(* Title: HOL/Cardinals/Fun_More.thy
Author: Andrei Popescu, TU Muenchen
Copyright 2012
More on injections, bijections and inverses.
*)
header {* More on Injections, Bijections and Inverses *}
theory Fun_More
imports Fun_More_LFP
begin
subsection {* Purely functional properties *}
(* unused *)
(*1*)lemma notIn_Un_bij_betw2:
assumes NIN: "b \<notin> A" and NIN': "b' \<notin> A'" and
BIJ: "bij_betw f A A'"
shows "bij_betw f (A \<union> {b}) (A' \<union> {b'}) = (f b = b')"
proof
assume "f b = b'"
thus "bij_betw f (A \<union> {b}) (A' \<union> {b'})"
using assms notIn_Un_bij_betw[of b A f A'] by auto
next
assume *: "bij_betw f (A \<union> {b}) (A' \<union> {b'})"
hence "f b \<in> A' \<union> {b'}"
unfolding bij_betw_def by auto
moreover
{assume "f b \<in> A'"
then obtain b1 where 1: "b1 \<in> A" and 2: "f b1 = f b" using BIJ
by (auto simp add: bij_betw_def)
hence "b = b1" using *
by (auto simp add: bij_betw_def inj_on_def)
with 1 NIN have False by auto
}
ultimately show "f b = b'" by blast
qed
(* unused *)
(*1*)lemma bij_betw_ball:
assumes BIJ: "bij_betw f A B"
shows "(\<forall>b \<in> B. phi b) = (\<forall>a \<in> A. phi(f a))"
using assms unfolding bij_betw_def inj_on_def by blast
(* unused *)
(*1*)lemma bij_betw_diff_singl:
assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
shows "bij_betw f (A - {a}) (A' - {f a})"
proof-
let ?B = "A - {a}" let ?B' = "A' - {f a}"
have "f a \<in> A'" using IN BIJ unfolding bij_betw_def by blast
hence "a \<notin> ?B \<and> f a \<notin> ?B' \<and> A = ?B \<union> {a} \<and> A' = ?B' \<union> {f a}"
using IN by blast
thus ?thesis using notIn_Un_bij_betw3[of a ?B f ?B'] BIJ by simp
qed
subsection {* Properties involving finite and infinite sets *}
(*3*)lemma inj_on_image_Pow:
assumes "inj_on f A"
shows "inj_on (image f) (Pow A)"
unfolding Pow_def inj_on_def proof(clarsimp)
fix X Y assume *: "X \<le> A" and **: "Y \<le> A" and
***: "f ` X = f ` Y"
show "X = Y"
proof(auto)
fix x assume ****: "x \<in> X"
with *** obtain y where "y \<in> Y \<and> f x = f y" by blast
with **** * ** assms show "x \<in> Y"
unfolding inj_on_def by auto
next
fix y assume ****: "y \<in> Y"
with *** obtain x where "x \<in> X \<and> f x = f y" by atomize_elim force
with **** * ** assms show "y \<in> X"
unfolding inj_on_def by auto
qed
qed
(*2*)lemma bij_betw_image_Pow:
assumes "bij_betw f A B"
shows "bij_betw (image f) (Pow A) (Pow B)"
using assms unfolding bij_betw_def
by (auto simp add: inj_on_image_Pow image_Pow_surj)
(* unused *)
(*1*)lemma bij_betw_inv_into_RIGHT:
assumes BIJ: "bij_betw f A A'" and SUB: "B' \<le> A'"
shows "f `((inv_into A f)`B') = B'"
using assms
proof(auto simp add: bij_betw_inv_into_right)
let ?f' = "(inv_into A f)"
fix a' assume *: "a' \<in> B'"
hence "a' \<in> A'" using SUB by auto
hence "a' = f (?f' a')"
using BIJ by (auto simp add: bij_betw_inv_into_right)
thus "a' \<in> f ` (?f' ` B')" using * by blast
qed
(* unused *)
(*1*)lemma bij_betw_inv_into_RIGHT_LEFT:
assumes BIJ: "bij_betw f A A'" and SUB: "B' \<le> A'" and
IM: "(inv_into A f) ` B' = B"
shows "f ` B = B'"
proof-
have "f`((inv_into A f)` B') = B'"
using assms bij_betw_inv_into_RIGHT[of f A A' B'] by auto
thus ?thesis using IM by auto
qed
(* unused *)
(*2*)lemma bij_betw_inv_into_twice:
assumes "bij_betw f A A'"
shows "\<forall>a \<in> A. inv_into A' (inv_into A f) a = f a"
proof
let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'"
have 1: "bij_betw ?f' A' A" using assms
by (auto simp add: bij_betw_inv_into)
fix a assume *: "a \<in> A"
then obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
using 1 unfolding bij_betw_def by force
hence "?f'' a = a'"
using * 1 3 by (auto simp add: bij_betw_inv_into_left)
moreover have "f a = a'" using assms 2 3
by (auto simp add: bij_betw_inv_into_right)
ultimately show "?f'' a = f a" by simp
qed
subsection {* Properties involving Hilbert choice *}
(*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
subsection {* Other facts *}
(*3*)lemma atLeastLessThan_injective:
assumes "{0 ..< m::nat} = {0 ..< n}"
shows "m = n"
proof-
{assume "m < n"
hence "m \<in> {0 ..< n}" by auto
hence "{0 ..< m} < {0 ..< n}" by auto
hence False using assms by blast
}
moreover
{assume "n < m"
hence "n \<in> {0 ..< m}" by auto
hence "{0 ..< n} < {0 ..< m}" by auto
hence False using assms by blast
}
ultimately show ?thesis by force
qed
(*2*)lemma atLeastLessThan_injective2:
"bij_betw f {0 ..< m::nat} {0 ..< n} \<Longrightarrow> m = n"
using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
card_atLeastLessThan[of m] card_atLeastLessThan[of n]
bij_betw_iff_card[of "{0 ..< m}" "{0 ..< n}"] by auto
(* unused *)
(*2*)lemma atLeastLessThan_less_eq3:
"(\<exists>f. inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}) = (m \<le> n)"
using atLeastLessThan_less_eq2
proof(auto)
assume "m \<le> n"
hence "inj_on id {0..<m} \<and> id ` {0..<m} \<subseteq> {0..<n}" unfolding inj_on_def by force
thus "\<exists>f. inj_on f {0..<m} \<and> f ` {0..<m} \<subseteq> {0..<n}" by blast
qed
(* unused *)
(*3*)lemma atLeastLessThan_less:
"({0..<m} < {0..<n}) = ((m::nat) < n)"
proof-
have "({0..<m} < {0..<n}) = ({0..<m} \<le> {0..<n} \<and> {0..<m} ~= {0..<n})"
using subset_iff_psubset_eq by blast
also have "\<dots> = (m \<le> n \<and> m ~= n)"
using atLeastLessThan_less_eq atLeastLessThan_injective by blast
also have "\<dots> = (m < n)" by auto
finally show ?thesis .
qed
end