Adapted to Isabelle2016

Isabelle2015 is still supported, but support for Isabelle2014 has been
discontinued.

.gitignore

@@ -157,8 +157,7 @@ pvsbin/
 /lib/isabelle/map/
 /lib/isabelle/real/
 /lib/isabelle/set/
-/lib/isabelle/Tools/why3
-/lib/isabelle/Why3_Number.thy
+/lib/isabelle/Why3_BV.thy
 /lib/isabelle/why3.ML
 /lib/isabelle/last_build
 /lib/isabelle/bv

Makefile.in

@@ -1144,7 +1144,7 @@ clean::
 
 ifeq (@enable_isabelle_libs@,yes)
 
-ISABELLEVERSIONSPECIFIC=why3.ML Tools/why3 Why3_Number.thy
+ISABELLEVERSIONSPECIFIC=why3.ML Why3_BV.thy
 
 ISABELLEVERSIONSPECIFICTARGETS=$(addprefix lib/isabelle/, $(ISABELLEVERSIONSPECIFIC))
 ISABELLEVERSIONSPECIFICSOURCES=$(addsuffix .@ISABELLEVERSION@, $(ISABELLEVERSIONSPECIFICTARGETS))

configure.in

@@ -618,21 +618,21 @@ else
         ISABELLEDETECTEDVERSION=`$ISABELLE version | sed -n -e 's|Isabelle\([[^:]]*\).*$|\1|p' `
 
         case $ISABELLEDETECTEDVERSION in
-          2014*)
+          2015*)
 	        enable_isabelle_support=yes
-                ISABELLEVERSION=2014
+                ISABELLEVERSION=2015
                 AC_MSG_RESULT($ISABELLEDETECTEDVERSION)
                 ;;
-          2015*)
+          2016*)
 	        enable_isabelle_support=yes
-                ISABELLEVERSION=2015
+                ISABELLEVERSION=2016
                 AC_MSG_RESULT($ISABELLEDETECTEDVERSION)
                 ;;
           *)
                 AC_MSG_RESULT($ISABELLEDETECTEDVERSION)
                 enable_isabelle_support=no
-	        AC_MSG_WARN(You need Isabelle 2014 or 2015; Isabelle discarded)
-                reason_isabelle_support=" (need version >= 2014)"
+	        AC_MSG_WARN(You need Isabelle 2015 or later; Isabelle discarded)
+                reason_isabelle_support=" (need version >= 2015)"
                 ;;
         esac
     fi

doc/isabelle.tex

@@ -11,9 +11,9 @@ using ``Edit'' action in \texttt{why3ide}.
 
 \subsection{Installation}
 
-You need version Isabelle2014 or Isabelle2015. Former versions are not
-supported. We assume below that your version is 2015, please replace
-2015 by 2014 otherwise.
+You need version Isabelle2015 or Isabelle2016. Former versions are not
+supported. We assume below that your version is 2016, please replace
+2016 by 2015 otherwise.
 
 Isabelle must be installed before compiling \why. After compilation
 and installation of \why, you must manually add the path
@@ -22,7 +22,7 @@ and installation of \why, you must manually add the path
 \end{verbatim}
 into either the user file
 \begin{verbatim}
-.isabelle/Isabelle2015/etc/components
+.isabelle/Isabelle2016/etc/components
 \end{verbatim}
 or the system-wide file
 \begin{verbatim}

drivers/isabelle-2014.gen

-theory number.Parity
-  syntax predicate even "<app><const name=\"Parity.even_odd_class.even\"/>%1</app>"
-  syntax predicate odd  "<app><const name=\"Parity.even_odd_class.odd\"/>%1</app>"
-end

drivers/isabelle-2015.gen

-theory number.Parity
-  syntax predicate even "<app><const name=\"Parity.semiring_parity_class.even\"/>%1</app>"
-  syntax predicate odd  "<app><const name=\"Parity.semiring_parity_class.odd\"/>%1</app>"
+theory algebra.Field
+  syntax function inv "<app><const name=\"Fields.inverse_class.inverse\"/>%1</app>"
+  syntax function (/) "<app><const name=\"Fields.inverse_class.divide\"/>%1%2</app>"
 end

drivers/isabelle-2016.gen

+theory algebra.Field
+  syntax function inv "<app><const name=\"Fields.inverse_class.inverse\"/>%1</app>"
+  syntax function (/) "<app><const name=\"Rings.divide_class.divide\"/>%1%2</app>"
+end

drivers/isabelle-common.gen

@@ -150,6 +150,11 @@ theory set.Fset
   syntax function cardinal "<app><const name=\"Nat.semiring_1_class.of_nat\"><fun><type name=\"Nat.nat\"/><type name=\"Int.int\"/></fun></const><app><const name=\"FSet.fcard\"/>%1</app></app>"
 end
 
+theory number.Parity
+  syntax predicate even "<app><const name=\"Parity.semiring_parity_class.even\"/>%1</app>"
+  syntax predicate odd  "<app><const name=\"Parity.semiring_parity_class.odd\"/>%1</app>"
+end
+
 theory number.Divisibility
   syntax predicate divides "<app><const name=\"Rings.dvd_class.dvd\"/>%1%2</app>"
 end
@@ -166,11 +171,6 @@ theory number.Coprime
   syntax predicate coprime "<app><const name=\"GCD.gcd_class.coprime\"/>%1%2</app>"
 end
 
-theory algebra.Field
-  syntax function inv "<app><const name=\"Fields.inverse_class.inverse\"/>%1</app>"
-  syntax function (/) "<app><const name=\"Fields.inverse_class.divide\"/>%1%2</app>"
-end
-
 theory real.Real
   syntax function zero "<number val=\"0\"><type name=\"Real.real\"/></number>"
   syntax function one  "<number val=\"1\"><type name=\"Real.real\"/></number>"

drivers/isabelle2014-realize.drv → drivers/isabelle2016-realize.drv

@@ -7,4 +7,4 @@ filename "%t.xml"
 transformation "inline_trivial"
 
 import "isabelle-common.gen"
-import "isabelle-2014.gen"
+import "isabelle-2016.gen"

drivers/isabelle2014.drv → drivers/isabelle2016.drv

@@ -8,7 +8,7 @@ transformation "inline_trivial"
 transformation "eliminate_builtin"
 
 import "isabelle-common.gen"
-import "isabelle-2014.gen"
+import "isabelle-2016.gen"
 
 
 transformation "simplify_trivial_quantification_in_goal"

lib/isabelle/Tools/why3.2014

-#!/usr/bin/env bash
-#
-# DESCRIPTION: process files generated by Why3
-
-
-## diagnostics
-
-PRG="$(basename "$0")"
-
-function usage()
-{
-  echo
-  echo "Usage: isabelle $PRG [OPTIONS] WHY3_FILE"
-  echo
-  echo "  Options are:"
-  echo "    -b           batch mode"
-  echo "    -i NAME      interactive mode using interface NAME"
-  echo
-  echo "Process files generated by Why3."
-  exit 1
-}
-
-function fail()
-{
-  echo "$1" >&2
-  exit 2
-}
-
-
-## utilities
-
-function make_theory()
-{
-  BNAME=`basename "$1"`
-
-  if [ ! -e "$1.thy" ]; then
-    echo -e "theory $BNAME\nimports Why3\nbegin\n\nwhy3_open \"$BNAME.xml\"\n" > "$1.thy"
-    sed \
-      -e 's/<lemma name="\([^"]*\)"[^>]*>/why3_vc \1\n\n/g' \
-      -e 's/<[^l][^>]*>//g' \
-      "$1.xml" >> "$1.thy"
-    echo -e "why3_end\n\nend" >> "$1.thy"
-  fi
-}
-
-
-## process command line
-
-while getopts "bi:" OPT
-do
-  case "$OPT" in
-    b)
-      BATCH=true
-      ;;
-    i)
-      INTERACTIVE=true
-      INTERFACE="$OPTARG"
-      ;;
-    \?)
-      usage
-      ;;
-  esac
-done
-
-
-## main
-
-shift $(($OPTIND - 1))
-
-[ "$#" != 1 ] && usage
-
-NAME=`dirname "$1"`/`basename "$1" .xml`
-
-if [ "$BATCH" = true ]; then
-  "$ISABELLE_PROCESS" -e "use_thy \"$NAME\";" -rq Why3
-elif [ "$INTERACTIVE" = true ]; then
-  make_theory "$NAME"
-  case "$INTERFACE" in
-    emacs)
-      "$ISABELLE_TOOL" emacs -L Why3 "${NAME}.thy"
-      ;;
-    jedit)
-      if [ -f "$JEDIT_SETTINGS/$WHY3_JEDIT_SERVER" ]; then
-        "$ISABELLE_TOOL" java -jar "$(jvmpath "$JEDIT_HOME/dist/jedit.jar")" \
-          "-settings=$(jvmpath "$JEDIT_SETTINGS")" "-server=$WHY3_JEDIT_SERVER" \
-          -reuseview -wait "$(jvmpath "${NAME}.thy")"
-      else
-        "$ISABELLE_TOOL" jedit -l Why3 "${NAME}.thy"
-      fi
-      ;;
-    *)
-      fail "Unknown Isabelle interface: \"$INTERFACE\""
-      ;;
-  esac
-else
-  usage
-fi

lib/isabelle/Why3_Number.thy.2015 → lib/isabelle/Why3_Number.thy

@@ -232,7 +232,7 @@ proof (induct n rule: less_induct)
   next
     case False
     with `1 < n` obtain k where "k dvd n" "k \<noteq> 1" "k \<noteq> n"
-      by (auto simp add: prime_nat_def)
+      by (auto simp add: Primes.prime_def)
     with `1 < n` have "k \<le> n" by (simp add: dvd_imp_le)
     with `k \<noteq> n` have "k < n" by simp
     moreover from `k dvd n` `1 < n` have "k \<noteq> 0" by (rule_tac notI) simp
@@ -263,7 +263,7 @@ proof
       with `2 \<le> p` obtain d
         where d: "prime d" "d * d \<le> p" "d dvd p"
         by (auto simp add: int_dvd_iff le_nat_iff int_mult)
-      from `prime d` have "2 \<le> d" by auto
+      from `prime d` have "2 \<le> d" by (simp add: prime_ge_2_nat)
       then have "2 \<le> int d" by simp
       with `2 \<le> int d` have "2 * 2 \<le> int d * int d"
         by (rule mult_mono) simp_all
@@ -275,14 +275,14 @@ qed
 why3_vc even_prime
 proof -
   from `prime (nat p)` have "0 \<le> p" by (simp add: prime_def)
-  from `prime (nat p)` have "2 \<le> nat p" by auto
+  from `prime (nat p)` have "2 \<le> nat p" by (simp add: prime_ge_2_nat)
   with `prime (nat p)` `even p` `0 \<le> p` show ?thesis
     by (auto simp add: order_le_less prime_odd_nat even_nat_iff [symmetric])
 qed
 
 why3_vc odd_prime
 proof -
-  from `prime (nat p)` have "2 \<le> nat p" by auto
+  from `prime (nat p)` have "2 \<le> nat p" by (simp add: prime_ge_2_nat)
   with `prime (nat p)` `3 \<le> p` show ?thesis
     by (auto simp add: order_le_less prime_odd_nat even_nat_iff [symmetric])
 qed

lib/isabelle/Why3_Number.thy.2014

-theory Why3_Number
-imports
-  Why3_Int
-  "~~/src/HOL/Number_Theory/Primes"
-begin
-
-section {* Parity properties *}
-
-why3_open "number/Parity.xml"
-
-why3_vc even_def by (simp add: even_equiv_def)
-
-why3_vc odd_def by (simp add: odd_equiv_def)
-
-why3_vc even_or_odd by auto
-
-why3_vc even_not_odd using assms by simp
-
-why3_vc odd_not_even using assms by simp
-
-why3_vc even_odd using assms by simp
-
-why3_vc odd_even using assms by simp
-
-why3_vc even_even using assms by simp
-
-why3_vc odd_odd using assms by simp
-
-why3_vc even_2k by simp
-
-why3_vc odd_2k1 by simp
-
-why3_end
-
-
-section {* Divisibility *}
-
-why3_open "number/Divisibility.xml"
-
-why3_vc divides_def by (simp add: dvd_def mult.commute)
-
-why3_vc divides_refl by simp
-
-why3_vc divides_1_n by simp
-
-why3_vc divides_0 by simp
-
-why3_vc divides_left using assms by simp
-
-why3_vc divides_right using assms by simp
-
-why3_vc divides_oppr using assms by simp
-
-why3_vc divides_oppl using assms by simp
-
-why3_vc divides_oppr_rev using assms by simp
-
-why3_vc divides_oppl_rev using assms by simp
-
-why3_vc divides_plusr using assms by simp
-
-why3_vc divides_minusr using assms by simp
-
-why3_vc divides_multl using assms by simp
-
-why3_vc divides_multr using assms by simp
-
-why3_vc divides_factorl by simp
-
-why3_vc divides_factorr by simp
-
-why3_vc divides_n_1 using assms by auto
-
-why3_vc divides_antisym
-  using assms
-  by (auto dest: zdvd_antisym_abs)
-
-why3_vc divides_trans using assms by (rule dvd_trans)
-
-why3_vc divides_bounds using assms by (simp add: dvd_imp_le_int)
-
-why3_vc mod_divides_euclidean
-  using assms
-  by (auto simp add: emod_def split add: split_if_asm)
-
-why3_vc divides_mod_euclidean
-  using assms
-  by (simp add: emod_def dvd_eq_mod_eq_0 zabs_def zmod_zminus2_eq_if)
-
-why3_vc mod_divides_computer
-  using assms
-  by (auto simp add: cmod_def zabs_def sgn_0_0 zmod_zminus1_eq_if
-    not_sym [OF less_imp_neq [OF pos_mod_bound]]
-    split add: split_if_asm)
-
-why3_vc divides_mod_computer
-  using assms
-  by (simp add: cmod_def dvd_eq_mod_eq_0 zabs_def
-    zmod_zminus1_eq_if zmod_zminus2_eq_if)
-
-why3_vc even_divides by (rule even_iff_2_dvd)
-
-why3_vc odd_divides by (simp add: even_iff_2_dvd)
-
-why3_end
-
-
-section {* Greatest Common Divisor *}
-
-why3_open "number/Gcd.xml"
-
-why3_vc gcd_nonneg by simp
-
-why3_vc gcd_def1 by simp
-
-why3_vc gcd_def2 by simp
-
-why3_vc gcd_def3 using assms by (rule gcd_greatest_int)
-
-why3_vc gcd_unique using assms
-  by (simp add: gcd_unique_int [symmetric])
-
-why3_vc Comm by (rule gcd_commute_int)
-
-why3_vc Assoc by (rule gcd_assoc_int)
-
-why3_vc gcd_0_pos using assms by simp
-
-why3_vc gcd_0_neg using assms by simp
-
-why3_vc gcd_opp by simp
-
-why3_vc gcd_euclid
-  using gcd_add_mult_int [of a "- q" b]
-  by (simp add: sign_simps)
-
-why3_vc Gcd_computer_mod
-  using assms gcd_add_mult_int [of b "- 1" "a mod b"]
-  by (simp add: cmod_def zabs_def gcd_red_int [symmetric] sgn_if sign_simps)
-    (simp add: zmod_zminus2_eq_if gcd_red_int [of a b])
-
-why3_vc Gcd_euclidean_mod
-  using assms gcd_add_mult_int [of b "- 1" "a mod b"]
-  by (simp add: emod_def zabs_def gcd_red_int [symmetric] sign_simps)
-    (simp add: zmod_zminus2_eq_if gcd_red_int [of a b])
-
-why3_vc gcd_mult using assms
-  by (simp add: gcd_mult_distrib_int [symmetric])
-
-why3_end
-
-
-section {* Prime numbers *}
-
-why3_open "number/Prime.xml"
-
-why3_vc prime_def
-  unfolding prime_def
-proof
-  assume "1 < nat p \<and> (\<forall>m. m dvd nat p \<longrightarrow> m = 1 \<or> m = nat p)"
-  then have "1 < p" and H: "\<And>m. m \<ge> 0 \<Longrightarrow> m dvd p \<Longrightarrow> m = 1 \<or> m = p"
-    by (auto simp add: dvd_int_iff)
-  show "2 \<le> p \<and> (\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p)"
-  proof
-    from `1 < p` show "2 \<le> p" by simp
-
-    show "\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p"
-    proof (intro strip)
-      fix n
-      assume "1 < n \<and> n < p"
-      with H [of n] show "\<not> n dvd p" by auto
-    qed
-  qed
-next
-  assume "2 \<le> p \<and> (\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p)"
-  then have "2 \<le> p" and H: "\<And>n. 1 < n \<Longrightarrow> n < nat p \<Longrightarrow> \<not> n dvd p"
-    by auto
-  show "1 < nat p \<and> (\<forall>m. m dvd nat p \<longrightarrow> m = 1 \<or> m = nat p)"
-  proof
-    from `2 \<le> p` show "1 < nat p" by simp
-
-    show "\<forall>m. m dvd nat p \<longrightarrow> m = 1 \<or> m = nat p"
-    proof (intro strip)
-      fix m
-      assume "m dvd nat p"
-      with `2 \<le> p` have "1 \<le> m" by (cases "m = 0") auto
-      show "m = 1 \<or> m = nat p"
-      proof (cases "m = 1")
-        case False
-        show ?thesis
-        proof (cases "m = nat p")
-          case False
-          from `2 \<le> p` `m dvd nat p` have "m \<le> nat p" by (simp add: dvd_imp_le)
-          with False `m \<noteq> 1` `1 \<le> m` `m dvd nat p` H show ?thesis by (simp add: int_dvd_iff)
-        qed simp
-      qed simp
-    qed
-  qed
-qed
-
-why3_vc not_prime_1 by simp
-
-why3_vc prime_2 by simp
-
-why3_vc prime_3 by simp
-
-why3_vc prime_divisors
-  using assms
-  by (auto simp add: prime_int_altdef dest: spec [of _ "\<bar>d\<bar>"])
-
-lemma small_divisors_aux:
-  "1 < (n::nat) \<Longrightarrow> n < p \<Longrightarrow> n dvd p \<Longrightarrow> \<exists>d. prime d \<and> d * d \<le> p \<and> d dvd p"
-proof (induct n rule: less_induct)
-  case (less n)
-  then obtain m where "p = n * m" by (auto simp add: dvd_def)
-  show ?case
-  proof (cases "prime n")
-    case True
-    show ?thesis
-    proof (cases "n \<le> m")
-      case True
-      with `p = n * m` `prime n`
-      show ?thesis by auto
-    next
-      case False
-      then have "m < n" by simp
-      moreover from `n < p` `p = n * m` have "1 < m" by simp
-      moreover from `1 < n` `n < p` `p = n * m` have "m < p" by simp
-      moreover from `p = n * m` have "m dvd p" by simp
-      ultimately show ?thesis by (rule less)
-    qed
-  next
-    case False
-    with `1 < n` obtain k where "k dvd n" "k \<noteq> 1" "k \<noteq> n"
-      by (auto simp add: prime_nat_def)
-    with `1 < n` have "k \<le> n" by (simp add: dvd_imp_le)
-    with `k \<noteq> n` have "k < n" by simp
-    moreover from `k dvd n` `1 < n` have "k \<noteq> 0" by (rule_tac notI) simp
-    with `k \<noteq> 1` have "1 < k" by simp
-    moreover from `k < n` `n < p` have "k < p" by simp
-    moreover from `k dvd n` `n dvd p` have "k dvd p" by (rule dvd.order_trans)
-    ultimately show ?thesis by (rule less)
-  qed
-qed
-
-why3_vc small_divisors
-  unfolding prime_def
-proof
-  show "2 \<le> p" by fact
-
-  show "\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p"
-  proof (intro strip)
-    fix n
-    assume "1 < n \<and> n < p"
-    show "\<not> n dvd p"
-    proof
-      assume "n dvd p"
-      with `1 < n \<and> n < p`
-      have "1 < nat n" "nat n < nat p" "nat n dvd nat p"
-        by (simp_all add: transfer_nat_int_relations)
-      then have "\<exists>d. prime d \<and> d * d \<le> nat p \<and> d dvd (nat p)"
-        by (rule small_divisors_aux)
-      with `2 \<le> p` obtain d
-        where d: "prime d" "d * d \<le> p" "d dvd p"
-        by (auto simp add: int_dvd_iff le_nat_iff int_mult)
-      from `prime d` have "2 \<le> d" by auto
-      then have "2 \<le> int d" by simp
-      with `2 \<le> int d` have "2 * 2 \<le> int d * int d"
-        by (rule mult_mono) simp_all
-      with d assms `2 \<le> d` show False by auto
-    qed
-  qed
-qed
-
-why3_vc even_prime
-proof -
-  from `prime (nat p)` have "0 \<le> p" by (simp add: prime_def)
-  from `prime (nat p)` have "2 \<le> nat p" by auto
-  with `prime (nat p)` `even p` `0 \<le> p` show ?thesis
-    by (auto simp add: order_le_less prime_odd_nat pos_int_even_equiv_nat_even)
-qed
-
-why3_vc odd_prime
-proof -
-  from `prime (nat p)` have "2 \<le> nat p" by auto
-  with `prime (nat p)` `3 \<le> p` show ?thesis
-    by (auto simp add: order_le_less prime_odd_nat pos_int_even_equiv_nat_even)
-qed
-
-why3_end
-
-
-section {* Coprime numbers *}
-
-why3_open "number/Coprime.xml"
-
-why3_vc coprime_def ..
-
-why3_vc prime_coprime
-proof -
-  have "(\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p) =
-    (\<forall>n. 1 \<le> n \<and> n < p \<longrightarrow> coprime n p)"
-  proof
-    assume H: "\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p"
-    show "\<forall>n. 1 \<le> n \<and> n < p \<longrightarrow> coprime n p"
-    proof (intro strip)
-      fix n
-      assume H': "1 \<le> n \<and> n < p"
-      {
-        fix d
-        assume "0 \<le> d" "d dvd n" "d dvd p"
-        with H' have "d \<noteq> 0" by auto
-        have "d = 1"
-        proof (rule ccontr)
-          assume "d \<noteq> 1"
-          with `0 \<le> d` `d \<noteq> 0` have "1 < d" by simp
-          moreover from `d dvd p` H' have "d \<le> p" by (auto dest: zdvd_imp_le)
-          moreover from `d dvd n` H' have "d \<noteq> p" by (auto dest: zdvd_imp_le)
-          ultimately show False using `d dvd p` H by auto
-        qed
-      }
-      then show "coprime n p"
-        by (auto simp add: coprime_int)
-    qed
-  next
-    assume H: "\<forall>n. 1 \<le> n \<and> n < p \<longrightarrow> coprime n p"
-    show "\<forall>n. 1 < n \<and> n < p \<longrightarrow> \<not> n dvd p"
-    proof (intro strip notI)
-      fix n
-      assume H': "1 < n \<and> n < p" "n dvd p"
-      then have "1 \<le> n \<and> n < p" by simp
-      with H have "coprime n p" by simp
-      with H' show False by (simp add: coprime_int)
-    qed
-  qed
-  then show ?thesis by (simp add: prime_def)
-qed
-
-why3_vc Gauss
-proof -
-  from assms
-  have "coprime a b" "a dvd c * b" by (simp_all add: mult.commute)
-  then show ?thesis by (rule coprime_dvd_mult_int)
-qed
-
-why3_vc Euclid
-proof -
-  from `p dvd a * b` have "nat \<bar>p\<bar> dvd nat \<bar>a\<bar> * nat \<bar>b\<bar>"
-    by (simp add: dvd_int_iff abs_mult nat_mult_distrib)
-  moreover from `prime (nat p)` have "prime (nat \<bar>p\<bar>)"
-    by (simp add: prime_def)
-  ultimately show ?thesis by (simp add: dvd_int_iff)
-qed
-
-why3_vc gcd_coprime
-proof -
-  have "gcd a (b * c) = gcd (b * c) a" by (simp add: gcd_commute_int)
-  also from assms have "coprime b a" by (simp add: gcd_commute_int)
-  then have "gcd (b * c) a = gcd c a" by (simp add: gcd_mult_cancel_int)
-  finally show ?thesis by (simp add: gcd_commute_int)
-qed
-
-why3_end
-
-end

lib/isabelle/Why3_Real.thy

@@ -181,9 +181,9 @@ subsection {* Rounding towards zero *}
 
 why3_vc Real_of_truncate