src/HOL/ZF/Games.thy
author haftmann
Tue Nov 30 17:19:11 2010 +0100 (2010-11-30)
changeset 40824 f5a0cb45d2a5
parent 39910 10097e0a9dbd
child 41528 276078f01ada
permissions -rw-r--r--
adapted fragile proof
krauss@35440
     1
(*  Title:      HOL/ZF/MainZF.thy/Games.thy
obua@19203
     2
    Author:     Steven Obua
obua@19203
     3
wenzelm@32960
     4
An application of HOLZF: Partizan Games.  See "Partizan Games in
wenzelm@32960
     5
Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
obua@19203
     6
*)
obua@19203
     7
obua@19203
     8
theory Games 
obua@19203
     9
imports MainZF
obua@19203
    10
begin
obua@19203
    11
haftmann@35416
    12
definition fixgames :: "ZF set \<Rightarrow> ZF set" where
obua@19203
    13
  "fixgames A \<equiv> { Opair l r | l r. explode l \<subseteq> A & explode r \<subseteq> A}"
haftmann@35416
    14
haftmann@35416
    15
definition games_lfp :: "ZF set" where
obua@19203
    16
  "games_lfp \<equiv> lfp fixgames"
haftmann@35416
    17
haftmann@35416
    18
definition games_gfp :: "ZF set" where
obua@19203
    19
  "games_gfp \<equiv> gfp fixgames"
obua@19203
    20
obua@19203
    21
lemma mono_fixgames: "mono (fixgames)"
obua@19203
    22
  apply (auto simp add: mono_def fixgames_def)
obua@19203
    23
  apply (rule_tac x=l in exI)
obua@19203
    24
  apply (rule_tac x=r in exI)
obua@19203
    25
  apply auto
obua@19203
    26
  done
obua@19203
    27
obua@19203
    28
lemma games_lfp_unfold: "games_lfp = fixgames games_lfp"
obua@19203
    29
  by (auto simp add: def_lfp_unfold games_lfp_def mono_fixgames)
obua@19203
    30
obua@19203
    31
lemma games_gfp_unfold: "games_gfp = fixgames games_gfp"
obua@19203
    32
  by (auto simp add: def_gfp_unfold games_gfp_def mono_fixgames)
obua@19203
    33
obua@19203
    34
lemma games_lfp_nonempty: "Opair Empty Empty \<in> games_lfp"
obua@19203
    35
proof -
obua@19203
    36
  have "fixgames {} \<subseteq> games_lfp" 
obua@19203
    37
    apply (subst games_lfp_unfold)
obua@19203
    38
    apply (simp add: mono_fixgames[simplified mono_def, rule_format])
obua@19203
    39
    done
obua@19203
    40
  moreover have "fixgames {} = {Opair Empty Empty}"
obua@19203
    41
    by (simp add: fixgames_def explode_Empty)
obua@19203
    42
  finally show ?thesis
obua@19203
    43
    by auto
obua@19203
    44
qed
obua@19203
    45
haftmann@35416
    46
definition left_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" where
obua@19203
    47
  "left_option g opt \<equiv> (Elem opt (Fst g))"
haftmann@35416
    48
haftmann@35416
    49
definition right_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" where
obua@19203
    50
  "right_option g opt \<equiv> (Elem opt (Snd g))"
haftmann@35416
    51
haftmann@35416
    52
definition is_option_of :: "(ZF * ZF) set" where
obua@19203
    53
  "is_option_of \<equiv> { (opt, g) | opt g. g \<in> games_gfp \<and> (left_option g opt \<or> right_option g opt) }"
obua@19203
    54
obua@19203
    55
lemma games_lfp_subset_gfp: "games_lfp \<subseteq> games_gfp"
obua@19203
    56
proof -
obua@19203
    57
  have "games_lfp \<subseteq> fixgames games_lfp" 
obua@19203
    58
    by (simp add: games_lfp_unfold[symmetric])
obua@19203
    59
  then show ?thesis
obua@19203
    60
    by (simp add: games_gfp_def gfp_upperbound)
obua@19203
    61
qed
obua@19203
    62
obua@19203
    63
lemma games_option_stable: 
obua@19203
    64
  assumes fixgames: "games = fixgames games"
obua@19203
    65
  and g: "g \<in> games"
obua@19203
    66
  and opt: "left_option g opt \<or> right_option g opt"
obua@19203
    67
  shows "opt \<in> games"
obua@19203
    68
proof -
obua@19203
    69
  from g fixgames have "g \<in> fixgames games" by auto
obua@19203
    70
  then have "\<exists> l r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games"
obua@19203
    71
    by (simp add: fixgames_def)
obua@19203
    72
  then obtain l where "\<exists> r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
obua@19203
    73
  then obtain r where lr: "g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
obua@19203
    74
  with opt show ?thesis
obua@19203
    75
    by (auto intro: Elem_explode_in simp add: left_option_def right_option_def Fst Snd)
obua@19203
    76
qed
obua@19203
    77
    
obua@19203
    78
lemma option2elem: "(opt,g) \<in> is_option_of  \<Longrightarrow> \<exists> u v. Elem opt u \<and> Elem u v \<and> Elem v g"
obua@19203
    79
  apply (simp add: is_option_of_def)
obua@19203
    80
  apply (subgoal_tac "(g \<in> games_gfp) = (g \<in> (fixgames games_gfp))")
obua@19203
    81
  prefer 2
obua@19203
    82
  apply (simp add: games_gfp_unfold[symmetric])
obua@19203
    83
  apply (auto simp add: fixgames_def left_option_def right_option_def Fst Snd)
obua@19203
    84
  apply (rule_tac x=l in exI, insert Elem_Opair_exists, blast)
obua@19203
    85
  apply (rule_tac x=r in exI, insert Elem_Opair_exists, blast) 
obua@19203
    86
  done
obua@19203
    87
obua@19203
    88
lemma is_option_of_subset_is_Elem_of: "is_option_of \<subseteq> (is_Elem_of^+)"
obua@19203
    89
proof -
obua@19203
    90
  {
obua@19203
    91
    fix opt
obua@19203
    92
    fix g
obua@19203
    93
    assume "(opt, g) \<in> is_option_of"
obua@19203
    94
    then have "\<exists> u v. (opt, u) \<in> (is_Elem_of^+) \<and> (u,v) \<in> (is_Elem_of^+) \<and> (v,g) \<in> (is_Elem_of^+)" 
obua@19203
    95
      apply -
obua@19203
    96
      apply (drule option2elem)
obua@19203
    97
      apply (auto simp add: r_into_trancl' is_Elem_of_def)
obua@19203
    98
      done
obua@19203
    99
    then have "(opt, g) \<in> (is_Elem_of^+)"
obua@19203
   100
      by (blast intro: trancl_into_rtrancl trancl_rtrancl_trancl)
obua@19203
   101
  } 
obua@19203
   102
  then show ?thesis by auto
obua@19203
   103
qed
obua@19203
   104
obua@19203
   105
lemma wfzf_is_option_of: "wfzf is_option_of"
obua@19203
   106
proof - 
obua@19203
   107
  have "wfzf (is_Elem_of^+)" by (simp add: wfzf_trancl wfzf_is_Elem_of)
obua@19203
   108
  then show ?thesis 
obua@19203
   109
    apply (rule wfzf_subset)
obua@19203
   110
    apply (rule is_option_of_subset_is_Elem_of)
obua@19203
   111
    done
obua@19203
   112
  qed
obua@19203
   113
  
obua@19203
   114
lemma games_gfp_imp_lfp: "g \<in> games_gfp \<longrightarrow> g \<in> games_lfp"
obua@19203
   115
proof -
obua@19203
   116
  have unfold_gfp: "\<And> x. x \<in> games_gfp \<Longrightarrow> x \<in> (fixgames games_gfp)" 
obua@19203
   117
    by (simp add: games_gfp_unfold[symmetric])
obua@19203
   118
  have unfold_lfp: "\<And> x. (x \<in> games_lfp) = (x \<in> (fixgames games_lfp))"
obua@19203
   119
    by (simp add: games_lfp_unfold[symmetric])
obua@19203
   120
  show ?thesis
obua@19203
   121
    apply (rule wf_induct[OF wfzf_implies_wf[OF wfzf_is_option_of]])
obua@19203
   122
    apply (auto simp add: is_option_of_def)
obua@19203
   123
    apply (drule_tac unfold_gfp)
obua@19203
   124
    apply (simp add: fixgames_def)
obua@19203
   125
    apply (auto simp add: left_option_def Fst right_option_def Snd)
obua@19203
   126
    apply (subgoal_tac "explode l \<subseteq> games_lfp")
obua@19203
   127
    apply (subgoal_tac "explode r \<subseteq> games_lfp")
obua@19203
   128
    apply (subst unfold_lfp)
obua@19203
   129
    apply (auto simp add: fixgames_def)
obua@19203
   130
    apply (simp_all add: explode_Elem Elem_explode_in)
obua@19203
   131
    done
obua@19203
   132
qed
obua@19203
   133
obua@19203
   134
theorem games_lfp_eq_gfp: "games_lfp = games_gfp"
obua@19203
   135
  apply (auto simp add: games_gfp_imp_lfp)
obua@19203
   136
  apply (insert games_lfp_subset_gfp)
obua@19203
   137
  apply auto
obua@19203
   138
  done
obua@19203
   139
obua@19203
   140
theorem unique_games: "(g = fixgames g) = (g = games_lfp)"
obua@19203
   141
proof -
obua@19203
   142
  {
obua@19203
   143
    fix g 
obua@19203
   144
    assume g: "g = fixgames g"
obua@19203
   145
    from g have "fixgames g \<subseteq> g" by auto
obua@19203
   146
    then have l:"games_lfp \<subseteq> g" 
obua@19203
   147
      by (simp add: games_lfp_def lfp_lowerbound)
obua@19203
   148
    from g have "g \<subseteq> fixgames g" by auto
obua@19203
   149
    then have u:"g \<subseteq> games_gfp" 
obua@19203
   150
      by (simp add: games_gfp_def gfp_upperbound)
obua@19203
   151
    from l u games_lfp_eq_gfp[symmetric] have "g = games_lfp"
obua@19203
   152
      by auto
obua@19203
   153
  }
obua@19203
   154
  note games = this
obua@19203
   155
  show ?thesis
obua@19203
   156
    apply (rule iff[rule_format])
obua@19203
   157
    apply (erule games)
obua@19203
   158
    apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   159
    done
obua@19203
   160
qed
obua@19203
   161
obua@19203
   162
lemma games_lfp_option_stable: 
obua@19203
   163
  assumes g: "g \<in> games_lfp"
obua@19203
   164
  and opt: "left_option g opt \<or> right_option g opt"
obua@19203
   165
  shows "opt \<in> games_lfp"
obua@19203
   166
  apply (rule games_option_stable[where g=g])
obua@19203
   167
  apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   168
  apply (simp_all add: prems)
obua@19203
   169
  done
obua@19203
   170
obua@19203
   171
lemma is_option_of_imp_games:
obua@19203
   172
  assumes hyp: "(opt, g) \<in> is_option_of"
obua@19203
   173
  shows "opt \<in> games_lfp \<and> g \<in> games_lfp"
obua@19203
   174
proof -
obua@19203
   175
  from hyp have g_game: "g \<in> games_lfp" 
obua@19203
   176
    by (simp add: is_option_of_def games_lfp_eq_gfp)
obua@19203
   177
  from hyp have "left_option g opt \<or> right_option g opt"
obua@19203
   178
    by (auto simp add: is_option_of_def)
obua@19203
   179
  with g_game games_lfp_option_stable[OF g_game, OF this] show ?thesis
obua@19203
   180
    by auto
obua@19203
   181
qed
obua@19203
   182
 
obua@19203
   183
lemma games_lfp_represent: "x \<in> games_lfp \<Longrightarrow> \<exists> l r. x = Opair l r"
obua@19203
   184
  apply (rule exI[where x="Fst x"])
obua@19203
   185
  apply (rule exI[where x="Snd x"])
obua@19203
   186
  apply (subgoal_tac "x \<in> (fixgames games_lfp)")
obua@19203
   187
  apply (simp add: fixgames_def)
obua@19203
   188
  apply (auto simp add: Fst Snd)
obua@19203
   189
  apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   190
  done
obua@19203
   191
obua@19203
   192
typedef game = games_lfp
obua@19203
   193
  by (blast intro: games_lfp_nonempty)
obua@19203
   194
haftmann@35416
   195
definition left_options :: "game \<Rightarrow> game zet" where
obua@19203
   196
  "left_options g \<equiv> zimage Abs_game (zexplode (Fst (Rep_game g)))"
haftmann@35416
   197
haftmann@35416
   198
definition right_options :: "game \<Rightarrow> game zet" where
obua@19203
   199
  "right_options g \<equiv> zimage Abs_game (zexplode (Snd (Rep_game g)))"
haftmann@35416
   200
haftmann@35416
   201
definition options :: "game \<Rightarrow> game zet" where
obua@19203
   202
  "options g \<equiv> zunion (left_options g) (right_options g)"
haftmann@35416
   203
haftmann@35416
   204
definition Game :: "game zet \<Rightarrow> game zet \<Rightarrow> game" where
obua@19203
   205
  "Game L R \<equiv> Abs_game (Opair (zimplode (zimage Rep_game L)) (zimplode (zimage Rep_game R)))"
obua@19203
   206
  
obua@19203
   207
lemma Repl_Rep_game_Abs_game: "\<forall> e. Elem e z \<longrightarrow> e \<in> games_lfp \<Longrightarrow> Repl z (Rep_game o Abs_game) = z"
obua@19203
   208
  apply (subst Ext)
obua@19203
   209
  apply (simp add: Repl)
obua@19203
   210
  apply auto
obua@19203
   211
  apply (subst Abs_game_inverse, simp_all add: game_def)
obua@19203
   212
  apply (rule_tac x=za in exI)
obua@19203
   213
  apply (subst Abs_game_inverse, simp_all add: game_def)
obua@19203
   214
  done
obua@19203
   215
obua@19203
   216
lemma game_split: "g = Game (left_options g) (right_options g)"
obua@19203
   217
proof -
obua@19203
   218
  have "\<exists> l r. Rep_game g = Opair l r"
obua@19203
   219
    apply (insert Rep_game[of g])
obua@19203
   220
    apply (simp add: game_def games_lfp_represent)
obua@19203
   221
    done
obua@19203
   222
  then obtain l r where lr: "Rep_game g = Opair l r" by auto
obua@19203
   223
  have partizan_g: "Rep_game g \<in> games_lfp" 
obua@19203
   224
    apply (insert Rep_game[of g])
obua@19203
   225
    apply (simp add: game_def)
obua@19203
   226
    done
obua@19203
   227
  have "\<forall> e. Elem e l \<longrightarrow> left_option (Rep_game g) e"
obua@19203
   228
    by (simp add: lr left_option_def Fst)
obua@19203
   229
  then have partizan_l: "\<forall> e. Elem e l \<longrightarrow> e \<in> games_lfp"
obua@19203
   230
    apply auto
obua@19203
   231
    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
obua@19203
   232
    apply auto
obua@19203
   233
    done
obua@19203
   234
  have "\<forall> e. Elem e r \<longrightarrow> right_option (Rep_game g) e"
obua@19203
   235
    by (simp add: lr right_option_def Snd)
obua@19203
   236
  then have partizan_r: "\<forall> e. Elem e r \<longrightarrow> e \<in> games_lfp"
obua@19203
   237
    apply auto
obua@19203
   238
    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
obua@19203
   239
    apply auto
obua@19203
   240
    done   
obua@19203
   241
  let ?L = "zimage (Abs_game) (zexplode l)"
obua@19203
   242
  let ?R = "zimage (Abs_game) (zexplode r)"
obua@19203
   243
  have L:"?L = left_options g"
obua@19203
   244
    by (simp add: left_options_def lr Fst)
obua@19203
   245
  have R:"?R = right_options g"
obua@19203
   246
    by (simp add: right_options_def lr Snd)
obua@19203
   247
  have "g = Game ?L ?R"
obua@19203
   248
    apply (simp add: Game_def Rep_game_inject[symmetric] comp_zimage_eq zimage_zexplode_eq zimplode_zexplode)
obua@19203
   249
    apply (simp add: Repl_Rep_game_Abs_game partizan_l partizan_r)
obua@19203
   250
    apply (subst Abs_game_inverse)
obua@19203
   251
    apply (simp_all add: lr[symmetric] Rep_game) 
obua@19203
   252
    done
obua@19203
   253
  then show ?thesis
obua@19203
   254
    by (simp add: L R)
obua@19203
   255
qed
obua@19203
   256
obua@19203
   257
lemma Opair_in_games_lfp: 
obua@19203
   258
  assumes l: "explode l \<subseteq> games_lfp"
obua@19203
   259
  and r: "explode r \<subseteq> games_lfp"
obua@19203
   260
  shows "Opair l r \<in> games_lfp"
obua@19203
   261
proof -
obua@19203
   262
  note f = unique_games[of games_lfp, simplified]
obua@19203
   263
  show ?thesis
obua@19203
   264
    apply (subst f)
obua@19203
   265
    apply (simp add: fixgames_def)
obua@19203
   266
    apply (rule exI[where x=l])
obua@19203
   267
    apply (rule exI[where x=r])
obua@19203
   268
    apply (auto simp add: l r)
obua@19203
   269
    done
obua@19203
   270
qed
obua@19203
   271
obua@19203
   272
lemma left_options[simp]: "left_options (Game l r) = l"
obua@19203
   273
  apply (simp add: left_options_def Game_def)
obua@19203
   274
  apply (subst Abs_game_inverse)
obua@19203
   275
  apply (simp add: game_def)
obua@19203
   276
  apply (rule Opair_in_games_lfp)
obua@19203
   277
  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
obua@19203
   278
  apply (simp add: Fst zexplode_zimplode comp_zimage_eq)
obua@19203
   279
  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
obua@19203
   280
  done
obua@19203
   281
obua@19203
   282
lemma right_options[simp]: "right_options (Game l r) = r"
obua@19203
   283
  apply (simp add: right_options_def Game_def)
obua@19203
   284
  apply (subst Abs_game_inverse)
obua@19203
   285
  apply (simp add: game_def)
obua@19203
   286
  apply (rule Opair_in_games_lfp)
obua@19203
   287
  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
obua@19203
   288
  apply (simp add: Snd zexplode_zimplode comp_zimage_eq)
obua@19203
   289
  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
obua@19203
   290
  done  
obua@19203
   291
obua@19203
   292
lemma Game_ext: "(Game l1 r1 = Game l2 r2) = ((l1 = l2) \<and> (r1 = r2))"
obua@19203
   293
  apply auto
obua@19203
   294
  apply (subst left_options[where l=l1 and r=r1,symmetric])
obua@19203
   295
  apply (subst left_options[where l=l2 and r=r2,symmetric])
obua@19203
   296
  apply simp
obua@19203
   297
  apply (subst right_options[where l=l1 and r=r1,symmetric])
obua@19203
   298
  apply (subst right_options[where l=l2 and r=r2,symmetric])
obua@19203
   299
  apply simp
obua@19203
   300
  done
obua@19203
   301
haftmann@35416
   302
definition option_of :: "(game * game) set" where
obua@19203
   303
  "option_of \<equiv> image (\<lambda> (option, g). (Abs_game option, Abs_game g)) is_option_of"
obua@19203
   304
obua@19203
   305
lemma option_to_is_option_of: "((option, g) \<in> option_of) = ((Rep_game option, Rep_game g) \<in> is_option_of)"
obua@19203
   306
  apply (auto simp add: option_of_def)
obua@19203
   307
  apply (subst Abs_game_inverse)
obua@19203
   308
  apply (simp add: is_option_of_imp_games game_def)
obua@19203
   309
  apply (subst Abs_game_inverse)
obua@19203
   310
  apply (simp add: is_option_of_imp_games game_def)
obua@19203
   311
  apply simp
obua@19203
   312
  apply (auto simp add: Bex_def image_def)  
obua@19203
   313
  apply (rule exI[where x="Rep_game option"])
obua@19203
   314
  apply (rule exI[where x="Rep_game g"])
obua@19203
   315
  apply (simp add: Rep_game_inverse)
obua@19203
   316
  done
obua@19203
   317
  
obua@19203
   318
lemma wf_is_option_of: "wf is_option_of"
obua@19203
   319
  apply (rule wfzf_implies_wf)
obua@19203
   320
  apply (simp add: wfzf_is_option_of)
obua@19203
   321
  done
obua@19203
   322
obua@19203
   323
lemma wf_option_of[recdef_wf, simp, intro]: "wf option_of"
obua@19203
   324
proof -
obua@19203
   325
  have option_of: "option_of = inv_image is_option_of Rep_game"
nipkow@39302
   326
    apply (rule set_eqI)
obua@19203
   327
    apply (case_tac "x")
krauss@19769
   328
    by (simp add: option_to_is_option_of) 
obua@19203
   329
  show ?thesis
obua@19203
   330
    apply (simp add: option_of)
obua@19203
   331
    apply (auto intro: wf_inv_image wf_is_option_of)
obua@19203
   332
    done
obua@19203
   333
qed
obua@19203
   334
  
obua@19203
   335
lemma right_option_is_option[simp, intro]: "zin x (right_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   336
  by (simp add: options_def zunion)
obua@19203
   337
obua@19203
   338
lemma left_option_is_option[simp, intro]: "zin x (left_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   339
  by (simp add: options_def zunion)
obua@19203
   340
obua@19203
   341
lemma zin_options[simp, intro]: "zin x (options g) \<Longrightarrow> (x, g) \<in> option_of"
obua@19203
   342
  apply (simp add: options_def zunion left_options_def right_options_def option_of_def 
obua@19203
   343
    image_def is_option_of_def zimage_iff zin_zexplode_eq) 
obua@19203
   344
  apply (cases g)
obua@19203
   345
  apply (cases x)
obua@19203
   346
  apply (auto simp add: Abs_game_inverse games_lfp_eq_gfp[symmetric] game_def 
obua@19203
   347
    right_option_def[symmetric] left_option_def[symmetric])
obua@19203
   348
  done
obua@19203
   349
krauss@35440
   350
function
obua@19203
   351
  neg_game :: "game \<Rightarrow> game"
krauss@35440
   352
where
krauss@35440
   353
  [simp del]: "neg_game g = Game (zimage neg_game (right_options g)) (zimage neg_game (left_options g))"
krauss@35440
   354
by auto
krauss@35440
   355
termination by (relation "option_of") auto
obua@19203
   356
obua@19203
   357
lemma "neg_game (neg_game g) = g"
obua@19203
   358
  apply (induct g rule: neg_game.induct)
obua@19203
   359
  apply (subst neg_game.simps)+
obua@19203
   360
  apply (simp add: right_options left_options comp_zimage_eq)
obua@19203
   361
  apply (subgoal_tac "zimage (neg_game o neg_game) (left_options g) = left_options g")
obua@19203
   362
  apply (subgoal_tac "zimage (neg_game o neg_game) (right_options g) = right_options g")
obua@19203
   363
  apply (auto simp add: game_split[symmetric])
obua@19203
   364
  apply (auto simp add: zet_ext_eq zimage_iff)
obua@19203
   365
  done
obua@19203
   366
krauss@35440
   367
function
obua@19203
   368
  ge_game :: "(game * game) \<Rightarrow> bool" 
krauss@35440
   369
where
krauss@35440
   370
  [simp del]: "ge_game (G, H) = (\<forall> x. if zin x (right_options G) then (
obua@19203
   371
                          if zin x (left_options H) then \<not> (ge_game (H, x) \<or> (ge_game (x, G))) 
obua@19203
   372
                                                    else \<not> (ge_game (H, x)))
obua@19203
   373
                          else (if zin x (left_options H) then \<not> (ge_game (x, G)) else True))"
krauss@35440
   374
by auto
krauss@35440
   375
termination by (relation "(gprod_2_1 option_of)") 
krauss@35440
   376
 (simp, auto simp: gprod_2_1_def)
obua@19203
   377
wenzelm@26304
   378
lemma ge_game_eq: "ge_game (G, H) = (\<forall> x. (zin x (right_options G) \<longrightarrow> \<not> ge_game (H, x)) \<and> (zin x (left_options H) \<longrightarrow> \<not> ge_game (x, G)))"
obua@19203
   379
  apply (subst ge_game.simps[where G=G and H=H])
obua@19203
   380
  apply (auto)
obua@19203
   381
  done
obua@19203
   382
obua@19203
   383
lemma ge_game_leftright_refl[rule_format]: 
obua@19203
   384
  "\<forall> y. (zin y (right_options x) \<longrightarrow> \<not> ge_game (x, y)) \<and> (zin y (left_options x) \<longrightarrow> \<not> (ge_game (y, x))) \<and> ge_game (x, x)"
obua@19203
   385
proof (induct x rule: wf_induct[OF wf_option_of]) 
obua@19203
   386
  case (1 "g")
obua@19203
   387
  { 
obua@19203
   388
    fix y
obua@19203
   389
    assume y: "zin y (right_options g)"
obua@19203
   390
    have "\<not> ge_game (g, y)"
obua@19203
   391
    proof -
obua@19203
   392
      have "(y, g) \<in> option_of" by (auto intro: y)
obua@19203
   393
      with 1 have "ge_game (y, y)" by auto
wenzelm@26304
   394
      with y show ?thesis by (subst ge_game_eq, auto)
obua@19203
   395
    qed
obua@19203
   396
  }
obua@19203
   397
  note right = this
obua@19203
   398
  { 
obua@19203
   399
    fix y
obua@19203
   400
    assume y: "zin y (left_options g)"
obua@19203
   401
    have "\<not> ge_game (y, g)"
obua@19203
   402
    proof -
obua@19203
   403
      have "(y, g) \<in> option_of" by (auto intro: y)
obua@19203
   404
      with 1 have "ge_game (y, y)" by auto
wenzelm@26304
   405
      with y show ?thesis by (subst ge_game_eq, auto)
obua@19203
   406
    qed
obua@19203
   407
  } 
obua@19203
   408
  note left = this
obua@19203
   409
  from left right show ?case
wenzelm@26304
   410
    by (auto, subst ge_game_eq, auto)
obua@19203
   411
qed
obua@19203
   412
obua@19203
   413
lemma ge_game_refl: "ge_game (x,x)" by (simp add: ge_game_leftright_refl)
obua@19203
   414
obua@19203
   415
lemma "\<forall> y. (zin y (right_options x) \<longrightarrow> \<not> ge_game (x, y)) \<and> (zin y (left_options x) \<longrightarrow> \<not> (ge_game (y, x))) \<and> ge_game (x, x)"
obua@19203
   416
proof (induct x rule: wf_induct[OF wf_option_of]) 
obua@19203
   417
  case (1 "g")  
obua@19203
   418
  show ?case
obua@19203
   419
  proof (auto)
obua@19203
   420
    {case (goal1 y) 
obua@19203
   421
      from goal1 have "(y, g) \<in> option_of" by (auto)
obua@19203
   422
      with 1 have "ge_game (y, y)" by auto
obua@19203
   423
      with goal1 have "\<not> ge_game (g, y)" 
wenzelm@32960
   424
        by (subst ge_game_eq, auto)
obua@19203
   425
      with goal1 show ?case by auto}
obua@19203
   426
    note right = this
obua@19203
   427
    {case (goal2 y)
obua@19203
   428
      from goal2 have "(y, g) \<in> option_of" by (auto)
obua@19203
   429
      with 1 have "ge_game (y, y)" by auto
obua@19203
   430
      with goal2 have "\<not> ge_game (y, g)" 
wenzelm@32960
   431
        by (subst ge_game_eq, auto)
obua@19203
   432
      with goal2 show ?case by auto}
obua@19203
   433
    note left = this
obua@19203
   434
    {case goal3
obua@19203
   435
      from left right show ?case
wenzelm@32960
   436
        by (subst ge_game_eq, auto)
obua@19203
   437
    }
obua@19203
   438
  qed
obua@19203
   439
qed
wenzelm@32960
   440
        
haftmann@35416
   441
definition eq_game :: "game \<Rightarrow> game \<Rightarrow> bool" where
obua@19203
   442
  "eq_game G H \<equiv> ge_game (G, H) \<and> ge_game (H, G)" 
obua@19203
   443
obua@19203
   444
lemma eq_game_sym: "(eq_game G H) = (eq_game H G)"
obua@19203
   445
  by (auto simp add: eq_game_def)
obua@19203
   446
obua@19203
   447
lemma eq_game_refl: "eq_game G G"
obua@19203
   448
  by (simp add: ge_game_refl eq_game_def)
obua@19203
   449
berghofe@23771
   450
lemma induct_game: "(\<And>x. \<forall>y. (y, x) \<in> lprod option_of \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
berghofe@23771
   451
  by (erule wf_induct[OF wf_lprod[OF wf_option_of]])
obua@19203
   452
obua@19203
   453
lemma ge_game_trans:
obua@19203
   454
  assumes "ge_game (x, y)" "ge_game (y, z)" 
obua@19203
   455
  shows "ge_game (x, z)"
obua@19203
   456
proof -  
obua@19203
   457
  { 
obua@19203
   458
    fix a
obua@19203
   459
    have "\<forall> x y z. a = [x,y,z] \<longrightarrow> ge_game (x,y) \<longrightarrow> ge_game (y,z) \<longrightarrow> ge_game (x, z)"
obua@19203
   460
    proof (induct a rule: induct_game)
obua@19203
   461
      case (1 a)
obua@19203
   462
      show ?case
obua@19203
   463
      proof (rule allI | rule impI)+
wenzelm@32960
   464
        case (goal1 x y z)
wenzelm@32960
   465
        show ?case
wenzelm@32960
   466
        proof -
wenzelm@32960
   467
          { fix xr
obua@19203
   468
            assume xr:"zin xr (right_options x)"
wenzelm@32960
   469
            assume "ge_game (z, xr)"
wenzelm@32960
   470
            have "ge_game (y, xr)"
wenzelm@32960
   471
              apply (rule 1[rule_format, where y="[y,z,xr]"])
wenzelm@32960
   472
              apply (auto intro: xr lprod_3_1 simp add: prems)
wenzelm@32960
   473
              done
wenzelm@32960
   474
            moreover from xr have "\<not> ge_game (y, xr)"
wenzelm@32960
   475
              by (simp add: goal1(2)[simplified ge_game_eq[of x y], rule_format, of xr, simplified xr])
wenzelm@32960
   476
            ultimately have "False" by auto      
wenzelm@32960
   477
          }
wenzelm@32960
   478
          note xr = this
wenzelm@32960
   479
          { fix zl
wenzelm@32960
   480
            assume zl:"zin zl (left_options z)"
wenzelm@32960
   481
            assume "ge_game (zl, x)"
wenzelm@32960
   482
            have "ge_game (zl, y)"
wenzelm@32960
   483
              apply (rule 1[rule_format, where y="[zl,x,y]"])
wenzelm@32960
   484
              apply (auto intro: zl lprod_3_2 simp add: prems)
wenzelm@32960
   485
              done
wenzelm@32960
   486
            moreover from zl have "\<not> ge_game (zl, y)"
wenzelm@32960
   487
              by (simp add: goal1(3)[simplified ge_game_eq[of y z], rule_format, of zl, simplified zl])
wenzelm@32960
   488
            ultimately have "False" by auto
wenzelm@32960
   489
          }
wenzelm@32960
   490
          note zl = this
wenzelm@32960
   491
          show ?thesis
wenzelm@32960
   492
            by (auto simp add: ge_game_eq[of x z] intro: xr zl)
wenzelm@32960
   493
        qed
obua@19203
   494
      qed
obua@19203
   495
    qed
obua@19203
   496
  } 
obua@19203
   497
  note trans = this[of "[x, y, z]", simplified, rule_format]    
obua@19203
   498
  with prems show ?thesis by blast
obua@19203
   499
qed
obua@19203
   500
obua@19203
   501
lemma eq_game_trans: "eq_game a b \<Longrightarrow> eq_game b c \<Longrightarrow> eq_game a c"
obua@19203
   502
  by (auto simp add: eq_game_def intro: ge_game_trans)
obua@19203
   503
haftmann@35416
   504
definition zero_game :: game
haftmann@35416
   505
 where  "zero_game \<equiv> Game zempty zempty"
obua@19203
   506
krauss@35440
   507
function 
krauss@35440
   508
  plus_game :: "game \<Rightarrow> game \<Rightarrow> game"
krauss@35440
   509
where
krauss@35440
   510
  [simp del]: "plus_game G H = Game (zunion (zimage (\<lambda> g. plus_game g H) (left_options G))
krauss@35440
   511
                                   (zimage (\<lambda> h. plus_game G h) (left_options H)))
krauss@35440
   512
                           (zunion (zimage (\<lambda> g. plus_game g H) (right_options G))
krauss@35440
   513
                                   (zimage (\<lambda> h. plus_game G h) (right_options H)))"
krauss@35440
   514
by auto
krauss@35440
   515
termination by (relation "gprod_2_2 option_of")
krauss@35440
   516
  (simp, auto simp: gprod_2_2_def)
obua@19203
   517
krauss@35440
   518
lemma plus_game_comm: "plus_game G H = plus_game H G"
obua@19203
   519
proof (induct G H rule: plus_game.induct)
obua@19203
   520
  case (1 G H)
obua@19203
   521
  show ?case
obua@19203
   522
    by (auto simp add: 
obua@19203
   523
      plus_game.simps[where G=G and H=H] 
obua@19203
   524
      plus_game.simps[where G=H and H=G]
obua@19203
   525
      Game_ext zet_ext_eq zunion zimage_iff prems)
obua@19203
   526
qed
obua@19203
   527
obua@19203
   528
lemma game_ext_eq: "(G = H) = (left_options G = left_options H \<and> right_options G = right_options H)"
obua@19203
   529
proof -
obua@19203
   530
  have "(G = H) = (Game (left_options G) (right_options G) = Game (left_options H) (right_options H))"
obua@19203
   531
    by (simp add: game_split[symmetric])
obua@19203
   532
  then show ?thesis by auto
obua@19203
   533
qed
obua@19203
   534
obua@19203
   535
lemma left_zero_game[simp]: "left_options (zero_game) = zempty"
obua@19203
   536
  by (simp add: zero_game_def)
obua@19203
   537
obua@19203
   538
lemma right_zero_game[simp]: "right_options (zero_game) = zempty"
obua@19203
   539
  by (simp add: zero_game_def)
obua@19203
   540
krauss@35440
   541
lemma plus_game_zero_right[simp]: "plus_game G zero_game = G"
obua@19203
   542
proof -
obua@19203
   543
  { 
obua@19203
   544
    fix G H
krauss@35440
   545
    have "H = zero_game \<longrightarrow> plus_game G H = G "
obua@19203
   546
    proof (induct G H rule: plus_game.induct, rule impI)
obua@19203
   547
      case (goal1 G H)
obua@19203
   548
      note induct_hyp = prems[simplified goal1, simplified] and prems
obua@19203
   549
      show ?case
wenzelm@32960
   550
        apply (simp only: plus_game.simps[where G=G and H=H])
wenzelm@32960
   551
        apply (simp add: game_ext_eq prems)
wenzelm@32960
   552
        apply (auto simp add: 
krauss@35440
   553
          zimage_cong[where f = "\<lambda> g. plus_game g zero_game" and g = "id"] 
wenzelm@32960
   554
          induct_hyp)
wenzelm@32960
   555
        done
obua@19203
   556
    qed
obua@19203
   557
  }
obua@19203
   558
  then show ?thesis by auto
obua@19203
   559
qed
obua@19203
   560
krauss@35440
   561
lemma plus_game_zero_left: "plus_game zero_game G = G"
obua@19203
   562
  by (simp add: plus_game_comm)
obua@19203
   563
obua@19203
   564
lemma left_imp_options[simp]: "zin opt (left_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   565
  by (simp add: options_def zunion)
obua@19203
   566
obua@19203
   567
lemma right_imp_options[simp]: "zin opt (right_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   568
  by (simp add: options_def zunion)
obua@19203
   569
obua@19203
   570
lemma left_options_plus: 
krauss@35440
   571
  "left_options (plus_game u v) =  zunion (zimage (\<lambda>g. plus_game g v) (left_options u)) (zimage (\<lambda>h. plus_game u h) (left_options v))" 
obua@19203
   572
  by (subst plus_game.simps, simp)
obua@19203
   573
obua@19203
   574
lemma right_options_plus:
krauss@35440
   575
  "right_options (plus_game u v) =  zunion (zimage (\<lambda>g. plus_game g v) (right_options u)) (zimage (\<lambda>h. plus_game u h) (right_options v))"
obua@19203
   576
  by (subst plus_game.simps, simp)
obua@19203
   577
wenzelm@32960
   578
lemma left_options_neg: "left_options (neg_game u) = zimage neg_game (right_options u)"  
obua@19203
   579
  by (subst neg_game.simps, simp)
obua@19203
   580
obua@19203
   581
lemma right_options_neg: "right_options (neg_game u) = zimage neg_game (left_options u)"
obua@19203
   582
  by (subst neg_game.simps, simp)
obua@19203
   583
  
krauss@35440
   584
lemma plus_game_assoc: "plus_game (plus_game F G) H = plus_game F (plus_game G H)"
obua@19203
   585
proof -
obua@19203
   586
  { 
obua@19203
   587
    fix a
krauss@35440
   588
    have "\<forall> F G H. a = [F, G, H] \<longrightarrow> plus_game (plus_game F G) H = plus_game F (plus_game G H)"
obua@19203
   589
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   590
      case (goal1 x F G H)
krauss@35440
   591
      let ?L = "plus_game (plus_game F G) H"
krauss@35440
   592
      let ?R = "plus_game F (plus_game G H)"
obua@19203
   593
      note options_plus = left_options_plus right_options_plus
obua@19203
   594
      {
wenzelm@32960
   595
        fix opt
wenzelm@32960
   596
        note hyp = goal1(1)[simplified goal1(2), rule_format] 
krauss@35440
   597
        have F: "zin opt (options F)  \<Longrightarrow> plus_game (plus_game opt G) H = plus_game opt (plus_game G H)"
wenzelm@32960
   598
          by (blast intro: hyp lprod_3_3)
krauss@35440
   599
        have G: "zin opt (options G) \<Longrightarrow> plus_game (plus_game F opt) H = plus_game F (plus_game opt H)"
wenzelm@32960
   600
          by (blast intro: hyp lprod_3_4)
krauss@35440
   601
        have H: "zin opt (options H) \<Longrightarrow> plus_game (plus_game F G) opt = plus_game F (plus_game G opt)" 
wenzelm@32960
   602
          by (blast intro: hyp lprod_3_5)
wenzelm@32960
   603
        note F and G and H
obua@19203
   604
      }
obua@19203
   605
      note induct_hyp = this
obua@19203
   606
      have "left_options ?L = left_options ?R \<and> right_options ?L = right_options ?R"
wenzelm@32960
   607
        by (auto simp add: 
krauss@35440
   608
          plus_game.simps[where G="plus_game F G" and H=H]
krauss@35440
   609
          plus_game.simps[where G="F" and H="plus_game G H"] 
wenzelm@32960
   610
          zet_ext_eq zunion zimage_iff options_plus
wenzelm@32960
   611
          induct_hyp left_imp_options right_imp_options)
obua@19203
   612
      then show ?case
wenzelm@32960
   613
        by (simp add: game_ext_eq)
obua@19203
   614
    qed
obua@19203
   615
  }
obua@19203
   616
  then show ?thesis by auto
obua@19203
   617
qed
obua@19203
   618
krauss@35440
   619
lemma neg_plus_game: "neg_game (plus_game G H) = plus_game (neg_game G) (neg_game H)"
obua@19203
   620
proof (induct G H rule: plus_game.induct)
obua@19203
   621
  case (1 G H)
obua@19203
   622
  note opt_ops = 
obua@19203
   623
    left_options_plus right_options_plus 
obua@19203
   624
    left_options_neg right_options_neg  
obua@19203
   625
  show ?case
obua@19203
   626
    by (auto simp add: opt_ops
krauss@35440
   627
      neg_game.simps[of "plus_game G H"]
obua@19203
   628
      plus_game.simps[of "neg_game G" "neg_game H"]
obua@19203
   629
      Game_ext zet_ext_eq zunion zimage_iff prems)
obua@19203
   630
qed
obua@19203
   631
krauss@35440
   632
lemma eq_game_plus_inverse: "eq_game (plus_game x (neg_game x)) zero_game"
obua@19203
   633
proof (induct x rule: wf_induct[OF wf_option_of])
obua@19203
   634
  case (goal1 x)
obua@19203
   635
  { fix y
obua@19203
   636
    assume "zin y (options x)"
krauss@35440
   637
    then have "eq_game (plus_game y (neg_game y)) zero_game"
obua@19203
   638
      by (auto simp add: prems)
obua@19203
   639
  }
obua@19203
   640
  note ihyp = this
obua@19203
   641
  {
obua@19203
   642
    fix y
obua@19203
   643
    assume y: "zin y (right_options x)"
krauss@35440
   644
    have "\<not> (ge_game (zero_game, plus_game y (neg_game x)))"
obua@19203
   645
      apply (subst ge_game.simps, simp)
krauss@35440
   646
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   647
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
obua@19203
   648
      apply (auto simp add: left_options_plus left_options_neg zunion zimage_iff intro: prems)
obua@19203
   649
      done
obua@19203
   650
  }
obua@19203
   651
  note case1 = this
obua@19203
   652
  {
obua@19203
   653
    fix y
obua@19203
   654
    assume y: "zin y (left_options x)"
krauss@35440
   655
    have "\<not> (ge_game (zero_game, plus_game x (neg_game y)))"
obua@19203
   656
      apply (subst ge_game.simps, simp)
krauss@35440
   657
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   658
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
obua@19203
   659
      apply (auto simp add: left_options_plus zunion zimage_iff intro: prems)
obua@19203
   660
      done
obua@19203
   661
  }
obua@19203
   662
  note case2 = this
obua@19203
   663
  {
obua@19203
   664
    fix y
obua@19203
   665
    assume y: "zin y (left_options x)"
krauss@35440
   666
    have "\<not> (ge_game (plus_game y (neg_game x), zero_game))"
obua@19203
   667
      apply (subst ge_game.simps, simp)
krauss@35440
   668
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   669
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
obua@19203
   670
      apply (auto simp add: right_options_plus right_options_neg zunion zimage_iff intro: prems)
obua@19203
   671
      done
obua@19203
   672
  }
obua@19203
   673
  note case3 = this
obua@19203
   674
  {
obua@19203
   675
    fix y
obua@19203
   676
    assume y: "zin y (right_options x)"
krauss@35440
   677
    have "\<not> (ge_game (plus_game x (neg_game y), zero_game))"
obua@19203
   678
      apply (subst ge_game.simps, simp)
krauss@35440
   679
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   680
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
obua@19203
   681
      apply (auto simp add: right_options_plus zunion zimage_iff intro: prems)
obua@19203
   682
      done
obua@19203
   683
  }
obua@19203
   684
  note case4 = this
obua@19203
   685
  show ?case
obua@19203
   686
    apply (simp add: eq_game_def)
krauss@35440
   687
    apply (simp add: ge_game.simps[of "plus_game x (neg_game x)" "zero_game"])
krauss@35440
   688
    apply (simp add: ge_game.simps[of "zero_game" "plus_game x (neg_game x)"])
obua@19203
   689
    apply (simp add: right_options_plus left_options_plus right_options_neg left_options_neg zunion zimage_iff)
obua@19203
   690
    apply (auto simp add: case1 case2 case3 case4)
obua@19203
   691
    done
obua@19203
   692
qed
obua@19203
   693
krauss@35440
   694
lemma ge_plus_game_left: "ge_game (y,z) = ge_game (plus_game x y, plus_game x z)"
obua@19203
   695
proof -
obua@19203
   696
  { fix a
krauss@35440
   697
    have "\<forall> x y z. a = [x,y,z] \<longrightarrow> ge_game (y,z) = ge_game (plus_game x y, plus_game x z)"
obua@19203
   698
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   699
      case (goal1 a x y z)
obua@19203
   700
      note induct_hyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   701
      { 
krauss@35440
   702
        assume hyp: "ge_game(plus_game x y, plus_game x z)"
wenzelm@32960
   703
        have "ge_game (y, z)"
wenzelm@32960
   704
        proof -
wenzelm@32960
   705
          { fix yr
wenzelm@32960
   706
            assume yr: "zin yr (right_options y)"
krauss@35440
   707
            from hyp have "\<not> (ge_game (plus_game x z, plus_game x yr))"
krauss@35440
   708
              by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"]
wenzelm@32960
   709
                right_options_plus zunion zimage_iff intro: yr)
wenzelm@32960
   710
            then have "\<not> (ge_game (z, yr))"
wenzelm@32960
   711
              apply (subst induct_hyp[where y="[x, z, yr]", of "x" "z" "yr"])
wenzelm@32960
   712
              apply (simp_all add: yr lprod_3_6)
wenzelm@32960
   713
              done
wenzelm@32960
   714
          }
wenzelm@32960
   715
          note yr = this
wenzelm@32960
   716
          { fix zl
wenzelm@32960
   717
            assume zl: "zin zl (left_options z)"
krauss@35440
   718
            from hyp have "\<not> (ge_game (plus_game x zl, plus_game x y))"
krauss@35440
   719
              by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"]
wenzelm@32960
   720
                left_options_plus zunion zimage_iff intro: zl)
wenzelm@32960
   721
            then have "\<not> (ge_game (zl, y))"
wenzelm@32960
   722
              apply (subst goal1(1)[rule_format, where y="[x, zl, y]", of "x" "zl" "y"])
wenzelm@32960
   723
              apply (simp_all add: goal1(2) zl lprod_3_7)
wenzelm@32960
   724
              done
wenzelm@32960
   725
          }     
wenzelm@32960
   726
          note zl = this
wenzelm@32960
   727
          show "ge_game (y, z)"
wenzelm@32960
   728
            apply (subst ge_game_eq)
wenzelm@32960
   729
            apply (auto simp add: yr zl)
wenzelm@32960
   730
            done
wenzelm@32960
   731
        qed      
obua@19203
   732
      }
obua@19203
   733
      note right_imp_left = this
obua@19203
   734
      {
wenzelm@32960
   735
        assume yz: "ge_game (y, z)"
wenzelm@32960
   736
        {
wenzelm@32960
   737
          fix x'
wenzelm@32960
   738
          assume x': "zin x' (right_options x)"
krauss@35440
   739
          assume hyp: "ge_game (plus_game x z, plus_game x' y)"
krauss@35440
   740
          then have n: "\<not> (ge_game (plus_game x' y, plus_game x' z))"
krauss@35440
   741
            by (auto simp add: ge_game_eq[of "plus_game x z" "plus_game x' y"] 
wenzelm@32960
   742
              right_options_plus zunion zimage_iff intro: x')
krauss@35440
   743
          have t: "ge_game (plus_game x' y, plus_game x' z)"
wenzelm@32960
   744
            apply (subst induct_hyp[symmetric])
wenzelm@32960
   745
            apply (auto intro: lprod_3_3 x' yz)
wenzelm@32960
   746
            done
wenzelm@32960
   747
          from n t have "False" by blast
wenzelm@32960
   748
        }    
wenzelm@32960
   749
        note case1 = this
wenzelm@32960
   750
        {
wenzelm@32960
   751
          fix x'
wenzelm@32960
   752
          assume x': "zin x' (left_options x)"
krauss@35440
   753
          assume hyp: "ge_game (plus_game x' z, plus_game x y)"
krauss@35440
   754
          then have n: "\<not> (ge_game (plus_game x' y, plus_game x' z))"
krauss@35440
   755
            by (auto simp add: ge_game_eq[of "plus_game x' z" "plus_game x y"] 
wenzelm@32960
   756
              left_options_plus zunion zimage_iff intro: x')
krauss@35440
   757
          have t: "ge_game (plus_game x' y, plus_game x' z)"
wenzelm@32960
   758
            apply (subst induct_hyp[symmetric])
wenzelm@32960
   759
            apply (auto intro: lprod_3_3 x' yz)
wenzelm@32960
   760
            done
wenzelm@32960
   761
          from n t have "False" by blast
wenzelm@32960
   762
        }
wenzelm@32960
   763
        note case3 = this
wenzelm@32960
   764
        {
wenzelm@32960
   765
          fix y'
wenzelm@32960
   766
          assume y': "zin y' (right_options y)"
krauss@35440
   767
          assume hyp: "ge_game (plus_game x z, plus_game x y')"
wenzelm@32960
   768
          then have "ge_game(z, y')"
wenzelm@32960
   769
            apply (subst induct_hyp[of "[x, z, y']" "x" "z" "y'"])
wenzelm@32960
   770
            apply (auto simp add: hyp lprod_3_6 y')
wenzelm@32960
   771
            done
wenzelm@32960
   772
          with yz have "ge_game (y, y')"
wenzelm@32960
   773
            by (blast intro: ge_game_trans)      
wenzelm@32960
   774
          with y' have "False" by (auto simp add: ge_game_leftright_refl)
wenzelm@32960
   775
        }
wenzelm@32960
   776
        note case2 = this
wenzelm@32960
   777
        {
wenzelm@32960
   778
          fix z'
wenzelm@32960
   779
          assume z': "zin z' (left_options z)"
krauss@35440
   780
          assume hyp: "ge_game (plus_game x z', plus_game x y)"
wenzelm@32960
   781
          then have "ge_game(z', y)"
wenzelm@32960
   782
            apply (subst induct_hyp[of "[x, z', y]" "x" "z'" "y"])
wenzelm@32960
   783
            apply (auto simp add: hyp lprod_3_7 z')
wenzelm@32960
   784
            done    
wenzelm@32960
   785
          with yz have "ge_game (z', z)"
wenzelm@32960
   786
            by (blast intro: ge_game_trans)      
wenzelm@32960
   787
          with z' have "False" by (auto simp add: ge_game_leftright_refl)
wenzelm@32960
   788
        }
wenzelm@32960
   789
        note case4 = this   
krauss@35440
   790
        have "ge_game(plus_game x y, plus_game x z)"
wenzelm@32960
   791
          apply (subst ge_game_eq)
wenzelm@32960
   792
          apply (auto simp add: right_options_plus left_options_plus zunion zimage_iff)
wenzelm@32960
   793
          apply (auto intro: case1 case2 case3 case4)
wenzelm@32960
   794
          done
obua@19203
   795
      }
obua@19203
   796
      note left_imp_right = this
obua@19203
   797
      show ?case by (auto intro: right_imp_left left_imp_right)
obua@19203
   798
    qed
obua@19203
   799
  }
obua@19203
   800
  note a = this[of "[x, y, z]"]
obua@19203
   801
  then show ?thesis by blast
obua@19203
   802
qed
obua@19203
   803
krauss@35440
   804
lemma ge_plus_game_right: "ge_game (y,z) = ge_game(plus_game y x, plus_game z x)"
obua@19203
   805
  by (simp add: ge_plus_game_left plus_game_comm)
obua@19203
   806
obua@19203
   807
lemma ge_neg_game: "ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   808
proof -
obua@19203
   809
  { fix a
obua@19203
   810
    have "\<forall> x y. a = [x, y] \<longrightarrow> ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   811
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   812
      case (goal1 a x y)
obua@19203
   813
      note ihyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   814
      { fix xl
wenzelm@32960
   815
        assume xl: "zin xl (left_options x)"
wenzelm@32960
   816
        have "ge_game (neg_game y, neg_game xl) = ge_game (xl, y)"
wenzelm@32960
   817
          apply (subst ihyp)
wenzelm@32960
   818
          apply (auto simp add: lprod_2_1 xl)
wenzelm@32960
   819
          done
obua@19203
   820
      }
obua@19203
   821
      note xl = this
obua@19203
   822
      { fix yr
wenzelm@32960
   823
        assume yr: "zin yr (right_options y)"
wenzelm@32960
   824
        have "ge_game (neg_game yr, neg_game x) = ge_game (x, yr)"
wenzelm@32960
   825
          apply (subst ihyp)
wenzelm@32960
   826
          apply (auto simp add: lprod_2_2 yr)
wenzelm@32960
   827
          done
obua@19203
   828
      }
obua@19203
   829
      note yr = this
obua@19203
   830
      show ?case
wenzelm@32960
   831
        by (auto simp add: ge_game_eq[of "neg_game x" "neg_game y"] ge_game_eq[of "y" "x"]
wenzelm@32960
   832
          right_options_neg left_options_neg zimage_iff  xl yr)
obua@19203
   833
    qed
obua@19203
   834
  }
obua@19203
   835
  note a = this[of "[x,y]"]
obua@19203
   836
  then show ?thesis by blast
obua@19203
   837
qed
obua@19203
   838
haftmann@35416
   839
definition eq_game_rel :: "(game * game) set" where
obua@19203
   840
  "eq_game_rel \<equiv> { (p, q) . eq_game p q }"
obua@19203
   841
obua@19203
   842
typedef Pg = "UNIV//eq_game_rel"
obua@19203
   843
  by (auto simp add: quotient_def)
obua@19203
   844
obua@19203
   845
lemma equiv_eq_game[simp]: "equiv UNIV eq_game_rel"
nipkow@30198
   846
  by (auto simp add: equiv_def refl_on_def sym_def trans_def eq_game_rel_def
obua@19203
   847
    eq_game_sym intro: eq_game_refl eq_game_trans)
obua@19203
   848
haftmann@25764
   849
instantiation Pg :: "{ord, zero, plus, minus, uminus}"
haftmann@25764
   850
begin
haftmann@25764
   851
haftmann@25764
   852
definition
haftmann@25764
   853
  Pg_zero_def: "0 = Abs_Pg (eq_game_rel `` {zero_game})"
haftmann@25764
   854
haftmann@25764
   855
definition
haftmann@25764
   856
  Pg_le_def: "G \<le> H \<longleftrightarrow> (\<exists> g h. g \<in> Rep_Pg G \<and> h \<in> Rep_Pg H \<and> ge_game (h, g))"
haftmann@25764
   857
haftmann@25764
   858
definition
haftmann@25764
   859
  Pg_less_def: "G < H \<longleftrightarrow> G \<le> H \<and> G \<noteq> (H::Pg)"
obua@19203
   860
haftmann@25764
   861
definition
haftmann@39910
   862
  Pg_minus_def: "- G = the_elem (\<Union> g \<in> Rep_Pg G. {Abs_Pg (eq_game_rel `` {neg_game g})})"
haftmann@25764
   863
haftmann@25764
   864
definition
haftmann@39910
   865
  Pg_plus_def: "G + H = the_elem (\<Union> g \<in> Rep_Pg G. \<Union> h \<in> Rep_Pg H. {Abs_Pg (eq_game_rel `` {plus_game g h})})"
haftmann@25764
   866
haftmann@25764
   867
definition
haftmann@25764
   868
  Pg_diff_def: "G - H = G + (- (H::Pg))"
haftmann@25764
   869
haftmann@25764
   870
instance ..
haftmann@25764
   871
haftmann@25764
   872
end
obua@19203
   873
obua@19203
   874
lemma Rep_Abs_eq_Pg[simp]: "Rep_Pg (Abs_Pg (eq_game_rel `` {g})) = eq_game_rel `` {g}"
obua@19203
   875
  apply (subst Abs_Pg_inverse)
obua@19203
   876
  apply (auto simp add: Pg_def quotient_def)
obua@19203
   877
  done
obua@19203
   878
obua@19203
   879
lemma char_Pg_le[simp]: "(Abs_Pg (eq_game_rel `` {g}) \<le> Abs_Pg (eq_game_rel `` {h})) = (ge_game (h, g))"
obua@19203
   880
  apply (simp add: Pg_le_def)
obua@19203
   881
  apply (auto simp add: eq_game_rel_def eq_game_def intro: ge_game_trans ge_game_refl)
obua@19203
   882
  done
obua@19203
   883
obua@19203
   884
lemma char_Pg_eq[simp]: "(Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {h})) = (eq_game g h)"
obua@19203
   885
  apply (simp add: Rep_Pg_inject [symmetric])
obua@19203
   886
  apply (subst eq_equiv_class_iff[of UNIV])
obua@19203
   887
  apply (simp_all)
obua@19203
   888
  apply (simp add: eq_game_rel_def)
obua@19203
   889
  done
obua@19203
   890
krauss@35440
   891
lemma char_Pg_plus[simp]: "Abs_Pg (eq_game_rel `` {g}) + Abs_Pg (eq_game_rel `` {h}) = Abs_Pg (eq_game_rel `` {plus_game g h})"
obua@19203
   892
proof -
krauss@35440
   893
  have "(\<lambda> g h. {Abs_Pg (eq_game_rel `` {plus_game g h})}) respects2 eq_game_rel" 
obua@19203
   894
    apply (simp add: congruent2_def)
obua@19203
   895
    apply (auto simp add: eq_game_rel_def eq_game_def)
haftmann@40824
   896
    apply (rule_tac y="plus_game a ba" in ge_game_trans)
obua@19203
   897
    apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+
haftmann@40824
   898
    apply (rule_tac y="plus_game b aa" in ge_game_trans)
obua@19203
   899
    apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+
obua@19203
   900
    done
obua@19203
   901
  then show ?thesis
obua@19203
   902
    by (simp add: Pg_plus_def UN_equiv_class2[OF equiv_eq_game equiv_eq_game]) 
obua@19203
   903
qed
obua@19203
   904
    
obua@19203
   905
lemma char_Pg_minus[simp]: "- Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {neg_game g})"
obua@19203
   906
proof -
obua@19203
   907
  have "(\<lambda> g. {Abs_Pg (eq_game_rel `` {neg_game g})}) respects eq_game_rel"
obua@19203
   908
    apply (simp add: congruent_def)
obua@19203
   909
    apply (auto simp add: eq_game_rel_def eq_game_def ge_neg_game)
obua@19203
   910
    done    
obua@19203
   911
  then show ?thesis
obua@19203
   912
    by (simp add: Pg_minus_def UN_equiv_class[OF equiv_eq_game])
obua@19203
   913
qed
obua@19203
   914
obua@19203
   915
lemma eq_Abs_Pg[rule_format, cases type: Pg]: "(\<forall> g. z = Abs_Pg (eq_game_rel `` {g}) \<longrightarrow> P) \<longrightarrow> P"
obua@19203
   916
  apply (cases z, simp)
obua@19203
   917
  apply (simp add: Rep_Pg_inject[symmetric])
obua@19203
   918
  apply (subst Abs_Pg_inverse, simp)
obua@19203
   919
  apply (auto simp add: Pg_def quotient_def)
obua@19203
   920
  done
obua@19203
   921
haftmann@35028
   922
instance Pg :: ordered_ab_group_add 
obua@19203
   923
proof
obua@19203
   924
  fix a b c :: Pg
obua@19203
   925
  show "a - b = a + (- b)" by (simp add: Pg_diff_def)
obua@19203
   926
  {
obua@19203
   927
    assume ab: "a \<le> b"
obua@19203
   928
    assume ba: "b \<le> a"
obua@19203
   929
    from ab ba show "a = b"
obua@19203
   930
      apply (cases a, cases b)
obua@19203
   931
      apply (simp add: eq_game_def)
obua@19203
   932
      done
obua@19203
   933
  }
haftmann@27679
   934
  then show "(a < b) = (a \<le> b \<and> \<not> b \<le> a)" by (auto simp add: Pg_less_def)
obua@19203
   935
  show "a + b = b + a"
obua@19203
   936
    apply (cases a, cases b)
obua@19203
   937
    apply (simp add: eq_game_def plus_game_comm)
obua@19203
   938
    done
obua@19203
   939
  show "a + b + c = a + (b + c)"
obua@19203
   940
    apply (cases a, cases b, cases c)
obua@19203
   941
    apply (simp add: eq_game_def plus_game_assoc)
obua@19203
   942
    done
obua@19203
   943
  show "0 + a = a"
obua@19203
   944
    apply (cases a)
obua@19203
   945
    apply (simp add: Pg_zero_def plus_game_zero_left)
obua@19203
   946
    done
obua@19203
   947
  show "- a + a = 0"
obua@19203
   948
    apply (cases a)
obua@19203
   949
    apply (simp add: Pg_zero_def eq_game_plus_inverse plus_game_comm)
obua@19203
   950
    done
obua@19203
   951
  show "a \<le> a"
obua@19203
   952
    apply (cases a)
obua@19203
   953
    apply (simp add: ge_game_refl)
obua@19203
   954
    done
obua@19203
   955
  {
obua@19203
   956
    assume ab: "a \<le> b"
obua@19203
   957
    assume bc: "b \<le> c"
obua@19203
   958
    from ab bc show "a \<le> c"
obua@19203
   959
      apply (cases a, cases b, cases c)
obua@19203
   960
      apply (auto intro: ge_game_trans)
obua@19203
   961
      done
obua@19203
   962
  }
obua@19203
   963
  {
obua@19203
   964
    assume ab: "a \<le> b"
obua@19203
   965
    from ab show "c + a \<le> c + b"
obua@19203
   966
      apply (cases a, cases b, cases c)
obua@19203
   967
      apply (simp add: ge_plus_game_left[symmetric])
obua@19203
   968
      done
obua@19203
   969
  }
obua@19203
   970
qed
obua@19203
   971
obua@19203
   972
end