(* Title: HOL/ZF/Games.thy Author: Steven Obua An application of HOLZF: Partizan Games. See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan *) theory Games imports MainZF begin definition fixgames :: "ZF set => ZF set" where "fixgames A ≡ { Opair l r | l r. explode l ⊆ A & explode r ⊆ A}" definition games_lfp :: "ZF set" where "games_lfp ≡ lfp fixgames" definition games_gfp :: "ZF set" where "games_gfp ≡ gfp fixgames" lemma mono_fixgames: "mono (fixgames)" apply (auto simp add: mono_def fixgames_def) apply (rule_tac x=l in exI) apply (rule_tac x=r in exI) apply auto done lemma games_lfp_unfold: "games_lfp = fixgames games_lfp" by (auto simp add: def_lfp_unfold games_lfp_def mono_fixgames) lemma games_gfp_unfold: "games_gfp = fixgames games_gfp" by (auto simp add: def_gfp_unfold games_gfp_def mono_fixgames) lemma games_lfp_nonempty: "Opair Empty Empty ∈ games_lfp" proof - have "fixgames {} ⊆ games_lfp" apply (subst games_lfp_unfold) apply (simp add: mono_fixgames[simplified mono_def, rule_format]) done moreover have "fixgames {} = {Opair Empty Empty}" by (simp add: fixgames_def explode_Empty) finally show ?thesis by auto qed definition left_option :: "ZF => ZF => bool" where "left_option g opt ≡ (Elem opt (Fst g))" definition right_option :: "ZF => ZF => bool" where "right_option g opt ≡ (Elem opt (Snd g))" definition is_option_of :: "(ZF * ZF) set" where "is_option_of ≡ { (opt, g) | opt g. g ∈ games_gfp ∧ (left_option g opt ∨ right_option g opt) }" lemma games_lfp_subset_gfp: "games_lfp ⊆ games_gfp" proof - have "games_lfp ⊆ fixgames games_lfp" by (simp add: games_lfp_unfold[symmetric]) then show ?thesis by (simp add: games_gfp_def gfp_upperbound) qed lemma games_option_stable: assumes fixgames: "games = fixgames games" and g: "g ∈ games" and opt: "left_option g opt ∨ right_option g opt" shows "opt ∈ games" proof - from g fixgames have "g ∈ fixgames games" by auto then have "∃ l r. g = Opair l r ∧ explode l ⊆ games ∧ explode r ⊆ games" by (simp add: fixgames_def) then obtain l where "∃ r. g = Opair l r ∧ explode l ⊆ games ∧ explode r ⊆ games" .. then obtain r where lr: "g = Opair l r ∧ explode l ⊆ games ∧ explode r ⊆ games" .. with opt show ?thesis by (auto intro: Elem_explode_in simp add: left_option_def right_option_def Fst Snd) qed lemma option2elem: "(opt,g) ∈ is_option_of ==> ∃ u v. Elem opt u ∧ Elem u v ∧ Elem v g" apply (simp add: is_option_of_def) apply (subgoal_tac "(g ∈ games_gfp) = (g ∈ (fixgames games_gfp))") prefer 2 apply (simp add: games_gfp_unfold[symmetric]) apply (auto simp add: fixgames_def left_option_def right_option_def Fst Snd) apply (rule_tac x=l in exI, insert Elem_Opair_exists, blast) apply (rule_tac x=r in exI, insert Elem_Opair_exists, blast) done lemma is_option_of_subset_is_Elem_of: "is_option_of ⊆ (is_Elem_of^+)" proof - { fix opt fix g assume "(opt, g) ∈ is_option_of" then have "∃ u v. (opt, u) ∈ (is_Elem_of^+) ∧ (u,v) ∈ (is_Elem_of^+) ∧ (v,g) ∈ (is_Elem_of^+)" apply - apply (drule option2elem) apply (auto simp add: r_into_trancl' is_Elem_of_def) done then have "(opt, g) ∈ (is_Elem_of^+)" by (blast intro: trancl_into_rtrancl trancl_rtrancl_trancl) } then show ?thesis by auto qed lemma wfzf_is_option_of: "wfzf is_option_of" proof - have "wfzf (is_Elem_of^+)" by (simp add: wfzf_trancl wfzf_is_Elem_of) then show ?thesis apply (rule wfzf_subset) apply (rule is_option_of_subset_is_Elem_of) done qed lemma games_gfp_imp_lfp: "g ∈ games_gfp --> g ∈ games_lfp" proof - have unfold_gfp: "!! x. x ∈ games_gfp ==> x ∈ (fixgames games_gfp)" by (simp add: games_gfp_unfold[symmetric]) have unfold_lfp: "!! x. (x ∈ games_lfp) = (x ∈ (fixgames games_lfp))" by (simp add: games_lfp_unfold[symmetric]) show ?thesis apply (rule wf_induct[OF wfzf_implies_wf[OF wfzf_is_option_of]]) apply (auto simp add: is_option_of_def) apply (drule_tac unfold_gfp) apply (simp add: fixgames_def) apply (auto simp add: left_option_def Fst right_option_def Snd) apply (subgoal_tac "explode l ⊆ games_lfp") apply (subgoal_tac "explode r ⊆ games_lfp") apply (subst unfold_lfp) apply (auto simp add: fixgames_def) apply (simp_all add: explode_Elem Elem_explode_in) done qed theorem games_lfp_eq_gfp: "games_lfp = games_gfp" apply (auto simp add: games_gfp_imp_lfp) apply (insert games_lfp_subset_gfp) apply auto done theorem unique_games: "(g = fixgames g) = (g = games_lfp)" proof - { fix g assume g: "g = fixgames g" from g have "fixgames g ⊆ g" by auto then have l:"games_lfp ⊆ g" by (simp add: games_lfp_def lfp_lowerbound) from g have "g ⊆ fixgames g" by auto then have u:"g ⊆ games_gfp" by (simp add: games_gfp_def gfp_upperbound) from l u games_lfp_eq_gfp[symmetric] have "g = games_lfp" by auto } note games = this show ?thesis apply (rule iff[rule_format]) apply (erule games) apply (simp add: games_lfp_unfold[symmetric]) done qed lemma games_lfp_option_stable: assumes g: "g ∈ games_lfp" and opt: "left_option g opt ∨ right_option g opt" shows "opt ∈ games_lfp" apply (rule games_option_stable[where g=g]) apply (simp add: games_lfp_unfold[symmetric]) apply (simp_all add: assms) done lemma is_option_of_imp_games: assumes hyp: "(opt, g) ∈ is_option_of" shows "opt ∈ games_lfp ∧ g ∈ games_lfp" proof - from hyp have g_game: "g ∈ games_lfp" by (simp add: is_option_of_def games_lfp_eq_gfp) from hyp have "left_option g opt ∨ right_option g opt" by (auto simp add: is_option_of_def) with g_game games_lfp_option_stable[OF g_game, OF this] show ?thesis by auto qed lemma games_lfp_represent: "x ∈ games_lfp ==> ∃ l r. x = Opair l r" apply (rule exI[where x="Fst x"]) apply (rule exI[where x="Snd x"]) apply (subgoal_tac "x ∈ (fixgames games_lfp)") apply (simp add: fixgames_def) apply (auto simp add: Fst Snd) apply (simp add: games_lfp_unfold[symmetric]) done definition "game = games_lfp" typedef game = game unfolding game_def by (blast intro: games_lfp_nonempty) definition left_options :: "game => game zet" where "left_options g ≡ zimage Abs_game (zexplode (Fst (Rep_game g)))" definition right_options :: "game => game zet" where "right_options g ≡ zimage Abs_game (zexplode (Snd (Rep_game g)))" definition options :: "game => game zet" where "options g ≡ zunion (left_options g) (right_options g)" definition Game :: "game zet => game zet => game" where "Game L R ≡ Abs_game (Opair (zimplode (zimage Rep_game L)) (zimplode (zimage Rep_game R)))" lemma Repl_Rep_game_Abs_game: "∀ e. Elem e z --> e ∈ games_lfp ==> Repl z (Rep_game o Abs_game) = z" apply (subst Ext) apply (simp add: Repl) apply auto apply (subst Abs_game_inverse, simp_all add: game_def) apply (rule_tac x=za in exI) apply (subst Abs_game_inverse, simp_all add: game_def) done lemma game_split: "g = Game (left_options g) (right_options g)" proof - have "∃ l r. Rep_game g = Opair l r" apply (insert Rep_game[of g]) apply (simp add: game_def games_lfp_represent) done then obtain l r where lr: "Rep_game g = Opair l r" by auto have partizan_g: "Rep_game g ∈ games_lfp" apply (insert Rep_game[of g]) apply (simp add: game_def) done have "∀ e. Elem e l --> left_option (Rep_game g) e" by (simp add: lr left_option_def Fst) then have partizan_l: "∀ e. Elem e l --> e ∈ games_lfp" apply auto apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g]) apply auto done have "∀ e. Elem e r --> right_option (Rep_game g) e" by (simp add: lr right_option_def Snd) then have partizan_r: "∀ e. Elem e r --> e ∈ games_lfp" apply auto apply (rule games_lfp_option_stable[where g="Rep_game g", OF partizan_g]) apply auto done let ?L = "zimage (Abs_game) (zexplode l)" let ?R = "zimage (Abs_game) (zexplode r)" have L:"?L = left_options g" by (simp add: left_options_def lr Fst) have R:"?R = right_options g" by (simp add: right_options_def lr Snd) have "g = Game ?L ?R" apply (simp add: Game_def Rep_game_inject[symmetric] comp_zimage_eq zimage_zexplode_eq zimplode_zexplode) apply (simp add: Repl_Rep_game_Abs_game partizan_l partizan_r) apply (subst Abs_game_inverse) apply (simp_all add: lr[symmetric] Rep_game) done then show ?thesis by (simp add: L R) qed lemma Opair_in_games_lfp: assumes l: "explode l ⊆ games_lfp" and r: "explode r ⊆ games_lfp" shows "Opair l r ∈ games_lfp" proof - note f = unique_games[of games_lfp, simplified] show ?thesis apply (subst f) apply (simp add: fixgames_def) apply (rule exI[where x=l]) apply (rule exI[where x=r]) apply (auto simp add: l r) done qed lemma left_options[simp]: "left_options (Game l r) = l" apply (simp add: left_options_def Game_def) apply (subst Abs_game_inverse) apply (simp add: game_def) apply (rule Opair_in_games_lfp) apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def]) apply (simp add: Fst zexplode_zimplode comp_zimage_eq) apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse) done lemma right_options[simp]: "right_options (Game l r) = r" apply (simp add: right_options_def Game_def) apply (subst Abs_game_inverse) apply (simp add: game_def) apply (rule Opair_in_games_lfp) apply (auto simp add: explode_Elem Elem_zimplode zimage_iff Rep_game[simplified game_def]) apply (simp add: Snd zexplode_zimplode comp_zimage_eq) apply (simp add: zet_ext_eq zimage_iff Rep_game_inverse) done lemma Game_ext: "(Game l1 r1 = Game l2 r2) = ((l1 = l2) ∧ (r1 = r2))" apply auto apply (subst left_options[where l=l1 and r=r1,symmetric]) apply (subst left_options[where l=l2 and r=r2,symmetric]) apply simp apply (subst right_options[where l=l1 and r=r1,symmetric]) apply (subst right_options[where l=l2 and r=r2,symmetric]) apply simp done definition option_of :: "(game * game) set" where "option_of ≡ image (λ (option, g). (Abs_game option, Abs_game g)) is_option_of" lemma option_to_is_option_of: "((option, g) ∈ option_of) = ((Rep_game option, Rep_game g) ∈ is_option_of)" apply (auto simp add: option_of_def) apply (subst Abs_game_inverse) apply (simp add: is_option_of_imp_games game_def) apply (subst Abs_game_inverse) apply (simp add: is_option_of_imp_games game_def) apply simp apply (auto simp add: Bex_def image_def) apply (rule exI[where x="Rep_game option"]) apply (rule exI[where x="Rep_game g"]) apply (simp add: Rep_game_inverse) done lemma wf_is_option_of: "wf is_option_of" apply (rule wfzf_implies_wf) apply (simp add: wfzf_is_option_of) done lemma wf_option_of[simp, intro]: "wf option_of" proof - have option_of: "option_of = inv_image is_option_of Rep_game" apply (rule set_eqI) apply (case_tac "x") by (simp add: option_to_is_option_of) show ?thesis apply (simp add: option_of) apply (auto intro: wf_is_option_of) done qed lemma right_option_is_option[simp, intro]: "zin x (right_options g) ==> zin x (options g)" by (simp add: options_def zunion) lemma left_option_is_option[simp, intro]: "zin x (left_options g) ==> zin x (options g)" by (simp add: options_def zunion) lemma zin_options[simp, intro]: "zin x (options g) ==> (x, g) ∈ option_of" apply (simp add: options_def zunion left_options_def right_options_def option_of_def image_def is_option_of_def zimage_iff zin_zexplode_eq) apply (cases g) apply (cases x) apply (auto simp add: Abs_game_inverse games_lfp_eq_gfp[symmetric] game_def right_option_def[symmetric] left_option_def[symmetric]) done function neg_game :: "game => game" where [simp del]: "neg_game g = Game (zimage neg_game (right_options g)) (zimage neg_game (left_options g))" by auto termination by (relation "option_of") auto lemma "neg_game (neg_game g) = g" apply (induct g rule: neg_game.induct) apply (subst neg_game.simps)+ apply (simp add: comp_zimage_eq) apply (subgoal_tac "zimage (neg_game o neg_game) (left_options g) = left_options g") apply (subgoal_tac "zimage (neg_game o neg_game) (right_options g) = right_options g") apply (auto simp add: game_split[symmetric]) apply (auto simp add: zet_ext_eq zimage_iff) done function ge_game :: "(game * game) => bool" where [simp del]: "ge_game (G, H) = (∀ x. if zin x (right_options G) then ( if zin x (left_options H) then ¬ (ge_game (H, x) ∨ (ge_game (x, G))) else ¬ (ge_game (H, x))) else (if zin x (left_options H) then ¬ (ge_game (x, G)) else True))" by auto termination by (relation "(gprod_2_1 option_of)") (simp, auto simp: gprod_2_1_def) lemma ge_game_eq: "ge_game (G, H) = (∀ x. (zin x (right_options G) --> ¬ ge_game (H, x)) ∧ (zin x (left_options H) --> ¬ ge_game (x, G)))" apply (subst ge_game.simps[where G=G and H=H]) apply (auto) done lemma ge_game_leftright_refl[rule_format]: "∀ y. (zin y (right_options x) --> ¬ ge_game (x, y)) ∧ (zin y (left_options x) --> ¬ (ge_game (y, x))) ∧ ge_game (x, x)" proof (induct x rule: wf_induct[OF wf_option_of]) case (1 "g") { fix y assume y: "zin y (right_options g)" have "¬ ge_game (g, y)" proof - have "(y, g) ∈ option_of" by (auto intro: y) with 1 have "ge_game (y, y)" by auto with y show ?thesis by (subst ge_game_eq, auto) qed } note right = this { fix y assume y: "zin y (left_options g)" have "¬ ge_game (y, g)" proof - have "(y, g) ∈ option_of" by (auto intro: y) with 1 have "ge_game (y, y)" by auto with y show ?thesis by (subst ge_game_eq, auto) qed } note left = this from left right show ?case by (auto, subst ge_game_eq, auto) qed lemma ge_game_refl: "ge_game (x,x)" by (simp add: ge_game_leftright_refl) lemma "∀ y. (zin y (right_options x) --> ¬ ge_game (x, y)) ∧ (zin y (left_options x) --> ¬ (ge_game (y, x))) ∧ ge_game (x, x)" proof (induct x rule: wf_induct[OF wf_option_of]) case (1 "g") show ?case proof (auto) {case (goal1 y) from goal1 have "(y, g) ∈ option_of" by (auto) with 1 have "ge_game (y, y)" by auto with goal1 have "¬ ge_game (g, y)" by (subst ge_game_eq, auto) with goal1 show ?case by auto} note right = this {case (goal2 y) from goal2 have "(y, g) ∈ option_of" by (auto) with 1 have "ge_game (y, y)" by auto with goal2 have "¬ ge_game (y, g)" by (subst ge_game_eq, auto) with goal2 show ?case by auto} note left = this {case goal3 from left right show ?case by (subst ge_game_eq, auto) } qed qed definition eq_game :: "game => game => bool" where "eq_game G H ≡ ge_game (G, H) ∧ ge_game (H, G)" lemma eq_game_sym: "(eq_game G H) = (eq_game H G)" by (auto simp add: eq_game_def) lemma eq_game_refl: "eq_game G G" by (simp add: ge_game_refl eq_game_def) lemma induct_game: "(!!x. ∀y. (y, x) ∈ lprod option_of --> P y ==> P x) ==> P a" by (erule wf_induct[OF wf_lprod[OF wf_option_of]]) lemma ge_game_trans: assumes "ge_game (x, y)" "ge_game (y, z)" shows "ge_game (x, z)" proof - { fix a have "∀ x y z. a = [x,y,z] --> ge_game (x,y) --> ge_game (y,z) --> ge_game (x, z)" proof (induct a rule: induct_game) case (1 a) show ?case proof (rule allI | rule impI)+ case (goal1 x y z) show ?case proof - { fix xr assume xr:"zin xr (right_options x)" assume a: "ge_game (z, xr)" have "ge_game (y, xr)" apply (rule 1[rule_format, where y="[y,z,xr]"]) apply (auto intro: xr lprod_3_1 simp add: goal1 a) done moreover from xr have "¬ ge_game (y, xr)" by (simp add: goal1(2)[simplified ge_game_eq[of x y], rule_format, of xr, simplified xr]) ultimately have "False" by auto } note xr = this { fix zl assume zl:"zin zl (left_options z)" assume a: "ge_game (zl, x)" have "ge_game (zl, y)" apply (rule 1[rule_format, where y="[zl,x,y]"]) apply (auto intro: zl lprod_3_2 simp add: goal1 a) done moreover from zl have "¬ ge_game (zl, y)" by (simp add: goal1(3)[simplified ge_game_eq[of y z], rule_format, of zl, simplified zl]) ultimately have "False" by auto } note zl = this show ?thesis by (auto simp add: ge_game_eq[of x z] intro: xr zl) qed qed qed } note trans = this[of "[x, y, z]", simplified, rule_format] with assms show ?thesis by blast qed lemma eq_game_trans: "eq_game a b ==> eq_game b c ==> eq_game a c" by (auto simp add: eq_game_def intro: ge_game_trans) definition zero_game :: game where "zero_game ≡ Game zempty zempty" function plus_game :: "game => game => game" where [simp del]: "plus_game G H = Game (zunion (zimage (λ g. plus_game g H) (left_options G)) (zimage (λ h. plus_game G h) (left_options H))) (zunion (zimage (λ g. plus_game g H) (right_options G)) (zimage (λ h. plus_game G h) (right_options H)))" by auto termination by (relation "gprod_2_2 option_of") (simp, auto simp: gprod_2_2_def) lemma plus_game_comm: "plus_game G H = plus_game H G" proof (induct G H rule: plus_game.induct) case (1 G H) show ?case by (auto simp add: plus_game.simps[where G=G and H=H] plus_game.simps[where G=H and H=G] Game_ext zet_ext_eq zunion zimage_iff 1) qed lemma game_ext_eq: "(G = H) = (left_options G = left_options H ∧ right_options G = right_options H)" proof - have "(G = H) = (Game (left_options G) (right_options G) = Game (left_options H) (right_options H))" by (simp add: game_split[symmetric]) then show ?thesis by auto qed lemma left_zero_game[simp]: "left_options (zero_game) = zempty" by (simp add: zero_game_def) lemma right_zero_game[simp]: "right_options (zero_game) = zempty" by (simp add: zero_game_def) lemma plus_game_zero_right[simp]: "plus_game G zero_game = G" proof - { fix G H have "H = zero_game --> plus_game G H = G " proof (induct G H rule: plus_game.induct, rule impI) case (goal1 G H) note induct_hyp = this[simplified goal1, simplified] and this show ?case apply (simp only: plus_game.simps[where G=G and H=H]) apply (simp add: game_ext_eq goal1) apply (auto simp add: zimage_cong [where f = "λ g. plus_game g zero_game" and g = "id"] induct_hyp) done qed } then show ?thesis by auto qed lemma plus_game_zero_left: "plus_game zero_game G = G" by (simp add: plus_game_comm) lemma left_imp_options[simp]: "zin opt (left_options g) ==> zin opt (options g)" by (simp add: options_def zunion) lemma right_imp_options[simp]: "zin opt (right_options g) ==> zin opt (options g)" by (simp add: options_def zunion) lemma left_options_plus: "left_options (plus_game u v) = zunion (zimage (λg. plus_game g v) (left_options u)) (zimage (λh. plus_game u h) (left_options v))" by (subst plus_game.simps, simp) lemma right_options_plus: "right_options (plus_game u v) = zunion (zimage (λg. plus_game g v) (right_options u)) (zimage (λh. plus_game u h) (right_options v))" by (subst plus_game.simps, simp) lemma left_options_neg: "left_options (neg_game u) = zimage neg_game (right_options u)" by (subst neg_game.simps, simp) lemma right_options_neg: "right_options (neg_game u) = zimage neg_game (left_options u)" by (subst neg_game.simps, simp) lemma plus_game_assoc: "plus_game (plus_game F G) H = plus_game F (plus_game G H)" proof - { fix a have "∀ F G H. a = [F, G, H] --> plus_game (plus_game F G) H = plus_game F (plus_game G H)" proof (induct a rule: induct_game, (rule impI | rule allI)+) case (goal1 x F G H) let ?L = "plus_game (plus_game F G) H" let ?R = "plus_game F (plus_game G H)" note options_plus = left_options_plus right_options_plus { fix opt note hyp = goal1(1)[simplified goal1(2), rule_format] have F: "zin opt (options F) ==> plus_game (plus_game opt G) H = plus_game opt (plus_game G H)" by (blast intro: hyp lprod_3_3) have G: "zin opt (options G) ==> plus_game (plus_game F opt) H = plus_game F (plus_game opt H)" by (blast intro: hyp lprod_3_4) have H: "zin opt (options H) ==> plus_game (plus_game F G) opt = plus_game F (plus_game G opt)" by (blast intro: hyp lprod_3_5) note F and G and H } note induct_hyp = this have "left_options ?L = left_options ?R ∧ right_options ?L = right_options ?R" by (auto simp add: plus_game.simps[where G="plus_game F G" and H=H] plus_game.simps[where G="F" and H="plus_game G H"] zet_ext_eq zunion zimage_iff options_plus induct_hyp left_imp_options right_imp_options) then show ?case by (simp add: game_ext_eq) qed } then show ?thesis by auto qed lemma neg_plus_game: "neg_game (plus_game G H) = plus_game (neg_game G) (neg_game H)" proof (induct G H rule: plus_game.induct) case (1 G H) note opt_ops = left_options_plus right_options_plus left_options_neg right_options_neg show ?case by (auto simp add: opt_ops neg_game.simps[of "plus_game G H"] plus_game.simps[of "neg_game G" "neg_game H"] Game_ext zet_ext_eq zunion zimage_iff 1) qed lemma eq_game_plus_inverse: "eq_game (plus_game x (neg_game x)) zero_game" proof (induct x rule: wf_induct[OF wf_option_of]) case (goal1 x) { fix y assume "zin y (options x)" then have "eq_game (plus_game y (neg_game y)) zero_game" by (auto simp add: goal1) } note ihyp = this { fix y assume y: "zin y (right_options x)" have "¬ (ge_game (zero_game, plus_game y (neg_game x)))" apply (subst ge_game.simps, simp) apply (rule exI[where x="plus_game y (neg_game y)"]) apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def]) apply (auto simp add: left_options_plus left_options_neg zunion zimage_iff intro: y) done } note case1 = this { fix y assume y: "zin y (left_options x)" have "¬ (ge_game (zero_game, plus_game x (neg_game y)))" apply (subst ge_game.simps, simp) apply (rule exI[where x="plus_game y (neg_game y)"]) apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def]) apply (auto simp add: left_options_plus zunion zimage_iff intro: y) done } note case2 = this { fix y assume y: "zin y (left_options x)" have "¬ (ge_game (plus_game y (neg_game x), zero_game))" apply (subst ge_game.simps, simp) apply (rule exI[where x="plus_game y (neg_game y)"]) apply (auto simp add: ihyp[of y, simplified y left_imp_options eq_game_def]) apply (auto simp add: right_options_plus right_options_neg zunion zimage_iff intro: y) done } note case3 = this { fix y assume y: "zin y (right_options x)" have "¬ (ge_game (plus_game x (neg_game y), zero_game))" apply (subst ge_game.simps, simp) apply (rule exI[where x="plus_game y (neg_game y)"]) apply (auto simp add: ihyp[of y, simplified y right_imp_options eq_game_def]) apply (auto simp add: right_options_plus zunion zimage_iff intro: y) done } note case4 = this show ?case apply (simp add: eq_game_def) apply (simp add: ge_game.simps[of "plus_game x (neg_game x)" "zero_game"]) apply (simp add: ge_game.simps[of "zero_game" "plus_game x (neg_game x)"]) apply (simp add: right_options_plus left_options_plus right_options_neg left_options_neg zunion zimage_iff) apply (auto simp add: case1 case2 case3 case4) done qed lemma ge_plus_game_left: "ge_game (y,z) = ge_game (plus_game x y, plus_game x z)" proof - { fix a have "∀ x y z. a = [x,y,z] --> ge_game (y,z) = ge_game (plus_game x y, plus_game x z)" proof (induct a rule: induct_game, (rule impI | rule allI)+) case (goal1 a x y z) note induct_hyp = goal1(1)[rule_format, simplified goal1(2)] { assume hyp: "ge_game(plus_game x y, plus_game x z)" have "ge_game (y, z)" proof - { fix yr assume yr: "zin yr (right_options y)" from hyp have "¬ (ge_game (plus_game x z, plus_game x yr))" by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"] right_options_plus zunion zimage_iff intro: yr) then have "¬ (ge_game (z, yr))" apply (subst induct_hyp[where y="[x, z, yr]", of "x" "z" "yr"]) apply (simp_all add: yr lprod_3_6) done } note yr = this { fix zl assume zl: "zin zl (left_options z)" from hyp have "¬ (ge_game (plus_game x zl, plus_game x y))" by (auto simp add: ge_game_eq[of "plus_game x y" "plus_game x z"] left_options_plus zunion zimage_iff intro: zl) then have "¬ (ge_game (zl, y))" apply (subst goal1(1)[rule_format, where y="[x, zl, y]", of "x" "zl" "y"]) apply (simp_all add: goal1(2) zl lprod_3_7) done } note zl = this show "ge_game (y, z)" apply (subst ge_game_eq) apply (auto simp add: yr zl) done qed } note right_imp_left = this { assume yz: "ge_game (y, z)" { fix x' assume x': "zin x' (right_options x)" assume hyp: "ge_game (plus_game x z, plus_game x' y)" then have n: "¬ (ge_game (plus_game x' y, plus_game x' z))" by (auto simp add: ge_game_eq[of "plus_game x z" "plus_game x' y"] right_options_plus zunion zimage_iff intro: x') have t: "ge_game (plus_game x' y, plus_game x' z)" apply (subst induct_hyp[symmetric]) apply (auto intro: lprod_3_3 x' yz) done from n t have "False" by blast } note case1 = this { fix x' assume x': "zin x' (left_options x)" assume hyp: "ge_game (plus_game x' z, plus_game x y)" then have n: "¬ (ge_game (plus_game x' y, plus_game x' z))" by (auto simp add: ge_game_eq[of "plus_game x' z" "plus_game x y"] left_options_plus zunion zimage_iff intro: x') have t: "ge_game (plus_game x' y, plus_game x' z)" apply (subst induct_hyp[symmetric]) apply (auto intro: lprod_3_3 x' yz) done from n t have "False" by blast } note case3 = this { fix y' assume y': "zin y' (right_options y)" assume hyp: "ge_game (plus_game x z, plus_game x y')" then have "ge_game(z, y')" apply (subst induct_hyp[of "[x, z, y']" "x" "z" "y'"]) apply (auto simp add: hyp lprod_3_6 y') done with yz have "ge_game (y, y')" by (blast intro: ge_game_trans) with y' have "False" by (auto simp add: ge_game_leftright_refl) } note case2 = this { fix z' assume z': "zin z' (left_options z)" assume hyp: "ge_game (plus_game x z', plus_game x y)" then have "ge_game(z', y)" apply (subst induct_hyp[of "[x, z', y]" "x" "z'" "y"]) apply (auto simp add: hyp lprod_3_7 z') done with yz have "ge_game (z', z)" by (blast intro: ge_game_trans) with z' have "False" by (auto simp add: ge_game_leftright_refl) } note case4 = this have "ge_game(plus_game x y, plus_game x z)" apply (subst ge_game_eq) apply (auto simp add: right_options_plus left_options_plus zunion zimage_iff) apply (auto intro: case1 case2 case3 case4) done } note left_imp_right = this show ?case by (auto intro: right_imp_left left_imp_right) qed } note a = this[of "[x, y, z]"] then show ?thesis by blast qed lemma ge_plus_game_right: "ge_game (y,z) = ge_game(plus_game y x, plus_game z x)" by (simp add: ge_plus_game_left plus_game_comm) lemma ge_neg_game: "ge_game (neg_game x, neg_game y) = ge_game (y, x)" proof - { fix a have "∀ x y. a = [x, y] --> ge_game (neg_game x, neg_game y) = ge_game (y, x)" proof (induct a rule: induct_game, (rule impI | rule allI)+) case (goal1 a x y) note ihyp = goal1(1)[rule_format, simplified goal1(2)] { fix xl assume xl: "zin xl (left_options x)" have "ge_game (neg_game y, neg_game xl) = ge_game (xl, y)" apply (subst ihyp) apply (auto simp add: lprod_2_1 xl) done } note xl = this { fix yr assume yr: "zin yr (right_options y)" have "ge_game (neg_game yr, neg_game x) = ge_game (x, yr)" apply (subst ihyp) apply (auto simp add: lprod_2_2 yr) done } note yr = this show ?case by (auto simp add: ge_game_eq[of "neg_game x" "neg_game y"] ge_game_eq[of "y" "x"] right_options_neg left_options_neg zimage_iff xl yr) qed } note a = this[of "[x,y]"] then show ?thesis by blast qed definition eq_game_rel :: "(game * game) set" where "eq_game_rel ≡ { (p, q) . eq_game p q }" definition "Pg = UNIV//eq_game_rel" typedef Pg = Pg unfolding Pg_def by (auto simp add: quotient_def) lemma equiv_eq_game[simp]: "equiv UNIV eq_game_rel" by (auto simp add: equiv_def refl_on_def sym_def trans_def eq_game_rel_def eq_game_sym intro: eq_game_refl eq_game_trans) instantiation Pg :: "{ord, zero, plus, minus, uminus}" begin definition Pg_zero_def: "0 = Abs_Pg (eq_game_rel `` {zero_game})" definition Pg_le_def: "G ≤ H <-> (∃ g h. g ∈ Rep_Pg G ∧ h ∈ Rep_Pg H ∧ ge_game (h, g))" definition Pg_less_def: "G < H <-> G ≤ H ∧ G ≠ (H::Pg)" definition Pg_minus_def: "- G = the_elem (\<Union> g ∈ Rep_Pg G. {Abs_Pg (eq_game_rel `` {neg_game g})})" definition Pg_plus_def: "G + H = the_elem (\<Union> g ∈ Rep_Pg G. \<Union> h ∈ Rep_Pg H. {Abs_Pg (eq_game_rel `` {plus_game g h})})" definition Pg_diff_def: "G - H = G + (- (H::Pg))" instance .. end lemma Rep_Abs_eq_Pg[simp]: "Rep_Pg (Abs_Pg (eq_game_rel `` {g})) = eq_game_rel `` {g}" apply (subst Abs_Pg_inverse) apply (auto simp add: Pg_def quotient_def) done lemma char_Pg_le[simp]: "(Abs_Pg (eq_game_rel `` {g}) ≤ Abs_Pg (eq_game_rel `` {h})) = (ge_game (h, g))" apply (simp add: Pg_le_def) apply (auto simp add: eq_game_rel_def eq_game_def intro: ge_game_trans ge_game_refl) done lemma char_Pg_eq[simp]: "(Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {h})) = (eq_game g h)" apply (simp add: Rep_Pg_inject [symmetric]) apply (subst eq_equiv_class_iff[of UNIV]) apply (simp_all) apply (simp add: eq_game_rel_def) done 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})" proof - have "(λ g h. {Abs_Pg (eq_game_rel `` {plus_game g h})}) respects2 eq_game_rel" apply (simp add: congruent2_def) apply (auto simp add: eq_game_rel_def eq_game_def) apply (rule_tac y="plus_game a ba" in ge_game_trans) apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+ apply (rule_tac y="plus_game b aa" in ge_game_trans) apply (simp add: ge_plus_game_left[symmetric] ge_plus_game_right[symmetric])+ done then show ?thesis by (simp add: Pg_plus_def UN_equiv_class2[OF equiv_eq_game equiv_eq_game]) qed lemma char_Pg_minus[simp]: "- Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {neg_game g})" proof - have "(λ g. {Abs_Pg (eq_game_rel `` {neg_game g})}) respects eq_game_rel" apply (simp add: congruent_def) apply (auto simp add: eq_game_rel_def eq_game_def ge_neg_game) done then show ?thesis by (simp add: Pg_minus_def UN_equiv_class[OF equiv_eq_game]) qed lemma eq_Abs_Pg[rule_format, cases type: Pg]: "(∀ g. z = Abs_Pg (eq_game_rel `` {g}) --> P) --> P" apply (cases z, simp) apply (simp add: Rep_Pg_inject[symmetric]) apply (subst Abs_Pg_inverse, simp) apply (auto simp add: Pg_def quotient_def) done instance Pg :: ordered_ab_group_add proof fix a b c :: Pg show "a - b = a + (- b)" by (simp add: Pg_diff_def) { assume ab: "a ≤ b" assume ba: "b ≤ a" from ab ba show "a = b" apply (cases a, cases b) apply (simp add: eq_game_def) done } then show "(a < b) = (a ≤ b ∧ ¬ b ≤ a)" by (auto simp add: Pg_less_def) show "a + b = b + a" apply (cases a, cases b) apply (simp add: eq_game_def plus_game_comm) done show "a + b + c = a + (b + c)" apply (cases a, cases b, cases c) apply (simp add: eq_game_def plus_game_assoc) done show "0 + a = a" apply (cases a) apply (simp add: Pg_zero_def plus_game_zero_left) done show "- a + a = 0" apply (cases a) apply (simp add: Pg_zero_def eq_game_plus_inverse plus_game_comm) done show "a ≤ a" apply (cases a) apply (simp add: ge_game_refl) done { assume ab: "a ≤ b" assume bc: "b ≤ c" from ab bc show "a ≤ c" apply (cases a, cases b, cases c) apply (auto intro: ge_game_trans) done } { assume ab: "a ≤ b" from ab show "c + a ≤ c + b" apply (cases a, cases b, cases c) apply (simp add: ge_plus_game_left[symmetric]) done } qed end