Theory RG_Examples

header {* \section{Examples} *}

theory RG_Examples imports RG_Syntax begin

lemmas definitions [simp]= stable_def Pre_def Rely_def Guar_def Post_def Com_def 

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

lemma le_less_trans2: "[|(j::nat)<k; i≤ j|] ==> i<k"
by simp

lemma add_le_less_mono: "[| (a::nat) < c; b≤d |] ==> a + b < c + d"
by simp

record Example1 =
  A :: "nat list"

lemma Example1: 
 "\<turnstile> COBEGIN
      SCHEME [0 ≤ i < n]
     (´A := ´A [i := 0], 
     \<lbrace> n < length ´A \<rbrace>, 
     \<lbrace> length ºA = length ªA ∧ ºA ! i = ªA ! i \<rbrace>, 
     \<lbrace> length ºA = length ªA ∧ (∀j<n. i ≠ j --> ºA ! j = ªA ! j) \<rbrace>, 
     \<lbrace> ´A ! i = 0 \<rbrace>) 
    COEND
 SAT [\<lbrace> n < length ´A \<rbrace>, \<lbrace> ºA = ªA \<rbrace>, \<lbrace> True \<rbrace>, \<lbrace> ∀i < n. ´A ! i = 0 \<rbrace>]"
apply(rule Parallel)
apply (auto intro!: Basic) 
done

lemma Example1_parameterized: 
"k < t ==>
  \<turnstile> COBEGIN 
    SCHEME [k*n≤i<(Suc k)*n] (´A:=´A[i:=0], 
   \<lbrace>t*n < length ´A\<rbrace>, 
   \<lbrace>t*n < length ºA ∧ length ºA=length ªA ∧ ºA!i = ªA!i\<rbrace>, 
   \<lbrace>t*n < length ºA ∧ length ºA=length ªA ∧ (∀j<length ºA . i≠j --> ºA!j = ªA!j)\<rbrace>, 
   \<lbrace>´A!i=0\<rbrace>) 
   COEND  
 SAT [\<lbrace>t*n < length ´A\<rbrace>, 
      \<lbrace>t*n < length ºA ∧ length ºA=length ªA ∧ (∀i<n. ºA!(k*n+i)=ªA!(k*n+i))\<rbrace>, 
      \<lbrace>t*n < length ºA ∧ length ºA=length ªA ∧ 
      (∀i<length ºA . (i<k*n --> ºA!i = ªA!i) ∧ ((Suc k)*n ≤ i--> ºA!i = ªA!i))\<rbrace>, 
      \<lbrace>∀i<n. ´A!(k*n+i) = 0\<rbrace>]"
apply(rule Parallel)
    apply auto
  apply(erule_tac x="k*n +i" in allE)
  apply(subgoal_tac "k*n+i <length (A b)")
   apply force
  apply(erule le_less_trans2) 
  apply(case_tac t,simp+)
  apply (simp add:add_commute)
  apply(simp add: add_le_mono)
apply(rule Basic)
   apply simp
   apply clarify
   apply (subgoal_tac "k*n+i< length (A x)")
    apply simp
   apply(erule le_less_trans2)
   apply(case_tac t,simp+)
   apply (simp add:add_commute)
   apply(rule add_le_mono, auto)
done


subsection {* Increment a Variable in Parallel *}

subsubsection {* Two components *}

record Example2 =
  x  :: nat
  c_0 :: nat
  c_1 :: nat

lemma Example2: 
 "\<turnstile>  COBEGIN
    (⟨ ´x:=´x+1;; ´c_0:=´c_0 + 1 ⟩, 
     \<lbrace>´x=´c_0 + ´c_1  ∧ ´c_0=0\<rbrace>, 
     \<lbrace>ºc_0 = ªc_0 ∧ 
        (ºx=ºc_0 + ºc_1 
        --> ªx = ªc_0 + ªc_1)\<rbrace>,  
     \<lbrace>ºc_1 = ªc_1 ∧ 
         (ºx=ºc_0 + ºc_1 
         --> ªx =ªc_0 + ªc_1)\<rbrace>,
     \<lbrace>´x=´c_0 + ´c_1 ∧ ´c_0=1 \<rbrace>)
  \<parallel>
      (⟨ ´x:=´x+1;; ´c_1:=´c_1+1 ⟩, 
     \<lbrace>´x=´c_0 + ´c_1 ∧ ´c_1=0 \<rbrace>, 
     \<lbrace>ºc_1 = ªc_1 ∧ 
        (ºx=ºc_0 + ºc_1 
        --> ªx = ªc_0 + ªc_1)\<rbrace>,  
     \<lbrace>ºc_0 = ªc_0 ∧ 
         (ºx=ºc_0 + ºc_1 
        --> ªx =ªc_0 + ªc_1)\<rbrace>,
     \<lbrace>´x=´c_0 + ´c_1 ∧ ´c_1=1\<rbrace>)
 COEND
 SAT [\<lbrace>´x=0 ∧ ´c_0=0 ∧ ´c_1=0\<rbrace>, 
      \<lbrace>ºx=ªx ∧  ºc_0= ªc_0 ∧ ºc_1=ªc_1\<rbrace>,
      \<lbrace>True\<rbrace>,
      \<lbrace>´x=2\<rbrace>]"
apply(rule Parallel)
   apply simp_all
   apply clarify
   apply(case_tac i)
    apply simp
    apply(rule conjI)
     apply clarify
     apply simp
    apply clarify
    apply simp
    apply(case_tac j,simp)
    apply simp
   apply simp
   apply(rule conjI)
    apply clarify
    apply simp
   apply clarify
   apply simp
   apply(subgoal_tac "j=0")
    apply (rotate_tac -1)
    apply (simp (asm_lr))
   apply arith
  apply clarify
  apply(case_tac i,simp,simp)
 apply clarify   
 apply simp
 apply(erule_tac x=0 in all_dupE)
 apply(erule_tac x=1 in allE,simp)
apply clarify
apply(case_tac i,simp)
 apply(rule Await)
  apply simp_all
 apply(clarify)
 apply(rule Seq)
  prefer 2
  apply(rule Basic)
   apply simp_all
  apply(rule subset_refl)
 apply(rule Basic)
 apply simp_all
 apply clarify
 apply simp
apply(rule Await)
 apply simp_all
apply(clarify)
apply(rule Seq)
 prefer 2
 apply(rule Basic)
  apply simp_all
 apply(rule subset_refl)
apply(auto intro!: Basic)
done

subsubsection {* Parameterized *}

lemma Example2_lemma2_aux: "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: 
  "j≤ s ==> (∑i::nat=0..<j. (b (s:=t)) i) = (∑i=0..<j. b i)"
apply(induct j)
 apply (simp_all cong:setsum_cong)
done

lemma Example2_lemma2: 
 "[|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

lemma Example2_lemma2_Suc0: "[|j<n; b j=0|] ==>
 Suc (∑i::nat=0..< n. b i)=(∑i=0..< n. (b (j:=Suc 0)) i)"
by(simp add:Example2_lemma2)

record Example2_parameterized =   
  C :: "nat => nat"
  y  :: nat

lemma Example2_parameterized: "0<n ==> 
  \<turnstile> COBEGIN SCHEME  [0≤i<n]
     (⟨ ´y:=´y+1;; ´C:=´C (i:=1) ⟩, 
     \<lbrace>´y=(∑i=0..<n. ´C i) ∧ ´C i=0\<rbrace>, 
     \<lbrace>ºC i = ªC i ∧ 
      (ºy=(∑i=0..<n. ºC i) --> ªy =(∑i=0..<n. ªC i))\<rbrace>,  
     \<lbrace>(∀j<n. i≠j --> ºC j = ªC j) ∧ 
       (ºy=(∑i=0..<n. ºC i) --> ªy =(∑i=0..<n. ªC i))\<rbrace>,
     \<lbrace>´y=(∑i=0..<n. ´C i) ∧ ´C i=1\<rbrace>) 
    COEND
 SAT [\<lbrace>´y=0 ∧ (∑i=0..<n. ´C i)=0 \<rbrace>, \<lbrace>ºC=ªC ∧ ºy=ªy\<rbrace>, \<lbrace>True\<rbrace>, \<lbrace>´y=n\<rbrace>]"
apply(rule Parallel)
apply force
apply force
apply(force)
apply clarify
apply simp
apply(simp cong:setsum_ivl_cong)
apply clarify
apply simp
apply(rule Await)
apply simp_all
apply clarify
apply(rule Seq)
prefer 2
apply(rule Basic)
apply(rule subset_refl)
apply simp+
apply(rule Basic)
apply simp
apply clarify
apply simp
apply(simp add:Example2_lemma2_Suc0 cong:if_cong)
apply simp+
done

subsection {* Find Least Element *}

text {* A previous lemma: *}

lemma mod_aux :"[|i < (n::nat); a mod n = i;  j < a + n; j mod n = i; a < j|] ==> False"
apply(subgoal_tac "a=a div n*n + a mod n" )
 prefer 2 apply (simp (no_asm_use))
apply(subgoal_tac "j=j div n*n + j mod n")
 prefer 2 apply (simp (no_asm_use))
apply simp
apply(subgoal_tac "a div n*n < j div n*n")
prefer 2 apply arith
apply(subgoal_tac "j div n*n < (a div n + 1)*n")
prefer 2 apply simp
apply (simp only:mult_less_cancel2)
apply arith
done

record Example3 =
  X :: "nat => nat"
  Y :: "nat => nat"

lemma Example3: "m mod n=0 ==> 
 \<turnstile> COBEGIN 
 SCHEME [0≤i<n]
 (WHILE (∀j<n. ´X i < ´Y j)  DO 
   IF P(B!(´X i)) THEN ´Y:=´Y (i:=´X i) 
   ELSE ´X:= ´X (i:=(´X i)+ n) FI 
  OD,
 \<lbrace>(´X i) mod n=i ∧ (∀j<´X i. j mod n=i --> ¬P(B!j)) ∧ (´Y i<m --> P(B!(´Y i)) ∧ ´Y i≤ m+i)\<rbrace>,
 \<lbrace>(∀j<n. i≠j --> ªY j ≤ ºY j) ∧ ºX i = ªX i ∧ 
   ºY i = ªY i\<rbrace>,
 \<lbrace>(∀j<n. i≠j --> ºX j = ªX j ∧ ºY j = ªY j) ∧   
   ªY i ≤ ºY i\<rbrace>,
 \<lbrace>(´X i) mod n=i ∧ (∀j<´X i. j mod n=i --> ¬P(B!j)) ∧ (´Y i<m --> P(B!(´Y i)) ∧ ´Y i≤ m+i) ∧ (∃j<n. ´Y j ≤ ´X i) \<rbrace>) 
 COEND
 SAT [\<lbrace> ∀i<n. ´X i=i ∧ ´Y i=m+i \<rbrace>,\<lbrace>ºX=ªX ∧ ºY=ªY\<rbrace>,\<lbrace>True\<rbrace>,
  \<lbrace>∀i<n. (´X i) mod n=i ∧ (∀j<´X i. j mod n=i --> ¬P(B!j)) ∧ 
    (´Y i<m --> P(B!(´Y i)) ∧ ´Y i≤ m+i) ∧ (∃j<n. ´Y j ≤ ´X i)\<rbrace>]"
apply(rule Parallel)
--{*5 subgoals left *}
apply force+
apply clarify
apply simp
apply(rule While)
    apply force
   apply force
  apply force
 apply(rule_tac pre'="\<lbrace> ´X i mod n = i ∧ (∀j. j<´X i --> j mod n = i --> ¬P(B!j)) ∧ (´Y i < n * q --> P (B!(´Y i))) ∧ ´X i<´Y i\<rbrace>" in Conseq)
     apply force
    apply(rule subset_refl)+
 apply(rule Cond)
    apply force
   apply(rule Basic)
      apply force
     apply fastsimp
    apply force
   apply force
  apply(rule Basic)
     apply simp
     apply clarify
     apply simp
     apply(case_tac "X x (j mod n)≤ j")
      apply(drule le_imp_less_or_eq)
      apply(erule disjE)
       apply(drule_tac j=j and n=n and i="j mod n" and a="X x (j mod n)" in mod_aux)
        apply assumption+
       apply simp+
    apply clarsimp
    apply fastsimp
apply force+
done

text {* Same but with a list as auxiliary variable: *}

record Example3_list =
  X :: "nat list"
  Y :: "nat list"

lemma Example3_list: "m mod n=0 ==> \<turnstile> (COBEGIN SCHEME [0≤i<n]
 (WHILE (∀j<n. ´X!i < ´Y!j)  DO 
     IF P(B!(´X!i)) THEN ´Y:=´Y[i:=´X!i] ELSE ´X:= ´X[i:=(´X!i)+ n] FI 
  OD,
 \<lbrace>n<length ´X ∧ n<length ´Y ∧ (´X!i) mod n=i ∧ (∀j<´X!i. j mod n=i --> ¬P(B!j)) ∧ (´Y!i<m --> P(B!(´Y!i)) ∧ ´Y!i≤ m+i)\<rbrace>,
 \<lbrace>(∀j<n. i≠j --> ªY!j ≤ ºY!j) ∧ ºX!i = ªX!i ∧ 
   ºY!i = ªY!i ∧ length ºX = length ªX ∧ length ºY = length ªY\<rbrace>,
 \<lbrace>(∀j<n. i≠j --> ºX!j = ªX!j ∧ ºY!j = ªY!j) ∧   
   ªY!i ≤ ºY!i ∧ length ºX = length ªX ∧ length ºY = length ªY\<rbrace>,
 \<lbrace>(´X!i) mod n=i ∧ (∀j<´X!i. j mod n=i --> ¬P(B!j)) ∧ (´Y!i<m --> P(B!(´Y!i)) ∧ ´Y!i≤ m+i) ∧ (∃j<n. ´Y!j ≤ ´X!i) \<rbrace>) COEND)
 SAT [\<lbrace>n<length ´X ∧ n<length ´Y ∧ (∀i<n. ´X!i=i ∧ ´Y!i=m+i) \<rbrace>,
      \<lbrace>ºX=ªX ∧ ºY=ªY\<rbrace>,
      \<lbrace>True\<rbrace>,
      \<lbrace>∀i<n. (´X!i) mod n=i ∧ (∀j<´X!i. j mod n=i --> ¬P(B!j)) ∧ 
        (´Y!i<m --> P(B!(´Y!i)) ∧ ´Y!i≤ m+i) ∧ (∃j<n. ´Y!j ≤ ´X!i)\<rbrace>]"
apply(rule Parallel)
--{* 5 subgoals left *}
apply force+
apply clarify
apply simp
apply(rule While)
    apply force
   apply force
  apply force
 apply(rule_tac pre'="\<lbrace>n<length ´X ∧ n<length ´Y ∧ ´X ! i mod n = i ∧ (∀j. j < ´X ! i --> j mod n = i --> ¬ P (B ! j)) ∧ (´Y ! i < n * q --> P (B ! (´Y ! i))) ∧ ´X!i<´Y!i\<rbrace>" in Conseq)
     apply force
    apply(rule subset_refl)+
 apply(rule Cond)
    apply force
   apply(rule Basic)
      apply force
     apply force
    apply force
   apply force
  apply(rule Basic)
     apply simp
     apply clarify
     apply simp
     apply(rule allI)
     apply(rule impI)+
     apply(case_tac "X x ! i≤ j")
      apply(drule le_imp_less_or_eq)
      apply(erule disjE)
       apply(drule_tac j=j and n=n and i=i and a="X x ! i" in mod_aux)
        apply assumption+
       apply simp
apply force+
done

end