# HG changeset patch
# User wenzelm
# Date 1446716399 -3600
# Node ID 5197a2ecb65850a97df9463e70274c98a4636a71
# Parent  a9599d3d761002c1877775ed1bfee705f56dd706
isabelle update_cartouches -c -t;

diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Approximation.thy
--- a/src/HOL/Decision_Procs/Approximation.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Approximation.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -18,7 +18,7 @@
 
 section "Horner Scheme"
 
-subsection \<open>Define auxiliary helper @{text horner} function\<close>
+subsection \<open>Define auxiliary helper \<open>horner\<close> function\<close>
 
 primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
 "horner F G 0 i k x       = 0" |
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Commutative_Ring_Complete.thy
--- a/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -101,7 +101,7 @@
     by (cases x) auto
 qed
 
-text \<open>mkPX conserves normalizedness (@{text "_cn"})\<close>
+text \<open>mkPX conserves normalizedness (\<open>_cn\<close>)\<close>
 lemma mkPX_cn:
   assumes "x \<noteq> 0"
     and "isnorm P"
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Cooper.thy
--- a/src/HOL/Decision_Procs/Cooper.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Cooper.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -18,7 +18,7 @@
 datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
   | Mul int num
 
-primrec num_size :: "num \<Rightarrow> nat" -- \<open>A size for num to make inductive proofs simpler\<close>
+primrec num_size :: "num \<Rightarrow> nat" \<comment> \<open>A size for num to make inductive proofs simpler\<close>
 where
   "num_size (C c) = 1"
 | "num_size (Bound n) = 1"
@@ -44,7 +44,7 @@
   | Closed nat | NClosed nat
 
 
-fun fmsize :: "fm \<Rightarrow> nat"  -- \<open>A size for fm\<close>
+fun fmsize :: "fm \<Rightarrow> nat"  \<comment> \<open>A size for fm\<close>
 where
   "fmsize (NOT p) = 1 + fmsize p"
 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
@@ -60,7 +60,7 @@
 lemma fmsize_pos: "fmsize p > 0"
   by (induct p rule: fmsize.induct) simp_all
 
-primrec Ifm :: "bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool"  -- \<open>Semantics of formulae (fm)\<close>
+primrec Ifm :: "bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool"  \<comment> \<open>Semantics of formulae (fm)\<close>
 where
   "Ifm bbs bs T \<longleftrightarrow> True"
 | "Ifm bbs bs F \<longleftrightarrow> False"
@@ -113,7 +113,7 @@
   by (induct p arbitrary: bs rule: prep.induct) auto
 
 
-fun qfree :: "fm \<Rightarrow> bool"  -- \<open>Quantifier freeness\<close>
+fun qfree :: "fm \<Rightarrow> bool"  \<comment> \<open>Quantifier freeness\<close>
 where
   "qfree (E p) \<longleftrightarrow> False"
 | "qfree (A p) \<longleftrightarrow> False"
@@ -127,7 +127,7 @@
 
 text \<open>Boundedness and substitution\<close>
 
-primrec numbound0 :: "num \<Rightarrow> bool"  -- \<open>a num is INDEPENDENT of Bound 0\<close>
+primrec numbound0 :: "num \<Rightarrow> bool"  \<comment> \<open>a num is INDEPENDENT of Bound 0\<close>
 where
   "numbound0 (C c) \<longleftrightarrow> True"
 | "numbound0 (Bound n) \<longleftrightarrow> n > 0"
@@ -142,7 +142,7 @@
   shows "Inum (b # bs) a = Inum (b' # bs) a"
   using nb by (induct a rule: num.induct) (auto simp add: gr0_conv_Suc)
 
-primrec bound0 :: "fm \<Rightarrow> bool" -- \<open>A Formula is independent of Bound 0\<close>
+primrec bound0 :: "fm \<Rightarrow> bool" \<comment> \<open>A Formula is independent of Bound 0\<close>
 where
   "bound0 T \<longleftrightarrow> True"
 | "bound0 F \<longleftrightarrow> True"
@@ -188,7 +188,7 @@
   "numbound0 a \<Longrightarrow> Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
   by (induct t rule: numsubst0.induct) (auto simp: nth_Cons' numbound0_I[where b="b" and b'="b'"])
 
-primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm"  -- \<open>substitue a num into a formula for Bound 0\<close>
+primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm"  \<comment> \<open>substitue a num into a formula for Bound 0\<close>
 where
   "subst0 t T = T"
 | "subst0 t F = F"
@@ -254,7 +254,7 @@
 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
   by (induct p) simp_all
 
-fun isatom :: "fm \<Rightarrow> bool"  -- \<open>test for atomicity\<close>
+fun isatom :: "fm \<Rightarrow> bool"  \<comment> \<open>test for atomicity\<close>
 where
   "isatom T \<longleftrightarrow> True"
 | "isatom F \<longleftrightarrow> True"
@@ -856,9 +856,9 @@
     (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
       simpfm simpfm_qf simp del: simpfm.simps)
 
-text \<open>Linearity for fm where Bound 0 ranges over @{text "\<int>"}\<close>
+text \<open>Linearity for fm where Bound 0 ranges over \<open>\<int>\<close>\<close>
 
-fun zsplit0 :: "num \<Rightarrow> int \<times> num"  -- \<open>splits the bounded from the unbounded part\<close>
+fun zsplit0 :: "num \<Rightarrow> int \<times> num"  \<comment> \<open>splits the bounded from the unbounded part\<close>
 where
   "zsplit0 (C c) = (0, C c)"
 | "zsplit0 (Bound n) = (if n = 0 then (1, C 0) else (0, Bound n))"
@@ -989,7 +989,7 @@
     by simp
 qed
 
-consts iszlfm :: "fm \<Rightarrow> bool"  -- \<open>Linearity test for fm\<close>
+consts iszlfm :: "fm \<Rightarrow> bool"  \<comment> \<open>Linearity test for fm\<close>
 recdef iszlfm "measure size"
   "iszlfm (And p q) \<longleftrightarrow> iszlfm p \<and> iszlfm q"
   "iszlfm (Or p q) \<longleftrightarrow> iszlfm p \<and> iszlfm q"
@@ -1006,7 +1006,7 @@
 lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p"
   by (induct p rule: iszlfm.induct) auto
 
-consts zlfm :: "fm \<Rightarrow> fm"  -- \<open>Linearity transformation for fm\<close>
+consts zlfm :: "fm \<Rightarrow> fm"  \<comment> \<open>Linearity transformation for fm\<close>
 recdef zlfm "measure fmsize"
   "zlfm (And p q) = And (zlfm p) (zlfm q)"
   "zlfm (Or p q) = Or (zlfm p) (zlfm q)"
@@ -1258,7 +1258,7 @@
   qed
 qed auto
 
-consts minusinf :: "fm \<Rightarrow> fm" -- \<open>Virtual substitution of @{text "-\<infinity>"}\<close>
+consts minusinf :: "fm \<Rightarrow> fm" \<comment> \<open>Virtual substitution of \<open>-\<infinity>\<close>\<close>
 recdef minusinf "measure size"
   "minusinf (And p q) = And (minusinf p) (minusinf q)"
   "minusinf (Or p q) = Or (minusinf p) (minusinf q)"
@@ -1273,7 +1273,7 @@
 lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)"
   by (induct p rule: minusinf.induct) auto
 
-consts plusinf :: "fm \<Rightarrow> fm"  -- \<open>Virtual substitution of @{text "+\<infinity>"}\<close>
+consts plusinf :: "fm \<Rightarrow> fm"  \<comment> \<open>Virtual substitution of \<open>+\<infinity>\<close>\<close>
 recdef plusinf "measure size"
   "plusinf (And p q) = And (plusinf p) (plusinf q)"
   "plusinf (Or p q) = Or (plusinf p) (plusinf q)"
@@ -1285,7 +1285,7 @@
   "plusinf (Ge  (CN 0 c e)) = T"
   "plusinf p = p"
 
-consts \<delta> :: "fm \<Rightarrow> int"  -- \<open>Compute @{text "lcm {d| N\<^sup>? Dvd c*x+t \<in> p}"}\<close>
+consts \<delta> :: "fm \<Rightarrow> int"  \<comment> \<open>Compute \<open>lcm {d| N\<^sup>? Dvd c*x+t \<in> p}\<close>\<close>
 recdef \<delta> "measure size"
   "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)"
   "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)"
@@ -1293,7 +1293,7 @@
   "\<delta> (NDvd i (CN 0 c e)) = i"
   "\<delta> p = 1"
 
-consts d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool"  -- \<open>check if a given l divides all the ds above\<close>
+consts d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool"  \<comment> \<open>check if a given l divides all the ds above\<close>
 recdef d_\<delta> "measure size"
   "d_\<delta> (And p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)"
   "d_\<delta> (Or p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)"
@@ -1354,7 +1354,7 @@
 qed simp_all
 
 
-consts a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm"  -- \<open>adjust the coefficients of a formula\<close>
+consts a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm"  \<comment> \<open>adjust the coefficients of a formula\<close>
 recdef a_\<beta> "measure size"
   "a_\<beta> (And p q) = (\<lambda>k. And (a_\<beta> p k) (a_\<beta> q k))"
   "a_\<beta> (Or p q) = (\<lambda>k. Or (a_\<beta> p k) (a_\<beta> q k))"
@@ -1368,7 +1368,7 @@
   "a_\<beta> (NDvd i (CN 0 c e))=(\<lambda>k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
   "a_\<beta> p = (\<lambda>k. p)"
 
-consts d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool"  -- \<open>test if all coeffs c of c divide a given l\<close>
+consts d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool"  \<comment> \<open>test if all coeffs c of c divide a given l\<close>
 recdef d_\<beta> "measure size"
   "d_\<beta> (And p q) = (\<lambda>k. (d_\<beta> p k) \<and> (d_\<beta> q k))"
   "d_\<beta> (Or p q) = (\<lambda>k. (d_\<beta> p k) \<and> (d_\<beta> q k))"
@@ -1382,7 +1382,7 @@
   "d_\<beta> (NDvd i (CN 0 c e))=(\<lambda>k. c dvd k)"
   "d_\<beta> p = (\<lambda>k. True)"
 
-consts \<zeta> :: "fm \<Rightarrow> int"  -- \<open>computes the lcm of all coefficients of x\<close>
+consts \<zeta> :: "fm \<Rightarrow> int"  \<comment> \<open>computes the lcm of all coefficients of x\<close>
 recdef \<zeta> "measure size"
   "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)"
   "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)"
@@ -1434,7 +1434,7 @@
   "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
   "mirror p = p"
 
-text \<open>Lemmas for the correctness of @{text "\<sigma>_\<rho>"}\<close>
+text \<open>Lemmas for the correctness of \<open>\<sigma>_\<rho>\<close>\<close>
 
 lemma dvd1_eq1:
   fixes x :: int
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Dense_Linear_Order.thy
--- a/src/HOL/Decision_Procs/Dense_Linear_Order.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Dense_Linear_Order.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -32,7 +32,7 @@
 lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)"
   by simp
 
-text\<open>Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"}\<close>
+text\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
 lemma minf_lt[no_atp]:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
 lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
@@ -44,7 +44,7 @@
 lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
 lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
 
-text\<open>Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"}\<close>
+text\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
 lemma pinf_gt[no_atp]:  "\<exists>z. \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
 lemma pinf_lt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
@@ -452,7 +452,7 @@
 lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y"
   using gt_ex by auto
 
-text \<open>Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"}\<close>
+text \<open>Theorems for \<open>\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
 lemma pinf_conj[no_atp]:
   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
     and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
@@ -516,7 +516,7 @@
   using lt_ex by auto
 
 
-text \<open>Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"}\<close>
+text \<open>Theorems for \<open>\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
 lemma minf_conj[no_atp]:
   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Ferrack.thy
--- a/src/HOL/Decision_Procs/Ferrack.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Ferrack.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -7,7 +7,7 @@
   "~~/src/HOL/Library/Code_Target_Numeral" "~~/src/HOL/Library/Old_Recdef"
 begin
 
-section \<open>Quantifier elimination for @{text "\<real> (0, 1, +, <)"}\<close>
+section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, <)\<close>\<close>
 
   (*********************************************************************************)
   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/MIR.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -7,7 +7,7 @@
   "~~/src/HOL/Library/Code_Target_Numeral" "~~/src/HOL/Library/Old_Recdef"
 begin
 
-section \<open>Quantifier elimination for @{text "\<real> (0, 1, +, floor, <)"}\<close>
+section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, floor, <)\<close>\<close>
 
 declare real_of_int_floor_cancel [simp del]
 
@@ -1450,8 +1450,8 @@
   by (induct p rule: qelim.induct) (auto simp del: simpfm.simps)
 
 
-text \<open>The @{text "\<int>"} Part\<close>
-text\<open>Linearity for fm where Bound 0 ranges over @{text "\<int>"}\<close>
+text \<open>The \<open>\<int>\<close> Part\<close>
+text\<open>Linearity for fm where Bound 0 ranges over \<open>\<int>\<close>\<close>
 
 function zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*) where
   "zsplit0 (C c) = (0,C c)"
@@ -1955,10 +1955,10 @@
   ultimately show ?case by blast
 qed auto
 
-text\<open>plusinf : Virtual substitution of @{text "+\<infinity>"}
-       minusinf: Virtual substitution of @{text "-\<infinity>"}
-       @{text "\<delta>"} Compute lcm @{text "d| Dvd d  c*x+t \<in> p"}
-       @{text "d_\<delta>"} checks if a given l divides all the ds above\<close>
+text\<open>plusinf : Virtual substitution of \<open>+\<infinity>\<close>
+       minusinf: Virtual substitution of \<open>-\<infinity>\<close>
+       \<open>\<delta>\<close> Compute lcm \<open>d| Dvd d  c*x+t \<in> p\<close>
+       \<open>d_\<delta>\<close> checks if a given l divides all the ds above\<close>
 
 fun minusinf:: "fm \<Rightarrow> fm" where
   "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
@@ -3294,9 +3294,9 @@
   using lp
   by (induct p rule: mirror.induct) (simp_all add: split_def image_Un)
   
-text \<open>The @{text "\<real>"} part\<close>
-
-text\<open>Linearity for fm where Bound 0 ranges over @{text "\<real>"}\<close>
+text \<open>The \<open>\<real>\<close> part\<close>
+
+text\<open>Linearity for fm where Bound 0 ranges over \<open>\<real>\<close>\<close>
 consts
   isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
 recdef isrlfm "measure size"
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
--- a/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -60,7 +60,7 @@
 | "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)"
 | "tmboundslt n (Mul i a) = tmboundslt n a"
 
-primrec tmbound0 :: "tm \<Rightarrow> bool"  -- \<open>a tm is INDEPENDENT of Bound 0\<close>
+primrec tmbound0 :: "tm \<Rightarrow> bool"  \<comment> \<open>a tm is INDEPENDENT of Bound 0\<close>
 where
   "tmbound0 (CP c) = True"
 | "tmbound0 (Bound n) = (n>0)"
@@ -76,7 +76,7 @@
   using nb
   by (induct a rule: tm.induct) auto
 
-primrec tmbound :: "nat \<Rightarrow> tm \<Rightarrow> bool"  -- \<open>a tm is INDEPENDENT of Bound n\<close>
+primrec tmbound :: "nat \<Rightarrow> tm \<Rightarrow> bool"  \<comment> \<open>a tm is INDEPENDENT of Bound n\<close>
 where
   "tmbound n (CP c) = True"
 | "tmbound n (Bound m) = (n \<noteq> m)"
@@ -626,7 +626,7 @@
 | "boundslt n (E p) = boundslt (Suc n) p"
 | "boundslt n (A p) = boundslt (Suc n) p"
 
-fun bound0:: "fm \<Rightarrow> bool"  -- \<open>a Formula is independent of Bound 0\<close>
+fun bound0:: "fm \<Rightarrow> bool"  \<comment> \<open>a Formula is independent of Bound 0\<close>
 where
   "bound0 T = True"
 | "bound0 F = True"
@@ -647,7 +647,7 @@
   using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
   by (induct p rule: bound0.induct) auto
 
-primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool"  -- \<open>a Formula is independent of Bound n\<close>
+primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool"  \<comment> \<open>a Formula is independent of Bound n\<close>
 where
   "bound m T = True"
 | "bound m F = True"
@@ -897,7 +897,7 @@
 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
   by (induct p) simp_all
 
-fun isatom :: "fm \<Rightarrow> bool"  -- \<open>test for atomicity\<close>
+fun isatom :: "fm \<Rightarrow> bool"  \<comment> \<open>test for atomicity\<close>
 where
   "isatom T = True"
 | "isatom F = True"
@@ -1643,7 +1643,7 @@
 
 subsection \<open>Core Procedure\<close>
 
-fun minusinf:: "fm \<Rightarrow> fm"  -- \<open>Virtual substitution of -\<infinity>\<close>
+fun minusinf:: "fm \<Rightarrow> fm"  \<comment> \<open>Virtual substitution of -\<infinity>\<close>
 where
   "minusinf (And p q) = conj (minusinf p) (minusinf q)"
 | "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
@@ -1653,7 +1653,7 @@
 | "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
 | "minusinf p = p"
 
-fun plusinf:: "fm \<Rightarrow> fm"  -- \<open>Virtual substitution of +\<infinity>\<close>
+fun plusinf:: "fm \<Rightarrow> fm"  \<comment> \<open>Virtual substitution of +\<infinity>\<close>
 where
   "plusinf (And p q) = conj (plusinf p) (plusinf q)"
 | "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Polynomial_List.thy
--- a/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -81,15 +81,15 @@
   obtains q where "poly p2 = poly (p1 *** q)"
   using assms by (auto simp add: divides_def)
 
--- \<open>order of a polynomial\<close>
+\<comment> \<open>order of a polynomial\<close>
 definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat"
   where "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ (Suc n)) divides p))"
 
--- \<open>degree of a polynomial\<close>
+\<comment> \<open>degree of a polynomial\<close>
 definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
   where "degree p = length (pnormalize p) - 1"
 
--- \<open>squarefree polynomials --- NB with respect to real roots only\<close>
+\<comment> \<open>squarefree polynomials --- NB with respect to real roots only\<close>
 definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
   where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
 
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -30,7 +30,7 @@
 | "polysize (Pw p n) = 1 + polysize p"
 | "polysize (CN c n p) = 4 + polysize c + polysize p"
 
-primrec polybound0:: "poly \<Rightarrow> bool" -- \<open>a poly is INDEPENDENT of Bound 0\<close>
+primrec polybound0:: "poly \<Rightarrow> bool" \<comment> \<open>a poly is INDEPENDENT of Bound 0\<close>
 where
   "polybound0 (C c) \<longleftrightarrow> True"
 | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
@@ -41,7 +41,7 @@
 | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
 | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
 
-primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- \<open>substitute a poly into a poly for Bound 0\<close>
+primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" \<comment> \<open>substitute a poly into a poly for Bound 0\<close>
 where
   "polysubst0 t (C c) = C c"
 | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
diff -r a9599d3d7610 -r 5197a2ecb658 src/HOL/Decision_Procs/ex/Approximation_Quickcheck_Ex.thy
--- a/src/HOL/Decision_Procs/ex/Approximation_Quickcheck_Ex.thy	Thu Nov 05 10:39:49 2015 +0100
+++ b/src/HOL/Decision_Procs/ex/Approximation_Quickcheck_Ex.thy	Thu Nov 05 10:39:59 2015 +0100
@@ -31,7 +31,7 @@
   shows "x > 1 \<Longrightarrow> x \<le> 2 ^ 20 * log 2 x + 1 \<and> (sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
   using [[quickcheck_approximation_custom_seed = 1]]
   using [[quickcheck_approximation_epsilon = 0.00000001]]
-    --\<open>avoids spurious counterexamples in approximate computation of @{term "(sin x)\<^sup>2 + (cos x)\<^sup>2"}
+    \<comment>\<open>avoids spurious counterexamples in approximate computation of @{term "(sin x)\<^sup>2 + (cos x)\<^sup>2"}
       and therefore avoids expensive failing attempts for certification\<close>
   quickcheck[approximation, expect=counterexample, size=20]
   oops