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 auto 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 auto done end