(*  Title:      HOL/ZF/LProd.thy

     Author:     Steven Obua

     Introduces the lprod relation.

     See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan

 *)

 theory LProd 

 imports "~~/src/HOL/Library/Multiset"

 begin

 inductive_set

   lprod :: "('a * 'a) set \<Rightarrow> ('a list * 'a list) set"

   for R :: "('a * 'a) set"

 where

   lprod_single[intro!]: "(a, b) \<in> R \<Longrightarrow> ([a], [b]) \<in> lprod R"

 | lprod_list[intro!]: "(ah@at, bh@bt) \<in> lprod R \<Longrightarrow> (a,b) \<in> R \<or> a = b \<Longrightarrow> (ah@a#at, bh@b#bt) \<in> lprod R"

 lemma "(as,bs) \<in> lprod R \<Longrightarrow> length as = length bs"

   apply (induct as bs rule: lprod.induct)

   apply auto

   done

 lemma "(as, bs) \<in> lprod R \<Longrightarrow> 1 \<le> length as \<and> 1 \<le> length bs"

   apply (induct as bs rule: lprod.induct)

   apply auto

   done

 lemma lprod_subset_elem: "(as, bs) \<in> lprod S \<Longrightarrow> S \<subseteq> R \<Longrightarrow> (as, bs) \<in> lprod R"

   apply (induct as bs rule: lprod.induct)

   apply (auto)

   done

 lemma lprod_subset: "S \<subseteq> R \<Longrightarrow> lprod S \<subseteq> lprod R"

   by (auto intro: lprod_subset_elem)

 lemma lprod_implies_mult: "(as, bs) \<in> lprod R \<Longrightarrow> trans R \<Longrightarrow> (multiset_of as, multiset_of bs) \<in> mult R"

 proof (induct as bs rule: lprod.induct)

   case (lprod_single a b)

   note step = one_step_implies_mult[

     where r=R and I="{#}" and K="{#a#}" and J="{#b#}", simplified]    

   show ?case by (auto intro: lprod_single step)

 next

   case (lprod_list ah at bh bt a b)

   then have transR: "trans R" by auto

   have as: "multiset_of (ah @ a # at) = multiset_of (ah @ at) + {#a#}" (is "_ = ?ma + _")

     by (simp add: algebra_simps)

   have bs: "multiset_of (bh @ b # bt) = multiset_of (bh @ bt) + {#b#}" (is "_ = ?mb + _")

     by (simp add: algebra_simps)

   from lprod_list have "(?ma, ?mb) \<in> mult R"

     by auto

   with mult_implies_one_step[OF transR] have 

     "\<exists>I J K. ?mb = I + J \<and> ?ma = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> R)"

     by blast

   then obtain I J K where 

     decomposed: "?mb = I + J \<and> ?ma = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> R)"

     by blast   

   show ?case

   proof (cases "a = b")

     case True    

     have "((I + {#b#}) + K, (I + {#b#}) + J) \<in> mult R"

       apply (rule one_step_implies_mult[OF transR])

       apply (auto simp add: decomposed)

       done

     then show ?thesis

       apply (simp only: as bs)

       apply (simp only: decomposed True)

       apply (simp add: algebra_simps)

       done

   next

     case False

     from False lprod_list have False: "(a, b) \<in> R" by blast

     have "(I + (K + {#a#}), I + (J + {#b#})) \<in> mult R"

       apply (rule one_step_implies_mult[OF transR])

       apply (auto simp add: False decomposed)

       done

     then show ?thesis

       apply (simp only: as bs)

       apply (simp only: decomposed)

       apply (simp add: algebra_simps)

       done

   qed

 qed

 lemma wf_lprod[simp,intro]:

   assumes wf_R: "wf R"

   shows "wf (lprod R)"

 proof -

   have subset: "lprod (R^+) \<subseteq> inv_image (mult (R^+)) multiset_of"

     by (auto simp add: lprod_implies_mult trans_trancl)

   note lprodtrancl = wf_subset[OF wf_inv_image[where r="mult (R^+)" and f="multiset_of", 

     OF wf_mult[OF wf_trancl[OF wf_R]]], OF subset]

   note lprod = wf_subset[OF lprodtrancl, where p="lprod R", OF lprod_subset, simplified]

   show ?thesis by (auto intro: lprod)

 qed

 definition gprod_2_2 :: "('a * 'a) set \<Rightarrow> (('a * 'a) * ('a * 'a)) set" where

   "gprod_2_2 R \<equiv> { ((a,b), (c,d)) . (a = c \<and> (b,d) \<in> R) \<or> (b = d \<and> (a,c) \<in> R) }"

 definition gprod_2_1 :: "('a * 'a) set \<Rightarrow> (('a * 'a) * ('a * 'a)) set" where

   "gprod_2_1 R \<equiv>  { ((a,b), (c,d)) . (a = d \<and> (b,c) \<in> R) \<or> (b = c \<and> (a,d) \<in> R) }"

 lemma lprod_2_3: "(a, b) \<in> R \<Longrightarrow> ([a, c], [b, c]) \<in> lprod R"

   by (auto intro: lprod_list[where a=c and b=c and 

     ah = "[a]" and at = "[]" and bh="[b]" and bt="[]", simplified]) 

 lemma lprod_2_4: "(a, b) \<in> R \<Longrightarrow> ([c, a], [c, b]) \<in> lprod R"

   by (auto intro: lprod_list[where a=c and b=c and 

     ah = "[]" and at = "[a]" and bh="[]" and bt="[b]", simplified])

 lemma lprod_2_1: "(a, b) \<in> R \<Longrightarrow> ([c, a], [b, c]) \<in> lprod R"

   by (auto intro: lprod_list[where a=c and b=c and 

     ah = "[]" and at = "[a]" and bh="[b]" and bt="[]", simplified]) 

 lemma lprod_2_2: "(a, b) \<in> R \<Longrightarrow> ([a, c], [c, b]) \<in> lprod R"

   by (auto intro: lprod_list[where a=c and b=c and 

     ah = "[a]" and at = "[]" and bh="[]" and bt="[b]", simplified])

 lemma [simp, intro]:

   assumes wfR: "wf R" shows "wf (gprod_2_1 R)"

 proof -

   have "gprod_2_1 R \<subseteq> inv_image (lprod R) (\<lambda> (a,b). [a,b])"

     by (auto simp add: gprod_2_1_def lprod_2_1 lprod_2_2)

   with wfR show ?thesis

     by (rule_tac wf_subset, auto)

 qed

 lemma [simp, intro]:

   assumes wfR: "wf R" shows "wf (gprod_2_2 R)"

 proof -

   have "gprod_2_2 R \<subseteq> inv_image (lprod R) (\<lambda> (a,b). [a,b])"

     by (auto simp add: gprod_2_2_def lprod_2_3 lprod_2_4)

   with wfR show ?thesis

     by (rule_tac wf_subset, auto)

 qed

 lemma lprod_3_1: assumes "(x', x) \<in> R" shows "([y, z, x'], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="y" and b="y" and ah="[]" and at="[z,x']" and bh="[x]" and bt="[z]", simplified])

   apply (auto simp add: lprod_2_1 assms)

   done

 lemma lprod_3_2: assumes "(z',z) \<in> R" shows "([z', x, y], [x,y,z]) \<in> lprod R"

   apply (rule lprod_list[where a="y" and b="y" and ah="[z',x]" and at="[]" and bh="[x]" and bt="[z]", simplified])

   apply (auto simp add: lprod_2_2 assms)

   done

 lemma lprod_3_3: assumes xr: "(xr, x) \<in> R" shows "([xr, y, z], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="y" and b="y" and ah="[xr]" and at="[z]" and bh="[x]" and bt="[z]", simplified])

   apply (simp add: xr lprod_2_3)

   done

 lemma lprod_3_4: assumes yr: "(yr, y) \<in> R" shows "([x, yr, z], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="x" and b="x" and ah="[]" and at="[yr,z]" and bh="[]" and bt="[y,z]", simplified])

   apply (simp add: yr lprod_2_3)

   done

 lemma lprod_3_5: assumes zr: "(zr, z) \<in> R" shows "([x, y, zr], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="x" and b="x" and ah="[]" and at="[y,zr]" and bh="[]" and bt="[y,z]", simplified])

   apply (simp add: zr lprod_2_4)

   done

 lemma lprod_3_6: assumes y': "(y', y) \<in> R" shows "([x, z, y'], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="z" and b="z" and ah="[x]" and at="[y']" and bh="[x,y]" and bt="[]", simplified])

   apply (simp add: y' lprod_2_4)

   done

 lemma lprod_3_7: assumes z': "(z',z) \<in> R" shows "([x, z', y], [x, y, z]) \<in> lprod R"

   apply (rule lprod_list[where a="y" and b="y" and ah="[x, z']" and at="[]" and bh="[x]" and bt="[z]", simplified])

   apply (simp add: z' lprod_2_4)

   done

 definition perm :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> bool" where

    "perm f A \<equiv> inj_on f A \<and> f ` A = A"

 lemma "((as,bs) \<in> lprod R) = 

   (\<exists> f. perm f {0 ..< (length as)} \<and> 

   (\<forall> j. j < length as \<longrightarrow> ((nth as j, nth bs (f j)) \<in> R \<or> (nth as j = nth bs (f j)))) \<and> 

   (\<exists> i. i < length as \<and> (nth as i, nth bs (f i)) \<in> R))"

 oops

 lemma "trans R \<Longrightarrow> (ah@a#at, bh@b#bt) \<in> lprod R \<Longrightarrow> (b, a) \<in> R \<or> a = b \<Longrightarrow> (ah@at, bh@bt) \<in> lprod R" 

 oops

 end