inject_exp[of p "i+j" k] by simp 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Number_Theory/Residues.thy
--- a/src/HOL/Number_Theory/Residues.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Number_Theory/Residues.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -65,7 +65,7 @@
   apply (rule comm_monoid)
   apply (unfold R_def residue_ring_def, auto)
   apply (subst mod_add_eq [symmetric])
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   apply (subst mod_mult_right_eq [symmetric])
   apply (simp add: field_simps)
   done
@@ -105,7 +105,7 @@
   apply auto
   apply (metis invertible_coprime_int)
   apply (subst (asm) coprime_iff_invertible'_int)
-  apply (auto simp add: cong_int_def mult_commute)
+  apply (auto simp add: cong_int_def mult.commute)
   done
 
 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
@@ -120,7 +120,7 @@
   apply auto
   apply (subgoal_tac "y mod m = - x mod m")
   apply simp
-  apply (metis minus_add_cancel mod_mult_self1 mult_commute)
+  apply (metis minus_add_cancel mod_mult_self1 mult.commute)
   done
 
 lemma finite [iff]: "finite (carrier R)"
@@ -159,7 +159,7 @@
   apply (insert m_gt_one)
   apply (induct n)
   apply (auto simp add: nat_pow_def one_cong)
-  apply (metis mult_commute mult_cong)
+  apply (metis mult.commute mult_cong)
   done
 
 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
@@ -196,7 +196,7 @@
 
 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
   apply (simp add: res_one_eq res_neg_eq)
-  apply (metis add_commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
+  apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
             zero_neq_one zmod_zminus1_eq_if)
   done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Euler.thy
--- a/src/HOL/Old_Number_Theory/Euler.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Euler.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -148,7 +148,7 @@
                    b = "x * (a * MultInv p x)" and
                    c = "a * (x * MultInv p x)" in  zcong_trans, force)
   apply (frule_tac p = p and x = x in MultInv_prop2, auto)
-apply (metis StandardRes_SRStar_prop3 mult_1_right mult_commute zcong_sym zcong_zmult_prop1)
+apply (metis StandardRes_SRStar_prop3 mult_1_right mult.commute zcong_sym zcong_zmult_prop1)
   apply (auto simp add: mult_ac)
   done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Factorization.thy
--- a/src/HOL/Old_Number_Theory/Factorization.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Factorization.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -68,7 +68,7 @@
 
 lemma prod_append: "prod (xs @ ys) = prod xs * prod ys"
   apply (induct xs)
-   apply (simp_all add: mult_assoc)
+   apply (simp_all add: mult.assoc)
   done
 
 lemma prod_xy_prod:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Fib.thy
--- a/src/HOL/Old_Number_Theory/Fib.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Fib.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -106,7 +106,7 @@
   apply (case_tac m)
    apply simp 
   apply (simp add: fib_add)
-  apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
+  apply (simp add: add.commute gcd_non_0 [OF fib_Suc_gr_0])
   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
   done
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Gauss.thy
--- a/src/HOL/Old_Number_Theory/Gauss.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Gauss.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -342,7 +342,7 @@
     apply (elim zcong_trans)
     by (simp only: zcong_refl)
   also have "y * a + ya * a = a * (y + ya)"
-    by (simp add: distrib_left mult_commute)
+    by (simp add: distrib_left mult.commute)
   finally have "[a * (y + ya) = 0] (mod p)" .
   with p_prime a_nonzero zcong_zprime_prod_zero [of p a "y + ya"]
     p_a_relprime
@@ -428,7 +428,7 @@
   also have "setprod uminus E = (setprod id E) * (-1)^(card E)"
     using finite_E by (induct set: finite) auto
   then have "setprod uminus E = (-1) ^ (card E) * (setprod id E)"
-    by (simp add: mult_commute)
+    by (simp add: mult.commute)
   with two show ?thesis
     by simp
 qed
@@ -443,7 +443,7 @@
   "[a ^ nat((p - 1) div 2) = (-1) ^ (card E)] (mod p)"
 proof -
   have "[setprod id A = setprod id F * setprod id D](mod p)"
-    by (auto simp add: prod_D_F_eq_prod_A mult_commute cong del:setprod.cong)
+    by (auto simp add: prod_D_F_eq_prod_A mult.commute cong del:setprod.cong)
   then have "[setprod id A = ((-1)^(card E) * setprod id E) *
       setprod id D] (mod p)"
     apply (rule zcong_trans)
@@ -452,7 +452,7 @@
   then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)"
     apply (rule zcong_trans)
     apply (insert C_prod_eq_D_times_E, erule subst)
-    apply (subst mult_assoc, auto)
+    apply (subst mult.assoc, auto)
     done
   then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)"
     apply (rule zcong_trans)
@@ -473,7 +473,7 @@
   then have "[setprod id A = ((-1)^(card E) * a^(card A) *
       setprod id A)](mod p)"
     apply (rule zcong_trans)
-    apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant mult_assoc)
+    apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant mult.assoc)
     done
   then have a: "[setprod id A * (-1)^(card E) =
       ((-1)^(card E) * a^(card A) * setprod id A * (-1)^(card E))](mod p)"
@@ -481,7 +481,7 @@
   then have "[setprod id A * (-1)^(card E) = setprod id A *
       (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
     apply (rule zcong_trans)
-    apply (simp add: a mult_commute mult_left_commute)
+    apply (simp add: a mult.commute mult.left_commute)
     done
   then have "[setprod id A * (-1)^(card E) = setprod id A *
       a^(card A)](mod p)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Int2.thy
--- a/src/HOL/Old_Number_Theory/Int2.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Int2.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -269,7 +269,7 @@
 
 lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
      (MultInv p j) * a] (mod p)"
-  by (auto simp add: mult_assoc zcong_scalar2)
+  by (auto simp add: mult.assoc zcong_scalar2)
 
 lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p));
     [(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |]
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/IntPrimes.thy
--- a/src/HOL/Old_Number_Theory/IntPrimes.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/IntPrimes.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -93,7 +93,7 @@
   apply (simp add: zgcd_greatest_iff)
   apply (rule_tac n = k in zrelprime_zdvd_zmult)
    prefer 2
-   apply (simp add: mult_commute)
+   apply (simp add: mult.commute)
   apply (metis zgcd_1 zgcd_commute zgcd_left_commute)
   done
 
@@ -142,7 +142,7 @@
     "[a = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
   apply (rule_tac b = "b * c" in zcong_trans)
    apply (unfold zcong_def)
-  apply (metis right_diff_distrib dvd_mult mult_commute)
+  apply (metis right_diff_distrib dvd_mult mult.commute)
   apply (metis right_diff_distrib dvd_mult)
   done
 
@@ -165,7 +165,7 @@
     apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
      prefer 4
      apply (simp add: zdvd_reduce)
-    apply (simp_all add: left_diff_distrib mult_commute right_diff_distrib)
+    apply (simp_all add: left_diff_distrib mult.commute right_diff_distrib)
   done
 
 lemma zcong_cancel:
@@ -188,7 +188,7 @@
 lemma zcong_cancel2:
   "0 \<le> m ==>
     zgcd k m = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
-  by (simp add: mult_commute zcong_cancel)
+  by (simp add: mult.commute zcong_cancel)
 
 lemma zcong_zgcd_zmult_zmod:
   "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd m n = 1
@@ -197,9 +197,9 @@
   apply (subgoal_tac "m dvd n * ka")
    apply (subgoal_tac "m dvd ka")
     apply (case_tac [2] "0 \<le> ka")
-  apply (metis dvd_mult_div_cancel dvd_refl dvd_mult_left mult_commute zrelprime_zdvd_zmult)
-  apply (metis abs_dvd_iff abs_of_nonneg add_0 zgcd_0_left zgcd_commute zgcd_zadd_zmult zgcd_zdvd_zgcd_zmult zgcd_zmult_distrib2_abs mult_1_right mult_commute)
-  apply (metis mult_le_0_iff  zdvd_mono zdvd_mult_cancel dvd_triv_left zero_le_mult_iff order_antisym linorder_linear order_refl mult_commute zrelprime_zdvd_zmult)
+  apply (metis dvd_mult_div_cancel dvd_refl dvd_mult_left mult.commute zrelprime_zdvd_zmult)
+  apply (metis abs_dvd_iff abs_of_nonneg add_0 zgcd_0_left zgcd_commute zgcd_zadd_zmult zgcd_zdvd_zgcd_zmult zgcd_zmult_distrib2_abs mult_1_right mult.commute)
+  apply (metis mult_le_0_iff  zdvd_mono zdvd_mult_cancel dvd_triv_left zero_le_mult_iff order_antisym linorder_linear order_refl mult.commute zrelprime_zdvd_zmult)
   apply (metis dvd_triv_left)
   done
 
@@ -208,7 +208,7 @@
     a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
   apply (unfold zcong_def dvd_def, auto)
   apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
-  apply (metis diff_add_cancel mod_pos_pos_trivial add_0 add_commute zmod_eq_0_iff mod_add_right_eq)
+  apply (metis diff_add_cancel mod_pos_pos_trivial add_0 add.commute zmod_eq_0_iff mod_add_right_eq)
   done
 
 lemma zcong_square_zless:
@@ -261,7 +261,7 @@
   apply (rule_tac m = m in zcong_zmod_aux)
   apply (rule trans)
    apply (rule_tac [2] k = m and m = "a div m - b div m" in zdvd_reduce)
-  apply (simp add: add_commute)
+  apply (simp add: add.commute)
   done
 
 lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
@@ -341,14 +341,14 @@
 lemma xzgcda_linear_aux1:
   "(a - r * b) * m + (c - r * d) * (n::int) =
    (a * m + c * n) - r * (b * m + d * n)"
-  by (simp add: left_diff_distrib distrib_left mult_assoc)
+  by (simp add: left_diff_distrib distrib_left mult.assoc)
 
 lemma xzgcda_linear_aux2:
   "r' = s' * m + t' * n ==> r = s * m + t * n
     ==> (r' mod r) = (s' - (r' div r) * s) * m + (t' - (r' div r) * t) * (n::int)"
   apply (rule trans)
    apply (rule_tac [2] xzgcda_linear_aux1 [symmetric])
-  apply (simp add: eq_diff_eq mult_commute)
+  apply (simp add: eq_diff_eq mult.commute)
   done
 
 lemma order_le_neq_implies_less: "(x::'a::order) \<le> y ==> x \<noteq> y ==> x < y"
@@ -391,7 +391,7 @@
    prefer 2
    apply simp
   apply (unfold zcong_def)
-  apply (simp (no_asm) add: mult_commute)
+  apply (simp (no_asm) add: mult.commute)
   done
 
 lemma zcong_lineq_unique:
@@ -407,7 +407,7 @@
   apply (cut_tac a = a and n = n in zcong_lineq_ex, auto)
   apply (rule_tac x = "x * b mod n" in exI, safe)
     apply (simp_all (no_asm_simp))
-  apply (metis zcong_scalar zcong_zmod mod_mult_right_eq mult_1 mult_assoc)
+  apply (metis zcong_scalar zcong_zmod mod_mult_right_eq mult_1 mult.assoc)
   done
 
 end
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Legacy_GCD.thy
--- a/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -161,7 +161,7 @@
     apply (rule_tac n = k in relprime_dvd_mult)
      apply (simp add: gcd_assoc)
      apply (simp add: gcd_commute)
-    apply (simp_all add: mult_commute)
+    apply (simp_all add: mult.commute)
   done
 
 
@@ -173,19 +173,19 @@
 lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
 proof -
   have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
-  also have "... = gcd (n + m) m" by (simp add: add_commute)
+  also have "... = gcd (n + m) m" by (simp add: add.commute)
   also have "... = gcd n m" by simp
   also have  "... = gcd m n" by (rule gcd_commute)
   finally show ?thesis .
 qed
 
 lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
-  apply (subst add_commute)
+  apply (subst add.commute)
   apply (rule gcd_add2)
   done
 
 lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
-  by (induct k) (simp_all add: add_assoc)
+  by (induct k) (simp_all add: add.assoc)
 
 lemma gcd_dvd_prod: "gcd m n dvd m * n" 
   using mult_dvd_mono [of 1] by auto
@@ -351,7 +351,7 @@
           hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" 
             by (simp only: diff_add_assoc[OF dble, of d, symmetric])
           hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
-            by (simp only: diff_mult_distrib2 add_commute mult_ac)
+            by (simp only: diff_mult_distrib2 add.commute mult_ac)
           hence ?thesis using H(1,2)
             apply -
             apply (rule exI[where x=d], simp)
@@ -374,7 +374,7 @@
   hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" 
     by (algebra add: diff_mult_distrib)
   hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" 
-    by (simp add: k mult_assoc)
+    by (simp add: k mult.assoc)
   thus ?thesis by blast
 qed
 
@@ -392,7 +392,7 @@
 qed
 
 lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
-by(simp add: gcd_mult_distrib2 mult_commute)
+by(simp add: gcd_mult_distrib2 mult.commute)
 
 lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
   (is "?lhs \<longleftrightarrow> ?rhs")
@@ -405,7 +405,7 @@
     hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" 
       by (simp only: diff_mult_distrib)
     hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
-      by (simp add: k[symmetric] mult_assoc)
+      by (simp add: k[symmetric] mult.assoc)
     hence ?lhs by blast}
   moreover
   {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
@@ -426,7 +426,7 @@
 qed
 
 lemma gcd_mult': "gcd b (a * b) = b"
-by (simp add: mult_commute[of a b]) 
+by (simp add: mult.commute[of a b]) 
 
 lemma gcd_add: "gcd(a + b) b = gcd a b" 
   "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Pocklington.thy
--- a/src/HOL/Old_Number_Theory/Pocklington.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Pocklington.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -120,7 +120,7 @@
 
 lemma cong_mult_rcancel: assumes an: "coprime a n" and axy:"[x*a = y*a] (mod n)"
   shows "[x = y] (mod n)"
-  using cong_mult_lcancel[OF an axy[unfolded mult_commute[of _a]]] .
+  using cong_mult_lcancel[OF an axy[unfolded mult.commute[of _a]]] .
 
 lemma cong_refl: "[x = x] (mod n)" by (simp add: modeq_def)
 
@@ -177,13 +177,13 @@
 proof
   assume H: "?rhs" from cong_mult[OF cong_refl[of a n] H] show ?lhs .
 next
-  assume H: "?lhs" hence H': "[x*a = y*a] (mod n)" by (simp add: mult_commute)
+  assume H: "?lhs" hence H': "[x*a = y*a] (mod n)" by (simp add: mult.commute)
   from cong_mult_rcancel[OF an H'] show ?rhs  .
 qed
 
 lemma cong_mult_rcancel_eq: assumes an: "coprime a n"
   shows "[x * a = y * a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
-using cong_mult_lcancel_eq[OF an, of x y] by (simp add: mult_commute)
+using cong_mult_lcancel_eq[OF an, of x y] by (simp add: mult.commute)
 
 lemma cong_add_lcancel_eq: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
   by (simp add: nat_mod)
@@ -358,7 +358,7 @@
   also have "\<dots> = m mod a" by (simp add: mod_mult2_eq)
   finally have th1: "[?w = m] (mod a)" by (simp add: modeq_def)
   from xq12(2) have "?w mod b = ((n + q2 * b) mod (a*b)) mod b" by simp
-  also have "\<dots> = ((n + q2 * b) mod (b*a)) mod b" by (simp add: mult_commute)
+  also have "\<dots> = ((n + q2 * b) mod (b*a)) mod b" by (simp add: mult.commute)
   also have "\<dots> = n mod b" by (simp add: mod_mult2_eq)
   finally have th2: "[?w = n] (mod b)" by (simp add: modeq_def)
   {fix y assume H: "y < a*b" "[y = m] (mod a)" "[y = n] (mod b)"
@@ -638,7 +638,7 @@
       also have "[(\<Prod>i\<in>?S. ?h i) = ?P] (mod n)"
         using eq0 fS an by (simp add: setprod_def modeq_def)
       finally show "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
-        unfolding cardfS mult_commute[of ?P "a^ (card ?S)"]
+        unfolding cardfS mult.commute[of ?P "a^ (card ?S)"]
           nproduct_cmul[OF fS, symmetric] mult_1_right by simp
     qed
     from cong_mult_lcancel[OF nP Paphi] have ?thesis . }
@@ -856,7 +856,7 @@
     from H have onz: "?o \<noteq> 0" by (simp add: ord_eq_0)
     hence op: "?o > 0" by simp
     from mod_div_equality[of d "ord n a"] lh
-    have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: modeq_def mult_commute)
+    have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: modeq_def mult.commute)
     hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
       by (simp add: modeq_def power_mult[symmetric] power_add[symmetric])
     hence th: "[a^?r = 1] (mod n)"
@@ -964,7 +964,7 @@
         from H[rule_format, of d] h d have "d = 1" by blast}
       moreover
       {assume h: "e\<^sup>2 \<le> n"
-        from e have "e dvd n" unfolding dvd_def by (simp add: mult_commute)
+        from e have "e dvd n" unfolding dvd_def by (simp add: mult.commute)
         with H[rule_format, of e] h have "e=1" by simp
         with e have "d = n" by simp}
       ultimately have "d=1 \<or> d=n"  by blast}
@@ -1231,7 +1231,7 @@
     from prime_ge_2[OF p(1)] have p01: "p \<noteq> 0" "p \<noteq> 1" "p =Suc(p - 1)" by arith+
     from div_mult1_eq[of r q p] p(2)
     have eq1: "r* (q div p) = (n - 1) div p"
-      unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult_commute)
+      unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult.commute)
     have ath: "\<And>a (b::nat). a <= b \<Longrightarrow> a \<noteq> 0 ==> 1 <= a \<and> 1 <= b" by arith
     from n0 have n00: "n \<noteq> 0" by arith
     from mod_le[OF n00]
@@ -1246,7 +1246,7 @@
       with eq0 have "a^ (n - 1) = (n*s)^p"
         by (simp add: power_mult[symmetric])
       hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
-      also have "\<dots> = 0" by (simp add: mult_assoc)
+      also have "\<dots> = 0" by (simp add: mult.assoc)
       finally have False by simp }
       then have th11: "a ^ ((n - 1) div p) mod n \<noteq> 0" by auto
     have th1: "[a ^ ((n - 1) div p) mod n = a ^ ((n - 1) div p)] (mod n)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Primes.thy
--- a/src/HOL/Old_Number_Theory/Primes.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Primes.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -108,7 +108,7 @@
 
 declare nat_mult_dvd_cancel_disj[presburger]
 lemma nat_mult_dvd_cancel_disj'[presburger]: 
-  "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger 
+  "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult.commute[of m k] mult.commute[of n k] by presburger 
 
 lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
   by presburger
@@ -120,7 +120,7 @@
 lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
 proof(auto simp add: dvd_def)
   fix k assume H: "0 < r" "r < n" "q * n + r = n * k"
-  from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute)
+  from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult.commute)
   {assume "k - q = 0" with r H(1) have False by simp}
   moreover
   {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto
@@ -186,7 +186,7 @@
 using coprime_def gcd_bezout by auto
 
 lemma coprime_divprod: "d dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
-  using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)
+  using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult.commute)
 
 lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
 lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
@@ -205,7 +205,7 @@
     from bezout_gcd_strong[OF az, of b] 
     obtain x y where xy: "a*x = b*y + ?g" by blast
     from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps)
-    hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
+    hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult.assoc)
     hence "a'*x = (b'*y + 1)"
       by (simp only: nat_mult_eq_cancel1[OF z']) 
     hence "a'*x - b'*y = 1" by simp
@@ -296,7 +296,7 @@
       using z by auto 
     then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
       using z ab'' by (simp only: power_mult_distrib[symmetric] 
-        diff_mult_distrib2 mult_assoc[symmetric])
+        diff_mult_distrib2 mult.assoc[symmetric])
     hence  ?thesis by blast }
   ultimately show ?thesis by blast
 qed
@@ -334,7 +334,7 @@
     from ab'(1) have "a' dvd a"  unfolding dvd_def by blast  
     with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
     from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
-    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
     with z have th_1: "a' dvd b'*c" by simp
     from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . 
     from ab' have "a = ?g*a'" by algebra
@@ -355,11 +355,11 @@
     from gcd_coprime_exists[OF z] 
     obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
     from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
-    hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)
+    hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult.commute)
     with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
     have "a' dvd a'^n" by (simp add: m)
     with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
-    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
     from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
     have "a' dvd b'" .
     hence "a'*?g dvd b'*?g" by simp
@@ -372,7 +372,7 @@
 proof-
   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
     unfolding dvd_def by blast
-  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
   hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   from n' k show ?thesis unfolding dvd_def by auto
@@ -609,7 +609,7 @@
     have "p dvd a \<or> p dvd b" .
     moreover
     {assume pa: "p dvd a"
-      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+      have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute)
       from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
       with prime_coprime[OF p, of b] b 
       have cpb: "coprime b p" using coprime_commute by blast 
@@ -618,7 +618,7 @@
       from coprime_divprod[OF pnba pnb] have ?thesis by blast }
     moreover
     {assume pb: "p dvd b"
-      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+      have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute)
       from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
       with prime_coprime[OF p, of a] a
       have cpb: "coprime a p" using coprime_commute by blast 
@@ -632,7 +632,7 @@
 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume H: "?lhs"
-  hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
+  hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult.commute)
   thus ?rhs by auto
 next
   assume ?rhs then show ?lhs by auto
@@ -705,7 +705,7 @@
       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
       have kb: "coprime k a" by (simp add: coprime_commute) 
       from H(3) l k pn0 n have kbln: "k * a = l ^ n" 
-        by (simp add: power_mult_distrib mult_commute)
+        by (simp add: power_mult_distrib mult.commute)
       from H(1)[rule_format, OF lc kb kbln]
       obtain r s where rs: "k = r ^n" "a = s^n" by blast
       from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
@@ -772,7 +772,7 @@
   moreover 
   {assume px: "p dvd y"
     then obtain d where d: "y = p*d" unfolding dvd_def by blast
-    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
+    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult.commute)
     hence th: "d*x = p^k" using p0 by simp
     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
     with d have "y = p^Suc i" by simp 
@@ -800,7 +800,7 @@
   shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
 proof
   assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" 
-    unfolding dvd_def  apply (auto simp add: mult_commute) by blast
+    unfolding dvd_def  apply (auto simp add: mult.commute) by blast
   from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
   from prime_ge_2[OF p] have p1: "p > 1" by arith
   from e ij have "p^(i + j) = p^k" by (simp add: power_add)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
--- a/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -22,7 +22,7 @@
   from finite_A have "a * setsum id A = setsum (%x. a * x) A"
     by (auto simp add: setsum_const_mult id_def)
   also have "setsum (%x. a * x) = setsum (%x. x * a)"
-    by (auto simp add: mult_commute)
+    by (auto simp add: mult.commute)
   also have "setsum (%x. x * a) A = setsum id B"
     by (simp add: B_def setsum.reindex [OF inj_on_xa_A])
   also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/WilsonBij.thy
--- a/src/HOL/Old_Number_Theory/WilsonBij.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/WilsonBij.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -207,7 +207,7 @@
 
 lemma reciP_sym: "zprime p ==> symP (reciR p)"
   apply (unfold reciR_def symP_def)
-  apply (simp add: mult_commute)
+  apply (simp add: mult.commute)
   apply auto
   done
 
@@ -234,7 +234,7 @@
     apply (subst setprod.insert)
       apply (auto simp add: fin_bijER)
   apply (subgoal_tac "zcong ((a * b) * \<Prod>A) (1 * 1) p")
-   apply (simp add: mult_assoc)
+   apply (simp add: mult.assoc)
   apply (rule zcong_zmult)
    apply auto
   done
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Old_Number_Theory/WilsonRuss.thy
--- a/src/HOL/Old_Number_Theory/WilsonRuss.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Old_Number_Theory/WilsonRuss.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -256,7 +256,7 @@
       apply (subgoal_tac [5]
         "zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")
        prefer 5
-       apply (simp add: mult_assoc)
+       apply (simp add: mult.assoc)
       apply (rule_tac [5] zcong_zmult)
        apply (rule_tac [5] inv_is_inv)
          apply (tactic "clarify_tac @{context} 4")
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Power.thy
--- a/src/HOL/Power.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Power.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -53,7 +53,7 @@
 
 lemma power_commutes:
   "a ^ n * a = a * a ^ n"
-  by (induct n) (simp_all add: mult_assoc)
+  by (induct n) (simp_all add: mult.assoc)
 
 lemma power_Suc2:
   "a ^ Suc n = a ^ n * a"
@@ -71,11 +71,11 @@
   by (simp add: numeral_2_eq_2)
 
 lemma power3_eq_cube: "a ^ 3 = a * a * a"
-  by (simp add: numeral_3_eq_3 mult_assoc)
+  by (simp add: numeral_3_eq_3 mult.assoc)
 
 lemma power_even_eq:
   "a ^ (2 * n) = (a ^ n)\<^sup>2"
-  by (subst mult_commute) (simp add: power_mult)
+  by (subst mult.commute) (simp add: power_mult)
 
 lemma power_odd_eq:
   "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
@@ -88,7 +88,7 @@
 lemma power_numeral_odd:
   "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
   unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
-  unfolding power_Suc power_add Let_def mult_assoc ..
+  unfolding power_Suc power_add Let_def mult.assoc ..
 
 lemma funpow_times_power:
   "(times x ^^ f x) = times (x ^ f x)"
@@ -100,7 +100,7 @@
   with Suc have "n = g x" by simp
   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
   moreover from Suc g_def have "f x = g x + 1" by simp
-  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc)
+  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
 qed
 
 end
@@ -197,7 +197,7 @@
   case 0 show ?case by simp
 next
   case (Suc n) then show ?case
-    by (simp del: power_Suc add: power_Suc2 mult_assoc)
+    by (simp del: power_Suc add: power_Suc2 mult.assoc)
 qed
 
 lemma power_minus_Bit0:
@@ -626,7 +626,7 @@
 lemma power2_diff:
   fixes x y :: "'a::comm_ring_1"
   shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
-  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
+  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult.commute)
 
 lemma power_0_Suc [simp]:
   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Presburger.thy
--- a/src/HOL/Presburger.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Presburger.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -54,8 +54,8 @@
   "(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
   "\<forall>x k. F = F"
 apply (auto elim!: dvdE simp add: algebra_simps)
-unfolding mult_assoc [symmetric] distrib_right [symmetric] left_diff_distrib [symmetric]
-unfolding dvd_def mult_commute [of d] 
+unfolding mult.assoc [symmetric] distrib_right [symmetric] left_diff_distrib [symmetric]
+unfolding dvd_def mult.commute [of d] 
 by auto
 
 subsection{* The A and B sets *}
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Bochner_Integration.thy
--- a/src/HOL/Probability/Bochner_Integration.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Bochner_Integration.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -743,7 +743,7 @@
   finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
 
   have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) + ereal (norm (s i x)) \<partial>M)"
-    by (auto intro!: nn_integral_mono) (metis add_commute norm_triangle_sub)
+    by (auto intro!: nn_integral_mono) (metis add.commute norm_triangle_sub)
   also have "\<dots> = (\<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) + (\<integral>\<^sup>+x. norm (s i x) \<partial>M)"
     by (rule nn_integral_add) auto
   also have "\<dots> < \<infinity>"
@@ -779,7 +779,7 @@
            (auto intro: s simple_bochner_integrable_compose2)
       also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s n x)) + norm (f x) \<partial>M)"
         by (auto intro!: nn_integral_mono)
-           (metis add_commute norm_minus_commute norm_triangle_sub)
+           (metis add.commute norm_minus_commute norm_triangle_sub)
       also have "\<dots> = ?t n" 
         by (rule nn_integral_add) auto
       finally show "norm (?s n) \<le> ?t n" .
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Convolution.thy
--- a/src/HOL/Probability/Convolution.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Convolution.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -66,7 +66,7 @@
   apply (subst nn_integral_indicator[symmetric], simp)
   apply (subst nn_integral_convolution, 
         auto intro!: borel_measurable_indicator borel_measurable_indicator' convolution_finite)+
-  by (rule nn_integral_cong)+ (auto simp: semigroup_add_class.add_assoc)
+  by (rule nn_integral_cong)+ (auto simp: semigroup_add_class.add.assoc)
 
 lemma convolution_emeasure_3':
   assumes [simp, measurable]:"A \<in> sets borel"
@@ -96,7 +96,7 @@
   have "emeasure (M \<star> N) A = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator A (x + y) \<partial>N \<partial>M" by (auto intro!: convolution_emeasure')
   also have "... = \<integral>\<^sup>+x. \<integral>\<^sup>+y. (\<lambda>(x,y). indicator A (x + y)) (x, y) \<partial>N \<partial>M" by (auto intro!: nn_integral_cong)
   also have "... = \<integral>\<^sup>+y. \<integral>\<^sup>+x. (\<lambda>(x,y). indicator A (x + y)) (x, y) \<partial>M \<partial>N" by (rule Fubini[symmetric]) simp
-  also have "... = emeasure (N \<star> M) A" by (auto intro!: nn_integral_cong simp: add_commute convolution_emeasure')
+  also have "... = emeasure (N \<star> M) A" by (auto intro!: nn_integral_cong simp: add.commute convolution_emeasure')
   finally show "emeasure (M \<star> N) A = emeasure (N \<star> M) A" by simp
 qed
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Distributions.thy
--- a/src/HOL/Probability/Distributions.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Distributions.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -865,7 +865,7 @@
     proof (subst nn_integral_FTC_atLeast)
       have "((\<lambda>a. - (exp (- (a\<^sup>2 * (1 + s\<^sup>2))) / (2 + 2 * s\<^sup>2))) ---> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
         apply (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_compose[OF filterlim_uminus_at_bot_at_top])
-        apply (subst mult_commute)         
+        apply (subst mult.commute)         
         apply (auto intro!: filterlim_tendsto_pos_mult_at_top
                             filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident] 
                     simp: add_pos_nonneg  power2_eq_square add_nonneg_eq_0_iff)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Finite_Product_Measure.thy
--- a/src/HOL/Probability/Finite_Product_Measure.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Finite_Product_Measure.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -643,7 +643,7 @@
       using E by (subst insert) (auto intro!: setprod.cong)
     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
-      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod.cong)
+      using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] setprod.cong)
     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
       using insert(1,2) J E by (intro setprod.mono_neutral_right) auto
     finally show "?\<mu> ?p = \<dots>" .
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Independent_Family.thy
--- a/src/HOL/Probability/Independent_Family.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Independent_Family.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1301,7 +1301,7 @@
 unfolding indep_var_def
 proof -
   have *: "(\<lambda>\<omega>. X1 \<omega> * X2 \<omega>) = (\<lambda>\<omega>. \<Prod>i\<in>UNIV. (case_bool X1 X2 i \<omega>))"
-    by (simp add: UNIV_bool mult_commute)
+    by (simp add: UNIV_bool mult.commute)
   have **: "(\<lambda> _. borel) = case_bool borel borel"
     by (rule ext, metis (full_types) bool.simps(3) bool.simps(4))
   show ?eq
@@ -1310,7 +1310,7 @@
     apply (auto)
     apply (subst **, subst indep_var_def [symmetric], rule assms)
     apply (simp split: bool.split add: assms)
-    by (simp add: UNIV_bool mult_commute)
+    by (simp add: UNIV_bool mult.commute)
   show ?int
     apply (subst *)
     apply (rule indep_vars_integrable)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Infinite_Product_Measure.thy
--- a/src/HOL/Probability/Infinite_Product_Measure.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -565,7 +565,7 @@
   also have "emeasure S ?E = (\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j))"
     using J by (intro emeasure_PiM_emb) simp_all
   also have "(\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j)) * (\<Prod>j\<in>J - (J \<inter> {..<i}). emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))"
-    by (subst mult_commute) (auto simp: J setprod.subset_diff[symmetric])
+    by (subst mult.commute) (auto simp: J setprod.subset_diff[symmetric])
   finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" .
 qed simp_all
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Lebesgue_Measure.thy
--- a/src/HOL/Probability/Lebesgue_Measure.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Lebesgue_Measure.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1080,7 +1080,7 @@
       by (auto intro: continuous_at_imp_continuous_on)
   qed simp
   then show ?thesis
-    by (auto simp: mult_commute)
+    by (auto simp: mult.commute)
 qed
 
 text {*
@@ -1160,7 +1160,7 @@
     using cont by (intro continuous_at_imp_continuous_on) auto
   show ?has ?eq
     using has_bochner_integral_FTC_Icc[OF `a \<le> b` 1 2] integral_FTC_Icc[OF `a \<le> b` 1 2]
-    by (auto simp: mult_commute)
+    by (auto simp: mult.commute)
 qed
 
 lemma nn_integral_FTC_atLeast:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy
--- a/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -614,7 +614,7 @@
     using f by (intro simple_function_partition) auto
   also have "\<dots> = c * integral\<^sup>S M f"
     using f unfolding simple_integral_def
-    by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult_assoc Int_def conj_commute)
+    by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult.assoc Int_def conj_commute)
   finally show ?thesis .
 qed
 
@@ -1129,7 +1129,7 @@
 lemma nn_integral_multc:
   assumes "f \<in> borel_measurable M" "0 \<le> c"
   shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>N M f * c"
-  unfolding mult_commute[of _ c] nn_integral_cmult[OF assms] by simp
+  unfolding mult.commute[of _ c] nn_integral_cmult[OF assms] by simp
 
 lemma nn_integral_indicator[simp]:
   "A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Probability/Sigma_Algebra.thy
--- a/src/HOL/Probability/Sigma_Algebra.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -830,11 +830,11 @@
       case 0 show ?case by simp
     next
       case (Suc m) thus ?case
-        by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)
+        by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans)
     qed
   }
   hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
-    by (metis add_commute le_add_diff_inverse nat_less_le)
+    by (metis add.commute le_add_diff_inverse nat_less_le)
   thus ?thesis
     by (auto simp add: disjoint_family_on_def)
       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Proofs/Extraction/Euclid.thy
--- a/src/HOL/Proofs/Extraction/Euclid.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Proofs/Extraction/Euclid.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -114,12 +114,12 @@
     assume "m \<le> n"
     with `0 < m` have "m dvd fact n" by (rule Suc)
     then have "m dvd (fact n * Suc n)" by (rule dvd_mult2)
-    then show ?thesis by (simp add: mult_commute)
+    then show ?thesis by (simp add: mult.commute)
   next
     assume "m = Suc n"
     then have "m dvd (fact n * Suc n)"
       by (auto intro: dvdI simp: mult_ac)
-    then show ?thesis by (simp add: mult_commute)
+    then show ?thesis by (simp add: mult.commute)
   qed
 qed
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Proofs/Extraction/Greatest_Common_Divisor.thy
--- a/src/HOL/Proofs/Extraction/Greatest_Common_Divisor.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Proofs/Extraction/Greatest_Common_Divisor.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -39,7 +39,7 @@
       by iprover
     have mn: "Suc m < n" by (rule 1)
     from h1 h1' h2' Suc have "k * (m1 * q + r1) = n" 
-      by (simp add: add_mult_distrib2 nat_mult_assoc [symmetric])
+      by (simp add: add_mult_distrib2 mult.assoc [symmetric])
     moreover have "\<And>l l1 l2. l * l1 = n \<Longrightarrow> l * l2 = Suc m \<Longrightarrow> l \<le> k"
     proof -
       fix l l1 l2
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Proofs/Lambda/ListApplication.thy
--- a/src/HOL/Proofs/Lambda/ListApplication.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Proofs/Lambda/ListApplication.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -110,7 +110,7 @@
     prefer 2
     apply (erule allE, erule mp, rule refl)
    apply simp
-   apply (simp only: foldl_conv_fold add_commute fold_plus_listsum_rev)
+   apply (simp only: foldl_conv_fold add.commute fold_plus_listsum_rev)
    apply (fastforce simp add: listsum_map_remove1)
   apply clarify
   apply (rule assms)
@@ -126,7 +126,7 @@
   apply (rule le_imp_less_Suc)
   apply (rule trans_le_add1)
   apply (rule trans_le_add2)
-  apply (simp only: foldl_conv_fold add_commute fold_plus_listsum_rev)
+  apply (simp only: foldl_conv_fold add.commute fold_plus_listsum_rev)
   apply (simp add: member_le_listsum_nat)
   done
 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Quotient_Examples/Int_Pow.thy
--- a/src/HOL/Quotient_Examples/Int_Pow.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Quotient_Examples/Int_Pow.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -53,7 +53,7 @@
 proof (cases "m\<ge>n")
   have [simp]: "a \<in> carrier G" using `a \<in> _` by (rule Units_closed)
   case True
-    then obtain k where *:"m = k + n" and **:"m = n + k" by (metis Nat.le_iff_add nat_add_commute)
+    then obtain k where *:"m = k + n" and **:"m = n + k" by (metis Nat.le_iff_add add.commute)
     have "a (^) (m::nat) \<otimes> inv (a (^) (n::nat)) = a (^) k"
       using * by (auto simp add: nat_pow_mult[symmetric] m_assoc)
     also have "\<dots> = inv (a (^) n) \<otimes> a (^) m"
@@ -63,7 +63,7 @@
   have [simp]: "a \<in> carrier G" using `a \<in> _` by (rule Units_closed)
   case False
     then obtain k where *:"n = k + m" and **:"n = m + k" 
-      by (metis Nat.le_iff_add nat_add_commute nat_le_linear)
+      by (metis Nat.le_iff_add add.commute nat_le_linear)
     have "a (^) (m::nat) \<otimes> inv (a (^) (n::nat)) = inv(a (^) k)"
       using * by (auto simp add: nat_pow_mult[symmetric] m_assoc[symmetric] inv_mult_units)
     also have "\<dots> = inv (a (^) n) \<otimes> a (^) m"
@@ -86,7 +86,7 @@
 proof(cases "b\<ge>c")
   have [simp]: "a \<in> carrier G" using `a \<in> _` by (rule Units_closed)
   case True
-    then obtain n where "b = n + c" by (metis Nat.le_iff_add nat_add_commute)
+    then obtain n where "b = n + c" by (metis Nat.le_iff_add add.commute)
     then have "d = n + e" using eq by arith
     from `b = _` have "a (^) b \<otimes> inv (a (^) c) = a (^) n" 
       by (auto simp add: nat_pow_mult[symmetric] m_assoc)
@@ -96,7 +96,7 @@
 next
   have [simp]: "a \<in> carrier G" using `a \<in> _` by (rule Units_closed)
   case False
-    then obtain n where "c = n + b" by (metis Nat.le_iff_add nat_add_commute nat_le_linear)
+    then obtain n where "c = n + b" by (metis Nat.le_iff_add add.commute nat_le_linear)
     then have "e = n + d" using eq by arith
     from `c = _` have "a (^) b \<otimes> inv (a (^) c) = inv (a (^) n)" 
       by (auto simp add: nat_pow_mult[symmetric] m_assoc[symmetric] inv_mult_units)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Quotient_Examples/Quotient_Rat.thy
--- a/src/HOL/Quotient_Examples/Quotient_Rat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Quotient_Examples/Quotient_Rat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -41,14 +41,14 @@
   "times_rat_raw (a :: int, b :: int) (c, d) = (a * c, b * d)"
 
 quotient_definition
-  "(op *) :: (rat \<Rightarrow> rat \<Rightarrow> rat)" is times_rat_raw by (auto simp add: mult_assoc mult_left_commute)
+  "(op *) :: (rat \<Rightarrow> rat \<Rightarrow> rat)" is times_rat_raw by (auto simp add: mult.assoc mult.left_commute)
 
 fun plus_rat_raw where
   "plus_rat_raw (a :: int, b :: int) (c, d) = (a * d + c * b, b * d)"
 
 quotient_definition
   "(op +) :: (rat \<Rightarrow> rat \<Rightarrow> rat)" is plus_rat_raw 
-  by (auto simp add: mult_commute mult_left_commute int_distrib(2))
+  by (auto simp add: mult.commute mult.left_commute int_distrib(2))
 
 fun uminus_rat_raw where
   "uminus_rat_raw (a :: int, b :: int) = (-a, b)"
@@ -78,13 +78,13 @@
     have "a * f * b * f * d * d \<le> e * b * b * f * d * d" using le nz(2)
       by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
     then have "c * f * f * d * (b * b) \<le> e * f * d * d * (b * b)" using eq
-      by (metis (no_types) mult_assoc mult_commute)
+      by (metis (no_types) mult.assoc mult.commute)
     then have "c * f * f * d \<le> e * f * d * d" using b2
       by (metis leD linorder_le_less_linear mult_strict_right_mono)
     then have "c * f * f * d * h * h \<le> e * f * d * d * h * h" using nz(4)
       by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
     then have "c * h * (d * h) * (f * f) \<le> g * d * (d * h) * (f * f)" using eq
-      by (metis (no_types) mult_assoc mult_commute)
+      by (metis (no_types) mult.assoc mult.commute)
     then have "c * h * (d * h) \<le> g * d * (d * h)" using f2
       by (metis leD linorder_le_less_linear mult_strict_right_mono)
   }
@@ -128,7 +128,7 @@
   show "1 * a = a"
     by partiality_descending auto
   show "a + b + c = a + (b + c)"
-    by partiality_descending (auto simp add: mult_commute distrib_left)
+    by partiality_descending (auto simp add: mult.commute distrib_left)
   show "a + b = b + a"
     by partiality_descending auto
   show "0 + a = a"
@@ -138,7 +138,7 @@
   show "a - b = a + - b"
     by (simp add: minus_rat_def)
   show "(a + b) * c = a * c + b * c"
-    by partiality_descending (auto simp add: mult_commute distrib_left)
+    by partiality_descending (auto simp add: mult.commute distrib_left)
   show "(0 :: rat) \<noteq> (1 :: rat)"
     by partiality_descending auto
 qed
@@ -167,7 +167,7 @@
   "rat_inverse_raw (a :: int, b :: int) = (if a = 0 then (0, 1) else (b, a))"
 
 quotient_definition
-  "inverse :: rat \<Rightarrow> rat" is rat_inverse_raw by (force simp add: mult_commute)
+  "inverse :: rat \<Rightarrow> rat" is rat_inverse_raw by (force simp add: mult.commute)
 
 definition
   divide_rat_def: "q / r = q * inverse (r::rat)"
@@ -194,7 +194,7 @@
   {
     assume "q \<le> r" and "r \<le> s"
     then show "q \<le> s"
-    proof (partiality_descending, auto simp add: mult_assoc[symmetric])
+    proof (partiality_descending, auto simp add: mult.assoc[symmetric])
       fix a b c d e f :: int
       assume nz: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
       then have d2: "d * d > 0"
@@ -220,9 +220,9 @@
     show "q \<le> q" by partiality_descending auto
     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
       unfolding less_rat_def
-      by partiality_descending (auto simp add: le_less mult_commute)
+      by partiality_descending (auto simp add: le_less mult.commute)
     show "q \<le> r \<or> r \<le> q"
-      by partiality_descending (auto simp add: mult_commute linorder_linear)
+      by partiality_descending (auto simp add: mult.commute linorder_linear)
   }
 qed
 
@@ -232,25 +232,25 @@
 proof
   fix q r s :: rat
   show "q \<le> r ==> s + q \<le> s + r"
-  proof (partiality_descending, auto simp add: algebra_simps, simp add: mult_assoc[symmetric])
+  proof (partiality_descending, auto simp add: algebra_simps, simp add: mult.assoc[symmetric])
     fix a b c d e :: int
     assume "e \<noteq> 0"
     then have e2: "e * e > 0"
       by (metis linorder_neqE_linordered_idom mult_eq_0_iff not_square_less_zero)
       assume "a * b * d * d \<le> b * b * c * d"
       then show "a * b * d * d * e * e * e * e \<le> b * b * c * d * e * e * e * e"
-        using e2 by (metis comm_mult_left_mono mult_commute linorder_le_cases
+        using e2 by (metis comm_mult_left_mono mult.commute linorder_le_cases
           mult_left_mono_neg)
     qed
   show "q < r ==> 0 < s ==> s * q < s * r" unfolding less_rat_def
-    proof (partiality_descending, auto simp add: algebra_simps, simp add: mult_assoc[symmetric])
+    proof (partiality_descending, auto simp add: algebra_simps, simp add: mult.assoc[symmetric])
     fix a b c d e f :: int
     assume a: "e \<noteq> 0" "f \<noteq> 0" "0 \<le> e * f" "a * b * d * d \<le> b * b * c * d"
     have "a * b * d * d * (e * f) \<le> b * b * c * d * (e * f)" using a
       by (simp add: mult_right_mono)
     then show "a * b * d * d * e * f * f * f \<le> b * b * c * d * e * f * f * f"
-      by (simp add: mult_assoc[symmetric]) (metis a(3) comm_mult_left_mono
-        mult_commute mult_left_mono_neg zero_le_mult_iff)
+      by (simp add: mult.assoc[symmetric]) (metis a(3) comm_mult_left_mono
+        mult.commute mult_left_mono_neg zero_le_mult_iff)
   qed
   show "\<exists>z. r \<le> of_int z"
     unfolding of_int_rat
@@ -258,7 +258,7 @@
     fix a b :: int
     assume "b \<noteq> 0"
     then have "a * b \<le> (a div b + 1) * b * b"
-      by (metis mult_commute div_mult_self1_is_id less_int_def linorder_le_cases zdiv_mono1 zdiv_mono1_neg zle_add1_eq_le)
+      by (metis mult.commute div_mult_self1_is_id less_int_def linorder_le_cases zdiv_mono1 zdiv_mono1_neg zle_add1_eq_le)
     then show "\<exists>z\<Colon>int. a * b \<le> z * b * b" by auto
   qed
 qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Rat.thy
--- a/src/HOL/Rat.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Rat.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -76,7 +76,7 @@
     by (simp add: dvd_div_mult_self)
   with `b \<noteq> 0` have "?b \<noteq> 0" by auto
   from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
-    by (simp add: eq_rat dvd_div_mult mult_commute [of a])
+    by (simp add: eq_rat dvd_div_mult mult.commute [of a])
   from `b \<noteq> 0` have coprime: "coprime ?a ?b"
     by (auto intro: div_gcd_coprime_int)
   show C proof (cases "b > 0")
@@ -156,7 +156,7 @@
 
 lift_definition inverse_rat :: "rat \<Rightarrow> rat"
   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
-  by (auto simp add: mult_commute)
+  by (auto simp add: mult.commute)
 
 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
   by transfer simp
@@ -254,7 +254,7 @@
   moreover have "b div gcd a b * gcd a b = b"
     by (rule dvd_div_mult_self) simp
   ultimately have "b div gcd a b \<noteq> 0" by auto
-  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
+  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult.commute [of a])
 qed
 
 definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
@@ -274,7 +274,7 @@
     with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
   qed
   from assms show ?thesis
-    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
+    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_times split: if_splits intro: aux)
 qed
 
 lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Real.thy
--- a/src/HOL/Real.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Real.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -412,7 +412,7 @@
 
 lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
   unfolding realrel_def mult_diff_mult
-  by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add
+  by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add
     vanishes_mult_bounded cauchy_imp_bounded simp_thms)
 
 lift_definition inverse_real :: "real \<Rightarrow> real"
@@ -812,7 +812,7 @@
 
   have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
     apply (simp add: divide_less_eq)
-    apply (subst mult_commute)
+    apply (subst mult.commute)
     apply (frule_tac y=y in ex_less_of_nat_mult)
     apply clarify
     apply (rule_tac x=n in exI)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -57,7 +57,7 @@
 
 lemma scale_left_commute:
   "scale a (scale b x) = scale b (scale a x)"
-by (simp add: mult_commute)
+by (simp add: mult.commute)
 
 lemma scale_zero_left [simp]: "scale 0 x = 0"
   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
@@ -255,7 +255,7 @@
 by (simp add: of_real_def scaleR_left_diff_distrib)
 
 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
-by (simp add: of_real_def mult_commute)
+by (simp add: of_real_def mult.commute)
 
 lemma of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
   by (induct s rule: infinite_finite_induct) auto
@@ -1171,7 +1171,7 @@
 lemma sgn_mult:
   fixes x y :: "'a::real_normed_div_algebra"
   shows "sgn (x * y) = sgn x * sgn y"
-by (simp add: sgn_div_norm norm_mult mult_commute)
+by (simp add: sgn_div_norm norm_mult mult.commute)
 
 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
 by (simp add: sgn_div_norm divide_inverse)
@@ -1387,7 +1387,7 @@
       also have "\<dots> \<le> (norm x * Kg) * Kf"
         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
-        by (rule mult_assoc)
+        by (rule mult.assoc)
       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
     qed
   qed
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Rings.thy
--- a/src/HOL/Rings.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Rings.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -144,7 +144,7 @@
 proof -
   from assms obtain v where "b = a * v" by (auto elim!: dvdE)
   moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
-  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
+  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
   then show ?thesis ..
 qed
 
@@ -160,10 +160,10 @@
 by (auto intro!: dvdI)
 
 lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
-by (auto intro!: mult_left_commute dvdI elim!: dvdE)
+by (auto intro!: mult.left_commute dvdI elim!: dvdE)
 
 lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
-  apply (subst mult_commute)
+  apply (subst mult.commute)
   apply (erule dvd_mult)
   done
 
@@ -185,7 +185,7 @@
 qed
 
 lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
-by (simp add: dvd_def mult_assoc, blast)
+by (simp add: dvd_def mult.assoc, blast)
 
 lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
   unfolding mult_ac [of a] by (rule dvd_mult_left)
@@ -684,7 +684,7 @@
   fix a b c :: 'a
   assume "a \<le> b" "0 \<le> c"
   thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
-  thus "a * c \<le> b * c" by (simp only: mult_commute)
+  thus "a * c \<le> b * c" by (simp only: mult.commute)
 qed
 
 end
@@ -707,7 +707,7 @@
   fix a b c :: 'a
   assume "a < b" "0 < c"
   thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
-  thus "a * c < b * c" by (simp only: mult_commute)
+  thus "a * c < b * c" by (simp only: mult.commute)
 qed
 
 subclass ordered_cancel_comm_semiring
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Set_Interval.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1480,7 +1480,7 @@
 lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
   apply (subgoal_tac "k = 0 | 0 < k", auto)
   apply (induct "n")
-  apply (simp_all add: setsum_add_nat_ivl add_commute atLeast0LessThan[symmetric])
+  apply (simp_all add: setsum_add_nat_ivl add.commute atLeast0LessThan[symmetric])
   done
 
 subsection{* Shifting bounds *}
@@ -1524,14 +1524,14 @@
     by (rule IH)
   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
-    by (rule add_assoc)
+    by (rule add.assoc)
   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
     by (rule setsum_atMost_Suc [symmetric])
   finally show ?case .
 qed
 
 lemma setsum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"
-  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add_commute)
+  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)
 
 lemma setsum_Suc_diff:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
@@ -1608,7 +1608,7 @@
   also from ngt1
   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
-       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
+       (simp add:  mult_ac trans [OF add.commute of_nat_Suc [symmetric]])
   finally show ?thesis
     unfolding mult_2 by (simp add: algebra_simps)
 next
diff -r de51a86fc903 -r cc97b347b301 src/HOL/String.thy
--- a/src/HOL/String.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/String.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -269,7 +269,7 @@
 lemma nibble_of_nat_of_char_div_16:
   "nibble_of_nat (nat_of_char c div 16) = (case c of Char x y \<Rightarrow> x)"
   by (cases c)
-    (simp add: nat_of_char_def add_commute nat_of_nibble_less_16)
+    (simp add: nat_of_char_def add.commute nat_of_nibble_less_16)
 
 lemma nibble_of_nat_nat_of_char:
   "nibble_of_nat (nat_of_char c) = (case c of Char x y \<Rightarrow> y)"
@@ -279,7 +279,7 @@
     by (simp add: nibble_of_nat_mod_16)
   then have "nibble_of_nat (nat_of_nibble y + nat_of_nibble x * 16) = y"
     by (simp only: nibble_of_nat_mod_16)
-  with Char show ?thesis by (simp add: nat_of_char_def add_commute)
+  with Char show ?thesis by (simp add: nat_of_char_def add.commute)
 qed
 
 code_datatype Char -- {* drop case certificate for char *}
@@ -303,7 +303,7 @@
   have "n mod 256 = 16 * (n div 16 mod 16) + n mod 16" by simp
   then show ?thesis
     by (simp add: char_of_nat_def enum_char_product_enum_nibble card_UNIV_nibble
-      card_UNIV_length_enum [symmetric] nibble_of_nat_def product_nth add_commute)
+      card_UNIV_length_enum [symmetric] nibble_of_nat_def product_nth add.commute)
 qed
 
 lemma char_of_nat_of_char [simp]:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Tools/numeral_simprocs.ML
--- a/src/HOL/Tools/numeral_simprocs.ML	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Tools/numeral_simprocs.ML	Fri Jul 04 20:18:47 2014 +0200
@@ -227,7 +227,7 @@
 
 (*combine unary minus with numeric literals, however nested within a product*)
 val mult_minus_simps =
-    [@{thm mult_assoc}, @{thm minus_mult_right}, @{thm minus_mult_commute}];
+    [@{thm mult.assoc}, @{thm minus_mult_right}, @{thm minus_mult_commute}];
 
 val norm_ss1 =
   simpset_of (put_simpset num_ss @{context}
@@ -719,7 +719,7 @@
            @{thm diff_conv_add_uminus}, @{thm "minus_divide_left"},
            @{thm "add_divide_distrib"} RS sym,
            @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
-           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
+           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult.commute}))))   
            (@{thm field_divide_inverse} RS sym)]
 
 val field_comp_ss =
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Transcendental.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -61,7 +61,7 @@
   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
     by (simp only: Suc)
   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
-    by (simp only: mult_left_commute)
+    by (simp only: mult.left_commute)
   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
     by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
   finally show ?case .
@@ -128,7 +128,7 @@
     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
       by (simp add: x_neq_0)
     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
-      by (simp only: mult_assoc)
+      by (simp only: mult.assoc)
     finally show "norm (norm (f n * z ^ n)) \<le>
                   K * norm (z ^ n) * inverse (norm (x ^ n))"
       by (simp add: mult_le_cancel_right x_neq_0)
@@ -145,7 +145,7 @@
     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
       using x_neq_0
       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
-                    power_inverse norm_power mult_assoc)
+                    power_inverse norm_power mult.assoc)
   qed
   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
     by (rule summable_comparison_test)
@@ -469,10 +469,10 @@
           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
   apply (simp add: right_diff_distrib diff_divide_distrib h)
-  apply (simp add: mult_assoc [symmetric])
+  apply (simp add: mult.assoc [symmetric])
   apply (cases "n", simp)
   apply (simp add: lemma_realpow_diff_sumr2 h
-                   right_diff_distrib [symmetric] mult_assoc
+                   right_diff_distrib [symmetric] mult.assoc
               del: power_Suc setsum_lessThan_Suc of_nat_Suc)
   apply (subst lemma_realpow_rev_sumr)
   apply (subst sumr_diff_mult_const2)
@@ -483,7 +483,7 @@
   apply (clarify)
   apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
               del: setsum_lessThan_Suc power_Suc)
-  apply (subst mult_assoc [symmetric], subst power_add [symmetric])
+  apply (subst mult.assoc [symmetric], subst power_add [symmetric])
   apply (simp add: mult_ac)
   done
 
@@ -508,7 +508,7 @@
   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
         norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
-    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult_commute norm_mult)
+    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
   proof (rule mult_right_mono [OF _ norm_ge_zero])
     from norm_ge_zero 2 have K: "0 \<le> K"
@@ -530,7 +530,7 @@
       done
   qed
   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
-    by (simp only: mult_assoc)
+    by (simp only: mult.assoc)
   finally show ?thesis .
 qed
 
@@ -638,9 +638,9 @@
       by (rule order_le_less_trans)
     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
-      apply (simp only: norm_mult mult_assoc)
+      apply (simp only: norm_mult mult.assoc)
       apply (rule mult_left_mono [OF _ norm_ge_zero])
-      apply (simp add: mult_assoc [symmetric])
+      apply (simp add: mult.assoc [symmetric])
       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
       done
   qed
@@ -797,7 +797,7 @@
     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
       unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
-      apply (subst (5) add_commute)
+      apply (subst (5) add.commute)
       by (rule abs_triangle_ineq)
     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
       using abs_triangle_ineq4 by auto
@@ -877,7 +877,7 @@
               unfolding abs_mult[symmetric] by auto
           qed
           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
-            unfolding abs_mult mult_assoc[symmetric] by algebra
+            unfolding abs_mult mult.assoc[symmetric] by algebra
           finally show ?thesis .
         qed
       }
@@ -900,7 +900,7 @@
             unfolding real_norm_def abs_mult
             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
         qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
-        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
+        from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
         show "summable (?f x)" by auto
       }
       show "summable (?f' x0)"
@@ -984,7 +984,7 @@
 
 lemma exp_fdiffs:
       "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
-  by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
+  by (simp add: diffs_def mult.assoc [symmetric] real_of_nat_def of_nat_mult
         del: mult_Suc of_nat_Suc)
 
 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
@@ -1132,7 +1132,7 @@
   by simp
 
 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
-  by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
+  by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute)
 
 text {* Strict monotonicity of exponential. *}
 
@@ -1824,7 +1824,7 @@
   by (simp add: powr_def)
 
 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
-  by (simp add: powr_powr mult_commute)
+  by (simp add: powr_powr mult.commute)
 
 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
   by (simp add: powr_def exp_minus [symmetric])
@@ -2006,7 +2006,7 @@
 by(simp add: root_powr_inverse ln_powr)
 
 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
-  by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult_commute)
+  by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
 
 lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
 by(simp add: log_def ln_root)
@@ -2078,7 +2078,7 @@
   unfolding powr_def exp_inj_iff by simp
 
 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
-  by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult_commute 
+  by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute 
             order.strict_trans2 powr_gt_zero zero_less_one)
 
 lemma ln_powr_bound2:
@@ -2339,7 +2339,7 @@
 qed
 
 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
-  by (subst add_commute, rule sin_cos_squared_add)
+  by (subst add.commute, rule sin_cos_squared_add)
 
 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
   using sin_cos_squared_add2 [unfolded power2_eq_square] .
@@ -2422,13 +2422,13 @@
   using sin_add [of x "- y"] by simp
 
 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
-  by (simp add: sin_diff mult_commute)
+  by (simp add: sin_diff mult.commute)
 
 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
   using cos_add [of x "- y"] by simp
 
 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
-  by (simp add: cos_diff mult_commute)
+  by (simp add: cos_diff mult.commute)
 
 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
   using sin_add [where x=x and y=x] by simp
@@ -2567,7 +2567,7 @@
     note *** = this
     have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
     from ** show ?thesis by (rule sumr_pos_lt_pair)
-      (simp add: divide_inverse mult_assoc [symmetric] ***)
+      (simp add: divide_inverse mult.assoc [symmetric] ***)
   qed
   ultimately have "0 < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
     by (rule order_less_trans)
@@ -2681,13 +2681,13 @@
   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
 
 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
-  by (metis cos_npi mult_commute)
+  by (metis cos_npi mult.commute)
 
 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
 
 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
-  by (simp add: mult_commute [of pi])
+  by (simp add: mult.commute [of pi])
 
 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
   by (simp add: cos_double)
@@ -2785,7 +2785,7 @@
 apply (cut_tac y="-y" in cos_total, simp) apply simp
 apply (erule ex1E)
 apply (rule_tac a = "x - (pi/2)" in ex1I)
-apply (simp (no_asm) add: add_assoc)
+apply (simp (no_asm) add: add.assoc)
 apply (rotate_tac 3)
 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
 done
@@ -3023,7 +3023,7 @@
   apply (auto simp add: divide_inverse)
   apply (rule mult_pos_pos)
   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
-  apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
+  apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
   done
 
 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
@@ -3321,7 +3321,7 @@
   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
   apply (auto dest: field_power_not_zero
           simp add: power_mult_distrib distrib_right power_divide tan_def
-                    mult_assoc power_inverse [symmetric])
+                    mult.assoc power_inverse [symmetric])
   done
 
 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
@@ -3559,11 +3559,11 @@
     by (auto simp add: algebra_simps sin_add)
   thus ?thesis
     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
-                  mult_commute [of pi])
+                  mult.commute [of pi])
 qed
 
 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
-  by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
+  by (simp add: cos_double mult.assoc power_add [symmetric] numeral_2_eq_2)
 
 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
@@ -3571,7 +3571,7 @@
   done
 
 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
-  by (auto simp add: mult_assoc)
+  by (auto simp add: mult.assoc)
 
 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
@@ -3771,14 +3771,14 @@
     next
       fix x :: real
       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
-      from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
+      from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]]
       show "f sums x" unfolding sums_def by auto
     qed
     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
   } note sums_even = this
 
   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
-    unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
+    unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
     by auto
 
   {
@@ -3858,7 +3858,7 @@
             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
               unfolding real_norm_def[symmetric] by (rule geometric_sums)
             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
-              unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
+              unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
               using sums_unique unfolding inverse_eq_divide by auto
             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Word/Misc_Numeric.thy
--- a/src/HOL/Word/Misc_Numeric.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Word/Misc_Numeric.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -25,7 +25,7 @@
 lemma emep1:
   fixes n d :: int
   shows "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
-  by (auto simp add: pos_zmod_mult_2 add_commute even_equiv_def)
+  by (auto simp add: pos_zmod_mult_2 add.commute even_equiv_def)
 
 lemma int_mod_ge:
   "a < n \<Longrightarrow> 0 < (n :: int) \<Longrightarrow> a \<le> a mod n"
@@ -63,7 +63,7 @@
 lemma p1mod22k:
   fixes b :: int
   shows "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
-  by (simp add: p1mod22k' add_commute)
+  by (simp add: p1mod22k' add.commute)
 
 lemma int_mod_lem: 
   "(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Word/Word.thy
--- a/src/HOL/Word/Word.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Word/Word.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -1753,7 +1753,7 @@
    prefer 2 
    apply (insert uint_range' [of s])[1]
    apply arith
-  apply (drule add_commute [THEN xtr1])
+  apply (drule add.commute [THEN xtr1])
   apply (simp add: diff_less_eq [symmetric])
   apply (drule less_le_mult)
    apply arith
@@ -2161,7 +2161,7 @@
   "a div b * b \<le> (a::nat)"
   using gt_or_eq_0 [of b]
   apply (rule disjE)
-   apply (erule xtr4 [OF thd mult_commute])
+   apply (erule xtr4 [OF thd mult.commute])
   apply clarsimp
   done
 
@@ -2921,7 +2921,7 @@
   apply (unfold shiftr_def)
   apply (induct "n")
    apply simp
-  apply (simp add: shiftr1_div_2 mult_commute
+  apply (simp add: shiftr1_div_2 mult.commute
                    zdiv_zmult2_eq [symmetric])
   done
 
@@ -2929,7 +2929,7 @@
   apply (unfold sshiftr_def)
   apply (induct "n")
    apply simp
-  apply (simp add: sshiftr1_div_2 mult_commute
+  apply (simp add: sshiftr1_div_2 mult.commute
                    zdiv_zmult2_eq [symmetric])
   done
 
@@ -3731,7 +3731,7 @@
   apply (clarsimp simp: word_ubin.eq_norm nth_bintr word_size split: prod.splits)
   apply (drule bin_nth_split)
   apply safe
-       apply (simp_all add: add_commute)
+       apply (simp_all add: add.commute)
    apply (erule bin_nth_uint_imp)+
   done
 
@@ -3895,7 +3895,7 @@
 
 lemma foldl_eq_foldr:
   "foldl op + x xs = foldr op + (x # xs) (0 :: 'a :: comm_monoid_add)" 
-  by (induct xs arbitrary: x) (auto simp add : add_assoc)
+  by (induct xs arbitrary: x) (auto simp add : add.assoc)
 
 lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]
 
@@ -3935,7 +3935,7 @@
 
 (* alternative proof of word_rcat_rsplit *)
 lemmas tdle = iffD2 [OF split_div_lemma refl, THEN conjunct1] 
-lemmas dtle = xtr4 [OF tdle mult_commute]
+lemmas dtle = xtr4 [OF tdle mult.commute]
 
 lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w"
   apply (rule word_eqI)
@@ -3959,7 +3959,7 @@
   fixes n::nat
   shows "0 < n \<Longrightarrow> (k * n + m) div n = m div n + k"
   and   "(k * n + m) mod n = m mod n"
-  by (auto simp: add_commute)
+  by (auto simp: add.commute)
 
 lemma word_rsplit_rcat_size [OF refl]:
   "word_rcat (ws :: 'a :: len word list) = frcw \<Longrightarrow> 
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Word/WordBitwise.thy
--- a/src/HOL/Word/WordBitwise.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Word/WordBitwise.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -306,7 +306,7 @@
 
   show ?case
     using Cons
-    apply (simp add: trans [OF of_bl_append add_commute]
+    apply (simp add: trans [OF of_bl_append add.commute]
                      rbl_mul_simps rbl_word_plus'
                      Cons.hyps distrib_right mult_bit
                      shiftl rbl_shiftl)
diff -r de51a86fc903 -r cc97b347b301 src/HOL/Word/Word_Miscellaneous.thy
--- a/src/HOL/Word/Word_Miscellaneous.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/Word/Word_Miscellaneous.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -146,14 +146,14 @@
 
 lemma emep1:
   "even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1"
-  apply (simp add: add_commute)
+  apply (simp add: add.commute)
   apply (safe dest!: even_equiv_def [THEN iffD1])
   apply (subst pos_zmod_mult_2)
    apply arith
   apply (simp add: mod_mult_mult1)
  done
 
-lemmas eme1p = emep1 [simplified add_commute]
+lemmas eme1p = emep1 [simplified add.commute]
 
 lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith
 
@@ -187,9 +187,9 @@
 lemmas iszero_minus = trans [THEN trans,
   OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric]]
 
-lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add_commute]
+lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add.commute]
 
-lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2]]
+lemmas add_diff_cancel2 = add.commute [THEN diff_eq_eq [THEN iffD2]]
 
 lemmas rdmods [symmetric] = mod_minus_eq
   mod_diff_left_eq mod_diff_right_eq mod_add_left_eq
@@ -296,13 +296,13 @@
   "a + b = c + d ==> 
    a >= c & b <= d | a <= c & b >= (d :: nat)" by arith
 
-lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels]
+lemmas pl_pl_rels' = add.commute [THEN [2] trans, THEN pl_pl_rels]
 
 lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))"  by arith
 
 lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b"  by arith
 
-lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm]
+lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]
 
 lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right]
 lemmas dtle = xtr3 [OF dme [symmetric] le_add1]
@@ -341,7 +341,7 @@
    apply clarsimp
   apply clarify
   apply (cases "b > 0")
-   apply (drule mult_commute [THEN xtr1])
+   apply (drule mult.commute [THEN xtr1])
    apply (frule (1) td_gal_lt [THEN iffD1])
    apply (clarsimp simp: le_simps)
    apply (rule mult_div_cancel [THEN [2] xtr4])
diff -r de51a86fc903 -r cc97b347b301 src/HOL/ex/Adhoc_Overloading_Examples.thy
--- a/src/HOL/ex/Adhoc_Overloading_Examples.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/ex/Adhoc_Overloading_Examples.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -211,7 +211,7 @@
   apply default
   unfolding permute_perm_def
   apply simp
-  apply (simp only: diff_conv_add_uminus minus_add add_assoc)
+  apply (simp only: diff_conv_add_uminus minus_add add.assoc)
   done
 
 text {*Permuting functions.*}
diff -r de51a86fc903 -r cc97b347b301 src/HOL/ex/Dedekind_Real.thy
--- a/src/HOL/ex/Dedekind_Real.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/ex/Dedekind_Real.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -229,7 +229,7 @@
 lemma preal_add_commute: "(x::preal) + y = y + x"
 apply (unfold preal_add_def add_set_def)
 apply (rule_tac f = Abs_preal in arg_cong)
-apply (force simp add: add_commute)
+apply (force simp add: add.commute)
 done
 
 text{*Lemmas for proving that addition of two positive reals gives
@@ -344,7 +344,7 @@
 lemma preal_mult_commute: "(x::preal) * y = y * x"
 apply (unfold preal_mult_def mult_set_def)
 apply (rule_tac f = Abs_preal in arg_cong)
-apply (force simp add: mult_commute)
+apply (force simp add: mult.commute)
 done
 
 text{*Multiplication of two positive reals gives a positive real.*}
@@ -420,7 +420,7 @@
     show "\<exists>y'\<in>B. z = (z/y) * y'"
     proof
       show "z = (z/y)*y"
-        by (simp add: divide_inverse mult_commute [of y] mult_assoc
+        by (simp add: divide_inverse mult.commute [of y] mult.assoc
                       order_less_imp_not_eq2)
       show "y \<in> B" by fact
     qed
@@ -501,7 +501,7 @@
         show "\<exists>v'\<in>A. x = (x / v) * v'"
         proof
           show "x = (x/v)*v"
-            by (simp add: divide_inverse mult_assoc vpos
+            by (simp add: divide_inverse mult.assoc vpos
                           order_less_imp_not_eq2)
           show "v \<in> A" by fact
         qed
@@ -550,7 +550,7 @@
   have cpos: "0 < ?c"
     by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
   show "a * d + b * e = ?c * (d + e)"
-    by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
+    by (simp add: divide_inverse mult.assoc order_less_imp_not_eq2)
   show "?c \<in> Rep_preal w"
   proof (cases rule: linorder_le_cases)
     assume "a \<le> b"
@@ -777,7 +777,7 @@
     qed
     hence "y < r" by simp
     with ypos have  dless: "?d < (r * ?d)/y"
-      by (simp add: pos_less_divide_eq mult_commute [of ?d]
+      by (simp add: pos_less_divide_eq mult.commute [of ?d]
                     mult_strict_right_mono dpos)
     have "r + ?d < r*x"
     proof -
@@ -826,10 +826,10 @@
     show "x/u < x/r" using xpos upos rpos
       by (simp add: divide_inverse mult_less_cancel_left rless) 
     show "inverse (x / r) \<notin> Rep_preal R" using notin
-      by (simp add: divide_inverse mult_commute) 
+      by (simp add: divide_inverse mult.commute) 
     show "u \<in> Rep_preal R" by (rule u) 
     show "x = x / u * u" using upos 
-      by (simp add: divide_inverse mult_commute) 
+      by (simp add: divide_inverse mult.commute) 
   qed
 qed
 
@@ -1284,9 +1284,9 @@
 lemma real_add_congruent2_lemma:
      "[|a + ba = aa + b; ab + bc = ac + bb|]
       ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
-apply (simp add: add_assoc)
-apply (rule add_left_commute [of ab, THEN ssubst])
-apply (simp add: add_assoc [symmetric])
+apply (simp add: add.assoc)
+apply (rule add.left_commute [of ab, THEN ssubst])
+apply (simp add: add.assoc [symmetric])
 apply (simp add: add_ac)
 done
 
@@ -1305,7 +1305,7 @@
 lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
 proof -
   have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
-    by (auto simp add: congruent_def add_commute) 
+    by (auto simp add: congruent_def add.commute) 
   thus ?thesis
     by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
 qed
@@ -1314,13 +1314,13 @@
 proof
   fix x y z :: real
   show "(x + y) + z = x + (y + z)"
-    by (cases x, cases y, cases z, simp add: real_add add_assoc)
+    by (cases x, cases y, cases z, simp add: real_add add.assoc)
   show "x + y = y + x"
-    by (cases x, cases y, simp add: real_add add_commute)
+    by (cases x, cases y, simp add: real_add add.commute)
   show "0 + x = x"
     by (cases x, simp add: real_add real_zero_def add_ac)
   show "- x + x = 0"
-    by (cases x, simp add: real_minus real_add real_zero_def add_commute)
+    by (cases x, simp add: real_minus real_add real_zero_def add.commute)
   show "x - y = x + - y"
     by (simp add: real_diff_def)
 qed
@@ -1332,9 +1332,9 @@
      "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
           x * x1 + y * y1 + (x * y2 + y * x2) =
           x * x2 + y * y2 + (x * y1 + y * x1)"
-apply (simp add: add_left_commute add_assoc [symmetric])
-apply (simp add: add_assoc distrib_left [symmetric])
-apply (simp add: add_commute)
+apply (simp add: add.left_commute add.assoc [symmetric])
+apply (simp add: add.assoc distrib_left [symmetric])
+apply (simp add: add.commute)
 done
 
 lemma real_mult_congruent2:
@@ -1343,7 +1343,7 @@
           { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
      respects2 realrel"
 apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
-apply (simp add: mult_commute add_commute)
+apply (simp add: mult.commute add.commute)
 apply (auto simp add: real_mult_congruent2_lemma)
 done
 
@@ -1393,7 +1393,7 @@
 subsection {* Inverse and Division *}
 
 lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
-by (simp add: real_zero_def add_commute)
+by (simp add: real_zero_def add.commute)
 
 text{*Instead of using an existential quantifier and constructing the inverse
 within the proof, we could define the inverse explicitly.*}
@@ -1416,8 +1416,8 @@
 apply (drule real_mult_inverse_left_ex, safe)
 apply (rule theI, assumption, rename_tac z)
 apply (subgoal_tac "(z * x) * y = z * (x * y)")
-apply (simp add: mult_commute)
-apply (rule mult_assoc)
+apply (simp add: mult.commute)
+apply (rule mult.assoc)
 done
 
 
@@ -1510,7 +1510,7 @@
 apply (cases x, cases y) 
 apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
                       add_ac)
-apply (simp_all add: add_assoc [symmetric])
+apply (simp_all add: add.assoc [symmetric])
 done
 
 lemma real_add_left_mono: 
@@ -1591,7 +1591,7 @@
 apply (simp add: real_of_preal_def real_zero_def, cases x)
 apply (auto simp add: real_minus add_ac)
 apply (cut_tac x = xa and y = y in linorder_less_linear)
-apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
+apply (auto dest!: less_add_left_Ex simp add: add.assoc [symmetric])
 done
 
 lemma real_of_preal_leD:
diff -r de51a86fc903 -r cc97b347b301 src/HOL/ex/Gauge_Integration.thy
--- a/src/HOL/ex/Gauge_Integration.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/ex/Gauge_Integration.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -510,9 +510,9 @@
 next
   case False
   then have "inverse (z - x) * (f z - f x - f' x * (z - x)) = (f z - f x) / (z - x) - f' x"
-    apply (subst mult_commute)
+    apply (subst mult.commute)
     apply (simp add: left_diff_distrib)
-    apply (simp add: mult_assoc divide_inverse)
+    apply (simp add: mult.assoc divide_inverse)
     apply (simp add: ring_distribs)
     done
   moreover from False `\<bar>z - x\<bar> < s` have "\<bar>(f z - f x) / (z - x) - f' x\<bar> < e / 2"
diff -r de51a86fc903 -r cc97b347b301 src/HOL/ex/ThreeDivides.thy
--- a/src/HOL/ex/ThreeDivides.thy	Fri Jul 04 20:07:08 2014 +0200
+++ b/src/HOL/ex/ThreeDivides.thy	Fri Jul 04 20:18:47 2014 +0200
@@ -88,7 +88,7 @@
 lemma three_divs_1:
   fixes D :: "nat \<Rightarrow> nat"
   shows "3 dvd (\<Sum>x<nd. D x * (10^x - 1))"
-  by (subst nat_mult_commute, rule div_sum) (simp add: three_divs_0 [simplified])
+  by (subst mult.commute, rule div_sum) (simp add: three_divs_0 [simplified])
 
 text {* Using lemmas @{text "digit_diff_split"} and 
 @{text "three_divs_1"} we now prove the following lemma.