Theory OG_Examples

theory OG_Examples
imports OG_Syntax
header {* \section{Examples} *}

theory OG_Examples imports OG_Syntax begin

subsection {* Mutual Exclusion *}

subsubsection {* Peterson's Algorithm I*}

text {* Eike Best. "Semantics of Sequential and Parallel Programs", page 217. *}

record Petersons_mutex_1 =
pr1 :: nat
pr2 :: nat
in1 :: bool
in2 :: bool
hold :: nat

lemma Petersons_mutex_1:
"\<parallel>- \<lbrace>´pr1=0 ∧ ¬´in1 ∧ ´pr2=0 ∧ ¬´in2 \<rbrace>
COBEGIN \<lbrace>´pr1=0 ∧ ¬´in1\<rbrace>
WHILE True INV \<lbrace>´pr1=0 ∧ ¬´in1\<rbrace>
DO
\<lbrace>´pr1=0 ∧ ¬´in1\<rbrace> ⟨ ´in1:=True,,´pr1:=1 ⟩;;
\<lbrace>´pr1=1 ∧ ´in1\<rbrace> ⟨ ´hold:=1,,´pr1:=2 ⟩;;
\<lbrace>´pr1=2 ∧ ´in1 ∧ (´hold=1 ∨ ´hold=2 ∧ ´pr2=2)\<rbrace>
AWAIT (¬´in2 ∨ ¬(´hold=1)) THEN ´pr1:=3 END;;
\<lbrace>´pr1=3 ∧ ´in1 ∧ (´hold=1 ∨ ´hold=2 ∧ ´pr2=2)\<rbrace>
⟨´in1:=False,,´pr1:=0⟩
OD \<lbrace>´pr1=0 ∧ ¬´in1\<rbrace>
\<parallel>
\<lbrace>´pr2=0 ∧ ¬´in2\<rbrace>
WHILE True INV \<lbrace>´pr2=0 ∧ ¬´in2\<rbrace>
DO
\<lbrace>´pr2=0 ∧ ¬´in2\<rbrace> ⟨ ´in2:=True,,´pr2:=1 ⟩;;
\<lbrace>´pr2=1 ∧ ´in2\<rbrace> ⟨ ´hold:=2,,´pr2:=2 ⟩;;
\<lbrace>´pr2=2 ∧ ´in2 ∧ (´hold=2 ∨ (´hold=1 ∧ ´pr1=2))\<rbrace>
AWAIT (¬´in1 ∨ ¬(´hold=2)) THEN ´pr2:=3 END;;
\<lbrace>´pr2=3 ∧ ´in2 ∧ (´hold=2 ∨ (´hold=1 ∧ ´pr1=2))\<rbrace>
⟨´in2:=False,,´pr2:=0⟩
OD \<lbrace>´pr2=0 ∧ ¬´in2\<rbrace>
COEND
\<lbrace>´pr1=0 ∧ ¬´in1 ∧ ´pr2=0 ∧ ¬´in2\<rbrace>"

apply oghoare
--{* 104 verification conditions. *}
apply auto
done

subsubsection {*Peterson's Algorithm II: A Busy Wait Solution *}

text {* Apt and Olderog. "Verification of sequential and concurrent Programs", page 282. *}

record Busy_wait_mutex =
flag1 :: bool
flag2 :: bool
turn :: nat
after1 :: bool
after2 :: bool

lemma Busy_wait_mutex:
"\<parallel>- \<lbrace>True\<rbrace>
´flag1:=False,, ´flag2:=False,,
COBEGIN \<lbrace>¬´flag1\<rbrace>
WHILE True
INV \<lbrace>¬´flag1\<rbrace>
DO \<lbrace>¬´flag1\<rbrace> ⟨ ´flag1:=True,,´after1:=False ⟩;;
\<lbrace>´flag1 ∧ ¬´after1\<rbrace> ⟨ ´turn:=1,,´after1:=True ⟩;;
\<lbrace>´flag1 ∧ ´after1 ∧ (´turn=1 ∨ ´turn=2)\<rbrace>
WHILE ¬(´flag2 --> ´turn=2)
INV \<lbrace>´flag1 ∧ ´after1 ∧ (´turn=1 ∨ ´turn=2)\<rbrace>
DO \<lbrace>´flag1 ∧ ´after1 ∧ (´turn=1 ∨ ´turn=2)\<rbrace> SKIP OD;;
\<lbrace>´flag1 ∧ ´after1 ∧ (´flag2 ∧ ´after2 --> ´turn=2)\<rbrace>
´flag1:=False
OD
\<lbrace>False\<rbrace>
\<parallel>
\<lbrace>¬´flag2\<rbrace>
WHILE True
INV \<lbrace>¬´flag2\<rbrace>
DO \<lbrace>¬´flag2\<rbrace> ⟨ ´flag2:=True,,´after2:=False ⟩;;
\<lbrace>´flag2 ∧ ¬´after2\<rbrace> ⟨ ´turn:=2,,´after2:=True ⟩;;
\<lbrace>´flag2 ∧ ´after2 ∧ (´turn=1 ∨ ´turn=2)\<rbrace>
WHILE ¬(´flag1 --> ´turn=1)
INV \<lbrace>´flag2 ∧ ´after2 ∧ (´turn=1 ∨ ´turn=2)\<rbrace>
DO \<lbrace>´flag2 ∧ ´after2 ∧ (´turn=1 ∨ ´turn=2)\<rbrace> SKIP OD;;
\<lbrace>´flag2 ∧ ´after2 ∧ (´flag1 ∧ ´after1 --> ´turn=1)\<rbrace>
´flag2:=False
OD
\<lbrace>False\<rbrace>
COEND
\<lbrace>False\<rbrace>"

apply oghoare
--{* 122 vc *}
apply auto
done

subsubsection {* Peterson's Algorithm III: A Solution using Semaphores *}

record Semaphores_mutex =
out :: bool
who :: nat

lemma Semaphores_mutex:
"\<parallel>- \<lbrace>i≠j\<rbrace>
´out:=True ,,
COBEGIN \<lbrace>i≠j\<rbrace>
WHILE True INV \<lbrace>i≠j\<rbrace>
DO \<lbrace>i≠j\<rbrace> AWAIT ´out THEN ´out:=False,, ´who:=i END;;
\<lbrace>¬´out ∧ ´who=i ∧ i≠j\<rbrace> ´out:=True OD
\<lbrace>False\<rbrace>
\<parallel>
\<lbrace>i≠j\<rbrace>
WHILE True INV \<lbrace>i≠j\<rbrace>
DO \<lbrace>i≠j\<rbrace> AWAIT ´out THEN ´out:=False,,´who:=j END;;
\<lbrace>¬´out ∧ ´who=j ∧ i≠j\<rbrace> ´out:=True OD
\<lbrace>False\<rbrace>
COEND
\<lbrace>False\<rbrace>"

apply oghoare
--{* 38 vc *}
apply auto
done

subsubsection {* Peterson's Algorithm III: Parameterized version: *}

lemma Semaphores_parameterized_mutex:
"0<n ==> \<parallel>- \<lbrace>True\<rbrace>
´out:=True ,,
COBEGIN
SCHEME [0≤ i< n]
\<lbrace>True\<rbrace>
WHILE True INV \<lbrace>True\<rbrace>
DO \<lbrace>True\<rbrace> AWAIT ´out THEN ´out:=False,, ´who:=i END;;
\<lbrace>¬´out ∧ ´who=i\<rbrace> ´out:=True OD
\<lbrace>False\<rbrace>
COEND
\<lbrace>False\<rbrace>"

apply oghoare
--{* 20 vc *}
apply auto
done

subsubsection{* The Ticket Algorithm *}

record Ticket_mutex =
num :: nat
nextv :: nat
turn :: "nat list"
index :: nat

lemma Ticket_mutex:
"[| 0<n; I=«n=length ´turn ∧ 0<´nextv ∧ (∀k l. k<n ∧ l<n ∧ k≠l
--> ´turn!k < ´num ∧ (´turn!k =0 ∨ ´turn!k≠´turn!l))» |]
==> \<parallel>- \<lbrace>n=length ´turn\<rbrace>
´index:= 0,,
WHILE ´index < n INV \<lbrace>n=length ´turn ∧ (∀i<´index. ´turn!i=0)\<rbrace>
DO ´turn:= ´turn[´index:=0],, ´index:=´index +1 OD,,
´num:=1 ,, ´nextv:=1 ,,
COBEGIN
SCHEME [0≤ i< n]
\<lbrace>´I\<rbrace>
WHILE True INV \<lbrace>´I\<rbrace>
DO \<lbrace>´I\<rbrace> ⟨ ´turn :=´turn[i:=´num],, ´num:=´num+1 ⟩;;
\<lbrace>´I\<rbrace> WAIT ´turn!i=´nextv END;;
\<lbrace>´I ∧ ´turn!i=´nextv\<rbrace> ´nextv:=´nextv+1
OD
\<lbrace>False\<rbrace>
COEND
\<lbrace>False\<rbrace>"

apply oghoare
--{* 35 vc *}
apply simp_all
--{* 21 vc *}
apply(tactic {* ALLGOALS (clarify_tac @{context}) *})
--{* 11 vc *}
apply simp_all
apply(tactic {* ALLGOALS (clarify_tac @{context}) *})
--{* 10 subgoals left *}
apply(erule less_SucE)
apply simp
apply simp
--{* 9 subgoals left *}
apply(case_tac "i=k")
apply force
apply simp
apply(case_tac "i=l")
apply force
apply force
--{* 8 subgoals left *}
prefer 8
apply force
apply force
--{* 6 subgoals left *}
prefer 6
apply(erule_tac x=i in allE)
apply fastforce
--{* 5 subgoals left *}
prefer 5
apply(case_tac [!] "j=k")
--{* 10 subgoals left *}
apply simp_all
apply(erule_tac x=k in allE)
apply force
--{* 9 subgoals left *}
apply(case_tac "j=l")
apply simp
apply(erule_tac x=k in allE)
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply force
apply(erule_tac x=k in allE)
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply force
--{* 8 subgoals left *}
apply force
apply(case_tac "j=l")
apply simp
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply force
apply force
apply force
--{* 5 subgoals left *}
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply(case_tac "j=l")
apply force
apply force
apply force
--{* 3 subgoals left *}
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply(case_tac "j=l")
apply force
apply force
apply force
--{* 1 subgoals left *}
apply(erule_tac x=k in allE)
apply(erule_tac x=l in allE)
apply(case_tac "j=l")
apply force
apply force
done

subsection{* Parallel Zero Search *}

text {* Synchronized Zero Search. Zero-6 *}

text {*Apt and Olderog. "Verification of sequential and concurrent Programs" page 294: *}

record Zero_search =
turn :: nat
found :: bool
x :: nat
y :: nat

lemma Zero_search:
"[|I1= « a≤´x ∧ (´found --> (a<´x ∧ f(´x)=0) ∨ (´y≤a ∧ f(´y)=0))
∧ (¬´found ∧ a<´ x --> f(´x)≠0) » ;
I2= «´y≤a+1 ∧ (´found --> (a<´x ∧ f(´x)=0) ∨ (´y≤a ∧ f(´y)=0))
∧ (¬´found ∧ ´y≤a --> f(´y)≠0) » |] ==>
\<parallel>- \<lbrace>∃ u. f(u)=0\<rbrace>
´turn:=1,, ´found:= False,,
´x:=a,, ´y:=a+1 ,,
COBEGIN \<lbrace>´I1\<rbrace>
WHILE ¬´found
INV \<lbrace>´I1\<rbrace>
DO \<lbrace>a≤´x ∧ (´found --> ´y≤a ∧ f(´y)=0) ∧ (a<´x --> f(´x)≠0)\<rbrace>
WAIT ´turn=1 END;;
\<lbrace>a≤´x ∧ (´found --> ´y≤a ∧ f(´y)=0) ∧ (a<´x --> f(´x)≠0)\<rbrace>
´turn:=2;;
\<lbrace>a≤´x ∧ (´found --> ´y≤a ∧ f(´y)=0) ∧ (a<´x --> f(´x)≠0)\<rbrace>
⟨ ´x:=´x+1,,
IF f(´x)=0 THEN ´found:=True ELSE SKIP FI⟩
OD;;
\<lbrace>´I1 ∧ ´found\<rbrace>
´turn:=2
\<lbrace>´I1 ∧ ´found\<rbrace>
\<parallel>
\<lbrace>´I2\<rbrace>
WHILE ¬´found
INV \<lbrace>´I2\<rbrace>
DO \<lbrace>´y≤a+1 ∧ (´found --> a<´x ∧ f(´x)=0) ∧ (´y≤a --> f(´y)≠0)\<rbrace>
WAIT ´turn=2 END;;
\<lbrace>´y≤a+1 ∧ (´found --> a<´x ∧ f(´x)=0) ∧ (´y≤a --> f(´y)≠0)\<rbrace>
´turn:=1;;
\<lbrace>´y≤a+1 ∧ (´found --> a<´x ∧ f(´x)=0) ∧ (´y≤a --> f(´y)≠0)\<rbrace>
⟨ ´y:=(´y - 1),,
IF f(´y)=0 THEN ´found:=True ELSE SKIP FI⟩
OD;;
\<lbrace>´I2 ∧ ´found\<rbrace>
´turn:=1
\<lbrace>´I2 ∧ ´found\<rbrace>
COEND
\<lbrace>f(´x)=0 ∨ f(´y)=0\<rbrace>"

apply oghoare
--{* 98 verification conditions *}
apply auto
--{* auto takes about 3 minutes !! *}
done

text {* Easier Version: without AWAIT. Apt and Olderog. page 256: *}

lemma Zero_Search_2:
"[|I1=« a≤´x ∧ (´found --> (a<´x ∧ f(´x)=0) ∨ (´y≤a ∧ f(´y)=0))
∧ (¬´found ∧ a<´x --> f(´x)≠0)»;
I2= «´y≤a+1 ∧ (´found --> (a<´x ∧ f(´x)=0) ∨ (´y≤a ∧ f(´y)=0))
∧ (¬´found ∧ ´y≤a --> f(´y)≠0)»|] ==>
\<parallel>- \<lbrace>∃u. f(u)=0\<rbrace>
´found:= False,,
´x:=a,, ´y:=a+1,,
COBEGIN \<lbrace>´I1\<rbrace>
WHILE ¬´found
INV \<lbrace>´I1\<rbrace>
DO \<lbrace>a≤´x ∧ (´found --> ´y≤a ∧ f(´y)=0) ∧ (a<´x --> f(´x)≠0)\<rbrace>
⟨ ´x:=´x+1,,IF f(´x)=0 THEN ´found:=True ELSE SKIP FI⟩
OD
\<lbrace>´I1 ∧ ´found\<rbrace>
\<parallel>
\<lbrace>´I2\<rbrace>
WHILE ¬´found
INV \<lbrace>´I2\<rbrace>
DO \<lbrace>´y≤a+1 ∧ (´found --> a<´x ∧ f(´x)=0) ∧ (´y≤a --> f(´y)≠0)\<rbrace>
⟨ ´y:=(´y - 1),,IF f(´y)=0 THEN ´found:=True ELSE SKIP FI⟩
OD
\<lbrace>´I2 ∧ ´found\<rbrace>
COEND
\<lbrace>f(´x)=0 ∨ f(´y)=0\<rbrace>"

apply oghoare
--{* 20 vc *}
apply auto
--{* auto takes aprox. 2 minutes. *}
done

subsection {* Producer/Consumer *}

subsubsection {* Previous lemmas *}

lemma nat_lemma2: "[| b = m*(n::nat) + t; a = s*n + u; t=u; b-a < n |] ==> m ≤ s"
proof -
assume "b = m*(n::nat) + t" "a = s*n + u" "t=u"
hence "(m - s) * n = b - a" by (simp add: diff_mult_distrib)
also assume "… < n"
finally have "m - s < 1" by simp
thus ?thesis by arith
qed

lemma mod_lemma: "[| (c::nat) ≤ a; a < b; b - c < n |] ==> b mod n ≠ a mod n"
apply(subgoal_tac "b=b div n*n + b mod n" )
prefer 2 apply (simp add: mod_div_equality [symmetric])
apply(subgoal_tac "a=a div n*n + a mod n")
prefer 2
apply(simp add: mod_div_equality [symmetric])
apply(subgoal_tac "b - a ≤ b - c")
prefer 2 apply arith
apply(drule le_less_trans)
back
apply assumption
apply(frule less_not_refl2)
apply(drule less_imp_le)
apply (drule_tac m = "a" and k = n in div_le_mono)
apply(safe)
apply(frule_tac b = "b" and a = "a" and n = "n" in nat_lemma2, assumption, assumption)
apply assumption
apply(drule order_antisym, assumption)
apply(rotate_tac -3)
apply(simp)
done


subsubsection {* Producer/Consumer Algorithm *}

record Producer_consumer =
ins :: nat
outs :: nat
li :: nat
lj :: nat
vx :: nat
vy :: nat
buffer :: "nat list"
b :: "nat list"

text {* The whole proof takes aprox. 4 minutes. *}

lemma Producer_consumer:
"[|INIT= «0<length a ∧ 0<length ´buffer ∧ length ´b=length a» ;
I= «(∀k<´ins. ´outs≤k --> (a ! k) = ´buffer ! (k mod (length ´buffer))) ∧
´outs≤´ins ∧ ´ins-´outs≤length ´buffer» ;
I1= «´I ∧ ´li≤length a» ;
p1= «´I1 ∧ ´li=´ins» ;
I2 = «´I ∧ (∀k<´lj. (a ! k)=(´b ! k)) ∧ ´lj≤length a» ;
p2 = «´I2 ∧ ´lj=´outs» |] ==>
\<parallel>- \<lbrace>´INIT\<rbrace>
´ins:=0,, ´outs:=0,, ´li:=0,, ´lj:=0,,
COBEGIN \<lbrace>´p1 ∧ ´INIT\<rbrace>
WHILE ´li <length a
INV \<lbrace>´p1 ∧ ´INIT\<rbrace>
DO \<lbrace>´p1 ∧ ´INIT ∧ ´li<length a\<rbrace>
´vx:= (a ! ´li);;
\<lbrace>´p1 ∧ ´INIT ∧ ´li<length a ∧ ´vx=(a ! ´li)\<rbrace>
WAIT ´ins-´outs < length ´buffer END;;
\<lbrace>´p1 ∧ ´INIT ∧ ´li<length a ∧ ´vx=(a ! ´li)
∧ ´ins-´outs < length ´buffer\<rbrace>
´buffer:=(list_update ´buffer (´ins mod (length ´buffer)) ´vx);;
\<lbrace>´p1 ∧ ´INIT ∧ ´li<length a
∧ (a ! ´li)=(´buffer ! (´ins mod (length ´buffer)))
∧ ´ins-´outs <length ´buffer\<rbrace>
´ins:=´ins+1;;
\<lbrace>´I1 ∧ ´INIT ∧ (´li+1)=´ins ∧ ´li<length a\<rbrace>
´li:=´li+1
OD
\<lbrace>´p1 ∧ ´INIT ∧ ´li=length a\<rbrace>
\<parallel>
\<lbrace>´p2 ∧ ´INIT\<rbrace>
WHILE ´lj < length a
INV \<lbrace>´p2 ∧ ´INIT\<rbrace>
DO \<lbrace>´p2 ∧ ´lj<length a ∧ ´INIT\<rbrace>
WAIT ´outs<´ins END;;
\<lbrace>´p2 ∧ ´lj<length a ∧ ´outs<´ins ∧ ´INIT\<rbrace>
´vy:=(´buffer ! (´outs mod (length ´buffer)));;
\<lbrace>´p2 ∧ ´lj<length a ∧ ´outs<´ins ∧ ´vy=(a ! ´lj) ∧ ´INIT\<rbrace>
´outs:=´outs+1;;
\<lbrace>´I2 ∧ (´lj+1)=´outs ∧ ´lj<length a ∧ ´vy=(a ! ´lj) ∧ ´INIT\<rbrace>
´b:=(list_update ´b ´lj ´vy);;
\<lbrace>´I2 ∧ (´lj+1)=´outs ∧ ´lj<length a ∧ (a ! ´lj)=(´b ! ´lj) ∧ ´INIT\<rbrace>
´lj:=´lj+1
OD
\<lbrace>´p2 ∧ ´lj=length a ∧ ´INIT\<rbrace>
COEND
\<lbrace> ∀k<length a. (a ! k)=(´b ! k)\<rbrace>"

apply oghoare
--{* 138 vc *}
apply(tactic {* ALLGOALS (clarify_tac @{context}) *})
--{* 112 subgoals left *}
apply(simp_all (no_asm))
--{* 97 subgoals left *}
apply(tactic {*ALLGOALS (conjI_Tac (K all_tac)) *})
--{* 930 subgoals left *}
apply(tactic {* ALLGOALS (clarify_tac @{context}) *})
--{* 99 subgoals left *}
apply(simp_all (*asm_lr*) only:length_0_conv [THEN sym])
--{* 20 subgoals left *}
apply (simp_all (*asm_lr*) del:length_0_conv length_greater_0_conv add: nth_list_update mod_lemma)
--{* 9 subgoals left *}
apply (force simp add:less_Suc_eq)
apply(drule sym)
apply (force simp add:less_Suc_eq)+
done

subsection {* Parameterized Examples *}

subsubsection {* Set Elements of an Array to Zero *}

record Example1 =
a :: "nat => nat"

lemma Example1:
"\<parallel>- \<lbrace>True\<rbrace>
COBEGIN SCHEME [0≤i<n] \<lbrace>True\<rbrace> ´a:=´a (i:=0) \<lbrace>´a i=0\<rbrace> COEND
\<lbrace>∀i < n. ´a i = 0\<rbrace>"

apply oghoare
apply simp_all
done

text {* Same example with lists as auxiliary variables. *}
record Example1_list =
A :: "nat list"
lemma Example1_list:
"\<parallel>- \<lbrace>n < length ´A\<rbrace>
COBEGIN
SCHEME [0≤i<n] \<lbrace>n < length ´A\<rbrace> ´A:=´A[i:=0] \<lbrace>´A!i=0\<rbrace>
COEND
\<lbrace>∀i < n. ´A!i = 0\<rbrace>"

apply oghoare
apply force+
done

subsubsection {* Increment a Variable in Parallel *}

text {* First some lemmas about summation properties. *}
(*
lemma Example2_lemma1: "!!b. j<n ==> (∑i::nat<n. b i) = (0::nat) ==> b j = 0 "
apply(induct n)
apply simp_all
apply(force simp add: less_Suc_eq)
done
*)

lemma Example2_lemma2_aux: "!!b. j<n ==>
(∑i=0..<n. (b i::nat)) =
(∑i=0..<j. b i) + b j + (∑i=0..<n-(Suc j) . b (Suc j + i))"

apply(induct n)
apply simp_all
apply(simp add:less_Suc_eq)
apply(auto)
apply(subgoal_tac "n - j = Suc(n- Suc j)")
apply simp
apply arith
done

lemma Example2_lemma2_aux2:
"!!b. j≤ s ==> (∑i::nat=0..<j. (b (s:=t)) i) = (∑i=0..<j. b i)"
apply(induct j)
apply simp_all
done

lemma Example2_lemma2:
"!!b. [|j<n; b j=0|] ==> Suc (∑i::nat=0..<n. b i)=(∑i=0..<n. (b (j := Suc 0)) i)"
apply(frule_tac b="(b (j:=(Suc 0)))" in Example2_lemma2_aux)
apply(erule_tac t="setsum (b(j := (Suc 0))) {0..<n}" in ssubst)
apply(frule_tac b=b in Example2_lemma2_aux)
apply(erule_tac t="setsum b {0..<n}" in ssubst)
apply(subgoal_tac "Suc (setsum b {0..<j} + b j + (∑i=0..<n - Suc j. b (Suc j + i)))=(setsum b {0..<j} + Suc (b j) + (∑i=0..<n - Suc j. b (Suc j + i)))")
apply(rotate_tac -1)
apply(erule ssubst)
apply(subgoal_tac "j≤j")
apply(drule_tac b="b" and t="(Suc 0)" in Example2_lemma2_aux2)
apply(rotate_tac -1)
apply(erule ssubst)
apply simp_all
done


record Example2 =
c :: "nat => nat"
x :: nat

lemma Example_2: "0<n ==>
\<parallel>- \<lbrace>´x=0 ∧ (∑i=0..<n. ´c i)=0\<rbrace>
COBEGIN
SCHEME [0≤i<n]
\<lbrace>´x=(∑i=0..<n. ´c i) ∧ ´c i=0\<rbrace>
⟨ ´x:=´x+(Suc 0),, ´c:=´c (i:=(Suc 0)) ⟩
\<lbrace>´x=(∑i=0..<n. ´c i) ∧ ´c i=(Suc 0)\<rbrace>
COEND
\<lbrace>´x=n\<rbrace>"

apply oghoare
apply (simp_all cong del: strong_setsum_cong)
apply (tactic {* ALLGOALS (clarify_tac @{context}) *})
apply (simp_all cong del: strong_setsum_cong)
apply(erule (1) Example2_lemma2)
apply(erule (1) Example2_lemma2)
apply(erule (1) Example2_lemma2)
apply(simp)
done

end