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products/Sources/formale Sprachen/Isabelle/HOL/Hoare_Parallel/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 14 kB

Quelle RG_Examples.thy Sprache: Isabelle

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) j\ \ i
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:
"\ COBEGIN
      SCHEME [0 ≤ i < n]
     (🍋A := 🍋A [i := 0],
     { n < length 🍋A },
     { length ♂A = length ♀A ∧ ♂A ! i = ♀A ! i },
     { length ♂A = length ♀A ∧ (∀j<n. i ≠ j ⟶ ♂A ! j = ♀A ! j) },
     { 🍋A ! i = 0 })
    COEND
SAT [{ n < length 🍋A }, { ♂A = ♀A }, { True }, { ∀i < n. 🍋A ! i = 0 }]"
apply(rule Parallel)
apply (auto intro!: Basic)
done

lemma Example1_parameterized:
"k < t \
  ⊨ COBEGIN
    SCHEME [k*n≤i<(Suc k)*n] (🍋A:=🍋A[i:=0],
   {t*n < length 🍋A},
   {t*n < length ♂A ∧ length ♂A=length ♀A ∧ ♂A!i = ♀A!i},
   {t*n < length ♂A ∧ length ♂A=length ♀A ∧ (∀j<length ♂A . i≠j ⟶ ♂A!j = ♀A!j)},
   {🍋A!i=0})
   COEND
SAT [{t*n < length 🍋A},
      {t*n < length ♂A ∧ length ♂A=length ♀A ∧ (∀i<n. ♂A!(k*n+i)=♀A!(k*n+i))},
      {t*n < length ♂A ∧ length ♂A=length ♀A ∧
      (∀i<length ♂A . (i<k*n ⟶ ♂A!i = ♀A!i) ∧ ((Suc k)*n ≤ i⟶ ♂A!i = ♀A!i))},
      {∀i<n. 🍋A!(k*n+i) = 0}]"
apply(rule Parallel)
    apply auto
  apply(erule_tac x="k*n +i" in allE)
  apply(subgoal_tac "k*n+i )
   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:
"\ COBEGIN
    (⟨ 🍋x:=🍋x+1;; 🍋c_0:=🍋c_0 + 1 ⟩,
     {🍋x=🍋c_0 + 🍋c_1  ∧ 🍋c_0=0},
     {♂c_0 = ♀c_0 ∧
        (♂x=♂c_0 + ♂c_1
        ⟶ ♀x = ♀c_0 + ♀c_1)},
     {♂c_1 = ♀c_1 ∧
         (♂x=♂c_0 + ♂c_1
         ⟶ ♀x =♀c_0 + ♀c_1)},
     {🍋x=🍋c_0 + 🍋c_1 ∧ 🍋c_0=1 })
  ∥
      (⟨ 🍋x:=🍋x+1;; 🍋c_1:=🍋c_1+1 ⟩,
     {🍋x=🍋c_0 + 🍋c_1 ∧ 🍋c_1=0 },
     {♂c_1 = ♀c_1 ∧
        (♂x=♂c_0 + ♂c_1
        ⟶ ♀x = ♀c_0 + ♀c_1)},
     {♂c_0 = ♀c_0 ∧
         (♂x=♂c_0 + ♂c_1
        ⟶ ♀x =♀c_0 + ♀c_1)},
     {🍋x=🍋c_0 + 🍋c_1 ∧ 🍋c_1=1})
COEND
SAT [{🍋x=0 ∧ 🍋c_0=0 ∧ 🍋c_1=0},
      {♂x=♀x ∧  ♂c_0= ♀c_0 ∧ ♂c_1=♀c_1},
      {True},
      {🍋x=2}]"
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 simp
   apply(rule conjI)
    apply clarify
    apply simp
   apply clarify
   apply simp
   apply(subgoal_tac "x=0")
    apply simp
   apply arith
  apply clarify
  apply(case_tac xa, 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
(∑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..i=0..
  by (induct j) simp_all

lemma Example2_lemma2:
"\j \ Suc (\i::nat=0..i=0..
apply(frule_tac b="(b (j:=(Suc 0)))" in Example2_lemma2_aux)
apply(erule_tac  t="sum (b(j := (Suc 0))) {0.. in ssubst)
apply(frule_tac b=b in Example2_lemma2_aux)
apply(erule_tac  t="sum b {0.. in ssubst)
apply(subgoal_tac "Suc (sum b {0..i=0..i=0..)
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 \
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
  ⊨ COBEGIN SCHEME  [0≤i<n]
     (⟨ 🍋y:=🍋y+1;; 🍋C:=🍋C (i:=1) ⟩,
     {🍋y=(∑i=0..<n. 🍋C i) ∧ 🍋C i=0},
     {♂C i = ♀C i ∧
      (♂y=(∑i=0..<n. ♂C i) ⟶ ♀y =(∑i=0..<n. ♀C i))},
     {(∀j<n. i≠j ⟶ ♂C j = ♀C j) ∧
       (♂y=(∑i=0..<n. ♂C i) ⟶ ♀y =(∑i=0..<n. ♀C i))},
     {🍋y=(∑i=0..<n. 🍋C i) ∧ 🍋C i=1})
    COEND
SAT [{🍋y=0 ∧ (∑i=0..<n. 🍋C i)=0 }, {♂C=♀C ∧ ♂y=♀y}, {True}, {🍋y=n}]"
apply(rule Parallel)
apply force
apply force
apply(force)
apply clarify
apply simp
apply simp
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_all
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 \
⊨ 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,
{(🍋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. i≠j ⟶ ♀Y j ≤ ♂Y j) ∧ ♂X i = ♀X i ∧
   ♂Y i = ♀Y i},
{(∀j<n. i≠j ⟶ ♂X j = ♀X j ∧ ♂Y j = ♀Y j) ∧
   ♀Y i ≤ ♂Y i},
{(🍋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) })
COEND
SAT [{ ∀i<n. 🍋X i=i ∧ 🍋Y i=m+i },{♂X=♀X ∧ ♂Y=♀Y},{True},
  {∀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)}]"
apply(rule Parallel)
🍋 ‹5 subgoals left›
apply force+
apply clarify
apply simp
apply(rule While)
    apply force
   apply force
    apply force
  apply (erule dvdE)
apply(rule_tac pre'="\ \X i mod n = i \ (\j. j<\X i \ j mod n = i \ \P(B!j)) \ (\Y i < n * k \ P (B!(\Y i))) \ \X i<\Y i\" in Conseq)
     apply force
    apply(rule subset_refl)+
apply(rule Cond)
    apply force
   apply(rule Basic)
      apply force
     apply fastforce
    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 \ \ (COBEGIN SCHEME [0\i
(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,
{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)},
{(∀j<n. i≠j ⟶ ♀Y!j ≤ ♂Y!j) ∧ ♂X!i = ♀X!i ∧
   ♂Y!i = ♀Y!i ∧ length ♂X = length ♀X ∧ length ♂Y = length ♀Y},
{(∀j<n. i≠j ⟶ ♂X!j = ♀X!j ∧ ♂Y!j = ♀Y!j) ∧
   ♀Y!i ≤ ♂Y!i ∧ length ♂X = length ♀X ∧ length ♂Y = 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) ∧ (∃j<n. 🍋Y!j ≤ 🍋X!i) }) COEND)
SAT [{n<length 🍋X ∧ n<length 🍋Y ∧ (∀i<n. 🍋X!i=i ∧ 🍋Y!i=m+i) },
      {♂X=♀X ∧ ♂Y=♀Y},
      {True},
      {∀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)}]"
  apply (rule Parallel)
apply (auto cong del: image_cong_simp)
apply force
apply (rule While)
    apply force
   apply force
  apply force
  apply (erule dvdE)
apply(rule_tac pre'="\nX \ nY \ \X ! i mod n = i \ (\j. j < \X ! i \ j mod n = i \ \ P (B ! j)) \ (\Y ! i < n * k \ P (B ! (\Y ! i))) \ \X!i<\Y!i\" 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

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