src/HOL/ZF/Games.thy
author wenzelm
Wed Nov 30 16:27:10 2011 +0100 (2011-11-30)
changeset 45694 4a8743618257
parent 44011 f67c93f52d13
child 46555 c2b5900988e2
permissions -rw-r--r--
prefer typedef without extra definition and alternative name;
tuned proofs;
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(*  Title:      HOL/ZF/Games.thy
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    Author:     Steven Obua
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An application of HOLZF: Partizan Games.  See "Partizan Games in
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Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
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*)
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theory Games 
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imports MainZF
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begin
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definition fixgames :: "ZF set \<Rightarrow> ZF set" where
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  "fixgames A \<equiv> { Opair l r | l r. explode l \<subseteq> A & explode r \<subseteq> A}"
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definition games_lfp :: "ZF set" where
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  "games_lfp \<equiv> lfp fixgames"
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definition games_gfp :: "ZF set" where
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  "games_gfp \<equiv> gfp fixgames"
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lemma mono_fixgames: "mono (fixgames)"
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  apply (auto simp add: mono_def fixgames_def)
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  apply (rule_tac x=l in exI)
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  apply (rule_tac x=r in exI)
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  apply auto
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  done
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lemma games_lfp_unfold: "games_lfp = fixgames games_lfp"
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  by (auto simp add: def_lfp_unfold games_lfp_def mono_fixgames)
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lemma games_gfp_unfold: "games_gfp = fixgames games_gfp"
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  by (auto simp add: def_gfp_unfold games_gfp_def mono_fixgames)
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lemma games_lfp_nonempty: "Opair Empty Empty \<in> games_lfp"
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proof -
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  have "fixgames {} \<subseteq> games_lfp" 
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    apply (subst games_lfp_unfold)
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    apply (simp add: mono_fixgames[simplified mono_def, rule_format])
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    done
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  moreover have "fixgames {} = {Opair Empty Empty}"
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    by (simp add: fixgames_def explode_Empty)
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  finally show ?thesis
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    by auto
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qed
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definition left_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" where
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  "left_option g opt \<equiv> (Elem opt (Fst g))"
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definition right_option :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" where
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  "right_option g opt \<equiv> (Elem opt (Snd g))"
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definition is_option_of :: "(ZF * ZF) set" where
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  "is_option_of \<equiv> { (opt, g) | opt g. g \<in> games_gfp \<and> (left_option g opt \<or> right_option g opt) }"
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lemma games_lfp_subset_gfp: "games_lfp \<subseteq> games_gfp"
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proof -
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  have "games_lfp \<subseteq> fixgames games_lfp" 
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    by (simp add: games_lfp_unfold[symmetric])
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  then show ?thesis
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    by (simp add: games_gfp_def gfp_upperbound)
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qed
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lemma games_option_stable: 
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  assumes fixgames: "games = fixgames games"
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  and g: "g \<in> games"
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  and opt: "left_option g opt \<or> right_option g opt"
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  shows "opt \<in> games"
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proof -
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  from g fixgames have "g \<in> fixgames games" by auto
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  then have "\<exists> l r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games"
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    by (simp add: fixgames_def)
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  then obtain l where "\<exists> r. g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
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  then obtain r where lr: "g = Opair l r \<and> explode l \<subseteq> games \<and> explode r \<subseteq> games" ..
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  with opt show ?thesis
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    by (auto intro: Elem_explode_in simp add: left_option_def right_option_def Fst Snd)
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qed
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lemma option2elem: "(opt,g) \<in> is_option_of  \<Longrightarrow> \<exists> u v. Elem opt u \<and> Elem u v \<and> Elem v g"
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  apply (simp add: is_option_of_def)
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  apply (subgoal_tac "(g \<in> games_gfp) = (g \<in> (fixgames games_gfp))")
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  prefer 2
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  apply (simp add: games_gfp_unfold[symmetric])
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  apply (auto simp add: fixgames_def left_option_def right_option_def Fst Snd)
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  apply (rule_tac x=l in exI, insert Elem_Opair_exists, blast)
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  apply (rule_tac x=r in exI, insert Elem_Opair_exists, blast) 
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  done
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lemma is_option_of_subset_is_Elem_of: "is_option_of \<subseteq> (is_Elem_of^+)"
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proof -
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  {
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    fix opt
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    fix g
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    assume "(opt, g) \<in> is_option_of"
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    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^+)" 
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      apply -
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      apply (drule option2elem)
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      apply (auto simp add: r_into_trancl' is_Elem_of_def)
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      done
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    then have "(opt, g) \<in> (is_Elem_of^+)"
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      by (blast intro: trancl_into_rtrancl trancl_rtrancl_trancl)
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  } 
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  then show ?thesis by auto
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qed
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lemma wfzf_is_option_of: "wfzf is_option_of"
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proof - 
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  have "wfzf (is_Elem_of^+)" by (simp add: wfzf_trancl wfzf_is_Elem_of)
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  then show ?thesis 
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    apply (rule wfzf_subset)
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    apply (rule is_option_of_subset_is_Elem_of)
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    done
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  qed
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lemma games_gfp_imp_lfp: "g \<in> games_gfp \<longrightarrow> g \<in> games_lfp"
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proof -
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  have unfold_gfp: "\<And> x. x \<in> games_gfp \<Longrightarrow> x \<in> (fixgames games_gfp)" 
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    by (simp add: games_gfp_unfold[symmetric])
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  have unfold_lfp: "\<And> x. (x \<in> games_lfp) = (x \<in> (fixgames games_lfp))"
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    by (simp add: games_lfp_unfold[symmetric])
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  show ?thesis
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    apply (rule wf_induct[OF wfzf_implies_wf[OF wfzf_is_option_of]])
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    apply (auto simp add: is_option_of_def)
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    apply (drule_tac unfold_gfp)
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    apply (simp add: fixgames_def)
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    apply (auto simp add: left_option_def Fst right_option_def Snd)
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    apply (subgoal_tac "explode l \<subseteq> games_lfp")
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    apply (subgoal_tac "explode r \<subseteq> games_lfp")
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    apply (subst unfold_lfp)
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    apply (auto simp add: fixgames_def)
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    apply (simp_all add: explode_Elem Elem_explode_in)
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    done
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qed
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theorem games_lfp_eq_gfp: "games_lfp = games_gfp"
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  apply (auto simp add: games_gfp_imp_lfp)
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  apply (insert games_lfp_subset_gfp)
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  apply auto
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  done
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theorem unique_games: "(g = fixgames g) = (g = games_lfp)"
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proof -
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  {
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    fix g 
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    assume g: "g = fixgames g"
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    from g have "fixgames g \<subseteq> g" by auto
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    then have l:"games_lfp \<subseteq> g" 
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      by (simp add: games_lfp_def lfp_lowerbound)
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    from g have "g \<subseteq> fixgames g" by auto
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    then have u:"g \<subseteq> games_gfp" 
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      by (simp add: games_gfp_def gfp_upperbound)
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    from l u games_lfp_eq_gfp[symmetric] have "g = games_lfp"
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      by auto
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  }
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  note games = this
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  show ?thesis
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    apply (rule iff[rule_format])
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    apply (erule games)
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    apply (simp add: games_lfp_unfold[symmetric])
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    done
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qed
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lemma games_lfp_option_stable: 
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  assumes g: "g \<in> games_lfp"
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  and opt: "left_option g opt \<or> right_option g opt"
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  shows "opt \<in> games_lfp"
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  apply (rule games_option_stable[where g=g])
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  apply (simp add: games_lfp_unfold[symmetric])
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  apply (simp_all add: assms)
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  done
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lemma is_option_of_imp_games:
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  assumes hyp: "(opt, g) \<in> is_option_of"
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  shows "opt \<in> games_lfp \<and> g \<in> games_lfp"
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proof -
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  from hyp have g_game: "g \<in> games_lfp" 
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    by (simp add: is_option_of_def games_lfp_eq_gfp)
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  from hyp have "left_option g opt \<or> right_option g opt"
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    by (auto simp add: is_option_of_def)
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  with g_game games_lfp_option_stable[OF g_game, OF this] show ?thesis
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    by auto
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qed
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lemma games_lfp_represent: "x \<in> games_lfp \<Longrightarrow> \<exists> l r. x = Opair l r"
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  apply (rule exI[where x="Fst x"])
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  apply (rule exI[where x="Snd x"])
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  apply (subgoal_tac "x \<in> (fixgames games_lfp)")
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  apply (simp add: fixgames_def)
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  apply (auto simp add: Fst Snd)
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  apply (simp add: games_lfp_unfold[symmetric])
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  done
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definition "game = games_lfp"
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typedef (open) game = game
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  unfolding game_def by (blast intro: games_lfp_nonempty)
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definition left_options :: "game \<Rightarrow> game zet" where
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  "left_options g \<equiv> zimage Abs_game (zexplode (Fst (Rep_game g)))"
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definition right_options :: "game \<Rightarrow> game zet" where
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  "right_options g \<equiv> zimage Abs_game (zexplode (Snd (Rep_game g)))"
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definition options :: "game \<Rightarrow> game zet" where
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  "options g \<equiv> zunion (left_options g) (right_options g)"
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definition Game :: "game zet \<Rightarrow> game zet \<Rightarrow> game" where
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  "Game L R \<equiv> Abs_game (Opair (zimplode (zimage Rep_game L)) (zimplode (zimage Rep_game R)))"
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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"
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  apply (subst Ext)
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  apply (simp add: Repl)
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  apply auto
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  apply (subst Abs_game_inverse, simp_all add: game_def)
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  apply (rule_tac x=za in exI)
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  apply (subst Abs_game_inverse, simp_all add: game_def)
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  done
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lemma game_split: "g = Game (left_options g) (right_options g)"
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proof -
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  have "\<exists> l r. Rep_game g = Opair l r"
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    apply (insert Rep_game[of g])
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    apply (simp add: game_def games_lfp_represent)
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    done
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  then obtain l r where lr: "Rep_game g = Opair l r" by auto
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  have partizan_g: "Rep_game g \<in> games_lfp" 
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    apply (insert Rep_game[of g])
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    apply (simp add: game_def)
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    done
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  have "\<forall> e. Elem e l \<longrightarrow> left_option (Rep_game g) e"
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    by (simp add: lr left_option_def Fst)
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  then have partizan_l: "\<forall> e. Elem e l \<longrightarrow> e \<in> games_lfp"
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    apply auto
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    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
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    apply auto
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    done
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  have "\<forall> e. Elem e r \<longrightarrow> right_option (Rep_game g) e"
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    by (simp add: lr right_option_def Snd)
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  then have partizan_r: "\<forall> e. Elem e r \<longrightarrow> e \<in> games_lfp"
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    apply auto
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    apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g])
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    apply auto
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    done   
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  let ?L = "zimage (Abs_game) (zexplode l)"
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  let ?R = "zimage (Abs_game) (zexplode r)"
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  have L:"?L = left_options g"
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    by (simp add: left_options_def lr Fst)
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  have R:"?R = right_options g"
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    by (simp add: right_options_def lr Snd)
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  have "g = Game ?L ?R"
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    apply (simp add: Game_def Rep_game_inject[symmetric] comp_zimage_eq zimage_zexplode_eq zimplode_zexplode)
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    apply (simp add: Repl_Rep_game_Abs_game partizan_l partizan_r)
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    apply (subst Abs_game_inverse)
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    apply (simp_all add: lr[symmetric] Rep_game) 
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    done
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  then show ?thesis
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    by (simp add: L R)
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qed
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lemma Opair_in_games_lfp: 
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  assumes l: "explode l \<subseteq> games_lfp"
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  and r: "explode r \<subseteq> games_lfp"
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  shows "Opair l r \<in> games_lfp"
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proof -
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  note f = unique_games[of games_lfp, simplified]
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  show ?thesis
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    apply (subst f)
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    apply (simp add: fixgames_def)
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    apply (rule exI[where x=l])
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    apply (rule exI[where x=r])
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    apply (auto simp add: l r)
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    done
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qed
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lemma left_options[simp]: "left_options (Game l r) = l"
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  apply (simp add: left_options_def Game_def)
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  apply (subst Abs_game_inverse)
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  apply (simp add: game_def)
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  apply (rule Opair_in_games_lfp)
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  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
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  apply (simp add: Fst zexplode_zimplode comp_zimage_eq)
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  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
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  done
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lemma right_options[simp]: "right_options (Game l r) = r"
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  apply (simp add: right_options_def Game_def)
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  apply (subst Abs_game_inverse)
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  apply (simp add: game_def)
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  apply (rule Opair_in_games_lfp)
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  apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def])
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  apply (simp add: Snd zexplode_zimplode comp_zimage_eq)
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  apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse)
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  done  
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lemma Game_ext: "(Game l1 r1 = Game l2 r2) = ((l1 = l2) \<and> (r1 = r2))"
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  apply auto
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  apply (subst left_options[where l=l1 and r=r1,symmetric])
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  apply (subst left_options[where l=l2 and r=r2,symmetric])
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  apply simp
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  apply (subst right_options[where l=l1 and r=r1,symmetric])
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  apply (subst right_options[where l=l2 and r=r2,symmetric])
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  apply simp
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  done
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definition option_of :: "(game * game) set" where
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  "option_of \<equiv> image (\<lambda> (option, g). (Abs_game option, Abs_game g)) is_option_of"
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lemma option_to_is_option_of: "((option, g) \<in> option_of) = ((Rep_game option, Rep_game g) \<in> is_option_of)"
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  apply (auto simp add: option_of_def)
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  apply (subst Abs_game_inverse)
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  apply (simp add: is_option_of_imp_games game_def)
obua@19203
   311
  apply (subst Abs_game_inverse)
obua@19203
   312
  apply (simp add: is_option_of_imp_games game_def)
obua@19203
   313
  apply simp
obua@19203
   314
  apply (auto simp add: Bex_def image_def)  
obua@19203
   315
  apply (rule exI[where x="Rep_game option"])
obua@19203
   316
  apply (rule exI[where x="Rep_game g"])
obua@19203
   317
  apply (simp add: Rep_game_inverse)
obua@19203
   318
  done
obua@19203
   319
  
obua@19203
   320
lemma wf_is_option_of: "wf is_option_of"
obua@19203
   321
  apply (rule wfzf_implies_wf)
obua@19203
   322
  apply (simp add: wfzf_is_option_of)
obua@19203
   323
  done
obua@19203
   324
krauss@44011
   325
lemma wf_option_of[simp, intro]: "wf option_of"
obua@19203
   326
proof -
obua@19203
   327
  have option_of: "option_of = inv_image is_option_of Rep_game"
nipkow@39302
   328
    apply (rule set_eqI)
obua@19203
   329
    apply (case_tac "x")
krauss@19769
   330
    by (simp add: option_to_is_option_of) 
obua@19203
   331
  show ?thesis
obua@19203
   332
    apply (simp add: option_of)
obua@19203
   333
    apply (auto intro: wf_inv_image wf_is_option_of)
obua@19203
   334
    done
obua@19203
   335
qed
obua@19203
   336
  
obua@19203
   337
lemma right_option_is_option[simp, intro]: "zin x (right_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   338
  by (simp add: options_def zunion)
obua@19203
   339
obua@19203
   340
lemma left_option_is_option[simp, intro]: "zin x (left_options g) \<Longrightarrow> zin x (options g)"
obua@19203
   341
  by (simp add: options_def zunion)
obua@19203
   342
obua@19203
   343
lemma zin_options[simp, intro]: "zin x (options g) \<Longrightarrow> (x, g) \<in> option_of"
obua@19203
   344
  apply (simp add: options_def zunion left_options_def right_options_def option_of_def 
obua@19203
   345
    image_def is_option_of_def zimage_iff zin_zexplode_eq) 
obua@19203
   346
  apply (cases g)
obua@19203
   347
  apply (cases x)
obua@19203
   348
  apply (auto simp add: Abs_game_inverse games_lfp_eq_gfp[symmetric] game_def 
obua@19203
   349
    right_option_def[symmetric] left_option_def[symmetric])
obua@19203
   350
  done
obua@19203
   351
krauss@35440
   352
function
obua@19203
   353
  neg_game :: "game \<Rightarrow> game"
krauss@35440
   354
where
krauss@35440
   355
  [simp del]: "neg_game g = Game (zimage neg_game (right_options g)) (zimage neg_game (left_options g))"
krauss@35440
   356
by auto
krauss@35440
   357
termination by (relation "option_of") auto
obua@19203
   358
obua@19203
   359
lemma "neg_game (neg_game g) = g"
obua@19203
   360
  apply (induct g rule: neg_game.induct)
obua@19203
   361
  apply (subst neg_game.simps)+
obua@19203
   362
  apply (simp add: right_options left_options comp_zimage_eq)
obua@19203
   363
  apply (subgoal_tac "zimage (neg_game o neg_game) (left_options g) = left_options g")
obua@19203
   364
  apply (subgoal_tac "zimage (neg_game o neg_game) (right_options g) = right_options g")
obua@19203
   365
  apply (auto simp add: game_split[symmetric])
obua@19203
   366
  apply (auto simp add: zet_ext_eq zimage_iff)
obua@19203
   367
  done
obua@19203
   368
krauss@35440
   369
function
obua@19203
   370
  ge_game :: "(game * game) \<Rightarrow> bool" 
krauss@35440
   371
where
krauss@35440
   372
  [simp del]: "ge_game (G, H) = (\<forall> x. if zin x (right_options G) then (
obua@19203
   373
                          if zin x (left_options H) then \<not> (ge_game (H, x) \<or> (ge_game (x, G))) 
obua@19203
   374
                                                    else \<not> (ge_game (H, x)))
obua@19203
   375
                          else (if zin x (left_options H) then \<not> (ge_game (x, G)) else True))"
krauss@35440
   376
by auto
krauss@35440
   377
termination by (relation "(gprod_2_1 option_of)") 
krauss@35440
   378
 (simp, auto simp: gprod_2_1_def)
obua@19203
   379
wenzelm@26304
   380
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
   381
  apply (subst ge_game.simps[where G=G and H=H])
obua@19203
   382
  apply (auto)
obua@19203
   383
  done
obua@19203
   384
obua@19203
   385
lemma ge_game_leftright_refl[rule_format]: 
obua@19203
   386
  "\<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
   387
proof (induct x rule: wf_induct[OF wf_option_of]) 
obua@19203
   388
  case (1 "g")
obua@19203
   389
  { 
obua@19203
   390
    fix y
obua@19203
   391
    assume y: "zin y (right_options g)"
obua@19203
   392
    have "\<not> ge_game (g, y)"
obua@19203
   393
    proof -
obua@19203
   394
      have "(y, g) \<in> option_of" by (auto intro: y)
obua@19203
   395
      with 1 have "ge_game (y, y)" by auto
wenzelm@26304
   396
      with y show ?thesis by (subst ge_game_eq, auto)
obua@19203
   397
    qed
obua@19203
   398
  }
obua@19203
   399
  note right = this
obua@19203
   400
  { 
obua@19203
   401
    fix y
obua@19203
   402
    assume y: "zin y (left_options g)"
obua@19203
   403
    have "\<not> ge_game (y, g)"
obua@19203
   404
    proof -
obua@19203
   405
      have "(y, g) \<in> option_of" by (auto intro: y)
obua@19203
   406
      with 1 have "ge_game (y, y)" by auto
wenzelm@26304
   407
      with y show ?thesis by (subst ge_game_eq, auto)
obua@19203
   408
    qed
obua@19203
   409
  } 
obua@19203
   410
  note left = this
obua@19203
   411
  from left right show ?case
wenzelm@26304
   412
    by (auto, subst ge_game_eq, auto)
obua@19203
   413
qed
obua@19203
   414
obua@19203
   415
lemma ge_game_refl: "ge_game (x,x)" by (simp add: ge_game_leftright_refl)
obua@19203
   416
obua@19203
   417
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
   418
proof (induct x rule: wf_induct[OF wf_option_of]) 
obua@19203
   419
  case (1 "g")  
obua@19203
   420
  show ?case
obua@19203
   421
  proof (auto)
obua@19203
   422
    {case (goal1 y) 
obua@19203
   423
      from goal1 have "(y, g) \<in> option_of" by (auto)
obua@19203
   424
      with 1 have "ge_game (y, y)" by auto
obua@19203
   425
      with goal1 have "\<not> ge_game (g, y)" 
wenzelm@32960
   426
        by (subst ge_game_eq, auto)
obua@19203
   427
      with goal1 show ?case by auto}
obua@19203
   428
    note right = this
obua@19203
   429
    {case (goal2 y)
obua@19203
   430
      from goal2 have "(y, g) \<in> option_of" by (auto)
obua@19203
   431
      with 1 have "ge_game (y, y)" by auto
obua@19203
   432
      with goal2 have "\<not> ge_game (y, g)" 
wenzelm@32960
   433
        by (subst ge_game_eq, auto)
obua@19203
   434
      with goal2 show ?case by auto}
obua@19203
   435
    note left = this
obua@19203
   436
    {case goal3
obua@19203
   437
      from left right show ?case
wenzelm@32960
   438
        by (subst ge_game_eq, auto)
obua@19203
   439
    }
obua@19203
   440
  qed
obua@19203
   441
qed
wenzelm@32960
   442
        
haftmann@35416
   443
definition eq_game :: "game \<Rightarrow> game \<Rightarrow> bool" where
obua@19203
   444
  "eq_game G H \<equiv> ge_game (G, H) \<and> ge_game (H, G)" 
obua@19203
   445
obua@19203
   446
lemma eq_game_sym: "(eq_game G H) = (eq_game H G)"
obua@19203
   447
  by (auto simp add: eq_game_def)
obua@19203
   448
obua@19203
   449
lemma eq_game_refl: "eq_game G G"
obua@19203
   450
  by (simp add: ge_game_refl eq_game_def)
obua@19203
   451
berghofe@23771
   452
lemma induct_game: "(\<And>x. \<forall>y. (y, x) \<in> lprod option_of \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
berghofe@23771
   453
  by (erule wf_induct[OF wf_lprod[OF wf_option_of]])
obua@19203
   454
obua@19203
   455
lemma ge_game_trans:
obua@19203
   456
  assumes "ge_game (x, y)" "ge_game (y, z)" 
obua@19203
   457
  shows "ge_game (x, z)"
obua@19203
   458
proof -  
obua@19203
   459
  { 
obua@19203
   460
    fix a
obua@19203
   461
    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
   462
    proof (induct a rule: induct_game)
obua@19203
   463
      case (1 a)
obua@19203
   464
      show ?case
obua@19203
   465
      proof (rule allI | rule impI)+
wenzelm@32960
   466
        case (goal1 x y z)
wenzelm@32960
   467
        show ?case
wenzelm@32960
   468
        proof -
wenzelm@32960
   469
          { fix xr
obua@19203
   470
            assume xr:"zin xr (right_options x)"
wenzelm@41528
   471
            assume a: "ge_game (z, xr)"
wenzelm@32960
   472
            have "ge_game (y, xr)"
wenzelm@32960
   473
              apply (rule 1[rule_format, where y="[y,z,xr]"])
wenzelm@41528
   474
              apply (auto intro: xr lprod_3_1 simp add: goal1 a)
wenzelm@32960
   475
              done
wenzelm@32960
   476
            moreover from xr have "\<not> ge_game (y, xr)"
wenzelm@32960
   477
              by (simp add: goal1(2)[simplified ge_game_eq[of x y], rule_format, of xr, simplified xr])
wenzelm@32960
   478
            ultimately have "False" by auto      
wenzelm@32960
   479
          }
wenzelm@32960
   480
          note xr = this
wenzelm@32960
   481
          { fix zl
wenzelm@32960
   482
            assume zl:"zin zl (left_options z)"
wenzelm@41528
   483
            assume a: "ge_game (zl, x)"
wenzelm@32960
   484
            have "ge_game (zl, y)"
wenzelm@32960
   485
              apply (rule 1[rule_format, where y="[zl,x,y]"])
wenzelm@41528
   486
              apply (auto intro: zl lprod_3_2 simp add: goal1 a)
wenzelm@32960
   487
              done
wenzelm@32960
   488
            moreover from zl have "\<not> ge_game (zl, y)"
wenzelm@32960
   489
              by (simp add: goal1(3)[simplified ge_game_eq[of y z], rule_format, of zl, simplified zl])
wenzelm@32960
   490
            ultimately have "False" by auto
wenzelm@32960
   491
          }
wenzelm@32960
   492
          note zl = this
wenzelm@32960
   493
          show ?thesis
wenzelm@32960
   494
            by (auto simp add: ge_game_eq[of x z] intro: xr zl)
wenzelm@32960
   495
        qed
obua@19203
   496
      qed
obua@19203
   497
    qed
obua@19203
   498
  } 
obua@19203
   499
  note trans = this[of "[x, y, z]", simplified, rule_format]    
wenzelm@41528
   500
  with assms show ?thesis by blast
obua@19203
   501
qed
obua@19203
   502
obua@19203
   503
lemma eq_game_trans: "eq_game a b \<Longrightarrow> eq_game b c \<Longrightarrow> eq_game a c"
obua@19203
   504
  by (auto simp add: eq_game_def intro: ge_game_trans)
obua@19203
   505
haftmann@35416
   506
definition zero_game :: game
haftmann@35416
   507
 where  "zero_game \<equiv> Game zempty zempty"
obua@19203
   508
krauss@35440
   509
function 
krauss@35440
   510
  plus_game :: "game \<Rightarrow> game \<Rightarrow> game"
krauss@35440
   511
where
krauss@35440
   512
  [simp del]: "plus_game G H = Game (zunion (zimage (\<lambda> g. plus_game g H) (left_options G))
krauss@35440
   513
                                   (zimage (\<lambda> h. plus_game G h) (left_options H)))
krauss@35440
   514
                           (zunion (zimage (\<lambda> g. plus_game g H) (right_options G))
krauss@35440
   515
                                   (zimage (\<lambda> h. plus_game G h) (right_options H)))"
krauss@35440
   516
by auto
krauss@35440
   517
termination by (relation "gprod_2_2 option_of")
krauss@35440
   518
  (simp, auto simp: gprod_2_2_def)
obua@19203
   519
krauss@35440
   520
lemma plus_game_comm: "plus_game G H = plus_game H G"
obua@19203
   521
proof (induct G H rule: plus_game.induct)
obua@19203
   522
  case (1 G H)
obua@19203
   523
  show ?case
obua@19203
   524
    by (auto simp add: 
obua@19203
   525
      plus_game.simps[where G=G and H=H] 
obua@19203
   526
      plus_game.simps[where G=H and H=G]
wenzelm@41528
   527
      Game_ext zet_ext_eq zunion zimage_iff 1)
obua@19203
   528
qed
obua@19203
   529
obua@19203
   530
lemma game_ext_eq: "(G = H) = (left_options G = left_options H \<and> right_options G = right_options H)"
obua@19203
   531
proof -
obua@19203
   532
  have "(G = H) = (Game (left_options G) (right_options G) = Game (left_options H) (right_options H))"
obua@19203
   533
    by (simp add: game_split[symmetric])
obua@19203
   534
  then show ?thesis by auto
obua@19203
   535
qed
obua@19203
   536
obua@19203
   537
lemma left_zero_game[simp]: "left_options (zero_game) = zempty"
obua@19203
   538
  by (simp add: zero_game_def)
obua@19203
   539
obua@19203
   540
lemma right_zero_game[simp]: "right_options (zero_game) = zempty"
obua@19203
   541
  by (simp add: zero_game_def)
obua@19203
   542
krauss@35440
   543
lemma plus_game_zero_right[simp]: "plus_game G zero_game = G"
obua@19203
   544
proof -
obua@19203
   545
  { 
obua@19203
   546
    fix G H
krauss@35440
   547
    have "H = zero_game \<longrightarrow> plus_game G H = G "
obua@19203
   548
    proof (induct G H rule: plus_game.induct, rule impI)
obua@19203
   549
      case (goal1 G H)
wenzelm@41528
   550
      note induct_hyp = this[simplified goal1, simplified] and this
obua@19203
   551
      show ?case
wenzelm@32960
   552
        apply (simp only: plus_game.simps[where G=G and H=H])
wenzelm@41528
   553
        apply (simp add: game_ext_eq goal1)
wenzelm@32960
   554
        apply (auto simp add: 
krauss@35440
   555
          zimage_cong[where f = "\<lambda> g. plus_game g zero_game" and g = "id"] 
wenzelm@32960
   556
          induct_hyp)
wenzelm@32960
   557
        done
obua@19203
   558
    qed
obua@19203
   559
  }
obua@19203
   560
  then show ?thesis by auto
obua@19203
   561
qed
obua@19203
   562
krauss@35440
   563
lemma plus_game_zero_left: "plus_game zero_game G = G"
obua@19203
   564
  by (simp add: plus_game_comm)
obua@19203
   565
obua@19203
   566
lemma left_imp_options[simp]: "zin opt (left_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   567
  by (simp add: options_def zunion)
obua@19203
   568
obua@19203
   569
lemma right_imp_options[simp]: "zin opt (right_options g) \<Longrightarrow> zin opt (options g)"
obua@19203
   570
  by (simp add: options_def zunion)
obua@19203
   571
obua@19203
   572
lemma left_options_plus: 
krauss@35440
   573
  "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
   574
  by (subst plus_game.simps, simp)
obua@19203
   575
obua@19203
   576
lemma right_options_plus:
krauss@35440
   577
  "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
   578
  by (subst plus_game.simps, simp)
obua@19203
   579
wenzelm@32960
   580
lemma left_options_neg: "left_options (neg_game u) = zimage neg_game (right_options u)"  
obua@19203
   581
  by (subst neg_game.simps, simp)
obua@19203
   582
obua@19203
   583
lemma right_options_neg: "right_options (neg_game u) = zimage neg_game (left_options u)"
obua@19203
   584
  by (subst neg_game.simps, simp)
obua@19203
   585
  
krauss@35440
   586
lemma plus_game_assoc: "plus_game (plus_game F G) H = plus_game F (plus_game G H)"
obua@19203
   587
proof -
obua@19203
   588
  { 
obua@19203
   589
    fix a
krauss@35440
   590
    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
   591
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   592
      case (goal1 x F G H)
krauss@35440
   593
      let ?L = "plus_game (plus_game F G) H"
krauss@35440
   594
      let ?R = "plus_game F (plus_game G H)"
obua@19203
   595
      note options_plus = left_options_plus right_options_plus
obua@19203
   596
      {
wenzelm@32960
   597
        fix opt
wenzelm@32960
   598
        note hyp = goal1(1)[simplified goal1(2), rule_format] 
krauss@35440
   599
        have F: "zin opt (options F)  \<Longrightarrow> plus_game (plus_game opt G) H = plus_game opt (plus_game G H)"
wenzelm@32960
   600
          by (blast intro: hyp lprod_3_3)
krauss@35440
   601
        have G: "zin opt (options G) \<Longrightarrow> plus_game (plus_game F opt) H = plus_game F (plus_game opt H)"
wenzelm@32960
   602
          by (blast intro: hyp lprod_3_4)
krauss@35440
   603
        have H: "zin opt (options H) \<Longrightarrow> plus_game (plus_game F G) opt = plus_game F (plus_game G opt)" 
wenzelm@32960
   604
          by (blast intro: hyp lprod_3_5)
wenzelm@32960
   605
        note F and G and H
obua@19203
   606
      }
obua@19203
   607
      note induct_hyp = this
obua@19203
   608
      have "left_options ?L = left_options ?R \<and> right_options ?L = right_options ?R"
wenzelm@32960
   609
        by (auto simp add: 
krauss@35440
   610
          plus_game.simps[where G="plus_game F G" and H=H]
krauss@35440
   611
          plus_game.simps[where G="F" and H="plus_game G H"] 
wenzelm@32960
   612
          zet_ext_eq zunion zimage_iff options_plus
wenzelm@32960
   613
          induct_hyp left_imp_options right_imp_options)
obua@19203
   614
      then show ?case
wenzelm@32960
   615
        by (simp add: game_ext_eq)
obua@19203
   616
    qed
obua@19203
   617
  }
obua@19203
   618
  then show ?thesis by auto
obua@19203
   619
qed
obua@19203
   620
krauss@35440
   621
lemma neg_plus_game: "neg_game (plus_game G H) = plus_game (neg_game G) (neg_game H)"
obua@19203
   622
proof (induct G H rule: plus_game.induct)
obua@19203
   623
  case (1 G H)
obua@19203
   624
  note opt_ops = 
obua@19203
   625
    left_options_plus right_options_plus 
obua@19203
   626
    left_options_neg right_options_neg  
obua@19203
   627
  show ?case
obua@19203
   628
    by (auto simp add: opt_ops
krauss@35440
   629
      neg_game.simps[of "plus_game G H"]
obua@19203
   630
      plus_game.simps[of "neg_game G" "neg_game H"]
wenzelm@41528
   631
      Game_ext zet_ext_eq zunion zimage_iff 1)
obua@19203
   632
qed
obua@19203
   633
krauss@35440
   634
lemma eq_game_plus_inverse: "eq_game (plus_game x (neg_game x)) zero_game"
obua@19203
   635
proof (induct x rule: wf_induct[OF wf_option_of])
obua@19203
   636
  case (goal1 x)
obua@19203
   637
  { fix y
obua@19203
   638
    assume "zin y (options x)"
krauss@35440
   639
    then have "eq_game (plus_game y (neg_game y)) zero_game"
wenzelm@41528
   640
      by (auto simp add: goal1)
obua@19203
   641
  }
obua@19203
   642
  note ihyp = this
obua@19203
   643
  {
obua@19203
   644
    fix y
obua@19203
   645
    assume y: "zin y (right_options x)"
krauss@35440
   646
    have "\<not> (ge_game (zero_game, plus_game y (neg_game x)))"
obua@19203
   647
      apply (subst ge_game.simps, simp)
krauss@35440
   648
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   649
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
wenzelm@41528
   650
      apply (auto simp add: left_options_plus left_options_neg zunion zimage_iff intro: y)
obua@19203
   651
      done
obua@19203
   652
  }
obua@19203
   653
  note case1 = this
obua@19203
   654
  {
obua@19203
   655
    fix y
obua@19203
   656
    assume y: "zin y (left_options x)"
krauss@35440
   657
    have "\<not> (ge_game (zero_game, plus_game x (neg_game y)))"
obua@19203
   658
      apply (subst ge_game.simps, simp)
krauss@35440
   659
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   660
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
wenzelm@41528
   661
      apply (auto simp add: left_options_plus zunion zimage_iff intro: y)
obua@19203
   662
      done
obua@19203
   663
  }
obua@19203
   664
  note case2 = this
obua@19203
   665
  {
obua@19203
   666
    fix y
obua@19203
   667
    assume y: "zin y (left_options x)"
krauss@35440
   668
    have "\<not> (ge_game (plus_game y (neg_game x), zero_game))"
obua@19203
   669
      apply (subst ge_game.simps, simp)
krauss@35440
   670
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   671
      apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def])
wenzelm@41528
   672
      apply (auto simp add: right_options_plus right_options_neg zunion zimage_iff intro: y)
obua@19203
   673
      done
obua@19203
   674
  }
obua@19203
   675
  note case3 = this
obua@19203
   676
  {
obua@19203
   677
    fix y
obua@19203
   678
    assume y: "zin y (right_options x)"
krauss@35440
   679
    have "\<not> (ge_game (plus_game x (neg_game y), zero_game))"
obua@19203
   680
      apply (subst ge_game.simps, simp)
krauss@35440
   681
      apply (rule exI[where x="plus_game y (neg_game y)"])
obua@19203
   682
      apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def])
wenzelm@41528
   683
      apply (auto simp add: right_options_plus zunion zimage_iff intro: y)
obua@19203
   684
      done
obua@19203
   685
  }
obua@19203
   686
  note case4 = this
obua@19203
   687
  show ?case
obua@19203
   688
    apply (simp add: eq_game_def)
krauss@35440
   689
    apply (simp add: ge_game.simps[of "plus_game x (neg_game x)" "zero_game"])
krauss@35440
   690
    apply (simp add: ge_game.simps[of "zero_game" "plus_game x (neg_game x)"])
obua@19203
   691
    apply (simp add: right_options_plus left_options_plus right_options_neg left_options_neg zunion zimage_iff)
obua@19203
   692
    apply (auto simp add: case1 case2 case3 case4)
obua@19203
   693
    done
obua@19203
   694
qed
obua@19203
   695
krauss@35440
   696
lemma ge_plus_game_left: "ge_game (y,z) = ge_game (plus_game x y, plus_game x z)"
obua@19203
   697
proof -
obua@19203
   698
  { fix a
krauss@35440
   699
    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
   700
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   701
      case (goal1 a x y z)
obua@19203
   702
      note induct_hyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   703
      { 
krauss@35440
   704
        assume hyp: "ge_game(plus_game x y, plus_game x z)"
wenzelm@32960
   705
        have "ge_game (y, z)"
wenzelm@32960
   706
        proof -
wenzelm@32960
   707
          { fix yr
wenzelm@32960
   708
            assume yr: "zin yr (right_options y)"
krauss@35440
   709
            from hyp have "\<not> (ge_game (plus_game x z, plus_game x yr))"
krauss@35440
   710
              by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"]
wenzelm@32960
   711
                right_options_plus zunion zimage_iff intro: yr)
wenzelm@32960
   712
            then have "\<not> (ge_game (z, yr))"
wenzelm@32960
   713
              apply (subst induct_hyp[where y="[x, z, yr]", of "x" "z" "yr"])
wenzelm@32960
   714
              apply (simp_all add: yr lprod_3_6)
wenzelm@32960
   715
              done
wenzelm@32960
   716
          }
wenzelm@32960
   717
          note yr = this
wenzelm@32960
   718
          { fix zl
wenzelm@32960
   719
            assume zl: "zin zl (left_options z)"
krauss@35440
   720
            from hyp have "\<not> (ge_game (plus_game x zl, plus_game x y))"
krauss@35440
   721
              by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"]
wenzelm@32960
   722
                left_options_plus zunion zimage_iff intro: zl)
wenzelm@32960
   723
            then have "\<not> (ge_game (zl, y))"
wenzelm@32960
   724
              apply (subst goal1(1)[rule_format, where y="[x, zl, y]", of "x" "zl" "y"])
wenzelm@32960
   725
              apply (simp_all add: goal1(2) zl lprod_3_7)
wenzelm@32960
   726
              done
wenzelm@32960
   727
          }     
wenzelm@32960
   728
          note zl = this
wenzelm@32960
   729
          show "ge_game (y, z)"
wenzelm@32960
   730
            apply (subst ge_game_eq)
wenzelm@32960
   731
            apply (auto simp add: yr zl)
wenzelm@32960
   732
            done
wenzelm@32960
   733
        qed      
obua@19203
   734
      }
obua@19203
   735
      note right_imp_left = this
obua@19203
   736
      {
wenzelm@32960
   737
        assume yz: "ge_game (y, z)"
wenzelm@32960
   738
        {
wenzelm@32960
   739
          fix x'
wenzelm@32960
   740
          assume x': "zin x' (right_options x)"
krauss@35440
   741
          assume hyp: "ge_game (plus_game x z, plus_game x' y)"
krauss@35440
   742
          then have n: "\<not> (ge_game (plus_game x' y, plus_game x' z))"
krauss@35440
   743
            by (auto simp add: ge_game_eq[of "plus_game x z" "plus_game x' y"] 
wenzelm@32960
   744
              right_options_plus zunion zimage_iff intro: x')
krauss@35440
   745
          have t: "ge_game (plus_game x' y, plus_game x' z)"
wenzelm@32960
   746
            apply (subst induct_hyp[symmetric])
wenzelm@32960
   747
            apply (auto intro: lprod_3_3 x' yz)
wenzelm@32960
   748
            done
wenzelm@32960
   749
          from n t have "False" by blast
wenzelm@32960
   750
        }    
wenzelm@32960
   751
        note case1 = this
wenzelm@32960
   752
        {
wenzelm@32960
   753
          fix x'
wenzelm@32960
   754
          assume x': "zin x' (left_options x)"
krauss@35440
   755
          assume hyp: "ge_game (plus_game x' z, plus_game x y)"
krauss@35440
   756
          then have n: "\<not> (ge_game (plus_game x' y, plus_game x' z))"
krauss@35440
   757
            by (auto simp add: ge_game_eq[of "plus_game x' z" "plus_game x y"] 
wenzelm@32960
   758
              left_options_plus zunion zimage_iff intro: x')
krauss@35440
   759
          have t: "ge_game (plus_game x' y, plus_game x' z)"
wenzelm@32960
   760
            apply (subst induct_hyp[symmetric])
wenzelm@32960
   761
            apply (auto intro: lprod_3_3 x' yz)
wenzelm@32960
   762
            done
wenzelm@32960
   763
          from n t have "False" by blast
wenzelm@32960
   764
        }
wenzelm@32960
   765
        note case3 = this
wenzelm@32960
   766
        {
wenzelm@32960
   767
          fix y'
wenzelm@32960
   768
          assume y': "zin y' (right_options y)"
krauss@35440
   769
          assume hyp: "ge_game (plus_game x z, plus_game x y')"
wenzelm@32960
   770
          then have "ge_game(z, y')"
wenzelm@32960
   771
            apply (subst induct_hyp[of "[x, z, y']" "x" "z" "y'"])
wenzelm@32960
   772
            apply (auto simp add: hyp lprod_3_6 y')
wenzelm@32960
   773
            done
wenzelm@32960
   774
          with yz have "ge_game (y, y')"
wenzelm@32960
   775
            by (blast intro: ge_game_trans)      
wenzelm@32960
   776
          with y' have "False" by (auto simp add: ge_game_leftright_refl)
wenzelm@32960
   777
        }
wenzelm@32960
   778
        note case2 = this
wenzelm@32960
   779
        {
wenzelm@32960
   780
          fix z'
wenzelm@32960
   781
          assume z': "zin z' (left_options z)"
krauss@35440
   782
          assume hyp: "ge_game (plus_game x z', plus_game x y)"
wenzelm@32960
   783
          then have "ge_game(z', y)"
wenzelm@32960
   784
            apply (subst induct_hyp[of "[x, z', y]" "x" "z'" "y"])
wenzelm@32960
   785
            apply (auto simp add: hyp lprod_3_7 z')
wenzelm@32960
   786
            done    
wenzelm@32960
   787
          with yz have "ge_game (z', z)"
wenzelm@32960
   788
            by (blast intro: ge_game_trans)      
wenzelm@32960
   789
          with z' have "False" by (auto simp add: ge_game_leftright_refl)
wenzelm@32960
   790
        }
wenzelm@32960
   791
        note case4 = this   
krauss@35440
   792
        have "ge_game(plus_game x y, plus_game x z)"
wenzelm@32960
   793
          apply (subst ge_game_eq)
wenzelm@32960
   794
          apply (auto simp add: right_options_plus left_options_plus zunion zimage_iff)
wenzelm@32960
   795
          apply (auto intro: case1 case2 case3 case4)
wenzelm@32960
   796
          done
obua@19203
   797
      }
obua@19203
   798
      note left_imp_right = this
obua@19203
   799
      show ?case by (auto intro: right_imp_left left_imp_right)
obua@19203
   800
    qed
obua@19203
   801
  }
obua@19203
   802
  note a = this[of "[x, y, z]"]
obua@19203
   803
  then show ?thesis by blast
obua@19203
   804
qed
obua@19203
   805
krauss@35440
   806
lemma ge_plus_game_right: "ge_game (y,z) = ge_game(plus_game y x, plus_game z x)"
obua@19203
   807
  by (simp add: ge_plus_game_left plus_game_comm)
obua@19203
   808
obua@19203
   809
lemma ge_neg_game: "ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   810
proof -
obua@19203
   811
  { fix a
obua@19203
   812
    have "\<forall> x y. a = [x, y] \<longrightarrow> ge_game (neg_game x, neg_game y) = ge_game (y, x)"
obua@19203
   813
    proof (induct a rule: induct_game, (rule impI | rule allI)+)
obua@19203
   814
      case (goal1 a x y)
obua@19203
   815
      note ihyp = goal1(1)[rule_format, simplified goal1(2)]
obua@19203
   816
      { fix xl
wenzelm@32960
   817
        assume xl: "zin xl (left_options x)"
wenzelm@32960
   818
        have "ge_game (neg_game y, neg_game xl) = ge_game (xl, y)"
wenzelm@32960
   819
          apply (subst ihyp)
wenzelm@32960
   820
          apply (auto simp add: lprod_2_1 xl)
wenzelm@32960
   821
          done
obua@19203
   822
      }
obua@19203
   823
      note xl = this
obua@19203
   824
      { fix yr
wenzelm@32960
   825
        assume yr: "zin yr (right_options y)"
wenzelm@32960
   826
        have "ge_game (neg_game yr, neg_game x) = ge_game (x, yr)"
wenzelm@32960
   827
          apply (subst ihyp)
wenzelm@32960
   828
          apply (auto simp add: lprod_2_2 yr)
wenzelm@32960
   829
          done
obua@19203
   830
      }
obua@19203
   831
      note yr = this
obua@19203
   832
      show ?case
wenzelm@32960
   833
        by (auto simp add: ge_game_eq[of "neg_game x" "neg_game y"] ge_game_eq[of "y" "x"]
wenzelm@32960
   834
          right_options_neg left_options_neg zimage_iff  xl yr)
obua@19203
   835
    qed
obua@19203
   836
  }
obua@19203
   837
  note a = this[of "[x,y]"]
obua@19203
   838
  then show ?thesis by blast
obua@19203
   839
qed
obua@19203
   840
haftmann@35416
   841
definition eq_game_rel :: "(game * game) set" where
obua@19203
   842
  "eq_game_rel \<equiv> { (p, q) . eq_game p q }"
obua@19203
   843
wenzelm@45694
   844
definition "Pg = UNIV//eq_game_rel"
wenzelm@45694
   845
wenzelm@45694
   846
typedef (open) Pg = Pg
wenzelm@45694
   847
  unfolding Pg_def by (auto simp add: quotient_def)
obua@19203
   848
obua@19203
   849
lemma equiv_eq_game[simp]: "equiv UNIV eq_game_rel"
nipkow@30198
   850
  by (auto simp add: equiv_def refl_on_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@39910
   866
  Pg_minus_def: "- G = the_elem (\<Union> g \<in> Rep_Pg G. {Abs_Pg (eq_game_rel `` {neg_game g})})"
haftmann@25764
   867
haftmann@25764
   868
definition
haftmann@39910
   869
  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
   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
krauss@35440
   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 -
krauss@35440
   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)
haftmann@40824
   900
    apply (rule_tac y="plus_game a ba" in ge_game_trans)
obua@19203
   901
    apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+
haftmann@40824
   902
    apply (rule_tac y="plus_game b aa" 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
haftmann@35028
   926
instance Pg :: ordered_ab_group_add 
obua@19203
   927
proof
obua@19203
   928
  fix a b c :: Pg
obua@19203
   929
  show "a - b = a + (- b)" by (simp add: Pg_diff_def)
obua@19203
   930
  {
obua@19203
   931
    assume ab: "a \<le> b"
obua@19203
   932
    assume ba: "b \<le> a"
obua@19203
   933
    from ab ba show "a = b"
obua@19203
   934
      apply (cases a, cases b)
obua@19203
   935
      apply (simp add: eq_game_def)
obua@19203
   936
      done
obua@19203
   937
  }
haftmann@27679
   938
  then show "(a < b) = (a \<le> b \<and> \<not> b \<le> a)" by (auto simp add: Pg_less_def)
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