src/HOL/ZF/Games.thy
author wenzelm
Mon Mar 17 18:37:05 2008 +0100 (2008-03-17)
changeset 26304 02fbd0e7954a
parent 25764 878c37886eed
child 27679 6392b92c3536
permissions -rw-r--r--
avoid rebinding of existing facts;
obua@19203
     1
(*  Title:      HOL/ZF/Games.thy
obua@19203
     2
    ID:         $Id$
obua@19203
     3
    Author:     Steven Obua
obua@19203
     4
obua@19203
     5
    An application of HOLZF: Partizan Games.
obua@19203
     6
    See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
obua@19203
     7
*)
obua@19203
     8
obua@19203
     9
theory Games 
obua@19203
    10
imports MainZF
obua@19203
    11
begin
obua@19203
    12
obua@19203
    13
constdefs
obua@19203
    14
  fixgames :: "ZF set \<Rightarrow> ZF set"
obua@19203
    15
  "fixgames A \<equiv> { Opair l r | l r. explode l \<subseteq> A & explode r \<subseteq> A}"
obua@19203
    16
  games_lfp :: "ZF set"
obua@19203
    17
  "games_lfp \<equiv> lfp fixgames"
obua@19203
    18
  games_gfp :: "ZF set"
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
obua@19203
    46
constdefs
obua@19203
    47
  left_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"
obua@19203
    48
  "left_option g opt \<equiv> (Elem opt (Fst g))"
obua@19203
    49
  right_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"
obua@19203
    50
  "right_option g opt \<equiv> (Elem opt (Snd g))"
obua@19203
    51
  is_option_of :: "(ZF * ZF) set"
obua@19203
    52
  "is_option_of \<equiv> { (opt, g) | opt g. g \<in> games_gfp \<and> (left_option g opt \<or> right_option g opt) }"
obua@19203
    53
obua@19203
    54
lemma games_lfp_subset_gfp: "games_lfp \<subseteq> games_gfp"
obua@19203
    55
proof -
obua@19203
    56
  have "games_lfp \<subseteq> fixgames games_lfp" 
obua@19203
    57
    by (simp add: games_lfp_unfold[symmetric])
obua@19203
    58
  then show ?thesis
obua@19203
    59
    by (simp add: games_gfp_def gfp_upperbound)
obua@19203
    60
qed
obua@19203
    61
obua@19203
    62
lemma games_option_stable: 
obua@19203
    63
  assumes fixgames: "games = fixgames games"
obua@19203
    64
  and g: "g \<in> games"
obua@19203
    65
  and opt: "left_option g opt \<or> right_option g opt"
obua@19203
    66
  shows "opt \<in> games"
obua@19203
    67
proof -
obua@19203
    68
  from g fixgames have "g \<in> fixgames games" by auto
obua@19203
    69
  then have "\<exists> l r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games"
obua@19203
    70
    by (simp add: fixgames_def)
obua@19203
    71
  then obtain l where "\<exists> r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
obua@19203
    72
  then obtain r where lr: "g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
obua@19203
    73
  with opt show ?thesis
obua@19203
    74
    by (auto intro: Elem_explode_in simp add: left_option_def right_option_def Fst Snd)
obua@19203
    75
qed
obua@19203
    76
    
obua@19203
    77
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
    78
  apply (simp add: is_option_of_def)
obua@19203
    79
  apply (subgoal_tac "(g \<in> games_gfp) = (g \<in> (fixgames games_gfp))")
obua@19203
    80
  prefer 2
obua@19203
    81
  apply (simp add: games_gfp_unfold[symmetric])
obua@19203
    82
  apply (auto simp add: fixgames_def left_option_def right_option_def Fst Snd)
obua@19203
    83
  apply (rule_tac x=l in exI, insert Elem_Opair_exists, blast)
obua@19203
    84
  apply (rule_tac x=r in exI, insert Elem_Opair_exists, blast) 
obua@19203
    85
  done
obua@19203
    86
obua@19203
    87
lemma is_option_of_subset_is_Elem_of: "is_option_of \<subseteq> (is_Elem_of^+)"
obua@19203
    88
proof -
obua@19203
    89
  {
obua@19203
    90
    fix opt
obua@19203
    91
    fix g
obua@19203
    92
    assume "(opt, g) \<in> is_option_of"
obua@19203
    93
    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
    94
      apply -
obua@19203
    95
      apply (drule option2elem)
obua@19203
    96
      apply (auto simp add: r_into_trancl' is_Elem_of_def)
obua@19203
    97
      done
obua@19203
    98
    then have "(opt, g) \<in> (is_Elem_of^+)"
obua@19203
    99
      by (blast intro: trancl_into_rtrancl trancl_rtrancl_trancl)
obua@19203
   100
  } 
obua@19203
   101
  then show ?thesis by auto
obua@19203
   102
qed
obua@19203
   103
obua@19203
   104
lemma wfzf_is_option_of: "wfzf is_option_of"
obua@19203
   105
proof - 
obua@19203
   106
  have "wfzf (is_Elem_of^+)" by (simp add: wfzf_trancl wfzf_is_Elem_of)
obua@19203
   107
  then show ?thesis 
obua@19203
   108
    apply (rule wfzf_subset)
obua@19203
   109
    apply (rule is_option_of_subset_is_Elem_of)
obua@19203
   110
    done
obua@19203
   111
  qed
obua@19203
   112
  
obua@19203
   113
lemma games_gfp_imp_lfp: "g \<in> games_gfp \<longrightarrow> g \<in> games_lfp"
obua@19203
   114
proof -
obua@19203
   115
  have unfold_gfp: "\<And> x. x \<in> games_gfp \<Longrightarrow> x \<in> (fixgames games_gfp)" 
obua@19203
   116
    by (simp add: games_gfp_unfold[symmetric])
obua@19203
   117
  have unfold_lfp: "\<And> x. (x \<in> games_lfp) = (x \<in> (fixgames games_lfp))"
obua@19203
   118
    by (simp add: games_lfp_unfold[symmetric])
obua@19203
   119
  show ?thesis
obua@19203
   120
    apply (rule wf_induct[OF wfzf_implies_wf[OF wfzf_is_option_of]])
obua@19203
   121
    apply (auto simp add: is_option_of_def)
obua@19203
   122
    apply (drule_tac unfold_gfp)
obua@19203
   123
    apply (simp add: fixgames_def)
obua@19203
   124
    apply (auto simp add: left_option_def Fst right_option_def Snd)
obua@19203
   125
    apply (subgoal_tac "explode l \<subseteq> games_lfp")
obua@19203
   126
    apply (subgoal_tac "explode r \<subseteq> games_lfp")
obua@19203
   127
    apply (subst unfold_lfp)
obua@19203
   128
    apply (auto simp add: fixgames_def)
obua@19203
   129
    apply (simp_all add: explode_Elem Elem_explode_in)
obua@19203
   130
    done
obua@19203
   131
qed
obua@19203
   132
obua@19203
   133
theorem games_lfp_eq_gfp: "games_lfp = games_gfp"
obua@19203
   134
  apply (auto simp add: games_gfp_imp_lfp)
obua@19203
   135
  apply (insert games_lfp_subset_gfp)
obua@19203
   136
  apply auto
obua@19203
   137
  done
obua@19203
   138
obua@19203
   139
theorem unique_games: "(g = fixgames g) = (g = games_lfp)"
obua@19203
   140
proof -
obua@19203
   141
  {
obua@19203
   142
    fix g 
obua@19203
   143
    assume g: "g = fixgames g"
obua@19203
   144
    from g have "fixgames g \<subseteq> g" by auto
obua@19203
   145
    then have l:"games_lfp \<subseteq> g" 
obua@19203
   146
      by (simp add: games_lfp_def lfp_lowerbound)
obua@19203
   147
    from g have "g \<subseteq> fixgames g" by auto
obua@19203
   148
    then have u:"g \<subseteq> games_gfp" 
obua@19203
   149
      by (simp add: games_gfp_def gfp_upperbound)
obua@19203
   150
    from l u games_lfp_eq_gfp[symmetric] have "g = games_lfp"
obua@19203
   151
      by auto
obua@19203
   152
  }
obua@19203
   153
  note games = this
obua@19203
   154
  show ?thesis
obua@19203
   155
    apply (rule iff[rule_format])
obua@19203
   156
    apply (erule games)
obua@19203
   157
    apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   158
    done
obua@19203
   159
qed
obua@19203
   160
obua@19203
   161
lemma games_lfp_option_stable: 
obua@19203
   162
  assumes g: "g \<in> games_lfp"
obua@19203
   163
  and opt: "left_option g opt \<or> right_option g opt"
obua@19203
   164
  shows "opt \<in> games_lfp"
obua@19203
   165
  apply (rule games_option_stable[where g=g])
obua@19203
   166
  apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   167
  apply (simp_all add: prems)
obua@19203
   168
  done
obua@19203
   169
obua@19203
   170
lemma is_option_of_imp_games:
obua@19203
   171
  assumes hyp: "(opt, g) \<in> is_option_of"
obua@19203
   172
  shows "opt \<in> games_lfp \<and> g \<in> games_lfp"
obua@19203
   173
proof -
obua@19203
   174
  from hyp have g_game: "g \<in> games_lfp" 
obua@19203
   175
    by (simp add: is_option_of_def games_lfp_eq_gfp)
obua@19203
   176
  from hyp have "left_option g opt \<or> right_option g opt"
obua@19203
   177
    by (auto simp add: is_option_of_def)
obua@19203
   178
  with g_game games_lfp_option_stable[OF g_game, OF this] show ?thesis
obua@19203
   179
    by auto
obua@19203
   180
qed
obua@19203
   181
 
obua@19203
   182
lemma games_lfp_represent: "x \<in> games_lfp \<Longrightarrow> \<exists> l r. x = Opair l r"
obua@19203
   183
  apply (rule exI[where x="Fst x"])
obua@19203
   184
  apply (rule exI[where x="Snd x"])
obua@19203
   185
  apply (subgoal_tac "x \<in> (fixgames games_lfp)")
obua@19203
   186
  apply (simp add: fixgames_def)
obua@19203
   187
  apply (auto simp add: Fst Snd)
obua@19203
   188
  apply (simp add: games_lfp_unfold[symmetric])
obua@19203
   189
  done
obua@19203
   190
obua@19203
   191
typedef game = games_lfp
obua@19203
   192
  by (blast intro: games_lfp_nonempty)
obua@19203
   193
obua@19203
   194
constdefs
obua@19203
   195
  left_options :: "game \<Rightarrow> game zet"
obua@19203
   196
  "left_options g \<equiv> zimage Abs_game (zexplode (Fst (Rep_game g)))"
obua@19203
   197
  right_options :: "game \<Rightarrow> game zet"
obua@19203
   198
  "right_options g \<equiv> zimage Abs_game (zexplode (Snd (Rep_game g)))"
obua@19203
   199
  options :: "game \<Rightarrow> game zet"
obua@19203
   200
  "options g \<equiv> zunion (left_options g) (right_options g)"
obua@19203
   201
  Game :: "game zet \<Rightarrow> game zet \<Rightarrow> game"
obua@19203
   202
  "Game L R \<equiv> Abs_game (Opair (zimplode (zimage Rep_game L)) (zimplode (zimage Rep_game R)))"
obua@19203
   203
  
obua@19203
   204
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
   205
  apply (subst Ext)
obua@19203
   206
  apply (simp add: Repl)
obua@19203
   207
  apply auto
obua@19203
   208
  apply (subst Abs_game_inverse, simp_all add: game_def)
obua@19203
   209
  apply (rule_tac x=za in exI)
obua@19203
   210
  apply (subst Abs_game_inverse, simp_all add: game_def)
obua@19203
   211
  done
obua@19203
   212
obua@19203
   213
lemma game_split: "g = Game (left_options g) (right_options g)"
obua@19203
   214
proof -
obua@19203
   215
  have "\<exists> l r. Rep_game g = Opair l r"
obua@19203
   216
    apply (insert Rep_game[of g])
obua@19203
   217
    apply (simp add: game_def games_lfp_represent)
obua@19203
   218
    done
obua@19203
   219
  then obtain l r where lr: "Rep_game g = Opair l r" by auto
obua@19203
   220
  have partizan_g: "Rep_game g \<in> games_lfp" 
obua@19203
   221
    apply (insert Rep_game[of g])
obua@19203
   222
    apply (simp add: game_def)
obua@19203
   223
    done
obua@19203
   224
  have "\<forall> e. Elem e l \<longrightarrow> left_option (Rep_game g) e"
obua@19203
   225
    by (simp add: lr left_option_def Fst)
obua@19203
   226
  then have partizan_l: "\<forall> e. Elem e l \<longrightarrow> e \<in> games_lfp"
obua@19203
   227
    apply auto
obua@19203
   228
    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
obua@19203
   229
    apply auto
obua@19203
   230
    done
obua@19203
   231
  have "\<forall> e. Elem e r \<longrightarrow> right_option (Rep_game g) e"
obua@19203
   232
    by (simp add: lr right_option_def Snd)
obua@19203
   233
  then have partizan_r: "\<forall> e. Elem e r \<longrightarrow> e \<in> games_lfp"
obua@19203
   234
    apply auto
obua@19203
   235
    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
obua@19203
   236
    apply auto
obua@19203
   237
    done   
obua@19203
   238
  let ?L = "zimage (Abs_game) (zexplode l)"
obua@19203
   239
  let ?R = "zimage (Abs_game) (zexplode r)"
obua@19203
   240
  have L:"?L = left_options g"
obua@19203
   241
    by (simp add: left_options_def lr Fst)
obua@19203
   242
  have R:"?R = right_options g"
obua@19203
   243
    by (simp add: right_options_def lr Snd)
obua@19203
   244
  have "g = Game ?L ?R"
obua@19203
   245
    apply (simp add: Game_def Rep_game_inject[symmetric] comp_zimage_eq zimage_zexplode_eq zimplode_zexplode)
obua@19203
   246
    apply (simp add: Repl_Rep_game_Abs_game partizan_l partizan_r)
obua@19203
   247
    apply (subst Abs_game_inverse)
obua@19203
   248
    apply (simp_all add: lr[symmetric] Rep_game) 
obua@19203
   249
    done
obua@19203
   250
  then show ?thesis
obua@19203
   251
    by (simp add: L R)
obua@19203
   252
qed
obua@19203
   253
obua@19203
   254
lemma Opair_in_games_lfp: 
obua@19203
   255
  assumes l: "explode l \<subseteq> games_lfp"
obua@19203
   256
  and r: "explode r \<subseteq> games_lfp"
obua@19203
   257
  shows "Opair l r \<in> games_lfp"
obua@19203
   258
proof -
obua@19203
   259
  note f = unique_games[of games_lfp, simplified]
obua@19203
   260
  show ?thesis
obua@19203
   261
    apply (subst f)
obua@19203
   262
    apply (simp add: fixgames_def)
obua@19203
   263
    apply (rule exI[where x=l])
obua@19203
   264
    apply (rule exI[where x=r])
obua@19203
   265
    apply (auto simp add: l r)
obua@19203
   266
    done
obua@19203
   267
qed
obua@19203
   268
obua@19203
   269
lemma left_options[simp]: "left_options (Game l r) = l"
obua@19203
   270
  apply (simp add: left_options_def Game_def)
obua@19203
   271
  apply (subst Abs_game_inverse)
obua@19203
   272
  apply (simp add: game_def)
obua@19203
   273
  apply (rule Opair_in_games_lfp)
obua@19203
   274
  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
obua@19203
   275
  apply (simp add: Fst zexplode_zimplode comp_zimage_eq)
obua@19203
   276
  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
obua@19203
   277
  done
obua@19203
   278
obua@19203
   279
lemma right_options[simp]: "right_options (Game l r) = r"
obua@19203
   280
  apply (simp add: right_options_def Game_def)
obua@19203
   281
  apply (subst Abs_game_inverse)
obua@19203
   282
  apply (simp add: game_def)
obua@19203
   283
  apply (rule Opair_in_games_lfp)
obua@19203
   284
  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
obua@19203
   285
  apply (simp add: Snd zexplode_zimplode comp_zimage_eq)
obua@19203
   286
  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
obua@19203
   287
  done  
obua@19203
   288
obua@19203
   289
lemma Game_ext: "(Game l1 r1 = Game l2 r2) = ((l1 = l2) \<and> (r1 = r2))"
obua@19203
   290
  apply auto
obua@19203
   291
  apply (subst left_options[where l=l1 and r=r1,symmetric])
obua@19203
   292
  apply (subst left_options[where l=l2 and r=r2,symmetric])
obua@19203
   293
  apply simp
obua@19203
   294
  apply (subst right_options[where l=l1 and r=r1,symmetric])
obua@19203
   295
  apply (subst right_options[where l=l2 and r=r2,symmetric])
obua@19203
   296
  apply simp
obua@19203
   297
  done
obua@19203
   298
obua@19203
   299
constdefs
obua@19203
   300
  option_of :: "(game * game) set"
obua@19203
   301
  "option_of \<equiv> image (\<lambda> (option, g). (Abs_game option, Abs_game g)) is_option_of"
obua@19203
   302
obua@19203
   303
lemma option_to_is_option_of: "((option, g) \<in> option_of) = ((Rep_game option, Rep_game g) \<in> is_option_of)"
obua@19203
   304
  apply (auto simp add: option_of_def)
obua@19203
   305
  apply (subst Abs_game_inverse)
obua@19203
   306
  apply (simp add: is_option_of_imp_games game_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 simp
obua@19203
   310
  apply (auto simp add: Bex_def image_def)  
obua@19203
   311
  apply (rule exI[where x="Rep_game option"])
obua@19203
   312
  apply (rule exI[where x="Rep_game g"])
obua@19203
   313
  apply (simp add: Rep_game_inverse)
obua@19203
   314
  done
obua@19203
   315
  
obua@19203
   316
lemma wf_is_option_of: "wf is_option_of"
obua@19203
   317
  apply (rule wfzf_implies_wf)
obua@19203
   318
  apply (simp add: wfzf_is_option_of)
obua@19203
   319
  done
obua@19203
   320
obua@19203
   321
lemma wf_option_of[recdef_wf, simp, intro]: "wf option_of"
obua@19203
   322
proof -
obua@19203
   323
  have option_of: "option_of = inv_image is_option_of Rep_game"
obua@19203
   324
    apply (rule set_ext)
obua@19203
   325
    apply (case_tac "x")
krauss@19769
   326
    by (simp add: option_to_is_option_of) 
obua@19203
   327
  show ?thesis
obua@19203
   328
    apply (simp add: option_of)
obua@19203
   329
    apply (auto intro: wf_inv_image wf_is_option_of)
obua@19203
   330
    done
obua@19203
   331
qed
obua@19203
   332
  
obua@19203
   333
lemma right_option_is_option[simp, intro]: "zin x (right_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   334
  by (simp add: options_def zunion)
obua@19203
   335
obua@19203
   336
lemma left_option_is_option[simp, intro]: "zin x (left_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   337
  by (simp add: options_def zunion)
obua@19203
   338
obua@19203
   339
lemma zin_options[simp, intro]: "zin x (options g) \<Longrightarrow> (x, g) \<in> option_of"
obua@19203
   340
  apply (simp add: options_def zunion left_options_def right_options_def option_of_def 
obua@19203
   341
    image_def is_option_of_def zimage_iff zin_zexplode_eq) 
obua@19203
   342
  apply (cases g)
obua@19203
   343
  apply (cases x)
obua@19203
   344
  apply (auto simp add: Abs_game_inverse games_lfp_eq_gfp[symmetric] game_def 
obua@19203
   345
    right_option_def[symmetric] left_option_def[symmetric])
obua@19203
   346
  done
obua@19203
   347
obua@19203
   348
consts
obua@19203
   349
  neg_game :: "game \<Rightarrow> game"
obua@19203
   350
obua@19203
   351
recdef neg_game "option_of"
obua@19203
   352
  "neg_game g = Game (zimage neg_game (right_options g)) (zimage neg_game (left_options g))"
obua@19203
   353
obua@19203
   354
declare neg_game.simps[simp del]
obua@19203
   355
obua@19203
   356
lemma "neg_game (neg_game g) = g"
obua@19203
   357
  apply (induct g rule: neg_game.induct)
obua@19203
   358
  apply (subst neg_game.simps)+
obua@19203
   359
  apply (simp add: right_options left_options comp_zimage_eq)
obua@19203
   360
  apply (subgoal_tac "zimage (neg_game o neg_game) (left_options g) = left_options g")
obua@19203
   361
  apply (subgoal_tac "zimage (neg_game o neg_game) (right_options g) = right_options g")
obua@19203
   362
  apply (auto simp add: game_split[symmetric])
obua@19203
   363
  apply (auto simp add: zet_ext_eq zimage_iff)
obua@19203
   364
  done
obua@19203
   365
obua@19203
   366
consts
obua@19203
   367
  ge_game :: "(game * game) \<Rightarrow> bool" 
obua@19203
   368
berghofe@23771
   369
recdef ge_game "(gprod_2_1 option_of)"
obua@19203
   370
  "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))"
obua@19203
   374
(hints simp: gprod_2_1_def)
obua@19203
   375
obua@19203
   376
declare ge_game.simps [simp del]
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@26304
   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@26304
   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@26304
   436
	by (subst ge_game_eq, auto)
obua@19203
   437
    }
obua@19203
   438
  qed
obua@19203
   439
qed
obua@19203
   440
	
obua@19203
   441
constdefs
obua@19203
   442
  eq_game :: "game \<Rightarrow> game \<Rightarrow> bool"
obua@19203
   443
  "eq_game G H \<equiv> ge_game (G, H) \<and> ge_game (H, G)" 
obua@19203
   444
obua@19203
   445
lemma eq_game_sym: "(eq_game G H) = (eq_game H G)"
obua@19203
   446
  by (auto simp add: eq_game_def)
obua@19203
   447
obua@19203
   448
lemma eq_game_refl: "eq_game G G"
obua@19203
   449
  by (simp add: ge_game_refl eq_game_def)
obua@19203
   450
berghofe@23771
   451
lemma induct_game: "(\<And>x. \<forall>y. (y, x) \<in> lprod option_of \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
berghofe@23771
   452
  by (erule wf_induct[OF wf_lprod[OF wf_option_of]])
obua@19203
   453
obua@19203
   454
lemma ge_game_trans:
obua@19203
   455
  assumes "ge_game (x, y)" "ge_game (y, z)" 
obua@19203
   456
  shows "ge_game (x, z)"
obua@19203
   457
proof -  
obua@19203
   458
  { 
obua@19203
   459
    fix a
obua@19203
   460
    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
   461
    proof (induct a rule: induct_game)
obua@19203
   462
      case (1 a)
obua@19203
   463
      show ?case
obua@19203
   464
      proof (rule allI | rule impI)+
obua@19203
   465
	case (goal1 x y z)
obua@19203
   466
	show ?case
obua@19203
   467
	proof -
obua@19203
   468
	  { fix xr
obua@19203
   469
            assume xr:"zin xr (right_options x)"
obua@19203
   470
	    assume "ge_game (z, xr)"
obua@19203
   471
	    have "ge_game (y, xr)"
obua@19203
   472
	      apply (rule 1[rule_format, where y="[y,z,xr]"])
obua@19203
   473
	      apply (auto intro: xr lprod_3_1 simp add: prems)
obua@19203
   474
	      done
obua@19203
   475
	    moreover from xr have "\<not> ge_game (y, xr)"
wenzelm@26304
   476
	      by (simp add: goal1(2)[simplified ge_game_eq[of x y], rule_format, of xr, simplified xr])
obua@19203
   477
	    ultimately have "False" by auto      
obua@19203
   478
	  }
obua@19203
   479
	  note xr = this
obua@19203
   480
	  { fix zl
obua@19203
   481
	    assume zl:"zin zl (left_options z)"
obua@19203
   482
	    assume "ge_game (zl, x)"
obua@19203
   483
	    have "ge_game (zl, y)"
obua@19203
   484
	      apply (rule 1[rule_format, where y="[zl,x,y]"])
obua@19203
   485
	      apply (auto intro: zl lprod_3_2 simp add: prems)
obua@19203
   486
	      done
obua@19203
   487
	    moreover from zl have "\<not> ge_game (zl, y)"
wenzelm@26304
   488
	      by (simp add: goal1(3)[simplified ge_game_eq[of y z], rule_format, of zl, simplified zl])
obua@19203
   489
	    ultimately have "False" by auto
obua@19203
   490
	  }
obua@19203
   491
	  note zl = this
obua@19203
   492
	  show ?thesis
wenzelm@26304
   493
	    by (auto simp add: ge_game_eq[of x z] intro: xr zl)
obua@19203
   494
	qed
obua@19203
   495
      qed
obua@19203
   496
    qed
obua@19203
   497
  } 
obua@19203
   498
  note trans = this[of "[x, y, z]", simplified, rule_format]    
obua@19203
   499
  with prems show ?thesis by blast
obua@19203
   500
qed
obua@19203
   501
obua@19203
   502
lemma eq_game_trans: "eq_game a b \<Longrightarrow> eq_game b c \<Longrightarrow> eq_game a c"
obua@19203
   503
  by (auto simp add: eq_game_def intro: ge_game_trans)
obua@19203
   504
obua@19203
   505
constdefs
obua@19203
   506
  zero_game :: game
obua@19203
   507
  "zero_game \<equiv> Game zempty zempty"
obua@19203
   508
obua@19203
   509
consts 
obua@19203
   510
  plus_game :: "game * game \<Rightarrow> game"
obua@19203
   511
berghofe@23771
   512
recdef plus_game "gprod_2_2 option_of"
obua@19203
   513
  "plus_game (G, H) = Game (zunion (zimage (\<lambda> g. plus_game (g, H)) (left_options G))
obua@19203
   514
                                   (zimage (\<lambda> h. plus_game (G, h)) (left_options H)))
obua@19203
   515
                           (zunion (zimage (\<lambda> g. plus_game (g, H)) (right_options G))
obua@19203
   516
                                   (zimage (\<lambda> h. plus_game (G, h)) (right_options H)))"
obua@19203
   517
(hints simp add: gprod_2_2_def)
obua@19203
   518
obua@19203
   519
declare plus_game.simps[simp del]
obua@19203
   520
obua@19203
   521
lemma plus_game_comm: "plus_game (G, H) = plus_game (H, G)"
obua@19203
   522
proof (induct G H rule: plus_game.induct)
obua@19203
   523
  case (1 G H)
obua@19203
   524
  show ?case
obua@19203
   525
    by (auto simp add: 
obua@19203
   526
      plus_game.simps[where G=G and H=H] 
obua@19203
   527
      plus_game.simps[where G=H and H=G]
obua@19203
   528
      Game_ext zet_ext_eq zunion zimage_iff prems)
obua@19203
   529
qed
obua@19203
   530
obua@19203
   531
lemma game_ext_eq: "(G = H) = (left_options G = left_options H \<and> right_options G = right_options H)"
obua@19203
   532
proof -
obua@19203
   533
  have "(G = H) = (Game (left_options G) (right_options G) = Game (left_options H) (right_options H))"
obua@19203
   534
    by (simp add: game_split[symmetric])
obua@19203
   535
  then show ?thesis by auto
obua@19203
   536
qed
obua@19203
   537
obua@19203
   538
lemma left_zero_game[simp]: "left_options (zero_game) = zempty"
obua@19203
   539
  by (simp add: zero_game_def)
obua@19203
   540
obua@19203
   541
lemma right_zero_game[simp]: "right_options (zero_game) = zempty"
obua@19203
   542
  by (simp add: zero_game_def)
obua@19203
   543
obua@19203
   544
lemma plus_game_zero_right[simp]: "plus_game (G, zero_game) = G"
obua@19203
   545
proof -
obua@19203
   546
  { 
obua@19203
   547
    fix G H
obua@19203
   548
    have "H = zero_game \<longrightarrow> plus_game (G, H) = G "
obua@19203
   549
    proof (induct G H rule: plus_game.induct, rule impI)
obua@19203
   550
      case (goal1 G H)
obua@19203
   551
      note induct_hyp = prems[simplified goal1, simplified] and prems
obua@19203
   552
      show ?case
obua@19203
   553
	apply (simp only: plus_game.simps[where G=G and H=H])
obua@19203
   554
	apply (simp add: game_ext_eq prems)
obua@19203
   555
	apply (auto simp add: 
obua@19203
   556
	  zimage_cong[where f = "\<lambda> g. plus_game (g, zero_game)" and g = "id"] 
obua@19203
   557
	  induct_hyp)
obua@19203
   558
	done
obua@19203
   559
    qed
obua@19203
   560
  }
obua@19203
   561
  then show ?thesis by auto
obua@19203
   562
qed
obua@19203
   563
obua@19203
   564
lemma plus_game_zero_left: "plus_game (zero_game, G) = G"
obua@19203
   565
  by (simp add: plus_game_comm)
obua@19203
   566
obua@19203
   567
lemma left_imp_options[simp]: "zin opt (left_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   568
  by (simp add: options_def zunion)
obua@19203
   569
obua@19203
   570
lemma right_imp_options[simp]: "zin opt (right_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   571
  by (simp add: options_def zunion)
obua@19203
   572
obua@19203
   573
lemma left_options_plus: 
obua@19203
   574
  "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
   575
  by (subst plus_game.simps, simp)
obua@19203
   576
obua@19203
   577
lemma right_options_plus:
obua@19203
   578
  "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
   579
  by (subst plus_game.simps, simp)
obua@19203
   580
obua@19203
   581
lemma left_options_neg: "left_options (neg_game u) = zimage neg_game (right_options u)"	 
obua@19203
   582
  by (subst neg_game.simps, simp)
obua@19203
   583
obua@19203
   584
lemma right_options_neg: "right_options (neg_game u) = zimage neg_game (left_options u)"
obua@19203
   585
  by (subst neg_game.simps, simp)
obua@19203
   586
  
obua@19203
   587
lemma plus_game_assoc: "plus_game (plus_game (F, G), H) = plus_game (F, plus_game (G, H))"
obua@19203
   588
proof -
obua@19203
   589
  { 
obua@19203
   590
    fix a
obua@19203
   591
    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
   592
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   593
      case (goal1 x F G H)
obua@19203
   594
      let ?L = "plus_game (plus_game (F, G), H)"
obua@19203
   595
      let ?R = "plus_game (F, plus_game (G, H))"
obua@19203
   596
      note options_plus = left_options_plus right_options_plus
obua@19203
   597
      {
obua@19203
   598
	fix opt
obua@19203
   599
	note hyp = goal1(1)[simplified goal1(2), rule_format] 
obua@19203
   600
	have F: "zin opt (options F)  \<Longrightarrow> plus_game (plus_game (opt, G), H) = plus_game (opt, plus_game (G, H))"
obua@19203
   601
	  by (blast intro: hyp lprod_3_3)
obua@19203
   602
	have G: "zin opt (options G) \<Longrightarrow> plus_game (plus_game (F, opt), H) = plus_game (F, plus_game (opt, H))"
obua@19203
   603
	  by (blast intro: hyp lprod_3_4)
obua@19203
   604
	have H: "zin opt (options H) \<Longrightarrow> plus_game (plus_game (F, G), opt) = plus_game (F, plus_game (G, opt))" 
obua@19203
   605
	  by (blast intro: hyp lprod_3_5)
obua@19203
   606
	note F and G and H
obua@19203
   607
      }
obua@19203
   608
      note induct_hyp = this
obua@19203
   609
      have "left_options ?L = left_options ?R \<and> right_options ?L = right_options ?R"
obua@19203
   610
	by (auto simp add: 
obua@19203
   611
	  plus_game.simps[where G="plus_game (F,G)" and H=H]
obua@19203
   612
	  plus_game.simps[where G="F" and H="plus_game (G,H)"] 
obua@19203
   613
	  zet_ext_eq zunion zimage_iff options_plus
obua@19203
   614
	  induct_hyp left_imp_options right_imp_options)
obua@19203
   615
      then show ?case
obua@19203
   616
	by (simp add: game_ext_eq)
obua@19203
   617
    qed
obua@19203
   618
  }
obua@19203
   619
  then show ?thesis by auto
obua@19203
   620
qed
obua@19203
   621
obua@19203
   622
lemma neg_plus_game: "neg_game (plus_game (G, H)) = plus_game(neg_game G, neg_game H)"
obua@19203
   623
proof (induct G H rule: plus_game.induct)
obua@19203
   624
  case (1 G H)
obua@19203
   625
  note opt_ops = 
obua@19203
   626
    left_options_plus right_options_plus 
obua@19203
   627
    left_options_neg right_options_neg  
obua@19203
   628
  show ?case
obua@19203
   629
    by (auto simp add: opt_ops
obua@19203
   630
      neg_game.simps[of "plus_game (G,H)"]
obua@19203
   631
      plus_game.simps[of "neg_game G" "neg_game H"]
obua@19203
   632
      Game_ext zet_ext_eq zunion zimage_iff prems)
obua@19203
   633
qed
obua@19203
   634
obua@19203
   635
lemma eq_game_plus_inverse: "eq_game (plus_game (x, neg_game x)) zero_game"
obua@19203
   636
proof (induct x rule: wf_induct[OF wf_option_of])
obua@19203
   637
  case (goal1 x)
obua@19203
   638
  { fix y
obua@19203
   639
    assume "zin y (options x)"
obua@19203
   640
    then have "eq_game (plus_game (y, neg_game y)) zero_game"
obua@19203
   641
      by (auto simp add: prems)
obua@19203
   642
  }
obua@19203
   643
  note ihyp = this
obua@19203
   644
  {
obua@19203
   645
    fix y
obua@19203
   646
    assume y: "zin y (right_options x)"
obua@19203
   647
    have "\<not> (ge_game (zero_game, plus_game (y, neg_game x)))"
obua@19203
   648
      apply (subst ge_game.simps, simp)
obua@19203
   649
      apply (rule exI[where x="plus_game (y, neg_game y)"])
obua@19203
   650
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
obua@19203
   651
      apply (auto simp add: left_options_plus left_options_neg zunion zimage_iff intro: prems)
obua@19203
   652
      done
obua@19203
   653
  }
obua@19203
   654
  note case1 = this
obua@19203
   655
  {
obua@19203
   656
    fix y
obua@19203
   657
    assume y: "zin y (left_options x)"
obua@19203
   658
    have "\<not> (ge_game (zero_game, plus_game (x, neg_game y)))"
obua@19203
   659
      apply (subst ge_game.simps, simp)
obua@19203
   660
      apply (rule exI[where x="plus_game (y, neg_game y)"])
obua@19203
   661
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
obua@19203
   662
      apply (auto simp add: left_options_plus zunion zimage_iff intro: prems)
obua@19203
   663
      done
obua@19203
   664
  }
obua@19203
   665
  note case2 = this
obua@19203
   666
  {
obua@19203
   667
    fix y
obua@19203
   668
    assume y: "zin y (left_options x)"
obua@19203
   669
    have "\<not> (ge_game (plus_game (y, neg_game x), zero_game))"
obua@19203
   670
      apply (subst ge_game.simps, simp)
obua@19203
   671
      apply (rule exI[where x="plus_game (y, neg_game y)"])
obua@19203
   672
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
obua@19203
   673
      apply (auto simp add: right_options_plus right_options_neg zunion zimage_iff intro: prems)
obua@19203
   674
      done
obua@19203
   675
  }
obua@19203
   676
  note case3 = this
obua@19203
   677
  {
obua@19203
   678
    fix y
obua@19203
   679
    assume y: "zin y (right_options x)"
obua@19203
   680
    have "\<not> (ge_game (plus_game (x, neg_game y), zero_game))"
obua@19203
   681
      apply (subst ge_game.simps, simp)
obua@19203
   682
      apply (rule exI[where x="plus_game (y, neg_game y)"])
obua@19203
   683
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
obua@19203
   684
      apply (auto simp add: right_options_plus zunion zimage_iff intro: prems)
obua@19203
   685
      done
obua@19203
   686
  }
obua@19203
   687
  note case4 = this
obua@19203
   688
  show ?case
obua@19203
   689
    apply (simp add: eq_game_def)
obua@19203
   690
    apply (simp add: ge_game.simps[of "plus_game (x, neg_game x)" "zero_game"])
obua@19203
   691
    apply (simp add: ge_game.simps[of "zero_game" "plus_game (x, neg_game x)"])
obua@19203
   692
    apply (simp add: right_options_plus left_options_plus right_options_neg left_options_neg zunion zimage_iff)
obua@19203
   693
    apply (auto simp add: case1 case2 case3 case4)
obua@19203
   694
    done
obua@19203
   695
qed
obua@19203
   696
obua@19203
   697
lemma ge_plus_game_left: "ge_game (y,z) = ge_game(plus_game (x, y), plus_game (x, z))"
obua@19203
   698
proof -
obua@19203
   699
  { fix a
obua@19203
   700
    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
   701
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   702
      case (goal1 a x y z)
obua@19203
   703
      note induct_hyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   704
      { 
obua@19203
   705
	assume hyp: "ge_game(plus_game (x, y), plus_game (x, z))"
obua@19203
   706
	have "ge_game (y, z)"
obua@19203
   707
	proof -
obua@19203
   708
	  { fix yr
obua@19203
   709
	    assume yr: "zin yr (right_options y)"
obua@19203
   710
	    from hyp have "\<not> (ge_game (plus_game (x, z), plus_game (x, yr)))"
wenzelm@26304
   711
	      by (auto simp add: ge_game_eq[of "plus_game (x,y)" "plus_game(x,z)"]
obua@19203
   712
		right_options_plus zunion zimage_iff intro: yr)
obua@19203
   713
	    then have "\<not> (ge_game (z, yr))"
obua@19203
   714
	      apply (subst induct_hyp[where y="[x, z, yr]", of "x" "z" "yr"])
obua@19203
   715
	      apply (simp_all add: yr lprod_3_6)
obua@19203
   716
	      done
obua@19203
   717
	  }
obua@19203
   718
	  note yr = this
obua@19203
   719
	  { fix zl
obua@19203
   720
	    assume zl: "zin zl (left_options z)"
obua@19203
   721
	    from hyp have "\<not> (ge_game (plus_game (x, zl), plus_game (x, y)))"
wenzelm@26304
   722
	      by (auto simp add: ge_game_eq[of "plus_game (x,y)" "plus_game(x,z)"]
obua@19203
   723
		left_options_plus zunion zimage_iff intro: zl)
obua@19203
   724
	    then have "\<not> (ge_game (zl, y))"
obua@19203
   725
	      apply (subst goal1(1)[rule_format, where y="[x, zl, y]", of "x" "zl" "y"])
obua@19203
   726
	      apply (simp_all add: goal1(2) zl lprod_3_7)
obua@19203
   727
	      done
obua@19203
   728
	  }	
obua@19203
   729
	  note zl = this
obua@19203
   730
	  show "ge_game (y, z)"
wenzelm@26304
   731
	    apply (subst ge_game_eq)
obua@19203
   732
	    apply (auto simp add: yr zl)
obua@19203
   733
	    done
obua@19203
   734
	qed      
obua@19203
   735
      }
obua@19203
   736
      note right_imp_left = this
obua@19203
   737
      {
obua@19203
   738
	assume yz: "ge_game (y, z)"
obua@19203
   739
	{
obua@19203
   740
	  fix x'
obua@19203
   741
	  assume x': "zin x' (right_options x)"
obua@19203
   742
	  assume hyp: "ge_game (plus_game (x, z), plus_game (x', y))"
obua@19203
   743
	  then have n: "\<not> (ge_game (plus_game (x', y), plus_game (x', z)))"
wenzelm@26304
   744
	    by (auto simp add: ge_game_eq[of "plus_game (x,z)" "plus_game (x', y)"] 
obua@19203
   745
	      right_options_plus zunion zimage_iff intro: x')
obua@19203
   746
	  have t: "ge_game (plus_game (x', y), plus_game (x', z))"
obua@19203
   747
	    apply (subst induct_hyp[symmetric])
obua@19203
   748
	    apply (auto intro: lprod_3_3 x' yz)
obua@19203
   749
	    done
obua@19203
   750
	  from n t have "False" by blast
obua@19203
   751
	}    
obua@19203
   752
	note case1 = this
obua@19203
   753
	{
obua@19203
   754
	  fix x'
obua@19203
   755
	  assume x': "zin x' (left_options x)"
obua@19203
   756
	  assume hyp: "ge_game (plus_game (x', z), plus_game (x, y))"
obua@19203
   757
	  then have n: "\<not> (ge_game (plus_game (x', y), plus_game (x', z)))"
wenzelm@26304
   758
	    by (auto simp add: ge_game_eq[of "plus_game (x',z)" "plus_game (x, y)"] 
obua@19203
   759
	      left_options_plus zunion zimage_iff intro: x')
obua@19203
   760
	  have t: "ge_game (plus_game (x', y), plus_game (x', z))"
obua@19203
   761
	    apply (subst induct_hyp[symmetric])
obua@19203
   762
	    apply (auto intro: lprod_3_3 x' yz)
obua@19203
   763
	    done
obua@19203
   764
	  from n t have "False" by blast
obua@19203
   765
	}
obua@19203
   766
	note case3 = this
obua@19203
   767
	{
obua@19203
   768
	  fix y'
obua@19203
   769
	  assume y': "zin y' (right_options y)"
obua@19203
   770
	  assume hyp: "ge_game (plus_game(x, z), plus_game (x, y'))"
obua@19203
   771
	  then have "ge_game(z, y')"
obua@19203
   772
	    apply (subst induct_hyp[of "[x, z, y']" "x" "z" "y'"])
obua@19203
   773
	    apply (auto simp add: hyp lprod_3_6 y')
obua@19203
   774
	    done
obua@19203
   775
	  with yz have "ge_game (y, y')"
obua@19203
   776
	    by (blast intro: ge_game_trans)      
obua@19203
   777
	  with y' have "False" by (auto simp add: ge_game_leftright_refl)
obua@19203
   778
	}
obua@19203
   779
	note case2 = this
obua@19203
   780
	{
obua@19203
   781
	  fix z'
obua@19203
   782
	  assume z': "zin z' (left_options z)"
obua@19203
   783
	  assume hyp: "ge_game (plus_game(x, z'), plus_game (x, y))"
obua@19203
   784
	  then have "ge_game(z', y)"
obua@19203
   785
	    apply (subst induct_hyp[of "[x, z', y]" "x" "z'" "y"])
obua@19203
   786
	    apply (auto simp add: hyp lprod_3_7 z')
obua@19203
   787
	    done    
obua@19203
   788
	  with yz have "ge_game (z', z)"
obua@19203
   789
	    by (blast intro: ge_game_trans)      
obua@19203
   790
	  with z' have "False" by (auto simp add: ge_game_leftright_refl)
obua@19203
   791
	}
obua@19203
   792
	note case4 = this   
obua@19203
   793
	have "ge_game(plus_game (x, y), plus_game (x, z))"
wenzelm@26304
   794
	  apply (subst ge_game_eq)
obua@19203
   795
	  apply (auto simp add: right_options_plus left_options_plus zunion zimage_iff)
obua@19203
   796
	  apply (auto intro: case1 case2 case3 case4)
obua@19203
   797
	  done
obua@19203
   798
      }
obua@19203
   799
      note left_imp_right = this
obua@19203
   800
      show ?case by (auto intro: right_imp_left left_imp_right)
obua@19203
   801
    qed
obua@19203
   802
  }
obua@19203
   803
  note a = this[of "[x, y, z]"]
obua@19203
   804
  then show ?thesis by blast
obua@19203
   805
qed
obua@19203
   806
obua@19203
   807
lemma ge_plus_game_right: "ge_game (y,z) = ge_game(plus_game (y, x), plus_game (z, x))"
obua@19203
   808
  by (simp add: ge_plus_game_left plus_game_comm)
obua@19203
   809
obua@19203
   810
lemma ge_neg_game: "ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   811
proof -
obua@19203
   812
  { fix a
obua@19203
   813
    have "\<forall> x y. a = [x, y] \<longrightarrow> ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   814
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   815
      case (goal1 a x y)
obua@19203
   816
      note ihyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   817
      { fix xl
obua@19203
   818
	assume xl: "zin xl (left_options x)"
obua@19203
   819
	have "ge_game (neg_game y, neg_game xl) = ge_game (xl, y)"
obua@19203
   820
	  apply (subst ihyp)
obua@19203
   821
	  apply (auto simp add: lprod_2_1 xl)
obua@19203
   822
	  done
obua@19203
   823
      }
obua@19203
   824
      note xl = this
obua@19203
   825
      { fix yr
obua@19203
   826
	assume yr: "zin yr (right_options y)"
obua@19203
   827
	have "ge_game (neg_game yr, neg_game x) = ge_game (x, yr)"
obua@19203
   828
	  apply (subst ihyp)
obua@19203
   829
	  apply (auto simp add: lprod_2_2 yr)
obua@19203
   830
	  done
obua@19203
   831
      }
obua@19203
   832
      note yr = this
obua@19203
   833
      show ?case
wenzelm@26304
   834
	by (auto simp add: ge_game_eq[of "neg_game x" "neg_game y"] ge_game_eq[of "y" "x"]
obua@19203
   835
	  right_options_neg left_options_neg zimage_iff  xl yr)
obua@19203
   836
    qed
obua@19203
   837
  }
obua@19203
   838
  note a = this[of "[x,y]"]
obua@19203
   839
  then show ?thesis by blast
obua@19203
   840
qed
obua@19203
   841
obua@19203
   842
constdefs 
obua@19203
   843
  eq_game_rel :: "(game * game) set"
obua@19203
   844
  "eq_game_rel \<equiv> { (p, q) . eq_game p q }"
obua@19203
   845
obua@19203
   846
typedef Pg = "UNIV//eq_game_rel"
obua@19203
   847
  by (auto simp add: quotient_def)
obua@19203
   848
obua@19203
   849
lemma equiv_eq_game[simp]: "equiv UNIV eq_game_rel"
obua@19203
   850
  by (auto simp add: equiv_def refl_def sym_def trans_def eq_game_rel_def
obua@19203
   851
    eq_game_sym intro: eq_game_refl eq_game_trans)
obua@19203
   852
haftmann@25764
   853
instantiation Pg :: "{ord, zero, plus, minus, uminus}"
haftmann@25764
   854
begin
haftmann@25764
   855
haftmann@25764
   856
definition
haftmann@25764
   857
  Pg_zero_def: "0 = Abs_Pg (eq_game_rel `` {zero_game})"
haftmann@25764
   858
haftmann@25764
   859
definition
haftmann@25764
   860
  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
   861
haftmann@25764
   862
definition
haftmann@25764
   863
  Pg_less_def: "G < H \<longleftrightarrow> G \<le> H \<and> G \<noteq> (H::Pg)"
obua@19203
   864
haftmann@25764
   865
definition
haftmann@25764
   866
  Pg_minus_def: "- G = contents (\<Union> g \<in> Rep_Pg G. {Abs_Pg (eq_game_rel `` {neg_game g})})"
haftmann@25764
   867
haftmann@25764
   868
definition
haftmann@25764
   869
  Pg_plus_def: "G + H = contents (\<Union> g \<in> Rep_Pg G. \<Union> h \<in> Rep_Pg H. {Abs_Pg (eq_game_rel `` {plus_game (g,h)})})"
haftmann@25764
   870
haftmann@25764
   871
definition
haftmann@25764
   872
  Pg_diff_def: "G - H = G + (- (H::Pg))"
haftmann@25764
   873
haftmann@25764
   874
instance ..
haftmann@25764
   875
haftmann@25764
   876
end
obua@19203
   877
obua@19203
   878
lemma Rep_Abs_eq_Pg[simp]: "Rep_Pg (Abs_Pg (eq_game_rel `` {g})) = eq_game_rel `` {g}"
obua@19203
   879
  apply (subst Abs_Pg_inverse)
obua@19203
   880
  apply (auto simp add: Pg_def quotient_def)
obua@19203
   881
  done
obua@19203
   882
obua@19203
   883
lemma char_Pg_le[simp]: "(Abs_Pg (eq_game_rel `` {g}) \<le> Abs_Pg (eq_game_rel `` {h})) = (ge_game (h, g))"
obua@19203
   884
  apply (simp add: Pg_le_def)
obua@19203
   885
  apply (auto simp add: eq_game_rel_def eq_game_def intro: ge_game_trans ge_game_refl)
obua@19203
   886
  done
obua@19203
   887
obua@19203
   888
lemma char_Pg_eq[simp]: "(Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {h})) = (eq_game g h)"
obua@19203
   889
  apply (simp add: Rep_Pg_inject [symmetric])
obua@19203
   890
  apply (subst eq_equiv_class_iff[of UNIV])
obua@19203
   891
  apply (simp_all)
obua@19203
   892
  apply (simp add: eq_game_rel_def)
obua@19203
   893
  done
obua@19203
   894
obua@19203
   895
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
   896
proof -
obua@19203
   897
  have "(\<lambda> g h. {Abs_Pg (eq_game_rel `` {plus_game (g, h)})}) respects2 eq_game_rel" 
obua@19203
   898
    apply (simp add: congruent2_def)
obua@19203
   899
    apply (auto simp add: eq_game_rel_def eq_game_def)
obua@19203
   900
    apply (rule_tac y="plus_game (y1, z2)" in ge_game_trans)
obua@19203
   901
    apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+
obua@19203
   902
    apply (rule_tac y="plus_game (z1, y2)" in ge_game_trans)
obua@19203
   903
    apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+
obua@19203
   904
    done
obua@19203
   905
  then show ?thesis
obua@19203
   906
    by (simp add: Pg_plus_def UN_equiv_class2[OF equiv_eq_game equiv_eq_game]) 
obua@19203
   907
qed
obua@19203
   908
    
obua@19203
   909
lemma char_Pg_minus[simp]: "- Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {neg_game g})"
obua@19203
   910
proof -
obua@19203
   911
  have "(\<lambda> g. {Abs_Pg (eq_game_rel `` {neg_game g})}) respects eq_game_rel"
obua@19203
   912
    apply (simp add: congruent_def)
obua@19203
   913
    apply (auto simp add: eq_game_rel_def eq_game_def ge_neg_game)
obua@19203
   914
    done    
obua@19203
   915
  then show ?thesis
obua@19203
   916
    by (simp add: Pg_minus_def UN_equiv_class[OF equiv_eq_game])
obua@19203
   917
qed
obua@19203
   918
obua@19203
   919
lemma eq_Abs_Pg[rule_format, cases type: Pg]: "(\<forall> g. z = Abs_Pg (eq_game_rel `` {g}) \<longrightarrow> P) \<longrightarrow> P"
obua@19203
   920
  apply (cases z, simp)
obua@19203
   921
  apply (simp add: Rep_Pg_inject[symmetric])
obua@19203
   922
  apply (subst Abs_Pg_inverse, simp)
obua@19203
   923
  apply (auto simp add: Pg_def quotient_def)
obua@19203
   924
  done
obua@19203
   925
obua@19203
   926
instance Pg :: pordered_ab_group_add 
obua@19203
   927
proof
obua@19203
   928
  fix a b c :: Pg
obua@19203
   929
  show "(a < b) = (a \<le> b \<and> a \<noteq> b)" by (simp add: Pg_less_def)
obua@19203
   930
  show "a - b = a + (- b)" by (simp add: Pg_diff_def)
obua@19203
   931
  {
obua@19203
   932
    assume ab: "a \<le> b"
obua@19203
   933
    assume ba: "b \<le> a"
obua@19203
   934
    from ab ba show "a = b"
obua@19203
   935
      apply (cases a, cases b)
obua@19203
   936
      apply (simp add: eq_game_def)
obua@19203
   937
      done
obua@19203
   938
  }
obua@19203
   939
  show "a + b = b + a"
obua@19203
   940
    apply (cases a, cases b)
obua@19203
   941
    apply (simp add: eq_game_def plus_game_comm)
obua@19203
   942
    done
obua@19203
   943
  show "a + b + c = a + (b + c)"
obua@19203
   944
    apply (cases a, cases b, cases c)
obua@19203
   945
    apply (simp add: eq_game_def plus_game_assoc)
obua@19203
   946
    done
obua@19203
   947
  show "0 + a = a"
obua@19203
   948
    apply (cases a)
obua@19203
   949
    apply (simp add: Pg_zero_def plus_game_zero_left)
obua@19203
   950
    done
obua@19203
   951
  show "- a + a = 0"
obua@19203
   952
    apply (cases a)
obua@19203
   953
    apply (simp add: Pg_zero_def eq_game_plus_inverse plus_game_comm)
obua@19203
   954
    done
obua@19203
   955
  show "a \<le> a"
obua@19203
   956
    apply (cases a)
obua@19203
   957
    apply (simp add: ge_game_refl)
obua@19203
   958
    done
obua@19203
   959
  {
obua@19203
   960
    assume ab: "a \<le> b"
obua@19203
   961
    assume bc: "b \<le> c"
obua@19203
   962
    from ab bc show "a \<le> c"
obua@19203
   963
      apply (cases a, cases b, cases c)
obua@19203
   964
      apply (auto intro: ge_game_trans)
obua@19203
   965
      done
obua@19203
   966
  }
obua@19203
   967
  {
obua@19203
   968
    assume ab: "a \<le> b"
obua@19203
   969
    from ab show "c + a \<le> c + b"
obua@19203
   970
      apply (cases a, cases b, cases c)
obua@19203
   971
      apply (simp add: ge_plus_game_left[symmetric])
obua@19203
   972
      done
obua@19203
   973
  }
obua@19203
   974
qed
obua@19203
   975
obua@19203
   976
end