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Commit 93445554 authored by MARCHE Claude's avatar MARCHE Claude
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Merge remote-tracking branch 'sberghofer/master'

parents 513e98dd b98ab16c
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.gitignore
View file @ 93445554
......@@ -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
View file @ 93445554
......@@ -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@, $(ISABELLEVERSIONSPECIFICTARGETS))
......
configure.in
View file @ 93445554
......@@ -618,21 +618,21 @@ else
ISABELLEDETECTEDVERSION=`$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
View file @ 93445554
......@@ -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 deleted 100644 → 0
View file @ 513e98dd
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
View file @ 93445554
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 0 → 100644
View file @ 93445554
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
View file @ 93445554
......@@ -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
View file @ 93445554
......@@ -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
View file @ 93445554
......@@ -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 deleted 100755 → 0
View file @ 513e98dd
#!/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_BV.thy → lib/isabelle/Why3_BV.thy.2015
View file @ 93445554
......@@ -240,11 +240,15 @@ where "rotate_left_bv v n \<equiv> word_rotl (unat n) v"
abbreviation (input) rotate_right_bv :: "'a::len0 word \<Rightarrow> 'a word \<Rightarrow> 'a word"
where "rotate_right_bv v n \<equiv> word_rotr (unat n) v"
definition eq_sub_bv :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
eq_sub_bv_defn: "eq_sub_bv a b i n =
(b AND (mask (unat n) << unat i) = a AND (mask (unat n) << unat i))"
type_synonym word8 = "8 word"
why3_open "bv/BV8.xml"
constants
zero=zero_class.zero
zeros=zero_class.zero
ones=max_word
bw_and=bitAND
bw_or=bitOR
......@@ -271,6 +275,7 @@ why3_open "bv/BV8.xml"
to_uint=uint
to_int=sint
of_int=of_int
eq_sub_bv=eq_sub_bv
types
t=word8
......@@ -280,7 +285,7 @@ why3_vc nth_out_of_bound
using assms
by (auto simp add: bv_nth_def test_bit_bin)
why3_vc Nth_zero by (simp add: bv_nth_def)
why3_vc Nth_zeros by (simp add: bv_nth_def)
why3_vc Nth_ones
using assms
......@@ -311,7 +316,7 @@ why3_vc Lsr_nth_high
by (simp add: bv_nth_def nth_shiftr)
(simp add: test_bit_bin nat_add_distrib [symmetric] nat_less_iff)
why3_vc lsr_zero by simp
why3_vc lsr_zeros by simp
why3_vc Asr_nth_low
using assms
......@@ -323,7 +328,7 @@ why3_vc Asr_nth_high
by (simp add: bv_nth_def nth_sshiftr word_size)
(simp add: nat_add_distrib [symmetric] le_nat_iff nat_less_iff)
why3_vc asr_zero by (simp add: sshiftr_def)
why3_vc asr_zeros by (simp add: sshiftr_def)
why3_vc Lsl_nth_high
using assms
......@@ -333,7 +338,7 @@ why3_vc Lsl_nth_low
using assms
by (simp add: bv_nth_def nth_shiftl nat_le_eq_zle)
why3_vc lsl_zero by simp
why3_vc lsl_zeros by simp
why3_vc to_uint_extensionality using assms by simp
......@@ -347,6 +352,9 @@ why3_vc to_uint_of_int
using assms
by (simp add: uint_in_range_def word_of_int uint_word_of_int mod_pos_pos_trivial)
why3_vc nth_bv_def
by (simp add: word_eq_iff word_ops_nth_size word_size nth_shiftr)
why3_vc Nth_bv_is_nth
by (simp add: bv_nth_def unat_def)
......@@ -354,7 +362,7 @@ why3_vc Nth_bv_is_nth2
using assms
by (simp add: bv_nth_def unat_def to_uint_of_int uint_in_range_def)
why3_vc Of_int_zero by simp
why3_vc Of_int_zeros by simp
why3_vc Of_int_ones by (simp add: max_word_eq)
......@@ -424,8 +432,10 @@ why3_vc Extensionality
by (simp add: eq_sub_def bv_nth_def word_eq_iff all_nat)
why3_vc eq_sub_equiv
by (simp add: eq_sub_equiv_aux eq_sub_def
eq_sub_bv_def [simplified of_int_1, folded mask_def] bv_nth_def)
by (simp add: eq_sub_equiv_aux eq_sub_def eq_sub_bv_defn bv_nth_def)
why3_vc eq_sub_bv_def
by (simp add: eq_sub_bv_defn mask_def)
why3_end
......@@ -434,7 +444,7 @@ type_synonym word16 = "16 word"
why3_open "bv/BV16.xml"
constants
zero=zero_class.zero
zeros=zero_class.zero
ones=max_word
bw_and=bitAND
bw_or=bitOR
......@@ -461,6 +471,7 @@ why3_open "bv/BV16.xml"
to_uint=uint
to_int=sint
of_int=of_int
eq_sub_bv=eq_sub_bv
types
t=word16
......@@ -470,7 +481,7 @@ why3_vc nth_out_of_bound
using assms
by (auto simp add: bv_nth_def test_bit_bin)
why3_vc Nth_zero by (simp add: bv_nth_def)
why3_vc Nth_zeros by (simp add: bv_nth_def)
why3_vc Nth_ones
using assms
......@@ -501,7 +512,7 @@ why3_vc Lsr_nth_high
by (simp add: bv_nth_def nth_shiftr)
(simp add: test_bit_bin nat_add_distrib [symmetric] nat_less_iff)
why3_vc lsr_zero by simp
why3_vc lsr_zeros by simp
why3_vc Asr_nth_low
using assms
......@@ -513,7 +524,7 @@ why3_vc Asr_nth_high
by (simp add: bv_nth_def nth_sshiftr word_size)
(simp add: nat_add_distrib [symmetric] le_nat_iff nat_less_iff)
why3_vc asr_zero by (simp add: sshiftr_def)
why3_vc asr_zeros by (simp add: sshiftr_def)
why3_vc Lsl_nth_high
using assms
......@@ -523,7 +534,7 @@ why3_vc Lsl_nth_low
using assms
by (simp add: bv_nth_def nth_shiftl nat_le_eq_zle)
why3_vc lsl_zero by simp
why3_vc lsl_zeros by simp
why3_vc to_uint_extensionality using assms by simp
......@@ -537,6 +548,9 @@ why3_vc to_uint_of_int
using assms
by (simp add: uint_in_range_def word_of_int uint_word_of_int mod_pos_pos_trivial)
why3_vc nth_bv_def
by (simp add: word_eq_iff word_ops_nth_size word_size nth_shiftr)
why3_vc Nth_bv_is_nth
by (simp add: bv_nth_def unat_def)
......@@ -544,7 +558,7 @@ why3_vc Nth_bv_is_nth2
using assms
by (simp add: bv_nth_def unat_def to_uint_of_int uint_in_range_def)
why3_vc Of_int_zero by simp
why3_vc Of_int_zeros by simp
why3_vc Of_int_ones by (simp add: max_word_eq)
......@@ -614,8 +628,10 @@ why3_vc Extensionality
by (simp add: eq_sub_def bv_nth_def word_eq_iff all_nat)
why3_vc eq_sub_equiv
by (simp add: eq_sub_equiv_aux eq_sub_def
eq_sub_bv_def [simplified of_int_1, folded mask_def] bv_nth_def)
by (simp add: eq_sub_equiv_aux eq_sub_def eq_sub_bv_defn bv_nth_def)
why3_vc eq_sub_bv_def
by (simp add: eq_sub_bv_defn mask_def)
why3_end
......@@ -624,7 +640,7 @@ type_synonym word32 = "32 word"
why3_open "bv/BV32.xml"
constants
zero=zero_class.zero
zeros=zero_class.zero
ones=max_word
bw_and=bitAND
bw_or=bitOR
......@@ -651,6 +667,7 @@ why3_open "bv/BV32.xml"
to_uint=uint
to_int=sint
of_int=of_int
eq_sub_bv=eq_sub_bv
types
t=word32
......@@ -660,7 +677,7 @@ why3_vc nth_out_of_bound
using assms
by (auto simp add: bv_nth_def test_bit_bin)
why3_vc Nth_zero by (simp add: bv_nth_def)
why3_vc Nth_zeros by (simp add: bv_nth_def)
why3_vc Nth_ones
using assms
......@@ -691,7 +708,7 @@ why3_vc Lsr_nth_high
by (simp add: bv_nth_def nth_shiftr)
(simp add: test_bit_bin nat_add_distrib [symmetric] nat_less_iff)
why3_vc lsr_zero by simp
why3_vc lsr_zeros by simp
why3_vc Asr_nth_low
using assms
......@@ -703,7 +720,7 @@ why3_vc Asr_nth_high
by (simp add: bv_nth_def nth_sshiftr word_size)
(simp add: nat_add_distrib [symmetric] le_nat_iff nat_less_iff)
why3_vc asr_zero by (simp add: sshiftr_def)
why3_vc asr_zeros by (simp add: sshiftr_def)
why3_vc Lsl_nth_high
using assms
......@@ -713,7 +730,7 @@ why3_vc Lsl_nth_low
using assms
by (simp add: bv_nth_def nth_shiftl nat_le_eq_zle)
why3_vc lsl_zero by simp
why3_vc lsl_zeros by simp
why3_vc to_uint_extensionality using assms by simp
......@@ -727,6 +744,9 @@ why3_vc to_uint_of_int
using assms
by (simp add: uint_in_range_def word_of_int uint_word_of_int mod_pos_pos_trivial)
why3_vc nth_bv_def
by (simp add: word_eq_iff word_ops_nth_size word_size nth_shiftr)
why3_vc Nth_bv_is_nth
by (simp add: bv_nth_def unat_def)
......@@ -734,7 +754,7 @@ why3_vc Nth_bv_is_nth2
using assms
by (simp add: bv_nth_def unat_def to_uint_of_int uint_in_range_def)
why3_vc Of_int_zero by simp
why3_vc Of_int_zeros by simp
why3_vc Of_int_ones by (simp add: max_word_eq)
......@@ -804,8 +824,10 @@ why3_vc Extensionality
by (simp add: eq_sub_def bv_nth_def word_eq_iff all_nat)
why3_vc eq_sub_equiv
by (simp add: eq_sub_equiv_aux eq_sub_def
eq_sub_bv_def [simplified of_int_1, folded mask_def] bv_nth_def)
by (simp add: eq_sub_equiv_aux eq_sub_def eq_sub_bv_defn bv_nth_def)
why3_vc eq_sub_bv_def
by (simp add: eq_sub_bv_defn mask_def)
why3_end
......@@ -814,7 +836,7 @@ type_synonym word64 = "64 word"
why3_open "bv/BV64.xml"
constants
zero=zero_class.zero
zeros=zero_class.zero
ones=max_word
bw_and=bitAND
bw_or=bitOR
......@@ -841,6 +863,7 @@ why3_open "bv/BV64.xml"
to_uint=uint
to_int=sint
of_int=of_int
eq_sub_bv=eq_sub_bv
types
t=word64
......@@ -850,7 +873,7 @@ why3_vc nth_out_of_bound
using assms
by (auto simp add: bv_nth_def test_bit_bin)
why3_vc Nth_zero by (simp add: bv_nth_def)
why3_vc Nth_zeros by (simp add: bv_nth_def)
why3_vc Nth_ones
using assms
......@@ -881,7 +904,7 @@ why3_vc Lsr_nth_high
by (simp add: bv_nth_def nth_shiftr)
(simp add: test_bit_bin nat_add_distrib [symmetric] nat_less_iff)
why3_vc lsr_zero by simp
why3_vc lsr_zeros by simp
why3_vc Asr_nth_low
using assms
......@@ -893,7 +916,7 @@ why3_vc Asr_nth_high
by (simp add: bv_nth_def nth_sshiftr word_size)
(simp add: nat_add_distrib [symmetric] le_nat_iff nat_less_iff)
why3_vc asr_zero by (simp add: sshiftr_def)
why3_vc asr_zeros by (simp add: sshiftr_def)
why3_vc Lsl_nth_high
using assms
......@@ -903,7 +926,7 @@ why3_vc Lsl_nth_low
using assms
by (simp add: bv_nth_def nth_shiftl nat_le_eq_zle)
why3_vc lsl_zero by simp
why3_vc lsl_zeros by simp
why3_vc to_uint_extensionality using assms by simp
......@@ -917,6 +940,9 @@ why3_vc to_uint_of_int
using assms
by (simp add: uint_in_range_def word_of_int uint_word_of_int mod_pos_pos_trivial)
why3_vc nth_bv_def
by (simp add: word_eq_iff word_ops_nth_size word_size nth_shiftr)
why3_vc Nth_bv_is_nth
by (simp add: bv_nth_def unat_def)
......@@ -924,7 +950,7 @@ why3_vc Nth_bv_is_nth2
using assms
by (simp add: bv_nth_def unat_def to_uint_of_int uint_in_range_def)
why3_vc Of_int_zero by simp
why3_vc Of_int_zeros by simp
why3_vc Of_int_ones by (simp add: max_word_eq)
......@@ -994,8 +1020,10 @@ why3_vc Extensionality
by (simp add: eq_sub_def bv_nth_def word_eq_iff all_nat)
why3_vc eq_sub_equiv
by (simp add: eq_sub_equiv_aux eq_sub_def
eq_sub_bv_def [simplified of_int_1, folded mask_def] bv_nth_def)
by (simp add: eq_sub_equiv_aux eq_sub_def eq_sub_bv_defn bv_nth_def)
why3_vc eq_sub_bv_def
by (simp add: eq_sub_bv_defn mask_def)
why3_end
......
lib/isabelle/Why3_BV.thy.2016 0 → 100644
View file @ 93445554
This diff is collapsed.
lib/isabelle/Why3_Number.thy.2015 → lib/isabelle/Why3_Number.thy
View file @ 93445554
......@@ -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 deleted 100644 → 0
View file @ 513e98dd
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