src/HOL/Ordinals_and_Cardinals/Fun_More_Base.thy

-(*  Title:      HOL/Ordinals_and_Cardinals/Fun_More_Base.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-More on injections, bijections and inverses (base).
-*)
-
-header {* More on Injections, Bijections and Inverses (Base) *}
-
-theory Fun_More_Base
-imports "~~/src/HOL/Library/Infinite_Set"
-begin
-
-
-text {* This section proves more facts (additional to those in @{text "Fun.thy"},
-@{text "Hilbert_Choice.thy"}, @{text "Finite_Set.thy"} and @{text "Infinite_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, auto)
-  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 infinite_super by (fast dest: finite_imageD)
-
-
-(*3*)lemma infinite_imp_bij_betw:
-assumes INF: "infinite 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"
-  have "infinite (A - {a})" using INF infinite_remove 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: "infinite 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 "infinite ?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_LEFT:
-assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A"
-shows "(inv_into A f)`(f ` B) = B"
-using assms unfolding bij_betw_def using inv_into_image_cancel by force
-
-
-(*1*)lemma bij_betw_inv_into_LEFT_RIGHT:
-assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A" and
-        IM: "f ` B = B'"
-shows "(inv_into A f) ` B' = B"
-using assms bij_betw_inv_into_LEFT[of f A A' B] by fast
-
-
-(*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)
-
-
-subsection {* Other facts  *}
-
-
-(*2*)lemma atLeastLessThan_less_eq:
-"({0..<m} \<le> {0..<n}) = ((m::nat) \<le> n)"
-unfolding ivl_subset by arith
-
-
-(*2*)lemma atLeastLessThan_less_eq2:
-assumes "inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}"
-shows "m \<le> n"
-using assms
-using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
-      card_atLeastLessThan[of m] card_atLeastLessThan[of n]
-      card_inj_on_le[of f "{0 ..< m}" "{0 ..< n}"] by auto
-
-
-
-end