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thys/turing_hoare.thy
changeset 99 fe7a257bdff4
parent 94 aeaf1374dc67
equal deleted inserted replaced
98:860f05037c36 99:fe7a257bdff4 
     7 
     7 
     8 definition 
     8 definition 
     9   assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
 
     9   assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
 
    10 where
 
    10 where
 
    11   "P \<mapsto> Q \<equiv> \<forall>l r. P (l, r) \<longrightarrow> Q (l, r)"
 
    11   "P \<mapsto> Q \<equiv> \<forall>l r. P (l, r) \<longrightarrow> Q (l, r)"
 
       
    12 
       
    13 lemma [intro, simp]:
 
       
    14   "P \<mapsto> P"
 
       
    15 unfolding assert_imp_def by simp
 
    12 
    16 
    13 fun 
    17 fun 
    14   holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)
 
    18   holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)
 
    15 where
 
    19 where
 
    16   "P holds_for (s, l, r) = P (l, r)"  
    20   "P holds_for (s, l, r) = P (l, r)"  
    55 
    59 
    56 
    60 
    57 
    61 
    58 
    62 
    59 text {*
 
    63 text {*
 
    60   {P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2   A well-formed
 
    64   {P} A {Q}   {Q} B {S}  A well-formed
 
    61   ---------------------------------------------------
 
    65   -----------------------------------------
 
    62   {P1} A |+| B {Q2}
 
    66   {P} A |+| B {S}
 
    63 *}
 
    67 *}
 
    64 
    68 
    65 
    69 
    66 lemma Hoare_plus_halt [case_names A_halt B_halt Imp A_wf]: 
    70 lemma Hoare_plus_halt [case_names A_halt B_halt A_wf]: 
    67   assumes A_halt : "{P1} A {Q1}"
 
    71   assumes A_halt : "{P} A {Q}"
 
    68   and B_halt : "{P2} B {Q2}"
 
    72   and B_halt : "{Q} B {S}"
 
    69   and a_imp: "Q1 \<mapsto> P2"
 
       
    70   and A_wf : "tm_wf (A, 0)"
 
    73   and A_wf : "tm_wf (A, 0)"
 
    71   shows "{P1} A |+| B {Q2}"
 
    74   shows "{P} A |+| B {S}"
 
    72 proof(rule Hoare_haltI)
 
    75 proof(rule Hoare_haltI)
 
    73   fix l r
 
    76   fix l r
 
    74   assume h: "P1 (l, r)"
 
    77   assume h: "P (l, r)"
 
    75   then obtain n1 l' r' 
    78   then obtain n1 l' r' 
    76     where "is_final (steps0 (1, l, r) A n1)"  
    79     where "is_final (steps0 (1, l, r) A n1)"  
    77       and a1: "Q1 holds_for (steps0 (1, l, r) A n1)"
 
    80       and a1: "Q holds_for (steps0 (1, l, r) A n1)"
 
    78       and a2: "steps0 (1, l, r) A n1 = (0, l', r')"
 
    81       and a2: "steps0 (1, l, r) A n1 = (0, l', r')"
 
    79     using A_halt unfolding Hoare_halt_def
 
    82     using A_halt unfolding Hoare_halt_def
 
    80     by (metis is_final_eq surj_pair) 
    83     by (metis is_final_eq surj_pair) 
    81   then obtain n2 
    84   then obtain n2 
    82     where "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
 
    85     where "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
 
    83     using A_wf by (rule_tac tm_comp_pre_halt_same) 
    86     using A_wf by (rule_tac tm_comp_pre_halt_same) 
    84   moreover
 
    87   moreover
 
    85   from a1 a2 a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
 
    88   from a1 a2 have "Q (l', r')" by (simp)
 
    86   then obtain n3 l'' r''
 
    89   then obtain n3 l'' r''
 
    87     where "is_final (steps0 (1, l', r') B n3)" 
    90     where "is_final (steps0 (1, l', r') B n3)" 
    88     and b1: "Q2 holds_for (steps0 (1, l', r') B n3)"
 
    91     and b1: "S holds_for (steps0 (1, l', r') B n3)"
 
    89     and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"
 
    92     and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"
 
    90     using B_halt unfolding Hoare_halt_def 
    93     using B_halt unfolding Hoare_halt_def 
    91     by (metis is_final_eq surj_pair) 
    94     by (metis is_final_eq surj_pair) 
    92   then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n3 = (0, l'', r'')"
 
    95   then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n3 = (0, l'', r'')"
 
    93     using A_wf by (rule_tac tm_comp_second_halt_same) 
    96     using A_wf by (rule_tac tm_comp_second_halt_same) 
    94   ultimately show 
    97   ultimately show 
    95     "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps0 (1, l, r) (A |+| B) n)"
 
    98     "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> S holds_for (steps0 (1, l, r) (A |+| B) n)"
 
    96     using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)
 
    99     using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)
 
    97 qed
 
   100 qed
 
    98 
   101 
    99 lemma Hoare_plus_halt_simple [case_names A_halt B_halt A_wf]: 
       
   100   assumes A_halt : "{P1} A {P2}"
 
       
   101   and B_halt : "{P2} B {P3}"
 
       
   102   and A_wf : "tm_wf (A, 0)"
 
       
   103   shows "{P1} A |+| B {P3}"
 
       
   104 by (rule Hoare_plus_halt[OF A_halt B_halt _ A_wf])
 
       
   105    (simp add: assert_imp_def)
 
       
   106 
       
   107 
       
   108 
       
   109 text {*
 
   102 text {*
 
   110   {P1} A {Q1}   {P2} B loops    Q1 \<mapsto> P2   A well-formed
 
   103   {P} A {Q}   {Q} B loops   A well-formed
 
   111   ------------------------------------------------------
 
   104   ------------------------------------------
 
   112           {P1} A |+| B  loops
 
   105           {P} A |+| B  loops
 
   113 *}
 
   106 *}
 
   114 
   107 
   115 lemma Hoare_plus_unhalt [case_names A_halt B_unhalt Imp A_wf]:
 
   108 lemma Hoare_plus_unhalt [case_names A_halt B_unhalt A_wf]:
 
   116   assumes A_halt: "{P1} A {Q1}"
 
   109   assumes A_halt: "{P} A {Q}"
 
   117   and B_uhalt: "{P2} B \<up>"
 
   110   and B_uhalt: "{Q} B \<up>"
 
   118   and a_imp: "Q1 \<mapsto> P2"
 
       
   119   and A_wf : "tm_wf (A, 0)"
 
   111   and A_wf : "tm_wf (A, 0)"
 
   120   shows "{P1} (A |+| B) \<up>"
 
   112   shows "{P} (A |+| B) \<up>"
 
   121 proof(rule_tac Hoare_unhaltI)
 
   113 proof(rule_tac Hoare_unhaltI)
 
   122   fix n l r 
   114   fix n l r 
   123   assume h: "P1 (l, r)"
 
   115   assume h: "P (l, r)"
 
   124   then obtain n1 l' r'
 
   116   then obtain n1 l' r'
 
   125     where a: "is_final (steps0 (1, l, r) A n1)" 
   117     where a: "is_final (steps0 (1, l, r) A n1)" 
   126     and b: "Q1 holds_for (steps0 (1, l, r) A n1)"
 
   118     and b: "Q holds_for (steps0 (1, l, r) A n1)"
 
   127     and c: "steps0 (1, l, r) A n1 = (0, l', r')"
 
   119     and c: "steps0 (1, l, r) A n1 = (0, l', r')"
 
   128     using A_halt unfolding Hoare_halt_def 
   120     using A_halt unfolding Hoare_halt_def 
   129     by (metis is_final_eq surj_pair) 
   121     by (metis is_final_eq surj_pair) 
   130   then obtain n2 where eq: "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
 
   122   then obtain n2 where eq: "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
 
   131     using A_wf by (rule_tac tm_comp_pre_halt_same)
 
   123     using A_wf by (rule_tac tm_comp_pre_halt_same)
 
   132   then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
 
   124   then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
 
   133   proof(cases "n2 \<le> n")
 
   125   proof(cases "n2 \<le> n")
 
   134     case True
 
   126     case True
 
   135     from b c a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
 
   127     from b c have "Q (l', r')" by simp
 
   136     then have "\<forall> n. \<not> is_final (steps0 (1, l', r') B n)  "
 
   128     then have "\<forall> n. \<not> is_final (steps0 (1, l', r') B n)  "
 
   137       using B_uhalt unfolding Hoare_unhalt_def by simp
 
   129       using B_uhalt unfolding Hoare_unhalt_def by simp
 
   138     then have "\<not> is_final (steps0 (Suc 0, l', r') B (n - n2))" by auto
 
   130     then have "\<not> is_final (steps0 (1, l', r') B (n - n2))" by auto
 
   139     then obtain s'' l'' r'' 
   131     then obtain s'' l'' r'' 
   140       where "steps0 (Suc 0, l', r') B (n - n2) = (s'', l'', r'')" 
   132       where "steps0 (1, l', r') B (n - n2) = (s'', l'', r'')" 
   141       and "\<not> is_final (s'', l'', r'')" by (metis surj_pair)
 
   133       and "\<not> is_final (s'', l'', r'')" by (metis surj_pair)
 
   142     then have "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - n2) = (s''+ length A div 2, l'', r'')"
 
   134     then have "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - n2) = (s''+ length A div 2, l'', r'')"
 
   143       using A_wf by (auto dest: tm_comp_second_same simp del: tm_wf.simps)
 
   135       using A_wf by (auto dest: tm_comp_second_same simp del: tm_wf.simps)
 
   144     then have "\<not> is_final (steps0 (1, l, r) (A |+| B) (n2 + (n  - n2)))"
 
   136     then have "\<not> is_final (steps0 (1, l, r) (A |+| B) (n2 + (n  - n2)))"
 
   145       using A_wf by (simp only: steps_add eq) (simp add: tm_wf.simps)
 
   137       using A_wf by (simp only: steps_add eq) (simp add: tm_wf.simps)
 
   153     with eq show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
 
   145     with eq show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
 
   154       by (simp add: not_is_final[where ?n1.0="n2"])
 
   146       by (simp add: not_is_final[where ?n1.0="n2"])
 
   155   qed
 
   147   qed
 
   156 qed
 
   148 qed
 
   157 
   149 
   158 lemma Hoare_plus_unhalt_simple [case_names A_halt B_unhalt A_wf]: 
   150 lemma Hoare_consequence:
 
   159  assumes A_halt: "{P1} A {P2}"
 
   151   assumes "P' \<mapsto> P" "{P} p {Q}" "Q \<mapsto> Q'"
 
   160   and B_uhalt: "{P2} B \<up>"
 
       
   161   and A_wf : "tm_wf (A, 0)"
 
       
   162   shows "{P1} (A |+| B) \<up>"
 
       
   163 by (rule Hoare_plus_unhalt[OF A_halt B_uhalt _ A_wf])
 
       
   164    (simp add: assert_imp_def)
 
       
   165 
       
   166 
       
   167 lemma Hoare_weaken:
 
       
   168   assumes a: "{P} p {Q}"
 
       
   169   and b: "P' \<mapsto> P" 
       
   170   and c: "Q \<mapsto> Q'"
 
       
   171   shows "{P'} p {Q'}"
 
   152   shows "{P'} p {Q'}"
 
   172 using assms
 
   153 using assms
 
   173 unfolding Hoare_halt_def assert_imp_def
 
   154 unfolding Hoare_halt_def assert_imp_def
 
   174 by (metis holds_for.simps surj_pair)
 
   155 by (metis holds_for.simps surj_pair)
 
   175 
   156
tm