--- a/src/ZF/Cardinal.thy Tue Sep 27 13:34:54 2022 +0200
+++ b/src/ZF/Cardinal.thy Tue Sep 27 16:51:35 2022 +0100
@@ -10,31 +10,31 @@
definition
(*least ordinal operator*)
Least :: "(i=>o) => i" (binder \<open>\<mu> \<close> 10) where
- "Least(P) == THE i. Ord(i) & P(i) & (\<forall>j. j<i \<longrightarrow> ~P(j))"
+ "Least(P) \<equiv> THE i. Ord(i) & P(i) & (\<forall>j. j<i \<longrightarrow> \<not>P(j))"
definition
eqpoll :: "[i,i] => o" (infixl \<open>\<approx>\<close> 50) where
- "A \<approx> B == \<exists>f. f \<in> bij(A,B)"
+ "A \<approx> B \<equiv> \<exists>f. f \<in> bij(A,B)"
definition
lepoll :: "[i,i] => o" (infixl \<open>\<lesssim>\<close> 50) where
- "A \<lesssim> B == \<exists>f. f \<in> inj(A,B)"
+ "A \<lesssim> B \<equiv> \<exists>f. f \<in> inj(A,B)"
definition
lesspoll :: "[i,i] => o" (infixl \<open>\<prec>\<close> 50) where
- "A \<prec> B == A \<lesssim> B & ~(A \<approx> B)"
+ "A \<prec> B \<equiv> A \<lesssim> B & \<not>(A \<approx> B)"
definition
cardinal :: "i=>i" (\<open>|_|\<close>) where
- "|A| == (\<mu> i. i \<approx> A)"
+ "|A| \<equiv> (\<mu> i. i \<approx> A)"
definition
Finite :: "i=>o" where
- "Finite(A) == \<exists>n\<in>nat. A \<approx> n"
+ "Finite(A) \<equiv> \<exists>n\<in>nat. A \<approx> n"
definition
Card :: "i=>o" where
- "Card(i) == (i = |i|)"
+ "Card(i) \<equiv> (i = |i|)"
subsection\<open>The Schroeder-Bernstein Theorem\<close>
@@ -47,7 +47,7 @@
lemma Banach_last_equation:
"g \<in> Y->X
- ==> g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) =
+ \<Longrightarrow> g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) =
X - lfp(X, %W. X - g``(Y - f``W))"
apply (rule_tac P = "%u. v = X-u" for v
in decomp_bnd_mono [THEN lfp_unfold, THEN ssubst])
@@ -55,7 +55,7 @@
done
lemma decomposition:
- "[| f \<in> X->Y; g \<in> Y->X |] ==>
+ "\<lbrakk>f \<in> X->Y; g \<in> Y->X\<rbrakk> \<Longrightarrow>
\<exists>XA XB YA YB. (XA \<inter> XB = 0) & (XA \<union> XB = X) &
(YA \<inter> YB = 0) & (YA \<union> YB = Y) &
f``XA=YA & g``YB=XB"
@@ -67,18 +67,18 @@
done
lemma schroeder_bernstein:
- "[| f \<in> inj(X,Y); g \<in> inj(Y,X) |] ==> \<exists>h. h \<in> bij(X,Y)"
+ "\<lbrakk>f \<in> inj(X,Y); g \<in> inj(Y,X)\<rbrakk> \<Longrightarrow> \<exists>h. h \<in> bij(X,Y)"
apply (insert decomposition [of f X Y g])
apply (simp add: inj_is_fun)
apply (blast intro!: restrict_bij bij_disjoint_Un intro: bij_converse_bij)
(* The instantiation of exI to @{term"restrict(f,XA) \<union> converse(restrict(g,YB))"}
- is forced by the context!! *)
+ is forced by the context\<And>*)
done
(** Equipollence is an equivalence relation **)
-lemma bij_imp_eqpoll: "f \<in> bij(A,B) ==> A \<approx> B"
+lemma bij_imp_eqpoll: "f \<in> bij(A,B) \<Longrightarrow> A \<approx> B"
apply (unfold eqpoll_def)
apply (erule exI)
done
@@ -86,20 +86,20 @@
(*A \<approx> A*)
lemmas eqpoll_refl = id_bij [THEN bij_imp_eqpoll, simp]
-lemma eqpoll_sym: "X \<approx> Y ==> Y \<approx> X"
+lemma eqpoll_sym: "X \<approx> Y \<Longrightarrow> Y \<approx> X"
apply (unfold eqpoll_def)
apply (blast intro: bij_converse_bij)
done
lemma eqpoll_trans [trans]:
- "[| X \<approx> Y; Y \<approx> Z |] ==> X \<approx> Z"
+ "\<lbrakk>X \<approx> Y; Y \<approx> Z\<rbrakk> \<Longrightarrow> X \<approx> Z"
apply (unfold eqpoll_def)
apply (blast intro: comp_bij)
done
(** Le-pollence is a partial ordering **)
-lemma subset_imp_lepoll: "X<=Y ==> X \<lesssim> Y"
+lemma subset_imp_lepoll: "X<=Y \<Longrightarrow> X \<lesssim> Y"
apply (unfold lepoll_def)
apply (rule exI)
apply (erule id_subset_inj)
@@ -109,35 +109,35 @@
lemmas le_imp_lepoll = le_imp_subset [THEN subset_imp_lepoll]
-lemma eqpoll_imp_lepoll: "X \<approx> Y ==> X \<lesssim> Y"
+lemma eqpoll_imp_lepoll: "X \<approx> Y \<Longrightarrow> X \<lesssim> Y"
by (unfold eqpoll_def bij_def lepoll_def, blast)
-lemma lepoll_trans [trans]: "[| X \<lesssim> Y; Y \<lesssim> Z |] ==> X \<lesssim> Z"
+lemma lepoll_trans [trans]: "\<lbrakk>X \<lesssim> Y; Y \<lesssim> Z\<rbrakk> \<Longrightarrow> X \<lesssim> Z"
apply (unfold lepoll_def)
apply (blast intro: comp_inj)
done
-lemma eq_lepoll_trans [trans]: "[| X \<approx> Y; Y \<lesssim> Z |] ==> X \<lesssim> Z"
+lemma eq_lepoll_trans [trans]: "\<lbrakk>X \<approx> Y; Y \<lesssim> Z\<rbrakk> \<Longrightarrow> X \<lesssim> Z"
by (blast intro: eqpoll_imp_lepoll lepoll_trans)
-lemma lepoll_eq_trans [trans]: "[| X \<lesssim> Y; Y \<approx> Z |] ==> X \<lesssim> Z"
+lemma lepoll_eq_trans [trans]: "\<lbrakk>X \<lesssim> Y; Y \<approx> Z\<rbrakk> \<Longrightarrow> X \<lesssim> Z"
by (blast intro: eqpoll_imp_lepoll lepoll_trans)
(*Asymmetry law*)
-lemma eqpollI: "[| X \<lesssim> Y; Y \<lesssim> X |] ==> X \<approx> Y"
+lemma eqpollI: "\<lbrakk>X \<lesssim> Y; Y \<lesssim> X\<rbrakk> \<Longrightarrow> X \<approx> Y"
apply (unfold lepoll_def eqpoll_def)
apply (elim exE)
apply (rule schroeder_bernstein, assumption+)
done
lemma eqpollE:
- "[| X \<approx> Y; [| X \<lesssim> Y; Y \<lesssim> X |] ==> P |] ==> P"
+ "\<lbrakk>X \<approx> Y; \<lbrakk>X \<lesssim> Y; Y \<lesssim> X\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast intro: eqpoll_imp_lepoll eqpoll_sym)
lemma eqpoll_iff: "X \<approx> Y \<longleftrightarrow> X \<lesssim> Y & Y \<lesssim> X"
by (blast intro: eqpollI elim!: eqpollE)
-lemma lepoll_0_is_0: "A \<lesssim> 0 ==> A = 0"
+lemma lepoll_0_is_0: "A \<lesssim> 0 \<Longrightarrow> A = 0"
apply (unfold lepoll_def inj_def)
apply (blast dest: apply_type)
done
@@ -149,20 +149,20 @@
by (blast intro: lepoll_0_is_0 lepoll_refl)
lemma Un_lepoll_Un:
- "[| A \<lesssim> B; C \<lesssim> D; B \<inter> D = 0 |] ==> A \<union> C \<lesssim> B \<union> D"
+ "\<lbrakk>A \<lesssim> B; C \<lesssim> D; B \<inter> D = 0\<rbrakk> \<Longrightarrow> A \<union> C \<lesssim> B \<union> D"
apply (unfold lepoll_def)
apply (blast intro: inj_disjoint_Un)
done
-(*A \<approx> 0 ==> A=0*)
+(*A \<approx> 0 \<Longrightarrow> A=0*)
lemmas eqpoll_0_is_0 = eqpoll_imp_lepoll [THEN lepoll_0_is_0]
lemma eqpoll_0_iff: "A \<approx> 0 \<longleftrightarrow> A=0"
by (blast intro: eqpoll_0_is_0 eqpoll_refl)
lemma eqpoll_disjoint_Un:
- "[| A \<approx> B; C \<approx> D; A \<inter> C = 0; B \<inter> D = 0 |]
- ==> A \<union> C \<approx> B \<union> D"
+ "\<lbrakk>A \<approx> B; C \<approx> D; A \<inter> C = 0; B \<inter> D = 0\<rbrakk>
+ \<Longrightarrow> A \<union> C \<approx> B \<union> D"
apply (unfold eqpoll_def)
apply (blast intro: bij_disjoint_Un)
done
@@ -170,16 +170,16 @@
subsection\<open>lesspoll: contributions by Krzysztof Grabczewski\<close>
-lemma lesspoll_not_refl: "~ (i \<prec> i)"
+lemma lesspoll_not_refl: "\<not> (i \<prec> i)"
by (simp add: lesspoll_def)
-lemma lesspoll_irrefl [elim!]: "i \<prec> i ==> P"
+lemma lesspoll_irrefl [elim!]: "i \<prec> i \<Longrightarrow> P"
by (simp add: lesspoll_def)
-lemma lesspoll_imp_lepoll: "A \<prec> B ==> A \<lesssim> B"
+lemma lesspoll_imp_lepoll: "A \<prec> B \<Longrightarrow> A \<lesssim> B"
by (unfold lesspoll_def, blast)
-lemma lepoll_well_ord: "[| A \<lesssim> B; well_ord(B,r) |] ==> \<exists>s. well_ord(A,s)"
+lemma lepoll_well_ord: "\<lbrakk>A \<lesssim> B; well_ord(B,r)\<rbrakk> \<Longrightarrow> \<exists>s. well_ord(A,s)"
apply (unfold lepoll_def)
apply (blast intro: well_ord_rvimage)
done
@@ -207,36 +207,36 @@
(** Variations on transitivity **)
lemma lesspoll_trans [trans]:
- "[| X \<prec> Y; Y \<prec> Z |] ==> X \<prec> Z"
+ "\<lbrakk>X \<prec> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
apply (unfold lesspoll_def)
apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
done
lemma lesspoll_trans1 [trans]:
- "[| X \<lesssim> Y; Y \<prec> Z |] ==> X \<prec> Z"
+ "\<lbrakk>X \<lesssim> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
apply (unfold lesspoll_def)
apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
done
lemma lesspoll_trans2 [trans]:
- "[| X \<prec> Y; Y \<lesssim> Z |] ==> X \<prec> Z"
+ "\<lbrakk>X \<prec> Y; Y \<lesssim> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
apply (unfold lesspoll_def)
apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
done
lemma eq_lesspoll_trans [trans]:
- "[| X \<approx> Y; Y \<prec> Z |] ==> X \<prec> Z"
+ "\<lbrakk>X \<approx> Y; Y \<prec> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
by (blast intro: eqpoll_imp_lepoll lesspoll_trans1)
lemma lesspoll_eq_trans [trans]:
- "[| X \<prec> Y; Y \<approx> Z |] ==> X \<prec> Z"
+ "\<lbrakk>X \<prec> Y; Y \<approx> Z\<rbrakk> \<Longrightarrow> X \<prec> Z"
by (blast intro: eqpoll_imp_lepoll lesspoll_trans2)
(** \<mu> -- the least number operator [from HOL/Univ.ML] **)
lemma Least_equality:
- "[| P(i); Ord(i); !!x. x<i ==> ~P(x) |] ==> (\<mu> x. P(x)) = i"
+ "\<lbrakk>P(i); Ord(i); \<And>x. x<i \<Longrightarrow> \<not>P(x)\<rbrakk> \<Longrightarrow> (\<mu> x. P(x)) = i"
apply (unfold Least_def)
apply (rule the_equality, blast)
apply (elim conjE)
@@ -254,7 +254,7 @@
case True thus ?thesis .
next
case False
- hence "\<And>x. x \<in> i \<Longrightarrow> ~P(x)" using step
+ hence "\<And>x. x \<in> i \<Longrightarrow> \<not>P(x)" using step
by blast
hence "(\<mu> a. P(a)) = i" using step
by (blast intro: Least_equality ltD)
@@ -278,7 +278,7 @@
case True thus ?thesis .
next
case False
- hence "\<And>x. x \<in> i \<Longrightarrow> ~ (\<mu> a. P(a)) \<le> i" using step
+ hence "\<And>x. x \<in> i \<Longrightarrow> \<not> (\<mu> a. P(a)) \<le> i" using step
by blast
hence "(\<mu> a. P(a)) = i" using step
by (blast elim: ltE intro: ltI Least_equality lt_trans1)
@@ -291,19 +291,19 @@
qed
(*\<mu> really is the smallest*)
-lemma less_LeastE: "[| P(i); i < (\<mu> x. P(x)) |] ==> Q"
+lemma less_LeastE: "\<lbrakk>P(i); i < (\<mu> x. P(x))\<rbrakk> \<Longrightarrow> Q"
apply (rule Least_le [THEN [2] lt_trans2, THEN lt_irrefl], assumption+)
apply (simp add: lt_Ord)
done
(*Easier to apply than LeastI: conclusion has only one occurrence of P*)
lemma LeastI2:
- "[| P(i); Ord(i); !!j. P(j) ==> Q(j) |] ==> Q(\<mu> j. P(j))"
+ "\<lbrakk>P(i); Ord(i); \<And>j. P(j) \<Longrightarrow> Q(j)\<rbrakk> \<Longrightarrow> Q(\<mu> j. P(j))"
by (blast intro: LeastI )
(*If there is no such P then \<mu> is vacuously 0*)
lemma Least_0:
- "[| ~ (\<exists>i. Ord(i) & P(i)) |] ==> (\<mu> x. P(x)) = 0"
+ "\<lbrakk>\<not> (\<exists>i. Ord(i) & P(i))\<rbrakk> \<Longrightarrow> (\<mu> x. P(x)) = 0"
apply (unfold Least_def)
apply (rule the_0, blast)
done
@@ -326,12 +326,12 @@
subsection\<open>Basic Properties of Cardinals\<close>
(*Not needed for simplification, but helpful below*)
-lemma Least_cong: "(!!y. P(y) \<longleftrightarrow> Q(y)) ==> (\<mu> x. P(x)) = (\<mu> x. Q(x))"
+lemma Least_cong: "(\<And>y. P(y) \<longleftrightarrow> Q(y)) \<Longrightarrow> (\<mu> x. P(x)) = (\<mu> x. Q(x))"
by simp
-(*Need AC to get @{term"X \<lesssim> Y ==> |X| \<le> |Y|"}; see well_ord_lepoll_imp_cardinal_le
+(*Need AC to get @{term"X \<lesssim> Y \<Longrightarrow> |X| \<le> |Y|"}; see well_ord_lepoll_imp_cardinal_le
Converse also requires AC, but see well_ord_cardinal_eqE*)
-lemma cardinal_cong: "X \<approx> Y ==> |X| = |Y|"
+lemma cardinal_cong: "X \<approx> Y \<Longrightarrow> |X| = |Y|"
apply (unfold eqpoll_def cardinal_def)
apply (rule Least_cong)
apply (blast intro: comp_bij bij_converse_bij)
@@ -345,7 +345,7 @@
by (best intro: LeastI Ord_ordertype ordermap_bij bij_converse_bij bij_imp_eqpoll r)
qed
-(* @{term"Ord(A) ==> |A| \<approx> A"} *)
+(* @{term"Ord(A) \<Longrightarrow> |A| \<approx> A"} *)
lemmas Ord_cardinal_eqpoll = well_ord_Memrel [THEN well_ord_cardinal_eqpoll]
lemma Ord_cardinal_idem: "Ord(A) \<Longrightarrow> ||A|| = |A|"
@@ -362,36 +362,36 @@
qed
lemma well_ord_cardinal_eqpoll_iff:
- "[| well_ord(X,r); well_ord(Y,s) |] ==> |X| = |Y| \<longleftrightarrow> X \<approx> Y"
+ "\<lbrakk>well_ord(X,r); well_ord(Y,s)\<rbrakk> \<Longrightarrow> |X| = |Y| \<longleftrightarrow> X \<approx> Y"
by (blast intro: cardinal_cong well_ord_cardinal_eqE)
(** Observations from Kunen, page 28 **)
-lemma Ord_cardinal_le: "Ord(i) ==> |i| \<le> i"
+lemma Ord_cardinal_le: "Ord(i) \<Longrightarrow> |i| \<le> i"
apply (unfold cardinal_def)
apply (erule eqpoll_refl [THEN Least_le])
done
-lemma Card_cardinal_eq: "Card(K) ==> |K| = K"
+lemma Card_cardinal_eq: "Card(K) \<Longrightarrow> |K| = K"
apply (unfold Card_def)
apply (erule sym)
done
-(* Could replace the @{term"~(j \<approx> i)"} by @{term"~(i \<preceq> j)"}. *)
-lemma CardI: "[| Ord(i); !!j. j<i ==> ~(j \<approx> i) |] ==> Card(i)"
+(* Could replace the @{term"\<not>(j \<approx> i)"} by @{term"\<not>(i \<preceq> j)"}. *)
+lemma CardI: "\<lbrakk>Ord(i); \<And>j. j<i \<Longrightarrow> \<not>(j \<approx> i)\<rbrakk> \<Longrightarrow> Card(i)"
apply (unfold Card_def cardinal_def)
apply (subst Least_equality)
apply (blast intro: eqpoll_refl)+
done
-lemma Card_is_Ord: "Card(i) ==> Ord(i)"
+lemma Card_is_Ord: "Card(i) \<Longrightarrow> Ord(i)"
apply (unfold Card_def cardinal_def)
apply (erule ssubst)
apply (rule Ord_Least)
done
-lemma Card_cardinal_le: "Card(K) ==> K \<le> |K|"
+lemma Card_cardinal_le: "Card(K) \<Longrightarrow> K \<le> |K|"
apply (simp (no_asm_simp) add: Card_is_Ord Card_cardinal_eq)
done
@@ -401,7 +401,7 @@
done
text\<open>The cardinals are the initial ordinals.\<close>
-lemma Card_iff_initial: "Card(K) \<longleftrightarrow> Ord(K) & (\<forall>j. j<K \<longrightarrow> ~ j \<approx> K)"
+lemma Card_iff_initial: "Card(K) \<longleftrightarrow> Ord(K) & (\<forall>j. j<K \<longrightarrow> \<not> j \<approx> K)"
proof -
{ fix j
assume K: "Card(K)" "j \<approx> K"
@@ -416,7 +416,7 @@
by (blast intro: CardI Card_is_Ord)
qed
-lemma lt_Card_imp_lesspoll: "[| Card(a); i<a |] ==> i \<prec> a"
+lemma lt_Card_imp_lesspoll: "\<lbrakk>Card(a); i<a\<rbrakk> \<Longrightarrow> i \<prec> a"
apply (unfold lesspoll_def)
apply (drule Card_iff_initial [THEN iffD1])
apply (blast intro!: leI [THEN le_imp_lepoll])
@@ -427,7 +427,7 @@
apply (blast elim!: ltE)
done
-lemma Card_Un: "[| Card(K); Card(L) |] ==> Card(K \<union> L)"
+lemma Card_Un: "\<lbrakk>Card(K); Card(L)\<rbrakk> \<Longrightarrow> Card(K \<union> L)"
apply (rule Ord_linear_le [of K L])
apply (simp_all add: subset_Un_iff [THEN iffD1] Card_is_Ord le_imp_subset
subset_Un_iff2 [THEN iffD1])
@@ -491,19 +491,19 @@
qed
text\<open>Since we have \<^term>\<open>|succ(nat)| \<le> |nat|\<close>, the converse of \<open>cardinal_mono\<close> fails!\<close>
-lemma cardinal_lt_imp_lt: "[| |i| < |j|; Ord(i); Ord(j) |] ==> i < j"
+lemma cardinal_lt_imp_lt: "\<lbrakk>|i| < |j|; Ord(i); Ord(j)\<rbrakk> \<Longrightarrow> i < j"
apply (rule Ord_linear2 [of i j], assumption+)
apply (erule lt_trans2 [THEN lt_irrefl])
apply (erule cardinal_mono)
done
-lemma Card_lt_imp_lt: "[| |i| < K; Ord(i); Card(K) |] ==> i < K"
+lemma Card_lt_imp_lt: "\<lbrakk>|i| < K; Ord(i); Card(K)\<rbrakk> \<Longrightarrow> i < K"
by (simp (no_asm_simp) add: cardinal_lt_imp_lt Card_is_Ord Card_cardinal_eq)
-lemma Card_lt_iff: "[| Ord(i); Card(K) |] ==> (|i| < K) \<longleftrightarrow> (i < K)"
+lemma Card_lt_iff: "\<lbrakk>Ord(i); Card(K)\<rbrakk> \<Longrightarrow> (|i| < K) \<longleftrightarrow> (i < K)"
by (blast intro: Card_lt_imp_lt Ord_cardinal_le [THEN lt_trans1])
-lemma Card_le_iff: "[| Ord(i); Card(K) |] ==> (K \<le> |i|) \<longleftrightarrow> (K \<le> i)"
+lemma Card_le_iff: "\<lbrakk>Ord(i); Card(K)\<rbrakk> \<Longrightarrow> (K \<le> |i|) \<longleftrightarrow> (K \<le> i)"
by (simp add: Card_lt_iff Card_is_Ord Ord_cardinal not_lt_iff_le [THEN iff_sym])
(*Can use AC or finiteness to discharge first premise*)
@@ -526,21 +526,21 @@
thus ?thesis by simp
qed
-lemma lepoll_cardinal_le: "[| A \<lesssim> i; Ord(i) |] ==> |A| \<le> i"
+lemma lepoll_cardinal_le: "\<lbrakk>A \<lesssim> i; Ord(i)\<rbrakk> \<Longrightarrow> |A| \<le> i"
apply (rule le_trans)
apply (erule well_ord_Memrel [THEN well_ord_lepoll_imp_cardinal_le], assumption)
apply (erule Ord_cardinal_le)
done
-lemma lepoll_Ord_imp_eqpoll: "[| A \<lesssim> i; Ord(i) |] ==> |A| \<approx> A"
+lemma lepoll_Ord_imp_eqpoll: "\<lbrakk>A \<lesssim> i; Ord(i)\<rbrakk> \<Longrightarrow> |A| \<approx> A"
by (blast intro: lepoll_cardinal_le well_ord_Memrel well_ord_cardinal_eqpoll dest!: lepoll_well_ord)
-lemma lesspoll_imp_eqpoll: "[| A \<prec> i; Ord(i) |] ==> |A| \<approx> A"
+lemma lesspoll_imp_eqpoll: "\<lbrakk>A \<prec> i; Ord(i)\<rbrakk> \<Longrightarrow> |A| \<approx> A"
apply (unfold lesspoll_def)
apply (blast intro: lepoll_Ord_imp_eqpoll)
done
-lemma cardinal_subset_Ord: "[|A<=i; Ord(i)|] ==> |A| \<subseteq> i"
+lemma cardinal_subset_Ord: "\<lbrakk>A<=i; Ord(i)\<rbrakk> \<Longrightarrow> |A| \<subseteq> i"
apply (drule subset_imp_lepoll [THEN lepoll_cardinal_le])
apply (auto simp add: lt_def)
apply (blast intro: Ord_trans)
@@ -549,7 +549,7 @@
subsection\<open>The finite cardinals\<close>
lemma cons_lepoll_consD:
- "[| cons(u,A) \<lesssim> cons(v,B); u\<notin>A; v\<notin>B |] ==> A \<lesssim> B"
+ "\<lbrakk>cons(u,A) \<lesssim> cons(v,B); u\<notin>A; v\<notin>B\<rbrakk> \<Longrightarrow> A \<lesssim> B"
apply (unfold lepoll_def inj_def, safe)
apply (rule_tac x = "\<lambda>x\<in>A. if f`x=v then f`u else f`x" in exI)
apply (rule CollectI)
@@ -562,13 +562,13 @@
apply blast
done
-lemma cons_eqpoll_consD: "[| cons(u,A) \<approx> cons(v,B); u\<notin>A; v\<notin>B |] ==> A \<approx> B"
+lemma cons_eqpoll_consD: "\<lbrakk>cons(u,A) \<approx> cons(v,B); u\<notin>A; v\<notin>B\<rbrakk> \<Longrightarrow> A \<approx> B"
apply (simp add: eqpoll_iff)
apply (blast intro: cons_lepoll_consD)
done
(*Lemma suggested by Mike Fourman*)
-lemma succ_lepoll_succD: "succ(m) \<lesssim> succ(n) ==> m \<lesssim> n"
+lemma succ_lepoll_succD: "succ(m) \<lesssim> succ(n) \<Longrightarrow> m \<lesssim> n"
apply (unfold succ_def)
apply (erule cons_lepoll_consD)
apply (rule mem_not_refl)+
@@ -576,7 +576,7 @@
lemma nat_lepoll_imp_le:
- "m \<in> nat ==> n \<in> nat \<Longrightarrow> m \<lesssim> n \<Longrightarrow> m \<le> n"
+ "m \<in> nat \<Longrightarrow> n \<in> nat \<Longrightarrow> m \<lesssim> n \<Longrightarrow> m \<le> n"
proof (induct m arbitrary: n rule: nat_induct)
case 0 thus ?case by (blast intro!: nat_0_le)
next
@@ -591,7 +591,7 @@
qed
qed
-lemma nat_eqpoll_iff: "[| m \<in> nat; n \<in> nat |] ==> m \<approx> n \<longleftrightarrow> m = n"
+lemma nat_eqpoll_iff: "\<lbrakk>m \<in> nat; n \<in> nat\<rbrakk> \<Longrightarrow> m \<approx> n \<longleftrightarrow> m = n"
apply (rule iffI)
apply (blast intro: nat_lepoll_imp_le le_anti_sym elim!: eqpollE)
apply (simp add: eqpoll_refl)
@@ -616,11 +616,11 @@
(*Part of Kunen's Lemma 10.6*)
-lemma succ_lepoll_natE: "[| succ(n) \<lesssim> n; n \<in> nat |] ==> P"
+lemma succ_lepoll_natE: "\<lbrakk>succ(n) \<lesssim> n; n \<in> nat\<rbrakk> \<Longrightarrow> P"
by (rule nat_lepoll_imp_le [THEN lt_irrefl], auto)
lemma nat_lepoll_imp_ex_eqpoll_n:
- "[| n \<in> nat; nat \<lesssim> X |] ==> \<exists>Y. Y \<subseteq> X & n \<approx> Y"
+ "\<lbrakk>n \<in> nat; nat \<lesssim> X\<rbrakk> \<Longrightarrow> \<exists>Y. Y \<subseteq> X & n \<approx> Y"
apply (unfold lepoll_def eqpoll_def)
apply (fast del: subsetI subsetCE
intro!: subset_SIs
@@ -649,22 +649,22 @@
qed
lemma lesspoll_succ_imp_lepoll:
- "[| A \<prec> succ(m); m \<in> nat |] ==> A \<lesssim> m"
+ "\<lbrakk>A \<prec> succ(m); m \<in> nat\<rbrakk> \<Longrightarrow> A \<lesssim> m"
apply (unfold lesspoll_def lepoll_def eqpoll_def bij_def)
apply (auto dest: inj_not_surj_succ)
done
-lemma lesspoll_succ_iff: "m \<in> nat ==> A \<prec> succ(m) \<longleftrightarrow> A \<lesssim> m"
+lemma lesspoll_succ_iff: "m \<in> nat \<Longrightarrow> A \<prec> succ(m) \<longleftrightarrow> A \<lesssim> m"
by (blast intro!: lepoll_imp_lesspoll_succ lesspoll_succ_imp_lepoll)
-lemma lepoll_succ_disj: "[| A \<lesssim> succ(m); m \<in> nat |] ==> A \<lesssim> m | A \<approx> succ(m)"
+lemma lepoll_succ_disj: "\<lbrakk>A \<lesssim> succ(m); m \<in> nat\<rbrakk> \<Longrightarrow> A \<lesssim> m | A \<approx> succ(m)"
apply (rule disjCI)
apply (rule lesspoll_succ_imp_lepoll)
prefer 2 apply assumption
apply (simp (no_asm_simp) add: lesspoll_def)
done
-lemma lesspoll_cardinal_lt: "[| A \<prec> i; Ord(i) |] ==> |A| < i"
+lemma lesspoll_cardinal_lt: "\<lbrakk>A \<prec> i; Ord(i)\<rbrakk> \<Longrightarrow> |A| < i"
apply (unfold lesspoll_def, clarify)
apply (frule lepoll_cardinal_le, assumption)
apply (blast intro: well_ord_Memrel well_ord_cardinal_eqpoll [THEN eqpoll_sym]
@@ -676,7 +676,7 @@
(*This implies Kunen's Lemma 10.6*)
lemma lt_not_lepoll:
- assumes n: "n<i" "n \<in> nat" shows "~ i \<lesssim> n"
+ assumes n: "n<i" "n \<in> nat" shows "\<not> i \<lesssim> n"
proof -
{ assume i: "i \<lesssim> n"
have "succ(n) \<lesssim> i" using n
@@ -700,8 +700,8 @@
thus ?thesis by (simp add: eqpoll_refl)
next
case gt
- hence "~ i \<lesssim> n" using n by (rule lt_not_lepoll)
- hence "~ i \<approx> n" using n by (blast intro: eqpoll_imp_lepoll)
+ hence "\<not> i \<lesssim> n" using n by (rule lt_not_lepoll)
+ hence "\<not> i \<approx> n" using n by (blast intro: eqpoll_imp_lepoll)
moreover have "i \<noteq> n" using \<open>n<i\<close> by auto
ultimately show ?thesis by blast
qed
@@ -710,7 +710,7 @@
proof -
{ fix i
assume i: "i < nat" "i \<approx> nat"
- hence "~ nat \<lesssim> i"
+ hence "\<not> nat \<lesssim> i"
by (simp add: lt_def lt_not_lepoll)
hence False using i
by (simp add: eqpoll_iff)
@@ -721,12 +721,12 @@
qed
(*Allows showing that |i| is a limit cardinal*)
-lemma nat_le_cardinal: "nat \<le> i ==> nat \<le> |i|"
+lemma nat_le_cardinal: "nat \<le> i \<Longrightarrow> nat \<le> |i|"
apply (rule Card_nat [THEN Card_cardinal_eq, THEN subst])
apply (erule cardinal_mono)
done
-lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
+lemma n_lesspoll_nat: "n \<in> nat \<Longrightarrow> n \<prec> nat"
by (blast intro: Ord_nat Card_nat ltI lt_Card_imp_lesspoll)
@@ -735,7 +735,7 @@
(*Congruence law for cons under equipollence*)
lemma cons_lepoll_cong:
- "[| A \<lesssim> B; b \<notin> B |] ==> cons(a,A) \<lesssim> cons(b,B)"
+ "\<lbrakk>A \<lesssim> B; b \<notin> B\<rbrakk> \<Longrightarrow> cons(a,A) \<lesssim> cons(b,B)"
apply (unfold lepoll_def, safe)
apply (rule_tac x = "\<lambda>y\<in>cons (a,A) . if y=a then b else f`y" in exI)
apply (rule_tac d = "%z. if z \<in> B then converse (f) `z else a" in lam_injective)
@@ -745,15 +745,15 @@
done
lemma cons_eqpoll_cong:
- "[| A \<approx> B; a \<notin> A; b \<notin> B |] ==> cons(a,A) \<approx> cons(b,B)"
+ "\<lbrakk>A \<approx> B; a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> cons(a,A) \<approx> cons(b,B)"
by (simp add: eqpoll_iff cons_lepoll_cong)
lemma cons_lepoll_cons_iff:
- "[| a \<notin> A; b \<notin> B |] ==> cons(a,A) \<lesssim> cons(b,B) \<longleftrightarrow> A \<lesssim> B"
+ "\<lbrakk>a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> cons(a,A) \<lesssim> cons(b,B) \<longleftrightarrow> A \<lesssim> B"
by (blast intro: cons_lepoll_cong cons_lepoll_consD)
lemma cons_eqpoll_cons_iff:
- "[| a \<notin> A; b \<notin> B |] ==> cons(a,A) \<approx> cons(b,B) \<longleftrightarrow> A \<approx> B"
+ "\<lbrakk>a \<notin> A; b \<notin> B\<rbrakk> \<Longrightarrow> cons(a,A) \<approx> cons(b,B) \<longleftrightarrow> A \<approx> B"
by (blast intro: cons_eqpoll_cong cons_eqpoll_consD)
lemma singleton_eqpoll_1: "{a} \<approx> 1"
@@ -766,7 +766,7 @@
apply (simp (no_asm) add: nat_into_Card [THEN Card_cardinal_eq])
done
-lemma not_0_is_lepoll_1: "A \<noteq> 0 ==> 1 \<lesssim> A"
+lemma not_0_is_lepoll_1: "A \<noteq> 0 \<Longrightarrow> 1 \<lesssim> A"
apply (erule not_emptyE)
apply (rule_tac a = "cons (x, A-{x}) " in subst)
apply (rule_tac [2] a = "cons(0,0)" and P= "%y. y \<lesssim> cons (x, A-{x})" in subst)
@@ -774,26 +774,26 @@
done
(*Congruence law for succ under equipollence*)
-lemma succ_eqpoll_cong: "A \<approx> B ==> succ(A) \<approx> succ(B)"
+lemma succ_eqpoll_cong: "A \<approx> B \<Longrightarrow> succ(A) \<approx> succ(B)"
apply (unfold succ_def)
apply (simp add: cons_eqpoll_cong mem_not_refl)
done
(*Congruence law for + under equipollence*)
-lemma sum_eqpoll_cong: "[| A \<approx> C; B \<approx> D |] ==> A+B \<approx> C+D"
+lemma sum_eqpoll_cong: "\<lbrakk>A \<approx> C; B \<approx> D\<rbrakk> \<Longrightarrow> A+B \<approx> C+D"
apply (unfold eqpoll_def)
apply (blast intro!: sum_bij)
done
(*Congruence law for * under equipollence*)
lemma prod_eqpoll_cong:
- "[| A \<approx> C; B \<approx> D |] ==> A*B \<approx> C*D"
+ "\<lbrakk>A \<approx> C; B \<approx> D\<rbrakk> \<Longrightarrow> A*B \<approx> C*D"
apply (unfold eqpoll_def)
apply (blast intro!: prod_bij)
done
lemma inj_disjoint_eqpoll:
- "[| f \<in> inj(A,B); A \<inter> B = 0 |] ==> A \<union> (B - range(f)) \<approx> B"
+ "\<lbrakk>f \<in> inj(A,B); A \<inter> B = 0\<rbrakk> \<Longrightarrow> A \<union> (B - range(f)) \<approx> B"
apply (unfold eqpoll_def)
apply (rule exI)
apply (rule_tac c = "%x. if x \<in> A then f`x else x"
@@ -814,7 +814,7 @@
text\<open>If \<^term>\<open>A\<close> has at most \<^term>\<open>n+1\<close> elements and \<^term>\<open>a \<in> A\<close>
then \<^term>\<open>A-{a}\<close> has at most \<^term>\<open>n\<close>.\<close>
lemma Diff_sing_lepoll:
- "[| a \<in> A; A \<lesssim> succ(n) |] ==> A - {a} \<lesssim> n"
+ "\<lbrakk>a \<in> A; A \<lesssim> succ(n)\<rbrakk> \<Longrightarrow> A - {a} \<lesssim> n"
apply (unfold succ_def)
apply (rule cons_lepoll_consD)
apply (rule_tac [3] mem_not_refl)
@@ -834,12 +834,12 @@
by (blast intro: cons_lepoll_consD mem_irrefl)
qed
-lemma Diff_sing_eqpoll: "[| a \<in> A; A \<approx> succ(n) |] ==> A - {a} \<approx> n"
+lemma Diff_sing_eqpoll: "\<lbrakk>a \<in> A; A \<approx> succ(n)\<rbrakk> \<Longrightarrow> A - {a} \<approx> n"
by (blast intro!: eqpollI
elim!: eqpollE
intro: Diff_sing_lepoll lepoll_Diff_sing)
-lemma lepoll_1_is_sing: "[| A \<lesssim> 1; a \<in> A |] ==> A = {a}"
+lemma lepoll_1_is_sing: "\<lbrakk>A \<lesssim> 1; a \<in> A\<rbrakk> \<Longrightarrow> A = {a}"
apply (frule Diff_sing_lepoll, assumption)
apply (drule lepoll_0_is_0)
apply (blast elim: equalityE)
@@ -854,12 +854,12 @@
done
lemma well_ord_Un:
- "[| well_ord(X,R); well_ord(Y,S) |] ==> \<exists>T. well_ord(X \<union> Y, T)"
+ "\<lbrakk>well_ord(X,R); well_ord(Y,S)\<rbrakk> \<Longrightarrow> \<exists>T. well_ord(X \<union> Y, T)"
by (erule well_ord_radd [THEN Un_lepoll_sum [THEN lepoll_well_ord]],
assumption)
(*Krzysztof Grabczewski*)
-lemma disj_Un_eqpoll_sum: "A \<inter> B = 0 ==> A \<union> B \<approx> A + B"
+lemma disj_Un_eqpoll_sum: "A \<inter> B = 0 \<Longrightarrow> A \<union> B \<approx> A + B"
apply (unfold eqpoll_def)
apply (rule_tac x = "\<lambda>a\<in>A \<union> B. if a \<in> A then Inl (a) else Inr (a)" in exI)
apply (rule_tac d = "%z. case (%x. x, %x. x, z)" in lam_bijective)
@@ -869,7 +869,7 @@
subsection \<open>Finite and infinite sets\<close>
-lemma eqpoll_imp_Finite_iff: "A \<approx> B ==> Finite(A) \<longleftrightarrow> Finite(B)"
+lemma eqpoll_imp_Finite_iff: "A \<approx> B \<Longrightarrow> Finite(A) \<longleftrightarrow> Finite(B)"
apply (unfold Finite_def)
apply (blast intro: eqpoll_trans eqpoll_sym)
done
@@ -879,7 +879,7 @@
apply (blast intro!: eqpoll_refl nat_0I)
done
-lemma Finite_cons: "Finite(x) ==> Finite(cons(y,x))"
+lemma Finite_cons: "Finite(x) \<Longrightarrow> Finite(cons(y,x))"
apply (unfold Finite_def)
apply (case_tac "y \<in> x")
apply (simp add: cons_absorb)
@@ -889,7 +889,7 @@
apply (simp (no_asm_simp) add: succ_def cons_eqpoll_cong mem_not_refl)
done
-lemma Finite_succ: "Finite(x) ==> Finite(succ(x))"
+lemma Finite_succ: "Finite(x) \<Longrightarrow> Finite(succ(x))"
apply (unfold succ_def)
apply (erule Finite_cons)
done
@@ -911,7 +911,7 @@
qed
lemma lesspoll_nat_is_Finite:
- "A \<prec> nat ==> Finite(A)"
+ "A \<prec> nat \<Longrightarrow> Finite(A)"
apply (unfold Finite_def)
apply (blast dest: ltD lesspoll_cardinal_lt
lesspoll_imp_eqpoll [THEN eqpoll_sym])
@@ -936,13 +936,13 @@
lemma Finite_succ_iff [iff]: "Finite(succ(x)) \<longleftrightarrow> Finite(x)"
by (simp add: succ_def)
-lemma Finite_Int: "Finite(A) | Finite(B) ==> Finite(A \<inter> B)"
+lemma Finite_Int: "Finite(A) | Finite(B) \<Longrightarrow> Finite(A \<inter> B)"
by (blast intro: subset_Finite)
lemmas Finite_Diff = Diff_subset [THEN subset_Finite]
lemma nat_le_infinite_Ord:
- "[| Ord(i); ~ Finite(i) |] ==> nat \<le> i"
+ "\<lbrakk>Ord(i); \<not> Finite(i)\<rbrakk> \<Longrightarrow> nat \<le> i"
apply (unfold Finite_def)
apply (erule Ord_nat [THEN [2] Ord_linear2])
prefer 2 apply assumption
@@ -950,19 +950,19 @@
done
lemma Finite_imp_well_ord:
- "Finite(A) ==> \<exists>r. well_ord(A,r)"
+ "Finite(A) \<Longrightarrow> \<exists>r. well_ord(A,r)"
apply (unfold Finite_def eqpoll_def)
apply (blast intro: well_ord_rvimage bij_is_inj well_ord_Memrel nat_into_Ord)
done
-lemma succ_lepoll_imp_not_empty: "succ(x) \<lesssim> y ==> y \<noteq> 0"
+lemma succ_lepoll_imp_not_empty: "succ(x) \<lesssim> y \<Longrightarrow> y \<noteq> 0"
by (fast dest!: lepoll_0_is_0)
-lemma eqpoll_succ_imp_not_empty: "x \<approx> succ(n) ==> x \<noteq> 0"
+lemma eqpoll_succ_imp_not_empty: "x \<approx> succ(n) \<Longrightarrow> x \<noteq> 0"
by (fast elim!: eqpoll_sym [THEN eqpoll_0_is_0, THEN succ_neq_0])
lemma Finite_Fin_lemma [rule_format]:
- "n \<in> nat ==> \<forall>A. (A\<approx>n & A \<subseteq> X) \<longrightarrow> A \<in> Fin(X)"
+ "n \<in> nat \<Longrightarrow> \<forall>A. (A\<approx>n & A \<subseteq> X) \<longrightarrow> A \<in> Fin(X)"
apply (induct_tac n)
apply (rule allI)
apply (fast intro!: Fin.emptyI dest!: eqpoll_imp_lepoll [THEN lepoll_0_is_0])
@@ -978,10 +978,10 @@
apply (simp add: cons_Diff)
done
-lemma Finite_Fin: "[| Finite(A); A \<subseteq> X |] ==> A \<in> Fin(X)"
+lemma Finite_Fin: "\<lbrakk>Finite(A); A \<subseteq> X\<rbrakk> \<Longrightarrow> A \<in> Fin(X)"
by (unfold Finite_def, blast intro: Finite_Fin_lemma)
-lemma Fin_lemma [rule_format]: "n \<in> nat ==> \<forall>A. A \<approx> n \<longrightarrow> A \<in> Fin(A)"
+lemma Fin_lemma [rule_format]: "n \<in> nat \<Longrightarrow> \<forall>A. A \<approx> n \<longrightarrow> A \<in> Fin(A)"
apply (induct_tac n)
apply (simp add: eqpoll_0_iff, clarify)
apply (subgoal_tac "\<exists>u. u \<in> A")
@@ -997,18 +997,18 @@
apply (blast intro: bij_converse_bij [THEN bij_is_fun, THEN apply_type])
done
-lemma Finite_into_Fin: "Finite(A) ==> A \<in> Fin(A)"
+lemma Finite_into_Fin: "Finite(A) \<Longrightarrow> A \<in> Fin(A)"
apply (unfold Finite_def)
apply (blast intro: Fin_lemma)
done
-lemma Fin_into_Finite: "A \<in> Fin(U) ==> Finite(A)"
+lemma Fin_into_Finite: "A \<in> Fin(U) \<Longrightarrow> Finite(A)"
by (fast intro!: Finite_0 Finite_cons elim: Fin_induct)
lemma Finite_Fin_iff: "Finite(A) \<longleftrightarrow> A \<in> Fin(A)"
by (blast intro: Finite_into_Fin Fin_into_Finite)
-lemma Finite_Un: "[| Finite(A); Finite(B) |] ==> Finite(A \<union> B)"
+lemma Finite_Un: "\<lbrakk>Finite(A); Finite(B)\<rbrakk> \<Longrightarrow> Finite(A \<union> B)"
by (blast intro!: Fin_into_Finite Fin_UnI
dest!: Finite_into_Fin
intro: Un_upper1 [THEN Fin_mono, THEN subsetD]
@@ -1018,7 +1018,7 @@
by (blast intro: subset_Finite Finite_Un)
text\<open>The converse must hold too.\<close>
-lemma Finite_Union: "[| \<forall>y\<in>X. Finite(y); Finite(X) |] ==> Finite(\<Union>(X))"
+lemma Finite_Union: "\<lbrakk>\<forall>y\<in>X. Finite(y); Finite(X)\<rbrakk> \<Longrightarrow> Finite(\<Union>(X))"
apply (simp add: Finite_Fin_iff)
apply (rule Fin_UnionI)
apply (erule Fin_induct, simp)
@@ -1027,15 +1027,15 @@
(* Induction principle for Finite(A), by Sidi Ehmety *)
lemma Finite_induct [case_names 0 cons, induct set: Finite]:
-"[| Finite(A); P(0);
- !! x B. [| Finite(B); x \<notin> B; P(B) |] ==> P(cons(x, B)) |]
- ==> P(A)"
+"\<lbrakk>Finite(A); P(0);
+ \<And>x B. \<lbrakk>Finite(B); x \<notin> B; P(B)\<rbrakk> \<Longrightarrow> P(cons(x, B))\<rbrakk>
+ \<Longrightarrow> P(A)"
apply (erule Finite_into_Fin [THEN Fin_induct])
apply (blast intro: Fin_into_Finite)+
done
-(*Sidi Ehmety. The contrapositive says ~Finite(A) ==> ~Finite(A-{a}) *)
-lemma Diff_sing_Finite: "Finite(A - {a}) ==> Finite(A)"
+(*Sidi Ehmety. The contrapositive says \<not>Finite(A) \<Longrightarrow> \<not>Finite(A-{a}) *)
+lemma Diff_sing_Finite: "Finite(A - {a}) \<Longrightarrow> Finite(A)"
apply (unfold Finite_def)
apply (case_tac "a \<in> A")
apply (subgoal_tac [2] "A-{a}=A", auto)
@@ -1046,8 +1046,8 @@
done
(*Sidi Ehmety. And the contrapositive of this says
- [| ~Finite(A); Finite(B) |] ==> ~Finite(A-B) *)
-lemma Diff_Finite [rule_format]: "Finite(B) ==> Finite(A-B) \<longrightarrow> Finite(A)"
+ \<lbrakk>\<not>Finite(A); Finite(B)\<rbrakk> \<Longrightarrow> \<not>Finite(A-B) *)
+lemma Diff_Finite [rule_format]: "Finite(B) \<Longrightarrow> Finite(A-B) \<longrightarrow> Finite(A)"
apply (erule Finite_induct, auto)
apply (case_tac "x \<in> A")
apply (subgoal_tac [2] "A-cons (x, B) = A - B")
@@ -1055,12 +1055,12 @@
apply (drule Diff_sing_Finite, auto)
done
-lemma Finite_RepFun: "Finite(A) ==> Finite(RepFun(A,f))"
+lemma Finite_RepFun: "Finite(A) \<Longrightarrow> Finite(RepFun(A,f))"
by (erule Finite_induct, simp_all)
lemma Finite_RepFun_iff_lemma [rule_format]:
- "[|Finite(x); !!x y. f(x)=f(y) ==> x=y|]
- ==> \<forall>A. x = RepFun(A,f) \<longrightarrow> Finite(A)"
+ "\<lbrakk>Finite(x); \<And>x y. f(x)=f(y) \<Longrightarrow> x=y\<rbrakk>
+ \<Longrightarrow> \<forall>A. x = RepFun(A,f) \<longrightarrow> Finite(A)"
apply (erule Finite_induct)
apply clarify
apply (case_tac "A=0", simp)
@@ -1078,15 +1078,15 @@
text\<open>I don't know why, but if the premise is expressed using meta-connectives
then the simplifier cannot prove it automatically in conditional rewriting.\<close>
lemma Finite_RepFun_iff:
- "(\<forall>x y. f(x)=f(y) \<longrightarrow> x=y) ==> Finite(RepFun(A,f)) \<longleftrightarrow> Finite(A)"
+ "(\<forall>x y. f(x)=f(y) \<longrightarrow> x=y) \<Longrightarrow> Finite(RepFun(A,f)) \<longleftrightarrow> Finite(A)"
by (blast intro: Finite_RepFun Finite_RepFun_iff_lemma [of _ f])
-lemma Finite_Pow: "Finite(A) ==> Finite(Pow(A))"
+lemma Finite_Pow: "Finite(A) \<Longrightarrow> Finite(Pow(A))"
apply (erule Finite_induct)
apply (simp_all add: Pow_insert Finite_Un Finite_RepFun)
done
-lemma Finite_Pow_imp_Finite: "Finite(Pow(A)) ==> Finite(A)"
+lemma Finite_Pow_imp_Finite: "Finite(Pow(A)) \<Longrightarrow> Finite(A)"
apply (subgoal_tac "Finite({{x} . x \<in> A})")
apply (simp add: Finite_RepFun_iff )
apply (blast intro: subset_Finite)
@@ -1103,7 +1103,7 @@
(*Krzysztof Grabczewski's proof that the converse of a finite, well-ordered
set is well-ordered. Proofs simplified by lcp. *)
-lemma nat_wf_on_converse_Memrel: "n \<in> nat ==> wf[n](converse(Memrel(n)))"
+lemma nat_wf_on_converse_Memrel: "n \<in> nat \<Longrightarrow> wf[n](converse(Memrel(n)))"
proof (induct n rule: nat_induct)
case 0 thus ?case by (blast intro: wf_onI)
next
@@ -1125,15 +1125,15 @@
qed
qed
-lemma nat_well_ord_converse_Memrel: "n \<in> nat ==> well_ord(n,converse(Memrel(n)))"
+lemma nat_well_ord_converse_Memrel: "n \<in> nat \<Longrightarrow> well_ord(n,converse(Memrel(n)))"
apply (frule Ord_nat [THEN Ord_in_Ord, THEN well_ord_Memrel])
apply (simp add: well_ord_def tot_ord_converse nat_wf_on_converse_Memrel)
done
lemma well_ord_converse:
- "[|well_ord(A,r);
- well_ord(ordertype(A,r), converse(Memrel(ordertype(A, r)))) |]
- ==> well_ord(A,converse(r))"
+ "\<lbrakk>well_ord(A,r);
+ well_ord(ordertype(A,r), converse(Memrel(ordertype(A, r))))\<rbrakk>
+ \<Longrightarrow> well_ord(A,converse(r))"
apply (rule well_ord_Int_iff [THEN iffD1])
apply (frule ordermap_bij [THEN bij_is_inj, THEN well_ord_rvimage], assumption)
apply (simp add: rvimage_converse converse_Int converse_prod
@@ -1153,16 +1153,16 @@
qed
lemma Finite_well_ord_converse:
- "[| Finite(A); well_ord(A,r) |] ==> well_ord(A,converse(r))"
+ "\<lbrakk>Finite(A); well_ord(A,r)\<rbrakk> \<Longrightarrow> well_ord(A,converse(r))"
apply (unfold Finite_def)
apply (rule well_ord_converse, assumption)
apply (blast dest: ordertype_eq_n intro!: nat_well_ord_converse_Memrel)
done
-lemma nat_into_Finite: "n \<in> nat ==> Finite(n)"
+lemma nat_into_Finite: "n \<in> nat \<Longrightarrow> Finite(n)"
by (auto simp add: Finite_def intro: eqpoll_refl)
-lemma nat_not_Finite: "~ Finite(nat)"
+lemma nat_not_Finite: "\<not> Finite(nat)"
proof -
{ fix n
assume n: "n \<in> nat" "nat \<approx> n"