(*  Title:      HOL/Cardinals/Fun_More_FP.thy

     Author:     Andrei Popescu, TU Muenchen

     Copyright   2012

 More on injections, bijections and inverses (FP).

 *)

 header {* More on Injections, Bijections and Inverses (FP) *}

 theory Fun_More_FP

 imports Hilbert_Choice

 begin

 text {* This section proves more facts (additional to those in @{text "Fun.thy"},

 @{text "Hilbert_Choice.thy"}, and @{text "Finite_Set.thy"}),

 mainly concerning injections, bijections, inverses and (numeric) cardinals of

 finite sets. *}

 subsection {* Purely functional properties  *}

 (*2*)lemma bij_betw_id_iff:

 "(A = B) = (bij_betw id A B)"

 by(simp add: bij_betw_def)

 (*2*)lemma bij_betw_byWitness:

 assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and

         RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and

         IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"

 shows "bij_betw f A A'"

 using assms

 proof(unfold bij_betw_def inj_on_def, safe)

   fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"

   have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp

   with ** show "a = b" by simp

 next

   fix a' assume *: "a' \<in> A'"

   hence "f' a' \<in> A" using IM2 by blast

   moreover

   have "a' = f(f' a')" using * RIGHT by simp

   ultimately show "a' \<in> f ` A" by blast

 qed

 (*3*)corollary notIn_Un_bij_betw:

 assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and

        BIJ: "bij_betw f A A'"

 shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"

 proof-

   have "bij_betw f {b} {f b}"

   unfolding bij_betw_def inj_on_def by simp

   with assms show ?thesis

   using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast

 qed

 (*1*)lemma notIn_Un_bij_betw3:

 assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"

 shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"

 proof

   assume "bij_betw f A A'"

   thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"

   using assms notIn_Un_bij_betw[of b A f A'] by blast

 next

   assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"

   have "f ` A = A'"

   proof(auto)

     fix a assume **: "a \<in> A"

     hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast

     moreover

     {assume "f a = f b"

      hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast

      with NIN ** have False by blast

     }

     ultimately show "f a \<in> A'" by blast

   next

     fix a' assume **: "a' \<in> A'"

     hence "a' \<in> f`(A \<union> {b})"

     using * by (auto simp add: bij_betw_def)

     then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast

     moreover

     {assume "a = b" with 1 ** NIN' have False by blast

     }

     ultimately have "a \<in> A" by blast

     with 1 show "a' \<in> f ` A" by blast

   qed

   thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast

 qed

 subsection {* Properties involving finite and infinite sets *}

 (*3*)lemma inj_on_finite:

 assumes "inj_on f A" "f ` A \<le> B" "finite B"

 shows "finite A"

 using assms by (metis finite_imageD finite_subset)

 (*3*)lemma infinite_imp_bij_betw:

 assumes INF: "\<not> finite A"

 shows "\<exists>h. bij_betw h A (A - {a})"

 proof(cases "a \<in> A")

   assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast

   thus ?thesis using bij_betw_id[of A] by auto

 next

   assume Case2: "a \<in> A"

 find_theorems "\<not> finite _"

   have "\<not> finite (A - {a})" using INF by auto

   with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"

   where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast

   obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast

   obtain A' where A'_def: "A' = g ` UNIV" by blast

   have temp: "\<forall>y. f y \<noteq> a" using 2 by blast

   have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"

   proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,

         case_tac "x = 0", auto simp add: 2)

     fix y  assume "a = (if y = 0 then a else f (Suc y))"

     thus "y = 0" using temp by (case_tac "y = 0", auto)

   next

     fix x y

     assume "f (Suc x) = (if y = 0 then a else f (Suc y))"

     thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)

   next

     fix n show "f (Suc n) \<in> A" using 2 by blast

   qed

   hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"

   using inj_on_imp_bij_betw[of g] unfolding A'_def by auto

   hence 5: "bij_betw (inv g) A' UNIV"

   by (auto simp add: bij_betw_inv_into)

   (*  *)

   obtain n where "g n = a" using 3 by auto

   hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"

   using 3 4 unfolding A'_def

   by clarify (rule bij_betw_subset, auto simp: image_set_diff)

   (*  *)

   obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast

   have 7: "bij_betw v UNIV (UNIV - {n})"

   proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)

     fix m1 m2 assume "v m1 = v m2"

     thus "m1 = m2"

     by(case_tac "m1 < n", case_tac "m2 < n",

        auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)

   next

     show "v ` UNIV = UNIV - {n}"

     proof(auto simp add: v_def)

       fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"

       {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto

        then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto

        with 71 have "n \<le> m'" by auto

        with 72 ** have False by auto

       }

       thus "m < n" by force

     qed

   qed

   (*  *)

   obtain h' where h'_def: "h' = g o v o (inv g)" by blast

   hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6

   by (auto simp add: bij_betw_trans)

   (*  *)

   obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast

   have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto

   hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto

   moreover

   {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto

    hence "bij_betw h  (A - A') (A - A')"

    using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto

   }

   moreover

   have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>

         ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"

   using 4 by blast

   ultimately have "bij_betw h A (A - {a})"

   using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp

   thus ?thesis by blast

 qed

 (*3*)lemma infinite_imp_bij_betw2:

 assumes INF: "\<not> finite A"

 shows "\<exists>h. bij_betw h A (A \<union> {a})"

 proof(cases "a \<in> A")

   assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast

   thus ?thesis using bij_betw_id[of A] by auto

 next

   let ?A' = "A \<union> {a}"

   assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast

   moreover have "\<not> finite ?A'" using INF by auto

   ultimately obtain f where "bij_betw f ?A' A"

   using infinite_imp_bij_betw[of ?A' a] by auto

   hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast

   thus ?thesis by auto

 qed

 subsection {* Properties involving Hilbert choice *}

 (*2*)lemma bij_betw_inv_into_left:

 assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"

 shows "(inv_into A f) (f a) = a"

 using assms unfolding bij_betw_def

 by clarify (rule inv_into_f_f)

 (*2*)lemma bij_betw_inv_into_right:

 assumes "bij_betw f A A'" "a' \<in> A'"

 shows "f(inv_into A f a') = a'"

 using assms unfolding bij_betw_def using f_inv_into_f by force

 (*1*)lemma bij_betw_inv_into_subset:

 assumes BIJ: "bij_betw f A A'" and

         SUB: "B \<le> A" and IM: "f ` B = B'"

 shows "bij_betw (inv_into A f) B' B"

 using assms unfolding bij_betw_def

 by (auto intro: inj_on_inv_into)

 end