# HG changeset patch
# User hoelzl
# Date 1384877272 -3600
# Node ID f7fef6b00bfea01143d95158fc3c7d564f57bed6
# Parent c76dec4df4d7d51c1718bc94e91d8fe808e5009d# Parent 237d5be57277e7b83a0328d9fa0da7a3682e7ee8
merged
diff -r c76dec4df4d7 -r f7fef6b00bfe NEWS
--- a/NEWS Mon Nov 18 17:15:01 2013 +0100
+++ b/NEWS Tue Nov 19 17:07:52 2013 +0100
@@ -18,6 +18,14 @@
even_zero_(nat|int) ~> even_zero
INCOMPATIBILITY.
+* Abolished neg_numeral.
+ * Canonical representation for minus one is "- 1".
+ * Canonical representation for other negative numbers is "- (numeral _)".
+ * When devising rules set for number calculation, consider the
+ following cases: 0, 1, numeral _, - 1, - numeral _.
+ * Syntax for negative numerals is mere input syntax.
+INCOMPATBILITY.
+
* Elimination of fact duplicates:
equals_zero_I ~> minus_unique
diff_eq_0_iff_eq ~> right_minus_eq
diff -r c76dec4df4d7 -r f7fef6b00bfe src/Doc/Datatypes/Datatypes.thy
--- a/src/Doc/Datatypes/Datatypes.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/Doc/Datatypes/Datatypes.thy Tue Nov 19 17:07:52 2013 +0100
@@ -350,7 +350,7 @@
custom names. In the example below, the familiar names @{text null}, @{text hd},
@{text tl}, @{text set}, @{text map}, and @{text list_all2}, override the
default names @{text is_Nil}, @{text un_Cons1}, @{text un_Cons2},
-@{text list_set}, @{text list_map}, and @{text list_rel}:
+@{text set_list}, @{text map_list}, and @{text rel_list}:
*}
(*<*)
@@ -363,7 +363,7 @@
Cons (infixr "#" 65)
hide_type list
- hide_const Nil Cons hd tl set map list_all2 list_case list_rec
+ hide_const Nil Cons hd tl set map list_all2
context early begin
(*>*)
@@ -501,7 +501,7 @@
reference manual \cite{isabelle-isar-ref}.
The optional names preceding the type variables allow to override the default
-names of the set functions (@{text t_set1}, \ldots, @{text t_setM}).
+names of the set functions (@{text set1_t}, \ldots, @{text setM_t}).
Inside a mutually recursive specification, all defined datatypes must
mention exactly the same type variables in the same order.
@@ -626,7 +626,7 @@
\begin{itemize}
\setlength{\itemsep}{0pt}
-\item \relax{Case combinator}: @{text t_case} (rendered using the familiar
+\item \relax{Case combinator}: @{text t.case_t} (rendered using the familiar
@{text case}--@{text of} syntax)
\item \relax{Discriminators}: @{text "t.is_C\<^sub>1"}, \ldots,
@@ -638,22 +638,22 @@
\phantom{\relax{Selectors:}} @{text t.un_C\<^sub>n1}$, \ldots, @{text t.un_C\<^sub>nk\<^sub>n}.
\item \relax{Set functions} (or \relax{natural transformations}):
-@{text t_set1}, \ldots, @{text t_setm}
-
-\item \relax{Map function} (or \relax{functorial action}): @{text t_map}
-
-\item \relax{Relator}: @{text t_rel}
-
-\item \relax{Iterator}: @{text t_fold}
-
-\item \relax{Recursor}: @{text t_rec}
+@{text set1_t}, \ldots, @{text t.setm_t}
+
+\item \relax{Map function} (or \relax{functorial action}): @{text t.map_t}
+
+\item \relax{Relator}: @{text t.rel_t}
+
+\item \relax{Iterator}: @{text t.fold_t}
+
+\item \relax{Recursor}: @{text t.rec_t}
\end{itemize}
\noindent
The case combinator, discriminators, and selectors are collectively called
\emph{destructors}. The prefix ``@{text "t."}'' is an optional component of the
-name and is normally hidden.
+names and is normally hidden.
*}
@@ -810,8 +810,8 @@
\item[@{text "t."}\hthm{sel\_split\_asm}\rm:] ~ \\
@{thm list.sel_split_asm[no_vars]}
-\item[@{text "t."}\hthm{case\_if}\rm:] ~ \\
-@{thm list.case_if[no_vars]}
+\item[@{text "t."}\hthm{case\_eq\_if}\rm:] ~ \\
+@{thm list.case_eq_if[no_vars]}
\end{description}
\end{indentblock}
@@ -914,7 +914,10 @@
is recommended to use @{command datatype_new_compat} or \keyw{rep\_datatype}
to register new-style datatypes as old-style datatypes.
-\item \emph{The recursor @{text "t_rec"} has a different signature for nested
+\item \emph{The constants @{text "t_case"} and @{text "t_rec"} are now called
+@{text "case_t"} and @{text "rec_t"}.
+
+\item \emph{The recursor @{text "rec_t"} has a different signature for nested
recursive datatypes.} In the old package, nested recursion through non-functions
was internally reduced to mutual recursion. This reduction was visible in the
type of the recursor, used by \keyw{primrec}. Recursion through functions was
@@ -1150,13 +1153,13 @@
\noindent
The next example features recursion through the @{text option} type. Although
@{text option} is not a new-style datatype, it is registered as a BNF with the
-map function @{const option_map}:
+map function @{const map_option}:
*}
primrec_new (*<*)(in early) (*>*)sum_btree :: "('a\<Colon>{zero,plus}) btree \<Rightarrow> 'a" where
"sum_btree (BNode a lt rt) =
- a + the_default 0 (option_map sum_btree lt) +
- the_default 0 (option_map sum_btree rt)"
+ a + the_default 0 (map_option sum_btree lt) +
+ the_default 0 (map_option sum_btree rt)"
text {*
\noindent
@@ -1552,9 +1555,9 @@
\begin{itemize}
\setlength{\itemsep}{0pt}
-\item \relax{Coiterator}: @{text t_unfold}
-
-\item \relax{Corecursor}: @{text t_corec}
+\item \relax{Coiterator}: @{text unfold_t}
+
+\item \relax{Corecursor}: @{text corec_t}
\end{itemize}
*}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Archimedean_Field.thy
--- a/src/HOL/Archimedean_Field.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Archimedean_Field.thy Tue Nov 19 17:07:52 2013 +0100
@@ -204,8 +204,8 @@
lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
using floor_of_int [of "numeral v"] by simp
-lemma floor_neg_numeral [simp]: "floor (neg_numeral v) = neg_numeral v"
- using floor_of_int [of "neg_numeral v"] by simp
+lemma floor_neg_numeral [simp]: "floor (- numeral v) = - numeral v"
+ using floor_of_int [of "- numeral v"] by simp
lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
by (simp add: le_floor_iff)
@@ -218,7 +218,7 @@
by (simp add: le_floor_iff)
lemma neg_numeral_le_floor [simp]:
- "neg_numeral v \<le> floor x \<longleftrightarrow> neg_numeral v \<le> x"
+ "- numeral v \<le> floor x \<longleftrightarrow> - numeral v \<le> x"
by (simp add: le_floor_iff)
lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
@@ -232,7 +232,7 @@
by (simp add: less_floor_iff)
lemma neg_numeral_less_floor [simp]:
- "neg_numeral v < floor x \<longleftrightarrow> neg_numeral v + 1 \<le> x"
+ "- numeral v < floor x \<longleftrightarrow> - numeral v + 1 \<le> x"
by (simp add: less_floor_iff)
lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
@@ -246,7 +246,7 @@
by (simp add: floor_le_iff)
lemma floor_le_neg_numeral [simp]:
- "floor x \<le> neg_numeral v \<longleftrightarrow> x < neg_numeral v + 1"
+ "floor x \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
by (simp add: floor_le_iff)
lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
@@ -260,7 +260,7 @@
by (simp add: floor_less_iff)
lemma floor_less_neg_numeral [simp]:
- "floor x < neg_numeral v \<longleftrightarrow> x < neg_numeral v"
+ "floor x < - numeral v \<longleftrightarrow> x < - numeral v"
by (simp add: floor_less_iff)
text {* Addition and subtraction of integers *}
@@ -272,10 +272,6 @@
"floor (x + numeral v) = floor x + numeral v"
using floor_add_of_int [of x "numeral v"] by simp
-lemma floor_add_neg_numeral [simp]:
- "floor (x + neg_numeral v) = floor x + neg_numeral v"
- using floor_add_of_int [of x "neg_numeral v"] by simp
-
lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
using floor_add_of_int [of x 1] by simp
@@ -286,10 +282,6 @@
"floor (x - numeral v) = floor x - numeral v"
using floor_diff_of_int [of x "numeral v"] by simp
-lemma floor_diff_neg_numeral [simp]:
- "floor (x - neg_numeral v) = floor x - neg_numeral v"
- using floor_diff_of_int [of x "neg_numeral v"] by simp
-
lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
using floor_diff_of_int [of x 1] by simp
@@ -353,8 +345,8 @@
lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
using ceiling_of_int [of "numeral v"] by simp
-lemma ceiling_neg_numeral [simp]: "ceiling (neg_numeral v) = neg_numeral v"
- using ceiling_of_int [of "neg_numeral v"] by simp
+lemma ceiling_neg_numeral [simp]: "ceiling (- numeral v) = - numeral v"
+ using ceiling_of_int [of "- numeral v"] by simp
lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
by (simp add: ceiling_le_iff)
@@ -367,7 +359,7 @@
by (simp add: ceiling_le_iff)
lemma ceiling_le_neg_numeral [simp]:
- "ceiling x \<le> neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v"
+ "ceiling x \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
by (simp add: ceiling_le_iff)
lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
@@ -381,7 +373,7 @@
by (simp add: ceiling_less_iff)
lemma ceiling_less_neg_numeral [simp]:
- "ceiling x < neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v - 1"
+ "ceiling x < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
by (simp add: ceiling_less_iff)
lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
@@ -395,7 +387,7 @@
by (simp add: le_ceiling_iff)
lemma neg_numeral_le_ceiling [simp]:
- "neg_numeral v \<le> ceiling x \<longleftrightarrow> neg_numeral v - 1 < x"
+ "- numeral v \<le> ceiling x \<longleftrightarrow> - numeral v - 1 < x"
by (simp add: le_ceiling_iff)
lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
@@ -409,7 +401,7 @@
by (simp add: less_ceiling_iff)
lemma neg_numeral_less_ceiling [simp]:
- "neg_numeral v < ceiling x \<longleftrightarrow> neg_numeral v < x"
+ "- numeral v < ceiling x \<longleftrightarrow> - numeral v < x"
by (simp add: less_ceiling_iff)
text {* Addition and subtraction of integers *}
@@ -421,10 +413,6 @@
"ceiling (x + numeral v) = ceiling x + numeral v"
using ceiling_add_of_int [of x "numeral v"] by simp
-lemma ceiling_add_neg_numeral [simp]:
- "ceiling (x + neg_numeral v) = ceiling x + neg_numeral v"
- using ceiling_add_of_int [of x "neg_numeral v"] by simp
-
lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
using ceiling_add_of_int [of x 1] by simp
@@ -435,10 +423,6 @@
"ceiling (x - numeral v) = ceiling x - numeral v"
using ceiling_diff_of_int [of x "numeral v"] by simp
-lemma ceiling_diff_neg_numeral [simp]:
- "ceiling (x - neg_numeral v) = ceiling x - neg_numeral v"
- using ceiling_diff_of_int [of x "neg_numeral v"] by simp
-
lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
using ceiling_diff_of_int [of x 1] by simp
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/BNF_Comp.thy
--- a/src/HOL/BNF/BNF_Comp.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/BNF_Comp.thy Tue Nov 19 17:07:52 2013 +0100
@@ -11,6 +11,9 @@
imports Basic_BNFs
begin
+lemma wpull_id: "wpull UNIV B1 B2 id id id id"
+unfolding wpull_def by simp
+
lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
by (rule ext) simp
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/BNF_Def.thy
--- a/src/HOL/BNF/BNF_Def.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/BNF_Def.thy Tue Nov 19 17:07:52 2013 +0100
@@ -190,9 +190,6 @@
lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"
unfolding vimage2p_def by -
-lemma vimage2pD: "vimage2p f g R x y \<Longrightarrow> R (f x) (g y)"
- unfolding vimage2p_def by -
-
lemma fun_rel_iff_leq_vimage2p: "(fun_rel R S) f g = (R \<le> vimage2p f g S)"
unfolding fun_rel_def vimage2p_def by auto
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/BNF_FP_Base.thy
--- a/src/HOL/BNF/BNF_FP_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/BNF_FP_Base.thy Tue Nov 19 17:07:52 2013 +0100
@@ -13,12 +13,6 @@
imports BNF_Comp Ctr_Sugar
begin
-lemma not_TrueE: "\<not> True \<Longrightarrow> P"
-by (erule notE, rule TrueI)
-
-lemma neq_eq_eq_contradict: "\<lbrakk>t \<noteq> u; s = t; s = u\<rbrakk> \<Longrightarrow> P"
-by fast
-
lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
by auto
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/BNF_GFP.thy
--- a/src/HOL/BNF/BNF_GFP.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/BNF_GFP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -15,14 +15,22 @@
"primcorec" :: thy_decl
begin
+lemma not_TrueE: "\<not> True \<Longrightarrow> P"
+by (erule notE, rule TrueI)
+
+lemma neq_eq_eq_contradict: "\<lbrakk>t \<noteq> u; s = t; s = u\<rbrakk> \<Longrightarrow> P"
+by fast
+
lemma sum_case_expand_Inr: "f o Inl = g \<Longrightarrow> f x = sum_case g (f o Inr) x"
by (auto split: sum.splits)
lemma sum_case_expand_Inr': "f o Inl = g \<Longrightarrow> h = f o Inr \<longleftrightarrow> sum_case g h = f"
-by (metis sum_case_o_inj(1,2) surjective_sum)
+apply rule
+ apply (rule ext, force split: sum.split)
+by (rule ext, metis sum_case_o_inj(2))
lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A"
-by auto
+by fast
lemma equiv_proj:
assumes e: "equiv A R" and "z \<in> R"
@@ -37,7 +45,6 @@
(* Operators: *)
definition image2 where "image2 A f g = {(f a, g a) | a. a \<in> A}"
-
lemma Id_onD: "(a, b) \<in> Id_on A \<Longrightarrow> a = b"
unfolding Id_on_def by simp
@@ -56,9 +63,6 @@
lemma Id_on_Gr: "Id_on A = Gr A id"
unfolding Id_on_def Gr_def by auto
-lemma Id_on_UNIV_I: "x = y \<Longrightarrow> (x, y) \<in> Id_on UNIV"
-unfolding Id_on_def by auto
-
lemma image2_eqI: "\<lbrakk>b = f x; c = g x; x \<in> A\<rbrakk> \<Longrightarrow> (b, c) \<in> image2 A f g"
unfolding image2_def by auto
@@ -77,6 +81,12 @@
lemma Gr_incl: "Gr A f \<subseteq> A <*> B \<longleftrightarrow> f ` A \<subseteq> B"
unfolding Gr_def by auto
+lemma subset_Collect_iff: "B \<subseteq> A \<Longrightarrow> (B \<subseteq> {x \<in> A. P x}) = (\<forall>x \<in> B. P x)"
+by blast
+
+lemma subset_CollectI: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> Q x \<Longrightarrow> P x) \<Longrightarrow> ({x \<in> B. Q x} \<subseteq> {x \<in> A. P x})"
+by blast
+
lemma in_rel_Collect_split_eq: "in_rel (Collect (split X)) = X"
unfolding fun_eq_iff by auto
@@ -130,9 +140,6 @@
"R \<subseteq> relInvImage UNIV (relImage R f) f"
unfolding relInvImage_def relImage_def by auto
-lemma equiv_Image: "equiv A R \<Longrightarrow> (\<And>a b. (a, b) \<in> R \<Longrightarrow> a \<in> A \<and> b \<in> A \<and> R `` {a} = R `` {b})"
-unfolding equiv_def refl_on_def Image_def by (auto intro: transD symD)
-
lemma relImage_proj:
assumes "equiv A R"
shows "relImage R (proj R) \<subseteq> Id_on (A//R)"
@@ -143,7 +150,7 @@
lemma relImage_relInvImage:
assumes "R \<subseteq> f ` A <*> f ` A"
shows "relImage (relInvImage A R f) f = R"
-using assms unfolding relImage_def relInvImage_def by fastforce
+using assms unfolding relImage_def relInvImage_def by fast
lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
by simp
@@ -255,13 +262,18 @@
shows "\<exists> a. a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
using assms unfolding wpull_def by blast
-lemma pickWP:
+lemma pickWP_raw:
assumes "wpull A B1 B2 f1 f2 p1 p2" and
"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
-shows "pickWP A p1 p2 b1 b2 \<in> A"
- "p1 (pickWP A p1 p2 b1 b2) = b1"
- "p2 (pickWP A p1 p2 b1 b2) = b2"
-unfolding pickWP_def using assms someI_ex[OF pickWP_pred] by fastforce+
+shows "pickWP A p1 p2 b1 b2 \<in> A
+ \<and> p1 (pickWP A p1 p2 b1 b2) = b1
+ \<and> p2 (pickWP A p1 p2 b1 b2) = b2"
+unfolding pickWP_def using assms someI_ex[OF pickWP_pred] by fastforce
+
+lemmas pickWP =
+ pickWP_raw[THEN conjunct1]
+ pickWP_raw[THEN conjunct2, THEN conjunct1]
+ pickWP_raw[THEN conjunct2, THEN conjunct2]
lemma Inl_Field_csum: "a \<in> Field r \<Longrightarrow> Inl a \<in> Field (r +c s)"
unfolding Field_card_of csum_def by auto
@@ -293,18 +305,12 @@
lemma image2pI: "R x y \<Longrightarrow> (image2p f g R) (f x) (g y)"
unfolding image2p_def by blast
-lemma image2p_eqI: "\<lbrakk>fx = f x; gy = g y; R x y\<rbrakk> \<Longrightarrow> (image2p f g R) fx gy"
- unfolding image2p_def by blast
-
lemma image2pE: "\<lbrakk>(image2p f g R) fx gy; (\<And>x y. fx = f x \<Longrightarrow> gy = g y \<Longrightarrow> R x y \<Longrightarrow> P)\<rbrakk> \<Longrightarrow> P"
unfolding image2p_def by blast
lemma fun_rel_iff_geq_image2p: "(fun_rel R S) f g = (image2p f g R \<le> S)"
unfolding fun_rel_def image2p_def by auto
-lemma convol_image_image2p: "<f o fst, g o snd> ` Collect (split R) \<subseteq> Collect (split (image2p f g R))"
- unfolding convol_def image2p_def by fastforce
-
lemma fun_rel_image2p: "(fun_rel R (image2p f g R)) f g"
unfolding fun_rel_def image2p_def by auto
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/BNF_Util.thy
--- a/src/HOL/BNF/BNF_Util.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/BNF_Util.thy Tue Nov 19 17:07:52 2013 +0100
@@ -9,15 +9,9 @@
header {* Library for Bounded Natural Functors *}
theory BNF_Util
-imports "../Cardinals/Cardinal_Arithmetic"
+imports "../Cardinals/Cardinal_Arithmetic_FP"
begin
-lemma subset_Collect_iff: "B \<subseteq> A \<Longrightarrow> (B \<subseteq> {x \<in> A. P x}) = (\<forall>x \<in> B. P x)"
-by blast
-
-lemma subset_CollectI: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> Q x \<Longrightarrow> P x) \<Longrightarrow> ({x \<in> B. Q x} \<subseteq> {x \<in> A. P x})"
-by blast
-
definition collect where
"collect F x = (\<Union>f \<in> F. f x)"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Basic_BNFs.thy
--- a/src/HOL/BNF/Basic_BNFs.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Basic_BNFs.thy Tue Nov 19 17:07:52 2013 +0100
@@ -13,14 +13,8 @@
imports BNF_Def
begin
-lemma wpull_id: "wpull UNIV B1 B2 id id id id"
-unfolding wpull_def by simp
-
lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
-lemma ctwo_card_order: "card_order ctwo"
-using Card_order_ctwo by (unfold ctwo_def Field_card_of)
-
lemma natLeq_cinfinite: "cinfinite natLeq"
unfolding cinfinite_def Field_natLeq by (rule nat_infinite)
@@ -62,11 +56,11 @@
proof -
show "sum_map id id = id" by (rule sum_map.id)
next
- fix f1 f2 g1 g2
+ fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"
show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
by (rule sum_map.comp[symmetric])
next
- fix x f1 f2 g1 g2
+ fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2
assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
thus "sum_map f1 f2 x = sum_map g1 g2 x"
@@ -76,11 +70,11 @@
case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
qed
next
- fix f1 f2
+ fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
show "setl o sum_map f1 f2 = image f1 o setl"
by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
next
- fix f1 f2
+ fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
show "setr o sum_map f1 f2 = image f2 o setr"
by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
next
@@ -88,13 +82,13 @@
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
- fix x
+ fix x :: "'o + 'p"
show "|setl x| \<le>o natLeq"
apply (rule ordLess_imp_ordLeq)
apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
by (simp add: setl_def split: sum.split)
next
- fix x
+ fix x :: "'o + 'p"
show "|setr x| \<le>o natLeq"
apply (rule ordLess_imp_ordLeq)
apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
@@ -229,22 +223,6 @@
thus ?thesis using that by fastforce
qed
-lemma card_of_bounded_range:
- "|{f :: 'd \<Rightarrow> 'a. range f \<subseteq> B}| \<le>o |Func (UNIV :: 'd set) B|" (is "|?LHS| \<le>o |?RHS|")
-proof -
- let ?f = "\<lambda>f. %x. if f x \<in> B then f x else undefined"
- have "inj_on ?f ?LHS" unfolding inj_on_def
- proof (unfold fun_eq_iff, safe)
- fix g :: "'d \<Rightarrow> 'a" and f :: "'d \<Rightarrow> 'a" and x
- assume "range f \<subseteq> B" "range g \<subseteq> B" and eq: "\<forall>x. ?f f x = ?f g x"
- hence "f x \<in> B" "g x \<in> B" by auto
- with eq have "Some (f x) = Some (g x)" by metis
- thus "f x = g x" by simp
- qed
- moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Func_def by fastforce
- ultimately show ?thesis using card_of_ordLeq by fast
-qed
-
bnf "'a \<Rightarrow> 'b"
map: "op \<circ>"
sets: range
@@ -275,7 +253,7 @@
next
fix f :: "'d => 'a"
have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
- also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
+ also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
finally show "|range f| \<le>o natLeq +c ?U" .
next
fix A B1 B2 f1 f2 p1 p2 assume p: "wpull A B1 B2 f1 f2 p1 p2"
@@ -294,7 +272,7 @@
(Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps subset_iff image_iff
- by auto (force, metis pair_collapse)
+ by auto (force, metis (no_types) pair_collapse)
qed
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Equiv_Relations_More.thy
--- a/src/HOL/BNF/Equiv_Relations_More.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Equiv_Relations_More.thy Tue Nov 19 17:07:52 2013 +0100
@@ -59,7 +59,7 @@
lemma in_quotient_imp_in_rel:
"\<lbrakk>equiv A r; X \<in> A//r; {x,y} \<subseteq> X\<rbrakk> \<Longrightarrow> (x,y) \<in> r"
-using quotient_eq_iff by fastforce
+using quotient_eq_iff[THEN iffD1] by fastforce
lemma in_quotient_imp_closed:
"\<lbrakk>equiv A r; X \<in> A//r; x \<in> X; (x,y) \<in> r\<rbrakk> \<Longrightarrow> y \<in> X"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Examples/Koenig.thy
--- a/src/HOL/BNF/Examples/Koenig.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Examples/Koenig.thy Tue Nov 19 17:07:52 2013 +0100
@@ -12,44 +12,33 @@
imports TreeFI Stream
begin
-(* selectors for streams *)
-lemma shd_def': "shd as = fst (stream_dtor as)"
-apply (case_tac as)
-apply (auto simp add: shd_def)
-by (simp add: Stream_def stream.dtor_ctor)
-
-lemma stl_def': "stl as = snd (stream_dtor as)"
-apply (case_tac as)
-apply (auto simp add: stl_def)
-by (simp add: Stream_def stream.dtor_ctor)
-
(* infinite trees: *)
coinductive infiniteTr where
-"\<lbrakk>tr' \<in> listF_set (sub tr); infiniteTr tr'\<rbrakk> \<Longrightarrow> infiniteTr tr"
+"\<lbrakk>tr' \<in> set_listF (sub tr); infiniteTr tr'\<rbrakk> \<Longrightarrow> infiniteTr tr"
lemma infiniteTr_strong_coind[consumes 1, case_names sub]:
assumes *: "phi tr" and
-**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr' \<or> infiniteTr tr'"
+**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> set_listF (sub tr). phi tr' \<or> infiniteTr tr'"
shows "infiniteTr tr"
using assms by (elim infiniteTr.coinduct) blast
lemma infiniteTr_coind[consumes 1, case_names sub, induct pred: infiniteTr]:
assumes *: "phi tr" and
-**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr'"
+**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> set_listF (sub tr). phi tr'"
shows "infiniteTr tr"
using assms by (elim infiniteTr.coinduct) blast
lemma infiniteTr_sub[simp]:
-"infiniteTr tr \<Longrightarrow> (\<exists> tr' \<in> listF_set (sub tr). infiniteTr tr')"
+"infiniteTr tr \<Longrightarrow> (\<exists> tr' \<in> set_listF (sub tr). infiniteTr tr')"
by (erule infiniteTr.cases) blast
primcorec konigPath where
"shd (konigPath t) = lab t"
-| "stl (konigPath t) = konigPath (SOME tr. tr \<in> listF_set (sub t) \<and> infiniteTr tr)"
+| "stl (konigPath t) = konigPath (SOME tr. tr \<in> set_listF (sub t) \<and> infiniteTr tr)"
(* proper paths in trees: *)
coinductive properPath where
-"\<lbrakk>shd as = lab tr; tr' \<in> listF_set (sub tr); properPath (stl as) tr'\<rbrakk> \<Longrightarrow>
+"\<lbrakk>shd as = lab tr; tr' \<in> set_listF (sub tr); properPath (stl as) tr'\<rbrakk> \<Longrightarrow>
properPath as tr"
lemma properPath_strong_coind[consumes 1, case_names shd_lab sub]:
@@ -57,7 +46,7 @@
**: "\<And> as tr. phi as tr \<Longrightarrow> shd as = lab tr" and
***: "\<And> as tr.
phi as tr \<Longrightarrow>
- \<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'"
+ \<exists> tr' \<in> set_listF (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'"
shows "properPath as tr"
using assms by (elim properPath.coinduct) blast
@@ -66,7 +55,7 @@
**: "\<And> as tr. phi as tr \<Longrightarrow> shd as = lab tr" and
***: "\<And> as tr.
phi as tr \<Longrightarrow>
- \<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr'"
+ \<exists> tr' \<in> set_listF (sub tr). phi (stl as) tr'"
shows "properPath as tr"
using properPath_strong_coind[of phi, OF * **] *** by blast
@@ -76,7 +65,7 @@
lemma properPath_sub:
"properPath as tr \<Longrightarrow>
- \<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'"
+ \<exists> tr' \<in> set_listF (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'"
by (erule properPath.cases) blast
(* prove the following by coinduction *)
@@ -88,10 +77,10 @@
assume "infiniteTr tr \<and> as = konigPath tr" hence "properPath as tr"
proof (coinduction arbitrary: tr as rule: properPath_coind)
case (sub tr as)
- let ?t = "SOME t'. t' \<in> listF_set (sub tr) \<and> infiniteTr t'"
- from sub have "\<exists>t' \<in> listF_set (sub tr). infiniteTr t'" by simp
- then have "\<exists>t'. t' \<in> listF_set (sub tr) \<and> infiniteTr t'" by blast
- then have "?t \<in> listF_set (sub tr) \<and> infiniteTr ?t" by (rule someI_ex)
+ let ?t = "SOME t'. t' \<in> set_listF (sub tr) \<and> infiniteTr t'"
+ from sub have "\<exists>t' \<in> set_listF (sub tr). infiniteTr t'" by simp
+ then have "\<exists>t'. t' \<in> set_listF (sub tr) \<and> infiniteTr t'" by blast
+ then have "?t \<in> set_listF (sub tr) \<and> infiniteTr ?t" by (rule someI_ex)
moreover have "stl (konigPath tr) = konigPath ?t" by simp
ultimately show ?case using sub by blast
qed simp
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Examples/ListF.thy
--- a/src/HOL/BNF/Examples/ListF.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Examples/ListF.thy Tue Nov 19 17:07:52 2013 +0100
@@ -62,7 +62,7 @@
"i < lengthh xs \<Longrightarrow> nthh (mapF f xs) i = f (nthh xs i)"
by (induct rule: nthh.induct) auto
-lemma nthh_listF_set[simp]: "i < lengthh xs \<Longrightarrow> nthh xs i \<in> listF_set xs"
+lemma nthh_listF_set[simp]: "i < lengthh xs \<Longrightarrow> nthh xs i \<in> set_listF xs"
by (induct rule: nthh.induct) auto
lemma NilF_iff[iff]: "(lengthh xs = 0) = (xs = NilF)"
@@ -105,7 +105,7 @@
qed simp
lemma list_set_nthh[simp]:
- "(x \<in> listF_set xs) \<Longrightarrow> (\<exists>i < lengthh xs. nthh xs i = x)"
+ "(x \<in> set_listF xs) \<Longrightarrow> (\<exists>i < lengthh xs. nthh xs i = x)"
by (induct xs) (auto, induct rule: nthh.induct, auto)
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Examples/Process.thy
--- a/src/HOL/BNF/Examples/Process.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Examples/Process.thy Tue Nov 19 17:07:52 2013 +0100
@@ -22,7 +22,7 @@
subsection {* Basic properties *}
declare
- pre_process_rel_def[simp]
+ rel_pre_process_def[simp]
sum_rel_def[simp]
prod_rel_def[simp]
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Examples/Stream.thy
--- a/src/HOL/BNF/Examples/Stream.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Examples/Stream.thy Tue Nov 19 17:07:52 2013 +0100
@@ -18,7 +18,7 @@
code_datatype Stream
lemma stream_case_cert:
- assumes "CASE \<equiv> stream_case c"
+ assumes "CASE \<equiv> case_stream c"
shows "CASE (a ## s) \<equiv> c a s"
using assms by simp_all
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/More_BNFs.thy
--- a/src/HOL/BNF/More_BNFs.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/More_BNFs.thy Tue Nov 19 17:07:52 2013 +0100
@@ -909,18 +909,18 @@
by (auto simp add: mmap_id0 mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
intro: mmap_cong wpull_mmap)
-inductive multiset_rel' where
-Zero: "multiset_rel' R {#} {#}"
+inductive rel_multiset' where
+Zero: "rel_multiset' R {#} {#}"
|
-Plus: "\<lbrakk>R a b; multiset_rel' R M N\<rbrakk> \<Longrightarrow> multiset_rel' R (M + {#a#}) (N + {#b#})"
+Plus: "\<lbrakk>R a b; rel_multiset' R M N\<rbrakk> \<Longrightarrow> rel_multiset' R (M + {#a#}) (N + {#b#})"
-lemma multiset_map_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
+lemma map_multiset_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)
-lemma multiset_map_Zero[simp]: "mmap f {#} = {#}" by simp
+lemma map_multiset_Zero[simp]: "mmap f {#} = {#}" by simp
-lemma multiset_rel_Zero: "multiset_rel R {#} {#}"
-unfolding multiset_rel_def Grp_def by auto
+lemma rel_multiset_Zero: "rel_multiset R {#} {#}"
+unfolding rel_multiset_def Grp_def by auto
declare multiset.count[simp]
declare Abs_multiset_inverse[simp]
@@ -928,7 +928,7 @@
declare union_preserves_multiset[simp]
-lemma multiset_map_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
+lemma map_multiset_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
proof (intro multiset_eqI, transfer fixing: f)
fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
assume "M1 \<in> multiset" "M2 \<in> multiset"
@@ -941,12 +941,12 @@
by (auto simp: setsum.distrib[symmetric])
qed
-lemma multiset_map_singl[simp]: "mmap f {#a#} = {#f a#}"
+lemma map_multiset_singl[simp]: "mmap f {#a#} = {#f a#}"
by transfer auto
-lemma multiset_rel_Plus:
-assumes ab: "R a b" and MN: "multiset_rel R M N"
-shows "multiset_rel R (M + {#a#}) (N + {#b#})"
+lemma rel_multiset_Plus:
+assumes ab: "R a b" and MN: "rel_multiset R M N"
+shows "rel_multiset R (M + {#a#}) (N + {#b#})"
proof-
{fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
@@ -956,13 +956,13 @@
}
thus ?thesis
using assms
- unfolding multiset_rel_def Grp_def by force
+ unfolding rel_multiset_def Grp_def by force
qed
-lemma multiset_rel'_imp_multiset_rel:
-"multiset_rel' R M N \<Longrightarrow> multiset_rel R M N"
-apply(induct rule: multiset_rel'.induct)
-using multiset_rel_Zero multiset_rel_Plus by auto
+lemma rel_multiset'_imp_rel_multiset:
+"rel_multiset' R M N \<Longrightarrow> rel_multiset R M N"
+apply(induct rule: rel_multiset'.induct)
+using rel_multiset_Zero rel_multiset_Plus by auto
lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
proof -
@@ -973,8 +973,7 @@
using finite_Collect_mem .
ultimately have fin: "finite {b. \<exists>a. f a = b \<and> a \<in># M}" by(rule finite_subset)
have i: "inj_on A ?B" unfolding inj_on_def A_def apply clarsimp
- by (metis (lifting, mono_tags) mem_Collect_eq rel_simps(54)
- setsum_gt_0_iff setsum_infinite)
+ by (metis (lifting, full_types) mem_Collect_eq neq0_conv setsum.neutral)
have 0: "\<And> b. 0 < setsum (count M) (A b) \<longleftrightarrow> (\<exists> a \<in> A b. count M a > 0)"
apply safe
apply (metis less_not_refl setsum_gt_0_iff setsum_infinite)
@@ -995,10 +994,10 @@
then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
qed
-lemma multiset_rel_mcard:
-assumes "multiset_rel R M N"
+lemma rel_multiset_mcard:
+assumes "rel_multiset R M N"
shows "mcard M = mcard N"
-using assms unfolding multiset_rel_def Grp_def by auto
+using assms unfolding rel_multiset_def Grp_def by auto
lemma multiset_induct2[case_names empty addL addR]:
assumes empty: "P {#} {#}"
@@ -1053,67 +1052,67 @@
qed
lemma msed_rel_invL:
-assumes "multiset_rel R (M + {#a#}) N"
-shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> multiset_rel R M N1"
+assumes "rel_multiset R (M + {#a#}) N"
+shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_multiset R M N1"
proof-
obtain K where KM: "mmap fst K = M + {#a#}"
and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
- unfolding multiset_rel_def Grp_def by auto
+ unfolding rel_multiset_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
using msed_map_invL[OF KN[unfolded K]] by auto
have Rab: "R a (snd ab)" using sK a unfolding K by auto
- have "multiset_rel R M N1" using sK K1M K1N1
- unfolding K multiset_rel_def Grp_def by auto
+ have "rel_multiset R M N1" using sK K1M K1N1
+ unfolding K rel_multiset_def Grp_def by auto
thus ?thesis using N Rab by auto
qed
lemma msed_rel_invR:
-assumes "multiset_rel R M (N + {#b#})"
-shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> multiset_rel R M1 N"
+assumes "rel_multiset R M (N + {#b#})"
+shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_multiset R M1 N"
proof-
obtain K where KN: "mmap snd K = N + {#b#}"
and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
- unfolding multiset_rel_def Grp_def by auto
+ unfolding rel_multiset_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
using msed_map_invL[OF KM[unfolded K]] by auto
have Rab: "R (fst ab) b" using sK b unfolding K by auto
- have "multiset_rel R M1 N" using sK K1N K1M1
- unfolding K multiset_rel_def Grp_def by auto
+ have "rel_multiset R M1 N" using sK K1N K1M1
+ unfolding K rel_multiset_def Grp_def by auto
thus ?thesis using M Rab by auto
qed
-lemma multiset_rel_imp_multiset_rel':
-assumes "multiset_rel R M N"
-shows "multiset_rel' R M N"
+lemma rel_multiset_imp_rel_multiset':
+assumes "rel_multiset R M N"
+shows "rel_multiset' R M N"
using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
case (less M)
- have c: "mcard M = mcard N" using multiset_rel_mcard[OF less.prems] .
+ have c: "mcard M = mcard N" using rel_multiset_mcard[OF less.prems] .
show ?case
proof(cases "M = {#}")
case True hence "N = {#}" using c by simp
- thus ?thesis using True multiset_rel'.Zero by auto
+ thus ?thesis using True rel_multiset'.Zero by auto
next
case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
- obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "multiset_rel R M1 N1"
+ obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_multiset R M1 N1"
using msed_rel_invL[OF less.prems[unfolded M]] by auto
- have "multiset_rel' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
- thus ?thesis using multiset_rel'.Plus[of R a b, OF R] unfolding M N by simp
+ have "rel_multiset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
+ thus ?thesis using rel_multiset'.Plus[of R a b, OF R] unfolding M N by simp
qed
qed
-lemma multiset_rel_multiset_rel':
-"multiset_rel R M N = multiset_rel' R M N"
-using multiset_rel_imp_multiset_rel' multiset_rel'_imp_multiset_rel by auto
+lemma rel_multiset_rel_multiset':
+"rel_multiset R M N = rel_multiset' R M N"
+using rel_multiset_imp_rel_multiset' rel_multiset'_imp_rel_multiset by auto
-(* The main end product for multiset_rel: inductive characterization *)
-theorems multiset_rel_induct[case_names empty add, induct pred: multiset_rel] =
- multiset_rel'.induct[unfolded multiset_rel_multiset_rel'[symmetric]]
+(* The main end product for rel_multiset: inductive characterization *)
+theorems rel_multiset_induct[case_names empty add, induct pred: rel_multiset] =
+ rel_multiset'.induct[unfolded rel_multiset_rel_multiset'[symmetric]]
@@ -1184,5 +1183,4 @@
qed
qed
-
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Tools/bnf_def.ML
--- a/src/HOL/BNF/Tools/bnf_def.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Tools/bnf_def.ML Tue Nov 19 17:07:52 2013 +0100
@@ -678,7 +678,7 @@
val def_qualify = Binding.conceal o Binding.qualify false (Binding.name_of bnf_b);
- fun mk_suffix_binding suf = Binding.suffix_name ("_" ^ suf) bnf_b;
+ fun mk_prefix_binding pre = Binding.prefix_name (pre ^ "_") bnf_b;
fun maybe_define user_specified (b, rhs) lthy =
let
@@ -703,7 +703,7 @@
lthy |> not (pointer_eq (lthy_old, lthy)) ? Local_Theory.restore;
val map_bind_def =
- (fn () => def_qualify (if Binding.is_empty map_b then mk_suffix_binding mapN else map_b),
+ (fn () => def_qualify (if Binding.is_empty map_b then mk_prefix_binding mapN else map_b),
map_rhs);
val set_binds_defs =
let
@@ -711,10 +711,10 @@
(case try (nth set_bs) (i - 1) of
SOME b => if Binding.is_empty b then get_b else K b
| NONE => get_b) #> def_qualify;
- val bs = if live = 1 then [set_name 1 (fn () => mk_suffix_binding setN)]
- else map (fn i => set_name i (fn () => mk_suffix_binding (mk_setN i))) (1 upto live);
+ val bs = if live = 1 then [set_name 1 (fn () => mk_prefix_binding setN)]
+ else map (fn i => set_name i (fn () => mk_prefix_binding (mk_setN i))) (1 upto live);
in bs ~~ set_rhss end;
- val bd_bind_def = (fn () => def_qualify (mk_suffix_binding bdN), bd_rhs);
+ val bd_bind_def = (fn () => def_qualify (mk_prefix_binding bdN), bd_rhs);
val ((((bnf_map_term, raw_map_def),
(bnf_set_terms, raw_set_defs)),
@@ -861,7 +861,7 @@
| SOME raw_rel => prep_term no_defs_lthy raw_rel);
val rel_bind_def =
- (fn () => def_qualify (if Binding.is_empty rel_b then mk_suffix_binding relN else rel_b),
+ (fn () => def_qualify (if Binding.is_empty rel_b then mk_prefix_binding relN else rel_b),
rel_rhs);
val wit_rhss =
@@ -873,8 +873,8 @@
val nwits = length wit_rhss;
val wit_binds_defs =
let
- val bs = if nwits = 1 then [fn () => def_qualify (mk_suffix_binding witN)]
- else map (fn i => fn () => def_qualify (mk_suffix_binding (mk_witN i))) (1 upto nwits);
+ val bs = if nwits = 1 then [fn () => def_qualify (mk_prefix_binding witN)]
+ else map (fn i => fn () => def_qualify (mk_prefix_binding (mk_witN i))) (1 upto nwits);
in bs ~~ wit_rhss end;
val (((bnf_rel_term, raw_rel_def), (bnf_wit_terms, raw_wit_defs)), (lthy, lthy_old)) =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
--- a/src/HOL/BNF/Tools/bnf_fp_def_sugar.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Tools/bnf_fp_def_sugar.ML Tue Nov 19 17:07:52 2013 +0100
@@ -544,10 +544,10 @@
val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
- fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
+ fun generate_iter pre (f_Tss, _, fss, xssss) ctor_iter =
let
val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
- val b = mk_binding suf;
+ val b = mk_binding pre;
val spec =
mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
mk_iter_body ctor_iter fss xssss);
@@ -563,10 +563,10 @@
val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
- fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
+ fun generate_coiter pre ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
let
val res_T = fold_rev (curry (op --->)) pf_Tss C_to_fpT;
- val b = mk_binding suf;
+ val b = mk_binding pre;
val spec =
mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
@@ -1356,7 +1356,7 @@
lthy |> Local_Theory.notes (anonymous_notes @ notes) |> snd)
end;
- fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
+ fun mk_binding pre = qualify false fp_b_name (Binding.prefix_name (pre ^ "_") fp_b);
fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
(((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Tools/bnf_gfp.ML
--- a/src/HOL/BNF/Tools/bnf_gfp.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Tools/bnf_gfp.ML Tue Nov 19 17:07:52 2013 +0100
@@ -74,7 +74,7 @@
val mk_internal_b = Binding.name #> Binding.prefix true b_name #> Binding.conceal;
fun mk_internal_bs name =
map (fn b =>
- Binding.prefix true b_name (Binding.suffix_name ("_" ^ name) b) |> Binding.conceal) bs;
+ Binding.prefix true b_name (Binding.prefix_name (name ^ "_") b) |> Binding.conceal) bs;
val external_bs = map2 (Binding.prefix false) b_names bs
|> note_all = false ? map Binding.conceal;
@@ -1695,7 +1695,7 @@
||>> mk_Frees "s" corec_sTs
||>> mk_Frees "P" (map2 mk_pred2T Ts Ts);
- fun dtor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtorN);
+ fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
val dtor_name = Binding.name_of o dtor_bind;
val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
@@ -1747,7 +1747,7 @@
val timer = time (timer "dtor definitions & thms");
- fun unfold_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtor_unfoldN);
+ fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_");
val unfold_name = Binding.name_of o unfold_bind;
val unfold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o unfold_bind;
@@ -1868,7 +1868,7 @@
Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;
- fun ctor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctorN);
+ fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
val ctor_name = Binding.name_of o ctor_bind;
val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
@@ -1939,7 +1939,7 @@
trans OF [mor RS unique, unfold_dtor]) unfold_unique_mor_thms unfold_dtor_thms
end;
- fun corec_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtor_corecN);
+ fun corec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_corecN ^ "_");
val corec_name = Binding.name_of o corec_bind;
val corec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o corec_bind;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/BNF/Tools/bnf_lfp.ML
--- a/src/HOL/BNF/Tools/bnf_lfp.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/BNF/Tools/bnf_lfp.ML Tue Nov 19 17:07:52 2013 +0100
@@ -44,7 +44,7 @@
val mk_internal_b = Binding.name #> Binding.prefix true b_name #> Binding.conceal;
fun mk_internal_bs name =
map (fn b =>
- Binding.prefix true b_name (Binding.suffix_name ("_" ^ name) b) |> Binding.conceal) bs;
+ Binding.prefix true b_name (Binding.prefix_name (name ^ "_") b) |> Binding.conceal) bs;
val external_bs = map2 (Binding.prefix false) b_names bs
|> note_all = false ? map Binding.conceal;
@@ -1021,7 +1021,7 @@
val phis = map2 retype_free (map mk_pred1T Ts) init_phis;
val phi2s = map2 retype_free (map2 mk_pred2T Ts Ts') init_phis;
- fun ctor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctorN);
+ fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
val ctor_name = Binding.name_of o ctor_bind;
val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
@@ -1080,7 +1080,7 @@
(mk_mor UNIVs ctors active_UNIVs ss (map (mk_nthN n fold_f) ks));
val foldx = HOLogic.choice_const foldT $ fold_fun;
- fun fold_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctor_foldN);
+ fun fold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctor_foldN ^ "_");
val fold_name = Binding.name_of o fold_bind;
val fold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o fold_bind;
@@ -1170,7 +1170,7 @@
Term.list_comb (mk_map_of_bnf Ds (passiveAs @ FTs) (passiveAs @ Ts) bnf,
map HOLogic.id_const passiveAs @ ctors)) Dss bnfs;
- fun dtor_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ dtorN);
+ fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
val dtor_name = Binding.name_of o dtor_bind;
val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
@@ -1243,7 +1243,7 @@
trans OF [mor RS unique, fold_ctor]) fold_unique_mor_thms fold_ctor_thms
end;
- fun rec_bind i = nth external_bs (i - 1) |> Binding.suffix_name ("_" ^ ctor_recN);
+ fun rec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctor_recN ^ "_");
val rec_name = Binding.name_of o rec_bind;
val rec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o rec_bind;
@@ -1354,7 +1354,7 @@
val cTs = map (SOME o certifyT lthy o TFree) induct_params;
val weak_ctor_induct_thms =
- let fun insts i = (replicate (i - 1) TrueI) @ (@{thm asm_rl} :: replicate (n - i) TrueI);
+ let fun insts i = (replicate (i - 1) TrueI) @ (asm_rl :: replicate (n - i) TrueI);
in map (fn i => (ctor_induct_thm OF insts i) RS mk_conjunctN n i) ks end;
val (ctor_induct2_thm, induct2_params) =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Cardinal_Arithmetic.thy
--- a/src/HOL/Cardinals/Cardinal_Arithmetic.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Cardinal_Arithmetic.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,270 +8,17 @@
header {* Cardinal Arithmetic *}
theory Cardinal_Arithmetic
-imports Cardinal_Order_Relation_Base
+imports Cardinal_Arithmetic_FP Cardinal_Order_Relation
begin
-text {*
- The following collection of lemmas should be seen as an user interface to the HOL Theory
- of cardinals. It is not expected to be complete in any sense, since its
- development was driven by demand arising from the development of the (co)datatype package.
-*}
-
-(*library candidate*)
-lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
-by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
-
-(*should supersede a weaker lemma from the library*)
-lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
-unfolding dir_image_def Field_def Range_def Domain_def by fastforce
-
-lemma card_order_dir_image:
- assumes bij: "bij f" and co: "card_order r"
- shows "card_order (dir_image r f)"
-proof -
- from assms have "Field (dir_image r f) = UNIV"
- using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
- moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
- with co have "Card_order (dir_image r f)"
- using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
- ultimately show ?thesis by auto
-qed
-
-(*library candidate*)
-lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
-by (rule card_order_on_ordIso)
-
-(*library candidate*)
-lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
-by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
-
-(*library candidate*)
-lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
-by (simp only: ordIso_refl card_of_Card_order)
-
-(*library candidate*)
-lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
-using card_order_on_Card_order[of UNIV r] by simp
-
-(*library candidate*)
-lemma card_of_Times_Plus_distrib:
- "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
-proof -
- let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
- have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
- thus ?thesis using card_of_ordIso by blast
-qed
-
-(*library candidate*)
-lemma Func_Times_Range:
- "|Func A (B <*> C)| =o |Func A B <*> Func A C|" (is "|?LHS| =o |?RHS|")
-proof -
- let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,
- \<lambda>x. if x \<in> A then snd (fg x) else undefined)"
- let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"
- have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def
- proof safe
- fix f g assume "f \<in> Func A B" "g \<in> Func A C"
- thus "(f, g) \<in> ?F ` Func A (B \<times> C)"
- by (intro image_eqI[of _ _ "?G (f, g)"]) (auto simp: Func_def)
- qed (auto simp: Func_def fun_eq_iff, metis pair_collapse)
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-subsection {* Zero *}
-
-definition czero where
- "czero = card_of {}"
-
-lemma czero_ordIso:
- "czero =o czero"
-using card_of_empty_ordIso by (simp add: czero_def)
-
-lemma card_of_ordIso_czero_iff_empty:
- "|A| =o (czero :: 'b rel) \<longleftrightarrow> A = ({} :: 'a set)"
-unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl card_of_empty_ordIso)
-
-(* A "not czero" Cardinal predicate *)
-abbreviation Cnotzero where
- "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
-
-(*helper*)
-lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
-by (metis Card_order_iff_ordIso_card_of czero_def)
-
-lemma czeroI:
- "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
-using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
-
-lemma czeroE:
- "r =o czero \<Longrightarrow> Field r = {}"
-unfolding czero_def
-by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
-
-lemma Cnotzero_mono:
- "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
-apply (rule ccontr)
-apply auto
-apply (drule czeroE)
-apply (erule notE)
-apply (erule czeroI)
-apply (drule card_of_mono2)
-apply (simp only: card_of_empty3)
-done
-
-subsection {* (In)finite cardinals *}
-
-definition cinfinite where
- "cinfinite r = infinite (Field r)"
-
-abbreviation Cinfinite where
- "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
-
-definition cfinite where
- "cfinite r = finite (Field r)"
-
-abbreviation Cfinite where
- "Cfinite r \<equiv> cfinite r \<and> Card_order r"
-
-lemma Cfinite_ordLess_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r <o s"
- unfolding cfinite_def cinfinite_def
- by (metis card_order_on_well_order_on finite_ordLess_infinite)
-
-lemma natLeq_ordLeq_cinfinite:
- assumes inf: "Cinfinite r"
- shows "natLeq \<le>o r"
-proof -
- from inf have "natLeq \<le>o |Field r|" by (simp add: cinfinite_def infinite_iff_natLeq_ordLeq)
- also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
- finally show ?thesis .
-qed
-
-lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
-unfolding cinfinite_def by (metis czeroE finite.emptyI)
-
-lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
-by (metis cinfinite_not_czero)
-
-lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
-by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
-
-lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
-by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
-
subsection {* Binary sum *}
-definition csum (infixr "+c" 65) where
- "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
-
-lemma Field_csum: "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
- unfolding csum_def Field_card_of by auto
-
-lemma Card_order_csum:
- "Card_order (r1 +c r2)"
-unfolding csum_def by (simp add: card_of_Card_order)
-
-lemma csum_Cnotzero1:
- "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
-unfolding csum_def
-by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
-
lemma csum_Cnotzero2:
"Cnotzero r2 \<Longrightarrow> Cnotzero (r1 +c r2)"
unfolding csum_def
by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
-lemma card_order_csum:
- assumes "card_order r1" "card_order r2"
- shows "card_order (r1 +c r2)"
-proof -
- have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
- thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
-qed
-
-lemma cinfinite_csum:
- "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
-unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
-
-lemma Cinfinite_csum:
- "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
-unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
-
-lemma Cinfinite_csum_strong:
- "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
-by (metis Cinfinite_csum)
-
-lemma Cinfinite_csum1:
- "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
-unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
-
-lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
-by (simp only: csum_def ordIso_Plus_cong)
-
-lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
-by (simp only: csum_def ordIso_Plus_cong1)
-
-lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
-by (simp only: csum_def ordIso_Plus_cong2)
-
-lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
-by (simp only: csum_def ordLeq_Plus_mono)
-
-lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
-by (simp only: csum_def ordLeq_Plus_mono1)
-
-lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
-by (simp only: csum_def ordLeq_Plus_mono2)
-
-lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
-by (simp only: csum_def Card_order_Plus1)
-
-lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
-by (simp only: csum_def Card_order_Plus2)
-
-lemma csum_com: "p1 +c p2 =o p2 +c p1"
-by (simp only: csum_def card_of_Plus_commute)
-
-lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
-by (simp only: csum_def Field_card_of card_of_Plus_assoc)
-
-lemma Cfinite_csum: "\<lbrakk>Cfinite r; Cfinite s\<rbrakk> \<Longrightarrow> Cfinite (r +c s)"
- unfolding cfinite_def csum_def Field_card_of using card_of_card_order_on by simp
-
-lemma csum_csum: "(r1 +c r2) +c (r3 +c r4) =o (r1 +c r3) +c (r2 +c r4)"
-proof -
- have "(r1 +c r2) +c (r3 +c r4) =o r1 +c r2 +c (r3 +c r4)"
- by (metis csum_assoc)
- also have "r1 +c r2 +c (r3 +c r4) =o r1 +c (r2 +c r3) +c r4"
- by (metis csum_assoc csum_cong2 ordIso_symmetric)
- also have "r1 +c (r2 +c r3) +c r4 =o r1 +c (r3 +c r2) +c r4"
- by (metis csum_com csum_cong1 csum_cong2)
- also have "r1 +c (r3 +c r2) +c r4 =o r1 +c r3 +c r2 +c r4"
- by (metis csum_assoc csum_cong2 ordIso_symmetric)
- also have "r1 +c r3 +c r2 +c r4 =o (r1 +c r3) +c (r2 +c r4)"
- by (metis csum_assoc ordIso_symmetric)
- finally show ?thesis .
-qed
-
-lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
-by (simp only: csum_def Field_card_of card_of_refl)
-
-lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
-using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
-
-
-subsection {* One *}
-
-definition cone where
- "cone = card_of {()}"
-
-lemma Card_order_cone: "Card_order cone"
-unfolding cone_def by (rule card_of_Card_order)
-
-lemma Cfinite_cone: "Cfinite cone"
- unfolding cfinite_def by (simp add: Card_order_cone)
-
lemma single_cone:
"|{x}| =o cone"
proof -
@@ -280,349 +27,37 @@
thus ?thesis unfolding cone_def using card_of_ordIso by blast
qed
-lemma cone_not_czero: "\<not> (cone =o czero)"
-unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
-
lemma cone_Cnotzero: "Cnotzero cone"
by (simp add: cone_not_czero Card_order_cone)
-lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
-unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
-
-
-subsection{* Two *}
-
-definition ctwo where
- "ctwo = |UNIV :: bool set|"
-
-lemma Card_order_ctwo: "Card_order ctwo"
-unfolding ctwo_def by (rule card_of_Card_order)
-
lemma cone_ordLeq_ctwo: "cone \<le>o ctwo"
unfolding cone_def ctwo_def card_of_ordLeq[symmetric] by auto
-lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
-using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
-unfolding czero_def ctwo_def by (metis UNIV_not_empty)
-
-lemma ctwo_Cnotzero: "Cnotzero ctwo"
-by (simp add: ctwo_not_czero Card_order_ctwo)
-
-
-subsection {* Family sum *}
-
-definition Csum where
- "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
-
-(* Similar setup to the one for SIGMA from theory Big_Operators: *)
-syntax "_Csum" ::
- "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
- ("(3CSUM _:_. _)" [0, 51, 10] 10)
-
-translations
- "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
-
-lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
-by (auto simp: Csum_def Field_card_of)
-
-(* NB: Always, under the cardinal operator,
-operations on sets are reduced automatically to operations on cardinals.
-This should make cardinal reasoning more direct and natural. *)
-
subsection {* Product *}
-definition cprod (infixr "*c" 80) where
- "r1 *c r2 = |Field r1 <*> Field r2|"
-
lemma Times_cprod: "|A \<times> B| =o |A| *c |B|"
by (simp only: cprod_def Field_card_of card_of_refl)
-lemma card_order_cprod:
- assumes "card_order r1" "card_order r2"
- shows "card_order (r1 *c r2)"
-proof -
- have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
- thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
-qed
-
-lemma Card_order_cprod: "Card_order (r1 *c r2)"
-by (simp only: cprod_def Field_card_of card_of_card_order_on)
-
lemma cprod_cong2: "p2 =o r2 \<Longrightarrow> q *c p2 =o q *c r2"
by (simp only: cprod_def ordIso_Times_cong2)
-lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
-by (simp only: cprod_def ordLeq_Times_mono1)
-
-lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
-by (simp only: cprod_def ordLeq_Times_mono2)
-
lemma ordLeq_cprod1: "\<lbrakk>Card_order p1; Cnotzero p2\<rbrakk> \<Longrightarrow> p1 \<le>o p1 *c p2"
unfolding cprod_def by (metis Card_order_Times1 czeroI)
-lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
-unfolding cprod_def by (metis Card_order_Times2 czeroI)
-
-lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
-by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
-
-lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
-by (metis cinfinite_mono ordLeq_cprod2)
-
-lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
-by (blast intro: cinfinite_cprod2 Card_order_cprod)
-
-lemma cprod_com: "p1 *c p2 =o p2 *c p1"
-by (simp only: cprod_def card_of_Times_commute)
-
-lemma card_of_Csum_Times:
- "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
-by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
-
-lemma card_of_Csum_Times':
- assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
- shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
-proof -
- from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
- with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
- hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
- also from * have "|I| *c |Field r| \<le>o |I| *c r"
- by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
- finally show ?thesis .
-qed
-
-lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
-unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
-
-lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
-unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
-
-lemma csum_absorb1':
- assumes card: "Card_order r2"
- and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
- shows "r2 +c r1 =o r2"
-by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
-
-lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
-by (rule csum_absorb1') auto
-
-lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
-unfolding cinfinite_def cprod_def
-by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
-
lemma cprod_infinite: "Cinfinite r \<Longrightarrow> r *c r =o r"
using cprod_infinite1' Cinfinite_Cnotzero ordLeq_refl by blast
subsection {* Exponentiation *}
-definition cexp (infixr "^c" 90) where
- "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
-
-lemma card_order_cexp:
- assumes "card_order r1" "card_order r2"
- shows "card_order (r1 ^c r2)"
-proof -
- have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
- thus ?thesis unfolding cexp_def Func_def by (simp add: card_of_card_order_on)
-qed
-
-lemma Card_order_cexp: "Card_order (r1 ^c r2)"
-unfolding cexp_def by (rule card_of_Card_order)
-
-lemma cexp_mono':
- assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
- and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
- shows "p1 ^c p2 \<le>o r1 ^c r2"
-proof(cases "Field p1 = {}")
- case True
- hence "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
- unfolding cone_def Field_card_of
- by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
- (metis Func_is_emp card_of_empty ex_in_conv)
- hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
- hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
- thus ?thesis
- proof (cases "Field p2 = {}")
- case True
- with n have "Field r2 = {}" .
- hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def by (auto intro: card_of_ordLeqI)
- thus ?thesis using `p1 ^c p2 \<le>o cone` ordLeq_transitive by auto
- next
- case False with True have "|Field (p1 ^c p2)| =o czero"
- unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
- thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
- by (simp add: card_of_empty)
- qed
-next
- case False
- have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
- using 1 2 by (auto simp: card_of_mono2)
- obtain f1 where f1: "f1 ` Field r1 = Field p1"
- using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
- obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
- using 2 unfolding card_of_ordLeq[symmetric] by blast
- have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
- unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
- have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
- using False by simp
- show ?thesis
- using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
-qed
-
-lemma cexp_mono:
- assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
- and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
- shows "p1 ^c p2 \<le>o r1 ^c r2"
- by (metis (full_types) "1" "2" card cexp_mono' czeroE czeroI n)
-
-lemma cexp_mono1:
- assumes 1: "p1 \<le>o r1" and q: "Card_order q"
- shows "p1 ^c q \<le>o r1 ^c q"
-using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
-
-lemma cexp_mono2':
- assumes 2: "p2 \<le>o r2" and q: "Card_order q"
- and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
- shows "q ^c p2 \<le>o q ^c r2"
-using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
-
-lemma cexp_mono2:
- assumes 2: "p2 \<le>o r2" and q: "Card_order q"
- and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
- shows "q ^c p2 \<le>o q ^c r2"
-using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
-
-lemma cexp_mono2_Cnotzero:
- assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
- shows "q ^c p2 \<le>o q ^c r2"
-by (metis assms cexp_mono2' czeroI)
-
-lemma cexp_cong:
- assumes 1: "p1 =o r1" and 2: "p2 =o r2"
- and Cr: "Card_order r2"
- and Cp: "Card_order p2"
- shows "p1 ^c p2 =o r1 ^c r2"
-proof -
- obtain f where "bij_betw f (Field p2) (Field r2)"
- using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
- hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
- have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
- and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
- using 0 Cr Cp czeroE czeroI by auto
- show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
- using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by blast
-qed
-
-lemma cexp_cong1:
- assumes 1: "p1 =o r1" and q: "Card_order q"
- shows "p1 ^c q =o r1 ^c q"
-by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
-
-lemma cexp_cong2:
- assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
- shows "q ^c p2 =o q ^c r2"
-by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
-
lemma cexp_czero: "r ^c czero =o cone"
unfolding cexp_def czero_def Field_card_of Func_empty by (rule single_cone)
-lemma cexp_cone:
- assumes "Card_order r"
- shows "r ^c cone =o r"
-proof -
- have "r ^c cone =o |Field r|"
- unfolding cexp_def cone_def Field_card_of Func_empty
- card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
- by (rule exI[of _ "\<lambda>f. f ()"]) auto
- also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
- finally show ?thesis .
-qed
-
-lemma cexp_cprod:
- assumes r1: "Card_order r1"
- shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
-proof -
- have "?L =o r1 ^c (r3 *c r2)"
- unfolding cprod_def cexp_def Field_card_of
- using card_of_Func_Times by(rule ordIso_symmetric)
- also have "r1 ^c (r3 *c r2) =o ?R"
- apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
- finally show ?thesis .
-qed
-
-lemma cexp_cprod_ordLeq:
- assumes r1: "Card_order r1" and r2: "Cinfinite r2"
- and r3: "Cnotzero r3" "r3 \<le>o r2"
- shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
-proof-
- have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
- also have "r1 ^c (r2 *c r3) =o ?R"
- apply(rule cexp_cong2)
- apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
- finally show ?thesis .
-qed
-
-lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
-by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
-
lemma Pow_cexp_ctwo:
"|Pow A| =o ctwo ^c |A|"
unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
-lemma ordLess_ctwo_cexp:
- assumes "Card_order r"
- shows "r <o ctwo ^c r"
-proof -
- have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
- also have "|Pow (Field r)| =o ctwo ^c r"
- unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
- finally show ?thesis .
-qed
-
-lemma ordLeq_cexp1:
- assumes "Cnotzero r" "Card_order q"
- shows "q \<le>o q ^c r"
-proof (cases "q =o (czero :: 'a rel)")
- case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
-next
- case False
- thus ?thesis
- apply -
- apply (rule ordIso_ordLeq_trans)
- apply (rule ordIso_symmetric)
- apply (rule cexp_cone)
- apply (rule assms(2))
- apply (rule cexp_mono2)
- apply (rule cone_ordLeq_Cnotzero)
- apply (rule assms(1))
- apply (rule assms(2))
- apply (rule notE)
- apply (rule cone_not_czero)
- apply assumption
- apply (rule Card_order_cone)
- done
-qed
-
-lemma ordLeq_cexp2:
- assumes "ctwo \<le>o q" "Card_order r"
- shows "r \<le>o q ^c r"
-proof (cases "r =o (czero :: 'a rel)")
- case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
-next
- case False thus ?thesis
- apply -
- apply (rule ordLess_imp_ordLeq)
- apply (rule ordLess_ordLeq_trans)
- apply (rule ordLess_ctwo_cexp)
- apply (rule assms(2))
- apply (rule cexp_mono1)
- apply (rule assms(1))
- apply (rule assms(2))
- done
-qed
-
lemma Cnotzero_cexp:
assumes "Cnotzero q" "Card_order r"
shows "Cnotzero (q ^c r)"
@@ -664,41 +99,7 @@
lemma Cinfinite_ctwo_cexp:
"Cinfinite r \<Longrightarrow> Cinfinite (ctwo ^c r)"
unfolding ctwo_def cexp_def cinfinite_def Field_card_of
-by (rule conjI, rule infinite_Func, auto, rule card_of_card_order_on)
-
-lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
-by (metis assms cinfinite_mono ordLeq_cexp2)
-
-lemma Cinfinite_cexp:
- "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
-by (simp add: cinfinite_cexp Card_order_cexp)
-
-lemma ctwo_ordLess_natLeq:
- "ctwo <o natLeq"
-unfolding ctwo_def using finite_iff_ordLess_natLeq finite_UNIV by fast
-
-lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
-by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
-
-lemma ctwo_ordLeq_Cinfinite:
- assumes "Cinfinite r"
- shows "ctwo \<le>o r"
-by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
-
-lemma Cinfinite_ordLess_cexp:
- assumes r: "Cinfinite r"
- shows "r <o r ^c r"
-proof -
- have "r <o ctwo ^c r" using r by (simp only: ordLess_ctwo_cexp)
- also have "ctwo ^c r \<le>o r ^c r"
- by (rule cexp_mono1[OF ctwo_ordLeq_Cinfinite]) (auto simp: r ctwo_not_czero Card_order_ctwo)
- finally show ?thesis .
-qed
-
-lemma infinite_ordLeq_cexp:
- assumes "Cinfinite r"
- shows "r \<le>o r ^c r"
-by (rule ordLess_imp_ordLeq[OF Cinfinite_ordLess_cexp[OF assms]])
+by (rule conjI, rule infinite_Func, auto)
lemma cone_ordLeq_iff_Field:
assumes "cone \<le>o r"
@@ -731,22 +132,6 @@
case False thus ?thesis using assms cexp_mono2' czeroI by metis
qed
-lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
-by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
-
-lemma UNION_Cinfinite_bound: "\<lbrakk>|I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |\<Union>i \<in> I. A i| \<le>o r"
-by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
-
-lemma csum_cinfinite_bound:
- assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
- shows "p +c q \<le>o r"
-proof -
- from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
- unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
- with assms show ?thesis unfolding cinfinite_def csum_def
- by (blast intro: card_of_Plus_ordLeq_infinite_Field)
-qed
-
lemma csum_cexp: "\<lbrakk>Cinfinite r1; Cinfinite r2; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow>
q ^c r1 +c q ^c r2 \<le>o q ^c (r1 +c r2)"
apply (rule csum_cinfinite_bound)
@@ -782,131 +167,30 @@
apply blast+
by (metis Cinfinite_cexp)
-lemma cprod_cinfinite_bound:
- assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
- shows "p *c q \<le>o r"
-proof -
- from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
- unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
- with assms show ?thesis unfolding cinfinite_def cprod_def
- by (blast intro: card_of_Times_ordLeq_infinite_Field)
-qed
-
-lemma cprod_csum_cexp:
- "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
-unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
-proof -
- let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
- have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
- by (auto simp: inj_on_def fun_eq_iff split: bool.split)
- moreover
- have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
- by (auto simp: Func_def)
- ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
-qed
-
-lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
-by (intro cprod_cinfinite_bound)
- (auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
-
-lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
- unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
-
-lemma cprod_cexp_csum_cexp_Cinfinite:
- assumes t: "Cinfinite t"
- shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
-proof -
- have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
- by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
- also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
- by (rule cexp_cprod[OF Card_order_csum])
- also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
- by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
- also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
- by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
- also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
- by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
- finally show ?thesis .
-qed
-
-lemma Cfinite_cexp_Cinfinite:
- assumes s: "Cfinite s" and t: "Cinfinite t"
- shows "s ^c t \<le>o ctwo ^c t"
-proof (cases "s \<le>o ctwo")
- case True thus ?thesis using t by (blast intro: cexp_mono1)
-next
- case False
- hence "ctwo \<le>o s" by (metis card_order_on_well_order_on ctwo_Cnotzero ordLeq_total s)
- hence "Cnotzero s" by (metis Cnotzero_mono ctwo_Cnotzero s)
- hence st: "Cnotzero (s *c t)" by (metis Cinfinite_cprod2 cinfinite_not_czero t)
- have "s ^c t \<le>o (ctwo ^c s) ^c t"
- using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
- also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
- by (blast intro: Card_order_ctwo cexp_cprod)
- also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
- using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
- finally show ?thesis .
-qed
-
-lemma csum_Cfinite_cexp_Cinfinite:
- assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
- shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
-proof (cases "Cinfinite r")
- case True
- hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
- hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
- also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
- finally show ?thesis .
-next
- case False
- with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
- hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
- hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
- also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
- by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
- finally show ?thesis .
-qed
-
lemma card_of_Sigma_ordLeq_Cinfinite:
"\<lbrakk>Cinfinite r; |I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r\<rbrakk> \<Longrightarrow> |SIGMA i : I. A i| \<le>o r"
unfolding cinfinite_def by (blast intro: card_of_Sigma_ordLeq_infinite_Field)
-(* cardSuc *)
-
-lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
-by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
-
-lemma cardSuc_UNION_Cinfinite:
- assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
- shows "EX i : Field (cardSuc r). B \<le> As i"
-using cardSuc_UNION assms unfolding cinfinite_def by blast
-
subsection {* Powerset *}
-definition cpow where "cpow r = |Pow (Field r)|"
-
-lemma card_order_cpow: "card_order r \<Longrightarrow> card_order (cpow r)"
-by (simp only: cpow_def Field_card_order Pow_UNIV card_of_card_order_on)
-
-lemma cpow_greater_eq: "Card_order r \<Longrightarrow> r \<le>o cpow r"
-by (rule ordLess_imp_ordLeq) (simp only: cpow_def Card_order_Pow)
-
lemma Card_order_cpow: "Card_order (cpow r)"
unfolding cpow_def by (rule card_of_Card_order)
-lemma Cinfinite_cpow: "Cinfinite r \<Longrightarrow> Cinfinite (cpow r)"
-unfolding cpow_def cinfinite_def by (metis Field_card_of card_of_Card_order infinite_Pow)
-
lemma cardSuc_ordLeq_cpow: "Card_order r \<Longrightarrow> cardSuc r \<le>o cpow r"
unfolding cpow_def by (metis Card_order_Pow cardSuc_ordLess_ordLeq card_of_Card_order)
lemma cpow_cexp_ctwo: "cpow r =o ctwo ^c r"
unfolding cpow_def ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+
subsection {* Lists *}
-definition clists where "clists r = |lists (Field r)|"
+text {*
+ The following collection of lemmas should be seen as an user interface to the HOL theory
+ of cardinals. It is not expected to be complete in any sense, since its
+ development was driven by demand arising from the development of the (co)datatype package.
+*}
lemma clists_Cinfinite: "Cinfinite r \<Longrightarrow> clists r =o r"
unfolding cinfinite_def clists_def by (blast intro: Card_order_lists_infinite)
@@ -915,6 +199,6 @@
unfolding clists_def by (rule card_of_Card_order)
lemma Cnotzero_clists: "Cnotzero (clists r)"
-by (simp add: clists_def card_of_ordIso_czero_iff_empty lists_not_empty) (rule card_of_Card_order)
+by (simp add: clists_def card_of_ordIso_czero_iff_empty lists_not_empty)
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Cardinal_Arithmetic_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Cardinal_Arithmetic_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,747 @@
+(* Title: HOL/Cardinals/Cardinal_Arithmetic_FP.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Cardinal arithmetic (FP).
+*)
+
+header {* Cardinal Arithmetic (FP) *}
+
+theory Cardinal_Arithmetic_FP
+imports Cardinal_Order_Relation_FP
+begin
+
+(*library candidate*)
+lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
+
+(*should supersede a weaker lemma from the library*)
+lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
+unfolding dir_image_def Field_def Range_def Domain_def by fast
+
+lemma card_order_dir_image:
+ assumes bij: "bij f" and co: "card_order r"
+ shows "card_order (dir_image r f)"
+proof -
+ from assms have "Field (dir_image r f) = UNIV"
+ using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
+ moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
+ with co have "Card_order (dir_image r f)"
+ using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
+ ultimately show ?thesis by auto
+qed
+
+(*library candidate*)
+lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
+by (rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
+by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
+by (simp only: ordIso_refl card_of_Card_order)
+
+(*library candidate*)
+lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
+using card_order_on_Card_order[of UNIV r] by simp
+
+(*library candidate*)
+lemma card_of_Times_Plus_distrib:
+ "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
+proof -
+ let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
+ have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+(*library candidate*)
+lemma Func_Times_Range:
+ "|Func A (B <*> C)| =o |Func A B <*> Func A C|" (is "|?LHS| =o |?RHS|")
+proof -
+ let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,
+ \<lambda>x. if x \<in> A then snd (fg x) else undefined)"
+ let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"
+ have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def
+ apply safe
+ apply (simp add: Func_def fun_eq_iff)
+ apply (metis (no_types) pair_collapse)
+ apply (auto simp: Func_def fun_eq_iff)[2]
+ proof -
+ fix f g assume "f \<in> Func A B" "g \<in> Func A C"
+ thus "(f, g) \<in> ?F ` Func A (B \<times> C)"
+ by (intro image_eqI[of _ _ "?G (f, g)"]) (auto simp: Func_def)
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+subsection {* Zero *}
+
+definition czero where
+ "czero = card_of {}"
+
+lemma czero_ordIso:
+ "czero =o czero"
+using card_of_empty_ordIso by (simp add: czero_def)
+
+lemma card_of_ordIso_czero_iff_empty:
+ "|A| =o (czero :: 'b rel) \<longleftrightarrow> A = ({} :: 'a set)"
+unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl card_of_empty_ordIso)
+
+(* A "not czero" Cardinal predicate *)
+abbreviation Cnotzero where
+ "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
+
+(*helper*)
+lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
+by (metis Card_order_iff_ordIso_card_of czero_def)
+
+lemma czeroI:
+ "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
+using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
+
+lemma czeroE:
+ "r =o czero \<Longrightarrow> Field r = {}"
+unfolding czero_def
+by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
+
+lemma Cnotzero_mono:
+ "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
+apply (rule ccontr)
+apply auto
+apply (drule czeroE)
+apply (erule notE)
+apply (erule czeroI)
+apply (drule card_of_mono2)
+apply (simp only: card_of_empty3)
+done
+
+subsection {* (In)finite cardinals *}
+
+definition cinfinite where
+ "cinfinite r = infinite (Field r)"
+
+abbreviation Cinfinite where
+ "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
+
+definition cfinite where
+ "cfinite r = finite (Field r)"
+
+abbreviation Cfinite where
+ "Cfinite r \<equiv> cfinite r \<and> Card_order r"
+
+lemma Cfinite_ordLess_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r <o s"
+ unfolding cfinite_def cinfinite_def
+ by (metis card_order_on_well_order_on finite_ordLess_infinite)
+
+lemma natLeq_ordLeq_cinfinite:
+ assumes inf: "Cinfinite r"
+ shows "natLeq \<le>o r"
+proof -
+ from inf have "natLeq \<le>o |Field r|" by (simp add: cinfinite_def infinite_iff_natLeq_ordLeq)
+ also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
+ finally show ?thesis .
+qed
+
+lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
+unfolding cinfinite_def by (metis czeroE finite.emptyI)
+
+lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
+by (metis cinfinite_not_czero)
+
+lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
+by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
+
+lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
+by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
+
+
+subsection {* Binary sum *}
+
+definition csum (infixr "+c" 65) where
+ "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
+
+lemma Field_csum: "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
+ unfolding csum_def Field_card_of by auto
+
+lemma Card_order_csum:
+ "Card_order (r1 +c r2)"
+unfolding csum_def by (simp add: card_of_Card_order)
+
+lemma csum_Cnotzero1:
+ "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
+unfolding csum_def
+by (metis Cnotzero_imp_not_empty Plus_eq_empty_conv card_of_Card_order card_of_ordIso_czero_iff_empty)
+
+lemma card_order_csum:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 +c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
+qed
+
+lemma cinfinite_csum:
+ "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
+
+lemma Cinfinite_csum1:
+ "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma Cinfinite_csum:
+ "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma Cinfinite_csum_strong:
+ "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
+by (metis Cinfinite_csum)
+
+lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
+by (simp only: csum_def ordIso_Plus_cong)
+
+lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
+by (simp only: csum_def ordIso_Plus_cong1)
+
+lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
+by (simp only: csum_def ordIso_Plus_cong2)
+
+lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
+by (simp only: csum_def ordLeq_Plus_mono)
+
+lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
+by (simp only: csum_def ordLeq_Plus_mono1)
+
+lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
+by (simp only: csum_def ordLeq_Plus_mono2)
+
+lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus1)
+
+lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus2)
+
+lemma csum_com: "p1 +c p2 =o p2 +c p1"
+by (simp only: csum_def card_of_Plus_commute)
+
+lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
+by (simp only: csum_def Field_card_of card_of_Plus_assoc)
+
+lemma Cfinite_csum: "\<lbrakk>Cfinite r; Cfinite s\<rbrakk> \<Longrightarrow> Cfinite (r +c s)"
+ unfolding cfinite_def csum_def Field_card_of using card_of_card_order_on by simp
+
+lemma csum_csum: "(r1 +c r2) +c (r3 +c r4) =o (r1 +c r3) +c (r2 +c r4)"
+proof -
+ have "(r1 +c r2) +c (r3 +c r4) =o r1 +c r2 +c (r3 +c r4)"
+ by (metis csum_assoc)
+ also have "r1 +c r2 +c (r3 +c r4) =o r1 +c (r2 +c r3) +c r4"
+ by (metis csum_assoc csum_cong2 ordIso_symmetric)
+ also have "r1 +c (r2 +c r3) +c r4 =o r1 +c (r3 +c r2) +c r4"
+ by (metis csum_com csum_cong1 csum_cong2)
+ also have "r1 +c (r3 +c r2) +c r4 =o r1 +c r3 +c r2 +c r4"
+ by (metis csum_assoc csum_cong2 ordIso_symmetric)
+ also have "r1 +c r3 +c r2 +c r4 =o (r1 +c r3) +c (r2 +c r4)"
+ by (metis csum_assoc ordIso_symmetric)
+ finally show ?thesis .
+qed
+
+lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
+by (simp only: csum_def Field_card_of card_of_refl)
+
+lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
+using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
+
+
+subsection {* One *}
+
+definition cone where
+ "cone = card_of {()}"
+
+lemma Card_order_cone: "Card_order cone"
+unfolding cone_def by (rule card_of_Card_order)
+
+lemma Cfinite_cone: "Cfinite cone"
+ unfolding cfinite_def by (simp add: Card_order_cone)
+
+lemma cone_not_czero: "\<not> (cone =o czero)"
+unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
+
+lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
+unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
+
+
+subsection{* Two *}
+
+definition ctwo where
+ "ctwo = |UNIV :: bool set|"
+
+lemma Card_order_ctwo: "Card_order ctwo"
+unfolding ctwo_def by (rule card_of_Card_order)
+
+lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
+using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
+unfolding czero_def ctwo_def by (metis UNIV_not_empty)
+
+lemma ctwo_Cnotzero: "Cnotzero ctwo"
+by (simp add: ctwo_not_czero Card_order_ctwo)
+
+
+subsection {* Family sum *}
+
+definition Csum where
+ "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
+
+(* Similar setup to the one for SIGMA from theory Big_Operators: *)
+syntax "_Csum" ::
+ "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
+ ("(3CSUM _:_. _)" [0, 51, 10] 10)
+
+translations
+ "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
+
+lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
+by (auto simp: Csum_def Field_card_of)
+
+(* NB: Always, under the cardinal operator,
+operations on sets are reduced automatically to operations on cardinals.
+This should make cardinal reasoning more direct and natural. *)
+
+
+subsection {* Product *}
+
+definition cprod (infixr "*c" 80) where
+ "r1 *c r2 = |Field r1 <*> Field r2|"
+
+lemma card_order_cprod:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 *c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
+qed
+
+lemma Card_order_cprod: "Card_order (r1 *c r2)"
+by (simp only: cprod_def Field_card_of card_of_card_order_on)
+
+lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
+by (simp only: cprod_def ordLeq_Times_mono1)
+
+lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
+by (simp only: cprod_def ordLeq_Times_mono2)
+
+lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
+unfolding cprod_def by (metis Card_order_Times2 czeroI)
+
+lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
+
+lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (metis cinfinite_mono ordLeq_cprod2)
+
+lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
+by (blast intro: cinfinite_cprod2 Card_order_cprod)
+
+lemma cprod_com: "p1 *c p2 =o p2 *c p1"
+by (simp only: cprod_def card_of_Times_commute)
+
+lemma card_of_Csum_Times:
+ "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
+by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
+
+lemma card_of_Csum_Times':
+ assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
+ shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
+proof -
+ from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
+ with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
+ hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
+ also from * have "|I| *c |Field r| \<le>o |I| *c r"
+ by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
+ finally show ?thesis .
+qed
+
+lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
+unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
+
+lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
+unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
+
+lemma csum_absorb1':
+ assumes card: "Card_order r2"
+ and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
+ shows "r2 +c r1 =o r2"
+by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
+
+lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
+by (rule csum_absorb1') auto
+
+
+subsection {* Exponentiation *}
+
+definition cexp (infixr "^c" 90) where
+ "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
+
+lemma Card_order_cexp: "Card_order (r1 ^c r2)"
+unfolding cexp_def by (rule card_of_Card_order)
+
+lemma cexp_mono':
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+proof(cases "Field p1 = {}")
+ case True
+ hence "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
+ unfolding cone_def Field_card_of
+ by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
+ (metis Func_is_emp card_of_empty ex_in_conv)
+ hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
+ hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
+ thus ?thesis
+ proof (cases "Field p2 = {}")
+ case True
+ with n have "Field r2 = {}" .
+ hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def by (auto intro: card_of_ordLeqI)
+ thus ?thesis using `p1 ^c p2 \<le>o cone` ordLeq_transitive by auto
+ next
+ case False with True have "|Field (p1 ^c p2)| =o czero"
+ unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
+ thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
+ by (simp add: card_of_empty)
+ qed
+next
+ case False
+ have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
+ using 1 2 by (auto simp: card_of_mono2)
+ obtain f1 where f1: "f1 ` Field r1 = Field p1"
+ using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
+ obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
+ using 2 unfolding card_of_ordLeq[symmetric] by blast
+ have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
+ unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
+ have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
+ using False by simp
+ show ?thesis
+ using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
+qed
+
+lemma cexp_mono:
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+ by (metis (full_types) "1" "2" card cexp_mono' czeroE czeroI n)
+
+lemma cexp_mono1:
+ assumes 1: "p1 \<le>o r1" and q: "Card_order q"
+ shows "p1 ^c q \<le>o r1 ^c q"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
+
+lemma cexp_mono2':
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
+
+lemma cexp_mono2:
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
+
+lemma cexp_mono2_Cnotzero:
+ assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+by (metis assms cexp_mono2' czeroI)
+
+lemma cexp_cong:
+ assumes 1: "p1 =o r1" and 2: "p2 =o r2"
+ and Cr: "Card_order r2"
+ and Cp: "Card_order p2"
+ shows "p1 ^c p2 =o r1 ^c r2"
+proof -
+ obtain f where "bij_betw f (Field p2) (Field r2)"
+ using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
+ hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
+ have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
+ and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
+ using 0 Cr Cp czeroE czeroI by auto
+ show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
+ using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by metis
+qed
+
+lemma cexp_cong1:
+ assumes 1: "p1 =o r1" and q: "Card_order q"
+ shows "p1 ^c q =o r1 ^c q"
+by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
+
+lemma cexp_cong2:
+ assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
+ shows "q ^c p2 =o q ^c r2"
+by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
+
+lemma cexp_cone:
+ assumes "Card_order r"
+ shows "r ^c cone =o r"
+proof -
+ have "r ^c cone =o |Field r|"
+ unfolding cexp_def cone_def Field_card_of Func_empty
+ card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
+ by (rule exI[of _ "\<lambda>f. f ()"]) auto
+ also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
+ finally show ?thesis .
+qed
+
+lemma cexp_cprod:
+ assumes r1: "Card_order r1"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
+proof -
+ have "?L =o r1 ^c (r3 *c r2)"
+ unfolding cprod_def cexp_def Field_card_of
+ using card_of_Func_Times by(rule ordIso_symmetric)
+ also have "r1 ^c (r3 *c r2) =o ?R"
+ apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
+ finally show ?thesis .
+qed
+
+lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
+unfolding cinfinite_def cprod_def
+by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
+
+lemma cexp_cprod_ordLeq:
+ assumes r1: "Card_order r1" and r2: "Cinfinite r2"
+ and r3: "Cnotzero r3" "r3 \<le>o r2"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
+proof-
+ have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
+ also have "r1 ^c (r2 *c r3) =o ?R"
+ apply(rule cexp_cong2)
+ apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
+ finally show ?thesis .
+qed
+
+lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
+by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
+
+lemma ordLess_ctwo_cexp:
+ assumes "Card_order r"
+ shows "r <o ctwo ^c r"
+proof -
+ have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
+ also have "|Pow (Field r)| =o ctwo ^c r"
+ unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+ finally show ?thesis .
+qed
+
+lemma ordLeq_cexp1:
+ assumes "Cnotzero r" "Card_order q"
+ shows "q \<le>o q ^c r"
+proof (cases "q =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False
+ thus ?thesis
+ apply -
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule ordIso_symmetric)
+ apply (rule cexp_cone)
+ apply (rule assms(2))
+ apply (rule cexp_mono2)
+ apply (rule cone_ordLeq_Cnotzero)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ apply (rule notE)
+ apply (rule cone_not_czero)
+ apply assumption
+ apply (rule Card_order_cone)
+ done
+qed
+
+lemma ordLeq_cexp2:
+ assumes "ctwo \<le>o q" "Card_order r"
+ shows "r \<le>o q ^c r"
+proof (cases "r =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False thus ?thesis
+ apply -
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule ordLess_ordLeq_trans)
+ apply (rule ordLess_ctwo_cexp)
+ apply (rule assms(2))
+ apply (rule cexp_mono1)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ done
+qed
+
+lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
+by (metis assms cinfinite_mono ordLeq_cexp2)
+
+lemma Cinfinite_cexp:
+ "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
+by (simp add: cinfinite_cexp Card_order_cexp)
+
+lemma ctwo_ordLess_natLeq: "ctwo <o natLeq"
+unfolding ctwo_def using finite_iff_ordLess_natLeq finite_UNIV by fast
+
+lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
+by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
+
+lemma ctwo_ordLeq_Cinfinite:
+ assumes "Cinfinite r"
+ shows "ctwo \<le>o r"
+by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
+
+lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
+by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
+
+lemma UNION_Cinfinite_bound: "\<lbrakk>|I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |\<Union>i \<in> I. A i| \<le>o r"
+by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
+
+lemma csum_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p +c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def csum_def
+ by (blast intro: card_of_Plus_ordLeq_infinite_Field)
+qed
+
+lemma cprod_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p *c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def cprod_def
+ by (blast intro: card_of_Times_ordLeq_infinite_Field)
+qed
+
+lemma cprod_csum_cexp:
+ "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
+unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
+proof -
+ let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
+ have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
+ by (auto simp: inj_on_def fun_eq_iff split: bool.split)
+ moreover
+ have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
+ by (auto simp: Func_def)
+ ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
+qed
+
+lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
+by (intro cprod_cinfinite_bound)
+ (auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
+
+lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
+ unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
+
+lemma cprod_cexp_csum_cexp_Cinfinite:
+ assumes t: "Cinfinite t"
+ shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
+proof -
+ have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
+ by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
+ also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
+ by (rule cexp_cprod[OF Card_order_csum])
+ also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
+ by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
+ also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
+ by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
+ also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
+ by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
+ finally show ?thesis .
+qed
+
+lemma Cfinite_cexp_Cinfinite:
+ assumes s: "Cfinite s" and t: "Cinfinite t"
+ shows "s ^c t \<le>o ctwo ^c t"
+proof (cases "s \<le>o ctwo")
+ case True thus ?thesis using t by (blast intro: cexp_mono1)
+next
+ case False
+ hence "ctwo \<le>o s" by (metis card_order_on_well_order_on ctwo_Cnotzero ordLeq_total s)
+ hence "Cnotzero s" by (metis Cnotzero_mono ctwo_Cnotzero s)
+ hence st: "Cnotzero (s *c t)" by (metis Cinfinite_cprod2 cinfinite_not_czero t)
+ have "s ^c t \<le>o (ctwo ^c s) ^c t"
+ using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
+ also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
+ by (blast intro: Card_order_ctwo cexp_cprod)
+ also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
+ using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
+ finally show ?thesis .
+qed
+
+lemma csum_Cfinite_cexp_Cinfinite:
+ assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
+ shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
+proof (cases "Cinfinite r")
+ case True
+ hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
+ hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
+ also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
+ finally show ?thesis .
+next
+ case False
+ with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
+ hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
+ hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
+ also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
+ by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
+ finally show ?thesis .
+qed
+
+lemma card_order_cexp:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 ^c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding cexp_def Func_def by (simp add: card_of_card_order_on)
+qed
+
+lemma Cinfinite_ordLess_cexp:
+ assumes r: "Cinfinite r"
+ shows "r <o r ^c r"
+proof -
+ have "r <o ctwo ^c r" using r by (simp only: ordLess_ctwo_cexp)
+ also have "ctwo ^c r \<le>o r ^c r"
+ by (rule cexp_mono1[OF ctwo_ordLeq_Cinfinite]) (auto simp: r ctwo_not_czero Card_order_ctwo)
+ finally show ?thesis .
+qed
+
+lemma infinite_ordLeq_cexp:
+ assumes "Cinfinite r"
+ shows "r \<le>o r ^c r"
+by (rule ordLess_imp_ordLeq[OF Cinfinite_ordLess_cexp[OF assms]])
+
+(* cardSuc *)
+
+lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
+by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
+
+lemma cardSuc_UNION_Cinfinite:
+ assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
+ shows "EX i : Field (cardSuc r). B \<le> As i"
+using cardSuc_UNION assms unfolding cinfinite_def by blast
+
+subsection {* Powerset *}
+
+definition cpow where "cpow r = |Pow (Field r)|"
+
+lemma card_order_cpow: "card_order r \<Longrightarrow> card_order (cpow r)"
+by (simp only: cpow_def Field_card_order Pow_UNIV card_of_card_order_on)
+
+lemma cpow_greater_eq: "Card_order r \<Longrightarrow> r \<le>o cpow r"
+by (rule ordLess_imp_ordLeq) (simp only: cpow_def Card_order_Pow)
+
+lemma Cinfinite_cpow: "Cinfinite r \<Longrightarrow> Cinfinite (cpow r)"
+unfolding cpow_def cinfinite_def by (metis Field_card_of card_of_Card_order infinite_Pow)
+
+subsection {* Lists *}
+
+definition clists where "clists r = |lists (Field r)|"
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Cardinal_Order_Relation.thy
--- a/src/HOL/Cardinals/Cardinal_Order_Relation.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Cardinal_Order_Relation.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,7 +8,7 @@
header {* Cardinal-Order Relations *}
theory Cardinal_Order_Relation
-imports Cardinal_Order_Relation_Base Constructions_on_Wellorders
+imports Cardinal_Order_Relation_FP Constructions_on_Wellorders
begin
declare
@@ -34,7 +34,6 @@
Card_order_singl_ordLeq[simp]
card_of_Pow[simp]
Card_order_Pow[simp]
- card_of_set_type[simp]
card_of_Plus1[simp]
Card_order_Plus1[simp]
card_of_Plus2[simp]
@@ -44,25 +43,19 @@
card_of_Plus_mono[simp]
card_of_Plus_cong2[simp]
card_of_Plus_cong[simp]
- card_of_Un1[simp]
- card_of_diff[simp]
card_of_Un_Plus_ordLeq[simp]
card_of_Times1[simp]
card_of_Times2[simp]
card_of_Times3[simp]
card_of_Times_mono1[simp]
card_of_Times_mono2[simp]
- card_of_Times_cong1[simp]
- card_of_Times_cong2[simp]
card_of_ordIso_finite[simp]
- finite_ordLess_infinite2[simp]
card_of_Times_same_infinite[simp]
card_of_Times_infinite_simps[simp]
card_of_Plus_infinite1[simp]
card_of_Plus_infinite2[simp]
card_of_Plus_ordLess_infinite[simp]
card_of_Plus_ordLess_infinite_Field[simp]
- card_of_lists_infinite[simp]
infinite_cartesian_product[simp]
cardSuc_Card_order[simp]
cardSuc_greater[simp]
@@ -143,6 +136,17 @@
subsection {* Cardinals versus set operations on arbitrary sets *}
+lemma card_of_set_type[simp]: "|UNIV::'a set| <o |UNIV::'a set set|"
+using card_of_Pow[of "UNIV::'a set"] by simp
+
+lemma card_of_Un1[simp]:
+shows "|A| \<le>o |A \<union> B| "
+using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
+lemma card_of_diff[simp]:
+shows "|A - B| \<le>o |A|"
+using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
+
lemma subset_ordLeq_strict:
assumes "A \<le> B" and "|A| <o |B|"
shows "A < B"
@@ -307,6 +311,16 @@
using card_of_Times3 card_of_Field_ordIso
ordIso_ordLeq_trans ordIso_symmetric by blast
+lemma card_of_Times_cong1[simp]:
+assumes "|A| =o |B|"
+shows "|A \<times> C| =o |B \<times> C|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
+
+lemma card_of_Times_cong2[simp]:
+assumes "|A| =o |B|"
+shows "|C \<times> A| =o |C \<times> B|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
+
lemma card_of_Times_mono[simp]:
assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
shows "|A \<times> C| \<le>o |B \<times> D|"
@@ -323,6 +337,11 @@
shows "|(Field r) \<times> C| =o |(Field r') \<times> C|"
using assms card_of_cong card_of_Times_cong1 by blast
+corollary ordIso_Times_cong2:
+assumes "r =o r'"
+shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
+using assms card_of_cong card_of_Times_cong2 by blast
+
lemma card_of_Times_cong[simp]:
assumes "|A| =o |B|" and "|C| =o |D|"
shows "|A \<times> C| =o |B \<times> D|"
@@ -501,11 +520,55 @@
using assms Plus_infinite_bij_betw[of "UNIV::'a set" g "UNIV::'b set"]
by auto
+lemma card_of_Un_infinite:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
+proof-
+ have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
+ moreover have "|A <+> B| =o |A|"
+ using assms by (metis card_of_Plus_infinite)
+ ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
+ hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
+ thus ?thesis using Un_commute[of B A] by auto
+qed
+
lemma card_of_Un_infinite_simps[simp]:
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |A \<union> B| =o |A|"
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |B \<union> A| =o |A|"
using card_of_Un_infinite by auto
+lemma card_of_Un_diff_infinite:
+assumes INF: "infinite A" and LESS: "|B| <o |A|"
+shows "|A - B| =o |A|"
+proof-
+ obtain C where C_def: "C = A - B" by blast
+ have "|A \<union> B| =o |A|"
+ using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
+ moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
+ ultimately have 1: "|C \<union> B| =o |A|" by auto
+ (* *)
+ {assume *: "|C| \<le>o |B|"
+ moreover
+ {assume **: "finite B"
+ hence "finite C"
+ using card_of_ordLeq_finite * by blast
+ hence False using ** INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite B" by auto
+ ultimately have False
+ using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
+ }
+ hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
+ {assume *: "finite C"
+ hence "finite B" using card_of_ordLeq_finite 2 by blast
+ hence False using * INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite C" by auto
+ hence "|C| =o |A|"
+ using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
+ thus ?thesis unfolding C_def .
+qed
+
corollary Card_order_Un_infinite:
assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
LEQ: "p \<le>o r"
@@ -597,6 +660,33 @@
thus ?thesis using 1 ordLess_ordIso_trans by blast
qed
+
+subsection {* Cardinals versus set operations involving infinite sets *}
+
+lemma finite_iff_cardOf_nat:
+"finite A = ( |A| <o |UNIV :: nat set| )"
+using infinite_iff_card_of_nat[of A]
+not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
+by (fastforce simp: card_of_Well_order)
+
+lemma finite_ordLess_infinite2[simp]:
+assumes "finite A" and "infinite B"
+shows "|A| <o |B|"
+using assms
+finite_ordLess_infinite[of "|A|" "|B|"]
+card_of_Well_order[of A] card_of_Well_order[of B]
+Field_card_of[of A] Field_card_of[of B] by auto
+
+lemma infinite_card_of_insert:
+assumes "infinite A"
+shows "|insert a A| =o |A|"
+proof-
+ have iA: "insert a A = A \<union> {a}" by simp
+ show ?thesis
+ using infinite_imp_bij_betw2[OF assms] unfolding iA
+ by (metis bij_betw_inv card_of_ordIso)
+qed
+
lemma card_of_Un_singl_ordLess_infinite1:
assumes "infinite B" and "|A| <o |B|"
shows "|{a} Un A| <o |B|"
@@ -616,7 +706,83 @@
qed
-subsection {* Cardinals versus lists *}
+subsection {* Cardinals versus lists *}
+
+text{* The next is an auxiliary operator, which shall be used for inductive
+proofs of facts concerning the cardinality of @{text "List"} : *}
+
+definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
+where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
+
+lemma lists_def2: "lists A = {l. set l \<le> A}"
+using in_listsI by blast
+
+lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
+unfolding lists_def2 nlists_def by blast
+
+lemma card_of_lists: "|A| \<le>o |lists A|"
+proof-
+ let ?h = "\<lambda> a. [a]"
+ have "inj_on ?h A \<and> ?h ` A \<le> lists A"
+ unfolding inj_on_def lists_def2 by auto
+ thus ?thesis by (metis card_of_ordLeq)
+qed
+
+lemma nlists_0: "nlists A 0 = {[]}"
+unfolding nlists_def by auto
+
+lemma nlists_not_empty:
+assumes "A \<noteq> {}"
+shows "nlists A n \<noteq> {}"
+proof(induct n, simp add: nlists_0)
+ fix n assume "nlists A n \<noteq> {}"
+ then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
+ hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
+ thus "nlists A (Suc n) \<noteq> {}" by auto
+qed
+
+lemma Nil_in_lists: "[] \<in> lists A"
+unfolding lists_def2 by auto
+
+lemma lists_not_empty: "lists A \<noteq> {}"
+using Nil_in_lists by blast
+
+lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+proof-
+ let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l"
+ have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
+ unfolding inj_on_def nlists_def by auto
+ moreover have "nlists A (Suc n) \<le> ?h ` ?B"
+ proof(auto)
+ fix l assume "l \<in> nlists A (Suc n)"
+ hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
+ then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
+ hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
+ thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto
+ qed
+ ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
+ unfolding bij_betw_def by auto
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+qed
+
+lemma card_of_nlists_infinite:
+assumes "infinite A"
+shows "|nlists A n| \<le>o |A|"
+proof(induct n)
+ have "A \<noteq> {}" using assms by auto
+ thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
+next
+ fix n assume IH: "|nlists A n| \<le>o |A|"
+ have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+ using card_of_nlists_Succ by blast
+ moreover
+ {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
+ hence "|A \<times> (nlists A n)| =o |A|"
+ using assms IH by (auto simp add: card_of_Times_infinite)
+ }
+ ultimately show "|nlists A (Suc n)| \<le>o |A|"
+ using ordIso_transitive ordIso_iff_ordLeq by blast
+qed
lemma Card_order_lists: "Card_order r \<Longrightarrow> r \<le>o |lists(Field r) |"
using card_of_lists card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
@@ -690,6 +856,22 @@
thus ?thesis using card_of_ordIso[of "lists A"] by auto
qed
+lemma card_of_lists_infinite[simp]:
+assumes "infinite A"
+shows "|lists A| =o |A|"
+proof-
+ have "|lists A| \<le>o |A|"
+ using assms
+ by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
+ infinite_iff_card_of_nat card_of_nlists_infinite)
+ thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
+qed
+
+lemma Card_order_lists_infinite:
+assumes "Card_order r" and "infinite(Field r)"
+shows "|lists(Field r)| =o r"
+using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
+
lemma ordIso_lists_cong:
assumes "r =o r'"
shows "|lists(Field r)| =o |lists(Field r')|"
@@ -827,13 +1009,22 @@
lemma Field_natLess: "Field natLess = (UNIV::nat set)"
by(unfold Field_def, auto)
+lemma natLeq_well_order_on: "well_order_on UNIV natLeq"
+using natLeq_Well_order Field_natLeq by auto
+
+lemma natLeq_wo_rel: "wo_rel natLeq"
+unfolding wo_rel_def using natLeq_Well_order .
+
lemma natLeq_ofilter_less: "ofilter natLeq {0 ..< n}"
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
- simp add: Field_natLeq, unfold rel.under_def, auto)
+ simp add: Field_natLeq, unfold rel.under_def, auto)
lemma natLeq_ofilter_leq: "ofilter natLeq {0 .. n}"
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
- simp add: Field_natLeq, unfold rel.under_def, auto)
+ simp add: Field_natLeq, unfold rel.under_def, auto)
+
+lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
+using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
lemma natLeq_ofilter_iff:
"ofilter natLeq A = (A = UNIV \<or> (\<exists>n. A = {0 ..< n}))"
@@ -900,7 +1091,7 @@
qed
-subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
+subsubsection {* "Backward compatibility" with the numeric cardinal operator for finite sets *}
lemma finite_card_of_iff_card:
assumes FIN: "finite A" and FIN': "finite B"
@@ -993,6 +1184,11 @@
shows "relChain r (\<lambda> i. under r i)"
using assms unfolding relChain_def by auto
+lemma card_of_infinite_diff_finite:
+assumes "infinite A" and "finite B"
+shows "|A - B| =o |A|"
+by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
+
lemma infinite_card_of_diff_singl:
assumes "infinite A"
shows "|A - {a}| =o |A|"
@@ -1110,6 +1306,30 @@
thus "f \<in> Pfunc A B" unfolding Func_option_def Pfunc_def by auto
qed
+lemma card_of_Func_mono:
+fixes A1 A2 :: "'a set" and B :: "'b set"
+assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
+shows "|Func A1 B| \<le>o |Func A2 B|"
+proof-
+ obtain bb where bb: "bb \<in> B" using B by auto
+ def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb)
+ else undefined"
+ show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
+ show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
+ fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
+ show "f = g"
+ proof(rule ext)
+ fix a show "f a = g a"
+ proof(cases "a \<in> A1")
+ case True
+ thus ?thesis using eq A12 unfolding F_def fun_eq_iff
+ by (elim allE[of _ a]) auto
+ qed(insert f g, unfold Func_def, fastforce)
+ qed
+ qed
+ qed(insert bb, unfold Func_def F_def, force)
+qed
+
lemma card_of_Func_option_mono:
fixes A1 A2 :: "'a set" and B :: "'b set"
assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
@@ -1178,4 +1398,18 @@
"|Func (UNIV::'a set) (UNIV::'b set)| =o |UNIV::('a \<Rightarrow> 'b) set|"
using card_of_Func_UNIV[of "UNIV::'b set"] by auto
+lemma ordLeq_Func:
+assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "|A| \<le>o |Func A B|"
+unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
+ let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
+ show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
+ show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
+qed
+
+lemma infinite_Func:
+assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "infinite (Func A B)"
+using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
+
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Cardinal_Order_Relation_Base.thy
--- a/src/HOL/Cardinals/Cardinal_Order_Relation_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2438 +0,0 @@
-(* Title: HOL/Cardinals/Cardinal_Order_Relation_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Cardinal-order relations (base).
-*)
-
-header {* Cardinal-Order Relations (Base) *}
-
-theory Cardinal_Order_Relation_Base
-imports Constructions_on_Wellorders_Base
-begin
-
-
-text{* In this section, we define cardinal-order relations to be minim well-orders
-on their field. Then we define the cardinal of a set to be {\em some} cardinal-order
-relation on that set, which will be unique up to order isomorphism. Then we study
-the connection between cardinals and:
-\begin{itemize}
-\item standard set-theoretic constructions: products,
-sums, unions, lists, powersets, set-of finite sets operator;
-\item finiteness and infiniteness (in particular, with the numeric cardinal operator
-for finite sets, @{text "card"}, from the theory @{text "Finite_Sets.thy"}).
-\end{itemize}
-%
-On the way, we define the canonical $\omega$ cardinal and finite cardinals. We also
-define (again, up to order isomorphism) the successor of a cardinal, and show that
-any cardinal admits a successor.
-
-Main results of this section are the existence of cardinal relations and the
-facts that, in the presence of infiniteness,
-most of the standard set-theoretic constructions (except for the powerset)
-{\em do not increase cardinality}. In particular, e.g., the set of words/lists over
-any infinite set has the same cardinality (hence, is in bijection) with that set.
-*}
-
-
-subsection {* Cardinal orders *}
-
-
-text{* A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the
-order-embedding relation, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
-strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
-
-definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
-where
-"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
-
-
-abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
-abbreviation "card_order r \<equiv> card_order_on UNIV r"
-
-
-lemma card_order_on_well_order_on:
-assumes "card_order_on A r"
-shows "well_order_on A r"
-using assms unfolding card_order_on_def by simp
-
-
-lemma card_order_on_Card_order:
-"card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
-unfolding card_order_on_def using rel.well_order_on_Field by blast
-
-
-text{* The existence of a cardinal relation on any given set (which will mean
-that any set has a cardinal) follows from two facts:
-\begin{itemize}
-\item Zermelo's theorem (proved in @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
-which states that on any given set there exists a well-order;
-\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
-such well-order, i.e., a cardinal order.
-\end{itemize}
-*}
-
-
-theorem card_order_on: "\<exists>r. card_order_on A r"
-proof-
- obtain R where R_def: "R = {r. well_order_on A r}" by blast
- have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
- using well_order_on[of A] R_def rel.well_order_on_Well_order by blast
- hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
- using exists_minim_Well_order[of R] by auto
- thus ?thesis using R_def unfolding card_order_on_def by auto
-qed
-
-
-lemma card_order_on_ordIso:
-assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
-shows "r =o r'"
-using assms unfolding card_order_on_def
-using ordIso_iff_ordLeq by blast
-
-
-lemma Card_order_ordIso:
-assumes CO: "Card_order r" and ISO: "r' =o r"
-shows "Card_order r'"
-using ISO unfolding ordIso_def
-proof(unfold card_order_on_def, auto)
- fix p' assume "well_order_on (Field r') p'"
- hence 0: "Well_order p' \<and> Field p' = Field r'"
- using rel.well_order_on_Well_order by blast
- obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
- using ISO unfolding ordIso_def by auto
- hence 3: "inj_on f (Field r') \<and> f ` (Field r') = Field r"
- by (auto simp add: iso_iff embed_inj_on)
- let ?p = "dir_image p' f"
- have 4: "p' =o ?p \<and> Well_order ?p"
- using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
- moreover have "Field ?p = Field r"
- using 0 3 by (auto simp add: dir_image_Field2 order_on_defs)
- ultimately have "well_order_on (Field r) ?p" by auto
- hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
- thus "r' \<le>o p'"
- using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
-qed
-
-
-lemma Card_order_ordIso2:
-assumes CO: "Card_order r" and ISO: "r =o r'"
-shows "Card_order r'"
-using assms Card_order_ordIso ordIso_symmetric by blast
-
-
-subsection {* Cardinal of a set *}
-
-
-text{* We define the cardinal of set to be {\em some} cardinal order on that set.
-We shall prove that this notion is unique up to order isomorphism, meaning
-that order isomorphism shall be the true identity of cardinals. *}
-
-
-definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )
-where "card_of A = (SOME r. card_order_on A r)"
-
-
-lemma card_of_card_order_on: "card_order_on A |A|"
-unfolding card_of_def by (auto simp add: card_order_on someI_ex)
-
-
-lemma card_of_well_order_on: "well_order_on A |A|"
-using card_of_card_order_on card_order_on_def by blast
-
-
-lemma Field_card_of: "Field |A| = A"
-using card_of_card_order_on[of A] unfolding card_order_on_def
-using rel.well_order_on_Field by blast
-
-
-lemma card_of_Card_order: "Card_order |A|"
-by (simp only: card_of_card_order_on Field_card_of)
-
-
-corollary ordIso_card_of_imp_Card_order:
-"r =o |A| \<Longrightarrow> Card_order r"
-using card_of_Card_order Card_order_ordIso by blast
-
-
-lemma card_of_Well_order: "Well_order |A|"
-using card_of_Card_order unfolding card_order_on_def by auto
-
-
-lemma card_of_refl: "|A| =o |A|"
-using card_of_Well_order ordIso_reflexive by blast
-
-
-lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"
-using card_of_card_order_on unfolding card_order_on_def by blast
-
-
-lemma card_of_ordIso:
-"(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
-proof(auto)
- fix f assume *: "bij_betw f A B"
- then obtain r where "well_order_on B r \<and> |A| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
- hence "|B| \<le>o |A|" using card_of_least
- ordLeq_ordIso_trans ordIso_symmetric by blast
- moreover
- {let ?g = "inv_into A f"
- have "bij_betw ?g B A" using * bij_betw_inv_into by blast
- then obtain r where "well_order_on A r \<and> |B| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
- hence "|A| \<le>o |B|" using card_of_least
- ordLeq_ordIso_trans ordIso_symmetric by blast
- }
- ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
-next
- assume "|A| =o |B|"
- then obtain f where "iso ( |A| ) ( |B| ) f"
- unfolding ordIso_def by auto
- hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
- thus "\<exists>f. bij_betw f A B" by auto
-qed
-
-
-lemma card_of_ordLeq:
-"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
-proof(auto)
- fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
- {assume "|B| <o |A|"
- hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
- then obtain g where "embed ( |B| ) ( |A| ) g"
- unfolding ordLeq_def by auto
- hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
- card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
- embed_Field[of "|B|" "|A|" g] by auto
- obtain h where "bij_betw h A B"
- using * ** 1 Cantor_Bernstein[of f] by fastforce
- hence "|A| =o |B|" using card_of_ordIso by blast
- hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
- }
- thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
- by (auto simp: card_of_Well_order)
-next
- assume *: "|A| \<le>o |B|"
- obtain f where "embed ( |A| ) ( |B| ) f"
- using * unfolding ordLeq_def by auto
- hence "inj_on f A \<and> f ` A \<le> B" using embed_inj_on[of "|A|" "|B|" f]
- card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
- embed_Field[of "|A|" "|B|" f] by auto
- thus "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
-qed
-
-
-lemma card_of_ordLeq2:
-"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
-using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
-
-
-lemma card_of_ordLess:
-"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
-proof-
- have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
- using card_of_ordLeq by blast
- also have "\<dots> = ( |B| <o |A| )"
- using card_of_Well_order[of A] card_of_Well_order[of B]
- not_ordLeq_iff_ordLess by blast
- finally show ?thesis .
-qed
-
-
-lemma card_of_ordLess2:
-"B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
-using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
-
-
-lemma card_of_ordIsoI:
-assumes "bij_betw f A B"
-shows "|A| =o |B|"
-using assms unfolding card_of_ordIso[symmetric] by auto
-
-
-lemma card_of_ordLeqI:
-assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
-shows "|A| \<le>o |B|"
-using assms unfolding card_of_ordLeq[symmetric] by auto
-
-
-lemma card_of_unique:
-"card_order_on A r \<Longrightarrow> r =o |A|"
-by (simp only: card_order_on_ordIso card_of_card_order_on)
-
-
-lemma card_of_mono1:
-"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
-using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
-
-
-lemma card_of_mono2:
-assumes "r \<le>o r'"
-shows "|Field r| \<le>o |Field r'|"
-proof-
- obtain f where
- 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
- using assms unfolding ordLeq_def
- by (auto simp add: rel.well_order_on_Well_order)
- hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
- by (auto simp add: embed_inj_on embed_Field)
- thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
-qed
-
-
-lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
-by (simp add: ordIso_iff_ordLeq card_of_mono2)
-
-
-lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
-using card_of_least card_of_well_order_on rel.well_order_on_Well_order by blast
-
-
-lemma card_of_Field_ordIso:
-assumes "Card_order r"
-shows "|Field r| =o r"
-proof-
- have "card_order_on (Field r) r"
- using assms card_order_on_Card_order by blast
- moreover have "card_order_on (Field r) |Field r|"
- using card_of_card_order_on by blast
- ultimately show ?thesis using card_order_on_ordIso by blast
-qed
-
-
-lemma Card_order_iff_ordIso_card_of:
-"Card_order r = (r =o |Field r| )"
-using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
-
-
-lemma Card_order_iff_ordLeq_card_of:
-"Card_order r = (r \<le>o |Field r| )"
-proof-
- have "Card_order r = (r =o |Field r| )"
- unfolding Card_order_iff_ordIso_card_of by simp
- also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
- unfolding ordIso_iff_ordLeq by simp
- also have "... = (r \<le>o |Field r| )"
- using card_of_Field_ordLess
- by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
- finally show ?thesis .
-qed
-
-
-lemma Card_order_iff_Restr_underS:
-assumes "Well_order r"
-shows "Card_order r = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o |Field r| )"
-using assms unfolding Card_order_iff_ordLeq_card_of
-using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
-
-
-lemma card_of_underS:
-assumes r: "Card_order r" and a: "a : Field r"
-shows "|rel.underS r a| <o r"
-proof-
- let ?A = "rel.underS r a" let ?r' = "Restr r ?A"
- have 1: "Well_order r"
- using r unfolding card_order_on_def by simp
- have "Well_order ?r'" using 1 Well_order_Restr by auto
- moreover have "card_order_on (Field ?r') |Field ?r'|"
- using card_of_card_order_on .
- ultimately have "|Field ?r'| \<le>o ?r'"
- unfolding card_order_on_def by simp
- moreover have "Field ?r' = ?A"
- using 1 wo_rel.underS_ofilter Field_Restr_ofilter
- unfolding wo_rel_def by fastforce
- ultimately have "|?A| \<le>o ?r'" by simp
- also have "?r' <o |Field r|"
- using 1 a r Card_order_iff_Restr_underS by blast
- also have "|Field r| =o r"
- using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
- finally show ?thesis .
-qed
-
-
-lemma ordLess_Field:
-assumes "r <o r'"
-shows "|Field r| <o r'"
-proof-
- have "well_order_on (Field r) r" using assms unfolding ordLess_def
- by (auto simp add: rel.well_order_on_Well_order)
- hence "|Field r| \<le>o r" using card_of_least by blast
- thus ?thesis using assms ordLeq_ordLess_trans by blast
-qed
-
-
-lemma internalize_card_of_ordLeq:
-"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
-proof
- assume "|A| \<le>o r"
- then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
- using internalize_ordLeq[of "|A|" r] by blast
- hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
- hence "|Field p| =o p" using card_of_Field_ordIso by blast
- hence "|A| =o |Field p| \<and> |Field p| \<le>o r"
- using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
- thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
-next
- assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
- thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
-qed
-
-
-lemma internalize_card_of_ordLeq2:
-"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
-using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
-
-
-
-subsection {* Cardinals versus set operations on arbitrary sets *}
-
-
-text{* Here we embark in a long journey of simple results showing
-that the standard set-theoretic operations are well-behaved w.r.t. the notion of
-cardinal -- essentially, this means that they preserve the ``cardinal identity"
-@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
-*}
-
-
-lemma card_of_empty: "|{}| \<le>o |A|"
-using card_of_ordLeq inj_on_id by blast
-
-
-lemma card_of_empty1:
-assumes "Well_order r \<or> Card_order r"
-shows "|{}| \<le>o r"
-proof-
- have "Well_order r" using assms unfolding card_order_on_def by auto
- hence "|Field r| <=o r"
- using assms card_of_Field_ordLess by blast
- moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
- ultimately show ?thesis using ordLeq_transitive by blast
-qed
-
-
-corollary Card_order_empty:
-"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
-
-
-lemma card_of_empty2:
-assumes LEQ: "|A| =o |{}|"
-shows "A = {}"
-using assms card_of_ordIso[of A] bij_betw_empty2 by blast
-
-
-lemma card_of_empty3:
-assumes LEQ: "|A| \<le>o |{}|"
-shows "A = {}"
-using assms
-by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
- ordLeq_Well_order_simp)
-
-
-lemma card_of_empty_ordIso:
-"|{}::'a set| =o |{}::'b set|"
-using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
-
-
-lemma card_of_image:
-"|f ` A| <=o |A|"
-proof(cases "A = {}", simp add: card_of_empty)
- assume "A ~= {}"
- hence "f ` A ~= {}" by auto
- thus "|f ` A| \<le>o |A|"
- using card_of_ordLeq2[of "f ` A" A] by auto
-qed
-
-
-lemma surj_imp_ordLeq:
-assumes "B <= f ` A"
-shows "|B| <=o |A|"
-proof-
- have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
- thus ?thesis using card_of_image ordLeq_transitive by blast
-qed
-
-
-lemma card_of_ordLeqI2:
-assumes "B \<subseteq> f ` A"
-shows "|B| \<le>o |A|"
-using assms by (metis surj_imp_ordLeq)
-
-
-lemma card_of_singl_ordLeq:
-assumes "A \<noteq> {}"
-shows "|{b}| \<le>o |A|"
-proof-
- obtain a where *: "a \<in> A" using assms by auto
- let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
- have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
- using * unfolding inj_on_def by auto
- thus ?thesis using card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_singl_ordLeq:
-"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
-using card_of_singl_ordLeq[of "Field r" b]
- card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
-
-
-lemma card_of_Pow: "|A| <o |Pow A|"
-using card_of_ordLess2[of "Pow A" A] Cantors_paradox[of A]
- Pow_not_empty[of A] by auto
-
-
-lemma infinite_Pow:
-assumes "infinite A"
-shows "infinite (Pow A)"
-proof-
- have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
- thus ?thesis by (metis assms finite_Pow_iff)
-qed
-
-
-corollary Card_order_Pow:
-"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
-using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
-
-
-corollary card_of_set_type: "|UNIV::'a set| <o |UNIV::'a set set|"
-using card_of_Pow[of "UNIV::'a set"] by simp
-
-
-lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
-proof-
- have "Inl ` A \<le> A <+> B" by auto
- thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_Plus1:
-"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
-using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
-proof-
- have "Inr ` B \<le> A <+> B" by auto
- thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_Plus2:
-"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
-using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
-proof-
- have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
- thus ?thesis using card_of_ordIso by auto
-qed
-
-
-lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
-proof-
- have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
- thus ?thesis using card_of_ordIso by auto
-qed
-
-
-lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
-proof-
- let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
- | Inr b \<Rightarrow> Inl b"
- have "bij_betw ?f (A <+> B) (B <+> A)"
- unfolding bij_betw_def inj_on_def by force
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Plus_assoc:
-fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
-shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
-proof -
- def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
- case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
- |Inr b \<Rightarrow> Inr (Inl b))
- |Inr c \<Rightarrow> Inr (Inr c)"
- have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
- proof
- fix x assume x: "x \<in> A <+> B <+> C"
- show "x \<in> f ` ((A <+> B) <+> C)"
- proof(cases x)
- case (Inl a)
- hence "a \<in> A" "x = f (Inl (Inl a))"
- using x unfolding f_def by auto
- thus ?thesis by auto
- next
- case (Inr bc) note 1 = Inr show ?thesis
- proof(cases bc)
- case (Inl b)
- hence "b \<in> B" "x = f (Inl (Inr b))"
- using x 1 unfolding f_def by auto
- thus ?thesis by auto
- next
- case (Inr c)
- hence "c \<in> C" "x = f (Inr c)"
- using x 1 unfolding f_def by auto
- thus ?thesis by auto
- qed
- qed
- qed
- hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
- unfolding bij_betw_def inj_on_def f_def by force
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Plus_mono1:
-assumes "|A| \<le>o |B|"
-shows "|A <+> C| \<le>o |B <+> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
- have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
- proof-
- {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
- "g d1 = g d2"
- hence "d1 = d2" using 1 unfolding inj_on_def
- by(case_tac d1, case_tac d2, auto simp add: g_def)
- }
- moreover
- {fix d assume "d \<in> A <+> C"
- hence "g d \<in> B <+> C" using 1
- by(case_tac d, auto simp add: g_def)
- }
- ultimately show ?thesis unfolding inj_on_def by auto
- qed
- thus ?thesis using card_of_ordLeq by metis
-qed
-
-
-corollary ordLeq_Plus_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
-using assms card_of_mono2 card_of_Plus_mono1 by blast
-
-
-lemma card_of_Plus_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C <+> A| \<le>o |C <+> B|"
-using assms card_of_Plus_mono1[of A B C]
- card_of_Plus_commute[of C A] card_of_Plus_commute[of B C]
- ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
-by blast
-
-
-corollary ordLeq_Plus_mono2:
-assumes "r \<le>o r'"
-shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
-using assms card_of_mono2 card_of_Plus_mono2 by blast
-
-
-lemma card_of_Plus_mono:
-assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
-shows "|A <+> C| \<le>o |B <+> D|"
-using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
- ordLeq_transitive[of "|A <+> C|"] by blast
-
-
-corollary ordLeq_Plus_mono:
-assumes "r \<le>o r'" and "p \<le>o p'"
-shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
-using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
-
-
-lemma card_of_Plus_cong1:
-assumes "|A| =o |B|"
-shows "|A <+> C| =o |B <+> C|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
-
-
-corollary ordIso_Plus_cong1:
-assumes "r =o r'"
-shows "|(Field r) <+> C| =o |(Field r') <+> C|"
-using assms card_of_cong card_of_Plus_cong1 by blast
-
-
-lemma card_of_Plus_cong2:
-assumes "|A| =o |B|"
-shows "|C <+> A| =o |C <+> B|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
-
-
-corollary ordIso_Plus_cong2:
-assumes "r =o r'"
-shows "|A <+> (Field r)| =o |A <+> (Field r')|"
-using assms card_of_cong card_of_Plus_cong2 by blast
-
-
-lemma card_of_Plus_cong:
-assumes "|A| =o |B|" and "|C| =o |D|"
-shows "|A <+> C| =o |B <+> D|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
-
-
-corollary ordIso_Plus_cong:
-assumes "r =o r'" and "p =o p'"
-shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
-using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
-
-
-lemma card_of_Un1:
-shows "|A| \<le>o |A \<union> B| "
-using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
-
-
-lemma card_of_diff:
-shows "|A - B| \<le>o |A|"
-using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
-
-
-lemma card_of_Un_Plus_ordLeq:
-"|A \<union> B| \<le>o |A <+> B|"
-proof-
- let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
- have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
- unfolding inj_on_def by auto
- thus ?thesis using card_of_ordLeq by blast
-qed
-
-
-lemma card_of_Times1:
-assumes "A \<noteq> {}"
-shows "|B| \<le>o |B \<times> A|"
-proof(cases "B = {}", simp add: card_of_empty)
- assume *: "B \<noteq> {}"
- have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
- thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
- card_of_ordLeq[of B "B \<times> A"] * by blast
-qed
-
-
-corollary Card_order_Times1:
-"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
-using card_of_Times1[of B] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
-proof-
- let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
- have "bij_betw ?f (A \<times> B) (B \<times> A)"
- unfolding bij_betw_def inj_on_def by auto
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Times2:
-assumes "A \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
-using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
- ordLeq_ordIso_trans by blast
-
-
-corollary Card_order_Times2:
-"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
-using card_of_Times2[of A] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
-using card_of_Times1[of A]
-by(cases "A = {}", simp add: card_of_empty, blast)
-
-
-lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
-proof-
- let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
- |Inr a \<Rightarrow> (a,False)"
- have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
- proof-
- {fix c1 and c2 assume "?f c1 = ?f c2"
- hence "c1 = c2"
- by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
- }
- moreover
- {fix c assume "c \<in> A <+> A"
- hence "?f c \<in> A \<times> (UNIV::bool set)"
- by(case_tac c, auto)
- }
- moreover
- {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
- have "(a,bl) \<in> ?f ` ( A <+> A)"
- proof(cases bl)
- assume bl hence "?f(Inl a) = (a,bl)" by auto
- thus ?thesis using * by force
- next
- assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
- thus ?thesis using * by force
- qed
- }
- ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
- qed
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Times_mono1:
-assumes "|A| \<le>o |B|"
-shows "|A \<times> C| \<le>o |B \<times> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>(a,c::'c). (f a,c))" by blast
- have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
- using 1 unfolding inj_on_def using g_def by auto
- thus ?thesis using card_of_ordLeq by metis
-qed
-
-
-corollary ordLeq_Times_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
-using assms card_of_mono2 card_of_Times_mono1 by blast
-
-
-lemma card_of_Times_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C \<times> A| \<le>o |C \<times> B|"
-using assms card_of_Times_mono1[of A B C]
- card_of_Times_commute[of C A] card_of_Times_commute[of B C]
- ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
-by blast
-
-
-corollary ordLeq_Times_mono2:
-assumes "r \<le>o r'"
-shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
-using assms card_of_mono2 card_of_Times_mono2 by blast
-
-
-lemma card_of_Times_cong1:
-assumes "|A| =o |B|"
-shows "|A \<times> C| =o |B \<times> C|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
-
-
-lemma card_of_Times_cong2:
-assumes "|A| =o |B|"
-shows "|C \<times> A| =o |C \<times> B|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
-
-
-corollary ordIso_Times_cong2:
-assumes "r =o r'"
-shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
-using assms card_of_cong card_of_Times_cong2 by blast
-
-
-lemma card_of_Sigma_mono1:
-assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
-shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
-proof-
- have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
- using assms by (auto simp add: card_of_ordLeq)
- with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
- obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by metis
- obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
- have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
- using 1 unfolding inj_on_def using g_def by force
- thus ?thesis using card_of_ordLeq by metis
-qed
-
-
-corollary card_of_Sigma_Times:
-"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
-using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
-
-
-lemma card_of_UNION_Sigma:
-"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
-using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
-
-
-lemma card_of_bool:
-assumes "a1 \<noteq> a2"
-shows "|UNIV::bool set| =o |{a1,a2}|"
-proof-
- let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
- have "bij_betw ?f UNIV {a1,a2}"
- proof-
- {fix bl1 and bl2 assume "?f bl1 = ?f bl2"
- hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
- }
- moreover
- {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)
- }
- moreover
- {fix a assume *: "a \<in> {a1,a2}"
- have "a \<in> ?f ` UNIV"
- proof(cases "a = a1")
- assume "a = a1"
- hence "?f True = a" by auto thus ?thesis by blast
- next
- assume "a \<noteq> a1" hence "a = a2" using * by auto
- hence "?f False = a" by auto thus ?thesis by blast
- qed
- }
- ultimately show ?thesis unfolding bij_betw_def inj_on_def
- by (metis image_subsetI order_eq_iff subsetI)
- qed
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Plus_Times_aux:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- LEQ: "|A| \<le>o |B|"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
- have 1: "|UNIV::bool set| \<le>o |A|"
- using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
- ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
- (* *)
- have "|A <+> B| \<le>o |B <+> B|"
- using LEQ card_of_Plus_mono1 by blast
- moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
- using card_of_Plus_Times_bool by blast
- moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
- using 1 by (simp add: card_of_Times_mono2)
- moreover have " |B \<times> A| =o |A \<times> B|"
- using card_of_Times_commute by blast
- ultimately show "|A <+> B| \<le>o |A \<times> B|"
- using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
- ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
- ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
- by blast
-qed
-
-
-lemma card_of_Plus_Times:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
- {assume "|A| \<le>o |B|"
- hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
- }
- moreover
- {assume "|B| \<le>o |A|"
- hence "|B <+> A| \<le>o |B \<times> A|"
- using assms by (auto simp add: card_of_Plus_Times_aux)
- hence ?thesis
- using card_of_Plus_commute card_of_Times_commute
- ordIso_ordLeq_trans ordLeq_ordIso_trans by metis
- }
- ultimately show ?thesis
- using card_of_Well_order[of A] card_of_Well_order[of B]
- ordLeq_total[of "|A|"] by metis
-qed
-
-
-lemma card_of_ordLeq_finite:
-assumes "|A| \<le>o |B|" and "finite B"
-shows "finite A"
-using assms unfolding ordLeq_def
-using embed_inj_on[of "|A|" "|B|"] embed_Field[of "|A|" "|B|"]
- Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
-
-
-lemma card_of_ordLeq_infinite:
-assumes "|A| \<le>o |B|" and "infinite A"
-shows "infinite B"
-using assms card_of_ordLeq_finite by auto
-
-
-lemma card_of_ordIso_finite:
-assumes "|A| =o |B|"
-shows "finite A = finite B"
-using assms unfolding ordIso_def iso_def[abs_def]
-by (auto simp: bij_betw_finite Field_card_of)
-
-
-lemma card_of_ordIso_finite_Field:
-assumes "Card_order r" and "r =o |A|"
-shows "finite(Field r) = finite A"
-using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
-
-
-subsection {* Cardinals versus set operations involving infinite sets *}
-
-
-text{* Here we show that, for infinite sets, most set-theoretic constructions
-do not increase the cardinality. The cornerstone for this is
-theorem @{text "Card_order_Times_same_infinite"}, which states that self-product
-does not increase cardinality -- the proof of this fact adapts a standard
-set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
-at page 47 in \cite{card-book}. Then everything else follows fairly easily. *}
-
-
-lemma infinite_iff_card_of_nat:
-"infinite A = ( |UNIV::nat set| \<le>o |A| )"
-by (auto simp add: infinite_iff_countable_subset card_of_ordLeq)
-
-
-lemma finite_iff_cardOf_nat:
-"finite A = ( |A| <o |UNIV :: nat set| )"
-using infinite_iff_card_of_nat[of A]
-not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
-by (fastforce simp: card_of_Well_order)
-
-lemma finite_ordLess_infinite2:
-assumes "finite A" and "infinite B"
-shows "|A| <o |B|"
-using assms
-finite_ordLess_infinite[of "|A|" "|B|"]
-card_of_Well_order[of A] card_of_Well_order[of B]
-Field_card_of[of A] Field_card_of[of B] by auto
-
-
-text{* The next two results correspond to the ZF fact that all infinite cardinals are
-limit ordinals: *}
-
-lemma Card_order_infinite_not_under:
-assumes CARD: "Card_order r" and INF: "infinite (Field r)"
-shows "\<not> (\<exists>a. Field r = rel.under r a)"
-proof(auto)
- have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
- using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
- fix a assume *: "Field r = rel.under r a"
- show False
- proof(cases "a \<in> Field r")
- assume Case1: "a \<notin> Field r"
- hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
- thus False using INF * by auto
- next
- let ?r' = "Restr r (rel.underS r a)"
- assume Case2: "a \<in> Field r"
- hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
- using 0 rel.Refl_under_underS rel.underS_notIn by fastforce
- have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
- using 0 wo_rel.underS_ofilter * 1 Case2 by auto
- hence "?r' <o r" using 0 using ofilter_ordLess by blast
- moreover
- have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
- using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
- ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
- moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
- ultimately have "|rel.underS r a| <o |rel.under r a|"
- using ordIso_symmetric ordLess_ordIso_trans by blast
- moreover
- {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
- using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
- hence "|rel.under r a| =o |rel.underS r a|" using card_of_ordIso by blast
- }
- ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
- qed
-qed
-
-
-lemma infinite_Card_order_limit:
-assumes r: "Card_order r" and "infinite (Field r)"
-and a: "a : Field r"
-shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
-proof-
- have "Field r \<noteq> rel.under r a"
- using assms Card_order_infinite_not_under by blast
- moreover have "rel.under r a \<le> Field r"
- using rel.under_Field .
- ultimately have "rel.under r a < Field r" by blast
- then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
- unfolding rel.under_def by blast
- moreover have ba: "b \<noteq> a"
- using 1 r unfolding card_order_on_def well_order_on_def
- linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
- ultimately have "(a,b) : r"
- using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
- total_on_def by blast
- thus ?thesis using 1 ba by auto
-qed
-
-
-theorem Card_order_Times_same_infinite:
-assumes CO: "Card_order r" and INF: "infinite(Field r)"
-shows "|Field r \<times> Field r| \<le>o r"
-proof-
- obtain phi where phi_def:
- "phi = (\<lambda>r::'a rel. Card_order r \<and> infinite(Field r) \<and>
- \<not> |Field r \<times> Field r| \<le>o r )" by blast
- have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
- unfolding phi_def card_order_on_def by auto
- have Ft: "\<not>(\<exists>r. phi r)"
- proof
- assume "\<exists>r. phi r"
- hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
- using temp1 by auto
- then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
- 3: "Card_order r \<and> Well_order r"
- using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
- let ?A = "Field r" let ?r' = "bsqr r"
- have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
- using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
- have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
- using card_of_Card_order card_of_Well_order by blast
- (* *)
- have "r <o |?A \<times> ?A|"
- using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
- moreover have "|?A \<times> ?A| \<le>o ?r'"
- using card_of_least[of "?A \<times> ?A"] 4 by auto
- ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
- then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
- unfolding ordLess_def embedS_def[abs_def]
- by (auto simp add: Field_bsqr)
- let ?B = "f ` ?A"
- have "|?A| =o |?B|"
- using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
- hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
- (* *)
- have "wo_rel.ofilter ?r' ?B"
- using 6 embed_Field_ofilter 3 4 by blast
- hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
- using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
- hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
- using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
- have "\<not> (\<exists>a. Field r = rel.under r a)"
- using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
- then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
- using temp2 3 bsqr_ofilter[of r ?B] by blast
- hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
- hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
- let ?r1 = "Restr r A1"
- have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
- moreover
- {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
- hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
- }
- ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
- (* *)
- have "infinite (Field r)" using 1 unfolding phi_def by simp
- hence "infinite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
- hence "infinite A1" using 9 infinite_super finite_cartesian_product by blast
- moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
- using card_of_Card_order[of A1] card_of_Well_order[of A1]
- by (simp add: Field_card_of)
- moreover have "\<not> r \<le>o | A1 |"
- using temp4 11 3 using not_ordLeq_iff_ordLess by blast
- ultimately have "infinite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
- by (simp add: card_of_card_order_on)
- hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
- using 2 unfolding phi_def by blast
- hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
- hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
- thus False using 11 not_ordLess_ordLeq by auto
- qed
- thus ?thesis using assms unfolding phi_def by blast
-qed
-
-
-corollary card_of_Times_same_infinite:
-assumes "infinite A"
-shows "|A \<times> A| =o |A|"
-proof-
- let ?r = "|A|"
- have "Field ?r = A \<and> Card_order ?r"
- using Field_card_of card_of_Card_order[of A] by fastforce
- hence "|A \<times> A| \<le>o |A|"
- using Card_order_Times_same_infinite[of ?r] assms by auto
- thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
-qed
-
-
-lemma card_of_Times_infinite:
-assumes INF: "infinite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
-shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
-proof-
- have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
- using assms by (simp add: card_of_Times1 card_of_Times2)
- moreover
- {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
- using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
- moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
- ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
- using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
- }
- ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
-qed
-
-
-corollary card_of_Times_infinite_simps:
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
-by (auto simp add: card_of_Times_infinite ordIso_symmetric)
-
-
-corollary Card_order_Times_infinite:
-assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
- NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
-shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
-proof-
- have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
- using assms by (simp add: card_of_Times_infinite card_of_mono2)
- thus ?thesis
- using assms card_of_Field_ordIso[of r]
- ordIso_transitive[of "|Field r \<times> Field p|"]
- ordIso_transitive[of _ "|Field r|"] by blast
-qed
-
-
-lemma card_of_Sigma_ordLeq_infinite:
-assumes INF: "infinite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|SIGMA i : I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
- have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
- using LEQ card_of_Sigma_Times by blast
- moreover have "|I \<times> B| =o |B|"
- using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
- ultimately show ?thesis using ordLeq_ordIso_trans by blast
-qed
-
-
-lemma card_of_Sigma_ordLeq_infinite_Field:
-assumes INF: "infinite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|SIGMA i : I. A i| \<le>o r"
-proof-
- let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
- hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
- hence "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
- card_of_Sigma_ordLeq_infinite by blast
- thus ?thesis using 1 ordLeq_ordIso_trans by blast
-qed
-
-
-lemma card_of_Times_ordLeq_infinite_Field:
-"\<lbrakk>infinite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
- \<Longrightarrow> |A <*> B| \<le>o r"
-by(simp add: card_of_Sigma_ordLeq_infinite_Field)
-
-
-lemma card_of_UNION_ordLeq_infinite:
-assumes INF: "infinite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|\<Union> i \<in> I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
- have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
- using card_of_UNION_Sigma by blast
- moreover have "|SIGMA i : I. A i| \<le>o |B|"
- using assms card_of_Sigma_ordLeq_infinite by blast
- ultimately show ?thesis using ordLeq_transitive by blast
-qed
-
-
-corollary card_of_UNION_ordLeq_infinite_Field:
-assumes INF: "infinite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|\<Union> i \<in> I. A i| \<le>o r"
-proof-
- let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
- hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
- hence "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ
- card_of_UNION_ordLeq_infinite by blast
- thus ?thesis using 1 ordLeq_ordIso_trans by blast
-qed
-
-
-lemma card_of_Plus_infinite1:
-assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|A <+> B| =o |A|"
-proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
- let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
- assume *: "B \<noteq> {}"
- then obtain b1 where 1: "b1 \<in> B" by blast
- show ?thesis
- proof(cases "B = {b1}")
- assume Case1: "B = {b1}"
- have 2: "bij_betw ?Inl A ((?Inl ` A))"
- unfolding bij_betw_def inj_on_def by auto
- hence 3: "infinite (?Inl ` A)"
- using INF bij_betw_finite[of ?Inl A] by blast
- let ?A' = "?Inl ` A \<union> {?Inr b1}"
- obtain g where "bij_betw g (?Inl ` A) ?A'"
- using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
- moreover have "?A' = A <+> B" using Case1 by blast
- ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
- hence "bij_betw (g o ?Inl) A (A <+> B)"
- using 2 by (auto simp add: bij_betw_trans)
- thus ?thesis using card_of_ordIso ordIso_symmetric by blast
- next
- assume Case2: "B \<noteq> {b1}"
- with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
- obtain f where "inj_on f B \<and> f ` B \<le> A"
- using LEQ card_of_ordLeq[of B] by fastforce
- with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
- unfolding inj_on_def by auto
- with 3 have "|A <+> B| \<le>o |A \<times> B|"
- by (auto simp add: card_of_Plus_Times)
- moreover have "|A \<times> B| =o |A|"
- using assms * by (simp add: card_of_Times_infinite_simps)
- ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
- thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
- qed
-qed
-
-
-lemma card_of_Plus_infinite2:
-assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|B <+> A| =o |A|"
-using assms card_of_Plus_commute card_of_Plus_infinite1
-ordIso_equivalence by blast
-
-
-lemma card_of_Plus_infinite:
-assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
-using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
-
-
-corollary Card_order_Plus_infinite:
-assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
- LEQ: "p \<le>o r"
-shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
-proof-
- have "| Field r <+> Field p | =o | Field r | \<and>
- | Field p <+> Field r | =o | Field r |"
- using assms by (simp add: card_of_Plus_infinite card_of_mono2)
- thus ?thesis
- using assms card_of_Field_ordIso[of r]
- ordIso_transitive[of "|Field r <+> Field p|"]
- ordIso_transitive[of _ "|Field r|"] by blast
-qed
-
-
-lemma card_of_Un_infinite:
-assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
-proof-
- have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
- moreover have "|A <+> B| =o |A|"
- using assms by (metis card_of_Plus_infinite)
- ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
- hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
- thus ?thesis using Un_commute[of B A] by auto
-qed
-
-
-lemma card_of_Un_diff_infinite:
-assumes INF: "infinite A" and LESS: "|B| <o |A|"
-shows "|A - B| =o |A|"
-proof-
- obtain C where C_def: "C = A - B" by blast
- have "|A \<union> B| =o |A|"
- using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
- moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
- ultimately have 1: "|C \<union> B| =o |A|" by auto
- (* *)
- {assume *: "|C| \<le>o |B|"
- moreover
- {assume **: "finite B"
- hence "finite C"
- using card_of_ordLeq_finite * by blast
- hence False using ** INF card_of_ordIso_finite 1 by blast
- }
- hence "infinite B" by auto
- ultimately have False
- using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
- }
- hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
- {assume *: "finite C"
- hence "finite B" using card_of_ordLeq_finite 2 by blast
- hence False using * INF card_of_ordIso_finite 1 by blast
- }
- hence "infinite C" by auto
- hence "|C| =o |A|"
- using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
- thus ?thesis unfolding C_def .
-qed
-
-
-lemma card_of_Plus_ordLess_infinite:
-assumes INF: "infinite C" and
- LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
-shows "|A <+> B| <o |C|"
-proof(cases "A = {} \<or> B = {}")
- assume Case1: "A = {} \<or> B = {}"
- hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
- using card_of_Plus_empty1 card_of_Plus_empty2 by blast
- hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
- using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
- thus ?thesis using LESS1 LESS2
- ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
- ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
-next
- assume Case2: "\<not>(A = {} \<or> B = {})"
- {assume *: "|C| \<le>o |A <+> B|"
- hence "infinite (A <+> B)" using INF card_of_ordLeq_finite by blast
- hence 1: "infinite A \<or> infinite B" using finite_Plus by blast
- {assume Case21: "|A| \<le>o |B|"
- hence "infinite B" using 1 card_of_ordLeq_finite by blast
- hence "|A <+> B| =o |B|" using Case2 Case21
- by (auto simp add: card_of_Plus_infinite)
- hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
- }
- moreover
- {assume Case22: "|B| \<le>o |A|"
- hence "infinite A" using 1 card_of_ordLeq_finite by blast
- hence "|A <+> B| =o |A|" using Case2 Case22
- by (auto simp add: card_of_Plus_infinite)
- hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
- }
- ultimately have False using ordLeq_total card_of_Well_order[of A]
- card_of_Well_order[of B] by blast
- }
- thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
- card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
-qed
-
-
-lemma card_of_Plus_ordLess_infinite_Field:
-assumes INF: "infinite (Field r)" and r: "Card_order r" and
- LESS1: "|A| <o r" and LESS2: "|B| <o r"
-shows "|A <+> B| <o r"
-proof-
- let ?C = "Field r"
- have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
- hence "|A| <o |?C|" "|B| <o |?C|"
- using LESS1 LESS2 ordLess_ordIso_trans by blast+
- hence "|A <+> B| <o |?C|" using INF
- card_of_Plus_ordLess_infinite by blast
- thus ?thesis using 1 ordLess_ordIso_trans by blast
-qed
-
-
-lemma infinite_card_of_insert:
-assumes "infinite A"
-shows "|insert a A| =o |A|"
-proof-
- have iA: "insert a A = A \<union> {a}" by simp
- show ?thesis
- using infinite_imp_bij_betw2[OF assms] unfolding iA
- by (metis bij_betw_inv card_of_ordIso)
-qed
-
-
-subsection {* Cardinals versus lists *}
-
-
-text{* The next is an auxiliary operator, which shall be used for inductive
-proofs of facts concerning the cardinality of @{text "List"} : *}
-
-definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
-where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
-
-
-lemma lists_def2: "lists A = {l. set l \<le> A}"
-using in_listsI by blast
-
-
-lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
-unfolding lists_def2 nlists_def by blast
-
-
-lemma card_of_lists: "|A| \<le>o |lists A|"
-proof-
- let ?h = "\<lambda> a. [a]"
- have "inj_on ?h A \<and> ?h ` A \<le> lists A"
- unfolding inj_on_def lists_def2 by auto
- thus ?thesis by (metis card_of_ordLeq)
-qed
-
-
-lemma nlists_0: "nlists A 0 = {[]}"
-unfolding nlists_def by auto
-
-
-lemma nlists_not_empty:
-assumes "A \<noteq> {}"
-shows "nlists A n \<noteq> {}"
-proof(induct n, simp add: nlists_0)
- fix n assume "nlists A n \<noteq> {}"
- then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
- hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
- thus "nlists A (Suc n) \<noteq> {}" by auto
-qed
-
-
-lemma Nil_in_lists: "[] \<in> lists A"
-unfolding lists_def2 by auto
-
-
-lemma lists_not_empty: "lists A \<noteq> {}"
-using Nil_in_lists by blast
-
-
-lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
-proof-
- let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l"
- have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
- unfolding inj_on_def nlists_def by auto
- moreover have "nlists A (Suc n) \<le> ?h ` ?B"
- proof(auto)
- fix l assume "l \<in> nlists A (Suc n)"
- hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
- then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
- hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
- thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto
- qed
- ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
- unfolding bij_betw_def by auto
- thus ?thesis using card_of_ordIso ordIso_symmetric by blast
-qed
-
-
-lemma card_of_nlists_infinite:
-assumes "infinite A"
-shows "|nlists A n| \<le>o |A|"
-proof(induct n)
- have "A \<noteq> {}" using assms by auto
- thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
-next
- fix n assume IH: "|nlists A n| \<le>o |A|"
- have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
- using card_of_nlists_Succ by blast
- moreover
- {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
- hence "|A \<times> (nlists A n)| =o |A|"
- using assms IH by (auto simp add: card_of_Times_infinite)
- }
- ultimately show "|nlists A (Suc n)| \<le>o |A|"
- using ordIso_transitive ordIso_iff_ordLeq by blast
-qed
-
-
-lemma card_of_lists_infinite:
-assumes "infinite A"
-shows "|lists A| =o |A|"
-proof-
- have "|lists A| \<le>o |A|"
- using assms
- by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
- infinite_iff_card_of_nat card_of_nlists_infinite)
- thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
-qed
-
-
-lemma Card_order_lists_infinite:
-assumes "Card_order r" and "infinite(Field r)"
-shows "|lists(Field r)| =o r"
-using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
-
-
-
-subsection {* The cardinal $\omega$ and the finite cardinals *}
-
-
-text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
-order relation on
-@{text "nat"}, that we abbreviate by @{text "natLeq"}. The finite cardinals
-shall be the restrictions of these relations to the numbers smaller than
-fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
-
-abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
-abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
-
-abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
-where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
-
-lemma infinite_cartesian_product:
-assumes "infinite A" "infinite B"
-shows "infinite (A \<times> B)"
-proof
- assume "finite (A \<times> B)"
- from assms(1) have "A \<noteq> {}" by auto
- with `finite (A \<times> B)` have "finite B" using finite_cartesian_productD2 by auto
- with assms(2) show False by simp
-qed
-
-
-
-subsubsection {* First as well-orders *}
-
-
-lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
-by(unfold Field_def, auto)
-
-
-lemma natLeq_Refl: "Refl natLeq"
-unfolding refl_on_def Field_def by auto
-
-
-lemma natLeq_trans: "trans natLeq"
-unfolding trans_def by auto
-
-
-lemma natLeq_Preorder: "Preorder natLeq"
-unfolding preorder_on_def
-by (auto simp add: natLeq_Refl natLeq_trans)
-
-
-lemma natLeq_antisym: "antisym natLeq"
-unfolding antisym_def by auto
-
-
-lemma natLeq_Partial_order: "Partial_order natLeq"
-unfolding partial_order_on_def
-by (auto simp add: natLeq_Preorder natLeq_antisym)
-
-
-lemma natLeq_Total: "Total natLeq"
-unfolding total_on_def by auto
-
-
-lemma natLeq_Linear_order: "Linear_order natLeq"
-unfolding linear_order_on_def
-by (auto simp add: natLeq_Partial_order natLeq_Total)
-
-
-lemma natLeq_natLess_Id: "natLess = natLeq - Id"
-by auto
-
-
-lemma natLeq_Well_order: "Well_order natLeq"
-unfolding well_order_on_def
-using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
-
-
-corollary natLeq_well_order_on: "well_order_on UNIV natLeq"
-using natLeq_Well_order Field_natLeq by auto
-
-
-lemma natLeq_wo_rel: "wo_rel natLeq"
-unfolding wo_rel_def using natLeq_Well_order .
-
-
-lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
-using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
-
-
-lemma closed_nat_set_iff:
-assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
-shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
-proof-
- {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
- moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
- ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
- using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
- have "A = {0 ..< n}"
- proof(auto simp add: 1)
- fix m assume *: "m \<in> A"
- {assume "n \<le> m" with assms * have "n \<in> A" by blast
- hence False using 1 by auto
- }
- thus "m < n" by fastforce
- qed
- hence "\<exists>n. A = {0 ..< n}" by blast
- }
- thus ?thesis by blast
-qed
-
-
-lemma Field_natLeq_on: "Field (natLeq_on n) = {0 ..< n}"
-unfolding Field_def by auto
-
-
-lemma natLeq_underS_less: "rel.underS natLeq n = {0 ..< n}"
-unfolding rel.underS_def by auto
-
-
-lemma Restr_natLeq: "Restr natLeq {0 ..< n} = natLeq_on n"
-by auto
-
-
-lemma Restr_natLeq2:
-"Restr natLeq (rel.underS natLeq n) = natLeq_on n"
-by (auto simp add: Restr_natLeq natLeq_underS_less)
-
-
-lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
-using Restr_natLeq[of n] natLeq_Well_order
- Well_order_Restr[of natLeq "{0..<n}"] by auto
-
-
-corollary natLeq_on_well_order_on: "well_order_on {0 ..< n} (natLeq_on n)"
-using natLeq_on_Well_order Field_natLeq_on by auto
-
-
-lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
-unfolding wo_rel_def using natLeq_on_Well_order .
-
-
-lemma natLeq_on_ofilter_less_eq:
-"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
-by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
- simp add: Field_natLeq_on, unfold rel.under_def, auto)
-
-
-lemma natLeq_on_ofilter_iff:
-"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
-proof(rule iffI)
- assume *: "wo_rel.ofilter (natLeq_on m) A"
- hence 1: "A \<le> {0..<m}"
- by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def Field_natLeq_on)
- hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
- using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def)
- hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
- thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
-next
- assume "(\<exists>n\<le>m. A = {0 ..< n})"
- thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
-qed
-
-
-
-subsubsection {* Then as cardinals *}
-
-
-lemma natLeq_Card_order: "Card_order natLeq"
-proof(auto simp add: natLeq_Well_order
- Card_order_iff_Restr_underS Restr_natLeq2, simp add: Field_natLeq)
- fix n have "finite(Field (natLeq_on n))"
- unfolding Field_natLeq_on by auto
- moreover have "infinite(UNIV::nat set)" by auto
- ultimately show "natLeq_on n <o |UNIV::nat set|"
- using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
- Field_card_of[of "UNIV::nat set"]
- card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
-qed
-
-
-corollary card_of_Field_natLeq:
-"|Field natLeq| =o natLeq"
-using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
- ordIso_symmetric[of natLeq] by blast
-
-
-corollary card_of_nat:
-"|UNIV::nat set| =o natLeq"
-using Field_natLeq card_of_Field_natLeq by auto
-
-
-corollary infinite_iff_natLeq_ordLeq:
-"infinite A = ( natLeq \<le>o |A| )"
-using infinite_iff_card_of_nat[of A] card_of_nat
- ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
-
-
-corollary finite_iff_ordLess_natLeq:
-"finite A = ( |A| <o natLeq)"
-using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
- card_of_Well_order natLeq_Well_order by blast
-
-
-lemma ordIso_natLeq_on_imp_finite:
-"|A| =o natLeq_on n \<Longrightarrow> finite A"
-unfolding ordIso_def iso_def[abs_def]
-by (auto simp: Field_natLeq_on bij_betw_finite Field_card_of)
-
-
-lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
-proof(unfold card_order_on_def,
- auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
- fix r assume "well_order_on {0..<n} r"
- thus "natLeq_on n \<le>o r"
- using finite_atLeastLessThan natLeq_on_well_order_on
- finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
-qed
-
-
-corollary card_of_Field_natLeq_on:
-"|Field (natLeq_on n)| =o natLeq_on n"
-using Field_natLeq_on natLeq_on_Card_order
- Card_order_iff_ordIso_card_of[of "natLeq_on n"]
- ordIso_symmetric[of "natLeq_on n"] by blast
-
-
-corollary card_of_less:
-"|{0 ..< n}| =o natLeq_on n"
-using Field_natLeq_on card_of_Field_natLeq_on by auto
-
-
-lemma natLeq_on_ordLeq_less_eq:
-"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
-proof
- assume "natLeq_on m \<le>o natLeq_on n"
- then obtain f where "inj_on f {0..<m} \<and> f ` {0..<m} \<le> {0..<n}"
- unfolding ordLeq_def using
- embed_inj_on[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on]
- embed_Field[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on] by blast
- thus "m \<le> n" using atLeastLessThan_less_eq2 by blast
-next
- assume "m \<le> n"
- hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
- hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
- thus "natLeq_on m \<le>o natLeq_on n"
- using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
-qed
-
-
-lemma natLeq_on_ordLeq_less:
-"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
-using not_ordLeq_iff_ordLess[of "natLeq_on m" "natLeq_on n"]
-natLeq_on_Well_order natLeq_on_ordLeq_less_eq by auto
-
-
-
-subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
-
-
-lemma finite_card_of_iff_card2:
-assumes FIN: "finite A" and FIN': "finite B"
-shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
-using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
-
-
-lemma finite_imp_card_of_natLeq_on:
-assumes "finite A"
-shows "|A| =o natLeq_on (card A)"
-proof-
- obtain h where "bij_betw h A {0 ..< card A}"
- using assms ex_bij_betw_finite_nat by blast
- thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
-qed
-
-
-lemma finite_iff_card_of_natLeq_on:
-"finite A = (\<exists>n. |A| =o natLeq_on n)"
-using finite_imp_card_of_natLeq_on[of A]
-by(auto simp add: ordIso_natLeq_on_imp_finite)
-
-
-
-subsection {* The successor of a cardinal *}
-
-
-text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
-being a successor cardinal of @{text "r"}. Although the definition does
-not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
-
-
-definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
-where
-"isCardSuc r r' \<equiv>
- Card_order r' \<and> r <o r' \<and>
- (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
-
-
-text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
-by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
-Again, the picked item shall be proved unique up to order-isomorphism. *}
-
-
-definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
-where
-"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
-
-
-lemma exists_minim_Card_order:
-"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
-unfolding card_order_on_def using exists_minim_Well_order by blast
-
-
-lemma exists_isCardSuc:
-assumes "Card_order r"
-shows "\<exists>r'. isCardSuc r r'"
-proof-
- let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
- have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
- by (simp add: card_of_Card_order Card_order_Pow)
- then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
- using exists_minim_Card_order[of ?R] by blast
- thus ?thesis unfolding isCardSuc_def by auto
-qed
-
-
-lemma cardSuc_isCardSuc:
-assumes "Card_order r"
-shows "isCardSuc r (cardSuc r)"
-unfolding cardSuc_def using assms
-by (simp add: exists_isCardSuc someI_ex)
-
-
-lemma cardSuc_Card_order:
-"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
-
-
-lemma cardSuc_greater:
-"Card_order r \<Longrightarrow> r <o cardSuc r"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
-
-
-lemma cardSuc_ordLeq:
-"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
-using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
-
-
-text{* The minimality property of @{text "cardSuc"} originally present in its definition
-is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
-
-lemma cardSuc_least_aux:
-"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
-
-
-text{* But from this we can infer general minimality: *}
-
-lemma cardSuc_least:
-assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
-shows "cardSuc r \<le>o r'"
-proof-
- let ?p = "cardSuc r"
- have 0: "Well_order ?p \<and> Well_order r'"
- using assms cardSuc_Card_order unfolding card_order_on_def by blast
- {assume "r' <o ?p"
- then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
- using internalize_ordLess[of r' ?p] by blast
- (* *)
- have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
- moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
- ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
- hence False using 2 not_ordLess_ordLeq by blast
- }
- thus ?thesis using 0 ordLess_or_ordLeq by blast
-qed
-
-
-lemma cardSuc_ordLess_ordLeq:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(r <o r') = (cardSuc r \<le>o r')"
-proof(auto simp add: assms cardSuc_least)
- assume "cardSuc r \<le>o r'"
- thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
-qed
-
-
-lemma cardSuc_ordLeq_ordLess:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(r' <o cardSuc r) = (r' \<le>o r)"
-proof-
- have "Well_order r \<and> Well_order r'"
- using assms unfolding card_order_on_def by auto
- moreover have "Well_order(cardSuc r)"
- using assms cardSuc_Card_order card_order_on_def by blast
- ultimately show ?thesis
- using assms cardSuc_ordLess_ordLeq[of r r']
- not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
-qed
-
-
-lemma cardSuc_mono_ordLeq:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
-using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
-
-
-lemma cardSuc_invar_ordIso:
-assumes CARD: "Card_order r" and CARD': "Card_order r'"
-shows "(cardSuc r =o cardSuc r') = (r =o r')"
-proof-
- have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
- using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
- thus ?thesis
- using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
- using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
-qed
-
-
-lemma cardSuc_natLeq_on_Suc:
-"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
-proof-
- obtain r r' p where r_def: "r = natLeq_on n" and
- r'_def: "r' = cardSuc(natLeq_on n)" and
- p_def: "p = natLeq_on(Suc n)" by blast
- (* Preliminary facts: *)
- have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
- using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
- hence WELL: "Well_order r \<and> Well_order r' \<and> Well_order p"
- unfolding card_order_on_def by force
- have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
- unfolding r_def p_def Field_natLeq_on by simp
- hence FIN: "finite (Field r)" by force
- have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
- hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
- hence LESS: "|Field r| <o |Field r'|"
- using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
- (* Main proof: *)
- have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
- using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
- moreover have "p \<le>o r'"
- proof-
- {assume "r' <o p"
- then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
- let ?q = "Restr p (f ` Field r')"
- have 1: "embed r' p f" using 0 unfolding embedS_def by force
- hence 2: "f ` Field r' < {0..<(Suc n)}"
- using WELL FIELD 0 by (auto simp add: embedS_iff)
- have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
- then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
- unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
- hence 4: "m \<le> n" using 2 by force
- (* *)
- have "bij_betw f (Field r') (f ` (Field r'))"
- using 1 WELL embed_inj_on unfolding bij_betw_def by force
- moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
- ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
- using bij_betw_same_card bij_betw_finite by metis
- hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
- hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
- hence False using LESS not_ordLess_ordLeq by auto
- }
- thus ?thesis using WELL CARD by (fastforce simp: not_ordLess_iff_ordLeq)
- qed
- ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def by blast
-qed
-
-
-lemma card_of_cardSuc_finite:
-"finite(Field(cardSuc |A| )) = finite A"
-proof
- assume *: "finite (Field (cardSuc |A| ))"
- have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
- using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
- hence "|A| \<le>o |Field(cardSuc |A| )|"
- using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
- ordLeq_ordIso_trans by blast
- thus "finite A" using * card_of_ordLeq_finite by blast
-next
- assume "finite A"
- then obtain n where "|A| =o natLeq_on n" using finite_iff_card_of_natLeq_on by blast
- hence "cardSuc |A| =o cardSuc(natLeq_on n)"
- using card_of_Card_order cardSuc_invar_ordIso natLeq_on_Card_order by blast
- hence "cardSuc |A| =o natLeq_on(Suc n)"
- using cardSuc_natLeq_on_Suc ordIso_transitive by blast
- hence "cardSuc |A| =o |{0..<(Suc n)}|" using card_of_less ordIso_equivalence by blast
- moreover have "|Field (cardSuc |A| ) | =o cardSuc |A|"
- using card_of_Field_ordIso cardSuc_Card_order card_of_Card_order by blast
- ultimately have "|Field (cardSuc |A| ) | =o |{0..<(Suc n)}|"
- using ordIso_equivalence by blast
- thus "finite (Field (cardSuc |A| ))"
- using card_of_ordIso_finite finite_atLeastLessThan by blast
-qed
-
-
-lemma cardSuc_finite:
-assumes "Card_order r"
-shows "finite (Field (cardSuc r)) = finite (Field r)"
-proof-
- let ?A = "Field r"
- have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
- hence "cardSuc |?A| =o cardSuc r" using assms
- by (simp add: card_of_Card_order cardSuc_invar_ordIso)
- moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
- by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
- moreover
- {have "|Field (cardSuc r) | =o cardSuc r"
- using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
- hence "cardSuc r =o |Field (cardSuc r) |"
- using ordIso_symmetric by blast
- }
- ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
- using ordIso_transitive by blast
- hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
- using card_of_ordIso_finite by blast
- thus ?thesis by (simp only: card_of_cardSuc_finite)
-qed
-
-
-lemma card_of_Plus_ordLeq_infinite_Field:
-assumes r: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and c: "Card_order r"
-shows "|A <+> B| \<le>o r"
-proof-
- let ?r' = "cardSuc r"
- have "Card_order ?r' \<and> infinite (Field ?r')" using assms
- by (simp add: cardSuc_Card_order cardSuc_finite)
- moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
- by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
- ultimately have "|A <+> B| <o ?r'"
- using card_of_Plus_ordLess_infinite_Field by blast
- thus ?thesis using c r
- by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
-qed
-
-
-lemma card_of_Un_ordLeq_infinite_Field:
-assumes C: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and "Card_order r"
-shows "|A Un B| \<le>o r"
-using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
-ordLeq_transitive by blast
-
-
-
-subsection {* Regular cardinals *}
-
-
-definition cofinal where
-"cofinal A r \<equiv>
- ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
-
-
-definition regular where
-"regular r \<equiv>
- ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
-
-
-definition relChain where
-"relChain r As \<equiv>
- ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
-
-lemma regular_UNION:
-assumes r: "Card_order r" "regular r"
-and As: "relChain r As"
-and Bsub: "B \<le> (UN i : Field r. As i)"
-and cardB: "|B| <o r"
-shows "EX i : Field r. B \<le> As i"
-proof-
- let ?phi = "%b j. j : Field r \<and> b : As j"
- have "ALL b : B. EX j. ?phi b j" using Bsub by blast
- then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
- using bchoice[of B ?phi] by blast
- let ?K = "f ` B"
- {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"
- have 2: "cofinal ?K r"
- unfolding cofinal_def proof auto
- fix i assume i: "i : Field r"
- with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast
- hence "i \<noteq> f b \<and> ~ (f b,i) : r"
- using As f unfolding relChain_def by auto
- hence "i \<noteq> f b \<and> (i, f b) : r" using r
- unfolding card_order_on_def well_order_on_def linear_order_on_def
- total_on_def using i f b by auto
- with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
- qed
- moreover have "?K \<le> Field r" using f by blast
- ultimately have "|?K| =o r" using 2 r unfolding regular_def by blast
- moreover
- {
- have "|?K| <=o |B|" using card_of_image .
- hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
- }
- ultimately have False using not_ordLess_ordIso by blast
- }
- thus ?thesis by blast
-qed
-
-
-lemma infinite_cardSuc_regular:
-assumes r_inf: "infinite (Field r)" and r_card: "Card_order r"
-shows "regular (cardSuc r)"
-proof-
- let ?r' = "cardSuc r"
- have r': "Card_order ?r'"
- "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
- using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
- show ?thesis
- unfolding regular_def proof auto
- fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
- hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
- also have 22: "|Field ?r'| =o ?r'"
- using r' by (simp add: card_of_Field_ordIso[of ?r'])
- finally have "|K| \<le>o ?r'" .
- moreover
- {let ?L = "UN j : K. rel.underS ?r' j"
- let ?J = "Field r"
- have rJ: "r =o |?J|"
- using r_card card_of_Field_ordIso ordIso_symmetric by blast
- assume "|K| <o ?r'"
- hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
- hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
- moreover
- {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
- using r' 1 by (auto simp: card_of_underS)
- hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
- using r' card_of_Card_order by blast
- hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
- using rJ ordLeq_ordIso_trans by blast
- }
- ultimately have "|?L| \<le>o |?J|"
- using r_inf card_of_UNION_ordLeq_infinite by blast
- hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
- hence "|?L| <o ?r'" using r' card_of_Card_order by blast
- moreover
- {
- have "Field ?r' \<le> ?L"
- using 2 unfolding rel.underS_def cofinal_def by auto
- hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
- hence "?r' \<le>o |?L|"
- using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
- }
- ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
- hence False using ordLess_irreflexive by blast
- }
- ultimately show "|K| =o ?r'"
- unfolding ordLeq_iff_ordLess_or_ordIso by blast
- qed
-qed
-
-lemma cardSuc_UNION:
-assumes r: "Card_order r" and "infinite (Field r)"
-and As: "relChain (cardSuc r) As"
-and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
-and cardB: "|B| <=o r"
-shows "EX i : Field (cardSuc r). B \<le> As i"
-proof-
- let ?r' = "cardSuc r"
- have "Card_order ?r' \<and> |B| <o ?r'"
- using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
- card_of_Card_order by blast
- moreover have "regular ?r'"
- using assms by(simp add: infinite_cardSuc_regular)
- ultimately show ?thesis
- using As Bsub cardB regular_UNION by blast
-qed
-
-
-subsection {* Others *}
-
-lemma card_of_infinite_diff_finite:
-assumes "infinite A" and "finite B"
-shows "|A - B| =o |A|"
-by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
-
-(* function space *)
-definition Func where
-"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
-
-lemma Func_empty:
-"Func {} B = {\<lambda>x. undefined}"
-unfolding Func_def by auto
-
-lemma Func_elim:
-assumes "g \<in> Func A B" and "a \<in> A"
-shows "\<exists> b. b \<in> B \<and> g a = b"
-using assms unfolding Func_def by (cases "g a = undefined") auto
-
-definition curr where
-"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
-
-lemma curr_in:
-assumes f: "f \<in> Func (A <*> B) C"
-shows "curr A f \<in> Func A (Func B C)"
-using assms unfolding curr_def Func_def by auto
-
-lemma curr_inj:
-assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
-shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
-proof safe
- assume c: "curr A f1 = curr A f2"
- show "f1 = f2"
- proof (rule ext, clarify)
- fix a b show "f1 (a, b) = f2 (a, b)"
- proof (cases "(a,b) \<in> A <*> B")
- case False
- thus ?thesis using assms unfolding Func_def by auto
- next
- case True hence a: "a \<in> A" and b: "b \<in> B" by auto
- thus ?thesis
- using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
- qed
- qed
-qed
-
-lemma curr_surj:
-assumes "g \<in> Func A (Func B C)"
-shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
-proof
- let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
- show "curr A ?f = g"
- proof (rule ext)
- fix a show "curr A ?f a = g a"
- proof (cases "a \<in> A")
- case False
- hence "g a = undefined" using assms unfolding Func_def by auto
- thus ?thesis unfolding curr_def using False by simp
- next
- case True
- obtain g1 where "g1 \<in> Func B C" and "g a = g1"
- using assms using Func_elim[OF assms True] by blast
- thus ?thesis using True unfolding Func_def curr_def by auto
- qed
- qed
- show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
-qed
-
-lemma bij_betw_curr:
-"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
-unfolding bij_betw_def inj_on_def image_def
-using curr_in curr_inj curr_surj by blast
-
-lemma card_of_Func_Times:
-"|Func (A <*> B) C| =o |Func A (Func B C)|"
-unfolding card_of_ordIso[symmetric]
-using bij_betw_curr by blast
-
-definition Func_map where
-"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
-
-lemma Func_map:
-assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
-shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
-using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
-
-lemma Func_non_emp:
-assumes "B \<noteq> {}"
-shows "Func A B \<noteq> {}"
-proof-
- obtain b where b: "b \<in> B" using assms by auto
- hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
- thus ?thesis by blast
-qed
-
-lemma Func_is_emp:
-"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
-proof
- assume L: ?L
- moreover {assume "A = {}" hence False using L Func_empty by auto}
- moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
- ultimately show ?R by blast
-next
- assume R: ?R
- moreover
- {fix f assume "f \<in> Func A B"
- moreover obtain a where "a \<in> A" using R by blast
- ultimately obtain b where "b \<in> B" unfolding Func_def by(cases "f a = undefined", force+)
- with R have False by auto
- }
- thus ?L by blast
-qed
-
-lemma Func_map_surj:
-assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
-and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
-shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
-proof(cases "B2 = {}")
- case True
- thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
-next
- case False note B2 = False
- show ?thesis
- proof safe
- fix h assume h: "h \<in> Func B2 B1"
- def j1 \<equiv> "inv_into A1 f1"
- have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
- then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
- {fix b2 assume b2: "b2 \<in> B2"
- hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
- moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
- ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
- } note kk = this
- obtain b22 where b22: "b22 \<in> B2" using B2 by auto
- def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
- have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
- have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
- using kk unfolding j2_def by auto
- def g \<equiv> "Func_map A2 j1 j2 h"
- have "Func_map B2 f1 f2 g = h"
- proof (rule ext)
- fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
- proof(cases "b2 \<in> B2")
- case True
- show ?thesis
- proof (cases "h b2 = undefined")
- case True
- hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
- show ?thesis using A2 f_inv_into_f[OF b1]
- unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
- qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
- auto intro: f_inv_into_f)
- qed(insert h, unfold Func_def Func_map_def, auto)
- qed
- moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
- using inv_into_into j2A2 B1 A2 inv_into_into
- unfolding j1_def image_def by fast+
- ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
- unfolding Func_map_def[abs_def] unfolding image_def by auto
- qed(insert B1 Func_map[OF _ _ A2(2)], auto)
-qed
-
-lemma card_of_Pow_Func:
-"|Pow A| =o |Func A (UNIV::bool set)|"
-proof-
- def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
- else undefined"
- have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
- unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
- fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
- thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
- next
- show "F ` Pow A = Func A UNIV"
- proof safe
- fix f assume f: "f \<in> Func A (UNIV::bool set)"
- show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
- let ?A1 = "{a \<in> A. f a = True}"
- show "f = F ?A1" unfolding F_def apply(rule ext)
- using f unfolding Func_def mem_Collect_eq by auto
- qed auto
- qed(unfold Func_def mem_Collect_eq F_def, auto)
- qed
- thus ?thesis unfolding card_of_ordIso[symmetric] by blast
-qed
-
-lemma card_of_Func_mono:
-fixes A1 A2 :: "'a set" and B :: "'b set"
-assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
-shows "|Func A1 B| \<le>o |Func A2 B|"
-proof-
- obtain bb where bb: "bb \<in> B" using B by auto
- def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb)
- else undefined"
- show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
- show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
- fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
- show "f = g"
- proof(rule ext)
- fix a show "f a = g a"
- proof(cases "a \<in> A1")
- case True
- thus ?thesis using eq A12 unfolding F_def fun_eq_iff
- by (elim allE[of _ a]) auto
- qed(insert f g, unfold Func_def, fastforce)
- qed
- qed
- qed(insert bb, unfold Func_def F_def, force)
-qed
-
-lemma ordLeq_Func:
-assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
-shows "|A| \<le>o |Func A B|"
-unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
- let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
- show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
- show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
-qed
-
-lemma infinite_Func:
-assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
-shows "infinite (Func A B)"
-using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
-
-lemma card_of_Func_UNIV:
-"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
-apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
- let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
- show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
- unfolding bij_betw_def inj_on_def proof safe
- fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
- hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
- then obtain f where f: "\<forall> a. h a = f a" by metis
- hence "range f \<subseteq> B" using h unfolding Func_def by auto
- thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
- qed(unfold Func_def fun_eq_iff, auto)
-qed
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Cardinal_Order_Relation_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Cardinal_Order_Relation_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,2174 @@
+(* Title: HOL/Cardinals/Cardinal_Order_Relation_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Cardinal-order relations (FP).
+*)
+
+header {* Cardinal-Order Relations (FP) *}
+
+theory Cardinal_Order_Relation_FP
+imports Constructions_on_Wellorders_FP
+begin
+
+
+text{* In this section, we define cardinal-order relations to be minim well-orders
+on their field. Then we define the cardinal of a set to be {\em some} cardinal-order
+relation on that set, which will be unique up to order isomorphism. Then we study
+the connection between cardinals and:
+\begin{itemize}
+\item standard set-theoretic constructions: products,
+sums, unions, lists, powersets, set-of finite sets operator;
+\item finiteness and infiniteness (in particular, with the numeric cardinal operator
+for finite sets, @{text "card"}, from the theory @{text "Finite_Sets.thy"}).
+\end{itemize}
+%
+On the way, we define the canonical $\omega$ cardinal and finite cardinals. We also
+define (again, up to order isomorphism) the successor of a cardinal, and show that
+any cardinal admits a successor.
+
+Main results of this section are the existence of cardinal relations and the
+facts that, in the presence of infiniteness,
+most of the standard set-theoretic constructions (except for the powerset)
+{\em do not increase cardinality}. In particular, e.g., the set of words/lists over
+any infinite set has the same cardinality (hence, is in bijection) with that set.
+*}
+
+
+subsection {* Cardinal orders *}
+
+
+text{* A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the
+order-embedding relation, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
+strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
+
+definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
+
+
+abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
+abbreviation "card_order r \<equiv> card_order_on UNIV r"
+
+
+lemma card_order_on_well_order_on:
+assumes "card_order_on A r"
+shows "well_order_on A r"
+using assms unfolding card_order_on_def by simp
+
+
+lemma card_order_on_Card_order:
+"card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
+unfolding card_order_on_def using rel.well_order_on_Field by blast
+
+
+text{* The existence of a cardinal relation on any given set (which will mean
+that any set has a cardinal) follows from two facts:
+\begin{itemize}
+\item Zermelo's theorem (proved in @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
+which states that on any given set there exists a well-order;
+\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
+such well-order, i.e., a cardinal order.
+\end{itemize}
+*}
+
+
+theorem card_order_on: "\<exists>r. card_order_on A r"
+proof-
+ obtain R where R_def: "R = {r. well_order_on A r}" by blast
+ have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
+ using well_order_on[of A] R_def rel.well_order_on_Well_order by blast
+ hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+ using exists_minim_Well_order[of R] by auto
+ thus ?thesis using R_def unfolding card_order_on_def by auto
+qed
+
+
+lemma card_order_on_ordIso:
+assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
+shows "r =o r'"
+using assms unfolding card_order_on_def
+using ordIso_iff_ordLeq by blast
+
+
+lemma Card_order_ordIso:
+assumes CO: "Card_order r" and ISO: "r' =o r"
+shows "Card_order r'"
+using ISO unfolding ordIso_def
+proof(unfold card_order_on_def, auto)
+ fix p' assume "well_order_on (Field r') p'"
+ hence 0: "Well_order p' \<and> Field p' = Field r'"
+ using rel.well_order_on_Well_order by blast
+ obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
+ using ISO unfolding ordIso_def by auto
+ hence 3: "inj_on f (Field r') \<and> f ` (Field r') = Field r"
+ by (auto simp add: iso_iff embed_inj_on)
+ let ?p = "dir_image p' f"
+ have 4: "p' =o ?p \<and> Well_order ?p"
+ using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
+ moreover have "Field ?p = Field r"
+ using 0 3 by (auto simp add: dir_image_Field2 order_on_defs)
+ ultimately have "well_order_on (Field r) ?p" by auto
+ hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
+ thus "r' \<le>o p'"
+ using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
+qed
+
+
+lemma Card_order_ordIso2:
+assumes CO: "Card_order r" and ISO: "r =o r'"
+shows "Card_order r'"
+using assms Card_order_ordIso ordIso_symmetric by blast
+
+
+subsection {* Cardinal of a set *}
+
+
+text{* We define the cardinal of set to be {\em some} cardinal order on that set.
+We shall prove that this notion is unique up to order isomorphism, meaning
+that order isomorphism shall be the true identity of cardinals. *}
+
+
+definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )
+where "card_of A = (SOME r. card_order_on A r)"
+
+
+lemma card_of_card_order_on: "card_order_on A |A|"
+unfolding card_of_def by (auto simp add: card_order_on someI_ex)
+
+
+lemma card_of_well_order_on: "well_order_on A |A|"
+using card_of_card_order_on card_order_on_def by blast
+
+
+lemma Field_card_of: "Field |A| = A"
+using card_of_card_order_on[of A] unfolding card_order_on_def
+using rel.well_order_on_Field by blast
+
+
+lemma card_of_Card_order: "Card_order |A|"
+by (simp only: card_of_card_order_on Field_card_of)
+
+
+corollary ordIso_card_of_imp_Card_order:
+"r =o |A| \<Longrightarrow> Card_order r"
+using card_of_Card_order Card_order_ordIso by blast
+
+
+lemma card_of_Well_order: "Well_order |A|"
+using card_of_Card_order unfolding card_order_on_def by auto
+
+
+lemma card_of_refl: "|A| =o |A|"
+using card_of_Well_order ordIso_reflexive by blast
+
+
+lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"
+using card_of_card_order_on unfolding card_order_on_def by blast
+
+
+lemma card_of_ordIso:
+"(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
+proof(auto)
+ fix f assume *: "bij_betw f A B"
+ then obtain r where "well_order_on B r \<and> |A| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|B| \<le>o |A|" using card_of_least
+ ordLeq_ordIso_trans ordIso_symmetric by blast
+ moreover
+ {let ?g = "inv_into A f"
+ have "bij_betw ?g B A" using * bij_betw_inv_into by blast
+ then obtain r where "well_order_on A r \<and> |B| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|A| \<le>o |B|" using card_of_least
+ ordLeq_ordIso_trans ordIso_symmetric by blast
+ }
+ ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
+next
+ assume "|A| =o |B|"
+ then obtain f where "iso ( |A| ) ( |B| ) f"
+ unfolding ordIso_def by auto
+ hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
+ thus "\<exists>f. bij_betw f A B" by auto
+qed
+
+
+lemma card_of_ordLeq:
+"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
+proof(auto)
+ fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
+ {assume "|B| <o |A|"
+ hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
+ then obtain g where "embed ( |B| ) ( |A| ) g"
+ unfolding ordLeq_def by auto
+ hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
+ card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
+ embed_Field[of "|B|" "|A|" g] by auto
+ obtain h where "bij_betw h A B"
+ using * ** 1 Cantor_Bernstein[of f] by fastforce
+ hence "|A| =o |B|" using card_of_ordIso by blast
+ hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
+ }
+ thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
+ by (auto simp: card_of_Well_order)
+next
+ assume *: "|A| \<le>o |B|"
+ obtain f where "embed ( |A| ) ( |B| ) f"
+ using * unfolding ordLeq_def by auto
+ hence "inj_on f A \<and> f ` A \<le> B" using embed_inj_on[of "|A|" "|B|" f]
+ card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
+ embed_Field[of "|A|" "|B|" f] by auto
+ thus "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
+qed
+
+
+lemma card_of_ordLeq2:
+"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
+using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
+
+
+lemma card_of_ordLess:
+"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
+proof-
+ have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
+ using card_of_ordLeq by blast
+ also have "\<dots> = ( |B| <o |A| )"
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ not_ordLeq_iff_ordLess by blast
+ finally show ?thesis .
+qed
+
+
+lemma card_of_ordLess2:
+"B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
+using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
+
+
+lemma card_of_ordIsoI:
+assumes "bij_betw f A B"
+shows "|A| =o |B|"
+using assms unfolding card_of_ordIso[symmetric] by auto
+
+
+lemma card_of_ordLeqI:
+assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
+shows "|A| \<le>o |B|"
+using assms unfolding card_of_ordLeq[symmetric] by auto
+
+
+lemma card_of_unique:
+"card_order_on A r \<Longrightarrow> r =o |A|"
+by (simp only: card_order_on_ordIso card_of_card_order_on)
+
+
+lemma card_of_mono1:
+"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
+using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
+
+
+lemma card_of_mono2:
+assumes "r \<le>o r'"
+shows "|Field r| \<le>o |Field r'|"
+proof-
+ obtain f where
+ 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
+ using assms unfolding ordLeq_def
+ by (auto simp add: rel.well_order_on_Well_order)
+ hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
+ by (auto simp add: embed_inj_on embed_Field)
+ thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
+by (simp add: ordIso_iff_ordLeq card_of_mono2)
+
+
+lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
+using card_of_least card_of_well_order_on rel.well_order_on_Well_order by blast
+
+
+lemma card_of_Field_ordIso:
+assumes "Card_order r"
+shows "|Field r| =o r"
+proof-
+ have "card_order_on (Field r) r"
+ using assms card_order_on_Card_order by blast
+ moreover have "card_order_on (Field r) |Field r|"
+ using card_of_card_order_on by blast
+ ultimately show ?thesis using card_order_on_ordIso by blast
+qed
+
+
+lemma Card_order_iff_ordIso_card_of:
+"Card_order r = (r =o |Field r| )"
+using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
+
+
+lemma Card_order_iff_ordLeq_card_of:
+"Card_order r = (r \<le>o |Field r| )"
+proof-
+ have "Card_order r = (r =o |Field r| )"
+ unfolding Card_order_iff_ordIso_card_of by simp
+ also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
+ unfolding ordIso_iff_ordLeq by simp
+ also have "... = (r \<le>o |Field r| )"
+ using card_of_Field_ordLess
+ by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
+ finally show ?thesis .
+qed
+
+
+lemma Card_order_iff_Restr_underS:
+assumes "Well_order r"
+shows "Card_order r = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o |Field r| )"
+using assms unfolding Card_order_iff_ordLeq_card_of
+using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
+
+
+lemma card_of_underS:
+assumes r: "Card_order r" and a: "a : Field r"
+shows "|rel.underS r a| <o r"
+proof-
+ let ?A = "rel.underS r a" let ?r' = "Restr r ?A"
+ have 1: "Well_order r"
+ using r unfolding card_order_on_def by simp
+ have "Well_order ?r'" using 1 Well_order_Restr by auto
+ moreover have "card_order_on (Field ?r') |Field ?r'|"
+ using card_of_card_order_on .
+ ultimately have "|Field ?r'| \<le>o ?r'"
+ unfolding card_order_on_def by simp
+ moreover have "Field ?r' = ?A"
+ using 1 wo_rel.underS_ofilter Field_Restr_ofilter
+ unfolding wo_rel_def by fastforce
+ ultimately have "|?A| \<le>o ?r'" by simp
+ also have "?r' <o |Field r|"
+ using 1 a r Card_order_iff_Restr_underS by blast
+ also have "|Field r| =o r"
+ using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
+ finally show ?thesis .
+qed
+
+
+lemma ordLess_Field:
+assumes "r <o r'"
+shows "|Field r| <o r'"
+proof-
+ have "well_order_on (Field r) r" using assms unfolding ordLess_def
+ by (auto simp add: rel.well_order_on_Well_order)
+ hence "|Field r| \<le>o r" using card_of_least by blast
+ thus ?thesis using assms ordLeq_ordLess_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq:
+"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
+proof
+ assume "|A| \<le>o r"
+ then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
+ using internalize_ordLeq[of "|A|" r] by blast
+ hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
+ hence "|Field p| =o p" using card_of_Field_ordIso by blast
+ hence "|A| =o |Field p| \<and> |Field p| \<le>o r"
+ using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
+ thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
+next
+ assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
+ thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq2:
+"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
+using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
+
+
+
+subsection {* Cardinals versus set operations on arbitrary sets *}
+
+
+text{* Here we embark in a long journey of simple results showing
+that the standard set-theoretic operations are well-behaved w.r.t. the notion of
+cardinal -- essentially, this means that they preserve the ``cardinal identity"
+@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
+*}
+
+
+lemma card_of_empty: "|{}| \<le>o |A|"
+using card_of_ordLeq inj_on_id by blast
+
+
+lemma card_of_empty1:
+assumes "Well_order r \<or> Card_order r"
+shows "|{}| \<le>o r"
+proof-
+ have "Well_order r" using assms unfolding card_order_on_def by auto
+ hence "|Field r| <=o r"
+ using assms card_of_Field_ordLess by blast
+ moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
+ ultimately show ?thesis using ordLeq_transitive by blast
+qed
+
+
+corollary Card_order_empty:
+"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
+
+
+lemma card_of_empty2:
+assumes LEQ: "|A| =o |{}|"
+shows "A = {}"
+using assms card_of_ordIso[of A] bij_betw_empty2 by blast
+
+
+lemma card_of_empty3:
+assumes LEQ: "|A| \<le>o |{}|"
+shows "A = {}"
+using assms
+by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
+ ordLeq_Well_order_simp)
+
+
+lemma card_of_empty_ordIso:
+"|{}::'a set| =o |{}::'b set|"
+using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
+
+
+lemma card_of_image:
+"|f ` A| <=o |A|"
+proof(cases "A = {}", simp add: card_of_empty)
+ assume "A ~= {}"
+ hence "f ` A ~= {}" by auto
+ thus "|f ` A| \<le>o |A|"
+ using card_of_ordLeq2[of "f ` A" A] by auto
+qed
+
+
+lemma surj_imp_ordLeq:
+assumes "B <= f ` A"
+shows "|B| <=o |A|"
+proof-
+ have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
+ thus ?thesis using card_of_image ordLeq_transitive by blast
+qed
+
+
+lemma card_of_ordLeqI2:
+assumes "B \<subseteq> f ` A"
+shows "|B| \<le>o |A|"
+using assms by (metis surj_imp_ordLeq)
+
+
+lemma card_of_singl_ordLeq:
+assumes "A \<noteq> {}"
+shows "|{b}| \<le>o |A|"
+proof-
+ obtain a where *: "a \<in> A" using assms by auto
+ let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
+ have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
+ using * unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by fast
+qed
+
+
+corollary Card_order_singl_ordLeq:
+"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
+using card_of_singl_ordLeq[of "Field r" b]
+ card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
+
+
+lemma card_of_Pow: "|A| <o |Pow A|"
+using card_of_ordLess2[of "Pow A" A] Cantors_paradox[of A]
+ Pow_not_empty[of A] by auto
+
+
+corollary Card_order_Pow:
+"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
+using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
+
+
+lemma infinite_Pow:
+assumes "infinite A"
+shows "infinite (Pow A)"
+proof-
+ have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
+ thus ?thesis by (metis assms finite_Pow_iff)
+qed
+
+
+lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
+proof-
+ have "Inl ` A \<le> A <+> B" by auto
+ thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus1:
+"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
+using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
+proof-
+ have "Inr ` B \<le> A <+> B" by auto
+ thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus2:
+"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
+using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
+proof-
+ have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by auto
+qed
+
+
+lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
+proof-
+ have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by auto
+qed
+
+
+lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
+proof-
+ let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
+ | Inr b \<Rightarrow> Inl b"
+ have "bij_betw ?f (A <+> B) (B <+> A)"
+ unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_assoc:
+fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
+shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
+proof -
+ def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
+ case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
+ |Inr b \<Rightarrow> Inr (Inl b))
+ |Inr c \<Rightarrow> Inr (Inr c)"
+ have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
+ proof
+ fix x assume x: "x \<in> A <+> B <+> C"
+ show "x \<in> f ` ((A <+> B) <+> C)"
+ proof(cases x)
+ case (Inl a)
+ hence "a \<in> A" "x = f (Inl (Inl a))"
+ using x unfolding f_def by auto
+ thus ?thesis by auto
+ next
+ case (Inr bc) note 1 = Inr show ?thesis
+ proof(cases bc)
+ case (Inl b)
+ hence "b \<in> B" "x = f (Inl (Inr b))"
+ using x 1 unfolding f_def by auto
+ thus ?thesis by auto
+ next
+ case (Inr c)
+ hence "c \<in> C" "x = f (Inr c)"
+ using x 1 unfolding f_def by auto
+ thus ?thesis by auto
+ qed
+ qed
+ qed
+ hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
+ unfolding bij_betw_def inj_on_def f_def by fastforce
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_mono1:
+assumes "|A| \<le>o |B|"
+shows "|A <+> C| \<le>o |B <+> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
+ have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
+ proof-
+ {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
+ "g d1 = g d2"
+ hence "d1 = d2" using 1 unfolding inj_on_def g_def by force
+ }
+ moreover
+ {fix d assume "d \<in> A <+> C"
+ hence "g d \<in> B <+> C" using 1
+ by(case_tac d, auto simp add: g_def)
+ }
+ ultimately show ?thesis unfolding inj_on_def by auto
+ qed
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary ordLeq_Plus_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
+using assms card_of_mono2 card_of_Plus_mono1 by blast
+
+
+lemma card_of_Plus_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C <+> A| \<le>o |C <+> B|"
+using assms card_of_Plus_mono1[of A B C]
+ card_of_Plus_commute[of C A] card_of_Plus_commute[of B C]
+ ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
+by blast
+
+
+corollary ordLeq_Plus_mono2:
+assumes "r \<le>o r'"
+shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
+using assms card_of_mono2 card_of_Plus_mono2 by blast
+
+
+lemma card_of_Plus_mono:
+assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+shows "|A <+> C| \<le>o |B <+> D|"
+using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
+ ordLeq_transitive[of "|A <+> C|"] by blast
+
+
+corollary ordLeq_Plus_mono:
+assumes "r \<le>o r'" and "p \<le>o p'"
+shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
+using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
+
+
+lemma card_of_Plus_cong1:
+assumes "|A| =o |B|"
+shows "|A <+> C| =o |B <+> C|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
+
+
+corollary ordIso_Plus_cong1:
+assumes "r =o r'"
+shows "|(Field r) <+> C| =o |(Field r') <+> C|"
+using assms card_of_cong card_of_Plus_cong1 by blast
+
+
+lemma card_of_Plus_cong2:
+assumes "|A| =o |B|"
+shows "|C <+> A| =o |C <+> B|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
+
+
+corollary ordIso_Plus_cong2:
+assumes "r =o r'"
+shows "|A <+> (Field r)| =o |A <+> (Field r')|"
+using assms card_of_cong card_of_Plus_cong2 by blast
+
+
+lemma card_of_Plus_cong:
+assumes "|A| =o |B|" and "|C| =o |D|"
+shows "|A <+> C| =o |B <+> D|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
+
+
+corollary ordIso_Plus_cong:
+assumes "r =o r'" and "p =o p'"
+shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
+using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
+
+
+lemma card_of_Un_Plus_ordLeq:
+"|A \<union> B| \<le>o |A <+> B|"
+proof-
+ let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+ have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
+ unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_Times1:
+assumes "A \<noteq> {}"
+shows "|B| \<le>o |B \<times> A|"
+proof(cases "B = {}", simp add: card_of_empty)
+ assume *: "B \<noteq> {}"
+ have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
+ thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
+ card_of_ordLeq[of B "B \<times> A"] * by blast
+qed
+
+
+lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
+proof-
+ let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
+ have "bij_betw ?f (A \<times> B) (B \<times> A)"
+ unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Times2:
+assumes "A \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
+using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
+ ordLeq_ordIso_trans by blast
+
+
+corollary Card_order_Times1:
+"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
+using card_of_Times1[of B] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+corollary Card_order_Times2:
+"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
+using card_of_Times2[of A] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
+using card_of_Times1[of A]
+by(cases "A = {}", simp add: card_of_empty, blast)
+
+
+lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
+proof-
+ let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
+ |Inr a \<Rightarrow> (a,False)"
+ have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
+ proof-
+ {fix c1 and c2 assume "?f c1 = ?f c2"
+ hence "c1 = c2"
+ by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
+ }
+ moreover
+ {fix c assume "c \<in> A <+> A"
+ hence "?f c \<in> A \<times> (UNIV::bool set)"
+ by(case_tac c, auto)
+ }
+ moreover
+ {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
+ have "(a,bl) \<in> ?f ` ( A <+> A)"
+ proof(cases bl)
+ assume bl hence "?f(Inl a) = (a,bl)" by auto
+ thus ?thesis using * by force
+ next
+ assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
+ thus ?thesis using * by force
+ qed
+ }
+ ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Times_mono1:
+assumes "|A| \<le>o |B|"
+shows "|A \<times> C| \<le>o |B \<times> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>(a,c::'c). (f a,c))" by blast
+ have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
+ using 1 unfolding inj_on_def using g_def by auto
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary ordLeq_Times_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
+using assms card_of_mono2 card_of_Times_mono1 by blast
+
+
+lemma card_of_Times_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C \<times> A| \<le>o |C \<times> B|"
+using assms card_of_Times_mono1[of A B C]
+ card_of_Times_commute[of C A] card_of_Times_commute[of B C]
+ ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
+by blast
+
+
+corollary ordLeq_Times_mono2:
+assumes "r \<le>o r'"
+shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
+using assms card_of_mono2 card_of_Times_mono2 by blast
+
+
+lemma card_of_Sigma_mono1:
+assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
+shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
+proof-
+ have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
+ using assms by (auto simp add: card_of_ordLeq)
+ with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
+ obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by metis
+ obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
+ have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
+ using 1 unfolding inj_on_def using g_def by force
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary card_of_Sigma_Times:
+"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
+using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
+
+
+lemma card_of_UNION_Sigma:
+"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
+
+
+lemma card_of_bool:
+assumes "a1 \<noteq> a2"
+shows "|UNIV::bool set| =o |{a1,a2}|"
+proof-
+ let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
+ have "bij_betw ?f UNIV {a1,a2}"
+ proof-
+ {fix bl1 and bl2 assume "?f bl1 = ?f bl2"
+ hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
+ }
+ moreover
+ {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)
+ }
+ moreover
+ {fix a assume *: "a \<in> {a1,a2}"
+ have "a \<in> ?f ` UNIV"
+ proof(cases "a = a1")
+ assume "a = a1"
+ hence "?f True = a" by auto thus ?thesis by blast
+ next
+ assume "a \<noteq> a1" hence "a = a2" using * by auto
+ hence "?f False = a" by auto thus ?thesis by blast
+ qed
+ }
+ ultimately show ?thesis unfolding bij_betw_def inj_on_def
+ by (metis image_subsetI order_eq_iff subsetI)
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_Times_aux:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ LEQ: "|A| \<le>o |B|"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ have 1: "|UNIV::bool set| \<le>o |A|"
+ using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
+ ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
+ (* *)
+ have "|A <+> B| \<le>o |B <+> B|"
+ using LEQ card_of_Plus_mono1 by blast
+ moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
+ using card_of_Plus_Times_bool by blast
+ moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
+ using 1 by (simp add: card_of_Times_mono2)
+ moreover have " |B \<times> A| =o |A \<times> B|"
+ using card_of_Times_commute by blast
+ ultimately show "|A <+> B| \<le>o |A \<times> B|"
+ using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
+ ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
+ ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
+ by blast
+qed
+
+
+lemma card_of_Plus_Times:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ {assume "|A| \<le>o |B|"
+ hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
+ }
+ moreover
+ {assume "|B| \<le>o |A|"
+ hence "|B <+> A| \<le>o |B \<times> A|"
+ using assms by (auto simp add: card_of_Plus_Times_aux)
+ hence ?thesis
+ using card_of_Plus_commute card_of_Times_commute
+ ordIso_ordLeq_trans ordLeq_ordIso_trans by metis
+ }
+ ultimately show ?thesis
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ ordLeq_total[of "|A|"] by metis
+qed
+
+
+lemma card_of_ordLeq_finite:
+assumes "|A| \<le>o |B|" and "finite B"
+shows "finite A"
+using assms unfolding ordLeq_def
+using embed_inj_on[of "|A|" "|B|"] embed_Field[of "|A|" "|B|"]
+ Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
+
+
+lemma card_of_ordLeq_infinite:
+assumes "|A| \<le>o |B|" and "infinite A"
+shows "infinite B"
+using assms card_of_ordLeq_finite by auto
+
+
+lemma card_of_ordIso_finite:
+assumes "|A| =o |B|"
+shows "finite A = finite B"
+using assms unfolding ordIso_def iso_def[abs_def]
+by (auto simp: bij_betw_finite Field_card_of)
+
+
+lemma card_of_ordIso_finite_Field:
+assumes "Card_order r" and "r =o |A|"
+shows "finite(Field r) = finite A"
+using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
+
+
+subsection {* Cardinals versus set operations involving infinite sets *}
+
+
+text{* Here we show that, for infinite sets, most set-theoretic constructions
+do not increase the cardinality. The cornerstone for this is
+theorem @{text "Card_order_Times_same_infinite"}, which states that self-product
+does not increase cardinality -- the proof of this fact adapts a standard
+set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
+at page 47 in \cite{card-book}. Then everything else follows fairly easily. *}
+
+
+lemma infinite_iff_card_of_nat:
+"infinite A = ( |UNIV::nat set| \<le>o |A| )"
+by (auto simp add: infinite_iff_countable_subset card_of_ordLeq)
+
+
+text{* The next two results correspond to the ZF fact that all infinite cardinals are
+limit ordinals: *}
+
+lemma Card_order_infinite_not_under:
+assumes CARD: "Card_order r" and INF: "infinite (Field r)"
+shows "\<not> (\<exists>a. Field r = rel.under r a)"
+proof(auto)
+ have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
+ using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
+ fix a assume *: "Field r = rel.under r a"
+ show False
+ proof(cases "a \<in> Field r")
+ assume Case1: "a \<notin> Field r"
+ hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
+ thus False using INF * by auto
+ next
+ let ?r' = "Restr r (rel.underS r a)"
+ assume Case2: "a \<in> Field r"
+ hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
+ using 0 rel.Refl_under_underS rel.underS_notIn by metis
+ have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
+ using 0 wo_rel.underS_ofilter * 1 Case2 by fast
+ hence "?r' <o r" using 0 using ofilter_ordLess by blast
+ moreover
+ have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
+ using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
+ ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
+ moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
+ ultimately have "|rel.underS r a| <o |rel.under r a|"
+ using ordIso_symmetric ordLess_ordIso_trans by blast
+ moreover
+ {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
+ using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
+ hence "|rel.under r a| =o |rel.underS r a|" using card_of_ordIso by blast
+ }
+ ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
+ qed
+qed
+
+
+lemma infinite_Card_order_limit:
+assumes r: "Card_order r" and "infinite (Field r)"
+and a: "a : Field r"
+shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
+proof-
+ have "Field r \<noteq> rel.under r a"
+ using assms Card_order_infinite_not_under by blast
+ moreover have "rel.under r a \<le> Field r"
+ using rel.under_Field .
+ ultimately have "rel.under r a < Field r" by blast
+ then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
+ unfolding rel.under_def by blast
+ moreover have ba: "b \<noteq> a"
+ using 1 r unfolding card_order_on_def well_order_on_def
+ linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
+ ultimately have "(a,b) : r"
+ using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def by blast
+ thus ?thesis using 1 ba by auto
+qed
+
+
+theorem Card_order_Times_same_infinite:
+assumes CO: "Card_order r" and INF: "infinite(Field r)"
+shows "|Field r \<times> Field r| \<le>o r"
+proof-
+ obtain phi where phi_def:
+ "phi = (\<lambda>r::'a rel. Card_order r \<and> infinite(Field r) \<and>
+ \<not> |Field r \<times> Field r| \<le>o r )" by blast
+ have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
+ unfolding phi_def card_order_on_def by auto
+ have Ft: "\<not>(\<exists>r. phi r)"
+ proof
+ assume "\<exists>r. phi r"
+ hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
+ using temp1 by auto
+ then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
+ 3: "Card_order r \<and> Well_order r"
+ using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
+ let ?A = "Field r" let ?r' = "bsqr r"
+ have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
+ using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
+ have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
+ using card_of_Card_order card_of_Well_order by blast
+ (* *)
+ have "r <o |?A \<times> ?A|"
+ using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
+ moreover have "|?A \<times> ?A| \<le>o ?r'"
+ using card_of_least[of "?A \<times> ?A"] 4 by auto
+ ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
+ then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
+ unfolding ordLess_def embedS_def[abs_def]
+ by (auto simp add: Field_bsqr)
+ let ?B = "f ` ?A"
+ have "|?A| =o |?B|"
+ using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
+ hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
+ (* *)
+ have "wo_rel.ofilter ?r' ?B"
+ using 6 embed_Field_ofilter 3 4 by blast
+ hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
+ using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
+ hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
+ using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
+ have "\<not> (\<exists>a. Field r = rel.under r a)"
+ using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
+ then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
+ using temp2 3 bsqr_ofilter[of r ?B] by blast
+ hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
+ hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
+ let ?r1 = "Restr r A1"
+ have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
+ moreover
+ {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
+ hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
+ }
+ ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
+ (* *)
+ have "infinite (Field r)" using 1 unfolding phi_def by simp
+ hence "infinite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
+ hence "infinite A1" using 9 infinite_super finite_cartesian_product by blast
+ moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
+ using card_of_Card_order[of A1] card_of_Well_order[of A1]
+ by (simp add: Field_card_of)
+ moreover have "\<not> r \<le>o | A1 |"
+ using temp4 11 3 using not_ordLeq_iff_ordLess by blast
+ ultimately have "infinite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
+ by (simp add: card_of_card_order_on)
+ hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
+ using 2 unfolding phi_def by blast
+ hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
+ hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
+ thus False using 11 not_ordLess_ordLeq by auto
+ qed
+ thus ?thesis using assms unfolding phi_def by blast
+qed
+
+
+corollary card_of_Times_same_infinite:
+assumes "infinite A"
+shows "|A \<times> A| =o |A|"
+proof-
+ let ?r = "|A|"
+ have "Field ?r = A \<and> Card_order ?r"
+ using Field_card_of card_of_Card_order[of A] by fastforce
+ hence "|A \<times> A| \<le>o |A|"
+ using Card_order_Times_same_infinite[of ?r] assms by auto
+ thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
+qed
+
+
+lemma card_of_Times_infinite:
+assumes INF: "infinite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
+shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
+proof-
+ have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
+ using assms by (simp add: card_of_Times1 card_of_Times2)
+ moreover
+ {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
+ using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
+ moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
+ ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
+ using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
+ }
+ ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
+qed
+
+
+corollary Card_order_Times_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
+shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
+proof-
+ have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
+ using assms by (simp add: card_of_Times_infinite card_of_mono2)
+ thus ?thesis
+ using assms card_of_Field_ordIso[of r]
+ ordIso_transitive[of "|Field r \<times> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+
+lemma card_of_Sigma_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|SIGMA i : I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
+ using LEQ card_of_Sigma_Times by blast
+ moreover have "|I \<times> B| =o |B|"
+ using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
+ ultimately show ?thesis using ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Sigma_ordLeq_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|SIGMA i : I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
+ card_of_Sigma_ordLeq_infinite by blast
+ thus ?thesis using 1 ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Times_ordLeq_infinite_Field:
+"\<lbrakk>infinite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
+ \<Longrightarrow> |A <*> B| \<le>o r"
+by(simp add: card_of_Sigma_ordLeq_infinite_Field)
+
+
+lemma card_of_Times_infinite_simps:
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
+by (auto simp add: card_of_Times_infinite ordIso_symmetric)
+
+
+lemma card_of_UNION_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|\<Union> i \<in> I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+ using card_of_UNION_Sigma by blast
+ moreover have "|SIGMA i : I. A i| \<le>o |B|"
+ using assms card_of_Sigma_ordLeq_infinite by blast
+ ultimately show ?thesis using ordLeq_transitive by blast
+qed
+
+
+corollary card_of_UNION_ordLeq_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|\<Union> i \<in> I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ
+ card_of_UNION_ordLeq_infinite by blast
+ thus ?thesis using 1 ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Plus_infinite1:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A <+> B| =o |A|"
+proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
+ let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
+ assume *: "B \<noteq> {}"
+ then obtain b1 where 1: "b1 \<in> B" by blast
+ show ?thesis
+ proof(cases "B = {b1}")
+ assume Case1: "B = {b1}"
+ have 2: "bij_betw ?Inl A ((?Inl ` A))"
+ unfolding bij_betw_def inj_on_def by auto
+ hence 3: "infinite (?Inl ` A)"
+ using INF bij_betw_finite[of ?Inl A] by blast
+ let ?A' = "?Inl ` A \<union> {?Inr b1}"
+ obtain g where "bij_betw g (?Inl ` A) ?A'"
+ using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
+ moreover have "?A' = A <+> B" using Case1 by blast
+ ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
+ hence "bij_betw (g o ?Inl) A (A <+> B)"
+ using 2 by (auto simp add: bij_betw_trans)
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+ next
+ assume Case2: "B \<noteq> {b1}"
+ with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
+ obtain f where "inj_on f B \<and> f ` B \<le> A"
+ using LEQ card_of_ordLeq[of B] by fastforce
+ with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
+ unfolding inj_on_def by auto
+ with 3 have "|A <+> B| \<le>o |A \<times> B|"
+ by (auto simp add: card_of_Plus_Times)
+ moreover have "|A \<times> B| =o |A|"
+ using assms * by (simp add: card_of_Times_infinite_simps)
+ ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
+ thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
+ qed
+qed
+
+
+lemma card_of_Plus_infinite2:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|B <+> A| =o |A|"
+using assms card_of_Plus_commute card_of_Plus_infinite1
+ordIso_equivalence by blast
+
+
+lemma card_of_Plus_infinite:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
+using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
+
+
+corollary Card_order_Plus_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ LEQ: "p \<le>o r"
+shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
+proof-
+ have "| Field r <+> Field p | =o | Field r | \<and>
+ | Field p <+> Field r | =o | Field r |"
+ using assms by (simp add: card_of_Plus_infinite card_of_mono2)
+ thus ?thesis
+ using assms card_of_Field_ordIso[of r]
+ ordIso_transitive[of "|Field r <+> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+
+subsection {* The cardinal $\omega$ and the finite cardinals *}
+
+
+text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
+order relation on
+@{text "nat"}, that we abbreviate by @{text "natLeq"}. The finite cardinals
+shall be the restrictions of these relations to the numbers smaller than
+fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
+
+abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
+abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
+
+abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
+where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
+
+lemma infinite_cartesian_product:
+assumes "infinite A" "infinite B"
+shows "infinite (A \<times> B)"
+proof
+ assume "finite (A \<times> B)"
+ from assms(1) have "A \<noteq> {}" by auto
+ with `finite (A \<times> B)` have "finite B" using finite_cartesian_productD2 by auto
+ with assms(2) show False by simp
+qed
+
+
+subsubsection {* First as well-orders *}
+
+
+lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
+by(unfold Field_def, auto)
+
+
+lemma natLeq_Refl: "Refl natLeq"
+unfolding refl_on_def Field_def by auto
+
+
+lemma natLeq_trans: "trans natLeq"
+unfolding trans_def by auto
+
+
+lemma natLeq_Preorder: "Preorder natLeq"
+unfolding preorder_on_def
+by (auto simp add: natLeq_Refl natLeq_trans)
+
+
+lemma natLeq_antisym: "antisym natLeq"
+unfolding antisym_def by auto
+
+
+lemma natLeq_Partial_order: "Partial_order natLeq"
+unfolding partial_order_on_def
+by (auto simp add: natLeq_Preorder natLeq_antisym)
+
+
+lemma natLeq_Total: "Total natLeq"
+unfolding total_on_def by auto
+
+
+lemma natLeq_Linear_order: "Linear_order natLeq"
+unfolding linear_order_on_def
+by (auto simp add: natLeq_Partial_order natLeq_Total)
+
+
+lemma natLeq_natLess_Id: "natLess = natLeq - Id"
+by auto
+
+
+lemma natLeq_Well_order: "Well_order natLeq"
+unfolding well_order_on_def
+using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
+
+
+lemma closed_nat_set_iff:
+assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
+shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+proof-
+ {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
+ moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
+ ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
+ using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
+ have "A = {0 ..< n}"
+ proof(auto simp add: 1)
+ fix m assume *: "m \<in> A"
+ {assume "n \<le> m" with assms * have "n \<in> A" by blast
+ hence False using 1 by auto
+ }
+ thus "m < n" by fastforce
+ qed
+ hence "\<exists>n. A = {0 ..< n}" by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma Field_natLeq_on: "Field (natLeq_on n) = {0 ..< n}"
+unfolding Field_def by auto
+
+
+lemma natLeq_underS_less: "rel.underS natLeq n = {0 ..< n}"
+unfolding rel.underS_def by auto
+
+
+lemma Restr_natLeq: "Restr natLeq {0 ..< n} = natLeq_on n"
+by force
+
+
+lemma Restr_natLeq2:
+"Restr natLeq (rel.underS natLeq n) = natLeq_on n"
+by (auto simp add: Restr_natLeq natLeq_underS_less)
+
+
+lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
+using Restr_natLeq[of n] natLeq_Well_order
+ Well_order_Restr[of natLeq "{0..<n}"] by auto
+
+
+corollary natLeq_on_well_order_on: "well_order_on {0 ..< n} (natLeq_on n)"
+using natLeq_on_Well_order Field_natLeq_on by auto
+
+
+lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
+unfolding wo_rel_def using natLeq_on_Well_order .
+
+
+lemma natLeq_on_ofilter_less_eq:
+"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
+apply (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def)
+apply (simp add: Field_natLeq_on)
+by (auto simp add: rel.under_def)
+
+lemma natLeq_on_ofilter_iff:
+"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
+proof(rule iffI)
+ assume *: "wo_rel.ofilter (natLeq_on m) A"
+ hence 1: "A \<le> {0..<m}"
+ by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def Field_natLeq_on)
+ hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
+ using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def)
+ hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+ thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
+next
+ assume "(\<exists>n\<le>m. A = {0 ..< n})"
+ thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
+qed
+
+
+
+subsubsection {* Then as cardinals *}
+
+
+lemma natLeq_Card_order: "Card_order natLeq"
+proof(auto simp add: natLeq_Well_order
+ Card_order_iff_Restr_underS Restr_natLeq2, simp add: Field_natLeq)
+ fix n have "finite(Field (natLeq_on n))"
+ unfolding Field_natLeq_on by auto
+ moreover have "infinite(UNIV::nat set)" by auto
+ ultimately show "natLeq_on n <o |UNIV::nat set|"
+ using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
+ Field_card_of[of "UNIV::nat set"]
+ card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
+qed
+
+
+corollary card_of_Field_natLeq:
+"|Field natLeq| =o natLeq"
+using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
+ ordIso_symmetric[of natLeq] by blast
+
+
+corollary card_of_nat:
+"|UNIV::nat set| =o natLeq"
+using Field_natLeq card_of_Field_natLeq by auto
+
+
+corollary infinite_iff_natLeq_ordLeq:
+"infinite A = ( natLeq \<le>o |A| )"
+using infinite_iff_card_of_nat[of A] card_of_nat
+ ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+
+
+corollary finite_iff_ordLess_natLeq:
+"finite A = ( |A| <o natLeq)"
+using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
+ card_of_Well_order natLeq_Well_order
+by auto
+
+lemma ordIso_natLeq_on_imp_finite:
+"|A| =o natLeq_on n \<Longrightarrow> finite A"
+unfolding ordIso_def iso_def[abs_def]
+by (auto simp: Field_natLeq_on bij_betw_finite Field_card_of)
+
+
+lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
+proof(unfold card_order_on_def,
+ auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
+ fix r assume "well_order_on {0..<n} r"
+ thus "natLeq_on n \<le>o r"
+ using finite_atLeastLessThan natLeq_on_well_order_on
+ finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
+qed
+
+
+corollary card_of_Field_natLeq_on:
+"|Field (natLeq_on n)| =o natLeq_on n"
+using Field_natLeq_on natLeq_on_Card_order
+ Card_order_iff_ordIso_card_of[of "natLeq_on n"]
+ ordIso_symmetric[of "natLeq_on n"] by blast
+
+
+corollary card_of_less:
+"|{0 ..< n}| =o natLeq_on n"
+using Field_natLeq_on card_of_Field_natLeq_on by auto
+
+
+lemma natLeq_on_ordLeq_less_eq:
+"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
+proof
+ assume "natLeq_on m \<le>o natLeq_on n"
+ then obtain f where "inj_on f {0..<m} \<and> f ` {0..<m} \<le> {0..<n}"
+ unfolding ordLeq_def using
+ embed_inj_on[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on]
+ embed_Field[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on] by blast
+ thus "m \<le> n" using atLeastLessThan_less_eq2 by blast
+next
+ assume "m \<le> n"
+ hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
+ hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
+ thus "natLeq_on m \<le>o natLeq_on n"
+ using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+qed
+
+
+lemma natLeq_on_ordLeq_less:
+"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
+using not_ordLeq_iff_ordLess[of "natLeq_on m" "natLeq_on n"]
+ natLeq_on_Well_order natLeq_on_ordLeq_less_eq
+by fastforce
+
+
+
+subsubsection {* "Backward compatibility" with the numeric cardinal operator for finite sets *}
+
+
+lemma finite_card_of_iff_card2:
+assumes FIN: "finite A" and FIN': "finite B"
+shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
+using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
+
+
+lemma finite_imp_card_of_natLeq_on:
+assumes "finite A"
+shows "|A| =o natLeq_on (card A)"
+proof-
+ obtain h where "bij_betw h A {0 ..< card A}"
+ using assms ex_bij_betw_finite_nat by blast
+ thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
+qed
+
+
+lemma finite_iff_card_of_natLeq_on:
+"finite A = (\<exists>n. |A| =o natLeq_on n)"
+using finite_imp_card_of_natLeq_on[of A]
+by(auto simp add: ordIso_natLeq_on_imp_finite)
+
+
+
+subsection {* The successor of a cardinal *}
+
+
+text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
+being a successor cardinal of @{text "r"}. Although the definition does
+not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
+
+
+definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
+where
+"isCardSuc r r' \<equiv>
+ Card_order r' \<and> r <o r' \<and>
+ (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
+
+
+text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
+by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
+Again, the picked item shall be proved unique up to order-isomorphism. *}
+
+
+definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
+where
+"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
+
+
+lemma exists_minim_Card_order:
+"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+unfolding card_order_on_def using exists_minim_Well_order by blast
+
+
+lemma exists_isCardSuc:
+assumes "Card_order r"
+shows "\<exists>r'. isCardSuc r r'"
+proof-
+ let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
+ have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
+ by (simp add: card_of_Card_order Card_order_Pow)
+ then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
+ using exists_minim_Card_order[of ?R] by blast
+ thus ?thesis unfolding isCardSuc_def by auto
+qed
+
+
+lemma cardSuc_isCardSuc:
+assumes "Card_order r"
+shows "isCardSuc r (cardSuc r)"
+unfolding cardSuc_def using assms
+by (simp add: exists_isCardSuc someI_ex)
+
+
+lemma cardSuc_Card_order:
+"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_greater:
+"Card_order r \<Longrightarrow> r <o cardSuc r"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_ordLeq:
+"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
+using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
+
+
+text{* The minimality property of @{text "cardSuc"} originally present in its definition
+is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
+
+lemma cardSuc_least_aux:
+"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+text{* But from this we can infer general minimality: *}
+
+lemma cardSuc_least:
+assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
+shows "cardSuc r \<le>o r'"
+proof-
+ let ?p = "cardSuc r"
+ have 0: "Well_order ?p \<and> Well_order r'"
+ using assms cardSuc_Card_order unfolding card_order_on_def by blast
+ {assume "r' <o ?p"
+ then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
+ using internalize_ordLess[of r' ?p] by blast
+ (* *)
+ have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
+ moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
+ ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
+ hence False using 2 not_ordLess_ordLeq by blast
+ }
+ thus ?thesis using 0 ordLess_or_ordLeq by blast
+qed
+
+
+lemma cardSuc_ordLess_ordLeq:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(r <o r') = (cardSuc r \<le>o r')"
+proof(auto simp add: assms cardSuc_least)
+ assume "cardSuc r \<le>o r'"
+ thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
+qed
+
+
+lemma cardSuc_ordLeq_ordLess:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(r' <o cardSuc r) = (r' \<le>o r)"
+proof-
+ have "Well_order r \<and> Well_order r'"
+ using assms unfolding card_order_on_def by auto
+ moreover have "Well_order(cardSuc r)"
+ using assms cardSuc_Card_order card_order_on_def by blast
+ ultimately show ?thesis
+ using assms cardSuc_ordLess_ordLeq[of r r']
+ not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
+qed
+
+
+lemma cardSuc_mono_ordLeq:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
+using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
+
+
+lemma cardSuc_invar_ordIso:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(cardSuc r =o cardSuc r') = (r =o r')"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
+ using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
+ thus ?thesis
+ using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
+ using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
+qed
+
+
+lemma cardSuc_natLeq_on_Suc:
+"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
+proof-
+ obtain r r' p where r_def: "r = natLeq_on n" and
+ r'_def: "r' = cardSuc(natLeq_on n)" and
+ p_def: "p = natLeq_on(Suc n)" by blast
+ (* Preliminary facts: *)
+ have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
+ using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
+ hence WELL: "Well_order r \<and> Well_order r' \<and> Well_order p"
+ unfolding card_order_on_def by force
+ have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
+ unfolding r_def p_def Field_natLeq_on by simp
+ hence FIN: "finite (Field r)" by force
+ have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
+ hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
+ hence LESS: "|Field r| <o |Field r'|"
+ using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
+ (* Main proof: *)
+ have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
+ using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
+ moreover have "p \<le>o r'"
+ proof-
+ {assume "r' <o p"
+ then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
+ let ?q = "Restr p (f ` Field r')"
+ have 1: "embed r' p f" using 0 unfolding embedS_def by force
+ hence 2: "f ` Field r' < {0..<(Suc n)}"
+ using WELL FIELD 0 by (auto simp add: embedS_iff)
+ have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
+ then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
+ unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
+ hence 4: "m \<le> n" using 2 by force
+ (* *)
+ have "bij_betw f (Field r') (f ` (Field r'))"
+ using 1 WELL embed_inj_on unfolding bij_betw_def by force
+ moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
+ ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
+ using bij_betw_same_card bij_betw_finite by metis
+ hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
+ hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
+ hence False using LESS not_ordLess_ordLeq by auto
+ }
+ thus ?thesis using WELL CARD by (fastforce simp: not_ordLess_iff_ordLeq)
+ qed
+ ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def by blast
+qed
+
+
+lemma card_of_cardSuc_finite:
+"finite(Field(cardSuc |A| )) = finite A"
+proof
+ assume *: "finite (Field (cardSuc |A| ))"
+ have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
+ using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
+ hence "|A| \<le>o |Field(cardSuc |A| )|"
+ using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
+ ordLeq_ordIso_trans by blast
+ thus "finite A" using * card_of_ordLeq_finite by blast
+next
+ assume "finite A"
+ then obtain n where "|A| =o natLeq_on n" using finite_iff_card_of_natLeq_on by blast
+ hence "cardSuc |A| =o cardSuc(natLeq_on n)"
+ using card_of_Card_order cardSuc_invar_ordIso natLeq_on_Card_order by blast
+ hence "cardSuc |A| =o natLeq_on(Suc n)"
+ using cardSuc_natLeq_on_Suc ordIso_transitive by blast
+ hence "cardSuc |A| =o |{0..<(Suc n)}|" using card_of_less ordIso_equivalence by blast
+ moreover have "|Field (cardSuc |A| ) | =o cardSuc |A|"
+ using card_of_Field_ordIso cardSuc_Card_order card_of_Card_order by blast
+ ultimately have "|Field (cardSuc |A| ) | =o |{0..<(Suc n)}|"
+ using ordIso_equivalence by blast
+ thus "finite (Field (cardSuc |A| ))"
+ using card_of_ordIso_finite finite_atLeastLessThan by blast
+qed
+
+
+lemma cardSuc_finite:
+assumes "Card_order r"
+shows "finite (Field (cardSuc r)) = finite (Field r)"
+proof-
+ let ?A = "Field r"
+ have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
+ hence "cardSuc |?A| =o cardSuc r" using assms
+ by (simp add: card_of_Card_order cardSuc_invar_ordIso)
+ moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
+ by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
+ moreover
+ {have "|Field (cardSuc r) | =o cardSuc r"
+ using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
+ hence "cardSuc r =o |Field (cardSuc r) |"
+ using ordIso_symmetric by blast
+ }
+ ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
+ using ordIso_transitive by blast
+ hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
+ using card_of_ordIso_finite by blast
+ thus ?thesis by (simp only: card_of_cardSuc_finite)
+qed
+
+
+lemma card_of_Plus_ordLess_infinite:
+assumes INF: "infinite C" and
+ LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+shows "|A <+> B| <o |C|"
+proof(cases "A = {} \<or> B = {}")
+ assume Case1: "A = {} \<or> B = {}"
+ hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
+ using card_of_Plus_empty1 card_of_Plus_empty2 by blast
+ hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
+ using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
+ thus ?thesis using LESS1 LESS2
+ ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
+ ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
+next
+ assume Case2: "\<not>(A = {} \<or> B = {})"
+ {assume *: "|C| \<le>o |A <+> B|"
+ hence "infinite (A <+> B)" using INF card_of_ordLeq_finite by blast
+ hence 1: "infinite A \<or> infinite B" using finite_Plus by blast
+ {assume Case21: "|A| \<le>o |B|"
+ hence "infinite B" using 1 card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |B|" using Case2 Case21
+ by (auto simp add: card_of_Plus_infinite)
+ hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ moreover
+ {assume Case22: "|B| \<le>o |A|"
+ hence "infinite A" using 1 card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |A|" using Case2 Case22
+ by (auto simp add: card_of_Plus_infinite)
+ hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ ultimately have False using ordLeq_total card_of_Well_order[of A]
+ card_of_Well_order[of B] by blast
+ }
+ thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
+ card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
+qed
+
+
+lemma card_of_Plus_ordLess_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LESS1: "|A| <o r" and LESS2: "|B| <o r"
+shows "|A <+> B| <o r"
+proof-
+ let ?C = "Field r"
+ have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|A| <o |?C|" "|B| <o |?C|"
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
+ hence "|A <+> B| <o |?C|" using INF
+ card_of_Plus_ordLess_infinite by blast
+ thus ?thesis using 1 ordLess_ordIso_trans by blast
+qed
+
+
+lemma card_of_Plus_ordLeq_infinite_Field:
+assumes r: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and c: "Card_order r"
+shows "|A <+> B| \<le>o r"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> infinite (Field ?r')" using assms
+ by (simp add: cardSuc_Card_order cardSuc_finite)
+ moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
+ by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+ ultimately have "|A <+> B| <o ?r'"
+ using card_of_Plus_ordLess_infinite_Field by blast
+ thus ?thesis using c r
+ by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+qed
+
+
+lemma card_of_Un_ordLeq_infinite_Field:
+assumes C: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and "Card_order r"
+shows "|A Un B| \<le>o r"
+using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
+ordLeq_transitive by fast
+
+
+
+subsection {* Regular cardinals *}
+
+
+definition cofinal where
+"cofinal A r \<equiv>
+ ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
+
+
+definition regular where
+"regular r \<equiv>
+ ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
+
+
+definition relChain where
+"relChain r As \<equiv>
+ ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
+
+lemma regular_UNION:
+assumes r: "Card_order r" "regular r"
+and As: "relChain r As"
+and Bsub: "B \<le> (UN i : Field r. As i)"
+and cardB: "|B| <o r"
+shows "EX i : Field r. B \<le> As i"
+proof-
+ let ?phi = "%b j. j : Field r \<and> b : As j"
+ have "ALL b : B. EX j. ?phi b j" using Bsub by blast
+ then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
+ using bchoice[of B ?phi] by blast
+ let ?K = "f ` B"
+ {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"
+ have 2: "cofinal ?K r"
+ unfolding cofinal_def proof auto
+ fix i assume i: "i : Field r"
+ with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast
+ hence "i \<noteq> f b \<and> ~ (f b,i) : r"
+ using As f unfolding relChain_def by auto
+ hence "i \<noteq> f b \<and> (i, f b) : r" using r
+ unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def using i f b by auto
+ with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
+ qed
+ moreover have "?K \<le> Field r" using f by blast
+ ultimately have "|?K| =o r" using 2 r unfolding regular_def by blast
+ moreover
+ {
+ have "|?K| <=o |B|" using card_of_image .
+ hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
+ }
+ ultimately have False using not_ordLess_ordIso by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma infinite_cardSuc_regular:
+assumes r_inf: "infinite (Field r)" and r_card: "Card_order r"
+shows "regular (cardSuc r)"
+proof-
+ let ?r' = "cardSuc r"
+ have r': "Card_order ?r'"
+ "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
+ using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
+ show ?thesis
+ unfolding regular_def proof auto
+ fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
+ hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
+ also have 22: "|Field ?r'| =o ?r'"
+ using r' by (simp add: card_of_Field_ordIso[of ?r'])
+ finally have "|K| \<le>o ?r'" .
+ moreover
+ {let ?L = "UN j : K. rel.underS ?r' j"
+ let ?J = "Field r"
+ have rJ: "r =o |?J|"
+ using r_card card_of_Field_ordIso ordIso_symmetric by blast
+ assume "|K| <o ?r'"
+ hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
+ hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
+ moreover
+ {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
+ using r' 1 by (auto simp: card_of_underS)
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
+ using r' card_of_Card_order by blast
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
+ using rJ ordLeq_ordIso_trans by blast
+ }
+ ultimately have "|?L| \<le>o |?J|"
+ using r_inf card_of_UNION_ordLeq_infinite by blast
+ hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
+ hence "|?L| <o ?r'" using r' card_of_Card_order by blast
+ moreover
+ {
+ have "Field ?r' \<le> ?L"
+ using 2 unfolding rel.underS_def cofinal_def by auto
+ hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
+ hence "?r' \<le>o |?L|"
+ using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
+ }
+ ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
+ hence False using ordLess_irreflexive by blast
+ }
+ ultimately show "|K| =o ?r'"
+ unfolding ordLeq_iff_ordLess_or_ordIso by blast
+ qed
+qed
+
+lemma cardSuc_UNION:
+assumes r: "Card_order r" and "infinite (Field r)"
+and As: "relChain (cardSuc r) As"
+and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
+and cardB: "|B| <=o r"
+shows "EX i : Field (cardSuc r). B \<le> As i"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> |B| <o ?r'"
+ using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
+ card_of_Card_order by blast
+ moreover have "regular ?r'"
+ using assms by(simp add: infinite_cardSuc_regular)
+ ultimately show ?thesis
+ using As Bsub cardB regular_UNION by blast
+qed
+
+
+subsection {* Others *}
+
+(* function space *)
+definition Func where
+"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
+
+lemma Func_empty:
+"Func {} B = {\<lambda>x. undefined}"
+unfolding Func_def by auto
+
+lemma Func_elim:
+assumes "g \<in> Func A B" and "a \<in> A"
+shows "\<exists> b. b \<in> B \<and> g a = b"
+using assms unfolding Func_def by (cases "g a = undefined") auto
+
+definition curr where
+"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
+
+lemma curr_in:
+assumes f: "f \<in> Func (A <*> B) C"
+shows "curr A f \<in> Func A (Func B C)"
+using assms unfolding curr_def Func_def by auto
+
+lemma curr_inj:
+assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
+shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
+proof safe
+ assume c: "curr A f1 = curr A f2"
+ show "f1 = f2"
+ proof (rule ext, clarify)
+ fix a b show "f1 (a, b) = f2 (a, b)"
+ proof (cases "(a,b) \<in> A <*> B")
+ case False
+ thus ?thesis using assms unfolding Func_def by auto
+ next
+ case True hence a: "a \<in> A" and b: "b \<in> B" by auto
+ thus ?thesis
+ using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
+ qed
+ qed
+qed
+
+lemma curr_surj:
+assumes "g \<in> Func A (Func B C)"
+shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
+proof
+ let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
+ show "curr A ?f = g"
+ proof (rule ext)
+ fix a show "curr A ?f a = g a"
+ proof (cases "a \<in> A")
+ case False
+ hence "g a = undefined" using assms unfolding Func_def by auto
+ thus ?thesis unfolding curr_def using False by simp
+ next
+ case True
+ obtain g1 where "g1 \<in> Func B C" and "g a = g1"
+ using assms using Func_elim[OF assms True] by blast
+ thus ?thesis using True unfolding Func_def curr_def by auto
+ qed
+ qed
+ show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
+qed
+
+lemma bij_betw_curr:
+"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
+unfolding bij_betw_def inj_on_def image_def
+apply (intro impI conjI ballI)
+apply (erule curr_inj[THEN iffD1], assumption+)
+apply auto
+apply (erule curr_in)
+using curr_surj by blast
+
+lemma card_of_Func_Times:
+"|Func (A <*> B) C| =o |Func A (Func B C)|"
+unfolding card_of_ordIso[symmetric]
+using bij_betw_curr by blast
+
+definition Func_map where
+"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
+
+lemma Func_map:
+assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
+shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
+using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
+
+lemma Func_non_emp:
+assumes "B \<noteq> {}"
+shows "Func A B \<noteq> {}"
+proof-
+ obtain b where b: "b \<in> B" using assms by auto
+ hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
+ thus ?thesis by blast
+qed
+
+lemma Func_is_emp:
+"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
+proof
+ assume L: ?L
+ moreover {assume "A = {}" hence False using L Func_empty by auto}
+ moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
+ ultimately show ?R by blast
+next
+ assume R: ?R
+ moreover
+ {fix f assume "f \<in> Func A B"
+ moreover obtain a where "a \<in> A" using R by blast
+ ultimately obtain b where "b \<in> B" unfolding Func_def by blast
+ with R have False by blast
+ }
+ thus ?L by blast
+qed
+
+lemma Func_map_surj:
+assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
+and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
+shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
+proof(cases "B2 = {}")
+ case True
+ thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
+next
+ case False note B2 = False
+ show ?thesis
+ proof safe
+ fix h assume h: "h \<in> Func B2 B1"
+ def j1 \<equiv> "inv_into A1 f1"
+ have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
+ then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
+ {fix b2 assume b2: "b2 \<in> B2"
+ hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
+ moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
+ ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
+ } note kk = this
+ obtain b22 where b22: "b22 \<in> B2" using B2 by auto
+ def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
+ have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
+ have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
+ using kk unfolding j2_def by auto
+ def g \<equiv> "Func_map A2 j1 j2 h"
+ have "Func_map B2 f1 f2 g = h"
+ proof (rule ext)
+ fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
+ proof(cases "b2 \<in> B2")
+ case True
+ show ?thesis
+ proof (cases "h b2 = undefined")
+ case True
+ hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
+ show ?thesis using A2 f_inv_into_f[OF b1]
+ unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
+ qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
+ auto intro: f_inv_into_f)
+ qed(insert h, unfold Func_def Func_map_def, auto)
+ qed
+ moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
+ using inv_into_into j2A2 B1 A2 inv_into_into
+ unfolding j1_def image_def by fast+
+ ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
+ unfolding Func_map_def[abs_def] unfolding image_def by auto
+ qed(insert B1 Func_map[OF _ _ A2(2)], auto)
+qed
+
+lemma card_of_Pow_Func:
+"|Pow A| =o |Func A (UNIV::bool set)|"
+proof-
+ def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
+ else undefined"
+ have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
+ unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
+ fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
+ thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
+ next
+ show "F ` Pow A = Func A UNIV"
+ proof safe
+ fix f assume f: "f \<in> Func A (UNIV::bool set)"
+ show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
+ let ?A1 = "{a \<in> A. f a = True}"
+ show "f = F ?A1" unfolding F_def apply(rule ext)
+ using f unfolding Func_def mem_Collect_eq by auto
+ qed auto
+ qed(unfold Func_def mem_Collect_eq F_def, auto)
+ qed
+ thus ?thesis unfolding card_of_ordIso[symmetric] by blast
+qed
+
+lemma card_of_Func_UNIV:
+"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
+apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
+ let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
+ show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
+ unfolding bij_betw_def inj_on_def proof safe
+ fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
+ hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
+ then obtain f where f: "\<forall> a. h a = f a" by metis
+ hence "range f \<subseteq> B" using h unfolding Func_def by auto
+ thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
+ qed(unfold Func_def fun_eq_iff, auto)
+qed
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Constructions_on_Wellorders.thy
--- a/src/HOL/Cardinals/Constructions_on_Wellorders.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,13 +8,11 @@
header {* Constructions on Wellorders *}
theory Constructions_on_Wellorders
-imports Constructions_on_Wellorders_Base Wellorder_Embedding
+imports Constructions_on_Wellorders_FP Wellorder_Embedding
begin
declare
ordLeq_Well_order_simp[simp]
- ordLess_Well_order_simp[simp]
- ordIso_Well_order_simp[simp]
not_ordLeq_iff_ordLess[simp]
not_ordLess_iff_ordLeq[simp]
@@ -88,7 +86,7 @@
by (auto simp add: ofilter_subset_embedS_iso)
-subsection {* Ordering the well-orders by existence of embeddings *}
+subsection {* Ordering the well-orders by existence of embeddings *}
corollary ordLeq_refl_on: "refl_on {r. Well_order r} ordLeq"
using ordLeq_reflexive unfolding ordLeq_def refl_on_def
@@ -113,6 +111,16 @@
corollary ordIso_equiv: "equiv {r. Well_order r} ordIso"
by (auto simp add: equiv_def ordIso_sym ordIso_refl_on ordIso_trans)
+lemma ordLess_Well_order_simp[simp]:
+assumes "r <o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLess_def by simp
+
+lemma ordIso_Well_order_simp[simp]:
+assumes "r =o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordIso_def by simp
+
lemma ordLess_irrefl: "irrefl ordLess"
by(unfold irrefl_def, auto simp add: ordLess_irreflexive)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Constructions_on_Wellorders_Base.thy
--- a/src/HOL/Cardinals/Constructions_on_Wellorders_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1633 +0,0 @@
-(* Title: HOL/Cardinals/Constructions_on_Wellorders_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Constructions on wellorders (base).
-*)
-
-header {* Constructions on Wellorders (Base) *}
-
-theory Constructions_on_Wellorders_Base
-imports Wellorder_Embedding_Base
-begin
-
-
-text {* In this section, we study basic constructions on well-orders, such as restriction to
-a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
-and bounded square. We also define between well-orders
-the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
-@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
-@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}). We study the
-connections between these relations, order filters, and the aforementioned constructions.
-A main result of this section is that @{text "<o"} is well-founded. *}
-
-
-subsection {* Restriction to a set *}
-
-
-abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
-where "Restr r A \<equiv> r Int (A \<times> A)"
-
-
-lemma Restr_subset:
-"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
-by blast
-
-
-lemma Restr_Field: "Restr r (Field r) = r"
-unfolding Field_def by auto
-
-
-lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
-unfolding refl_on_def Field_def by auto
-
-
-lemma antisym_Restr:
-"antisym r \<Longrightarrow> antisym(Restr r A)"
-unfolding antisym_def Field_def by auto
-
-
-lemma Total_Restr:
-"Total r \<Longrightarrow> Total(Restr r A)"
-unfolding total_on_def Field_def by auto
-
-
-lemma trans_Restr:
-"trans r \<Longrightarrow> trans(Restr r A)"
-unfolding trans_def Field_def by blast
-
-
-lemma Preorder_Restr:
-"Preorder r \<Longrightarrow> Preorder(Restr r A)"
-unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
-
-
-lemma Partial_order_Restr:
-"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
-unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
-
-
-lemma Linear_order_Restr:
-"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
-unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
-
-
-lemma Well_order_Restr:
-assumes "Well_order r"
-shows "Well_order(Restr r A)"
-proof-
- have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
- hence "wf(Restr r A - Id)" using assms
- using well_order_on_def wf_subset by blast
- thus ?thesis using assms unfolding well_order_on_def
- by (simp add: Linear_order_Restr)
-qed
-
-
-lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
-by (auto simp add: Field_def)
-
-
-lemma Refl_Field_Restr:
-"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
-by (auto simp add: refl_on_def Field_def)
-
-
-lemma Refl_Field_Restr2:
-"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
-by (auto simp add: Refl_Field_Restr)
-
-
-lemma well_order_on_Restr:
-assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
-shows "well_order_on A (Restr r A)"
-using assms
-using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
- order_on_defs[of "Field r" r] by auto
-
-
-subsection {* Order filters versus restrictions and embeddings *}
-
-
-lemma Field_Restr_ofilter:
-"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
-by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
-
-
-lemma ofilter_Restr_under:
-assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
-shows "rel.under (Restr r A) a = rel.under r a"
-using assms wo_rel_def
-proof(auto simp add: wo_rel.ofilter_def rel.under_def)
- fix b assume *: "a \<in> A" and "(b,a) \<in> r"
- hence "b \<in> rel.under r a \<and> a \<in> Field r"
- unfolding rel.under_def using Field_def by fastforce
- thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
-qed
-
-
-lemma ofilter_embed:
-assumes "Well_order r"
-shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
-proof
- assume *: "wo_rel.ofilter r A"
- show "A \<le> Field r \<and> embed (Restr r A) r id"
- proof(unfold embed_def, auto)
- fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
- by (auto simp add: wo_rel_def wo_rel.ofilter_def)
- next
- fix a assume "a \<in> Field (Restr r A)"
- thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
- by (simp add: ofilter_Restr_under Field_Restr_ofilter)
- qed
-next
- assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
- hence "Field(Restr r A) \<le> Field r"
- using assms embed_Field[of "Restr r A" r id] id_def
- Well_order_Restr[of r] by auto
- {fix a assume "a \<in> A"
- hence "a \<in> Field(Restr r A)" using * assms
- by (simp add: order_on_defs Refl_Field_Restr2)
- hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
- using * unfolding embed_def by auto
- hence "rel.under r a \<le> rel.under (Restr r A) a"
- unfolding bij_betw_def by auto
- also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
- also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
- finally have "rel.under r a \<le> A" .
- }
- thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
-qed
-
-
-lemma ofilter_Restr_Int:
-assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
-shows "wo_rel.ofilter (Restr r B) (A Int B)"
-proof-
- let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence Field: "Field ?rB = Field r Int B"
- using Refl_Field_Restr by blast
- have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- (* Main proof *)
- show ?thesis using WellB assms
- proof(auto simp add: wo_rel.ofilter_def rel.under_def)
- fix a assume "a \<in> A" and *: "a \<in> B"
- hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
- with * show "a \<in> Field ?rB" using Field by auto
- next
- fix a b assume "a \<in> A" and "(b,a) \<in> r"
- thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
- qed
-qed
-
-
-lemma ofilter_Restr_subset:
-assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
-shows "wo_rel.ofilter (Restr r B) A"
-proof-
- have "A Int B = A" using SUB by blast
- thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
-qed
-
-
-lemma ofilter_subset_embed:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence FieldA: "Field ?rA = Field r Int A"
- using Refl_Field_Restr by blast
- have FieldB: "Field ?rB = Field r Int B"
- using Refl Refl_Field_Restr by blast
- have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- (* Main proof *)
- show ?thesis
- proof
- assume *: "A \<le> B"
- hence "wo_rel.ofilter (Restr r B) A" using assms
- by (simp add: ofilter_Restr_subset)
- hence "embed (Restr ?rB A) (Restr r B) id"
- using WellB ofilter_embed[of "?rB" A] by auto
- thus "embed (Restr r A) (Restr r B) id"
- using * by (simp add: Restr_subset)
- next
- assume *: "embed (Restr r A) (Restr r B) id"
- {fix a assume **: "a \<in> A"
- hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
- with ** FieldA have "a \<in> Field ?rA" by auto
- hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
- hence "a \<in> B" using FieldB by auto
- }
- thus "A \<le> B" by blast
- qed
-qed
-
-
-lemma ofilter_subset_embedS_iso:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
- ((A = B) = (iso (Restr r A) (Restr r B) id))"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence "Field ?rA = Field r Int A"
- using Refl_Field_Restr by blast
- hence FieldA: "Field ?rA = A" using OFA Well
- by (auto simp add: wo_rel.ofilter_def)
- have "Field ?rB = Field r Int B"
- using Refl Refl_Field_Restr by blast
- hence FieldB: "Field ?rB = B" using OFB Well
- by (auto simp add: wo_rel.ofilter_def)
- (* Main proof *)
- show ?thesis unfolding embedS_def iso_def
- using assms ofilter_subset_embed[of r A B]
- FieldA FieldB bij_betw_id_iff[of A B] by auto
-qed
-
-
-lemma ofilter_subset_embedS:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A < B) = embedS (Restr r A) (Restr r B) id"
-using assms by (simp add: ofilter_subset_embedS_iso)
-
-
-lemma embed_implies_iso_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r' r f"
-shows "iso r' (Restr r (f ` (Field r'))) f"
-proof-
- let ?A' = "Field r'"
- let ?r'' = "Restr r (f ` ?A')"
- have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
- have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter by blast
- hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
- hence "bij_betw f ?A' (Field ?r'')"
- using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
- moreover
- {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
- unfolding Field_def by auto
- hence "compat r' ?r'' f"
- using assms embed_iff_compat_inj_on_ofilter
- unfolding compat_def by blast
- }
- ultimately show ?thesis using WELL' 0 iso_iff3 by blast
-qed
-
-
-subsection {* The strict inclusion on proper ofilters is well-founded *}
-
-
-definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
-where
-"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
- wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
-
-
-lemma wf_ofilterIncl:
-assumes WELL: "Well_order r"
-shows "wf(ofilterIncl r)"
-proof-
- have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
- hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
- let ?h = "(\<lambda> A. wo_rel.suc r A)"
- let ?rS = "r - Id"
- have "wf ?rS" using WELL by (simp add: order_on_defs)
- moreover
- have "compat (ofilterIncl r) ?rS ?h"
- proof(unfold compat_def ofilterIncl_def,
- intro allI impI, simp, elim conjE)
- fix A B
- assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
- **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
- then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
- 1: "A = rel.underS r a \<and> B = rel.underS r b"
- using Well by (auto simp add: wo_rel.ofilter_underS_Field)
- hence "a \<noteq> b" using *** by auto
- moreover
- have "(a,b) \<in> r" using 0 1 Lo ***
- by (auto simp add: rel.underS_incl_iff)
- moreover
- have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
- using Well 0 1 by (simp add: wo_rel.suc_underS)
- ultimately
- show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
- by simp
- qed
- ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
-qed
-
-
-
-subsection {* Ordering the well-orders by existence of embeddings *}
-
-
-text {* We define three relations between well-orders:
-\begin{itemize}
-\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
-\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
-\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
-\end{itemize}
-%
-The prefix "ord" and the index "o" in these names stand for "ordinal-like".
-These relations shall be proved to be inter-connected in a similar fashion as the trio
-@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
-*}
-
-
-definition ordLeq :: "('a rel * 'a' rel) set"
-where
-"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
-
-
-abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
-where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
-
-
-abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
-where "r \<le>o r' \<equiv> r <=o r'"
-
-
-definition ordLess :: "('a rel * 'a' rel) set"
-where
-"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
-
-abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
-where "r <o r' \<equiv> (r,r') \<in> ordLess"
-
-
-definition ordIso :: "('a rel * 'a' rel) set"
-where
-"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
-
-abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
-where "r =o r' \<equiv> (r,r') \<in> ordIso"
-
-
-lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
-
-lemma ordLeq_Well_order_simp:
-assumes "r \<le>o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordLeq_def by simp
-
-
-lemma ordLess_Well_order_simp:
-assumes "r <o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordLess_def by simp
-
-
-lemma ordIso_Well_order_simp:
-assumes "r =o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordIso_def by simp
-
-
-text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
-on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
-restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
-to @{text "'a rel rel"}. *}
-
-
-lemma ordLeq_reflexive:
-"Well_order r \<Longrightarrow> r \<le>o r"
-unfolding ordLeq_def using id_embed[of r] by blast
-
-
-lemma ordLeq_transitive[trans]:
-assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
-shows "r \<le>o r''"
-proof-
- obtain f and f'
- where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
- "embed r r' f" and "embed r' r'' f'"
- using * ** unfolding ordLeq_def by blast
- hence "embed r r'' (f' o f)"
- using comp_embed[of r r' f r'' f'] by auto
- thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
-qed
-
-
-lemma ordLeq_total:
-"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
-unfolding ordLeq_def using wellorders_totally_ordered by blast
-
-
-lemma ordIso_reflexive:
-"Well_order r \<Longrightarrow> r =o r"
-unfolding ordIso_def using id_iso[of r] by blast
-
-
-lemma ordIso_transitive[trans]:
-assumes *: "r =o r'" and **: "r' =o r''"
-shows "r =o r''"
-proof-
- obtain f and f'
- where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
- "iso r r' f" and 3: "iso r' r'' f'"
- using * ** unfolding ordIso_def by auto
- hence "iso r r'' (f' o f)"
- using comp_iso[of r r' f r'' f'] by auto
- thus "r =o r''" unfolding ordIso_def using 1 by auto
-qed
-
-
-lemma ordIso_symmetric:
-assumes *: "r =o r'"
-shows "r' =o r"
-proof-
- obtain f where 1: "Well_order r \<and> Well_order r'" and
- 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
- using * by (auto simp add: ordIso_def iso_def)
- let ?f' = "inv_into (Field r) f"
- have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
- using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
- thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
-qed
-
-
-lemma ordLeq_ordLess_trans[trans]:
-assumes "r \<le>o r'" and " r' <o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordLess_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordLess_def
- using embed_comp_embedS by blast
-qed
-
-
-lemma ordLess_ordLeq_trans[trans]:
-assumes "r <o r'" and " r' \<le>o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordLess_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordLess_def
- using embedS_comp_embed by blast
-qed
-
-
-lemma ordLeq_ordIso_trans[trans]:
-assumes "r \<le>o r'" and " r' =o r''"
-shows "r \<le>o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordIso_def
- using embed_comp_iso by blast
-qed
-
-
-lemma ordIso_ordLeq_trans[trans]:
-assumes "r =o r'" and " r' \<le>o r''"
-shows "r \<le>o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordIso_def
- using iso_comp_embed by blast
-qed
-
-
-lemma ordLess_ordIso_trans[trans]:
-assumes "r <o r'" and " r' =o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLess_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLess_def ordIso_def
- using embedS_comp_iso by blast
-qed
-
-
-lemma ordIso_ordLess_trans[trans]:
-assumes "r =o r'" and " r' <o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLess_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLess_def ordIso_def
- using iso_comp_embedS by blast
-qed
-
-
-lemma ordLess_not_embed:
-assumes "r <o r'"
-shows "\<not>(\<exists>f'. embed r' r f')"
-proof-
- obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
- 3: " \<not> bij_betw f (Field r) (Field r')"
- using assms unfolding ordLess_def by (auto simp add: embedS_def)
- {fix f' assume *: "embed r' r f'"
- hence "bij_betw f (Field r) (Field r')" using 1 2
- by (simp add: embed_bothWays_Field_bij_betw)
- with 3 have False by contradiction
- }
- thus ?thesis by blast
-qed
-
-
-lemma ordLess_Field:
-assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
-shows "\<not> (f`(Field r1) = Field r2)"
-proof-
- let ?A1 = "Field r1" let ?A2 = "Field r2"
- obtain g where
- 0: "Well_order r1 \<and> Well_order r2" and
- 1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
- using OL unfolding ordLess_def by (auto simp add: embedS_def)
- hence "\<forall>a \<in> ?A1. f a = g a"
- using 0 EMB embed_unique[of r1] by auto
- hence "\<not>(bij_betw f ?A1 ?A2)"
- using 1 bij_betw_cong[of ?A1] by blast
- moreover
- have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
- ultimately show ?thesis by (simp add: bij_betw_def)
-qed
-
-
-lemma ordLess_iff:
-"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
-proof
- assume *: "r <o r'"
- hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
- with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
- unfolding ordLess_def by auto
-next
- assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
- then obtain f where 1: "embed r r' f"
- using wellorders_totally_ordered[of r r'] by blast
- moreover
- {assume "bij_betw f (Field r) (Field r')"
- with * 1 have "embed r' r (inv_into (Field r) f) "
- using inv_into_Field_embed_bij_betw[of r r' f] by auto
- with * have False by blast
- }
- ultimately show "(r,r') \<in> ordLess"
- unfolding ordLess_def using * by (fastforce simp add: embedS_def)
-qed
-
-
-lemma ordLess_irreflexive: "\<not> r <o r"
-proof
- assume "r <o r"
- hence "Well_order r \<and> \<not>(\<exists>f. embed r r f)"
- unfolding ordLess_iff ..
- moreover have "embed r r id" using id_embed[of r] .
- ultimately show False by blast
-qed
-
-
-lemma ordLeq_iff_ordLess_or_ordIso:
-"r \<le>o r' = (r <o r' \<or> r =o r')"
-unfolding ordRels_def embedS_defs iso_defs by blast
-
-
-lemma ordIso_iff_ordLeq:
-"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
-proof
- assume "r =o r'"
- then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
- embed r r' f \<and> bij_betw f (Field r) (Field r')"
- unfolding ordIso_def iso_defs by auto
- hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
- by (simp add: inv_into_Field_embed_bij_betw)
- thus "r \<le>o r' \<and> r' \<le>o r"
- unfolding ordLeq_def using 1 by auto
-next
- assume "r \<le>o r' \<and> r' \<le>o r"
- then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
- embed r r' f \<and> embed r' r g"
- unfolding ordLeq_def by auto
- hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
- thus "r =o r'" unfolding ordIso_def using 1 by auto
-qed
-
-
-lemma not_ordLess_ordLeq:
-"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
-using ordLess_ordLeq_trans ordLess_irreflexive by blast
-
-
-lemma ordLess_or_ordLeq:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "r <o r' \<or> r' \<le>o r"
-proof-
- have "r \<le>o r' \<or> r' \<le>o r"
- using assms by (simp add: ordLeq_total)
- moreover
- {assume "\<not> r <o r' \<and> r \<le>o r'"
- hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
- hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
- }
- ultimately show ?thesis by blast
-qed
-
-
-lemma not_ordLess_ordIso:
-"r <o r' \<Longrightarrow> \<not> r =o r'"
-using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
-
-
-lemma not_ordLeq_iff_ordLess:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(\<not> r' \<le>o r) = (r <o r')"
-using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
-
-
-lemma not_ordLess_iff_ordLeq:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(\<not> r' <o r) = (r \<le>o r')"
-using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
-
-
-lemma ordLess_transitive[trans]:
-"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
-using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
-
-
-corollary ordLess_trans: "trans ordLess"
-unfolding trans_def using ordLess_transitive by blast
-
-
-lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
-
-
-lemma ordIso_imp_ordLeq:
-"r =o r' \<Longrightarrow> r \<le>o r'"
-using ordIso_iff_ordLeq by blast
-
-
-lemma ordLess_imp_ordLeq:
-"r <o r' \<Longrightarrow> r \<le>o r'"
-using ordLeq_iff_ordLess_or_ordIso by blast
-
-
-lemma ofilter_subset_ordLeq:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
-proof
- assume "A \<le> B"
- thus "Restr r A \<le>o Restr r B"
- unfolding ordLeq_def using assms
- Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
-next
- assume *: "Restr r A \<le>o Restr r B"
- then obtain f where "embed (Restr r A) (Restr r B) f"
- unfolding ordLeq_def by blast
- {assume "B < A"
- hence "Restr r B <o Restr r A"
- unfolding ordLess_def using assms
- Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
- hence False using * not_ordLess_ordLeq by blast
- }
- thus "A \<le> B" using OFA OFB WELL
- wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
-qed
-
-
-lemma ofilter_subset_ordLess:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A < B) = (Restr r A <o Restr r B)"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have 1: "Well_order ?rA \<and> Well_order ?rB"
- using WELL Well_order_Restr by blast
- have "(A < B) = (\<not> B \<le> A)" using assms
- wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
- also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
- using assms ofilter_subset_ordLeq by blast
- also have "\<dots> = (Restr r A <o Restr r B)"
- using 1 not_ordLeq_iff_ordLess by blast
- finally show ?thesis .
-qed
-
-
-lemma ofilter_ordLess:
-"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
-by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
- wo_rel_def Restr_Field)
-
-
-corollary underS_Restr_ordLess:
-assumes "Well_order r" and "Field r \<noteq> {}"
-shows "Restr r (rel.underS r a) <o r"
-proof-
- have "rel.underS r a < Field r" using assms
- by (simp add: rel.underS_Field3)
- thus ?thesis using assms
- by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
-qed
-
-
-lemma embed_ordLess_ofilterIncl:
-assumes
- OL12: "r1 <o r2" and OL23: "r2 <o r3" and
- EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
-shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
-proof-
- have OL13: "r1 <o r3"
- using OL12 OL23 using ordLess_transitive by auto
- let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A3 ="Field r3"
- obtain f12 g23 where
- 0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
- 1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
- 2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
- using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
- hence "\<forall>a \<in> ?A2. f23 a = g23 a"
- using EMB23 embed_unique[of r2 r3] by blast
- hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
- using 2 bij_betw_cong[of ?A2 f23 g23] by blast
- (* *)
- have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
- using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
- have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
- using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
- have 6: "wo_rel.ofilter r3 (f13 ` ?A1) \<and> f13 ` ?A1 \<noteq> ?A3"
- using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
- (* *)
- have "f12 ` ?A1 < ?A2"
- using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
- moreover have "inj_on f23 ?A2"
- using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
- ultimately
- have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
- moreover
- {have "embed r1 r3 (f23 o f12)"
- using 1 EMB23 0 by (auto simp add: comp_embed)
- hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
- using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
- hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
- }
- ultimately
- have "f13 ` ?A1 < f23 ` ?A2" by simp
- (* *)
- with 5 6 show ?thesis
- unfolding ofilterIncl_def by auto
-qed
-
-
-lemma ordLess_iff_ordIso_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
-proof(auto)
- fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
- hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
- thus "r' <o r" using ** ordIso_ordLess_trans by blast
-next
- assume "r' <o r"
- then obtain f where 1: "Well_order r \<and> Well_order r'" and
- 2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
- unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
- hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
- then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
- using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
- have "iso r' (Restr r (f ` (Field r'))) f"
- using embed_implies_iso_Restr 2 assms by blast
- moreover have "Well_order (Restr r (f ` (Field r')))"
- using WELL Well_order_Restr by blast
- ultimately have "r' =o Restr r (f ` (Field r'))"
- using WELL' unfolding ordIso_def by auto
- hence "r' =o Restr r (rel.underS r a)" using 4 by auto
- thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
-qed
-
-
-lemma internalize_ordLess:
-"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
-proof
- assume *: "r' <o r"
- hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
- with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
- using ordLess_iff_ordIso_Restr by blast
- let ?p = "Restr r (rel.underS r a)"
- have "wo_rel.ofilter r (rel.underS r a)" using 0
- by (simp add: wo_rel_def wo_rel.underS_ofilter)
- hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
- hence "Field ?p < Field r" using rel.underS_Field2 1 by fastforce
- moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
- ultimately
- show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
-next
- assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
- thus "r' <o r" using ordIso_ordLess_trans by blast
-qed
-
-
-lemma internalize_ordLeq:
-"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
-proof
- assume *: "r' \<le>o r"
- moreover
- {assume "r' <o r"
- then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
- using internalize_ordLess[of r' r] by blast
- hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- using ordLeq_iff_ordLess_or_ordIso by blast
- }
- moreover
- have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
- ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- using ordLeq_iff_ordLess_or_ordIso by blast
-next
- assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
-qed
-
-
-lemma ordLeq_iff_ordLess_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
-proof(auto)
- assume *: "r \<le>o r'"
- fix a assume "a \<in> Field r"
- hence "Restr r (rel.underS r a) <o r"
- using WELL underS_Restr_ordLess[of r] by blast
- thus "Restr r (rel.underS r a) <o r'"
- using * ordLess_ordLeq_trans by blast
-next
- assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
- {assume "r' <o r"
- then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
- using assms ordLess_iff_ordIso_Restr by blast
- hence False using * not_ordLess_ordIso ordIso_symmetric by blast
- }
- thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
-qed
-
-
-lemma finite_ordLess_infinite:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- FIN: "finite(Field r)" and INF: "infinite(Field r')"
-shows "r <o r'"
-proof-
- {assume "r' \<le>o r"
- then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
- unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
- hence False using finite_imageD finite_subset FIN INF by metis
- }
- thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
-qed
-
-
-lemma finite_well_order_on_ordIso:
-assumes FIN: "finite A" and
- WELL: "well_order_on A r" and WELL': "well_order_on A r'"
-shows "r =o r'"
-proof-
- have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
- using assms rel.well_order_on_Well_order by blast
- moreover
- have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
- \<longrightarrow> r =o r'"
- proof(clarify)
- fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
- have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
- using * ** rel.well_order_on_Well_order by blast
- assume "r \<le>o r'"
- then obtain f where 1: "embed r r' f" and
- "inj_on f A \<and> f ` A \<le> A"
- unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
- hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
- thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
- qed
- ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
-qed
-
-
-subsection{* @{text "<o"} is well-founded *}
-
-
-text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
-on the restricted type @{text "'a rel rel"}. We prove this by first showing that, for any set
-of well-orders all embedded in a fixed well-order, the function mapping each well-order
-in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
-{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
-
-
-definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
-where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
-
-
-lemma ord_to_filter_compat:
-"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
- (ofilterIncl r0)
- (ord_to_filter r0)"
-proof(unfold compat_def ord_to_filter_def, clarify)
- fix r1::"'a rel" and r2::"'a rel"
- let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A0 ="Field r0"
- let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
- let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
- assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
- hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
- by (auto simp add: ordLess_def embedS_def)
- hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
- thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
- using * ** by (simp add: embed_ordLess_ofilterIncl)
-qed
-
-
-theorem wf_ordLess: "wf ordLess"
-proof-
- {fix r0 :: "('a \<times> 'a) set"
- (* need to annotate here!*)
- let ?ordLess = "ordLess::('d rel * 'd rel) set"
- let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
- {assume Case1: "Well_order r0"
- hence "wf ?R"
- using wf_ofilterIncl[of r0]
- compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
- ord_to_filter_compat[of r0] by auto
- }
- moreover
- {assume Case2: "\<not> Well_order r0"
- hence "?R = {}" unfolding ordLess_def by auto
- hence "wf ?R" using wf_empty by simp
- }
- ultimately have "wf ?R" by blast
- }
- thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
-qed
-
-corollary exists_minim_Well_order:
-assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
-shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
-proof-
- obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
- using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
- equals0I[of R] by blast
- with not_ordLeq_iff_ordLess WELL show ?thesis by blast
-qed
-
-
-
-subsection {* Copy via direct images *}
-
-
-text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
-from @{text "Relation.thy"}. It is useful for transporting a well-order between
-different types. *}
-
-
-definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
-where
-"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
-
-
-lemma dir_image_Field:
-"Field(dir_image r f) \<le> f ` (Field r)"
-unfolding dir_image_def Field_def by auto
-
-
-lemma dir_image_minus_Id:
-"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
-unfolding inj_on_def Field_def dir_image_def by auto
-
-
-lemma Refl_dir_image:
-assumes "Refl r"
-shows "Refl(dir_image r f)"
-proof-
- {fix a' b'
- assume "(a',b') \<in> dir_image r f"
- then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
- unfolding dir_image_def by blast
- hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
- hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
- with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
- unfolding dir_image_def by auto
- }
- thus ?thesis
- by(unfold refl_on_def Field_def Domain_def Range_def, auto)
-qed
-
-
-lemma trans_dir_image:
-assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
-shows "trans(dir_image r f)"
-proof(unfold trans_def, auto)
- fix a' b' c'
- assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
- then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
- 2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
- unfolding dir_image_def by blast
- hence "b1 \<in> Field r \<and> b2 \<in> Field r"
- unfolding Field_def by auto
- hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
- hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
- thus "(a',c') \<in> dir_image r f"
- unfolding dir_image_def using 1 by auto
-qed
-
-
-lemma Preorder_dir_image:
-"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
-by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
-
-
-lemma antisym_dir_image:
-assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
-shows "antisym(dir_image r f)"
-proof(unfold antisym_def, auto)
- fix a' b'
- assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
- then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
- 2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
- 3: "{a1,a2,b1,b2} \<le> Field r"
- unfolding dir_image_def Field_def by blast
- hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
- hence "a1 = b2" using 2 AN unfolding antisym_def by auto
- thus "a' = b'" using 1 by auto
-qed
-
-
-lemma Partial_order_dir_image:
-"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
-by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
-
-
-lemma Total_dir_image:
-assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
-shows "Total(dir_image r f)"
-proof(unfold total_on_def, intro ballI impI)
- fix a' b'
- assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
- then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
- using dir_image_Field[of r f] by blast
- moreover assume "a' \<noteq> b'"
- ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
- hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
- thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
- using 1 unfolding dir_image_def by auto
-qed
-
-
-lemma Linear_order_dir_image:
-"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
-by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
-
-
-lemma wf_dir_image:
-assumes WF: "wf r" and INJ: "inj_on f (Field r)"
-shows "wf(dir_image r f)"
-proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
- fix A'::"'b set"
- assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
- obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
- have "A \<noteq> {} \<and> A \<le> Field r"
- using A_def dir_image_Field[of r f] SUB NE by blast
- then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
- using WF unfolding wf_eq_minimal2 by metis
- have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
- proof(clarify)
- fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
- obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
- 3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
- using ** unfolding dir_image_def Field_def by blast
- hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
- hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
- with 1 show False by auto
- qed
- thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
- using A_def 1 by blast
-qed
-
-
-lemma Well_order_dir_image:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
-using assms unfolding well_order_on_def
-using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
- dir_image_minus_Id[of f r]
- subset_inj_on[of f "Field r" "Field(r - Id)"]
- mono_Field[of "r - Id" r] by auto
-
-
-lemma dir_image_Field2:
-"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
-unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
-
-
-lemma dir_image_bij_betw:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
-unfolding bij_betw_def
-by (simp add: dir_image_Field2 order_on_defs)
-
-
-lemma dir_image_compat:
-"compat r (dir_image r f) f"
-unfolding compat_def dir_image_def by auto
-
-
-lemma dir_image_iso:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> iso r (dir_image r f) f"
-using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
-
-
-lemma dir_image_ordIso:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> r =o dir_image r f"
-unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
-
-
-lemma Well_order_iso_copy:
-assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
-shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
-proof-
- let ?r' = "dir_image r f"
- have 1: "A = Field r \<and> Well_order r"
- using WELL rel.well_order_on_Well_order by blast
- hence 2: "iso r ?r' f"
- using dir_image_iso using BIJ unfolding bij_betw_def by auto
- hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
- hence "Field ?r' = A'"
- using 1 BIJ unfolding bij_betw_def by auto
- moreover have "Well_order ?r'"
- using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
- ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
-qed
-
-
-
-subsection {* Bounded square *}
-
-
-text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
-order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
-following criteria (in this order):
-\begin{itemize}
-\item compare the maximums;
-\item compare the first components;
-\item compare the second components.
-\end{itemize}
-%
-The only application of this construction that we are aware of is
-at proving that the square of an infinite set has the same cardinal
-as that set. The essential property required there (and which is ensured by this
-construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
-in a product of proper filters on the original relation (assumed to be a well-order). *}
-
-
-definition bsqr :: "'a rel => ('a * 'a)rel"
-where
-"bsqr r = {((a1,a2),(b1,b2)).
- {a1,a2,b1,b2} \<le> Field r \<and>
- (a1 = b1 \<and> a2 = b2 \<or>
- (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id
- )}"
-
-
-lemma Field_bsqr:
-"Field (bsqr r) = Field r \<times> Field r"
-proof
- show "Field (bsqr r) \<le> Field r \<times> Field r"
- proof-
- {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
- moreover
- have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
- a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
- ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
- }
- thus ?thesis unfolding Field_def by force
- qed
-next
- show "Field r \<times> Field r \<le> Field (bsqr r)"
- proof(auto)
- fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
- hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
- thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
- qed
-qed
-
-
-lemma bsqr_Refl: "Refl(bsqr r)"
-by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
-
-
-lemma bsqr_Trans:
-assumes "Well_order r"
-shows "trans (bsqr r)"
-proof(unfold trans_def, auto)
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Trans: "trans r" using wo_rel.TRANS by auto
- have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
- hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
- (* Main proof *)
- fix a1 a2 b1 b2 c1 c2
- assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
- hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
- have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- using * unfolding bsqr_def by auto
- have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
- using ** unfolding bsqr_def by auto
- show "((a1,a2),(c1,c2)) \<in> bsqr r"
- proof-
- {assume Case1: "a1 = b1 \<and> a2 = b2"
- hence ?thesis using ** by simp
- }
- moreover
- {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
- {assume Case21: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
- using Case2 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
- hence ?thesis using Case2 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
- {assume Case31: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence ?thesis using Case3 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
- hence "(a1,c1) \<in> r - Id"
- using Case3 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
- hence ?thesis using Case3 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- {assume Case41: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence ?thesis using Case4 0 unfolding bsqr_def by force
- }
- moreover
- {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
- hence ?thesis using Case4 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
- hence "(a2,c2) \<in> r - Id"
- using Case4 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- ultimately show ?thesis using 0 1 by auto
- qed
-qed
-
-
-lemma bsqr_antisym:
-assumes "Well_order r"
-shows "antisym (bsqr r)"
-proof(unfold antisym_def, clarify)
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Trans: "trans r" using wo_rel.TRANS by auto
- have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
- hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
- hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
- using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
- (* Main proof *)
- fix a1 a2 b1 b2
- assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
- hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
- have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- using * unfolding bsqr_def by auto
- have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
- using ** unfolding bsqr_def by auto
- show "a1 = b1 \<and> a2 = b2"
- proof-
- {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
- {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case1 IrrS by blast
- }
- moreover
- {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
- hence False using Case1 by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
- {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case2 by auto
- }
- moreover
- {assume Case22: "(b1,a1) \<in> r - Id"
- hence False using Case2 IrrS by blast
- }
- moreover
- {assume Case23: "b1 = a1"
- hence False using Case2 by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- moreover
- {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case3 by auto
- }
- moreover
- {assume Case32: "(b1,a1) \<in> r - Id"
- hence False using Case3 by auto
- }
- moreover
- {assume Case33: "(b2,a2) \<in> r - Id"
- hence False using Case3 IrrS by blast
- }
- ultimately have ?thesis using 0 2 by auto
- }
- ultimately show ?thesis using 0 1 by blast
- qed
-qed
-
-
-lemma bsqr_Total:
-assumes "Well_order r"
-shows "Total(bsqr r)"
-proof-
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
- using wo_rel.TOTALS by auto
- (* Main proof *)
- {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
- hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
- using Field_bsqr by blast
- have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
- proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
- (* Why didn't clarsimp simp add: Well 0 do the same job? *)
- assume Case1: "(a1,a2) \<in> r"
- hence 1: "wo_rel.max2 r a1 a2 = a2"
- using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case11: "(b1,b2) \<in> r"
- hence 2: "wo_rel.max2 r b1 b2 = b2"
- using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 2 unfolding bsqr_def by auto
- next
- assume Case112: "a2 = b2"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
- next
- assume Case1122: "a1 = b1"
- thus ?thesis using Case112 by auto
- qed
- qed
- next
- assume Case12: "(b2,b1) \<in> r"
- hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
- assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 unfolding bsqr_def by auto
- next
- assume Case122: "a2 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
- next
- assume Case1222: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
- next
- assume Case12222: "a2 = b2"
- thus ?thesis using Case122 Case1222 by auto
- qed
- qed
- qed
- qed
- next
- assume Case2: "(a2,a1) \<in> r"
- hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case21: "(b1,b2) \<in> r"
- hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 unfolding bsqr_def by auto
- next
- assume Case212: "a1 = b2"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
- next
- assume Case2122: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
- next
- assume Case21222: "a2 = b2"
- thus ?thesis using Case2122 Case212 by auto
- qed
- qed
- qed
- next
- assume Case22: "(b2,b1) \<in> r"
- hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 3 unfolding bsqr_def by auto
- next
- assume Case222: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
- next
- assume Case2222: "a2 = b2"
- thus ?thesis using Case222 by auto
- qed
- qed
- qed
- qed
- }
- thus ?thesis unfolding total_on_def by fast
-qed
-
-
-lemma bsqr_Linear_order:
-assumes "Well_order r"
-shows "Linear_order(bsqr r)"
-unfolding order_on_defs
-using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
-
-
-lemma bsqr_Well_order:
-assumes "Well_order r"
-shows "Well_order(bsqr r)"
-using assms
-proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
- have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- using assms well_order_on_def Linear_order_Well_order_iff by blast
- fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
- hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
- (* *)
- obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
- have "M \<noteq> {}" using 1 M_def ** by auto
- moreover
- have "M \<le> Field r" unfolding M_def
- using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
- ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
- using 0 by blast
- (* *)
- obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
- have "A1 \<le> Field r" unfolding A1_def using 1 by auto
- moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
- ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
- using 0 by blast
- (* *)
- obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
- have "A2 \<le> Field r" unfolding A2_def using 1 by auto
- moreover have "A2 \<noteq> {}" unfolding A2_def
- using m_min a1_min unfolding A1_def M_def by blast
- ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
- using 0 by blast
- (* *)
- have 2: "wo_rel.max2 r a1 a2 = m"
- using a1_min a2_min unfolding A1_def A2_def by auto
- have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
- (* *)
- moreover
- {fix b1 b2 assume ***: "(b1,b2) \<in> D"
- hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
- have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
- using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
- have "((a1,a2),(b1,b2)) \<in> bsqr r"
- proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
- assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
- thus ?thesis unfolding bsqr_def using 4 5 by auto
- next
- assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
- hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
- hence 6: "(a1,b1) \<in> r" using a1_min by auto
- show ?thesis
- proof(cases "a1 = b1")
- assume Case21: "a1 \<noteq> b1"
- thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
- next
- assume Case22: "a1 = b1"
- hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
- hence 7: "(a2,b2) \<in> r" using a2_min by auto
- thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
- qed
- qed
- }
- (* *)
- ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
-qed
-
-
-lemma bsqr_max2:
-assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
-shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
-proof-
- have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
- using LEQ unfolding Field_def by auto
- hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
- hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
- using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
- moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
- using LEQ unfolding bsqr_def by auto
- ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
-qed
-
-
-lemma bsqr_ofilter:
-assumes WELL: "Well_order r" and
- OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
- NE: "\<not> (\<exists>a. Field r = rel.under r a)"
-shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
-proof-
- let ?r' = "bsqr r"
- have Well: "wo_rel r" using WELL wo_rel_def by blast
- hence Trans: "trans r" using wo_rel.TRANS by blast
- have Well': "Well_order ?r' \<and> wo_rel ?r'"
- using WELL bsqr_Well_order wo_rel_def by blast
- (* *)
- have "D < Field ?r'" unfolding Field_bsqr using SUB .
- with OF obtain a1 and a2 where
- "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
- using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
- hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
- let ?m = "wo_rel.max2 r a1 a2"
- have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
- proof(unfold 1)
- {fix b1 b2
- let ?n = "wo_rel.max2 r b1 b2"
- assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
- hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
- unfolding rel.underS_def by blast
- hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
- moreover
- {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
- hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
- hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
- using Well by (simp add: wo_rel.max2_greater)
- }
- ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
- using Trans trans_def[of r] by blast
- hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
- thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
- qed
- moreover have "wo_rel.ofilter r (rel.under r ?m)"
- using Well by (simp add: wo_rel.under_ofilter)
- moreover have "rel.under r ?m < Field r"
- using NE rel.under_Field[of r ?m] by blast
- ultimately show ?thesis by blast
-qed
-
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Constructions_on_Wellorders_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,1621 @@
+(* Title: HOL/Cardinals/Constructions_on_Wellorders_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Constructions on wellorders (FP).
+*)
+
+header {* Constructions on Wellorders (FP) *}
+
+theory Constructions_on_Wellorders_FP
+imports Wellorder_Embedding_FP
+begin
+
+
+text {* In this section, we study basic constructions on well-orders, such as restriction to
+a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
+and bounded square. We also define between well-orders
+the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
+@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
+@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}). We study the
+connections between these relations, order filters, and the aforementioned constructions.
+A main result of this section is that @{text "<o"} is well-founded. *}
+
+
+subsection {* Restriction to a set *}
+
+
+abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
+where "Restr r A \<equiv> r Int (A \<times> A)"
+
+
+lemma Restr_subset:
+"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
+by blast
+
+
+lemma Restr_Field: "Restr r (Field r) = r"
+unfolding Field_def by auto
+
+
+lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
+unfolding refl_on_def Field_def by auto
+
+
+lemma antisym_Restr:
+"antisym r \<Longrightarrow> antisym(Restr r A)"
+unfolding antisym_def Field_def by auto
+
+
+lemma Total_Restr:
+"Total r \<Longrightarrow> Total(Restr r A)"
+unfolding total_on_def Field_def by auto
+
+
+lemma trans_Restr:
+"trans r \<Longrightarrow> trans(Restr r A)"
+unfolding trans_def Field_def by blast
+
+
+lemma Preorder_Restr:
+"Preorder r \<Longrightarrow> Preorder(Restr r A)"
+unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
+
+
+lemma Partial_order_Restr:
+"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
+unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
+
+
+lemma Linear_order_Restr:
+"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
+unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
+
+
+lemma Well_order_Restr:
+assumes "Well_order r"
+shows "Well_order(Restr r A)"
+proof-
+ have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
+ hence "wf(Restr r A - Id)" using assms
+ using well_order_on_def wf_subset by blast
+ thus ?thesis using assms unfolding well_order_on_def
+ by (simp add: Linear_order_Restr)
+qed
+
+
+lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
+by (auto simp add: Field_def)
+
+
+lemma Refl_Field_Restr:
+"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
+unfolding refl_on_def Field_def by blast
+
+
+lemma Refl_Field_Restr2:
+"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: Refl_Field_Restr)
+
+
+lemma well_order_on_Restr:
+assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
+shows "well_order_on A (Restr r A)"
+using assms
+using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
+ order_on_defs[of "Field r" r] by auto
+
+
+subsection {* Order filters versus restrictions and embeddings *}
+
+
+lemma Field_Restr_ofilter:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
+
+
+lemma ofilter_Restr_under:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
+shows "rel.under (Restr r A) a = rel.under r a"
+using assms wo_rel_def
+proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix b assume *: "a \<in> A" and "(b,a) \<in> r"
+ hence "b \<in> rel.under r a \<and> a \<in> Field r"
+ unfolding rel.under_def using Field_def by fastforce
+ thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_embed:
+assumes "Well_order r"
+shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
+proof
+ assume *: "wo_rel.ofilter r A"
+ show "A \<le> Field r \<and> embed (Restr r A) r id"
+ proof(unfold embed_def, auto)
+ fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
+ by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ next
+ fix a assume "a \<in> Field (Restr r A)"
+ thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
+ by (simp add: ofilter_Restr_under Field_Restr_ofilter)
+ qed
+next
+ assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
+ hence "Field(Restr r A) \<le> Field r"
+ using assms embed_Field[of "Restr r A" r id] id_def
+ Well_order_Restr[of r] by auto
+ {fix a assume "a \<in> A"
+ hence "a \<in> Field(Restr r A)" using * assms
+ by (simp add: order_on_defs Refl_Field_Restr2)
+ hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
+ using * unfolding embed_def by auto
+ hence "rel.under r a \<le> rel.under (Restr r A) a"
+ unfolding bij_betw_def by auto
+ also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
+ also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
+ finally have "rel.under r a \<le> A" .
+ }
+ thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_Restr_Int:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter (Restr r B) (A Int B)"
+proof-
+ let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence Field: "Field ?rB = Field r Int B"
+ using Refl_Field_Restr by blast
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis using WellB assms
+ proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix a assume "a \<in> A" and *: "a \<in> B"
+ hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
+ with * show "a \<in> Field ?rB" using Field by auto
+ next
+ fix a b assume "a \<in> A" and "(b,a) \<in> r"
+ thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
+ qed
+qed
+
+
+lemma ofilter_Restr_subset:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
+shows "wo_rel.ofilter (Restr r B) A"
+proof-
+ have "A Int B = A" using SUB by blast
+ thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
+qed
+
+
+lemma ofilter_subset_embed:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence FieldA: "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ have FieldB: "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis
+ proof
+ assume *: "A \<le> B"
+ hence "wo_rel.ofilter (Restr r B) A" using assms
+ by (simp add: ofilter_Restr_subset)
+ hence "embed (Restr ?rB A) (Restr r B) id"
+ using WellB ofilter_embed[of "?rB" A] by auto
+ thus "embed (Restr r A) (Restr r B) id"
+ using * by (simp add: Restr_subset)
+ next
+ assume *: "embed (Restr r A) (Restr r B) id"
+ {fix a assume **: "a \<in> A"
+ hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
+ with ** FieldA have "a \<in> Field ?rA" by auto
+ hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
+ hence "a \<in> B" using FieldB by auto
+ }
+ thus "A \<le> B" by blast
+ qed
+qed
+
+
+lemma ofilter_subset_embedS_iso:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
+ ((A = B) = (iso (Restr r A) (Restr r B) id))"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ hence FieldA: "Field ?rA = A" using OFA Well
+ by (auto simp add: wo_rel.ofilter_def)
+ have "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ hence FieldB: "Field ?rB = B" using OFB Well
+ by (auto simp add: wo_rel.ofilter_def)
+ (* Main proof *)
+ show ?thesis unfolding embedS_def iso_def
+ using assms ofilter_subset_embed[of r A B]
+ FieldA FieldB bij_betw_id_iff[of A B] by auto
+qed
+
+
+lemma ofilter_subset_embedS:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = embedS (Restr r A) (Restr r B) id"
+using assms by (simp add: ofilter_subset_embedS_iso)
+
+
+lemma embed_implies_iso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r' r f"
+shows "iso r' (Restr r (f ` (Field r'))) f"
+proof-
+ let ?A' = "Field r'"
+ let ?r'' = "Restr r (f ` ?A')"
+ have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
+ have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter by blast
+ hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
+ hence "bij_betw f ?A' (Field ?r'')"
+ using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
+ moreover
+ {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
+ unfolding Field_def by auto
+ hence "compat r' ?r'' f"
+ using assms embed_iff_compat_inj_on_ofilter
+ unfolding compat_def by blast
+ }
+ ultimately show ?thesis using WELL' 0 iso_iff3 by blast
+qed
+
+
+subsection {* The strict inclusion on proper ofilters is well-founded *}
+
+
+definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
+where
+"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
+ wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
+
+
+lemma wf_ofilterIncl:
+assumes WELL: "Well_order r"
+shows "wf(ofilterIncl r)"
+proof-
+ have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
+ let ?h = "(\<lambda> A. wo_rel.suc r A)"
+ let ?rS = "r - Id"
+ have "wf ?rS" using WELL by (simp add: order_on_defs)
+ moreover
+ have "compat (ofilterIncl r) ?rS ?h"
+ proof(unfold compat_def ofilterIncl_def,
+ intro allI impI, simp, elim conjE)
+ fix A B
+ assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
+ **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
+ then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
+ 1: "A = rel.underS r a \<and> B = rel.underS r b"
+ using Well by (auto simp add: wo_rel.ofilter_underS_Field)
+ hence "a \<noteq> b" using *** by auto
+ moreover
+ have "(a,b) \<in> r" using 0 1 Lo ***
+ by (auto simp add: rel.underS_incl_iff)
+ moreover
+ have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
+ using Well 0 1 by (simp add: wo_rel.suc_underS)
+ ultimately
+ show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
+ by simp
+ qed
+ ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
+qed
+
+
+
+subsection {* Ordering the well-orders by existence of embeddings *}
+
+
+text {* We define three relations between well-orders:
+\begin{itemize}
+\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
+\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
+\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
+\end{itemize}
+%
+The prefix "ord" and the index "o" in these names stand for "ordinal-like".
+These relations shall be proved to be inter-connected in a similar fashion as the trio
+@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
+*}
+
+
+definition ordLeq :: "('a rel * 'a' rel) set"
+where
+"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
+
+
+abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
+where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
+
+
+abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
+where "r \<le>o r' \<equiv> r <=o r'"
+
+
+definition ordLess :: "('a rel * 'a' rel) set"
+where
+"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
+
+abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
+where "r <o r' \<equiv> (r,r') \<in> ordLess"
+
+
+definition ordIso :: "('a rel * 'a' rel) set"
+where
+"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
+
+abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
+where "r =o r' \<equiv> (r,r') \<in> ordIso"
+
+
+lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
+
+lemma ordLeq_Well_order_simp:
+assumes "r \<le>o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLeq_def by simp
+
+
+text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
+on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
+restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
+to @{text "'a rel rel"}. *}
+
+
+lemma ordLeq_reflexive:
+"Well_order r \<Longrightarrow> r \<le>o r"
+unfolding ordLeq_def using id_embed[of r] by blast
+
+
+lemma ordLeq_transitive[trans]:
+assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "embed r r' f" and "embed r' r'' f'"
+ using * ** unfolding ordLeq_def by blast
+ hence "embed r r'' (f' o f)"
+ using comp_embed[of r r' f r'' f'] by auto
+ thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
+qed
+
+
+lemma ordLeq_total:
+"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
+unfolding ordLeq_def using wellorders_totally_ordered by blast
+
+
+lemma ordIso_reflexive:
+"Well_order r \<Longrightarrow> r =o r"
+unfolding ordIso_def using id_iso[of r] by blast
+
+
+lemma ordIso_transitive[trans]:
+assumes *: "r =o r'" and **: "r' =o r''"
+shows "r =o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "iso r r' f" and 3: "iso r' r'' f'"
+ using * ** unfolding ordIso_def by auto
+ hence "iso r r'' (f' o f)"
+ using comp_iso[of r r' f r'' f'] by auto
+ thus "r =o r''" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma ordIso_symmetric:
+assumes *: "r =o r'"
+shows "r' =o r"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ using * by (auto simp add: ordIso_def iso_def)
+ let ?f' = "inv_into (Field r) f"
+ have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
+ using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
+ thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
+qed
+
+
+lemma ordLeq_ordLess_trans[trans]:
+assumes "r \<le>o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embed_comp_embedS by blast
+qed
+
+
+lemma ordLess_ordLeq_trans[trans]:
+assumes "r <o r'" and " r' \<le>o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embedS_comp_embed by blast
+qed
+
+
+lemma ordLeq_ordIso_trans[trans]:
+assumes "r \<le>o r'" and " r' =o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using embed_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLeq_trans[trans]:
+assumes "r =o r'" and " r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using iso_comp_embed by blast
+qed
+
+
+lemma ordLess_ordIso_trans[trans]:
+assumes "r <o r'" and " r' =o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using embedS_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLess_trans[trans]:
+assumes "r =o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using iso_comp_embedS by blast
+qed
+
+
+lemma ordLess_not_embed:
+assumes "r <o r'"
+shows "\<not>(\<exists>f'. embed r' r f')"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
+ 3: " \<not> bij_betw f (Field r) (Field r')"
+ using assms unfolding ordLess_def by (auto simp add: embedS_def)
+ {fix f' assume *: "embed r' r f'"
+ hence "bij_betw f (Field r) (Field r')" using 1 2
+ by (simp add: embed_bothWays_Field_bij_betw)
+ with 3 have False by contradiction
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma ordLess_Field:
+assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
+shows "\<not> (f`(Field r1) = Field r2)"
+proof-
+ let ?A1 = "Field r1" let ?A2 = "Field r2"
+ obtain g where
+ 0: "Well_order r1 \<and> Well_order r2" and
+ 1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
+ using OL unfolding ordLess_def by (auto simp add: embedS_def)
+ hence "\<forall>a \<in> ?A1. f a = g a"
+ using 0 EMB embed_unique[of r1] by auto
+ hence "\<not>(bij_betw f ?A1 ?A2)"
+ using 1 bij_betw_cong[of ?A1] by blast
+ moreover
+ have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
+ ultimately show ?thesis by (simp add: bij_betw_def)
+qed
+
+
+lemma ordLess_iff:
+"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
+proof
+ assume *: "r <o r'"
+ hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
+ with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ unfolding ordLess_def by auto
+next
+ assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ then obtain f where 1: "embed r r' f"
+ using wellorders_totally_ordered[of r r'] by blast
+ moreover
+ {assume "bij_betw f (Field r) (Field r')"
+ with * 1 have "embed r' r (inv_into (Field r) f) "
+ using inv_into_Field_embed_bij_betw[of r r' f] by auto
+ with * have False by blast
+ }
+ ultimately show "(r,r') \<in> ordLess"
+ unfolding ordLess_def using * by (fastforce simp add: embedS_def)
+qed
+
+
+lemma ordLess_irreflexive: "\<not> r <o r"
+proof
+ assume "r <o r"
+ hence "Well_order r \<and> \<not>(\<exists>f. embed r r f)"
+ unfolding ordLess_iff ..
+ moreover have "embed r r id" using id_embed[of r] .
+ ultimately show False by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_or_ordIso:
+"r \<le>o r' = (r <o r' \<or> r =o r')"
+unfolding ordRels_def embedS_defs iso_defs by blast
+
+
+lemma ordIso_iff_ordLeq:
+"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
+proof
+ assume "r =o r'"
+ then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ unfolding ordIso_def iso_defs by auto
+ hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
+ by (simp add: inv_into_Field_embed_bij_betw)
+ thus "r \<le>o r' \<and> r' \<le>o r"
+ unfolding ordLeq_def using 1 by auto
+next
+ assume "r \<le>o r' \<and> r' \<le>o r"
+ then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> embed r' r g"
+ unfolding ordLeq_def by auto
+ hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
+ thus "r =o r'" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma not_ordLess_ordLeq:
+"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
+using ordLess_ordLeq_trans ordLess_irreflexive by blast
+
+
+lemma ordLess_or_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r <o r' \<or> r' \<le>o r"
+proof-
+ have "r \<le>o r' \<or> r' \<le>o r"
+ using assms by (simp add: ordLeq_total)
+ moreover
+ {assume "\<not> r <o r' \<and> r \<le>o r'"
+ hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
+ hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma not_ordLess_ordIso:
+"r <o r' \<Longrightarrow> \<not> r =o r'"
+using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
+
+
+lemma not_ordLeq_iff_ordLess:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' \<le>o r) = (r <o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma not_ordLess_iff_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' <o r) = (r \<le>o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma ordLess_transitive[trans]:
+"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
+using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
+
+
+corollary ordLess_trans: "trans ordLess"
+unfolding trans_def using ordLess_transitive by blast
+
+
+lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
+
+
+lemma ordIso_imp_ordLeq:
+"r =o r' \<Longrightarrow> r \<le>o r'"
+using ordIso_iff_ordLeq by blast
+
+
+lemma ordLess_imp_ordLeq:
+"r <o r' \<Longrightarrow> r \<le>o r'"
+using ordLeq_iff_ordLess_or_ordIso by blast
+
+
+lemma ofilter_subset_ordLeq:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
+proof
+ assume "A \<le> B"
+ thus "Restr r A \<le>o Restr r B"
+ unfolding ordLeq_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
+next
+ assume *: "Restr r A \<le>o Restr r B"
+ then obtain f where "embed (Restr r A) (Restr r B) f"
+ unfolding ordLeq_def by blast
+ {assume "B < A"
+ hence "Restr r B <o Restr r A"
+ unfolding ordLess_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
+ hence False using * not_ordLess_ordLeq by blast
+ }
+ thus "A \<le> B" using OFA OFB WELL
+ wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
+qed
+
+
+lemma ofilter_subset_ordLess:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = (Restr r A <o Restr r B)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have 1: "Well_order ?rA \<and> Well_order ?rB"
+ using WELL Well_order_Restr by blast
+ have "(A < B) = (\<not> B \<le> A)" using assms
+ wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
+ also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
+ using assms ofilter_subset_ordLeq by blast
+ also have "\<dots> = (Restr r A <o Restr r B)"
+ using 1 not_ordLeq_iff_ordLess by blast
+ finally show ?thesis .
+qed
+
+
+lemma ofilter_ordLess:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
+by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
+ wo_rel_def Restr_Field)
+
+
+corollary underS_Restr_ordLess:
+assumes "Well_order r" and "Field r \<noteq> {}"
+shows "Restr r (rel.underS r a) <o r"
+proof-
+ have "rel.underS r a < Field r" using assms
+ by (simp add: rel.underS_Field3)
+ thus ?thesis using assms
+ by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
+qed
+
+
+lemma embed_ordLess_ofilterIncl:
+assumes
+ OL12: "r1 <o r2" and OL23: "r2 <o r3" and
+ EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
+shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
+proof-
+ have OL13: "r1 <o r3"
+ using OL12 OL23 using ordLess_transitive by auto
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A3 ="Field r3"
+ obtain f12 g23 where
+ 0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
+ 1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
+ 2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
+ using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
+ hence "\<forall>a \<in> ?A2. f23 a = g23 a"
+ using EMB23 embed_unique[of r2 r3] by blast
+ hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
+ using 2 bij_betw_cong[of ?A2 f23 g23] by blast
+ (* *)
+ have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
+ using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
+ using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 6: "wo_rel.ofilter r3 (f13 ` ?A1) \<and> f13 ` ?A1 \<noteq> ?A3"
+ using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
+ (* *)
+ have "f12 ` ?A1 < ?A2"
+ using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ moreover have "inj_on f23 ?A2"
+ using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
+ ultimately
+ have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
+ moreover
+ {have "embed r1 r3 (f23 o f12)"
+ using 1 EMB23 0 by (auto simp add: comp_embed)
+ hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
+ using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
+ hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
+ }
+ ultimately
+ have "f13 ` ?A1 < f23 ` ?A2" by simp
+ (* *)
+ with 5 6 show ?thesis
+ unfolding ofilterIncl_def by auto
+qed
+
+
+lemma ordLess_iff_ordIso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
+proof(auto)
+ fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
+ hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
+ thus "r' <o r" using ** ordIso_ordLess_trans by blast
+next
+ assume "r' <o r"
+ then obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
+ unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
+ hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
+ then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
+ using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
+ have "iso r' (Restr r (f ` (Field r'))) f"
+ using embed_implies_iso_Restr 2 assms by blast
+ moreover have "Well_order (Restr r (f ` (Field r')))"
+ using WELL Well_order_Restr by blast
+ ultimately have "r' =o Restr r (f ` (Field r'))"
+ using WELL' unfolding ordIso_def by auto
+ hence "r' =o Restr r (rel.underS r a)" using 4 by auto
+ thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
+qed
+
+
+lemma internalize_ordLess:
+"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
+proof
+ assume *: "r' <o r"
+ hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
+ with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
+ using ordLess_iff_ordIso_Restr by blast
+ let ?p = "Restr r (rel.underS r a)"
+ have "wo_rel.ofilter r (rel.underS r a)" using 0
+ by (simp add: wo_rel_def wo_rel.underS_ofilter)
+ hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
+ hence "Field ?p < Field r" using rel.underS_Field2 1 by fast
+ moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
+ ultimately
+ show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
+next
+ assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
+ thus "r' <o r" using ordIso_ordLess_trans by blast
+qed
+
+
+lemma internalize_ordLeq:
+"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
+proof
+ assume *: "r' \<le>o r"
+ moreover
+ {assume "r' <o r"
+ then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
+ using internalize_ordLess[of r' r] by blast
+ hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+ }
+ moreover
+ have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
+ ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+next
+ assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
+proof(auto)
+ assume *: "r \<le>o r'"
+ fix a assume "a \<in> Field r"
+ hence "Restr r (rel.underS r a) <o r"
+ using WELL underS_Restr_ordLess[of r] by blast
+ thus "Restr r (rel.underS r a) <o r'"
+ using * ordLess_ordLeq_trans by blast
+next
+ assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
+ {assume "r' <o r"
+ then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
+ using assms ordLess_iff_ordIso_Restr by blast
+ hence False using * not_ordLess_ordIso ordIso_symmetric by blast
+ }
+ thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
+qed
+
+
+lemma finite_ordLess_infinite:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ FIN: "finite(Field r)" and INF: "infinite(Field r')"
+shows "r <o r'"
+proof-
+ {assume "r' \<le>o r"
+ then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
+ unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
+ hence False using finite_imageD finite_subset FIN INF by metis
+ }
+ thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
+qed
+
+
+lemma finite_well_order_on_ordIso:
+assumes FIN: "finite A" and
+ WELL: "well_order_on A r" and WELL': "well_order_on A r'"
+shows "r =o r'"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using assms rel.well_order_on_Well_order by blast
+ moreover
+ have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
+ \<longrightarrow> r =o r'"
+ proof(clarify)
+ fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
+ have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using * ** rel.well_order_on_Well_order by blast
+ assume "r \<le>o r'"
+ then obtain f where 1: "embed r r' f" and
+ "inj_on f A \<and> f ` A \<le> A"
+ unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
+ hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
+ thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
+ qed
+ ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by metis
+qed
+
+
+subsection{* @{text "<o"} is well-founded *}
+
+
+text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
+on the restricted type @{text "'a rel rel"}. We prove this by first showing that, for any set
+of well-orders all embedded in a fixed well-order, the function mapping each well-order
+in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
+{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
+
+
+definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
+where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
+
+
+lemma ord_to_filter_compat:
+"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
+ (ofilterIncl r0)
+ (ord_to_filter r0)"
+proof(unfold compat_def ord_to_filter_def, clarify)
+ fix r1::"'a rel" and r2::"'a rel"
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A0 ="Field r0"
+ let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
+ let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
+ assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
+ hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
+ by (auto simp add: ordLess_def embedS_def)
+ hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
+ thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
+ using * ** by (simp add: embed_ordLess_ofilterIncl)
+qed
+
+
+theorem wf_ordLess: "wf ordLess"
+proof-
+ {fix r0 :: "('a \<times> 'a) set"
+ (* need to annotate here!*)
+ let ?ordLess = "ordLess::('d rel * 'd rel) set"
+ let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
+ {assume Case1: "Well_order r0"
+ hence "wf ?R"
+ using wf_ofilterIncl[of r0]
+ compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
+ ord_to_filter_compat[of r0] by auto
+ }
+ moreover
+ {assume Case2: "\<not> Well_order r0"
+ hence "?R = {}" unfolding ordLess_def by auto
+ hence "wf ?R" using wf_empty by simp
+ }
+ ultimately have "wf ?R" by blast
+ }
+ thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
+qed
+
+corollary exists_minim_Well_order:
+assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
+shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+proof-
+ obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
+ using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
+ equals0I[of R] by blast
+ with not_ordLeq_iff_ordLess WELL show ?thesis by blast
+qed
+
+
+
+subsection {* Copy via direct images *}
+
+
+text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
+from @{text "Relation.thy"}. It is useful for transporting a well-order between
+different types. *}
+
+
+definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
+where
+"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
+
+
+lemma dir_image_Field:
+"Field(dir_image r f) \<le> f ` (Field r)"
+unfolding dir_image_def Field_def by auto
+
+
+lemma dir_image_minus_Id:
+"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
+unfolding inj_on_def Field_def dir_image_def by auto
+
+
+lemma Refl_dir_image:
+assumes "Refl r"
+shows "Refl(dir_image r f)"
+proof-
+ {fix a' b'
+ assume "(a',b') \<in> dir_image r f"
+ then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
+ unfolding dir_image_def by blast
+ hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
+ hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
+ with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
+ unfolding dir_image_def by auto
+ }
+ thus ?thesis
+ by(unfold refl_on_def Field_def Domain_def Range_def, auto)
+qed
+
+
+lemma trans_dir_image:
+assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
+shows "trans(dir_image r f)"
+proof(unfold trans_def, auto)
+ fix a' b' c'
+ assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
+ then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
+ 2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
+ unfolding dir_image_def by blast
+ hence "b1 \<in> Field r \<and> b2 \<in> Field r"
+ unfolding Field_def by auto
+ hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
+ hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
+ thus "(a',c') \<in> dir_image r f"
+ unfolding dir_image_def using 1 by auto
+qed
+
+
+lemma Preorder_dir_image:
+"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
+by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
+
+
+lemma antisym_dir_image:
+assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
+shows "antisym(dir_image r f)"
+proof(unfold antisym_def, auto)
+ fix a' b'
+ assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
+ then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
+ 2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
+ 3: "{a1,a2,b1,b2} \<le> Field r"
+ unfolding dir_image_def Field_def by blast
+ hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
+ hence "a1 = b2" using 2 AN unfolding antisym_def by auto
+ thus "a' = b'" using 1 by auto
+qed
+
+
+lemma Partial_order_dir_image:
+"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
+by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
+
+
+lemma Total_dir_image:
+assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
+shows "Total(dir_image r f)"
+proof(unfold total_on_def, intro ballI impI)
+ fix a' b'
+ assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
+ then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
+ using dir_image_Field[of r f] by blast
+ moreover assume "a' \<noteq> b'"
+ ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
+ hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
+ thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
+ using 1 unfolding dir_image_def by auto
+qed
+
+
+lemma Linear_order_dir_image:
+"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
+by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
+
+
+lemma wf_dir_image:
+assumes WF: "wf r" and INJ: "inj_on f (Field r)"
+shows "wf(dir_image r f)"
+proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
+ fix A'::"'b set"
+ assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
+ obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
+ have "A \<noteq> {} \<and> A \<le> Field r"
+ using A_def dir_image_Field[of r f] SUB NE by blast
+ then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
+ using WF unfolding wf_eq_minimal2 by metis
+ have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
+ proof(clarify)
+ fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
+ obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
+ 3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
+ using ** unfolding dir_image_def Field_def by blast
+ hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
+ hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
+ with 1 show False by auto
+ qed
+ thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
+ using A_def 1 by blast
+qed
+
+
+lemma Well_order_dir_image:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
+using assms unfolding well_order_on_def
+using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
+ dir_image_minus_Id[of f r]
+ subset_inj_on[of f "Field r" "Field(r - Id)"]
+ mono_Field[of "r - Id" r] by auto
+
+
+lemma dir_image_Field2:
+"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
+unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
+
+
+lemma dir_image_bij_betw:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
+unfolding bij_betw_def
+by (simp add: dir_image_Field2 order_on_defs)
+
+
+lemma dir_image_compat:
+"compat r (dir_image r f) f"
+unfolding compat_def dir_image_def by auto
+
+
+lemma dir_image_iso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> iso r (dir_image r f) f"
+using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
+
+
+lemma dir_image_ordIso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
+
+
+lemma Well_order_iso_copy:
+assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
+shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
+proof-
+ let ?r' = "dir_image r f"
+ have 1: "A = Field r \<and> Well_order r"
+ using WELL rel.well_order_on_Well_order by blast
+ hence 2: "iso r ?r' f"
+ using dir_image_iso using BIJ unfolding bij_betw_def by auto
+ hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
+ hence "Field ?r' = A'"
+ using 1 BIJ unfolding bij_betw_def by auto
+ moreover have "Well_order ?r'"
+ using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
+ ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
+qed
+
+
+
+subsection {* Bounded square *}
+
+
+text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
+order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
+following criteria (in this order):
+\begin{itemize}
+\item compare the maximums;
+\item compare the first components;
+\item compare the second components.
+\end{itemize}
+%
+The only application of this construction that we are aware of is
+at proving that the square of an infinite set has the same cardinal
+as that set. The essential property required there (and which is ensured by this
+construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
+in a product of proper filters on the original relation (assumed to be a well-order). *}
+
+
+definition bsqr :: "'a rel => ('a * 'a)rel"
+where
+"bsqr r = {((a1,a2),(b1,b2)).
+ {a1,a2,b1,b2} \<le> Field r \<and>
+ (a1 = b1 \<and> a2 = b2 \<or>
+ (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id
+ )}"
+
+
+lemma Field_bsqr:
+"Field (bsqr r) = Field r \<times> Field r"
+proof
+ show "Field (bsqr r) \<le> Field r \<times> Field r"
+ proof-
+ {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
+ moreover
+ have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
+ a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
+ ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
+ }
+ thus ?thesis unfolding Field_def by force
+ qed
+next
+ show "Field r \<times> Field r \<le> Field (bsqr r)"
+ proof(auto)
+ fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
+ hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
+ thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
+ qed
+qed
+
+
+lemma bsqr_Refl: "Refl(bsqr r)"
+by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
+
+
+lemma bsqr_Trans:
+assumes "Well_order r"
+shows "trans (bsqr r)"
+proof(unfold trans_def, auto)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ (* Main proof *)
+ fix a1 a2 b1 b2 c1 c2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "((a1,a2),(c1,c2)) \<in> bsqr r"
+ proof-
+ {assume Case1: "a1 = b1 \<and> a2 = b2"
+ hence ?thesis using ** by simp
+ }
+ moreover
+ {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case21: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ using Case2 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
+ hence ?thesis using Case2 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case31: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence "(a1,c1) \<in> r - Id"
+ using Case3 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ {assume Case41: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by force
+ }
+ moreover
+ {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ hence "(a2,c2) \<in> r - Id"
+ using Case4 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by auto
+ qed
+qed
+
+
+lemma bsqr_antisym:
+assumes "Well_order r"
+shows "antisym (bsqr r)"
+proof(unfold antisym_def, clarify)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
+ using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
+ (* Main proof *)
+ fix a1 a2 b1 b2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "a1 = b1 \<and> a2 = b2"
+ proof-
+ {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case1 IrrS by blast
+ }
+ moreover
+ {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
+ hence False using Case1 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case2 by auto
+ }
+ moreover
+ {assume Case22: "(b1,a1) \<in> r - Id"
+ hence False using Case2 IrrS by blast
+ }
+ moreover
+ {assume Case23: "b1 = a1"
+ hence False using Case2 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ moreover
+ {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case32: "(b1,a1) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case33: "(b2,a2) \<in> r - Id"
+ hence False using Case3 IrrS by blast
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by blast
+ qed
+qed
+
+
+lemma bsqr_Total:
+assumes "Well_order r"
+shows "Total(bsqr r)"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+ using wo_rel.TOTALS by auto
+ (* Main proof *)
+ {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
+ hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
+ using Field_bsqr by blast
+ have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
+ proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
+ (* Why didn't clarsimp simp add: Well 0 do the same job? *)
+ assume Case1: "(a1,a2) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case11: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case112: "a2 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
+ next
+ assume Case1122: "a1 = b1"
+ thus ?thesis using Case112 by auto
+ qed
+ qed
+ next
+ assume Case12: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case122: "a2 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
+ next
+ assume Case1222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
+ next
+ assume Case12222: "a2 = b2"
+ thus ?thesis using Case122 Case1222 by auto
+ qed
+ qed
+ qed
+ qed
+ next
+ assume Case2: "(a2,a1) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case212: "a1 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
+ next
+ assume Case2122: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
+ next
+ assume Case21222: "a2 = b2"
+ thus ?thesis using Case2122 Case212 by auto
+ qed
+ qed
+ qed
+ next
+ assume Case22: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
+ next
+ assume Case2222: "a2 = b2"
+ thus ?thesis using Case222 by auto
+ qed
+ qed
+ qed
+ qed
+ }
+ thus ?thesis unfolding total_on_def by fast
+qed
+
+
+lemma bsqr_Linear_order:
+assumes "Well_order r"
+shows "Linear_order(bsqr r)"
+unfolding order_on_defs
+using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
+
+
+lemma bsqr_Well_order:
+assumes "Well_order r"
+shows "Well_order(bsqr r)"
+using assms
+proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
+ have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ using assms well_order_on_def Linear_order_Well_order_iff by blast
+ fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
+ hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
+ (* *)
+ obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
+ have "M \<noteq> {}" using 1 M_def ** by auto
+ moreover
+ have "M \<le> Field r" unfolding M_def
+ using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A1 \<le> Field r" unfolding A1_def using 1 by auto
+ moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
+ ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A2 \<le> Field r" unfolding A2_def using 1 by auto
+ moreover have "A2 \<noteq> {}" unfolding A2_def
+ using m_min a1_min unfolding A1_def M_def by blast
+ ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ have 2: "wo_rel.max2 r a1 a2 = m"
+ using a1_min a2_min unfolding A1_def A2_def by auto
+ have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
+ (* *)
+ moreover
+ {fix b1 b2 assume ***: "(b1,b2) \<in> D"
+ hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
+ have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+ using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
+ have "((a1,a2),(b1,b2)) \<in> bsqr r"
+ proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
+ assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
+ thus ?thesis unfolding bsqr_def using 4 5 by auto
+ next
+ assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
+ hence 6: "(a1,b1) \<in> r" using a1_min by auto
+ show ?thesis
+ proof(cases "a1 = b1")
+ assume Case21: "a1 \<noteq> b1"
+ thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
+ next
+ assume Case22: "a1 = b1"
+ hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
+ hence 7: "(a2,b2) \<in> r" using a2_min by auto
+ thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
+ qed
+ qed
+ }
+ (* *)
+ ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
+qed
+
+
+lemma bsqr_max2:
+assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
+shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+proof-
+ have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
+ using LEQ unfolding Field_def by auto
+ hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
+ using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ using LEQ unfolding bsqr_def by auto
+ ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
+qed
+
+
+lemma bsqr_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
+ NE: "\<not> (\<exists>a. Field r = rel.under r a)"
+shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
+proof-
+ let ?r' = "bsqr r"
+ have Well: "wo_rel r" using WELL wo_rel_def by blast
+ hence Trans: "trans r" using wo_rel.TRANS by blast
+ have Well': "Well_order ?r' \<and> wo_rel ?r'"
+ using WELL bsqr_Well_order wo_rel_def by blast
+ (* *)
+ have "D < Field ?r'" unfolding Field_bsqr using SUB .
+ with OF obtain a1 and a2 where
+ "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
+ using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
+ hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
+ let ?m = "wo_rel.max2 r a1 a2"
+ have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
+ proof(unfold 1)
+ {fix b1 b2
+ let ?n = "wo_rel.max2 r b1 b2"
+ assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
+ hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
+ unfolding rel.underS_def by blast
+ hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
+ moreover
+ {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
+ hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
+ using Well by (simp add: wo_rel.max2_greater)
+ }
+ ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
+ using Trans trans_def[of r] by blast
+ hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
+ thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
+ qed
+ moreover have "wo_rel.ofilter r (rel.under r ?m)"
+ using Well by (simp add: wo_rel.under_ofilter)
+ moreover have "rel.under r ?m < Field r"
+ using NE rel.under_Field[of r ?m] by blast
+ ultimately show ?thesis by blast
+qed
+
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Fun_More.thy
--- a/src/HOL/Cardinals/Fun_More.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Fun_More.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,7 +8,7 @@
header {* More on Injections, Bijections and Inverses *}
theory Fun_More
-imports Fun_More_Base
+imports Fun_More_FP
begin
@@ -132,6 +132,18 @@
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:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Fun_More_Base.thy
--- a/src/HOL/Cardinals/Fun_More_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,252 +0,0 @@
-(* Title: HOL/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, 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 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
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Fun_More_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Fun_More_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,239 @@
+(* 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 "~~/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, 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 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_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 fastforce
+
+
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Order_Relation_More.thy
--- a/src/HOL/Cardinals/Order_Relation_More.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Order_Relation_More.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,64 +8,70 @@
header {* Basics on Order-Like Relations *}
theory Order_Relation_More
-imports Order_Relation_More_Base
+imports Order_Relation_More_FP
begin
subsection {* The upper and lower bounds operators *}
-lemma (in rel) aboveS_subset_above: "aboveS a \<le> above a"
+context rel
+begin
+
+lemma aboveS_subset_above: "aboveS a \<le> above a"
by(auto simp add: aboveS_def above_def)
-lemma (in rel) AboveS_subset_Above: "AboveS A \<le> Above A"
+lemma AboveS_subset_Above: "AboveS A \<le> Above A"
by(auto simp add: AboveS_def Above_def)
-lemma (in rel) UnderS_disjoint: "A Int (UnderS A) = {}"
+lemma UnderS_disjoint: "A Int (UnderS A) = {}"
by(auto simp add: UnderS_def)
-lemma (in rel) aboveS_notIn: "a \<notin> aboveS a"
+lemma aboveS_notIn: "a \<notin> aboveS a"
by(auto simp add: aboveS_def)
-lemma (in rel) Refl_above_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> above a"
+lemma Refl_above_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> above a"
by(auto simp add: refl_on_def above_def)
-lemma (in rel) in_Above_under: "a \<in> Field r \<Longrightarrow> a \<in> Above (under a)"
+lemma in_Above_under: "a \<in> Field r \<Longrightarrow> a \<in> Above (under a)"
by(auto simp add: Above_def under_def)
-lemma (in rel) in_Under_above: "a \<in> Field r \<Longrightarrow> a \<in> Under (above a)"
+lemma in_Under_above: "a \<in> Field r \<Longrightarrow> a \<in> Under (above a)"
by(auto simp add: Under_def above_def)
-lemma (in rel) in_UnderS_aboveS: "a \<in> Field r \<Longrightarrow> a \<in> UnderS (aboveS a)"
+lemma in_UnderS_aboveS: "a \<in> Field r \<Longrightarrow> a \<in> UnderS (aboveS a)"
by(auto simp add: UnderS_def aboveS_def)
-lemma (in rel) subset_Above_Under: "B \<le> Field r \<Longrightarrow> B \<le> Above (Under B)"
+lemma UnderS_subset_Under: "UnderS A \<le> Under A"
+by(auto simp add: UnderS_def Under_def)
+
+lemma subset_Above_Under: "B \<le> Field r \<Longrightarrow> B \<le> Above (Under B)"
by(auto simp add: Above_def Under_def)
-lemma (in rel) subset_Under_Above: "B \<le> Field r \<Longrightarrow> B \<le> Under (Above B)"
+lemma subset_Under_Above: "B \<le> Field r \<Longrightarrow> B \<le> Under (Above B)"
by(auto simp add: Under_def Above_def)
-lemma (in rel) subset_AboveS_UnderS: "B \<le> Field r \<Longrightarrow> B \<le> AboveS (UnderS B)"
+lemma subset_AboveS_UnderS: "B \<le> Field r \<Longrightarrow> B \<le> AboveS (UnderS B)"
by(auto simp add: AboveS_def UnderS_def)
-lemma (in rel) subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS (AboveS B)"
+lemma subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS (AboveS B)"
by(auto simp add: UnderS_def AboveS_def)
-lemma (in rel) Under_Above_Galois:
+lemma Under_Above_Galois:
"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> Above C) = (C \<le> Under B)"
by(unfold Above_def Under_def, blast)
-lemma (in rel) UnderS_AboveS_Galois:
+lemma UnderS_AboveS_Galois:
"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> AboveS C) = (C \<le> UnderS B)"
by(unfold AboveS_def UnderS_def, blast)
-lemma (in rel) Refl_above_aboveS:
+lemma Refl_above_aboveS:
assumes REFL: "Refl r" and IN: "a \<in> Field r"
shows "above a = aboveS a \<union> {a}"
proof(unfold above_def aboveS_def, auto)
show "(a,a) \<in> r" using REFL IN refl_on_def[of _ r] by blast
qed
-lemma (in rel) Linear_order_under_aboveS_Field:
+lemma Linear_order_under_aboveS_Field:
assumes LIN: "Linear_order r" and IN: "a \<in> Field r"
shows "Field r = under a \<union> aboveS a"
proof(unfold under_def aboveS_def, auto)
@@ -88,7 +94,7 @@
using LIN order_on_defs[of "Field r" r] refl_on_def[of "Field r" r] by blast
qed
-lemma (in rel) Linear_order_underS_above_Field:
+lemma Linear_order_underS_above_Field:
assumes LIN: "Linear_order r" and IN: "a \<in> Field r"
shows "Field r = underS a \<union> above a"
proof(unfold underS_def above_def, auto)
@@ -111,19 +117,25 @@
using LIN order_on_defs[of "Field r" r] refl_on_def[of "Field r" r] by blast
qed
-lemma (in rel) under_empty: "a \<notin> Field r \<Longrightarrow> under a = {}"
+lemma under_empty: "a \<notin> Field r \<Longrightarrow> under a = {}"
unfolding Field_def under_def by auto
-lemma (in rel) above_Field: "above a \<le> Field r"
+lemma Under_Field: "Under A \<le> Field r"
+by(unfold Under_def Field_def, auto)
+
+lemma UnderS_Field: "UnderS A \<le> Field r"
+by(unfold UnderS_def Field_def, auto)
+
+lemma above_Field: "above a \<le> Field r"
by(unfold above_def Field_def, auto)
-lemma (in rel) aboveS_Field: "aboveS a \<le> Field r"
+lemma aboveS_Field: "aboveS a \<le> Field r"
by(unfold aboveS_def Field_def, auto)
-lemma (in rel) Above_Field: "Above A \<le> Field r"
+lemma Above_Field: "Above A \<le> Field r"
by(unfold Above_def Field_def, auto)
-lemma (in rel) Refl_under_Under:
+lemma Refl_under_Under:
assumes REFL: "Refl r" and NE: "A \<noteq> {}"
shows "Under A = (\<Inter> a \<in> A. under a)"
proof
@@ -147,7 +159,7 @@
qed
qed
-lemma (in rel) Refl_underS_UnderS:
+lemma Refl_underS_UnderS:
assumes REFL: "Refl r" and NE: "A \<noteq> {}"
shows "UnderS A = (\<Inter> a \<in> A. underS a)"
proof
@@ -171,7 +183,7 @@
qed
qed
-lemma (in rel) Refl_above_Above:
+lemma Refl_above_Above:
assumes REFL: "Refl r" and NE: "A \<noteq> {}"
shows "Above A = (\<Inter> a \<in> A. above a)"
proof
@@ -195,7 +207,7 @@
qed
qed
-lemma (in rel) Refl_aboveS_AboveS:
+lemma Refl_aboveS_AboveS:
assumes REFL: "Refl r" and NE: "A \<noteq> {}"
shows "AboveS A = (\<Inter> a \<in> A. aboveS a)"
proof
@@ -219,31 +231,31 @@
qed
qed
-lemma (in rel) under_Under_singl: "under a = Under {a}"
+lemma under_Under_singl: "under a = Under {a}"
by(unfold Under_def under_def, auto simp add: Field_def)
-lemma (in rel) underS_UnderS_singl: "underS a = UnderS {a}"
+lemma underS_UnderS_singl: "underS a = UnderS {a}"
by(unfold UnderS_def underS_def, auto simp add: Field_def)
-lemma (in rel) above_Above_singl: "above a = Above {a}"
+lemma above_Above_singl: "above a = Above {a}"
by(unfold Above_def above_def, auto simp add: Field_def)
-lemma (in rel) aboveS_AboveS_singl: "aboveS a = AboveS {a}"
+lemma aboveS_AboveS_singl: "aboveS a = AboveS {a}"
by(unfold AboveS_def aboveS_def, auto simp add: Field_def)
-lemma (in rel) Under_decr: "A \<le> B \<Longrightarrow> Under B \<le> Under A"
+lemma Under_decr: "A \<le> B \<Longrightarrow> Under B \<le> Under A"
by(unfold Under_def, auto)
-lemma (in rel) UnderS_decr: "A \<le> B \<Longrightarrow> UnderS B \<le> UnderS A"
+lemma UnderS_decr: "A \<le> B \<Longrightarrow> UnderS B \<le> UnderS A"
by(unfold UnderS_def, auto)
-lemma (in rel) Above_decr: "A \<le> B \<Longrightarrow> Above B \<le> Above A"
+lemma Above_decr: "A \<le> B \<Longrightarrow> Above B \<le> Above A"
by(unfold Above_def, auto)
-lemma (in rel) AboveS_decr: "A \<le> B \<Longrightarrow> AboveS B \<le> AboveS A"
+lemma AboveS_decr: "A \<le> B \<Longrightarrow> AboveS B \<le> AboveS A"
by(unfold AboveS_def, auto)
-lemma (in rel) under_incl_iff:
+lemma under_incl_iff:
assumes TRANS: "trans r" and REFL: "Refl r" and IN: "a \<in> Field r"
shows "(under a \<le> under b) = ((a,b) \<in> r)"
proof
@@ -259,7 +271,7 @@
by (auto simp add: under_def)
qed
-lemma (in rel) above_decr:
+lemma above_decr:
assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
shows "above b \<le> above a"
proof(unfold above_def, auto)
@@ -268,7 +280,7 @@
show "(a,x) \<in> r" by blast
qed
-lemma (in rel) aboveS_decr:
+lemma aboveS_decr:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
REL: "(a,b) \<in> r"
shows "aboveS b \<le> aboveS a"
@@ -282,7 +294,7 @@
show "(a,x) \<in> r" by blast
qed
-lemma (in rel) under_trans:
+lemma under_trans:
assumes TRANS: "trans r" and
IN1: "a \<in> under b" and IN2: "b \<in> under c"
shows "a \<in> under c"
@@ -294,7 +306,7 @@
thus ?thesis unfolding under_def by simp
qed
-lemma (in rel) under_underS_trans:
+lemma under_underS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> under b" and IN2: "b \<in> underS c"
shows "a \<in> underS c"
@@ -312,7 +324,7 @@
from 1 3 show ?thesis unfolding underS_def by simp
qed
-lemma (in rel) underS_under_trans:
+lemma underS_under_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> underS b" and IN2: "b \<in> under c"
shows "a \<in> underS c"
@@ -330,7 +342,7 @@
from 1 3 show ?thesis unfolding underS_def by simp
qed
-lemma (in rel) underS_underS_trans:
+lemma underS_underS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> underS b" and IN2: "b \<in> underS c"
shows "a \<in> underS c"
@@ -340,7 +352,7 @@
with assms under_underS_trans show ?thesis by auto
qed
-lemma (in rel) above_trans:
+lemma above_trans:
assumes TRANS: "trans r" and
IN1: "b \<in> above a" and IN2: "c \<in> above b"
shows "c \<in> above a"
@@ -352,7 +364,7 @@
thus ?thesis unfolding above_def by simp
qed
-lemma (in rel) above_aboveS_trans:
+lemma above_aboveS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "b \<in> above a" and IN2: "c \<in> aboveS b"
shows "c \<in> aboveS a"
@@ -370,7 +382,7 @@
from 1 3 show ?thesis unfolding aboveS_def by simp
qed
-lemma (in rel) aboveS_above_trans:
+lemma aboveS_above_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "b \<in> aboveS a" and IN2: "c \<in> above b"
shows "c \<in> aboveS a"
@@ -388,7 +400,7 @@
from 1 3 show ?thesis unfolding aboveS_def by simp
qed
-lemma (in rel) aboveS_aboveS_trans:
+lemma aboveS_aboveS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "b \<in> aboveS a" and IN2: "c \<in> aboveS b"
shows "c \<in> aboveS a"
@@ -398,7 +410,22 @@
with assms above_aboveS_trans show ?thesis by auto
qed
-lemma (in rel) underS_Under_trans:
+lemma under_Under_trans:
+assumes TRANS: "trans r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> Under C"
+shows "a \<in> Under C"
+proof-
+ have "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)"
+ using IN1 IN2 under_def Under_def by blast
+ hence "\<forall>c \<in> C. (a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ moreover
+ have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast
+ ultimately
+ show ?thesis unfolding Under_def by blast
+qed
+
+lemma underS_Under_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> underS b" and IN2: "b \<in> Under C"
shows "a \<in> UnderS C"
@@ -426,7 +453,7 @@
using Under_def by auto
qed
-lemma (in rel) underS_UnderS_trans:
+lemma underS_UnderS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> underS b" and IN2: "b \<in> UnderS C"
shows "a \<in> UnderS C"
@@ -437,7 +464,7 @@
show ?thesis by auto
qed
-lemma (in rel) above_Above_trans:
+lemma above_Above_trans:
assumes TRANS: "trans r" and
IN1: "a \<in> above b" and IN2: "b \<in> Above C"
shows "a \<in> Above C"
@@ -452,7 +479,7 @@
show ?thesis unfolding Above_def by auto
qed
-lemma (in rel) aboveS_Above_trans:
+lemma aboveS_Above_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> aboveS b" and IN2: "b \<in> Above C"
shows "a \<in> AboveS C"
@@ -480,7 +507,7 @@
using Above_def by auto
qed
-lemma (in rel) above_AboveS_trans:
+lemma above_AboveS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> above b" and IN2: "b \<in> AboveS C"
shows "a \<in> AboveS C"
@@ -508,7 +535,7 @@
using Above_def by auto
qed
-lemma (in rel) aboveS_AboveS_trans:
+lemma aboveS_AboveS_trans:
assumes TRANS: "trans r" and ANTISYM: "antisym r" and
IN1: "a \<in> aboveS b" and IN2: "b \<in> AboveS C"
shows "a \<in> AboveS C"
@@ -519,6 +546,35 @@
show ?thesis by auto
qed
+lemma under_UnderS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> UnderS C"
+shows "a \<in> UnderS C"
+proof-
+ from IN2 have "b \<in> Under C"
+ using UnderS_subset_Under[of C] by blast
+ with assms under_Under_trans
+ have "a \<in> Under C" by blast
+ (* *)
+ moreover
+ have "a \<notin> C"
+ proof
+ assume *: "a \<in> C"
+ have 1: "(a,b) \<in> r"
+ using IN1 under_def[of b] by auto
+ have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r"
+ using IN2 UnderS_def[of C] by blast
+ with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast
+ with 1 ANTISYM antisym_def[of r]
+ show False by blast
+ qed
+ (* *)
+ ultimately
+ show ?thesis unfolding UnderS_def Under_def by fast
+qed
+
+end (* context rel *)
+
subsection {* Properties depending on more than one relation *}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Order_Relation_More_Base.thy
--- a/src/HOL/Cardinals/Order_Relation_More_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,286 +0,0 @@
-(* Title: HOL/Cardinals/Order_Relation_More_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Basics on order-like relations (base).
-*)
-
-header {* Basics on Order-Like Relations (Base) *}
-
-theory Order_Relation_More_Base
-imports "~~/src/HOL/Library/Order_Relation"
-begin
-
-
-text{* In this section, we develop basic concepts and results pertaining
-to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
-total relations. The development is placed on top of the definitions
-from the theory @{text "Order_Relation"}. We also
-further define upper and lower bounds operators. *}
-
-
-locale rel = fixes r :: "'a rel"
-
-text{* The following context encompasses all this section, except
-for its last subsection. In other words, for the rest of this section except its last
-subsection, we consider a fixed relation @{text "r"}. *}
-
-context rel
-begin
-
-
-subsection {* Auxiliaries *}
-
-
-lemma refl_on_domain:
-"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(auto simp add: refl_on_def)
-
-
-corollary well_order_on_domain:
-"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(simp add: refl_on_domain order_on_defs)
-
-
-lemma well_order_on_Field:
-"well_order_on A r \<Longrightarrow> A = Field r"
-by(auto simp add: refl_on_def Field_def order_on_defs)
-
-
-lemma well_order_on_Well_order:
-"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
-using well_order_on_Field by simp
-
-
-lemma Total_subset_Id:
-assumes TOT: "Total r" and SUB: "r \<le> Id"
-shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
-proof-
- {assume "r \<noteq> {}"
- then obtain a b where 1: "(a,b) \<in> r" by fast
- hence "a = b" using SUB by blast
- hence 2: "(a,a) \<in> r" using 1 by simp
- {fix c d assume "(c,d) \<in> r"
- hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
- hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
- ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
- using TOT unfolding total_on_def by blast
- hence "a = c \<and> a = d" using SUB by blast
- }
- hence "r \<le> {(a,a)}" by auto
- with 2 have "\<exists>a. r = {(a,a)}" by blast
- }
- thus ?thesis by blast
-qed
-
-
-lemma Linear_order_in_diff_Id:
-assumes LI: "Linear_order r" and
- IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
-shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
-using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
-
-
-subsection {* The upper and lower bounds operators *}
-
-
-text{* Here we define upper (``above") and lower (``below") bounds operators.
-We think of @{text "r"} as a {\em non-strict} relation. The suffix ``S"
-at the names of some operators indicates that the bounds are strict -- e.g.,
-@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}).
-Capitalization of the first letter in the name reminds that the operator acts on sets, rather
-than on individual elements. *}
-
-definition under::"'a \<Rightarrow> 'a set"
-where "under a \<equiv> {b. (b,a) \<in> r}"
-
-definition underS::"'a \<Rightarrow> 'a set"
-where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition Under::"'a set \<Rightarrow> 'a set"
-where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
-
-definition UnderS::"'a set \<Rightarrow> 'a set"
-where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition above::"'a \<Rightarrow> 'a set"
-where "above a \<equiv> {b. (a,b) \<in> r}"
-
-definition aboveS::"'a \<Rightarrow> 'a set"
-where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
-
-definition Above::"'a set \<Rightarrow> 'a set"
-where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
-
-definition AboveS::"'a set \<Rightarrow> 'a set"
-where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
-(* *)
-
-text{* Note: In the definitions of @{text "Above[S]"} and @{text "Under[S]"},
- we bounded comprehension by @{text "Field r"} in order to properly cover
- the case of @{text "A"} being empty. *}
-
-
-lemma UnderS_subset_Under: "UnderS A \<le> Under A"
-by(auto simp add: UnderS_def Under_def)
-
-
-lemma underS_subset_under: "underS a \<le> under a"
-by(auto simp add: underS_def under_def)
-
-
-lemma underS_notIn: "a \<notin> underS a"
-by(simp add: underS_def)
-
-
-lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a"
-by(simp add: refl_on_def under_def)
-
-
-lemma AboveS_disjoint: "A Int (AboveS A) = {}"
-by(auto simp add: AboveS_def)
-
-
-lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)"
-by(auto simp add: AboveS_def underS_def)
-
-
-lemma Refl_under_underS:
-assumes "Refl r" "a \<in> Field r"
-shows "under a = underS a \<union> {a}"
-unfolding under_def underS_def
-using assms refl_on_def[of _ r] by fastforce
-
-
-lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}"
-by (auto simp: Field_def underS_def)
-
-
-lemma under_Field: "under a \<le> Field r"
-by(unfold under_def Field_def, auto)
-
-
-lemma underS_Field: "underS a \<le> Field r"
-by(unfold underS_def Field_def, auto)
-
-
-lemma underS_Field2:
-"a \<in> Field r \<Longrightarrow> underS a < Field r"
-using assms underS_notIn underS_Field by blast
-
-
-lemma underS_Field3:
-"Field r \<noteq> {} \<Longrightarrow> underS a < Field r"
-by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
-
-
-lemma Under_Field: "Under A \<le> Field r"
-by(unfold Under_def Field_def, auto)
-
-
-lemma UnderS_Field: "UnderS A \<le> Field r"
-by(unfold UnderS_def Field_def, auto)
-
-
-lemma AboveS_Field: "AboveS A \<le> Field r"
-by(unfold AboveS_def Field_def, auto)
-
-
-lemma under_incr:
-assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
-shows "under a \<le> under b"
-proof(unfold under_def, auto)
- fix x assume "(x,a) \<in> r"
- with REL TRANS trans_def[of r]
- show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incr:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
- REL: "(a,b) \<in> r"
-shows "underS a \<le> underS b"
-proof(unfold underS_def, auto)
- assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
- with ANTISYM antisym_def[of r] REL
- show False by blast
-next
- fix x assume "x \<noteq> a" "(x,a) \<in> r"
- with REL TRANS trans_def[of r]
- show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incl_iff:
-assumes LO: "Linear_order r" and
- INa: "a \<in> Field r" and INb: "b \<in> Field r"
-shows "(underS a \<le> underS b) = ((a,b) \<in> r)"
-proof
- assume "(a,b) \<in> r"
- thus "underS a \<le> underS b" using LO
- by (simp add: order_on_defs underS_incr)
-next
- assume *: "underS a \<le> underS b"
- {assume "a = b"
- hence "(a,b) \<in> r" using assms
- by (simp add: order_on_defs refl_on_def)
- }
- moreover
- {assume "a \<noteq> b \<and> (b,a) \<in> r"
- hence "b \<in> underS a" unfolding underS_def by blast
- hence "b \<in> underS b" using * by blast
- hence False by (simp add: underS_notIn)
- }
- ultimately
- show "(a,b) \<in> r" using assms
- order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
-qed
-
-
-lemma under_Under_trans:
-assumes TRANS: "trans r" and
- IN1: "a \<in> under b" and IN2: "b \<in> Under C"
-shows "a \<in> Under C"
-proof-
- have "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)"
- using IN1 IN2 under_def Under_def by blast
- hence "\<forall>c \<in> C. (a,c) \<in> r"
- using TRANS trans_def[of r] by blast
- moreover
- have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast
- ultimately
- show ?thesis unfolding Under_def by blast
-qed
-
-
-lemma under_UnderS_trans:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
- IN1: "a \<in> under b" and IN2: "b \<in> UnderS C"
-shows "a \<in> UnderS C"
-proof-
- from IN2 have "b \<in> Under C"
- using UnderS_subset_Under[of C] by blast
- with assms under_Under_trans
- have "a \<in> Under C" by blast
- (* *)
- moreover
- have "a \<notin> C"
- proof
- assume *: "a \<in> C"
- have 1: "(a,b) \<in> r"
- using IN1 under_def[of b] by auto
- have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r"
- using IN2 UnderS_def[of C] by blast
- with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast
- with 1 ANTISYM antisym_def[of r]
- show False by blast
- qed
- (* *)
- ultimately
- show ?thesis unfolding UnderS_def Under_def by fast
-qed
-
-
-end (* context rel *)
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Order_Relation_More_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Order_Relation_More_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,230 @@
+(* Title: HOL/Cardinals/Order_Relation_More_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Basics on order-like relations (FP).
+*)
+
+header {* Basics on Order-Like Relations (FP) *}
+
+theory Order_Relation_More_FP
+imports "~~/src/HOL/Library/Order_Relation"
+begin
+
+
+text{* In this section, we develop basic concepts and results pertaining
+to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
+total relations. The development is placed on top of the definitions
+from the theory @{text "Order_Relation"}. We also
+further define upper and lower bounds operators. *}
+
+
+locale rel = fixes r :: "'a rel"
+
+text{* The following context encompasses all this section, except
+for its last subsection. In other words, for the rest of this section except its last
+subsection, we consider a fixed relation @{text "r"}. *}
+
+context rel
+begin
+
+
+subsection {* Auxiliaries *}
+
+
+lemma refl_on_domain:
+"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(auto simp add: refl_on_def)
+
+
+corollary well_order_on_domain:
+"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(simp add: refl_on_domain order_on_defs)
+
+
+lemma well_order_on_Field:
+"well_order_on A r \<Longrightarrow> A = Field r"
+by(auto simp add: refl_on_def Field_def order_on_defs)
+
+
+lemma well_order_on_Well_order:
+"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
+using well_order_on_Field by simp
+
+
+lemma Total_subset_Id:
+assumes TOT: "Total r" and SUB: "r \<le> Id"
+shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
+proof-
+ {assume "r \<noteq> {}"
+ then obtain a b where 1: "(a,b) \<in> r" by fast
+ hence "a = b" using SUB by blast
+ hence 2: "(a,a) \<in> r" using 1 by simp
+ {fix c d assume "(c,d) \<in> r"
+ hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
+ hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
+ ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
+ using TOT unfolding total_on_def by blast
+ hence "a = c \<and> a = d" using SUB by blast
+ }
+ hence "r \<le> {(a,a)}" by auto
+ with 2 have "\<exists>a. r = {(a,a)}" by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma Linear_order_in_diff_Id:
+assumes LI: "Linear_order r" and
+ IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
+shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
+using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
+
+
+subsection {* The upper and lower bounds operators *}
+
+
+text{* Here we define upper (``above") and lower (``below") bounds operators.
+We think of @{text "r"} as a {\em non-strict} relation. The suffix ``S"
+at the names of some operators indicates that the bounds are strict -- e.g.,
+@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}).
+Capitalization of the first letter in the name reminds that the operator acts on sets, rather
+than on individual elements. *}
+
+definition under::"'a \<Rightarrow> 'a set"
+where "under a \<equiv> {b. (b,a) \<in> r}"
+
+definition underS::"'a \<Rightarrow> 'a set"
+where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
+
+definition Under::"'a set \<Rightarrow> 'a set"
+where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
+
+definition UnderS::"'a set \<Rightarrow> 'a set"
+where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
+
+definition above::"'a \<Rightarrow> 'a set"
+where "above a \<equiv> {b. (a,b) \<in> r}"
+
+definition aboveS::"'a \<Rightarrow> 'a set"
+where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
+
+definition Above::"'a set \<Rightarrow> 'a set"
+where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
+
+definition AboveS::"'a set \<Rightarrow> 'a set"
+where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
+(* *)
+
+text{* Note: In the definitions of @{text "Above[S]"} and @{text "Under[S]"},
+ we bounded comprehension by @{text "Field r"} in order to properly cover
+ the case of @{text "A"} being empty. *}
+
+
+lemma underS_subset_under: "underS a \<le> under a"
+by(auto simp add: underS_def under_def)
+
+
+lemma underS_notIn: "a \<notin> underS a"
+by(simp add: underS_def)
+
+
+lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a"
+by(simp add: refl_on_def under_def)
+
+
+lemma AboveS_disjoint: "A Int (AboveS A) = {}"
+by(auto simp add: AboveS_def)
+
+
+lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)"
+by(auto simp add: AboveS_def underS_def)
+
+
+lemma Refl_under_underS:
+assumes "Refl r" "a \<in> Field r"
+shows "under a = underS a \<union> {a}"
+unfolding under_def underS_def
+using assms refl_on_def[of _ r] by fastforce
+
+
+lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}"
+by (auto simp: Field_def underS_def)
+
+
+lemma under_Field: "under a \<le> Field r"
+by(unfold under_def Field_def, auto)
+
+
+lemma underS_Field: "underS a \<le> Field r"
+by(unfold underS_def Field_def, auto)
+
+
+lemma underS_Field2:
+"a \<in> Field r \<Longrightarrow> underS a < Field r"
+using assms underS_notIn underS_Field by blast
+
+
+lemma underS_Field3:
+"Field r \<noteq> {} \<Longrightarrow> underS a < Field r"
+by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
+
+
+lemma AboveS_Field: "AboveS A \<le> Field r"
+by(unfold AboveS_def Field_def, auto)
+
+
+lemma under_incr:
+assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
+shows "under a \<le> under b"
+proof(unfold under_def, auto)
+ fix x assume "(x,a) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(x,b) \<in> r" by blast
+qed
+
+
+lemma underS_incr:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ REL: "(a,b) \<in> r"
+shows "underS a \<le> underS b"
+proof(unfold underS_def, auto)
+ assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
+ with ANTISYM antisym_def[of r] REL
+ show False by blast
+next
+ fix x assume "x \<noteq> a" "(x,a) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(x,b) \<in> r" by blast
+qed
+
+
+lemma underS_incl_iff:
+assumes LO: "Linear_order r" and
+ INa: "a \<in> Field r" and INb: "b \<in> Field r"
+shows "(underS a \<le> underS b) = ((a,b) \<in> r)"
+proof
+ assume "(a,b) \<in> r"
+ thus "underS a \<le> underS b" using LO
+ by (simp add: order_on_defs underS_incr)
+next
+ assume *: "underS a \<le> underS b"
+ {assume "a = b"
+ hence "(a,b) \<in> r" using assms
+ by (simp add: order_on_defs refl_on_def)
+ }
+ moreover
+ {assume "a \<noteq> b \<and> (b,a) \<in> r"
+ hence "b \<in> underS a" unfolding underS_def by blast
+ hence "b \<in> underS b" using * by blast
+ hence False by (simp add: underS_notIn)
+ }
+ ultimately
+ show "(a,b) \<in> r" using assms
+ order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
+qed
+
+
+end (* context rel *)
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/README.txt
--- a/src/HOL/Cardinals/README.txt Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/README.txt Tue Nov 19 17:07:52 2013 +0100
@@ -89,15 +89,16 @@
Minor technicalities and naming issues:
---------------------------------------
-1. Most of the definitions and theorems are proved in files suffixed with
-"_Base". Bootstrapping considerations (for the (co)datatype package) made this
-division desirable.
+1. Most of the definitions and theorems are proved in files suffixed with "_FP".
+Bootstrapping considerations (for the (co)datatype package) made this division
+desirable.
-2. Even though we would have preferred to use "initial segment" instead of
-"order filter", we chose the latter to avoid terminological clash with
-the operator "init_seg_of" from Zorn.thy. The latter expresses a related, but different
-concept -- it considers a relation, rather than a set, as initial segment of a relation.
+2. Even though we would have preferred to use "initial segment" instead of
+"order filter", we chose the latter to avoid terminological clash with the
+operator "init_seg_of" from Zorn.thy. The latter expresses a related, but
+different concept -- it considers a relation, rather than a set, as initial
+segment of a relation.
3. We prefer to define the upper-bound operations under, underS,
@@ -148,7 +149,7 @@
Notes for anyone who would like to enrich these theories in the future
--------------------------------------------------------------------------------------
-Theory Fun_More (and Fun_More_Base):
+Theory Fun_More (and Fun_More_FP):
- Careful: "inj" is an abbreviation for "inj_on UNIV", while
"bij" is not an abreviation for "bij_betw UNIV UNIV", but
a defined constant; there is no "surj_betw", but only "surj".
@@ -166,7 +167,7 @@
- In subsection "Other facts":
-- Does the lemma "atLeastLessThan_injective" already exist anywhere?
-Theory Order_Relation_More (and Order_Relation_More_Base):
+Theory Order_Relation_More (and Order_Relation_More_FP):
- In subsection "Auxiliaries":
-- Recall the lemmas "order_on_defs", "Field_def", "Domain_def", "Range_def", "converse_def".
-- Recall that "refl_on r A" forces r to not be defined outside A.
@@ -181,16 +182,16 @@
abbreviation "Linear_order r ≡ linear_order_on (Field r) r"
abbreviation "Well_order r ≡ well_order_on (Field r) r"
-Theory Wellorder_Relation (and Wellorder_Relation_Base):
+Theory Wellorder_Relation (and Wellorder_Relation_FP):
- In subsection "Auxiliaries": recall lemmas "order_on_defs"
- In subsection "The notions of maximum, minimum, supremum, successor and order filter":
Should we define all constants from "wo_rel" in "rel" instead,
so that their outside definition not be conditional in "wo_rel r"?
-Theory Wellfounded_More (and Wellfounded_More_Base):
+Theory Wellfounded_More (and Wellfounded_More_FP):
Recall the lemmas "wfrec" and "wf_induct".
-Theory Wellorder_Embedding (and Wellorder_Embedding_Base):
+Theory Wellorder_Embedding (and Wellorder_Embedding_FP):
- Recall "inj_on_def" and "bij_betw_def".
- Line 5 in the proof of lemma embed_in_Field: we have to figure out for this and many other
situations: Why did it work without annotations to Refl_under_in?
@@ -200,7 +201,7 @@
making impossible to debug theorem instantiations.
- At lemma "embed_unique": If we add the attribute "rule format" at lemma, we get an error at qed.
-Theory Constructions_on_Wellorders (and Constructions_on_Wellorders_Base):
+Theory Constructions_on_Wellorders (and Constructions_on_Wellorders_FP):
- Some of the lemmas in this section are about more general kinds of relations than
well-orders, but it is not clear whether they are useful in such more general contexts.
- Recall that "equiv" does not have the "equiv_on" and "Equiv" versions,
@@ -208,7 +209,7 @@
- The lemmas "ord_trans" are not clearly useful together, as their employment within blast or auto
tends to diverge.
-Theory Cardinal_Order_Relation (and Cardinal_Order_Relation_Base):
+Theory Cardinal_Order_Relation (and Cardinal_Order_Relation_FP):
- Careful: if "|..|" meets an outer parehthesis, an extra space needs to be inserted, as in
"( |A| )".
- At lemma like ordLeq_Sigma_mono1: Not worth stating something like ordLeq_Sigma_mono2 --
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellfounded_More.thy
--- a/src/HOL/Cardinals/Wellfounded_More.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Wellfounded_More.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,7 +8,7 @@
header {* More on Well-Founded Relations *}
theory Wellfounded_More
-imports Wellfounded_More_Base Order_Relation_More
+imports Wellfounded_More_FP Order_Relation_More
begin
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellfounded_More_Base.thy
--- a/src/HOL/Cardinals/Wellfounded_More_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,194 +0,0 @@
-(* Title: HOL/Cardinals/Wellfounded_More_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-More on well-founded relations (base).
-*)
-
-header {* More on Well-Founded Relations (Base) *}
-
-theory Wellfounded_More_Base
-imports Order_Relation_More_Base "~~/src/HOL/Library/Wfrec"
-begin
-
-
-text {* This section contains some variations of results in the theory
-@{text "Wellfounded.thy"}:
-\begin{itemize}
-\item means for slightly more direct definitions by well-founded recursion;
-\item variations of well-founded induction;
-\item means for proving a linear order to be a well-order.
-\end{itemize} *}
-
-
-subsection {* Well-founded recursion via genuine fixpoints *}
-
-
-(*2*)lemma wfrec_fixpoint:
-fixes r :: "('a * 'a) set" and
- H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
-assumes WF: "wf r" and ADM: "adm_wf r H"
-shows "wfrec r H = H (wfrec r H)"
-proof(rule ext)
- fix x
- have "wfrec r H x = H (cut (wfrec r H) r x) x"
- using wfrec[of r H] WF by simp
- also
- {have "\<And> y. (y,x) : r \<Longrightarrow> (cut (wfrec r H) r x) y = (wfrec r H) y"
- by (auto simp add: cut_apply)
- hence "H (cut (wfrec r H) r x) x = H (wfrec r H) x"
- using ADM adm_wf_def[of r H] by auto
- }
- finally show "wfrec r H x = H (wfrec r H) x" .
-qed
-
-
-
-subsection {* Characterizations of well-founded-ness *}
-
-
-text {* A transitive relation is well-founded iff it is ``locally" well-founded,
-i.e., iff its restriction to the lower bounds of of any element is well-founded. *}
-
-(*3*)lemma trans_wf_iff:
-assumes "trans r"
-shows "wf r = (\<forall>a. wf(r Int (r^-1``{a} \<times> r^-1``{a})))"
-proof-
- obtain R where R_def: "R = (\<lambda> a. r Int (r^-1``{a} \<times> r^-1``{a}))" by blast
- {assume *: "wf r"
- {fix a
- have "wf(R a)"
- using * R_def wf_subset[of r "R a"] by auto
- }
- }
- (* *)
- moreover
- {assume *: "\<forall>a. wf(R a)"
- have "wf r"
- proof(unfold wf_def, clarify)
- fix phi a
- assume **: "\<forall>a. (\<forall>b. (b,a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
- obtain chi where chi_def: "chi = (\<lambda>b. (b,a) \<in> r \<longrightarrow> phi b)" by blast
- with * have "wf (R a)" by auto
- hence "(\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
- unfolding wf_def by blast
- moreover
- have "\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
- proof(auto simp add: chi_def R_def)
- fix b
- assume 1: "(b,a) \<in> r" and 2: "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
- hence "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
- using assms trans_def[of r] by blast
- thus "phi b" using ** by blast
- qed
- ultimately have "\<forall>b. chi b" by (rule mp)
- with ** chi_def show "phi a" by blast
- qed
- }
- ultimately show ?thesis using R_def by blast
-qed
-
-
-text {* The next lemma is a variation of @{text "wf_eq_minimal"} from Wellfounded,
-allowing one to assume the set included in the field. *}
-
-(*2*)lemma wf_eq_minimal2:
-"wf r = (\<forall>A. A <= Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r))"
-proof-
- let ?phi = "\<lambda> A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r)"
- have "wf r = (\<forall>A. ?phi A)"
- by (auto simp: ex_in_conv [THEN sym], erule wfE_min, assumption, blast)
- (rule wfI_min, metis)
- (* *)
- also have "(\<forall>A. ?phi A) = (\<forall>B \<le> Field r. ?phi B)"
- proof
- assume "\<forall>A. ?phi A"
- thus "\<forall>B \<le> Field r. ?phi B" by simp
- next
- assume *: "\<forall>B \<le> Field r. ?phi B"
- show "\<forall>A. ?phi A"
- proof(clarify)
- fix A::"'a set" assume **: "A \<noteq> {}"
- obtain B where B_def: "B = A Int (Field r)" by blast
- show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r"
- proof(cases "B = {}")
- assume Case1: "B = {}"
- obtain a where 1: "a \<in> A \<and> a \<notin> Field r"
- using ** Case1 unfolding B_def by blast
- hence "\<forall>a' \<in> A. (a',a) \<notin> r" using 1 unfolding Field_def by blast
- thus ?thesis using 1 by blast
- next
- assume Case2: "B \<noteq> {}" have 1: "B \<le> Field r" unfolding B_def by blast
- obtain a where 2: "a \<in> B \<and> (\<forall>a' \<in> B. (a',a) \<notin> r)"
- using Case2 1 * by blast
- have "\<forall>a' \<in> A. (a',a) \<notin> r"
- proof(clarify)
- fix a' assume "a' \<in> A" and **: "(a',a) \<in> r"
- hence "a' \<in> B" unfolding B_def Field_def by blast
- thus False using 2 ** by blast
- qed
- thus ?thesis using 2 unfolding B_def by blast
- qed
- qed
- qed
- finally show ?thesis by blast
-qed
-
-subsection {* Characterizations of well-founded-ness *}
-
-text {* The next lemma and its corollary enable one to prove that
-a linear order is a well-order in a way which is more standard than
-via well-founded-ness of the strict version of the relation. *}
-
-(*3*)
-lemma Linear_order_wf_diff_Id:
-assumes LI: "Linear_order r"
-shows "wf(r - Id) = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-proof(cases "r \<le> Id")
- assume Case1: "r \<le> Id"
- hence temp: "r - Id = {}" by blast
- hence "wf(r - Id)" by (simp add: temp)
- moreover
- {fix A assume *: "A \<le> Field r" and **: "A \<noteq> {}"
- obtain a where 1: "r = {} \<or> r = {(a,a)}" using LI
- unfolding order_on_defs using Case1 rel.Total_subset_Id by metis
- hence "A = {a} \<and> r = {(a,a)}" using * ** unfolding Field_def by blast
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" using 1 by blast
- }
- ultimately show ?thesis by blast
-next
- assume Case2: "\<not> r \<le> Id"
- hence 1: "Field r = Field(r - Id)" using Total_Id_Field LI
- unfolding order_on_defs by blast
- show ?thesis
- proof
- assume *: "wf(r - Id)"
- show "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- proof(clarify)
- fix A assume **: "A \<le> Field r" and ***: "A \<noteq> {}"
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
- using 1 * unfolding wf_eq_minimal2 by simp
- moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
- using rel.Linear_order_in_diff_Id[of r] ** LI by blast
- ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" by blast
- qed
- next
- assume *: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- show "wf(r - Id)"
- proof(unfold wf_eq_minimal2, clarify)
- fix A assume **: "A \<le> Field(r - Id)" and ***: "A \<noteq> {}"
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
- using 1 * by simp
- moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
- using rel.Linear_order_in_diff_Id[of r] ** LI mono_Field[of "r - Id" r] by blast
- ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id" by blast
- qed
- qed
-qed
-
-(*3*)corollary Linear_order_Well_order_iff:
-assumes "Linear_order r"
-shows "Well_order r = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-using assms unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellfounded_More_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellfounded_More_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,194 @@
+(* Title: HOL/Cardinals/Wellfounded_More_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on well-founded relations (FP).
+*)
+
+header {* More on Well-Founded Relations (FP) *}
+
+theory Wellfounded_More_FP
+imports Order_Relation_More_FP "~~/src/HOL/Library/Wfrec"
+begin
+
+
+text {* This section contains some variations of results in the theory
+@{text "Wellfounded.thy"}:
+\begin{itemize}
+\item means for slightly more direct definitions by well-founded recursion;
+\item variations of well-founded induction;
+\item means for proving a linear order to be a well-order.
+\end{itemize} *}
+
+
+subsection {* Well-founded recursion via genuine fixpoints *}
+
+
+(*2*)lemma wfrec_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H"
+shows "wfrec r H = H (wfrec r H)"
+proof(rule ext)
+ fix x
+ have "wfrec r H x = H (cut (wfrec r H) r x) x"
+ using wfrec[of r H] WF by simp
+ also
+ {have "\<And> y. (y,x) : r \<Longrightarrow> (cut (wfrec r H) r x) y = (wfrec r H) y"
+ by (auto simp add: cut_apply)
+ hence "H (cut (wfrec r H) r x) x = H (wfrec r H) x"
+ using ADM adm_wf_def[of r H] by auto
+ }
+ finally show "wfrec r H x = H (wfrec r H) x" .
+qed
+
+
+
+subsection {* Characterizations of well-founded-ness *}
+
+
+text {* A transitive relation is well-founded iff it is ``locally" well-founded,
+i.e., iff its restriction to the lower bounds of of any element is well-founded. *}
+
+(*3*)lemma trans_wf_iff:
+assumes "trans r"
+shows "wf r = (\<forall>a. wf(r Int (r^-1``{a} \<times> r^-1``{a})))"
+proof-
+ obtain R where R_def: "R = (\<lambda> a. r Int (r^-1``{a} \<times> r^-1``{a}))" by blast
+ {assume *: "wf r"
+ {fix a
+ have "wf(R a)"
+ using * R_def wf_subset[of r "R a"] by auto
+ }
+ }
+ (* *)
+ moreover
+ {assume *: "\<forall>a. wf(R a)"
+ have "wf r"
+ proof(unfold wf_def, clarify)
+ fix phi a
+ assume **: "\<forall>a. (\<forall>b. (b,a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
+ obtain chi where chi_def: "chi = (\<lambda>b. (b,a) \<in> r \<longrightarrow> phi b)" by blast
+ with * have "wf (R a)" by auto
+ hence "(\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
+ unfolding wf_def by blast
+ moreover
+ have "\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
+ proof(auto simp add: chi_def R_def)
+ fix b
+ assume 1: "(b,a) \<in> r" and 2: "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
+ hence "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
+ using assms trans_def[of r] by blast
+ thus "phi b" using ** by blast
+ qed
+ ultimately have "\<forall>b. chi b" by (rule mp)
+ with ** chi_def show "phi a" by blast
+ qed
+ }
+ ultimately show ?thesis using R_def by blast
+qed
+
+
+text {* The next lemma is a variation of @{text "wf_eq_minimal"} from Wellfounded,
+allowing one to assume the set included in the field. *}
+
+(*2*)lemma wf_eq_minimal2:
+"wf r = (\<forall>A. A <= Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r))"
+proof-
+ let ?phi = "\<lambda> A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r)"
+ have "wf r = (\<forall>A. ?phi A)"
+ by (auto simp: ex_in_conv [THEN sym], erule wfE_min, assumption, blast)
+ (rule wfI_min, metis)
+ (* *)
+ also have "(\<forall>A. ?phi A) = (\<forall>B \<le> Field r. ?phi B)"
+ proof
+ assume "\<forall>A. ?phi A"
+ thus "\<forall>B \<le> Field r. ?phi B" by simp
+ next
+ assume *: "\<forall>B \<le> Field r. ?phi B"
+ show "\<forall>A. ?phi A"
+ proof(clarify)
+ fix A::"'a set" assume **: "A \<noteq> {}"
+ obtain B where B_def: "B = A Int (Field r)" by blast
+ show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r"
+ proof(cases "B = {}")
+ assume Case1: "B = {}"
+ obtain a where 1: "a \<in> A \<and> a \<notin> Field r"
+ using ** Case1 unfolding B_def by blast
+ hence "\<forall>a' \<in> A. (a',a) \<notin> r" using 1 unfolding Field_def by blast
+ thus ?thesis using 1 by blast
+ next
+ assume Case2: "B \<noteq> {}" have 1: "B \<le> Field r" unfolding B_def by blast
+ obtain a where 2: "a \<in> B \<and> (\<forall>a' \<in> B. (a',a) \<notin> r)"
+ using Case2 1 * by blast
+ have "\<forall>a' \<in> A. (a',a) \<notin> r"
+ proof(clarify)
+ fix a' assume "a' \<in> A" and **: "(a',a) \<in> r"
+ hence "a' \<in> B" unfolding B_def Field_def by blast
+ thus False using 2 ** by blast
+ qed
+ thus ?thesis using 2 unfolding B_def by blast
+ qed
+ qed
+ qed
+ finally show ?thesis by blast
+qed
+
+subsection {* Characterizations of well-founded-ness *}
+
+text {* The next lemma and its corollary enable one to prove that
+a linear order is a well-order in a way which is more standard than
+via well-founded-ness of the strict version of the relation. *}
+
+(*3*)
+lemma Linear_order_wf_diff_Id:
+assumes LI: "Linear_order r"
+shows "wf(r - Id) = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
+proof(cases "r \<le> Id")
+ assume Case1: "r \<le> Id"
+ hence temp: "r - Id = {}" by blast
+ hence "wf(r - Id)" by (simp add: temp)
+ moreover
+ {fix A assume *: "A \<le> Field r" and **: "A \<noteq> {}"
+ obtain a where 1: "r = {} \<or> r = {(a,a)}" using LI
+ unfolding order_on_defs using Case1 rel.Total_subset_Id by metis
+ hence "A = {a} \<and> r = {(a,a)}" using * ** unfolding Field_def by blast
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" using 1 by blast
+ }
+ ultimately show ?thesis by blast
+next
+ assume Case2: "\<not> r \<le> Id"
+ hence 1: "Field r = Field(r - Id)" using Total_Id_Field LI
+ unfolding order_on_defs by blast
+ show ?thesis
+ proof
+ assume *: "wf(r - Id)"
+ show "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ proof(clarify)
+ fix A assume **: "A \<le> Field r" and ***: "A \<noteq> {}"
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
+ using 1 * unfolding wf_eq_minimal2 by simp
+ moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
+ using rel.Linear_order_in_diff_Id[of r] ** LI by blast
+ ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" by blast
+ qed
+ next
+ assume *: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ show "wf(r - Id)"
+ proof(unfold wf_eq_minimal2, clarify)
+ fix A assume **: "A \<le> Field(r - Id)" and ***: "A \<noteq> {}"
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
+ using 1 * by simp
+ moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
+ using rel.Linear_order_in_diff_Id[of r] ** LI mono_Field[of "r - Id" r] by blast
+ ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id" by blast
+ qed
+ qed
+qed
+
+(*3*)corollary Linear_order_Well_order_iff:
+assumes "Linear_order r"
+shows "Well_order r = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
+using assms unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Embedding.thy
--- a/src/HOL/Cardinals/Wellorder_Embedding.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Wellorder_Embedding.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,7 +8,7 @@
header {* Well-Order Embeddings *}
theory Wellorder_Embedding
-imports Wellorder_Embedding_Base Fun_More Wellorder_Relation
+imports Wellorder_Embedding_FP Fun_More Wellorder_Relation
begin
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Embedding_Base.thy
--- a/src/HOL/Cardinals/Wellorder_Embedding_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1146 +0,0 @@
-(* Title: HOL/Cardinals/Wellorder_Embedding_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Well-order embeddings (base).
-*)
-
-header {* Well-Order Embeddings (Base) *}
-
-theory Wellorder_Embedding_Base
-imports "~~/src/HOL/Library/Zorn" Fun_More_Base Wellorder_Relation_Base
-begin
-
-
-text{* In this section, we introduce well-order {\em embeddings} and {\em isomorphisms} and
-prove their basic properties. The notion of embedding is considered from the point
-of view of the theory of ordinals, and therefore requires the source to be injected
-as an {\em initial segment} (i.e., {\em order filter}) of the target. A main result
-of this section is the existence of embeddings (in one direction or another) between
-any two well-orders, having as a consequence the fact that, given any two sets on
-any two types, one is smaller than (i.e., can be injected into) the other. *}
-
-
-subsection {* Auxiliaries *}
-
-lemma UNION_inj_on_ofilter:
-assumes WELL: "Well_order r" and
- OF: "\<And> i. i \<in> I \<Longrightarrow> wo_rel.ofilter r (A i)" and
- INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
-shows "inj_on f (\<Union> i \<in> I. A i)"
-proof-
- have "wo_rel r" using WELL by (simp add: wo_rel_def)
- hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i <= A j \<or> A j <= A i"
- using wo_rel.ofilter_linord[of r] OF by blast
- with WELL INJ show ?thesis
- by (auto simp add: inj_on_UNION_chain)
-qed
-
-
-lemma under_underS_bij_betw:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- IN: "a \<in> Field r" and IN': "f a \<in> Field r'" and
- BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
-shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
-proof-
- have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
- unfolding rel.underS_def by auto
- moreover
- {have "Refl r \<and> Refl r'" using WELL WELL'
- by (auto simp add: order_on_defs)
- hence "rel.under r a = rel.underS r a \<union> {a} \<and>
- rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
- using IN IN' by(auto simp add: rel.Refl_under_underS)
- }
- ultimately show ?thesis
- using BIJ notIn_Un_bij_betw[of a "rel.underS r a" f "rel.underS r' (f a)"] by auto
-qed
-
-
-
-subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
-functions *}
-
-
-text{* Standardly, a function is an embedding of a well-order in another if it injectively and
-order-compatibly maps the former into an order filter of the latter.
-Here we opt for a more succinct definition (operator @{text "embed"}),
-asking that, for any element in the source, the function should be a bijection
-between the set of strict lower bounds of that element
-and the set of strict lower bounds of its image. (Later we prove equivalence with
-the standard definition -- lemma @{text "embed_iff_compat_inj_on_ofilter"}.)
-A {\em strict embedding} (operator @{text "embedS"}) is a non-bijective embedding
-and an isomorphism (operator @{text "iso"}) is a bijective embedding. *}
-
-
-definition embed :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
-where
-"embed r r' f \<equiv> \<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
-
-
-lemmas embed_defs = embed_def embed_def[abs_def]
-
-
-text {* Strict embeddings: *}
-
-definition embedS :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
-where
-"embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
-
-
-lemmas embedS_defs = embedS_def embedS_def[abs_def]
-
-
-definition iso :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
-where
-"iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
-
-
-lemmas iso_defs = iso_def iso_def[abs_def]
-
-
-definition compat :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
-where
-"compat r r' f \<equiv> \<forall>a b. (a,b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
-
-
-lemma compat_wf:
-assumes CMP: "compat r r' f" and WF: "wf r'"
-shows "wf r"
-proof-
- have "r \<le> inv_image r' f"
- unfolding inv_image_def using CMP
- by (auto simp add: compat_def)
- with WF show ?thesis
- using wf_inv_image[of r' f] wf_subset[of "inv_image r' f"] by auto
-qed
-
-
-lemma id_embed: "embed r r id"
-by(auto simp add: id_def embed_def bij_betw_def)
-
-
-lemma id_iso: "iso r r id"
-by(auto simp add: id_def embed_def iso_def bij_betw_def)
-
-
-lemma embed_in_Field:
-assumes WELL: "Well_order r" and
- EMB: "embed r r' f" and IN: "a \<in> Field r"
-shows "f a \<in> Field r'"
-proof-
- have Well: "wo_rel r"
- using WELL by (auto simp add: wo_rel_def)
- hence 1: "Refl r"
- by (auto simp add: wo_rel.REFL)
- hence "a \<in> rel.under r a" using IN rel.Refl_under_in by fastforce
- hence "f a \<in> rel.under r' (f a)"
- using EMB IN by (auto simp add: embed_def bij_betw_def)
- thus ?thesis unfolding Field_def
- by (auto simp: rel.under_def)
-qed
-
-
-lemma comp_embed:
-assumes WELL: "Well_order r" and
- EMB: "embed r r' f" and EMB': "embed r' r'' f'"
-shows "embed r r'' (f' o f)"
-proof(unfold embed_def, auto)
- fix a assume *: "a \<in> Field r"
- hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
- using embed_def[of r] EMB by auto
- moreover
- {have "f a \<in> Field r'"
- using EMB WELL * by (auto simp add: embed_in_Field)
- hence "bij_betw f' (rel.under r' (f a)) (rel.under r'' (f' (f a)))"
- using embed_def[of r'] EMB' by auto
- }
- ultimately
- show "bij_betw (f' \<circ> f) (rel.under r a) (rel.under r'' (f'(f a)))"
- by(auto simp add: bij_betw_trans)
-qed
-
-
-lemma comp_iso:
-assumes WELL: "Well_order r" and
- EMB: "iso r r' f" and EMB': "iso r' r'' f'"
-shows "iso r r'' (f' o f)"
-using assms unfolding iso_def
-by (auto simp add: comp_embed bij_betw_trans)
-
-
-text{* That @{text "embedS"} is also preserved by function composition shall be proved only later. *}
-
-
-lemma embed_Field:
-"\<lbrakk>Well_order r; embed r r' f\<rbrakk> \<Longrightarrow> f`(Field r) \<le> Field r'"
-by (auto simp add: embed_in_Field)
-
-
-lemma embed_preserves_ofilter:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and OF: "wo_rel.ofilter r A"
-shows "wo_rel.ofilter r' (f`A)"
-proof-
- (* Preliminary facts *)
- from WELL have Well: "wo_rel r" unfolding wo_rel_def .
- from WELL' have Well': "wo_rel r'" unfolding wo_rel_def .
- from OF have 0: "A \<le> Field r" by(auto simp add: Well wo_rel.ofilter_def)
- (* Main proof *)
- show ?thesis using Well' WELL EMB 0 embed_Field[of r r' f]
- proof(unfold wo_rel.ofilter_def, auto simp add: image_def)
- fix a b'
- assume *: "a \<in> A" and **: "b' \<in> rel.under r' (f a)"
- hence "a \<in> Field r" using 0 by auto
- hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
- using * EMB by (auto simp add: embed_def)
- hence "f`(rel.under r a) = rel.under r' (f a)"
- by (simp add: bij_betw_def)
- with ** image_def[of f "rel.under r a"] obtain b where
- 1: "b \<in> rel.under r a \<and> b' = f b" by blast
- hence "b \<in> A" using Well * OF
- by (auto simp add: wo_rel.ofilter_def)
- with 1 show "\<exists>b \<in> A. b' = f b" by blast
- qed
-qed
-
-
-lemma embed_Field_ofilter:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f"
-shows "wo_rel.ofilter r' (f`(Field r))"
-proof-
- have "wo_rel.ofilter r (Field r)"
- using WELL by (auto simp add: wo_rel_def wo_rel.Field_ofilter)
- with WELL WELL' EMB
- show ?thesis by (auto simp add: embed_preserves_ofilter)
-qed
-
-
-lemma embed_compat:
-assumes EMB: "embed r r' f"
-shows "compat r r' f"
-proof(unfold compat_def, clarify)
- fix a b
- assume *: "(a,b) \<in> r"
- hence 1: "b \<in> Field r" using Field_def[of r] by blast
- have "a \<in> rel.under r b"
- using * rel.under_def[of r] by simp
- hence "f a \<in> rel.under r' (f b)"
- using EMB embed_def[of r r' f]
- bij_betw_def[of f "rel.under r b" "rel.under r' (f b)"]
- image_def[of f "rel.under r b"] 1 by auto
- thus "(f a, f b) \<in> r'"
- by (auto simp add: rel.under_def)
-qed
-
-
-lemma embed_inj_on:
-assumes WELL: "Well_order r" and EMB: "embed r r' f"
-shows "inj_on f (Field r)"
-proof(unfold inj_on_def, clarify)
- (* Preliminary facts *)
- from WELL have Well: "wo_rel r" unfolding wo_rel_def .
- with wo_rel.TOTAL[of r]
- have Total: "Total r" by simp
- from Well wo_rel.REFL[of r]
- have Refl: "Refl r" by simp
- (* Main proof *)
- fix a b
- assume *: "a \<in> Field r" and **: "b \<in> Field r" and
- ***: "f a = f b"
- hence 1: "a \<in> Field r \<and> b \<in> Field r"
- unfolding Field_def by auto
- {assume "(a,b) \<in> r"
- hence "a \<in> rel.under r b \<and> b \<in> rel.under r b"
- using Refl by(auto simp add: rel.under_def refl_on_def)
- hence "a = b"
- using EMB 1 ***
- by (auto simp add: embed_def bij_betw_def inj_on_def)
- }
- moreover
- {assume "(b,a) \<in> r"
- hence "a \<in> rel.under r a \<and> b \<in> rel.under r a"
- using Refl by(auto simp add: rel.under_def refl_on_def)
- hence "a = b"
- using EMB 1 ***
- by (auto simp add: embed_def bij_betw_def inj_on_def)
- }
- ultimately
- show "a = b" using Total 1
- by (auto simp add: total_on_def)
-qed
-
-
-lemma embed_underS:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and IN: "a \<in> Field r"
-shows "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
-proof-
- have "bij_betw f (rel.under r a) (rel.under r' (f a))"
- using assms by (auto simp add: embed_def)
- moreover
- {have "f a \<in> Field r'" using assms embed_Field[of r r' f] by auto
- hence "rel.under r a = rel.underS r a \<union> {a} \<and>
- rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
- using assms by (auto simp add: order_on_defs rel.Refl_under_underS)
- }
- moreover
- {have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
- unfolding rel.underS_def by blast
- }
- ultimately show ?thesis
- by (auto simp add: notIn_Un_bij_betw3)
-qed
-
-
-lemma embed_iff_compat_inj_on_ofilter:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> wo_rel.ofilter r' (f`(Field r)))"
-using assms
-proof(auto simp add: embed_compat embed_inj_on embed_Field_ofilter,
- unfold embed_def, auto) (* get rid of one implication *)
- fix a
- assume *: "inj_on f (Field r)" and
- **: "compat r r' f" and
- ***: "wo_rel.ofilter r' (f`(Field r))" and
- ****: "a \<in> Field r"
- (* Preliminary facts *)
- have Well: "wo_rel r"
- using WELL wo_rel_def[of r] by simp
- hence Refl: "Refl r"
- using wo_rel.REFL[of r] by simp
- have Total: "Total r"
- using Well wo_rel.TOTAL[of r] by simp
- have Well': "wo_rel r'"
- using WELL' wo_rel_def[of r'] by simp
- hence Antisym': "antisym r'"
- using wo_rel.ANTISYM[of r'] by simp
- have "(a,a) \<in> r"
- using **** Well wo_rel.REFL[of r]
- refl_on_def[of _ r] by auto
- hence "(f a, f a) \<in> r'"
- using ** by(auto simp add: compat_def)
- hence 0: "f a \<in> Field r'"
- unfolding Field_def by auto
- have "f a \<in> f`(Field r)"
- using **** by auto
- hence 2: "rel.under r' (f a) \<le> f`(Field r)"
- using Well' *** wo_rel.ofilter_def[of r' "f`(Field r)"] by fastforce
- (* Main proof *)
- show "bij_betw f (rel.under r a) (rel.under r' (f a))"
- proof(unfold bij_betw_def, auto)
- show "inj_on f (rel.under r a)"
- using *
- by (auto simp add: rel.under_Field subset_inj_on)
- next
- fix b assume "b \<in> rel.under r a"
- thus "f b \<in> rel.under r' (f a)"
- unfolding rel.under_def using **
- by (auto simp add: compat_def)
- next
- fix b' assume *****: "b' \<in> rel.under r' (f a)"
- hence "b' \<in> f`(Field r)"
- using 2 by auto
- with Field_def[of r] obtain b where
- 3: "b \<in> Field r" and 4: "b' = f b" by auto
- have "(b,a): r"
- proof-
- {assume "(a,b) \<in> r"
- with ** 4 have "(f a, b'): r'"
- by (auto simp add: compat_def)
- with ***** Antisym' have "f a = b'"
- by(auto simp add: rel.under_def antisym_def)
- with 3 **** 4 * have "a = b"
- by(auto simp add: inj_on_def)
- }
- moreover
- {assume "a = b"
- hence "(b,a) \<in> r" using Refl **** 3
- by (auto simp add: refl_on_def)
- }
- ultimately
- show ?thesis using Total **** 3 by (fastforce simp add: total_on_def)
- qed
- with 4 show "b' \<in> f`(rel.under r a)"
- unfolding rel.under_def by auto
- qed
-qed
-
-
-lemma inv_into_ofilter_embed:
-assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and
- BIJ: "\<forall>b \<in> A. bij_betw f (rel.under r b) (rel.under r' (f b))" and
- IMAGE: "f ` A = Field r'"
-shows "embed r' r (inv_into A f)"
-proof-
- (* Preliminary facts *)
- have Well: "wo_rel r"
- using WELL wo_rel_def[of r] by simp
- have Refl: "Refl r"
- using Well wo_rel.REFL[of r] by simp
- have Total: "Total r"
- using Well wo_rel.TOTAL[of r] by simp
- (* Main proof *)
- have 1: "bij_betw f A (Field r')"
- proof(unfold bij_betw_def inj_on_def, auto simp add: IMAGE)
- fix b1 b2
- assume *: "b1 \<in> A" and **: "b2 \<in> A" and
- ***: "f b1 = f b2"
- have 11: "b1 \<in> Field r \<and> b2 \<in> Field r"
- using * ** Well OF by (auto simp add: wo_rel.ofilter_def)
- moreover
- {assume "(b1,b2) \<in> r"
- hence "b1 \<in> rel.under r b2 \<and> b2 \<in> rel.under r b2"
- unfolding rel.under_def using 11 Refl
- by (auto simp add: refl_on_def)
- hence "b1 = b2" using BIJ * ** ***
- by (auto simp add: bij_betw_def inj_on_def)
- }
- moreover
- {assume "(b2,b1) \<in> r"
- hence "b1 \<in> rel.under r b1 \<and> b2 \<in> rel.under r b1"
- unfolding rel.under_def using 11 Refl
- by (auto simp add: refl_on_def)
- hence "b1 = b2" using BIJ * ** ***
- by (auto simp add: bij_betw_def inj_on_def)
- }
- ultimately
- show "b1 = b2"
- using Total by (auto simp add: total_on_def)
- qed
- (* *)
- let ?f' = "(inv_into A f)"
- (* *)
- have 2: "\<forall>b \<in> A. bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
- proof(clarify)
- fix b assume *: "b \<in> A"
- hence "rel.under r b \<le> A"
- using Well OF by(auto simp add: wo_rel.ofilter_def)
- moreover
- have "f ` (rel.under r b) = rel.under r' (f b)"
- using * BIJ by (auto simp add: bij_betw_def)
- ultimately
- show "bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
- using 1 by (auto simp add: bij_betw_inv_into_subset)
- qed
- (* *)
- have 3: "\<forall>b' \<in> Field r'. bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
- proof(clarify)
- fix b' assume *: "b' \<in> Field r'"
- have "b' = f (?f' b')" using * 1
- by (auto simp add: bij_betw_inv_into_right)
- moreover
- {obtain b where 31: "b \<in> A" and "f b = b'" using IMAGE * by force
- hence "?f' b' = b" using 1 by (auto simp add: bij_betw_inv_into_left)
- with 31 have "?f' b' \<in> A" by auto
- }
- ultimately
- show "bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
- using 2 by auto
- qed
- (* *)
- thus ?thesis unfolding embed_def .
-qed
-
-
-lemma inv_into_underS_embed:
-assumes WELL: "Well_order r" and
- BIJ: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
- IN: "a \<in> Field r" and
- IMAGE: "f ` (rel.underS r a) = Field r'"
-shows "embed r' r (inv_into (rel.underS r a) f)"
-using assms
-by(auto simp add: wo_rel_def wo_rel.underS_ofilter inv_into_ofilter_embed)
-
-
-lemma inv_into_Field_embed:
-assumes WELL: "Well_order r" and EMB: "embed r r' f" and
- IMAGE: "Field r' \<le> f ` (Field r)"
-shows "embed r' r (inv_into (Field r) f)"
-proof-
- have "(\<forall>b \<in> Field r. bij_betw f (rel.under r b) (rel.under r' (f b)))"
- using EMB by (auto simp add: embed_def)
- moreover
- have "f ` (Field r) \<le> Field r'"
- using EMB WELL by (auto simp add: embed_Field)
- ultimately
- show ?thesis using assms
- by(auto simp add: wo_rel_def wo_rel.Field_ofilter inv_into_ofilter_embed)
-qed
-
-
-lemma inv_into_Field_embed_bij_betw:
-assumes WELL: "Well_order r" and
- EMB: "embed r r' f" and BIJ: "bij_betw f (Field r) (Field r')"
-shows "embed r' r (inv_into (Field r) f)"
-proof-
- have "Field r' \<le> f ` (Field r)"
- using BIJ by (auto simp add: bij_betw_def)
- thus ?thesis using assms
- by(auto simp add: inv_into_Field_embed)
-qed
-
-
-
-
-
-subsection {* Given any two well-orders, one can be embedded in the other *}
-
-
-text{* Here is an overview of the proof of of this fact, stated in theorem
-@{text "wellorders_totally_ordered"}:
-
- Fix the well-orders @{text "r::'a rel"} and @{text "r'::'a' rel"}.
- Attempt to define an embedding @{text "f::'a \<Rightarrow> 'a'"} from @{text "r"} to @{text "r'"} in the
- natural way by well-order recursion ("hoping" that @{text "Field r"} turns out to be smaller
- than @{text "Field r'"}), but also record, at the recursive step, in a function
- @{text "g::'a \<Rightarrow> bool"}, the extra information of whether @{text "Field r'"}
- gets exhausted or not.
-
- If @{text "Field r'"} does not get exhausted, then @{text "Field r"} is indeed smaller
- and @{text "f"} is the desired embedding from @{text "r"} to @{text "r'"}
- (lemma @{text "wellorders_totally_ordered_aux"}).
-
- Otherwise, it means that @{text "Field r'"} is the smaller one, and the inverse of
- (the "good" segment of) @{text "f"} is the desired embedding from @{text "r'"} to @{text "r"}
- (lemma @{text "wellorders_totally_ordered_aux2"}).
-*}
-
-
-lemma wellorders_totally_ordered_aux:
-fixes r ::"'a rel" and r'::"'a' rel" and
- f :: "'a \<Rightarrow> 'a'" and a::'a
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and IN: "a \<in> Field r" and
- IH: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
- NOT: "f ` (rel.underS r a) \<noteq> Field r'" and SUC: "f a = wo_rel.suc r' (f`(rel.underS r a))"
-shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
-proof-
- (* Preliminary facts *)
- have Well: "wo_rel r" using WELL unfolding wo_rel_def .
- hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
- have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
- have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
- have OF: "wo_rel.ofilter r (rel.underS r a)"
- by (auto simp add: Well wo_rel.underS_ofilter)
- hence UN: "rel.underS r a = (\<Union> b \<in> rel.underS r a. rel.under r b)"
- using Well wo_rel.ofilter_under_UNION[of r "rel.underS r a"] by blast
- (* Gather facts about elements of rel.underS r a *)
- {fix b assume *: "b \<in> rel.underS r a"
- hence t0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
- have t1: "b \<in> Field r"
- using * rel.underS_Field[of r a] by auto
- have t2: "f`(rel.under r b) = rel.under r' (f b)"
- using IH * by (auto simp add: bij_betw_def)
- hence t3: "wo_rel.ofilter r' (f`(rel.under r b))"
- using Well' by (auto simp add: wo_rel.under_ofilter)
- have "f`(rel.under r b) \<le> Field r'"
- using t2 by (auto simp add: rel.under_Field)
- moreover
- have "b \<in> rel.under r b"
- using t1 by(auto simp add: Refl rel.Refl_under_in)
- ultimately
- have t4: "f b \<in> Field r'" by auto
- have "f`(rel.under r b) = rel.under r' (f b) \<and>
- wo_rel.ofilter r' (f`(rel.under r b)) \<and>
- f b \<in> Field r'"
- using t2 t3 t4 by auto
- }
- hence bFact:
- "\<forall>b \<in> rel.underS r a. f`(rel.under r b) = rel.under r' (f b) \<and>
- wo_rel.ofilter r' (f`(rel.under r b)) \<and>
- f b \<in> Field r'" by blast
- (* *)
- have subField: "f`(rel.underS r a) \<le> Field r'"
- using bFact by blast
- (* *)
- have OF': "wo_rel.ofilter r' (f`(rel.underS r a))"
- proof-
- have "f`(rel.underS r a) = f`(\<Union> b \<in> rel.underS r a. rel.under r b)"
- using UN by auto
- also have "\<dots> = (\<Union> b \<in> rel.underS r a. f`(rel.under r b))" by blast
- also have "\<dots> = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))"
- using bFact by auto
- finally
- have "f`(rel.underS r a) = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))" .
- thus ?thesis
- using Well' bFact
- wo_rel.ofilter_UNION[of r' "rel.underS r a" "\<lambda> b. rel.under r' (f b)"] by fastforce
- qed
- (* *)
- have "f`(rel.underS r a) \<union> rel.AboveS r' (f`(rel.underS r a)) = Field r'"
- using Well' OF' by (auto simp add: wo_rel.ofilter_AboveS_Field)
- hence NE: "rel.AboveS r' (f`(rel.underS r a)) \<noteq> {}"
- using subField NOT by blast
- (* Main proof *)
- have INCL1: "f`(rel.underS r a) \<le> rel.underS r' (f a) "
- proof(auto)
- fix b assume *: "b \<in> rel.underS r a"
- have "f b \<noteq> f a \<and> (f b, f a) \<in> r'"
- using subField Well' SUC NE *
- wo_rel.suc_greater[of r' "f`(rel.underS r a)" "f b"] by auto
- thus "f b \<in> rel.underS r' (f a)"
- unfolding rel.underS_def by simp
- qed
- (* *)
- have INCL2: "rel.underS r' (f a) \<le> f`(rel.underS r a)"
- proof
- fix b' assume "b' \<in> rel.underS r' (f a)"
- hence "b' \<noteq> f a \<and> (b', f a) \<in> r'"
- unfolding rel.underS_def by simp
- thus "b' \<in> f`(rel.underS r a)"
- using Well' SUC NE OF'
- wo_rel.suc_ofilter_in[of r' "f ` rel.underS r a" b'] by auto
- qed
- (* *)
- have INJ: "inj_on f (rel.underS r a)"
- proof-
- have "\<forall>b \<in> rel.underS r a. inj_on f (rel.under r b)"
- using IH by (auto simp add: bij_betw_def)
- moreover
- have "\<forall>b. wo_rel.ofilter r (rel.under r b)"
- using Well by (auto simp add: wo_rel.under_ofilter)
- ultimately show ?thesis
- using WELL bFact UN
- UNION_inj_on_ofilter[of r "rel.underS r a" "\<lambda>b. rel.under r b" f]
- by auto
- qed
- (* *)
- have BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
- unfolding bij_betw_def
- using INJ INCL1 INCL2 by auto
- (* *)
- have "f a \<in> Field r'"
- using Well' subField NE SUC
- by (auto simp add: wo_rel.suc_inField)
- thus ?thesis
- using WELL WELL' IN BIJ under_underS_bij_betw[of r r' a f] by auto
-qed
-
-
-lemma wellorders_totally_ordered_aux2:
-fixes r ::"'a rel" and r'::"'a' rel" and
- f :: "'a \<Rightarrow> 'a'" and g :: "'a \<Rightarrow> bool" and a::'a
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
-MAIN1:
- "\<And> a. (False \<notin> g`(rel.underS r a) \<and> f`(rel.underS r a) \<noteq> Field r'
- \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True)
- \<and>
- (\<not>(False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')
- \<longrightarrow> g a = False)" and
-MAIN2: "\<And> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
- bij_betw f (rel.under r a) (rel.under r' (f a))" and
-Case: "a \<in> Field r \<and> False \<in> g`(rel.under r a)"
-shows "\<exists>f'. embed r' r f'"
-proof-
- have Well: "wo_rel r" using WELL unfolding wo_rel_def .
- hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
- have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
- have Antisym: "antisym r" using Well wo_rel.ANTISYM[of r] by auto
- have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
- (* *)
- have 0: "rel.under r a = rel.underS r a \<union> {a}"
- using Refl Case by(auto simp add: rel.Refl_under_underS)
- (* *)
- have 1: "g a = False"
- proof-
- {assume "g a \<noteq> False"
- with 0 Case have "False \<in> g`(rel.underS r a)" by blast
- with MAIN1 have "g a = False" by blast}
- thus ?thesis by blast
- qed
- let ?A = "{a \<in> Field r. g a = False}"
- let ?a = "(wo_rel.minim r ?A)"
- (* *)
- have 2: "?A \<noteq> {} \<and> ?A \<le> Field r" using Case 1 by blast
- (* *)
- have 3: "False \<notin> g`(rel.underS r ?a)"
- proof
- assume "False \<in> g`(rel.underS r ?a)"
- then obtain b where "b \<in> rel.underS r ?a" and 31: "g b = False" by auto
- hence 32: "(b,?a) \<in> r \<and> b \<noteq> ?a"
- by (auto simp add: rel.underS_def)
- hence "b \<in> Field r" unfolding Field_def by auto
- with 31 have "b \<in> ?A" by auto
- hence "(?a,b) \<in> r" using wo_rel.minim_least 2 Well by fastforce
- (* again: why worked without type annotations? *)
- with 32 Antisym show False
- by (auto simp add: antisym_def)
- qed
- have temp: "?a \<in> ?A"
- using Well 2 wo_rel.minim_in[of r ?A] by auto
- hence 4: "?a \<in> Field r" by auto
- (* *)
- have 5: "g ?a = False" using temp by blast
- (* *)
- have 6: "f`(rel.underS r ?a) = Field r'"
- using MAIN1[of ?a] 3 5 by blast
- (* *)
- have 7: "\<forall>b \<in> rel.underS r ?a. bij_betw f (rel.under r b) (rel.under r' (f b))"
- proof
- fix b assume as: "b \<in> rel.underS r ?a"
- moreover
- have "wo_rel.ofilter r (rel.underS r ?a)"
- using Well by (auto simp add: wo_rel.underS_ofilter)
- ultimately
- have "False \<notin> g`(rel.under r b)" using 3 Well by (subst (asm) wo_rel.ofilter_def) fast+
- moreover have "b \<in> Field r"
- unfolding Field_def using as by (auto simp add: rel.underS_def)
- ultimately
- show "bij_betw f (rel.under r b) (rel.under r' (f b))"
- using MAIN2 by auto
- qed
- (* *)
- have "embed r' r (inv_into (rel.underS r ?a) f)"
- using WELL WELL' 7 4 6 inv_into_underS_embed[of r ?a f r'] by auto
- thus ?thesis
- unfolding embed_def by blast
-qed
-
-
-theorem wellorders_totally_ordered:
-fixes r ::"'a rel" and r'::"'a' rel"
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(\<exists>f. embed r r' f) \<or> (\<exists>f'. embed r' r f')"
-proof-
- (* Preliminary facts *)
- have Well: "wo_rel r" using WELL unfolding wo_rel_def .
- hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
- have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
- have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
- (* Main proof *)
- obtain H where H_def: "H =
- (\<lambda>h a. if False \<notin> (snd o h)`(rel.underS r a) \<and> (fst o h)`(rel.underS r a) \<noteq> Field r'
- then (wo_rel.suc r' ((fst o h)`(rel.underS r a)), True)
- else (undefined, False))" by blast
- have Adm: "wo_rel.adm_wo r H"
- using Well
- proof(unfold wo_rel.adm_wo_def, clarify)
- fix h1::"'a \<Rightarrow> 'a' * bool" and h2::"'a \<Rightarrow> 'a' * bool" and x
- assume "\<forall>y\<in>rel.underS r x. h1 y = h2 y"
- hence "\<forall>y\<in>rel.underS r x. (fst o h1) y = (fst o h2) y \<and>
- (snd o h1) y = (snd o h2) y" by auto
- hence "(fst o h1)`(rel.underS r x) = (fst o h2)`(rel.underS r x) \<and>
- (snd o h1)`(rel.underS r x) = (snd o h2)`(rel.underS r x)"
- by (auto simp add: image_def)
- thus "H h1 x = H h2 x" by (simp add: H_def del: not_False_in_image_Ball)
- qed
- (* More constant definitions: *)
- obtain h::"'a \<Rightarrow> 'a' * bool" and f::"'a \<Rightarrow> 'a'" and g::"'a \<Rightarrow> bool"
- where h_def: "h = wo_rel.worec r H" and
- f_def: "f = fst o h" and g_def: "g = snd o h" by blast
- obtain test where test_def:
- "test = (\<lambda> a. False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')" by blast
- (* *)
- have *: "\<And> a. h a = H h a"
- using Adm Well wo_rel.worec_fixpoint[of r H] by (simp add: h_def)
- have Main1:
- "\<And> a. (test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
- (\<not>(test a) \<longrightarrow> g a = False)"
- proof- (* How can I prove this withou fixing a? *)
- fix a show "(test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
- (\<not>(test a) \<longrightarrow> g a = False)"
- using *[of a] test_def f_def g_def H_def by auto
- qed
- (* *)
- let ?phi = "\<lambda> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
- bij_betw f (rel.under r a) (rel.under r' (f a))"
- have Main2: "\<And> a. ?phi a"
- proof-
- fix a show "?phi a"
- proof(rule wo_rel.well_order_induct[of r ?phi],
- simp only: Well, clarify)
- fix a
- assume IH: "\<forall>b. b \<noteq> a \<and> (b,a) \<in> r \<longrightarrow> ?phi b" and
- *: "a \<in> Field r" and
- **: "False \<notin> g`(rel.under r a)"
- have 1: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))"
- proof(clarify)
- fix b assume ***: "b \<in> rel.underS r a"
- hence 0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
- moreover have "b \<in> Field r"
- using *** rel.underS_Field[of r a] by auto
- moreover have "False \<notin> g`(rel.under r b)"
- using 0 ** Trans rel.under_incr[of r b a] by auto
- ultimately show "bij_betw f (rel.under r b) (rel.under r' (f b))"
- using IH by auto
- qed
- (* *)
- have 21: "False \<notin> g`(rel.underS r a)"
- using ** rel.underS_subset_under[of r a] by auto
- have 22: "g`(rel.under r a) \<le> {True}" using ** by auto
- moreover have 23: "a \<in> rel.under r a"
- using Refl * by (auto simp add: rel.Refl_under_in)
- ultimately have 24: "g a = True" by blast
- have 2: "f`(rel.underS r a) \<noteq> Field r'"
- proof
- assume "f`(rel.underS r a) = Field r'"
- hence "g a = False" using Main1 test_def by blast
- with 24 show False using ** by blast
- qed
- (* *)
- have 3: "f a = wo_rel.suc r' (f`(rel.underS r a))"
- using 21 2 Main1 test_def by blast
- (* *)
- show "bij_betw f (rel.under r a) (rel.under r' (f a))"
- using WELL WELL' 1 2 3 *
- wellorders_totally_ordered_aux[of r r' a f] by auto
- qed
- qed
- (* *)
- let ?chi = "(\<lambda> a. a \<in> Field r \<and> False \<in> g`(rel.under r a))"
- show ?thesis
- proof(cases "\<exists>a. ?chi a")
- assume "\<not> (\<exists>a. ?chi a)"
- hence "\<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
- using Main2 by blast
- thus ?thesis unfolding embed_def by blast
- next
- assume "\<exists>a. ?chi a"
- then obtain a where "?chi a" by blast
- hence "\<exists>f'. embed r' r f'"
- using wellorders_totally_ordered_aux2[of r r' g f a]
- WELL WELL' Main1 Main2 test_def by blast
- thus ?thesis by blast
- qed
-qed
-
-
-subsection {* Uniqueness of embeddings *}
-
-
-text{* Here we show a fact complementary to the one from the previous subsection -- namely,
-that between any two well-orders there is {\em at most} one embedding, and is the one
-definable by the expected well-order recursive equation. As a consequence, any two
-embeddings of opposite directions are mutually inverse. *}
-
-
-lemma embed_determined:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and IN: "a \<in> Field r"
-shows "f a = wo_rel.suc r' (f`(rel.underS r a))"
-proof-
- have "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
- using assms by (auto simp add: embed_underS)
- hence "f`(rel.underS r a) = rel.underS r' (f a)"
- by (auto simp add: bij_betw_def)
- moreover
- {have "f a \<in> Field r'" using IN
- using EMB WELL embed_Field[of r r' f] by auto
- hence "f a = wo_rel.suc r' (rel.underS r' (f a))"
- using WELL' by (auto simp add: wo_rel_def wo_rel.suc_underS)
- }
- ultimately show ?thesis by simp
-qed
-
-
-lemma embed_unique:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMBf: "embed r r' f" and EMBg: "embed r r' g"
-shows "a \<in> Field r \<longrightarrow> f a = g a"
-proof(rule wo_rel.well_order_induct[of r], auto simp add: WELL wo_rel_def)
- fix a
- assume IH: "\<forall>b. b \<noteq> a \<and> (b,a): r \<longrightarrow> b \<in> Field r \<longrightarrow> f b = g b" and
- *: "a \<in> Field r"
- hence "\<forall>b \<in> rel.underS r a. f b = g b"
- unfolding rel.underS_def by (auto simp add: Field_def)
- hence "f`(rel.underS r a) = g`(rel.underS r a)" by force
- thus "f a = g a"
- using assms * embed_determined[of r r' f a] embed_determined[of r r' g a] by auto
-qed
-
-
-lemma embed_bothWays_inverse:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and EMB': "embed r' r f'"
-shows "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
-proof-
- have "embed r r (f' o f)" using assms
- by(auto simp add: comp_embed)
- moreover have "embed r r id" using assms
- by (auto simp add: id_embed)
- ultimately have "\<forall>a \<in> Field r. f'(f a) = a"
- using assms embed_unique[of r r "f' o f" id] id_def by auto
- moreover
- {have "embed r' r' (f o f')" using assms
- by(auto simp add: comp_embed)
- moreover have "embed r' r' id" using assms
- by (auto simp add: id_embed)
- ultimately have "\<forall>a' \<in> Field r'. f(f' a') = a'"
- using assms embed_unique[of r' r' "f o f'" id] id_def by auto
- }
- ultimately show ?thesis by blast
-qed
-
-
-lemma embed_bothWays_bij_betw:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and EMB': "embed r' r g"
-shows "bij_betw f (Field r) (Field r')"
-proof-
- let ?A = "Field r" let ?A' = "Field r'"
- have "embed r r (g o f) \<and> embed r' r' (f o g)"
- using assms by (auto simp add: comp_embed)
- hence 1: "(\<forall>a \<in> ?A. g(f a) = a) \<and> (\<forall>a' \<in> ?A'. f(g a') = a')"
- using WELL id_embed[of r] embed_unique[of r r "g o f" id]
- WELL' id_embed[of r'] embed_unique[of r' r' "f o g" id]
- id_def by auto
- have 2: "(\<forall>a \<in> ?A. f a \<in> ?A') \<and> (\<forall>a' \<in> ?A'. g a' \<in> ?A)"
- using assms embed_Field[of r r' f] embed_Field[of r' r g] by blast
- (* *)
- show ?thesis
- proof(unfold bij_betw_def inj_on_def, auto simp add: 2)
- fix a b assume *: "a \<in> ?A" "b \<in> ?A" and **: "f a = f b"
- have "a = g(f a) \<and> b = g(f b)" using * 1 by auto
- with ** show "a = b" by auto
- next
- fix a' assume *: "a' \<in> ?A'"
- hence "g a' \<in> ?A \<and> f(g a') = a'" using 1 2 by auto
- thus "a' \<in> f ` ?A" by force
- qed
-qed
-
-
-lemma embed_bothWays_iso:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and EMB': "embed r' r g"
-shows "iso r r' f"
-unfolding iso_def using assms by (auto simp add: embed_bothWays_bij_betw)
-
-
-subsection {* More properties of embeddings, strict embeddings and isomorphisms *}
-
-
-lemma embed_bothWays_Field_bij_betw:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r r' f" and EMB': "embed r' r f'"
-shows "bij_betw f (Field r) (Field r')"
-proof-
- have "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
- using assms by (auto simp add: embed_bothWays_inverse)
- moreover
- have "f`(Field r) \<le> Field r' \<and> f' ` (Field r') \<le> Field r"
- using assms by (auto simp add: embed_Field)
- ultimately
- show ?thesis using bij_betw_byWitness[of "Field r" f' f "Field r'"] by auto
-qed
-
-
-lemma embedS_comp_embed:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "embedS r r' f" and EMB': "embed r' r'' f'"
-shows "embedS r r'' (f' o f)"
-proof-
- let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
- have 1: "embed r r' f \<and> \<not> (bij_betw f (Field r) (Field r'))"
- using EMB by (auto simp add: embedS_def)
- hence 2: "embed r r'' ?g"
- using WELL EMB' comp_embed[of r r' f r'' f'] by auto
- moreover
- {assume "bij_betw ?g (Field r) (Field r'')"
- hence "embed r'' r ?h" using 2 WELL
- by (auto simp add: inv_into_Field_embed_bij_betw)
- hence "embed r' r (?h o f')" using WELL' EMB'
- by (auto simp add: comp_embed)
- hence "bij_betw f (Field r) (Field r')" using WELL WELL' 1
- by (auto simp add: embed_bothWays_Field_bij_betw)
- with 1 have False by blast
- }
- ultimately show ?thesis unfolding embedS_def by auto
-qed
-
-
-lemma embed_comp_embedS:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "embed r r' f" and EMB': "embedS r' r'' f'"
-shows "embedS r r'' (f' o f)"
-proof-
- let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
- have 1: "embed r' r'' f' \<and> \<not> (bij_betw f' (Field r') (Field r''))"
- using EMB' by (auto simp add: embedS_def)
- hence 2: "embed r r'' ?g"
- using WELL EMB comp_embed[of r r' f r'' f'] by auto
- moreover
- {assume "bij_betw ?g (Field r) (Field r'')"
- hence "embed r'' r ?h" using 2 WELL
- by (auto simp add: inv_into_Field_embed_bij_betw)
- hence "embed r'' r' (f o ?h)" using WELL'' EMB
- by (auto simp add: comp_embed)
- hence "bij_betw f' (Field r') (Field r'')" using WELL' WELL'' 1
- by (auto simp add: embed_bothWays_Field_bij_betw)
- with 1 have False by blast
- }
- ultimately show ?thesis unfolding embedS_def by auto
-qed
-
-
-lemma embed_comp_iso:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "embed r r' f" and EMB': "iso r' r'' f'"
-shows "embed r r'' (f' o f)"
-using assms unfolding iso_def
-by (auto simp add: comp_embed)
-
-
-lemma iso_comp_embed:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "iso r r' f" and EMB': "embed r' r'' f'"
-shows "embed r r'' (f' o f)"
-using assms unfolding iso_def
-by (auto simp add: comp_embed)
-
-
-lemma embedS_comp_iso:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "embedS r r' f" and EMB': "iso r' r'' f'"
-shows "embedS r r'' (f' o f)"
-using assms unfolding iso_def
-by (auto simp add: embedS_comp_embed)
-
-
-lemma iso_comp_embedS:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
- and EMB: "iso r r' f" and EMB': "embedS r' r'' f'"
-shows "embedS r r'' (f' o f)"
-using assms unfolding iso_def using embed_comp_embedS
-by (auto simp add: embed_comp_embedS)
-
-
-lemma embedS_Field:
-assumes WELL: "Well_order r" and EMB: "embedS r r' f"
-shows "f ` (Field r) < Field r'"
-proof-
- have "f`(Field r) \<le> Field r'" using assms
- by (auto simp add: embed_Field embedS_def)
- moreover
- {have "inj_on f (Field r)" using assms
- by (auto simp add: embedS_def embed_inj_on)
- hence "f`(Field r) \<noteq> Field r'" using EMB
- by (auto simp add: embedS_def bij_betw_def)
- }
- ultimately show ?thesis by blast
-qed
-
-
-lemma embedS_iff:
-assumes WELL: "Well_order r" and ISO: "embed r r' f"
-shows "embedS r r' f = (f ` (Field r) < Field r')"
-proof
- assume "embedS r r' f"
- thus "f ` Field r \<subset> Field r'"
- using WELL by (auto simp add: embedS_Field)
-next
- assume "f ` Field r \<subset> Field r'"
- hence "\<not> bij_betw f (Field r) (Field r')"
- unfolding bij_betw_def by blast
- thus "embedS r r' f" unfolding embedS_def
- using ISO by auto
-qed
-
-
-lemma iso_Field:
-"iso r r' f \<Longrightarrow> f ` (Field r) = Field r'"
-using assms by (auto simp add: iso_def bij_betw_def)
-
-
-lemma iso_iff:
-assumes "Well_order r"
-shows "iso r r' f = (embed r r' f \<and> f ` (Field r) = Field r')"
-proof
- assume "iso r r' f"
- thus "embed r r' f \<and> f ` (Field r) = Field r'"
- by (auto simp add: iso_Field iso_def)
-next
- assume *: "embed r r' f \<and> f ` Field r = Field r'"
- hence "inj_on f (Field r)" using assms by (auto simp add: embed_inj_on)
- with * have "bij_betw f (Field r) (Field r')"
- unfolding bij_betw_def by simp
- with * show "iso r r' f" unfolding iso_def by auto
-qed
-
-
-lemma iso_iff2:
-assumes "Well_order r"
-shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and>
- (\<forall>a \<in> Field r. \<forall>b \<in> Field r.
- (((a,b) \<in> r) = ((f a, f b) \<in> r'))))"
-using assms
-proof(auto simp add: iso_def)
- fix a b
- assume "embed r r' f"
- hence "compat r r' f" using embed_compat[of r] by auto
- moreover assume "(a,b) \<in> r"
- ultimately show "(f a, f b) \<in> r'" using compat_def[of r] by auto
-next
- let ?f' = "inv_into (Field r) f"
- assume "embed r r' f" and 1: "bij_betw f (Field r) (Field r')"
- hence "embed r' r ?f'" using assms
- by (auto simp add: inv_into_Field_embed_bij_betw)
- hence 2: "compat r' r ?f'" using embed_compat[of r'] by auto
- fix a b assume *: "a \<in> Field r" "b \<in> Field r" and **: "(f a,f b) \<in> r'"
- hence "?f'(f a) = a \<and> ?f'(f b) = b" using 1
- by (auto simp add: bij_betw_inv_into_left)
- thus "(a,b) \<in> r" using ** 2 compat_def[of r' r ?f'] by fastforce
-next
- assume *: "bij_betw f (Field r) (Field r')" and
- **: "\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')"
- have 1: "\<And> a. rel.under r a \<le> Field r \<and> rel.under r' (f a) \<le> Field r'"
- by (auto simp add: rel.under_Field)
- have 2: "inj_on f (Field r)" using * by (auto simp add: bij_betw_def)
- {fix a assume ***: "a \<in> Field r"
- have "bij_betw f (rel.under r a) (rel.under r' (f a))"
- proof(unfold bij_betw_def, auto)
- show "inj_on f (rel.under r a)"
- using 1 2 by (auto simp add: subset_inj_on)
- next
- fix b assume "b \<in> rel.under r a"
- hence "a \<in> Field r \<and> b \<in> Field r \<and> (b,a) \<in> r"
- unfolding rel.under_def by (auto simp add: Field_def Range_def Domain_def)
- with 1 ** show "f b \<in> rel.under r' (f a)"
- unfolding rel.under_def by auto
- next
- fix b' assume "b' \<in> rel.under r' (f a)"
- hence 3: "(b',f a) \<in> r'" unfolding rel.under_def by simp
- hence "b' \<in> Field r'" unfolding Field_def by auto
- with * obtain b where "b \<in> Field r \<and> f b = b'"
- unfolding bij_betw_def by force
- with 3 ** ***
- show "b' \<in> f ` (rel.under r a)" unfolding rel.under_def by blast
- qed
- }
- thus "embed r r' f" unfolding embed_def using * by auto
-qed
-
-
-lemma iso_iff3:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
-proof
- assume "iso r r' f"
- thus "bij_betw f (Field r) (Field r') \<and> compat r r' f"
- unfolding compat_def using WELL by (auto simp add: iso_iff2 Field_def)
-next
- have Well: "wo_rel r \<and> wo_rel r'" using WELL WELL'
- by (auto simp add: wo_rel_def)
- assume *: "bij_betw f (Field r) (Field r') \<and> compat r r' f"
- thus "iso r r' f"
- unfolding "compat_def" using assms
- proof(auto simp add: iso_iff2)
- fix a b assume **: "a \<in> Field r" "b \<in> Field r" and
- ***: "(f a, f b) \<in> r'"
- {assume "(b,a) \<in> r \<or> b = a"
- hence "(b,a): r"using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
- hence "(f b, f a) \<in> r'" using * unfolding compat_def by auto
- hence "f a = f b"
- using Well *** wo_rel.ANTISYM[of r'] antisym_def[of r'] by blast
- hence "a = b" using * ** unfolding bij_betw_def inj_on_def by auto
- hence "(a,b) \<in> r" using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
- }
- thus "(a,b) \<in> r"
- using Well ** wo_rel.TOTAL[of r] total_on_def[of _ r] by blast
- qed
-qed
-
-
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Embedding_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellorder_Embedding_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,1145 @@
+(* Title: HOL/Cardinals/Wellorder_Embedding_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order embeddings (FP).
+*)
+
+header {* Well-Order Embeddings (FP) *}
+
+theory Wellorder_Embedding_FP
+imports "~~/src/HOL/Library/Zorn" Fun_More_FP Wellorder_Relation_FP
+begin
+
+
+text{* In this section, we introduce well-order {\em embeddings} and {\em isomorphisms} and
+prove their basic properties. The notion of embedding is considered from the point
+of view of the theory of ordinals, and therefore requires the source to be injected
+as an {\em initial segment} (i.e., {\em order filter}) of the target. A main result
+of this section is the existence of embeddings (in one direction or another) between
+any two well-orders, having as a consequence the fact that, given any two sets on
+any two types, one is smaller than (i.e., can be injected into) the other. *}
+
+
+subsection {* Auxiliaries *}
+
+lemma UNION_inj_on_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "\<And> i. i \<in> I \<Longrightarrow> wo_rel.ofilter r (A i)" and
+ INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
+shows "inj_on f (\<Union> i \<in> I. A i)"
+proof-
+ have "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i <= A j \<or> A j <= A i"
+ using wo_rel.ofilter_linord[of r] OF by blast
+ with WELL INJ show ?thesis
+ by (auto simp add: inj_on_UNION_chain)
+qed
+
+
+lemma under_underS_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ IN: "a \<in> Field r" and IN': "f a \<in> Field r'" and
+ BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
+proof-
+ have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
+ unfolding rel.underS_def by auto
+ moreover
+ {have "Refl r \<and> Refl r'" using WELL WELL'
+ by (auto simp add: order_on_defs)
+ hence "rel.under r a = rel.underS r a \<union> {a} \<and>
+ rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
+ using IN IN' by(auto simp add: rel.Refl_under_underS)
+ }
+ ultimately show ?thesis
+ using BIJ notIn_Un_bij_betw[of a "rel.underS r a" f "rel.underS r' (f a)"] by auto
+qed
+
+
+
+subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
+functions *}
+
+
+text{* Standardly, a function is an embedding of a well-order in another if it injectively and
+order-compatibly maps the former into an order filter of the latter.
+Here we opt for a more succinct definition (operator @{text "embed"}),
+asking that, for any element in the source, the function should be a bijection
+between the set of strict lower bounds of that element
+and the set of strict lower bounds of its image. (Later we prove equivalence with
+the standard definition -- lemma @{text "embed_iff_compat_inj_on_ofilter"}.)
+A {\em strict embedding} (operator @{text "embedS"}) is a non-bijective embedding
+and an isomorphism (operator @{text "iso"}) is a bijective embedding. *}
+
+
+definition embed :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"embed r r' f \<equiv> \<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
+
+
+lemmas embed_defs = embed_def embed_def[abs_def]
+
+
+text {* Strict embeddings: *}
+
+definition embedS :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
+
+
+lemmas embedS_defs = embedS_def embedS_def[abs_def]
+
+
+definition iso :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
+
+
+lemmas iso_defs = iso_def iso_def[abs_def]
+
+
+definition compat :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"compat r r' f \<equiv> \<forall>a b. (a,b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
+
+
+lemma compat_wf:
+assumes CMP: "compat r r' f" and WF: "wf r'"
+shows "wf r"
+proof-
+ have "r \<le> inv_image r' f"
+ unfolding inv_image_def using CMP
+ by (auto simp add: compat_def)
+ with WF show ?thesis
+ using wf_inv_image[of r' f] wf_subset[of "inv_image r' f"] by auto
+qed
+
+
+lemma id_embed: "embed r r id"
+by(auto simp add: id_def embed_def bij_betw_def)
+
+
+lemma id_iso: "iso r r id"
+by(auto simp add: id_def embed_def iso_def bij_betw_def)
+
+
+lemma embed_in_Field:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "f a \<in> Field r'"
+proof-
+ have Well: "wo_rel r"
+ using WELL by (auto simp add: wo_rel_def)
+ hence 1: "Refl r"
+ by (auto simp add: wo_rel.REFL)
+ hence "a \<in> rel.under r a" using IN rel.Refl_under_in by fastforce
+ hence "f a \<in> rel.under r' (f a)"
+ using EMB IN by (auto simp add: embed_def bij_betw_def)
+ thus ?thesis unfolding Field_def
+ by (auto simp: rel.under_def)
+qed
+
+
+lemma comp_embed:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and EMB': "embed r' r'' f'"
+shows "embed r r'' (f' o f)"
+proof(unfold embed_def, auto)
+ fix a assume *: "a \<in> Field r"
+ hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using embed_def[of r] EMB by auto
+ moreover
+ {have "f a \<in> Field r'"
+ using EMB WELL * by (auto simp add: embed_in_Field)
+ hence "bij_betw f' (rel.under r' (f a)) (rel.under r'' (f' (f a)))"
+ using embed_def[of r'] EMB' by auto
+ }
+ ultimately
+ show "bij_betw (f' \<circ> f) (rel.under r a) (rel.under r'' (f'(f a)))"
+ by(auto simp add: bij_betw_trans)
+qed
+
+
+lemma comp_iso:
+assumes WELL: "Well_order r" and
+ EMB: "iso r r' f" and EMB': "iso r' r'' f'"
+shows "iso r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed bij_betw_trans)
+
+
+text{* That @{text "embedS"} is also preserved by function composition shall be proved only later. *}
+
+
+lemma embed_Field:
+"\<lbrakk>Well_order r; embed r r' f\<rbrakk> \<Longrightarrow> f`(Field r) \<le> Field r'"
+by (auto simp add: embed_in_Field)
+
+
+lemma embed_preserves_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and OF: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter r' (f`A)"
+proof-
+ (* Preliminary facts *)
+ from WELL have Well: "wo_rel r" unfolding wo_rel_def .
+ from WELL' have Well': "wo_rel r'" unfolding wo_rel_def .
+ from OF have 0: "A \<le> Field r" by(auto simp add: Well wo_rel.ofilter_def)
+ (* Main proof *)
+ show ?thesis using Well' WELL EMB 0 embed_Field[of r r' f]
+ proof(unfold wo_rel.ofilter_def, auto simp add: image_def)
+ fix a b'
+ assume *: "a \<in> A" and **: "b' \<in> rel.under r' (f a)"
+ hence "a \<in> Field r" using 0 by auto
+ hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using * EMB by (auto simp add: embed_def)
+ hence "f`(rel.under r a) = rel.under r' (f a)"
+ by (simp add: bij_betw_def)
+ with ** image_def[of f "rel.under r a"] obtain b where
+ 1: "b \<in> rel.under r a \<and> b' = f b" by blast
+ hence "b \<in> A" using Well * OF
+ by (auto simp add: wo_rel.ofilter_def)
+ with 1 show "\<exists>b \<in> A. b' = f b" by blast
+ qed
+qed
+
+
+lemma embed_Field_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f"
+shows "wo_rel.ofilter r' (f`(Field r))"
+proof-
+ have "wo_rel.ofilter r (Field r)"
+ using WELL by (auto simp add: wo_rel_def wo_rel.Field_ofilter)
+ with WELL WELL' EMB
+ show ?thesis by (auto simp add: embed_preserves_ofilter)
+qed
+
+
+lemma embed_compat:
+assumes EMB: "embed r r' f"
+shows "compat r r' f"
+proof(unfold compat_def, clarify)
+ fix a b
+ assume *: "(a,b) \<in> r"
+ hence 1: "b \<in> Field r" using Field_def[of r] by blast
+ have "a \<in> rel.under r b"
+ using * rel.under_def[of r] by simp
+ hence "f a \<in> rel.under r' (f b)"
+ using EMB embed_def[of r r' f]
+ bij_betw_def[of f "rel.under r b" "rel.under r' (f b)"]
+ image_def[of f "rel.under r b"] 1 by auto
+ thus "(f a, f b) \<in> r'"
+ by (auto simp add: rel.under_def)
+qed
+
+
+lemma embed_inj_on:
+assumes WELL: "Well_order r" and EMB: "embed r r' f"
+shows "inj_on f (Field r)"
+proof(unfold inj_on_def, clarify)
+ (* Preliminary facts *)
+ from WELL have Well: "wo_rel r" unfolding wo_rel_def .
+ with wo_rel.TOTAL[of r]
+ have Total: "Total r" by simp
+ from Well wo_rel.REFL[of r]
+ have Refl: "Refl r" by simp
+ (* Main proof *)
+ fix a b
+ assume *: "a \<in> Field r" and **: "b \<in> Field r" and
+ ***: "f a = f b"
+ hence 1: "a \<in> Field r \<and> b \<in> Field r"
+ unfolding Field_def by auto
+ {assume "(a,b) \<in> r"
+ hence "a \<in> rel.under r b \<and> b \<in> rel.under r b"
+ using Refl by(auto simp add: rel.under_def refl_on_def)
+ hence "a = b"
+ using EMB 1 ***
+ by (auto simp add: embed_def bij_betw_def inj_on_def)
+ }
+ moreover
+ {assume "(b,a) \<in> r"
+ hence "a \<in> rel.under r a \<and> b \<in> rel.under r a"
+ using Refl by(auto simp add: rel.under_def refl_on_def)
+ hence "a = b"
+ using EMB 1 ***
+ by (auto simp add: embed_def bij_betw_def inj_on_def)
+ }
+ ultimately
+ show "a = b" using Total 1
+ by (auto simp add: total_on_def)
+qed
+
+
+lemma embed_underS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+proof-
+ have "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using assms by (auto simp add: embed_def)
+ moreover
+ {have "f a \<in> Field r'" using assms embed_Field[of r r' f] by auto
+ hence "rel.under r a = rel.underS r a \<union> {a} \<and>
+ rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
+ using assms by (auto simp add: order_on_defs rel.Refl_under_underS)
+ }
+ moreover
+ {have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
+ unfolding rel.underS_def by blast
+ }
+ ultimately show ?thesis
+ by (auto simp add: notIn_Un_bij_betw3)
+qed
+
+
+lemma embed_iff_compat_inj_on_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> wo_rel.ofilter r' (f`(Field r)))"
+using assms
+proof(auto simp add: embed_compat embed_inj_on embed_Field_ofilter,
+ unfold embed_def, auto) (* get rid of one implication *)
+ fix a
+ assume *: "inj_on f (Field r)" and
+ **: "compat r r' f" and
+ ***: "wo_rel.ofilter r' (f`(Field r))" and
+ ****: "a \<in> Field r"
+ (* Preliminary facts *)
+ have Well: "wo_rel r"
+ using WELL wo_rel_def[of r] by simp
+ hence Refl: "Refl r"
+ using wo_rel.REFL[of r] by simp
+ have Total: "Total r"
+ using Well wo_rel.TOTAL[of r] by simp
+ have Well': "wo_rel r'"
+ using WELL' wo_rel_def[of r'] by simp
+ hence Antisym': "antisym r'"
+ using wo_rel.ANTISYM[of r'] by simp
+ have "(a,a) \<in> r"
+ using **** Well wo_rel.REFL[of r]
+ refl_on_def[of _ r] by auto
+ hence "(f a, f a) \<in> r'"
+ using ** by(auto simp add: compat_def)
+ hence 0: "f a \<in> Field r'"
+ unfolding Field_def by auto
+ have "f a \<in> f`(Field r)"
+ using **** by auto
+ hence 2: "rel.under r' (f a) \<le> f`(Field r)"
+ using Well' *** wo_rel.ofilter_def[of r' "f`(Field r)"] by fastforce
+ (* Main proof *)
+ show "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ proof(unfold bij_betw_def, auto)
+ show "inj_on f (rel.under r a)"
+ using * by (metis (no_types) rel.under_Field subset_inj_on)
+ next
+ fix b assume "b \<in> rel.under r a"
+ thus "f b \<in> rel.under r' (f a)"
+ unfolding rel.under_def using **
+ by (auto simp add: compat_def)
+ next
+ fix b' assume *****: "b' \<in> rel.under r' (f a)"
+ hence "b' \<in> f`(Field r)"
+ using 2 by auto
+ with Field_def[of r] obtain b where
+ 3: "b \<in> Field r" and 4: "b' = f b" by auto
+ have "(b,a): r"
+ proof-
+ {assume "(a,b) \<in> r"
+ with ** 4 have "(f a, b'): r'"
+ by (auto simp add: compat_def)
+ with ***** Antisym' have "f a = b'"
+ by(auto simp add: rel.under_def antisym_def)
+ with 3 **** 4 * have "a = b"
+ by(auto simp add: inj_on_def)
+ }
+ moreover
+ {assume "a = b"
+ hence "(b,a) \<in> r" using Refl **** 3
+ by (auto simp add: refl_on_def)
+ }
+ ultimately
+ show ?thesis using Total **** 3 by (fastforce simp add: total_on_def)
+ qed
+ with 4 show "b' \<in> f`(rel.under r a)"
+ unfolding rel.under_def by auto
+ qed
+qed
+
+
+lemma inv_into_ofilter_embed:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and
+ BIJ: "\<forall>b \<in> A. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ IMAGE: "f ` A = Field r'"
+shows "embed r' r (inv_into A f)"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r"
+ using WELL wo_rel_def[of r] by simp
+ have Refl: "Refl r"
+ using Well wo_rel.REFL[of r] by simp
+ have Total: "Total r"
+ using Well wo_rel.TOTAL[of r] by simp
+ (* Main proof *)
+ have 1: "bij_betw f A (Field r')"
+ proof(unfold bij_betw_def inj_on_def, auto simp add: IMAGE)
+ fix b1 b2
+ assume *: "b1 \<in> A" and **: "b2 \<in> A" and
+ ***: "f b1 = f b2"
+ have 11: "b1 \<in> Field r \<and> b2 \<in> Field r"
+ using * ** Well OF by (auto simp add: wo_rel.ofilter_def)
+ moreover
+ {assume "(b1,b2) \<in> r"
+ hence "b1 \<in> rel.under r b2 \<and> b2 \<in> rel.under r b2"
+ unfolding rel.under_def using 11 Refl
+ by (auto simp add: refl_on_def)
+ hence "b1 = b2" using BIJ * ** ***
+ by (simp add: bij_betw_def inj_on_def)
+ }
+ moreover
+ {assume "(b2,b1) \<in> r"
+ hence "b1 \<in> rel.under r b1 \<and> b2 \<in> rel.under r b1"
+ unfolding rel.under_def using 11 Refl
+ by (auto simp add: refl_on_def)
+ hence "b1 = b2" using BIJ * ** ***
+ by (simp add: bij_betw_def inj_on_def)
+ }
+ ultimately
+ show "b1 = b2"
+ using Total by (auto simp add: total_on_def)
+ qed
+ (* *)
+ let ?f' = "(inv_into A f)"
+ (* *)
+ have 2: "\<forall>b \<in> A. bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
+ proof(clarify)
+ fix b assume *: "b \<in> A"
+ hence "rel.under r b \<le> A"
+ using Well OF by(auto simp add: wo_rel.ofilter_def)
+ moreover
+ have "f ` (rel.under r b) = rel.under r' (f b)"
+ using * BIJ by (auto simp add: bij_betw_def)
+ ultimately
+ show "bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
+ using 1 by (auto simp add: bij_betw_inv_into_subset)
+ qed
+ (* *)
+ have 3: "\<forall>b' \<in> Field r'. bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
+ proof(clarify)
+ fix b' assume *: "b' \<in> Field r'"
+ have "b' = f (?f' b')" using * 1
+ by (auto simp add: bij_betw_inv_into_right)
+ moreover
+ {obtain b where 31: "b \<in> A" and "f b = b'" using IMAGE * by force
+ hence "?f' b' = b" using 1 by (auto simp add: bij_betw_inv_into_left)
+ with 31 have "?f' b' \<in> A" by auto
+ }
+ ultimately
+ show "bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
+ using 2 by auto
+ qed
+ (* *)
+ thus ?thesis unfolding embed_def .
+qed
+
+
+lemma inv_into_underS_embed:
+assumes WELL: "Well_order r" and
+ BIJ: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ IN: "a \<in> Field r" and
+ IMAGE: "f ` (rel.underS r a) = Field r'"
+shows "embed r' r (inv_into (rel.underS r a) f)"
+using assms
+by(auto simp add: wo_rel_def wo_rel.underS_ofilter inv_into_ofilter_embed)
+
+
+lemma inv_into_Field_embed:
+assumes WELL: "Well_order r" and EMB: "embed r r' f" and
+ IMAGE: "Field r' \<le> f ` (Field r)"
+shows "embed r' r (inv_into (Field r) f)"
+proof-
+ have "(\<forall>b \<in> Field r. bij_betw f (rel.under r b) (rel.under r' (f b)))"
+ using EMB by (auto simp add: embed_def)
+ moreover
+ have "f ` (Field r) \<le> Field r'"
+ using EMB WELL by (auto simp add: embed_Field)
+ ultimately
+ show ?thesis using assms
+ by(auto simp add: wo_rel_def wo_rel.Field_ofilter inv_into_ofilter_embed)
+qed
+
+
+lemma inv_into_Field_embed_bij_betw:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and BIJ: "bij_betw f (Field r) (Field r')"
+shows "embed r' r (inv_into (Field r) f)"
+proof-
+ have "Field r' \<le> f ` (Field r)"
+ using BIJ by (auto simp add: bij_betw_def)
+ thus ?thesis using assms
+ by(auto simp add: inv_into_Field_embed)
+qed
+
+
+
+
+
+subsection {* Given any two well-orders, one can be embedded in the other *}
+
+
+text{* Here is an overview of the proof of of this fact, stated in theorem
+@{text "wellorders_totally_ordered"}:
+
+ Fix the well-orders @{text "r::'a rel"} and @{text "r'::'a' rel"}.
+ Attempt to define an embedding @{text "f::'a \<Rightarrow> 'a'"} from @{text "r"} to @{text "r'"} in the
+ natural way by well-order recursion ("hoping" that @{text "Field r"} turns out to be smaller
+ than @{text "Field r'"}), but also record, at the recursive step, in a function
+ @{text "g::'a \<Rightarrow> bool"}, the extra information of whether @{text "Field r'"}
+ gets exhausted or not.
+
+ If @{text "Field r'"} does not get exhausted, then @{text "Field r"} is indeed smaller
+ and @{text "f"} is the desired embedding from @{text "r"} to @{text "r'"}
+ (lemma @{text "wellorders_totally_ordered_aux"}).
+
+ Otherwise, it means that @{text "Field r'"} is the smaller one, and the inverse of
+ (the "good" segment of) @{text "f"} is the desired embedding from @{text "r'"} to @{text "r"}
+ (lemma @{text "wellorders_totally_ordered_aux2"}).
+*}
+
+
+lemma wellorders_totally_ordered_aux:
+fixes r ::"'a rel" and r'::"'a' rel" and
+ f :: "'a \<Rightarrow> 'a'" and a::'a
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and IN: "a \<in> Field r" and
+ IH: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ NOT: "f ` (rel.underS r a) \<noteq> Field r'" and SUC: "f a = wo_rel.suc r' (f`(rel.underS r a))"
+shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ have OF: "wo_rel.ofilter r (rel.underS r a)"
+ by (auto simp add: Well wo_rel.underS_ofilter)
+ hence UN: "rel.underS r a = (\<Union> b \<in> rel.underS r a. rel.under r b)"
+ using Well wo_rel.ofilter_under_UNION[of r "rel.underS r a"] by blast
+ (* Gather facts about elements of rel.underS r a *)
+ {fix b assume *: "b \<in> rel.underS r a"
+ hence t0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
+ have t1: "b \<in> Field r"
+ using * rel.underS_Field[of r a] by auto
+ have t2: "f`(rel.under r b) = rel.under r' (f b)"
+ using IH * by (auto simp add: bij_betw_def)
+ hence t3: "wo_rel.ofilter r' (f`(rel.under r b))"
+ using Well' by (auto simp add: wo_rel.under_ofilter)
+ have "f`(rel.under r b) \<le> Field r'"
+ using t2 by (auto simp add: rel.under_Field)
+ moreover
+ have "b \<in> rel.under r b"
+ using t1 by(auto simp add: Refl rel.Refl_under_in)
+ ultimately
+ have t4: "f b \<in> Field r'" by auto
+ have "f`(rel.under r b) = rel.under r' (f b) \<and>
+ wo_rel.ofilter r' (f`(rel.under r b)) \<and>
+ f b \<in> Field r'"
+ using t2 t3 t4 by auto
+ }
+ hence bFact:
+ "\<forall>b \<in> rel.underS r a. f`(rel.under r b) = rel.under r' (f b) \<and>
+ wo_rel.ofilter r' (f`(rel.under r b)) \<and>
+ f b \<in> Field r'" by blast
+ (* *)
+ have subField: "f`(rel.underS r a) \<le> Field r'"
+ using bFact by blast
+ (* *)
+ have OF': "wo_rel.ofilter r' (f`(rel.underS r a))"
+ proof-
+ have "f`(rel.underS r a) = f`(\<Union> b \<in> rel.underS r a. rel.under r b)"
+ using UN by auto
+ also have "\<dots> = (\<Union> b \<in> rel.underS r a. f`(rel.under r b))" by blast
+ also have "\<dots> = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))"
+ using bFact by auto
+ finally
+ have "f`(rel.underS r a) = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))" .
+ thus ?thesis
+ using Well' bFact
+ wo_rel.ofilter_UNION[of r' "rel.underS r a" "\<lambda> b. rel.under r' (f b)"] by fastforce
+ qed
+ (* *)
+ have "f`(rel.underS r a) \<union> rel.AboveS r' (f`(rel.underS r a)) = Field r'"
+ using Well' OF' by (auto simp add: wo_rel.ofilter_AboveS_Field)
+ hence NE: "rel.AboveS r' (f`(rel.underS r a)) \<noteq> {}"
+ using subField NOT by blast
+ (* Main proof *)
+ have INCL1: "f`(rel.underS r a) \<le> rel.underS r' (f a) "
+ proof(auto)
+ fix b assume *: "b \<in> rel.underS r a"
+ have "f b \<noteq> f a \<and> (f b, f a) \<in> r'"
+ using subField Well' SUC NE *
+ wo_rel.suc_greater[of r' "f`(rel.underS r a)" "f b"] by force
+ thus "f b \<in> rel.underS r' (f a)"
+ unfolding rel.underS_def by simp
+ qed
+ (* *)
+ have INCL2: "rel.underS r' (f a) \<le> f`(rel.underS r a)"
+ proof
+ fix b' assume "b' \<in> rel.underS r' (f a)"
+ hence "b' \<noteq> f a \<and> (b', f a) \<in> r'"
+ unfolding rel.underS_def by simp
+ thus "b' \<in> f`(rel.underS r a)"
+ using Well' SUC NE OF'
+ wo_rel.suc_ofilter_in[of r' "f ` rel.underS r a" b'] by auto
+ qed
+ (* *)
+ have INJ: "inj_on f (rel.underS r a)"
+ proof-
+ have "\<forall>b \<in> rel.underS r a. inj_on f (rel.under r b)"
+ using IH by (auto simp add: bij_betw_def)
+ moreover
+ have "\<forall>b. wo_rel.ofilter r (rel.under r b)"
+ using Well by (auto simp add: wo_rel.under_ofilter)
+ ultimately show ?thesis
+ using WELL bFact UN
+ UNION_inj_on_ofilter[of r "rel.underS r a" "\<lambda>b. rel.under r b" f]
+ by auto
+ qed
+ (* *)
+ have BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+ unfolding bij_betw_def
+ using INJ INCL1 INCL2 by auto
+ (* *)
+ have "f a \<in> Field r'"
+ using Well' subField NE SUC
+ by (auto simp add: wo_rel.suc_inField)
+ thus ?thesis
+ using WELL WELL' IN BIJ under_underS_bij_betw[of r r' a f] by auto
+qed
+
+
+lemma wellorders_totally_ordered_aux2:
+fixes r ::"'a rel" and r'::"'a' rel" and
+ f :: "'a \<Rightarrow> 'a'" and g :: "'a \<Rightarrow> bool" and a::'a
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+MAIN1:
+ "\<And> a. (False \<notin> g`(rel.underS r a) \<and> f`(rel.underS r a) \<noteq> Field r'
+ \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True)
+ \<and>
+ (\<not>(False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')
+ \<longrightarrow> g a = False)" and
+MAIN2: "\<And> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
+ bij_betw f (rel.under r a) (rel.under r' (f a))" and
+Case: "a \<in> Field r \<and> False \<in> g`(rel.under r a)"
+shows "\<exists>f'. embed r' r f'"
+proof-
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Antisym: "antisym r" using Well wo_rel.ANTISYM[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ (* *)
+ have 0: "rel.under r a = rel.underS r a \<union> {a}"
+ using Refl Case by(auto simp add: rel.Refl_under_underS)
+ (* *)
+ have 1: "g a = False"
+ proof-
+ {assume "g a \<noteq> False"
+ with 0 Case have "False \<in> g`(rel.underS r a)" by blast
+ with MAIN1 have "g a = False" by blast}
+ thus ?thesis by blast
+ qed
+ let ?A = "{a \<in> Field r. g a = False}"
+ let ?a = "(wo_rel.minim r ?A)"
+ (* *)
+ have 2: "?A \<noteq> {} \<and> ?A \<le> Field r" using Case 1 by blast
+ (* *)
+ have 3: "False \<notin> g`(rel.underS r ?a)"
+ proof
+ assume "False \<in> g`(rel.underS r ?a)"
+ then obtain b where "b \<in> rel.underS r ?a" and 31: "g b = False" by auto
+ hence 32: "(b,?a) \<in> r \<and> b \<noteq> ?a"
+ by (auto simp add: rel.underS_def)
+ hence "b \<in> Field r" unfolding Field_def by auto
+ with 31 have "b \<in> ?A" by auto
+ hence "(?a,b) \<in> r" using wo_rel.minim_least 2 Well by fastforce
+ (* again: why worked without type annotations? *)
+ with 32 Antisym show False
+ by (auto simp add: antisym_def)
+ qed
+ have temp: "?a \<in> ?A"
+ using Well 2 wo_rel.minim_in[of r ?A] by auto
+ hence 4: "?a \<in> Field r" by auto
+ (* *)
+ have 5: "g ?a = False" using temp by blast
+ (* *)
+ have 6: "f`(rel.underS r ?a) = Field r'"
+ using MAIN1[of ?a] 3 5 by blast
+ (* *)
+ have 7: "\<forall>b \<in> rel.underS r ?a. bij_betw f (rel.under r b) (rel.under r' (f b))"
+ proof
+ fix b assume as: "b \<in> rel.underS r ?a"
+ moreover
+ have "wo_rel.ofilter r (rel.underS r ?a)"
+ using Well by (auto simp add: wo_rel.underS_ofilter)
+ ultimately
+ have "False \<notin> g`(rel.under r b)" using 3 Well by (subst (asm) wo_rel.ofilter_def) fast+
+ moreover have "b \<in> Field r"
+ unfolding Field_def using as by (auto simp add: rel.underS_def)
+ ultimately
+ show "bij_betw f (rel.under r b) (rel.under r' (f b))"
+ using MAIN2 by auto
+ qed
+ (* *)
+ have "embed r' r (inv_into (rel.underS r ?a) f)"
+ using WELL WELL' 7 4 6 inv_into_underS_embed[of r ?a f r'] by auto
+ thus ?thesis
+ unfolding embed_def by blast
+qed
+
+
+theorem wellorders_totally_ordered:
+fixes r ::"'a rel" and r'::"'a' rel"
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<exists>f. embed r r' f) \<or> (\<exists>f'. embed r' r f')"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ (* Main proof *)
+ obtain H where H_def: "H =
+ (\<lambda>h a. if False \<notin> (snd o h)`(rel.underS r a) \<and> (fst o h)`(rel.underS r a) \<noteq> Field r'
+ then (wo_rel.suc r' ((fst o h)`(rel.underS r a)), True)
+ else (undefined, False))" by blast
+ have Adm: "wo_rel.adm_wo r H"
+ using Well
+ proof(unfold wo_rel.adm_wo_def, clarify)
+ fix h1::"'a \<Rightarrow> 'a' * bool" and h2::"'a \<Rightarrow> 'a' * bool" and x
+ assume "\<forall>y\<in>rel.underS r x. h1 y = h2 y"
+ hence "\<forall>y\<in>rel.underS r x. (fst o h1) y = (fst o h2) y \<and>
+ (snd o h1) y = (snd o h2) y" by auto
+ hence "(fst o h1)`(rel.underS r x) = (fst o h2)`(rel.underS r x) \<and>
+ (snd o h1)`(rel.underS r x) = (snd o h2)`(rel.underS r x)"
+ by (auto simp add: image_def)
+ thus "H h1 x = H h2 x" by (simp add: H_def del: not_False_in_image_Ball)
+ qed
+ (* More constant definitions: *)
+ obtain h::"'a \<Rightarrow> 'a' * bool" and f::"'a \<Rightarrow> 'a'" and g::"'a \<Rightarrow> bool"
+ where h_def: "h = wo_rel.worec r H" and
+ f_def: "f = fst o h" and g_def: "g = snd o h" by blast
+ obtain test where test_def:
+ "test = (\<lambda> a. False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')" by blast
+ (* *)
+ have *: "\<And> a. h a = H h a"
+ using Adm Well wo_rel.worec_fixpoint[of r H] by (simp add: h_def)
+ have Main1:
+ "\<And> a. (test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
+ (\<not>(test a) \<longrightarrow> g a = False)"
+ proof- (* How can I prove this withou fixing a? *)
+ fix a show "(test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
+ (\<not>(test a) \<longrightarrow> g a = False)"
+ using *[of a] test_def f_def g_def H_def by auto
+ qed
+ (* *)
+ let ?phi = "\<lambda> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
+ bij_betw f (rel.under r a) (rel.under r' (f a))"
+ have Main2: "\<And> a. ?phi a"
+ proof-
+ fix a show "?phi a"
+ proof(rule wo_rel.well_order_induct[of r ?phi],
+ simp only: Well, clarify)
+ fix a
+ assume IH: "\<forall>b. b \<noteq> a \<and> (b,a) \<in> r \<longrightarrow> ?phi b" and
+ *: "a \<in> Field r" and
+ **: "False \<notin> g`(rel.under r a)"
+ have 1: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))"
+ proof(clarify)
+ fix b assume ***: "b \<in> rel.underS r a"
+ hence 0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
+ moreover have "b \<in> Field r"
+ using *** rel.underS_Field[of r a] by auto
+ moreover have "False \<notin> g`(rel.under r b)"
+ using 0 ** Trans rel.under_incr[of r b a] by auto
+ ultimately show "bij_betw f (rel.under r b) (rel.under r' (f b))"
+ using IH by auto
+ qed
+ (* *)
+ have 21: "False \<notin> g`(rel.underS r a)"
+ using ** rel.underS_subset_under[of r a] by auto
+ have 22: "g`(rel.under r a) \<le> {True}" using ** by auto
+ moreover have 23: "a \<in> rel.under r a"
+ using Refl * by (auto simp add: rel.Refl_under_in)
+ ultimately have 24: "g a = True" by blast
+ have 2: "f`(rel.underS r a) \<noteq> Field r'"
+ proof
+ assume "f`(rel.underS r a) = Field r'"
+ hence "g a = False" using Main1 test_def by blast
+ with 24 show False using ** by blast
+ qed
+ (* *)
+ have 3: "f a = wo_rel.suc r' (f`(rel.underS r a))"
+ using 21 2 Main1 test_def by blast
+ (* *)
+ show "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using WELL WELL' 1 2 3 *
+ wellorders_totally_ordered_aux[of r r' a f] by auto
+ qed
+ qed
+ (* *)
+ let ?chi = "(\<lambda> a. a \<in> Field r \<and> False \<in> g`(rel.under r a))"
+ show ?thesis
+ proof(cases "\<exists>a. ?chi a")
+ assume "\<not> (\<exists>a. ?chi a)"
+ hence "\<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using Main2 by blast
+ thus ?thesis unfolding embed_def by blast
+ next
+ assume "\<exists>a. ?chi a"
+ then obtain a where "?chi a" by blast
+ hence "\<exists>f'. embed r' r f'"
+ using wellorders_totally_ordered_aux2[of r r' g f a]
+ WELL WELL' Main1 Main2 test_def by fast
+ thus ?thesis by blast
+ qed
+qed
+
+
+subsection {* Uniqueness of embeddings *}
+
+
+text{* Here we show a fact complementary to the one from the previous subsection -- namely,
+that between any two well-orders there is {\em at most} one embedding, and is the one
+definable by the expected well-order recursive equation. As a consequence, any two
+embeddings of opposite directions are mutually inverse. *}
+
+
+lemma embed_determined:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "f a = wo_rel.suc r' (f`(rel.underS r a))"
+proof-
+ have "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+ using assms by (auto simp add: embed_underS)
+ hence "f`(rel.underS r a) = rel.underS r' (f a)"
+ by (auto simp add: bij_betw_def)
+ moreover
+ {have "f a \<in> Field r'" using IN
+ using EMB WELL embed_Field[of r r' f] by auto
+ hence "f a = wo_rel.suc r' (rel.underS r' (f a))"
+ using WELL' by (auto simp add: wo_rel_def wo_rel.suc_underS)
+ }
+ ultimately show ?thesis by simp
+qed
+
+
+lemma embed_unique:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMBf: "embed r r' f" and EMBg: "embed r r' g"
+shows "a \<in> Field r \<longrightarrow> f a = g a"
+proof(rule wo_rel.well_order_induct[of r], auto simp add: WELL wo_rel_def)
+ fix a
+ assume IH: "\<forall>b. b \<noteq> a \<and> (b,a): r \<longrightarrow> b \<in> Field r \<longrightarrow> f b = g b" and
+ *: "a \<in> Field r"
+ hence "\<forall>b \<in> rel.underS r a. f b = g b"
+ unfolding rel.underS_def by (auto simp add: Field_def)
+ hence "f`(rel.underS r a) = g`(rel.underS r a)" by force
+ thus "f a = g a"
+ using assms * embed_determined[of r r' f a] embed_determined[of r r' g a] by auto
+qed
+
+
+lemma embed_bothWays_inverse:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r f'"
+shows "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
+proof-
+ have "embed r r (f' o f)" using assms
+ by(auto simp add: comp_embed)
+ moreover have "embed r r id" using assms
+ by (auto simp add: id_embed)
+ ultimately have "\<forall>a \<in> Field r. f'(f a) = a"
+ using assms embed_unique[of r r "f' o f" id] id_def by auto
+ moreover
+ {have "embed r' r' (f o f')" using assms
+ by(auto simp add: comp_embed)
+ moreover have "embed r' r' id" using assms
+ by (auto simp add: id_embed)
+ ultimately have "\<forall>a' \<in> Field r'. f(f' a') = a'"
+ using assms embed_unique[of r' r' "f o f'" id] id_def by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma embed_bothWays_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r g"
+shows "bij_betw f (Field r) (Field r')"
+proof-
+ let ?A = "Field r" let ?A' = "Field r'"
+ have "embed r r (g o f) \<and> embed r' r' (f o g)"
+ using assms by (auto simp add: comp_embed)
+ hence 1: "(\<forall>a \<in> ?A. g(f a) = a) \<and> (\<forall>a' \<in> ?A'. f(g a') = a')"
+ using WELL id_embed[of r] embed_unique[of r r "g o f" id]
+ WELL' id_embed[of r'] embed_unique[of r' r' "f o g" id]
+ id_def by auto
+ have 2: "(\<forall>a \<in> ?A. f a \<in> ?A') \<and> (\<forall>a' \<in> ?A'. g a' \<in> ?A)"
+ using assms embed_Field[of r r' f] embed_Field[of r' r g] by blast
+ (* *)
+ show ?thesis
+ proof(unfold bij_betw_def inj_on_def, auto simp add: 2)
+ fix a b assume *: "a \<in> ?A" "b \<in> ?A" and **: "f a = f b"
+ have "a = g(f a) \<and> b = g(f b)" using * 1 by auto
+ with ** show "a = b" by auto
+ next
+ fix a' assume *: "a' \<in> ?A'"
+ hence "g a' \<in> ?A \<and> f(g a') = a'" using 1 2 by auto
+ thus "a' \<in> f ` ?A" by force
+ qed
+qed
+
+
+lemma embed_bothWays_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r g"
+shows "iso r r' f"
+unfolding iso_def using assms by (auto simp add: embed_bothWays_bij_betw)
+
+
+subsection {* More properties of embeddings, strict embeddings and isomorphisms *}
+
+
+lemma embed_bothWays_Field_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r f'"
+shows "bij_betw f (Field r) (Field r')"
+proof-
+ have "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
+ using assms by (auto simp add: embed_bothWays_inverse)
+ moreover
+ have "f`(Field r) \<le> Field r' \<and> f' ` (Field r') \<le> Field r"
+ using assms by (auto simp add: embed_Field)
+ ultimately
+ show ?thesis using bij_betw_byWitness[of "Field r" f' f "Field r'"] by auto
+qed
+
+
+lemma embedS_comp_embed:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embedS r r' f" and EMB': "embed r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+ let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
+ have 1: "embed r r' f \<and> \<not> (bij_betw f (Field r) (Field r'))"
+ using EMB by (auto simp add: embedS_def)
+ hence 2: "embed r r'' ?g"
+ using WELL EMB' comp_embed[of r r' f r'' f'] by auto
+ moreover
+ {assume "bij_betw ?g (Field r) (Field r'')"
+ hence "embed r'' r ?h" using 2 WELL
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence "embed r' r (?h o f')" using WELL' EMB'
+ by (auto simp add: comp_embed)
+ hence "bij_betw f (Field r) (Field r')" using WELL WELL' 1
+ by (auto simp add: embed_bothWays_Field_bij_betw)
+ with 1 have False by blast
+ }
+ ultimately show ?thesis unfolding embedS_def by auto
+qed
+
+
+lemma embed_comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embed r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+ let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
+ have 1: "embed r' r'' f' \<and> \<not> (bij_betw f' (Field r') (Field r''))"
+ using EMB' by (auto simp add: embedS_def)
+ hence 2: "embed r r'' ?g"
+ using WELL EMB comp_embed[of r r' f r'' f'] by auto
+ moreover
+ {assume "bij_betw ?g (Field r) (Field r'')"
+ hence "embed r'' r ?h" using 2 WELL
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence "embed r'' r' (f o ?h)" using WELL'' EMB
+ by (auto simp add: comp_embed)
+ hence "bij_betw f' (Field r') (Field r'')" using WELL' WELL'' 1
+ by (auto simp add: embed_bothWays_Field_bij_betw)
+ with 1 have False by blast
+ }
+ ultimately show ?thesis unfolding embedS_def by auto
+qed
+
+
+lemma embed_comp_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embed r r' f" and EMB': "iso r' r'' f'"
+shows "embed r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed)
+
+
+lemma iso_comp_embed:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "iso r r' f" and EMB': "embed r' r'' f'"
+shows "embed r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed)
+
+
+lemma embedS_comp_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embedS r r' f" and EMB': "iso r' r'' f'"
+shows "embedS r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: embedS_comp_embed)
+
+
+lemma iso_comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "iso r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+using assms unfolding iso_def using embed_comp_embedS
+by (auto simp add: embed_comp_embedS)
+
+
+lemma embedS_Field:
+assumes WELL: "Well_order r" and EMB: "embedS r r' f"
+shows "f ` (Field r) < Field r'"
+proof-
+ have "f`(Field r) \<le> Field r'" using assms
+ by (auto simp add: embed_Field embedS_def)
+ moreover
+ {have "inj_on f (Field r)" using assms
+ by (auto simp add: embedS_def embed_inj_on)
+ hence "f`(Field r) \<noteq> Field r'" using EMB
+ by (auto simp add: embedS_def bij_betw_def)
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma embedS_iff:
+assumes WELL: "Well_order r" and ISO: "embed r r' f"
+shows "embedS r r' f = (f ` (Field r) < Field r')"
+proof
+ assume "embedS r r' f"
+ thus "f ` Field r \<subset> Field r'"
+ using WELL by (auto simp add: embedS_Field)
+next
+ assume "f ` Field r \<subset> Field r'"
+ hence "\<not> bij_betw f (Field r) (Field r')"
+ unfolding bij_betw_def by blast
+ thus "embedS r r' f" unfolding embedS_def
+ using ISO by auto
+qed
+
+
+lemma iso_Field:
+"iso r r' f \<Longrightarrow> f ` (Field r) = Field r'"
+using assms by (auto simp add: iso_def bij_betw_def)
+
+
+lemma iso_iff:
+assumes "Well_order r"
+shows "iso r r' f = (embed r r' f \<and> f ` (Field r) = Field r')"
+proof
+ assume "iso r r' f"
+ thus "embed r r' f \<and> f ` (Field r) = Field r'"
+ by (auto simp add: iso_Field iso_def)
+next
+ assume *: "embed r r' f \<and> f ` Field r = Field r'"
+ hence "inj_on f (Field r)" using assms by (auto simp add: embed_inj_on)
+ with * have "bij_betw f (Field r) (Field r')"
+ unfolding bij_betw_def by simp
+ with * show "iso r r' f" unfolding iso_def by auto
+qed
+
+
+lemma iso_iff2:
+assumes "Well_order r"
+shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and>
+ (\<forall>a \<in> Field r. \<forall>b \<in> Field r.
+ (((a,b) \<in> r) = ((f a, f b) \<in> r'))))"
+using assms
+proof(auto simp add: iso_def)
+ fix a b
+ assume "embed r r' f"
+ hence "compat r r' f" using embed_compat[of r] by auto
+ moreover assume "(a,b) \<in> r"
+ ultimately show "(f a, f b) \<in> r'" using compat_def[of r] by auto
+next
+ let ?f' = "inv_into (Field r) f"
+ assume "embed r r' f" and 1: "bij_betw f (Field r) (Field r')"
+ hence "embed r' r ?f'" using assms
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence 2: "compat r' r ?f'" using embed_compat[of r'] by auto
+ fix a b assume *: "a \<in> Field r" "b \<in> Field r" and **: "(f a,f b) \<in> r'"
+ hence "?f'(f a) = a \<and> ?f'(f b) = b" using 1
+ by (auto simp add: bij_betw_inv_into_left)
+ thus "(a,b) \<in> r" using ** 2 compat_def[of r' r ?f'] by fastforce
+next
+ assume *: "bij_betw f (Field r) (Field r')" and
+ **: "\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')"
+ have 1: "\<And> a. rel.under r a \<le> Field r \<and> rel.under r' (f a) \<le> Field r'"
+ by (auto simp add: rel.under_Field)
+ have 2: "inj_on f (Field r)" using * by (auto simp add: bij_betw_def)
+ {fix a assume ***: "a \<in> Field r"
+ have "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ proof(unfold bij_betw_def, auto)
+ show "inj_on f (rel.under r a)"
+ using 1 2 by (metis subset_inj_on)
+ next
+ fix b assume "b \<in> rel.under r a"
+ hence "a \<in> Field r \<and> b \<in> Field r \<and> (b,a) \<in> r"
+ unfolding rel.under_def by (auto simp add: Field_def Range_def Domain_def)
+ with 1 ** show "f b \<in> rel.under r' (f a)"
+ unfolding rel.under_def by auto
+ next
+ fix b' assume "b' \<in> rel.under r' (f a)"
+ hence 3: "(b',f a) \<in> r'" unfolding rel.under_def by simp
+ hence "b' \<in> Field r'" unfolding Field_def by auto
+ with * obtain b where "b \<in> Field r \<and> f b = b'"
+ unfolding bij_betw_def by force
+ with 3 ** ***
+ show "b' \<in> f ` (rel.under r a)" unfolding rel.under_def by blast
+ qed
+ }
+ thus "embed r r' f" unfolding embed_def using * by auto
+qed
+
+
+lemma iso_iff3:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
+proof
+ assume "iso r r' f"
+ thus "bij_betw f (Field r) (Field r') \<and> compat r r' f"
+ unfolding compat_def using WELL by (auto simp add: iso_iff2 Field_def)
+next
+ have Well: "wo_rel r \<and> wo_rel r'" using WELL WELL'
+ by (auto simp add: wo_rel_def)
+ assume *: "bij_betw f (Field r) (Field r') \<and> compat r r' f"
+ thus "iso r r' f"
+ unfolding "compat_def" using assms
+ proof(auto simp add: iso_iff2)
+ fix a b assume **: "a \<in> Field r" "b \<in> Field r" and
+ ***: "(f a, f b) \<in> r'"
+ {assume "(b,a) \<in> r \<or> b = a"
+ hence "(b,a): r"using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
+ hence "(f b, f a) \<in> r'" using * unfolding compat_def by auto
+ hence "f a = f b"
+ using Well *** wo_rel.ANTISYM[of r'] antisym_def[of r'] by blast
+ hence "a = b" using * ** unfolding bij_betw_def inj_on_def by auto
+ hence "(a,b) \<in> r" using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
+ }
+ thus "(a,b) \<in> r"
+ using Well ** wo_rel.TOTAL[of r] total_on_def[of _ r] by blast
+ qed
+qed
+
+
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Relation.thy
--- a/src/HOL/Cardinals/Wellorder_Relation.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Cardinals/Wellorder_Relation.thy Tue Nov 19 17:07:52 2013 +0100
@@ -8,7 +8,7 @@
header {* Well-Order Relations *}
theory Wellorder_Relation
-imports Wellorder_Relation_Base Wellfounded_More
+imports Wellorder_Relation_FP Wellfounded_More
begin
context wo_rel
@@ -64,17 +64,7 @@
lemma minim_Under:
"\<lbrakk>B \<le> Field r; B \<noteq> {}\<rbrakk> \<Longrightarrow> minim B \<in> Under B"
-by(auto simp add: Under_def
-minim_in
-minim_inField
-minim_least
-under_ofilter
-underS_ofilter
-Field_ofilter
-ofilter_Under
-ofilter_UnderS
-ofilter_Un
-)
+by(auto simp add: Under_def minim_inField minim_least)
lemma equals_minim_Under:
"\<lbrakk>B \<le> Field r; a \<in> B; a \<in> Under B\<rbrakk>
@@ -410,7 +400,41 @@
qed
-subsubsection {* Properties of order filters *}
+subsubsection {* Properties of order filters *}
+
+lemma ofilter_Under[simp]:
+assumes "A \<le> Field r"
+shows "ofilter(Under A)"
+proof(unfold ofilter_def, auto)
+ fix x assume "x \<in> Under A"
+ thus "x \<in> Field r"
+ using Under_Field assms by auto
+next
+ fix a x
+ assume "a \<in> Under A" and "x \<in> under a"
+ thus "x \<in> Under A"
+ using TRANS under_Under_trans by auto
+qed
+
+lemma ofilter_UnderS[simp]:
+assumes "A \<le> Field r"
+shows "ofilter(UnderS A)"
+proof(unfold ofilter_def, auto)
+ fix x assume "x \<in> UnderS A"
+ thus "x \<in> Field r"
+ using UnderS_Field assms by auto
+next
+ fix a x
+ assume "a \<in> UnderS A" and "x \<in> under a"
+ thus "x \<in> UnderS A"
+ using TRANS ANTISYM under_UnderS_trans by auto
+qed
+
+lemma ofilter_Int[simp]: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A Int B)"
+unfolding ofilter_def by blast
+
+lemma ofilter_Un[simp]: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A \<union> B)"
+unfolding ofilter_def by blast
lemma ofilter_INTER:
"\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> ofilter(A i)\<rbrakk> \<Longrightarrow> ofilter (\<Inter> i \<in> I. A i)"
@@ -496,10 +520,6 @@
under_ofilter[simp]
underS_ofilter[simp]
Field_ofilter[simp]
- ofilter_Under[simp]
- ofilter_UnderS[simp]
- ofilter_Int[simp]
- ofilter_Un[simp]
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Relation_Base.thy
--- a/src/HOL/Cardinals/Wellorder_Relation_Base.thy Mon Nov 18 17:15:01 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,669 +0,0 @@
-(* Title: HOL/Cardinals/Wellorder_Relation_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Well-order relations (base).
-*)
-
-header {* Well-Order Relations (Base) *}
-
-theory Wellorder_Relation_Base
-imports Wellfounded_More_Base
-begin
-
-
-text{* In this section, we develop basic concepts and results pertaining
-to well-order relations. Note that we consider well-order relations
-as {\em non-strict relations},
-i.e., as containing the diagonals of their fields. *}
-
-
-locale wo_rel = rel + assumes WELL: "Well_order r"
-begin
-
-text{* The following context encompasses all this section. In other words,
-for the whole section, we consider a fixed well-order relation @{term "r"}. *}
-
-(* context wo_rel *)
-
-
-subsection {* Auxiliaries *}
-
-
-lemma REFL: "Refl r"
-using WELL order_on_defs[of _ r] by auto
-
-
-lemma TRANS: "trans r"
-using WELL order_on_defs[of _ r] by auto
-
-
-lemma ANTISYM: "antisym r"
-using WELL order_on_defs[of _ r] by auto
-
-
-lemma TOTAL: "Total r"
-using WELL order_on_defs[of _ r] by auto
-
-
-lemma TOTALS: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
-using REFL TOTAL refl_on_def[of _ r] total_on_def[of _ r] by force
-
-
-lemma LIN: "Linear_order r"
-using WELL well_order_on_def[of _ r] by auto
-
-
-lemma WF: "wf (r - Id)"
-using WELL well_order_on_def[of _ r] by auto
-
-
-lemma cases_Total:
-"\<And> phi a b. \<lbrakk>{a,b} <= Field r; ((a,b) \<in> r \<Longrightarrow> phi a b); ((b,a) \<in> r \<Longrightarrow> phi a b)\<rbrakk>
- \<Longrightarrow> phi a b"
-using TOTALS by auto
-
-
-lemma cases_Total3:
-"\<And> phi a b. \<lbrakk>{a,b} \<le> Field r; ((a,b) \<in> r - Id \<or> (b,a) \<in> r - Id \<Longrightarrow> phi a b);
- (a = b \<Longrightarrow> phi a b)\<rbrakk> \<Longrightarrow> phi a b"
-using TOTALS by auto
-
-
-subsection {* Well-founded induction and recursion adapted to non-strict well-order relations *}
-
-
-text{* Here we provide induction and recursion principles specific to {\em non-strict}
-well-order relations.
-Although minor variations of those for well-founded relations, they will be useful
-for doing away with the tediousness of
-having to take out the diagonal each time in order to switch to a well-founded relation. *}
-
-
-lemma well_order_induct:
-assumes IND: "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x"
-shows "P a"
-proof-
- have "\<And>x. \<forall>y. (y, x) \<in> r - Id \<longrightarrow> P y \<Longrightarrow> P x"
- using IND by blast
- thus "P a" using WF wf_induct[of "r - Id" P a] by blast
-qed
-
-
-definition
-worec :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
-where
-"worec F \<equiv> wfrec (r - Id) F"
-
-
-definition
-adm_wo :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
-where
-"adm_wo H \<equiv> \<forall>f g x. (\<forall>y \<in> underS x. f y = g y) \<longrightarrow> H f x = H g x"
-
-
-lemma worec_fixpoint:
-assumes ADM: "adm_wo H"
-shows "worec H = H (worec H)"
-proof-
- let ?rS = "r - Id"
- have "adm_wf (r - Id) H"
- unfolding adm_wf_def
- using ADM adm_wo_def[of H] underS_def by auto
- hence "wfrec ?rS H = H (wfrec ?rS H)"
- using WF wfrec_fixpoint[of ?rS H] by simp
- thus ?thesis unfolding worec_def .
-qed
-
-
-subsection {* The notions of maximum, minimum, supremum, successor and order filter *}
-
-
-text{*
-We define the successor {\em of a set}, and not of an element (the latter is of course
-a particular case). Also, we define the maximum {\em of two elements}, @{text "max2"},
-and the minimum {\em of a set}, @{text "minim"} -- we chose these variants since we
-consider them the most useful for well-orders. The minimum is defined in terms of the
-auxiliary relational operator @{text "isMinim"}. Then, supremum and successor are
-defined in terms of minimum as expected.
-The minimum is only meaningful for non-empty sets, and the successor is only
-meaningful for sets for which strict upper bounds exist.
-Order filters for well-orders are also known as ``initial segments". *}
-
-
-definition max2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
-where "max2 a b \<equiv> if (a,b) \<in> r then b else a"
-
-
-definition isMinim :: "'a set \<Rightarrow> 'a \<Rightarrow> bool"
-where "isMinim A b \<equiv> b \<in> A \<and> (\<forall>a \<in> A. (b,a) \<in> r)"
-
-definition minim :: "'a set \<Rightarrow> 'a"
-where "minim A \<equiv> THE b. isMinim A b"
-
-
-definition supr :: "'a set \<Rightarrow> 'a"
-where "supr A \<equiv> minim (Above A)"
-
-definition suc :: "'a set \<Rightarrow> 'a"
-where "suc A \<equiv> minim (AboveS A)"
-
-definition ofilter :: "'a set \<Rightarrow> bool"
-where
-"ofilter A \<equiv> (A \<le> Field r) \<and> (\<forall>a \<in> A. under a \<le> A)"
-
-
-subsubsection {* Properties of max2 *}
-
-
-lemma max2_greater_among:
-assumes "a \<in> Field r" and "b \<in> Field r"
-shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r \<and> max2 a b \<in> {a,b}"
-proof-
- {assume "(a,b) \<in> r"
- hence ?thesis using max2_def assms REFL refl_on_def
- by (auto simp add: refl_on_def)
- }
- moreover
- {assume "a = b"
- hence "(a,b) \<in> r" using REFL assms
- by (auto simp add: refl_on_def)
- }
- moreover
- {assume *: "a \<noteq> b \<and> (b,a) \<in> r"
- hence "(a,b) \<notin> r" using ANTISYM
- by (auto simp add: antisym_def)
- hence ?thesis using * max2_def assms REFL refl_on_def
- by (auto simp add: refl_on_def)
- }
- ultimately show ?thesis using assms TOTAL
- total_on_def[of "Field r" r] by blast
-qed
-
-
-lemma max2_greater:
-assumes "a \<in> Field r" and "b \<in> Field r"
-shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r"
-using assms by (auto simp add: max2_greater_among)
-
-
-lemma max2_among:
-assumes "a \<in> Field r" and "b \<in> Field r"
-shows "max2 a b \<in> {a, b}"
-using assms max2_greater_among[of a b] by simp
-
-
-lemma max2_equals1:
-assumes "a \<in> Field r" and "b \<in> Field r"
-shows "(max2 a b = a) = ((b,a) \<in> r)"
-using assms ANTISYM unfolding antisym_def using TOTALS
-by(auto simp add: max2_def max2_among)
-
-
-lemma max2_equals2:
-assumes "a \<in> Field r" and "b \<in> Field r"
-shows "(max2 a b = b) = ((a,b) \<in> r)"
-using assms ANTISYM unfolding antisym_def using TOTALS
-unfolding max2_def by auto
-
-
-subsubsection {* Existence and uniqueness for isMinim and well-definedness of minim *}
-
-
-lemma isMinim_unique:
-assumes MINIM: "isMinim B a" and MINIM': "isMinim B a'"
-shows "a = a'"
-proof-
- {have "a \<in> B"
- using MINIM isMinim_def by simp
- hence "(a',a) \<in> r"
- using MINIM' isMinim_def by simp
- }
- moreover
- {have "a' \<in> B"
- using MINIM' isMinim_def by simp
- hence "(a,a') \<in> r"
- using MINIM isMinim_def by simp
- }
- ultimately
- show ?thesis using ANTISYM antisym_def[of r] by blast
-qed
-
-
-lemma Well_order_isMinim_exists:
-assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
-shows "\<exists>b. isMinim B b"
-proof-
- from spec[OF WF[unfolded wf_eq_minimal[of "r - Id"]], of B] NE obtain b where
- *: "b \<in> B \<and> (\<forall>b'. b' \<noteq> b \<and> (b',b) \<in> r \<longrightarrow> b' \<notin> B)" by auto
- show ?thesis
- proof(simp add: isMinim_def, rule exI[of _ b], auto)
- show "b \<in> B" using * by simp
- next
- fix b' assume As: "b' \<in> B"
- hence **: "b \<in> Field r \<and> b' \<in> Field r" using As SUB * by auto
- (* *)
- from As * have "b' = b \<or> (b',b) \<notin> r" by auto
- moreover
- {assume "b' = b"
- hence "(b,b') \<in> r"
- using ** REFL by (auto simp add: refl_on_def)
- }
- moreover
- {assume "b' \<noteq> b \<and> (b',b) \<notin> r"
- hence "(b,b') \<in> r"
- using ** TOTAL by (auto simp add: total_on_def)
- }
- ultimately show "(b,b') \<in> r" by blast
- qed
-qed
-
-
-lemma minim_isMinim:
-assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
-shows "isMinim B (minim B)"
-proof-
- let ?phi = "(\<lambda> b. isMinim B b)"
- from assms Well_order_isMinim_exists
- obtain b where *: "?phi b" by blast
- moreover
- have "\<And> b'. ?phi b' \<Longrightarrow> b' = b"
- using isMinim_unique * by auto
- ultimately show ?thesis
- unfolding minim_def using theI[of ?phi b] by blast
-qed
-
-
-subsubsection{* Properties of minim *}
-
-
-lemma minim_in:
-assumes "B \<le> Field r" and "B \<noteq> {}"
-shows "minim B \<in> B"
-proof-
- from minim_isMinim[of B] assms
- have "isMinim B (minim B)" by simp
- thus ?thesis by (simp add: isMinim_def)
-qed
-
-
-lemma minim_inField:
-assumes "B \<le> Field r" and "B \<noteq> {}"
-shows "minim B \<in> Field r"
-proof-
- have "minim B \<in> B" using assms by (simp add: minim_in)
- thus ?thesis using assms by blast
-qed
-
-
-lemma minim_least:
-assumes SUB: "B \<le> Field r" and IN: "b \<in> B"
-shows "(minim B, b) \<in> r"
-proof-
- from minim_isMinim[of B] assms
- have "isMinim B (minim B)" by auto
- thus ?thesis by (auto simp add: isMinim_def IN)
-qed
-
-
-lemma equals_minim:
-assumes SUB: "B \<le> Field r" and IN: "a \<in> B" and
- LEAST: "\<And> b. b \<in> B \<Longrightarrow> (a,b) \<in> r"
-shows "a = minim B"
-proof-
- from minim_isMinim[of B] assms
- have "isMinim B (minim B)" by auto
- moreover have "isMinim B a" using IN LEAST isMinim_def by auto
- ultimately show ?thesis
- using isMinim_unique by auto
-qed
-
-
-subsubsection{* Properties of successor *}
-
-
-lemma suc_AboveS:
-assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}"
-shows "suc B \<in> AboveS B"
-proof(unfold suc_def)
- have "AboveS B \<le> Field r"
- using AboveS_Field by auto
- thus "minim (AboveS B) \<in> AboveS B"
- using assms by (simp add: minim_in)
-qed
-
-
-lemma suc_greater:
-assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}" and
- IN: "b \<in> B"
-shows "suc B \<noteq> b \<and> (b,suc B) \<in> r"
-proof-
- from assms suc_AboveS
- have "suc B \<in> AboveS B" by simp
- with IN AboveS_def show ?thesis by simp
-qed
-
-
-lemma suc_least_AboveS:
-assumes ABOVES: "a \<in> AboveS B"
-shows "(suc B,a) \<in> r"
-proof(unfold suc_def)
- have "AboveS B \<le> Field r"
- using AboveS_Field by auto
- thus "(minim (AboveS B),a) \<in> r"
- using assms minim_least by simp
-qed
-
-
-lemma suc_inField:
-assumes "B \<le> Field r" and "AboveS B \<noteq> {}"
-shows "suc B \<in> Field r"
-proof-
- have "suc B \<in> AboveS B" using suc_AboveS assms by simp
- thus ?thesis
- using assms AboveS_Field by auto
-qed
-
-
-lemma equals_suc_AboveS:
-assumes SUB: "B \<le> Field r" and ABV: "a \<in> AboveS B" and
- MINIM: "\<And> a'. a' \<in> AboveS B \<Longrightarrow> (a,a') \<in> r"
-shows "a = suc B"
-proof(unfold suc_def)
- have "AboveS B \<le> Field r"
- using AboveS_Field[of B] by auto
- thus "a = minim (AboveS B)"
- using assms equals_minim
- by simp
-qed
-
-
-lemma suc_underS:
-assumes IN: "a \<in> Field r"
-shows "a = suc (underS a)"
-proof-
- have "underS a \<le> Field r"
- using underS_Field by auto
- moreover
- have "a \<in> AboveS (underS a)"
- using in_AboveS_underS IN by auto
- moreover
- have "\<forall>a' \<in> AboveS (underS a). (a,a') \<in> r"
- proof(clarify)
- fix a'
- assume *: "a' \<in> AboveS (underS a)"
- hence **: "a' \<in> Field r"
- using AboveS_Field by auto
- {assume "(a,a') \<notin> r"
- hence "a' = a \<or> (a',a) \<in> r"
- using TOTAL IN ** by (auto simp add: total_on_def)
- moreover
- {assume "a' = a"
- hence "(a,a') \<in> r"
- using REFL IN ** by (auto simp add: refl_on_def)
- }
- moreover
- {assume "a' \<noteq> a \<and> (a',a) \<in> r"
- hence "a' \<in> underS a"
- unfolding underS_def by simp
- hence "a' \<notin> AboveS (underS a)"
- using AboveS_disjoint by blast
- with * have False by simp
- }
- ultimately have "(a,a') \<in> r" by blast
- }
- thus "(a, a') \<in> r" by blast
- qed
- ultimately show ?thesis
- using equals_suc_AboveS by auto
-qed
-
-
-subsubsection {* Properties of order filters *}
-
-
-lemma under_ofilter:
-"ofilter (under a)"
-proof(unfold ofilter_def under_def, auto simp add: Field_def)
- fix aa x
- assume "(aa,a) \<in> r" "(x,aa) \<in> r"
- thus "(x,a) \<in> r"
- using TRANS trans_def[of r] by blast
-qed
-
-
-lemma underS_ofilter:
-"ofilter (underS a)"
-proof(unfold ofilter_def underS_def under_def, auto simp add: Field_def)
- fix aa assume "(a, aa) \<in> r" "(aa, a) \<in> r" and DIFF: "aa \<noteq> a"
- thus False
- using ANTISYM antisym_def[of r] by blast
-next
- fix aa x
- assume "(aa,a) \<in> r" "aa \<noteq> a" "(x,aa) \<in> r"
- thus "(x,a) \<in> r"
- using TRANS trans_def[of r] by blast
-qed
-
-
-lemma Field_ofilter:
-"ofilter (Field r)"
-by(unfold ofilter_def under_def, auto simp add: Field_def)
-
-
-lemma ofilter_underS_Field:
-"ofilter A = ((\<exists>a \<in> Field r. A = underS a) \<or> (A = Field r))"
-proof
- assume "(\<exists>a\<in>Field r. A = underS a) \<or> A = Field r"
- thus "ofilter A"
- by (auto simp: underS_ofilter Field_ofilter)
-next
- assume *: "ofilter A"
- let ?One = "(\<exists>a\<in>Field r. A = underS a)"
- let ?Two = "(A = Field r)"
- show "?One \<or> ?Two"
- proof(cases ?Two, simp)
- let ?B = "(Field r) - A"
- let ?a = "minim ?B"
- assume "A \<noteq> Field r"
- moreover have "A \<le> Field r" using * ofilter_def by simp
- ultimately have 1: "?B \<noteq> {}" by blast
- hence 2: "?a \<in> Field r" using minim_inField[of ?B] by blast
- have 3: "?a \<in> ?B" using minim_in[of ?B] 1 by blast
- hence 4: "?a \<notin> A" by blast
- have 5: "A \<le> Field r" using * ofilter_def[of A] by auto
- (* *)
- moreover
- have "A = underS ?a"
- proof
- show "A \<le> underS ?a"
- proof(unfold underS_def, auto simp add: 4)
- fix x assume **: "x \<in> A"
- hence 11: "x \<in> Field r" using 5 by auto
- have 12: "x \<noteq> ?a" using 4 ** by auto
- have 13: "under x \<le> A" using * ofilter_def ** by auto
- {assume "(x,?a) \<notin> r"
- hence "(?a,x) \<in> r"
- using TOTAL total_on_def[of "Field r" r]
- 2 4 11 12 by auto
- hence "?a \<in> under x" using under_def by auto
- hence "?a \<in> A" using ** 13 by blast
- with 4 have False by simp
- }
- thus "(x,?a) \<in> r" by blast
- qed
- next
- show "underS ?a \<le> A"
- proof(unfold underS_def, auto)
- fix x
- assume **: "x \<noteq> ?a" and ***: "(x,?a) \<in> r"
- hence 11: "x \<in> Field r" using Field_def by fastforce
- {assume "x \<notin> A"
- hence "x \<in> ?B" using 11 by auto
- hence "(?a,x) \<in> r" using 3 minim_least[of ?B x] by blast
- hence False
- using ANTISYM antisym_def[of r] ** *** by auto
- }
- thus "x \<in> A" by blast
- qed
- qed
- ultimately have ?One using 2 by blast
- thus ?thesis by simp
- qed
-qed
-
-
-lemma ofilter_Under:
-assumes "A \<le> Field r"
-shows "ofilter(Under A)"
-proof(unfold ofilter_def, auto)
- fix x assume "x \<in> Under A"
- thus "x \<in> Field r"
- using Under_Field assms by auto
-next
- fix a x
- assume "a \<in> Under A" and "x \<in> under a"
- thus "x \<in> Under A"
- using TRANS under_Under_trans by auto
-qed
-
-
-lemma ofilter_UnderS:
-assumes "A \<le> Field r"
-shows "ofilter(UnderS A)"
-proof(unfold ofilter_def, auto)
- fix x assume "x \<in> UnderS A"
- thus "x \<in> Field r"
- using UnderS_Field assms by auto
-next
- fix a x
- assume "a \<in> UnderS A" and "x \<in> under a"
- thus "x \<in> UnderS A"
- using TRANS ANTISYM under_UnderS_trans by auto
-qed
-
-
-lemma ofilter_Int: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A Int B)"
-unfolding ofilter_def by blast
-
-
-lemma ofilter_Un: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A \<union> B)"
-unfolding ofilter_def by blast
-
-
-lemma ofilter_UNION:
-"(\<And> i. i \<in> I \<Longrightarrow> ofilter(A i)) \<Longrightarrow> ofilter (\<Union> i \<in> I. A i)"
-unfolding ofilter_def by blast
-
-
-lemma ofilter_under_UNION:
-assumes "ofilter A"
-shows "A = (\<Union> a \<in> A. under a)"
-proof
- have "\<forall>a \<in> A. under a \<le> A"
- using assms ofilter_def by auto
- thus "(\<Union> a \<in> A. under a) \<le> A" by blast
-next
- have "\<forall>a \<in> A. a \<in> under a"
- using REFL Refl_under_in assms ofilter_def by blast
- thus "A \<le> (\<Union> a \<in> A. under a)" by blast
-qed
-
-
-subsubsection{* Other properties *}
-
-
-lemma ofilter_linord:
-assumes OF1: "ofilter A" and OF2: "ofilter B"
-shows "A \<le> B \<or> B \<le> A"
-proof(cases "A = Field r")
- assume Case1: "A = Field r"
- hence "B \<le> A" using OF2 ofilter_def by auto
- thus ?thesis by simp
-next
- assume Case2: "A \<noteq> Field r"
- with ofilter_underS_Field OF1 obtain a where
- 1: "a \<in> Field r \<and> A = underS a" by auto
- show ?thesis
- proof(cases "B = Field r")
- assume Case21: "B = Field r"
- hence "A \<le> B" using OF1 ofilter_def by auto
- thus ?thesis by simp
- next
- assume Case22: "B \<noteq> Field r"
- with ofilter_underS_Field OF2 obtain b where
- 2: "b \<in> Field r \<and> B = underS b" by auto
- have "a = b \<or> (a,b) \<in> r \<or> (b,a) \<in> r"
- using 1 2 TOTAL total_on_def[of _ r] by auto
- moreover
- {assume "a = b" with 1 2 have ?thesis by auto
- }
- moreover
- {assume "(a,b) \<in> r"
- with underS_incr TRANS ANTISYM 1 2
- have "A \<le> B" by auto
- hence ?thesis by auto
- }
- moreover
- {assume "(b,a) \<in> r"
- with underS_incr TRANS ANTISYM 1 2
- have "B \<le> A" by auto
- hence ?thesis by auto
- }
- ultimately show ?thesis by blast
- qed
-qed
-
-
-lemma ofilter_AboveS_Field:
-assumes "ofilter A"
-shows "A \<union> (AboveS A) = Field r"
-proof
- show "A \<union> (AboveS A) \<le> Field r"
- using assms ofilter_def AboveS_Field by auto
-next
- {fix x assume *: "x \<in> Field r" and **: "x \<notin> A"
- {fix y assume ***: "y \<in> A"
- with ** have 1: "y \<noteq> x" by auto
- {assume "(y,x) \<notin> r"
- moreover
- have "y \<in> Field r" using assms ofilter_def *** by auto
- ultimately have "(x,y) \<in> r"
- using 1 * TOTAL total_on_def[of _ r] by auto
- with *** assms ofilter_def under_def have "x \<in> A" by auto
- with ** have False by contradiction
- }
- hence "(y,x) \<in> r" by blast
- with 1 have "y \<noteq> x \<and> (y,x) \<in> r" by auto
- }
- with * have "x \<in> AboveS A" unfolding AboveS_def by auto
- }
- thus "Field r \<le> A \<union> (AboveS A)" by blast
-qed
-
-
-lemma suc_ofilter_in:
-assumes OF: "ofilter A" and ABOVE_NE: "AboveS A \<noteq> {}" and
- REL: "(b,suc A) \<in> r" and DIFF: "b \<noteq> suc A"
-shows "b \<in> A"
-proof-
- have *: "suc A \<in> Field r \<and> b \<in> Field r"
- using WELL REL well_order_on_domain by auto
- {assume **: "b \<notin> A"
- hence "b \<in> AboveS A"
- using OF * ofilter_AboveS_Field by auto
- hence "(suc A, b) \<in> r"
- using suc_least_AboveS by auto
- hence False using REL DIFF ANTISYM *
- by (auto simp add: antisym_def)
- }
- thus ?thesis by blast
-qed
-
-
-
-end (* context wo_rel *)
-
-
-
-end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Cardinals/Wellorder_Relation_FP.thy
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellorder_Relation_FP.thy Tue Nov 19 17:07:52 2013 +0100
@@ -0,0 +1,631 @@
+(* Title: HOL/Cardinals/Wellorder_Relation_FP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order relations (FP).
+*)
+
+header {* Well-Order Relations (FP) *}
+
+theory Wellorder_Relation_FP
+imports Wellfounded_More_FP
+begin
+
+
+text{* In this section, we develop basic concepts and results pertaining
+to well-order relations. Note that we consider well-order relations
+as {\em non-strict relations},
+i.e., as containing the diagonals of their fields. *}
+
+
+locale wo_rel = rel + assumes WELL: "Well_order r"
+begin
+
+text{* The following context encompasses all this section. In other words,
+for the whole section, we consider a fixed well-order relation @{term "r"}. *}
+
+(* context wo_rel *)
+
+
+subsection {* Auxiliaries *}
+
+
+lemma REFL: "Refl r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TRANS: "trans r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma ANTISYM: "antisym r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TOTAL: "Total r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TOTALS: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+using REFL TOTAL refl_on_def[of _ r] total_on_def[of _ r] by force
+
+
+lemma LIN: "Linear_order r"
+using WELL well_order_on_def[of _ r] by auto
+
+
+lemma WF: "wf (r - Id)"
+using WELL well_order_on_def[of _ r] by auto
+
+
+lemma cases_Total:
+"\<And> phi a b. \<lbrakk>{a,b} <= Field r; ((a,b) \<in> r \<Longrightarrow> phi a b); ((b,a) \<in> r \<Longrightarrow> phi a b)\<rbrakk>
+ \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+lemma cases_Total3:
+"\<And> phi a b. \<lbrakk>{a,b} \<le> Field r; ((a,b) \<in> r - Id \<or> (b,a) \<in> r - Id \<Longrightarrow> phi a b);
+ (a = b \<Longrightarrow> phi a b)\<rbrakk> \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+subsection {* Well-founded induction and recursion adapted to non-strict well-order relations *}
+
+
+text{* Here we provide induction and recursion principles specific to {\em non-strict}
+well-order relations.
+Although minor variations of those for well-founded relations, they will be useful
+for doing away with the tediousness of
+having to take out the diagonal each time in order to switch to a well-founded relation. *}
+
+
+lemma well_order_induct:
+assumes IND: "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x"
+shows "P a"
+proof-
+ have "\<And>x. \<forall>y. (y, x) \<in> r - Id \<longrightarrow> P y \<Longrightarrow> P x"
+ using IND by blast
+ thus "P a" using WF wf_induct[of "r - Id" P a] by blast
+qed
+
+
+definition
+worec :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where
+"worec F \<equiv> wfrec (r - Id) F"
+
+
+definition
+adm_wo :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+"adm_wo H \<equiv> \<forall>f g x. (\<forall>y \<in> underS x. f y = g y) \<longrightarrow> H f x = H g x"
+
+
+lemma worec_fixpoint:
+assumes ADM: "adm_wo H"
+shows "worec H = H (worec H)"
+proof-
+ let ?rS = "r - Id"
+ have "adm_wf (r - Id) H"
+ unfolding adm_wf_def
+ using ADM adm_wo_def[of H] underS_def by auto
+ hence "wfrec ?rS H = H (wfrec ?rS H)"
+ using WF wfrec_fixpoint[of ?rS H] by simp
+ thus ?thesis unfolding worec_def .
+qed
+
+
+subsection {* The notions of maximum, minimum, supremum, successor and order filter *}
+
+
+text{*
+We define the successor {\em of a set}, and not of an element (the latter is of course
+a particular case). Also, we define the maximum {\em of two elements}, @{text "max2"},
+and the minimum {\em of a set}, @{text "minim"} -- we chose these variants since we
+consider them the most useful for well-orders. The minimum is defined in terms of the
+auxiliary relational operator @{text "isMinim"}. Then, supremum and successor are
+defined in terms of minimum as expected.
+The minimum is only meaningful for non-empty sets, and the successor is only
+meaningful for sets for which strict upper bounds exist.
+Order filters for well-orders are also known as ``initial segments". *}
+
+
+definition max2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+where "max2 a b \<equiv> if (a,b) \<in> r then b else a"
+
+
+definition isMinim :: "'a set \<Rightarrow> 'a \<Rightarrow> bool"
+where "isMinim A b \<equiv> b \<in> A \<and> (\<forall>a \<in> A. (b,a) \<in> r)"
+
+definition minim :: "'a set \<Rightarrow> 'a"
+where "minim A \<equiv> THE b. isMinim A b"
+
+
+definition supr :: "'a set \<Rightarrow> 'a"
+where "supr A \<equiv> minim (Above A)"
+
+definition suc :: "'a set \<Rightarrow> 'a"
+where "suc A \<equiv> minim (AboveS A)"
+
+definition ofilter :: "'a set \<Rightarrow> bool"
+where
+"ofilter A \<equiv> (A \<le> Field r) \<and> (\<forall>a \<in> A. under a \<le> A)"
+
+
+subsubsection {* Properties of max2 *}
+
+
+lemma max2_greater_among:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r \<and> max2 a b \<in> {a,b}"
+proof-
+ {assume "(a,b) \<in> r"
+ hence ?thesis using max2_def assms REFL refl_on_def
+ by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "a = b"
+ hence "(a,b) \<in> r" using REFL assms
+ by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume *: "a \<noteq> b \<and> (b,a) \<in> r"
+ hence "(a,b) \<notin> r" using ANTISYM
+ by (auto simp add: antisym_def)
+ hence ?thesis using * max2_def assms REFL refl_on_def
+ by (auto simp add: refl_on_def)
+ }
+ ultimately show ?thesis using assms TOTAL
+ total_on_def[of "Field r" r] by blast
+qed
+
+
+lemma max2_greater:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r"
+using assms by (auto simp add: max2_greater_among)
+
+
+lemma max2_among:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "max2 a b \<in> {a, b}"
+using assms max2_greater_among[of a b] by simp
+
+
+lemma max2_equals1:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(max2 a b = a) = ((b,a) \<in> r)"
+using assms ANTISYM unfolding antisym_def using TOTALS
+by(auto simp add: max2_def max2_among)
+
+
+lemma max2_equals2:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(max2 a b = b) = ((a,b) \<in> r)"
+using assms ANTISYM unfolding antisym_def using TOTALS
+unfolding max2_def by auto
+
+
+subsubsection {* Existence and uniqueness for isMinim and well-definedness of minim *}
+
+
+lemma isMinim_unique:
+assumes MINIM: "isMinim B a" and MINIM': "isMinim B a'"
+shows "a = a'"
+proof-
+ {have "a \<in> B"
+ using MINIM isMinim_def by simp
+ hence "(a',a) \<in> r"
+ using MINIM' isMinim_def by simp
+ }
+ moreover
+ {have "a' \<in> B"
+ using MINIM' isMinim_def by simp
+ hence "(a,a') \<in> r"
+ using MINIM isMinim_def by simp
+ }
+ ultimately
+ show ?thesis using ANTISYM antisym_def[of r] by blast
+qed
+
+
+lemma Well_order_isMinim_exists:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "\<exists>b. isMinim B b"
+proof-
+ from spec[OF WF[unfolded wf_eq_minimal[of "r - Id"]], of B] NE obtain b where
+ *: "b \<in> B \<and> (\<forall>b'. b' \<noteq> b \<and> (b',b) \<in> r \<longrightarrow> b' \<notin> B)" by auto
+ show ?thesis
+ proof(simp add: isMinim_def, rule exI[of _ b], auto)
+ show "b \<in> B" using * by simp
+ next
+ fix b' assume As: "b' \<in> B"
+ hence **: "b \<in> Field r \<and> b' \<in> Field r" using As SUB * by auto
+ (* *)
+ from As * have "b' = b \<or> (b',b) \<notin> r" by auto
+ moreover
+ {assume "b' = b"
+ hence "(b,b') \<in> r"
+ using ** REFL by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "b' \<noteq> b \<and> (b',b) \<notin> r"
+ hence "(b,b') \<in> r"
+ using ** TOTAL by (auto simp add: total_on_def)
+ }
+ ultimately show "(b,b') \<in> r" by blast
+ qed
+qed
+
+
+lemma minim_isMinim:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "isMinim B (minim B)"
+proof-
+ let ?phi = "(\<lambda> b. isMinim B b)"
+ from assms Well_order_isMinim_exists
+ obtain b where *: "?phi b" by blast
+ moreover
+ have "\<And> b'. ?phi b' \<Longrightarrow> b' = b"
+ using isMinim_unique * by auto
+ ultimately show ?thesis
+ unfolding minim_def using theI[of ?phi b] by blast
+qed
+
+
+subsubsection{* Properties of minim *}
+
+
+lemma minim_in:
+assumes "B \<le> Field r" and "B \<noteq> {}"
+shows "minim B \<in> B"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by simp
+ thus ?thesis by (simp add: isMinim_def)
+qed
+
+
+lemma minim_inField:
+assumes "B \<le> Field r" and "B \<noteq> {}"
+shows "minim B \<in> Field r"
+proof-
+ have "minim B \<in> B" using assms by (simp add: minim_in)
+ thus ?thesis using assms by blast
+qed
+
+
+lemma minim_least:
+assumes SUB: "B \<le> Field r" and IN: "b \<in> B"
+shows "(minim B, b) \<in> r"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by auto
+ thus ?thesis by (auto simp add: isMinim_def IN)
+qed
+
+
+lemma equals_minim:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> B" and
+ LEAST: "\<And> b. b \<in> B \<Longrightarrow> (a,b) \<in> r"
+shows "a = minim B"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by auto
+ moreover have "isMinim B a" using IN LEAST isMinim_def by auto
+ ultimately show ?thesis
+ using isMinim_unique by auto
+qed
+
+
+subsubsection{* Properties of successor *}
+
+
+lemma suc_AboveS:
+assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}"
+shows "suc B \<in> AboveS B"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field by auto
+ thus "minim (AboveS B) \<in> AboveS B"
+ using assms by (simp add: minim_in)
+qed
+
+
+lemma suc_greater:
+assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}" and
+ IN: "b \<in> B"
+shows "suc B \<noteq> b \<and> (b,suc B) \<in> r"
+proof-
+ from assms suc_AboveS
+ have "suc B \<in> AboveS B" by simp
+ with IN AboveS_def show ?thesis by simp
+qed
+
+
+lemma suc_least_AboveS:
+assumes ABOVES: "a \<in> AboveS B"
+shows "(suc B,a) \<in> r"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field by auto
+ thus "(minim (AboveS B),a) \<in> r"
+ using assms minim_least by simp
+qed
+
+
+lemma suc_inField:
+assumes "B \<le> Field r" and "AboveS B \<noteq> {}"
+shows "suc B \<in> Field r"
+proof-
+ have "suc B \<in> AboveS B" using suc_AboveS assms by simp
+ thus ?thesis
+ using assms AboveS_Field by auto
+qed
+
+
+lemma equals_suc_AboveS:
+assumes SUB: "B \<le> Field r" and ABV: "a \<in> AboveS B" and
+ MINIM: "\<And> a'. a' \<in> AboveS B \<Longrightarrow> (a,a') \<in> r"
+shows "a = suc B"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field[of B] by auto
+ thus "a = minim (AboveS B)"
+ using assms equals_minim
+ by simp
+qed
+
+
+lemma suc_underS:
+assumes IN: "a \<in> Field r"
+shows "a = suc (underS a)"
+proof-
+ have "underS a \<le> Field r"
+ using underS_Field by auto
+ moreover
+ have "a \<in> AboveS (underS a)"
+ using in_AboveS_underS IN by auto
+ moreover
+ have "\<forall>a' \<in> AboveS (underS a). (a,a') \<in> r"
+ proof(clarify)
+ fix a'
+ assume *: "a' \<in> AboveS (underS a)"
+ hence **: "a' \<in> Field r"
+ using AboveS_Field by auto
+ {assume "(a,a') \<notin> r"
+ hence "a' = a \<or> (a',a) \<in> r"
+ using TOTAL IN ** by (auto simp add: total_on_def)
+ moreover
+ {assume "a' = a"
+ hence "(a,a') \<in> r"
+ using REFL IN ** by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "a' \<noteq> a \<and> (a',a) \<in> r"
+ hence "a' \<in> underS a"
+ unfolding underS_def by simp
+ hence "a' \<notin> AboveS (underS a)"
+ using AboveS_disjoint by blast
+ with * have False by simp
+ }
+ ultimately have "(a,a') \<in> r" by blast
+ }
+ thus "(a, a') \<in> r" by blast
+ qed
+ ultimately show ?thesis
+ using equals_suc_AboveS by auto
+qed
+
+
+subsubsection {* Properties of order filters *}
+
+
+lemma under_ofilter:
+"ofilter (under a)"
+proof(unfold ofilter_def under_def, auto simp add: Field_def)
+ fix aa x
+ assume "(aa,a) \<in> r" "(x,aa) \<in> r"
+ thus "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+qed
+
+
+lemma underS_ofilter:
+"ofilter (underS a)"
+proof(unfold ofilter_def underS_def under_def, auto simp add: Field_def)
+ fix aa assume "(a, aa) \<in> r" "(aa, a) \<in> r" and DIFF: "aa \<noteq> a"
+ thus False
+ using ANTISYM antisym_def[of r] by blast
+next
+ fix aa x
+ assume "(aa,a) \<in> r" "aa \<noteq> a" "(x,aa) \<in> r"
+ thus "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+qed
+
+
+lemma Field_ofilter:
+"ofilter (Field r)"
+by(unfold ofilter_def under_def, auto simp add: Field_def)
+
+
+lemma ofilter_underS_Field:
+"ofilter A = ((\<exists>a \<in> Field r. A = underS a) \<or> (A = Field r))"
+proof
+ assume "(\<exists>a\<in>Field r. A = underS a) \<or> A = Field r"
+ thus "ofilter A"
+ by (auto simp: underS_ofilter Field_ofilter)
+next
+ assume *: "ofilter A"
+ let ?One = "(\<exists>a\<in>Field r. A = underS a)"
+ let ?Two = "(A = Field r)"
+ show "?One \<or> ?Two"
+ proof(cases ?Two, simp)
+ let ?B = "(Field r) - A"
+ let ?a = "minim ?B"
+ assume "A \<noteq> Field r"
+ moreover have "A \<le> Field r" using * ofilter_def by simp
+ ultimately have 1: "?B \<noteq> {}" by blast
+ hence 2: "?a \<in> Field r" using minim_inField[of ?B] by blast
+ have 3: "?a \<in> ?B" using minim_in[of ?B] 1 by blast
+ hence 4: "?a \<notin> A" by blast
+ have 5: "A \<le> Field r" using * ofilter_def[of A] by auto
+ (* *)
+ moreover
+ have "A = underS ?a"
+ proof
+ show "A \<le> underS ?a"
+ proof(unfold underS_def, auto simp add: 4)
+ fix x assume **: "x \<in> A"
+ hence 11: "x \<in> Field r" using 5 by auto
+ have 12: "x \<noteq> ?a" using 4 ** by auto
+ have 13: "under x \<le> A" using * ofilter_def ** by auto
+ {assume "(x,?a) \<notin> r"
+ hence "(?a,x) \<in> r"
+ using TOTAL total_on_def[of "Field r" r]
+ 2 4 11 12 by auto
+ hence "?a \<in> under x" using under_def by auto
+ hence "?a \<in> A" using ** 13 by blast
+ with 4 have False by simp
+ }
+ thus "(x,?a) \<in> r" by blast
+ qed
+ next
+ show "underS ?a \<le> A"
+ proof(unfold underS_def, auto)
+ fix x
+ assume **: "x \<noteq> ?a" and ***: "(x,?a) \<in> r"
+ hence 11: "x \<in> Field r" using Field_def by fastforce
+ {assume "x \<notin> A"
+ hence "x \<in> ?B" using 11 by auto
+ hence "(?a,x) \<in> r" using 3 minim_least[of ?B x] by blast
+ hence False
+ using ANTISYM antisym_def[of r] ** *** by auto
+ }
+ thus "x \<in> A" by blast
+ qed
+ qed
+ ultimately have ?One using 2 by blast
+ thus ?thesis by simp
+ qed
+qed
+
+
+lemma ofilter_UNION:
+"(\<And> i. i \<in> I \<Longrightarrow> ofilter(A i)) \<Longrightarrow> ofilter (\<Union> i \<in> I. A i)"
+unfolding ofilter_def by blast
+
+
+lemma ofilter_under_UNION:
+assumes "ofilter A"
+shows "A = (\<Union> a \<in> A. under a)"
+proof
+ have "\<forall>a \<in> A. under a \<le> A"
+ using assms ofilter_def by auto
+ thus "(\<Union> a \<in> A. under a) \<le> A" by blast
+next
+ have "\<forall>a \<in> A. a \<in> under a"
+ using REFL Refl_under_in assms ofilter_def by blast
+ thus "A \<le> (\<Union> a \<in> A. under a)" by blast
+qed
+
+
+subsubsection{* Other properties *}
+
+
+lemma ofilter_linord:
+assumes OF1: "ofilter A" and OF2: "ofilter B"
+shows "A \<le> B \<or> B \<le> A"
+proof(cases "A = Field r")
+ assume Case1: "A = Field r"
+ hence "B \<le> A" using OF2 ofilter_def by auto
+ thus ?thesis by simp
+next
+ assume Case2: "A \<noteq> Field r"
+ with ofilter_underS_Field OF1 obtain a where
+ 1: "a \<in> Field r \<and> A = underS a" by auto
+ show ?thesis
+ proof(cases "B = Field r")
+ assume Case21: "B = Field r"
+ hence "A \<le> B" using OF1 ofilter_def by auto
+ thus ?thesis by simp
+ next
+ assume Case22: "B \<noteq> Field r"
+ with ofilter_underS_Field OF2 obtain b where
+ 2: "b \<in> Field r \<and> B = underS b" by auto
+ have "a = b \<or> (a,b) \<in> r \<or> (b,a) \<in> r"
+ using 1 2 TOTAL total_on_def[of _ r] by auto
+ moreover
+ {assume "a = b" with 1 2 have ?thesis by auto
+ }
+ moreover
+ {assume "(a,b) \<in> r"
+ with underS_incr TRANS ANTISYM 1 2
+ have "A \<le> B" by auto
+ hence ?thesis by auto
+ }
+ moreover
+ {assume "(b,a) \<in> r"
+ with underS_incr TRANS ANTISYM 1 2
+ have "B \<le> A" by auto
+ hence ?thesis by auto
+ }
+ ultimately show ?thesis by blast
+ qed
+qed
+
+
+lemma ofilter_AboveS_Field:
+assumes "ofilter A"
+shows "A \<union> (AboveS A) = Field r"
+proof
+ show "A \<union> (AboveS A) \<le> Field r"
+ using assms ofilter_def AboveS_Field by auto
+next
+ {fix x assume *: "x \<in> Field r" and **: "x \<notin> A"
+ {fix y assume ***: "y \<in> A"
+ with ** have 1: "y \<noteq> x" by auto
+ {assume "(y,x) \<notin> r"
+ moreover
+ have "y \<in> Field r" using assms ofilter_def *** by auto
+ ultimately have "(x,y) \<in> r"
+ using 1 * TOTAL total_on_def[of _ r] by auto
+ with *** assms ofilter_def under_def have "x \<in> A" by auto
+ with ** have False by contradiction
+ }
+ hence "(y,x) \<in> r" by blast
+ with 1 have "y \<noteq> x \<and> (y,x) \<in> r" by auto
+ }
+ with * have "x \<in> AboveS A" unfolding AboveS_def by auto
+ }
+ thus "Field r \<le> A \<union> (AboveS A)" by blast
+qed
+
+
+lemma suc_ofilter_in:
+assumes OF: "ofilter A" and ABOVE_NE: "AboveS A \<noteq> {}" and
+ REL: "(b,suc A) \<in> r" and DIFF: "b \<noteq> suc A"
+shows "b \<in> A"
+proof-
+ have *: "suc A \<in> Field r \<and> b \<in> Field r"
+ using WELL REL well_order_on_domain by auto
+ {assume **: "b \<notin> A"
+ hence "b \<in> AboveS A"
+ using OF * ofilter_AboveS_Field by auto
+ hence "(suc A, b) \<in> r"
+ using suc_least_AboveS by auto
+ hence False using REL DIFF ANTISYM *
+ by (auto simp add: antisym_def)
+ }
+ thus ?thesis by blast
+qed
+
+
+
+end (* context wo_rel *)
+
+
+
+end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Code_Numeral.thy
--- a/src/HOL/Code_Numeral.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Code_Numeral.thy Tue Nov 19 17:07:52 2013 +0100
@@ -96,10 +96,6 @@
qed
lemma [transfer_rule]:
- "fun_rel HOL.eq pcr_integer (neg_numeral :: num \<Rightarrow> int) (neg_numeral :: num \<Rightarrow> integer)"
- by (unfold neg_numeral_def [abs_def]) transfer_prover
-
-lemma [transfer_rule]:
"fun_rel HOL.eq (fun_rel HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
by (unfold Num.sub_def [abs_def]) transfer_prover
@@ -147,10 +143,6 @@
"int_of_integer (numeral k) = numeral k"
by transfer rule
-lemma int_of_integer_neg_numeral [simp]:
- "int_of_integer (neg_numeral k) = neg_numeral k"
- by transfer rule
-
lemma int_of_integer_sub [simp]:
"int_of_integer (Num.sub k l) = Num.sub k l"
by transfer rule
@@ -253,11 +245,11 @@
definition Neg :: "num \<Rightarrow> integer"
where
- [simp, code_abbrev]: "Neg = neg_numeral"
+ [simp, code_abbrev]: "Neg n = - Pos n"
lemma [transfer_rule]:
- "fun_rel HOL.eq pcr_integer neg_numeral Neg"
- by simp transfer_prover
+ "fun_rel HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
+ by (simp add: Neg_def [abs_def]) transfer_prover
code_datatype "0::integer" Pos Neg
@@ -272,7 +264,7 @@
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
- by (transfer, simp only: neg_numeral_def numeral_Bit0 minus_add_distrib)+
+ by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
is "\<lambda>m n. numeral m - numeral n :: int"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Complex.thy
--- a/src/HOL/Complex.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Complex.thy Tue Nov 19 17:07:52 2013 +0100
@@ -108,7 +108,12 @@
definition complex_divide_def:
"x / (y\<Colon>complex) = x * inverse y"
-lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
+lemma Complex_eq_1 [simp]:
+ "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
+ by (simp add: complex_one_def)
+
+lemma Complex_eq_neg_1 [simp]:
+ "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
by (simp add: complex_one_def)
lemma complex_Re_one [simp]: "Re 1 = 1"
@@ -166,21 +171,21 @@
lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
using complex_Re_of_int [of "numeral v"] by simp
-lemma complex_Re_neg_numeral [simp]: "Re (neg_numeral v) = neg_numeral v"
- using complex_Re_of_int [of "neg_numeral v"] by simp
+lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v"
+ using complex_Re_of_int [of "- numeral v"] by simp
lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
using complex_Im_of_int [of "numeral v"] by simp
-lemma complex_Im_neg_numeral [simp]: "Im (neg_numeral v) = 0"
- using complex_Im_of_int [of "neg_numeral v"] by simp
+lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0"
+ using complex_Im_of_int [of "- numeral v"] by simp
lemma Complex_eq_numeral [simp]:
- "(Complex a b = numeral w) = (a = numeral w \<and> b = 0)"
+ "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
by (simp add: complex_eq_iff)
lemma Complex_eq_neg_numeral [simp]:
- "(Complex a b = neg_numeral w) = (a = neg_numeral w \<and> b = 0)"
+ "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
by (simp add: complex_eq_iff)
@@ -421,7 +426,7 @@
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
by (simp add: complex_eq_iff)
-lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> neg_numeral w"
+lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
by (simp add: complex_eq_iff)
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
@@ -508,7 +513,7 @@
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
by (simp add: complex_eq_iff)
-lemma complex_cnj_neg_numeral [simp]: "cnj (neg_numeral w) = neg_numeral w"
+lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
by (simp add: complex_eq_iff)
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Decision_Procs/Approximation.thy
--- a/src/HOL/Decision_Procs/Approximation.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Decision_Procs/Approximation.thy Tue Nov 19 17:07:52 2013 +0100
@@ -11,8 +11,10 @@
"~~/src/HOL/Library/Code_Target_Numeral"
begin
-declare powr_numeral[simp]
-declare powr_neg_numeral[simp]
+declare powr_one [simp]
+declare powr_numeral [simp]
+declare powr_neg_one [simp]
+declare powr_neg_numeral [simp]
section "Horner Scheme"
@@ -1261,8 +1263,8 @@
unfolding cos_periodic_int ..
also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
- by (simp only: minus_float.rep_eq real_of_int_minus real_of_one minus_one[symmetric]
- mult_minus_left mult_1_left) simp
+ by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
+ mult_minus_left mult_1_left) simp
also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
unfolding uminus_float.rep_eq cos_minus ..
also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
@@ -1306,7 +1308,7 @@
also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
using cos_monotone_0_pi'[OF lx_0 lx_le_x pi_x]
by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
- minus_one[symmetric] mult_minus_left mult_1_left) simp
+ mult_minus_left mult_1_left) simp
also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
using lb_cos[OF lx_0 pi_lx] by simp
finally show ?thesis unfolding u by (simp add: real_of_float_max)
@@ -2104,8 +2106,9 @@
lemma interpret_floatarith_num:
shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
and "interpret_floatarith (Num (Float 1 0)) vs = 1"
+ and "interpret_floatarith (Num (Float (- 1) 0)) vs = - 1"
and "interpret_floatarith (Num (Float (numeral a) 0)) vs = numeral a"
- and "interpret_floatarith (Num (Float (neg_numeral a) 0)) vs = neg_numeral a" by auto
+ and "interpret_floatarith (Num (Float (- numeral a) 0)) vs = - numeral a" by auto
subsection "Implement approximation function"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Decision_Procs/Cooper.thy
--- a/src/HOL/Decision_Procs/Cooper.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Decision_Procs/Cooper.thy Tue Nov 19 17:07:52 2013 +0100
@@ -2006,9 +2006,10 @@
| SOME n => @{code Bound} (@{code nat_of_integer} n))
| num_of_term vs @{term "0::int"} = @{code C} (@{code int_of_integer} 0)
| num_of_term vs @{term "1::int"} = @{code C} (@{code int_of_integer} 1)
+ | num_of_term vs @{term "- 1::int"} = @{code C} (@{code int_of_integer} (~ 1))
| num_of_term vs (@{term "numeral :: _ \<Rightarrow> int"} $ t) =
@{code C} (@{code int_of_integer} (HOLogic.dest_num t))
- | num_of_term vs (@{term "neg_numeral :: _ \<Rightarrow> int"} $ t) =
+ | num_of_term vs (@{term "- numeral :: _ \<Rightarrow> int"} $ t) =
@{code C} (@{code int_of_integer} (~(HOLogic.dest_num t)))
| num_of_term vs (Bound i) = @{code Bound} (@{code nat_of_integer} i)
| num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} $ t') = @{code Neg} (num_of_term vs t')
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Decision_Procs/MIR.thy Tue Nov 19 17:07:52 2013 +0100
@@ -3154,7 +3154,7 @@
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1"
by (simp only: algebra_simps)
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1"
- by (simp add: algebra_simps minus_one [symmetric] del: minus_one)
+ by (simp add: algebra_simps)
with nob have ?case by blast }
ultimately show ?case by blast
next
@@ -5549,6 +5549,7 @@
| num_of_term vs @{term "real (1::int)"} = mk_C 1
| num_of_term vs @{term "0::real"} = mk_C 0
| num_of_term vs @{term "1::real"} = mk_C 1
+ | num_of_term vs @{term "- 1::real"} = mk_C (~ 1)
| num_of_term vs (Bound i) = mk_Bound i
| num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
| num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
@@ -5561,7 +5562,7 @@
| _ => error "num_of_term: unsupported Multiplication")
| num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t')) =
mk_C (HOLogic.dest_num t')
- | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "neg_numeral :: _ \<Rightarrow> int"} $ t')) =
+ | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t')) =
mk_C (~ (HOLogic.dest_num t'))
| num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) =
@{code Floor} (num_of_term vs t')
@@ -5569,7 +5570,7 @@
@{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t')))
| num_of_term vs (@{term "numeral :: _ \<Rightarrow> real"} $ t') =
mk_C (HOLogic.dest_num t')
- | num_of_term vs (@{term "neg_numeral :: _ \<Rightarrow> real"} $ t') =
+ | num_of_term vs (@{term "- numeral :: _ \<Rightarrow> real"} $ t') =
mk_C (~ (HOLogic.dest_num t'))
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
@@ -5583,7 +5584,7 @@
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
mk_Dvd (HOLogic.dest_num t1, num_of_term vs t2)
- | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "neg_numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
+ | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
mk_Dvd (~ (HOLogic.dest_num t1), num_of_term vs t2)
| fm_of_term vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
@{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Tue Nov 19 17:07:52 2013 +0100
@@ -1256,12 +1256,10 @@
apply (case_tac n', simp, simp)
apply (case_tac n, simp, simp)
apply (case_tac n, case_tac n', simp add: Let_def)
- apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
- apply (auto simp add: polyadd_eq_const_degree)
+ apply (auto simp add: polyadd_eq_const_degree)[2]
apply (metis head_nz)
apply (metis head_nz)
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
- apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
done
lemma polymul_head_polyeq:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Divides.thy
--- a/src/HOL/Divides.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Divides.thy Tue Nov 19 17:07:52 2013 +0100
@@ -1919,10 +1919,9 @@
val zero = @{term "0 :: int"}
val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
- val simps = @{thms arith_simps} @ @{thms rel_simps} @
- map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
- fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
- (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));
+ val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]
+ fun prove ctxt goal = (writeln "prove"; Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
+ (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps)))));
fun binary_proc proc ctxt ct =
(case Thm.term_of ct of
_ $ t $ u =>
@@ -1945,23 +1944,23 @@
simproc_setup binary_int_div
("numeral m div numeral n :: int" |
- "numeral m div neg_numeral n :: int" |
- "neg_numeral m div numeral n :: int" |
- "neg_numeral m div neg_numeral n :: int") =
+ "numeral m div - numeral n :: int" |
+ "- numeral m div numeral n :: int" |
+ "- numeral m div - numeral n :: int") =
{* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
simproc_setup binary_int_mod
("numeral m mod numeral n :: int" |
- "numeral m mod neg_numeral n :: int" |
- "neg_numeral m mod numeral n :: int" |
- "neg_numeral m mod neg_numeral n :: int") =
+ "numeral m mod - numeral n :: int" |
+ "- numeral m mod numeral n :: int" |
+ "- numeral m mod - numeral n :: int") =
{* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
lemmas posDivAlg_eqn_numeral [simp] =
posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
lemmas negDivAlg_eqn_numeral [simp] =
- negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
+ negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w
text{*Special-case simplification *}
@@ -1973,14 +1972,14 @@
div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
lemmas div_pos_neg_1_numeral [simp] =
- div_pos_neg [OF zero_less_one, of "neg_numeral w",
+ div_pos_neg [OF zero_less_one, of "- numeral w",
OF neg_numeral_less_zero] for w
lemmas mod_pos_pos_1_numeral [simp] =
mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
lemmas mod_pos_neg_1_numeral [simp] =
- mod_pos_neg [OF zero_less_one, of "neg_numeral w",
+ mod_pos_neg [OF zero_less_one, of "- numeral w",
OF neg_numeral_less_zero] for w
lemmas posDivAlg_eqn_1_numeral [simp] =
@@ -2290,6 +2289,8 @@
shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
using assms unfolding divmod_int_rel_def by auto
+declaration {* K (Lin_Arith.add_simps @{thms uminus_numeral_One}) *}
+
lemma neg_divmod_int_rel_mult_2:
assumes "b \<le> 0"
assumes "divmod_int_rel (a + 1) b (q, r)"
@@ -2427,13 +2428,13 @@
lemma dvd_neg_numeral_left [simp]:
fixes y :: "'a::comm_ring_1"
- shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
- unfolding neg_numeral_def minus_dvd_iff ..
+ shows "(- numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
+ by (fact minus_dvd_iff)
lemma dvd_neg_numeral_right [simp]:
fixes x :: "'a::comm_ring_1"
- shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)"
- unfolding neg_numeral_def dvd_minus_iff ..
+ shows "x dvd (- numeral k) \<longleftrightarrow> x dvd (numeral k)"
+ by (fact dvd_minus_iff)
lemmas dvd_eq_mod_eq_0_numeral [simp] =
dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/GCD.thy
--- a/src/HOL/GCD.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/GCD.thy Tue Nov 19 17:07:52 2013 +0100
@@ -134,6 +134,14 @@
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
by (simp add: gcd_int_def)
+lemma gcd_neg_numeral_1_int [simp]:
+ "gcd (- numeral n :: int) x = gcd (numeral n) x"
+ by (fact gcd_neg1_int)
+
+lemma gcd_neg_numeral_2_int [simp]:
+ "gcd x (- numeral n :: int) = gcd x (numeral n)"
+ by (fact gcd_neg2_int)
+
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
by(simp add: gcd_int_def)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/IMP/Hoare_Examples.thy
--- a/src/HOL/IMP/Hoare_Examples.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/IMP/Hoare_Examples.thy Tue Nov 19 17:07:52 2013 +0100
@@ -2,17 +2,6 @@
theory Hoare_Examples imports Hoare begin
-text{* Improves proof automation for negative numerals: *}
-
-lemma add_neg1R[simp]:
- "x + -1 = x - (1 :: int)"
-by arith
-
-lemma add_neg_numeralR[simp]:
- "x + neg_numeral n = (x::'a::neg_numeral) - numeral(n)"
-by (simp only: diff_minus_eq_add[symmetric] minus_neg_numeral)
-
-
text{* Summing up the first @{text x} natural numbers in variable @{text y}. *}
fun sum :: "int \<Rightarrow> int" where
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Int.thy
--- a/src/HOL/Int.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Int.thy Tue Nov 19 17:07:52 2013 +0100
@@ -232,9 +232,8 @@
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
-lemma of_int_neg_numeral [simp, code_post]: "of_int (neg_numeral k) = neg_numeral k"
- unfolding neg_numeral_def neg_numeral_class.neg_numeral_def
- by (simp only: of_int_minus of_int_numeral)
+lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
+ by simp
lemma of_int_power:
"of_int (z ^ n) = of_int z ^ n"
@@ -370,7 +369,7 @@
by (simp add: nat_eq_iff)
lemma nat_neg_numeral [simp]:
- "nat (neg_numeral k) = 0"
+ "nat (- numeral k) = 0"
by simp
lemma nat_2: "nat 2 = Suc (Suc 0)"
@@ -511,13 +510,13 @@
lemma nonneg_int_cases:
assumes "0 \<le> k" obtains n where "k = int n"
- using assms by (cases k, simp, simp del: of_nat_Suc)
+ using assms by (rule nonneg_eq_int)
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..
-lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
+lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..
@@ -525,15 +524,15 @@
lemmas max_number_of [simp] =
max_def [of "numeral u" "numeral v"]
- max_def [of "numeral u" "neg_numeral v"]
- max_def [of "neg_numeral u" "numeral v"]
- max_def [of "neg_numeral u" "neg_numeral v"] for u v
+ max_def [of "numeral u" "- numeral v"]
+ max_def [of "- numeral u" "numeral v"]
+ max_def [of "- numeral u" "- numeral v"] for u v
lemmas min_number_of [simp] =
min_def [of "numeral u" "numeral v"]
- min_def [of "numeral u" "neg_numeral v"]
- min_def [of "neg_numeral u" "numeral v"]
- min_def [of "neg_numeral u" "neg_numeral v"] for u v
+ min_def [of "numeral u" "- numeral v"]
+ min_def [of "- numeral u" "numeral v"]
+ min_def [of "- numeral u" "- numeral v"] for u v
subsubsection {* Binary comparisons *}
@@ -1070,8 +1069,6 @@
by auto
qed
-ML_val {* @{const_name neg_numeral} *}
-
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
by (insert abs_zmult_eq_1 [of m n], arith)
@@ -1127,62 +1124,30 @@
inverse_eq_divide [of "numeral w"] for w
lemmas inverse_eq_divide_neg_numeral [simp] =
- inverse_eq_divide [of "neg_numeral w"] for w
+ inverse_eq_divide [of "- numeral w"] for w
text {*These laws simplify inequalities, moving unary minus from a term
into the literal.*}
-lemmas le_minus_iff_numeral [simp, no_atp] =
- le_minus_iff [of "numeral v"]
- le_minus_iff [of "neg_numeral v"] for v
-
-lemmas equation_minus_iff_numeral [simp, no_atp] =
- equation_minus_iff [of "numeral v"]
- equation_minus_iff [of "neg_numeral v"] for v
+lemmas equation_minus_iff_numeral [no_atp] =
+ equation_minus_iff [of "numeral v"] for v
-lemmas minus_less_iff_numeral [simp, no_atp] =
- minus_less_iff [of _ "numeral v"]
- minus_less_iff [of _ "neg_numeral v"] for v
+lemmas minus_equation_iff_numeral [no_atp] =
+ minus_equation_iff [of _ "numeral v"] for v
-lemmas minus_le_iff_numeral [simp, no_atp] =
- minus_le_iff [of _ "numeral v"]
- minus_le_iff [of _ "neg_numeral v"] for v
-
-lemmas minus_equation_iff_numeral [simp, no_atp] =
- minus_equation_iff [of _ "numeral v"]
- minus_equation_iff [of _ "neg_numeral v"] for v
-
-text{*To Simplify Inequalities Where One Side is the Constant 1*}
+lemmas le_minus_iff_numeral [no_atp] =
+ le_minus_iff [of "numeral v"] for v
-lemma less_minus_iff_1 [simp]:
- fixes b::"'b::linordered_idom"
- shows "(1 < - b) = (b < -1)"
-by auto
-
-lemma le_minus_iff_1 [simp]:
- fixes b::"'b::linordered_idom"
- shows "(1 \<le> - b) = (b \<le> -1)"
-by auto
-
-lemma equation_minus_iff_1 [simp]:
- fixes b::"'b::ring_1"
- shows "(1 = - b) = (b = -1)"
-by (subst equation_minus_iff, auto)
+lemmas minus_le_iff_numeral [no_atp] =
+ minus_le_iff [of _ "numeral v"] for v
-lemma minus_less_iff_1 [simp]:
- fixes a::"'b::linordered_idom"
- shows "(- a < 1) = (-1 < a)"
-by auto
+lemmas less_minus_iff_numeral [no_atp] =
+ less_minus_iff [of "numeral v"] for v
-lemma minus_le_iff_1 [simp]:
- fixes a::"'b::linordered_idom"
- shows "(- a \<le> 1) = (-1 \<le> a)"
-by auto
+lemmas minus_less_iff_numeral [no_atp] =
+ minus_less_iff [of _ "numeral v"] for v
-lemma minus_equation_iff_1 [simp]:
- fixes a::"'b::ring_1"
- shows "(- a = 1) = (a = -1)"
-by (subst minus_equation_iff, auto)
+-- {* FIXME maybe simproc *}
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
@@ -1197,27 +1162,28 @@
lemmas le_divide_eq_numeral1 [simp] =
pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
- neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
+ neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
lemmas divide_le_eq_numeral1 [simp] =
pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
- neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
+ neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
lemmas less_divide_eq_numeral1 [simp] =
pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
- neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
+ neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
lemmas divide_less_eq_numeral1 [simp] =
pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
- neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
+ neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
lemmas eq_divide_eq_numeral1 [simp] =
eq_divide_eq [of _ _ "numeral w"]
- eq_divide_eq [of _ _ "neg_numeral w"] for w
+ eq_divide_eq [of _ _ "- numeral w"] for w
lemmas divide_eq_eq_numeral1 [simp] =
divide_eq_eq [of _ "numeral w"]
- divide_eq_eq [of _ "neg_numeral w"] for w
+ divide_eq_eq [of _ "- numeral w"] for w
+
subsubsection{*Optional Simplification Rules Involving Constants*}
@@ -1225,27 +1191,27 @@
lemmas le_divide_eq_numeral =
le_divide_eq [of "numeral w"]
- le_divide_eq [of "neg_numeral w"] for w
+ le_divide_eq [of "- numeral w"] for w
lemmas divide_le_eq_numeral =
divide_le_eq [of _ _ "numeral w"]
- divide_le_eq [of _ _ "neg_numeral w"] for w
+ divide_le_eq [of _ _ "- numeral w"] for w
lemmas less_divide_eq_numeral =
less_divide_eq [of "numeral w"]
- less_divide_eq [of "neg_numeral w"] for w
+ less_divide_eq [of "- numeral w"] for w
lemmas divide_less_eq_numeral =
divide_less_eq [of _ _ "numeral w"]
- divide_less_eq [of _ _ "neg_numeral w"] for w
+ divide_less_eq [of _ _ "- numeral w"] for w
lemmas eq_divide_eq_numeral =
eq_divide_eq [of "numeral w"]
- eq_divide_eq [of "neg_numeral w"] for w
+ eq_divide_eq [of "- numeral w"] for w
lemmas divide_eq_eq_numeral =
divide_eq_eq [of _ _ "numeral w"]
- divide_eq_eq [of _ _ "neg_numeral w"] for w
+ divide_eq_eq [of _ _ "- numeral w"] for w
text{*Not good as automatic simprules because they cause case splits.*}
@@ -1257,21 +1223,20 @@
text{*Division By @{text "-1"}*}
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
- unfolding minus_one [symmetric]
unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
by simp
lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
- unfolding minus_one [symmetric] by (rule divide_minus_left)
+ by (fact divide_minus_left)
lemma half_gt_zero_iff:
- "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
-by auto
+ "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
+ by auto
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
- by simp
+ by (fact divide_numeral_1)
subsection {* The divides relation *}
@@ -1475,7 +1440,7 @@
[simp, code_abbrev]: "Pos = numeral"
definition Neg :: "num \<Rightarrow> int" where
- [simp, code_abbrev]: "Neg = neg_numeral"
+ [simp, code_abbrev]: "Neg n = - (Pos n)"
code_datatype "0::int" Pos Neg
@@ -1489,7 +1454,7 @@
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
- unfolding Pos_def Neg_def neg_numeral_def
+ unfolding Pos_def Neg_def
by (simp_all add: numeral_Bit0)
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
@@ -1505,12 +1470,10 @@
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
- apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def
- neg_numeral_def numeral_BitM)
+ apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
apply (simp_all only: algebra_simps minus_diff_eq)
apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
apply (simp_all only: minus_add add.assoc left_minus)
- apply (simp_all only: algebra_simps right_minus)
done
text {* Implementations *}
@@ -1606,10 +1569,10 @@
"nat (Int.Neg k) = 0"
"nat 0 = 0"
"nat (Int.Pos k) = nat_of_num k"
- by (simp_all add: nat_of_num_numeral nat_numeral)
+ by (simp_all add: nat_of_num_numeral)
lemma (in ring_1) of_int_code [code]:
- "of_int (Int.Neg k) = neg_numeral k"
+ "of_int (Int.Neg k) = - numeral k"
"of_int 0 = 0"
"of_int (Int.Pos k) = numeral k"
by simp_all
@@ -1653,7 +1616,7 @@
lemma int_power:
"int (m ^ n) = int m ^ n"
- by (rule of_nat_power)
+ by (fact of_nat_power)
lemmas zpower_int = int_power [symmetric]
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Binomial.thy
--- a/src/HOL/Library/Binomial.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Binomial.thy Tue Nov 19 17:07:52 2013 +0100
@@ -370,7 +370,7 @@
by auto
from False show ?thesis
by (simp add: pochhammer_def gbinomial_def field_simps
- eq setprod_timesf[symmetric] del: minus_one)
+ eq setprod_timesf[symmetric])
qed
lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
@@ -441,9 +441,9 @@
from eq[symmetric]
have ?thesis using kn
apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
- gbinomial_pochhammer field_simps pochhammer_Suc_setprod del: minus_one)
+ gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
- of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)
+ of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc)
unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
unfolding mult_assoc[symmetric]
unfolding setprod_timesf[symmetric]
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Bit.thy
--- a/src/HOL/Library/Bit.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Bit.thy Tue Nov 19 17:07:52 2013 +0100
@@ -147,11 +147,11 @@
text {* All numerals reduce to either 0 or 1. *}
-lemma bit_minus1 [simp]: "-1 = (1 :: bit)"
- by (simp only: minus_one [symmetric] uminus_bit_def)
+lemma bit_minus1 [simp]: "- 1 = (1 :: bit)"
+ by (simp only: uminus_bit_def)
-lemma bit_neg_numeral [simp]: "(neg_numeral w :: bit) = numeral w"
- by (simp only: neg_numeral_def uminus_bit_def)
+lemma bit_neg_numeral [simp]: "(- numeral w :: bit) = numeral w"
+ by (simp only: uminus_bit_def)
lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)"
by (simp only: numeral_Bit0 bit_add_self)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Code_Prolog.thy
--- a/src/HOL/Library/Code_Prolog.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Code_Prolog.thy Tue Nov 19 17:07:52 2013 +0100
@@ -12,10 +12,8 @@
section {* Setup for Numerals *}
-setup {* Predicate_Compile_Data.ignore_consts
- [@{const_name numeral}, @{const_name neg_numeral}] *}
+setup {* Predicate_Compile_Data.ignore_consts [@{const_name numeral}] *}
-setup {* Predicate_Compile_Data.keep_functions
- [@{const_name numeral}, @{const_name neg_numeral}] *}
+setup {* Predicate_Compile_Data.keep_functions [@{const_name numeral}] *}
end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Code_Real_Approx_By_Float.thy
--- a/src/HOL/Library/Code_Real_Approx_By_Float.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Code_Real_Approx_By_Float.thy Tue Nov 19 17:07:52 2013 +0100
@@ -169,7 +169,7 @@
by simp
lemma [code_unfold del]:
- "neg_numeral k \<equiv> (of_rat (neg_numeral k) :: real)"
+ "- numeral k \<equiv> (of_rat (- numeral k) :: real)"
by simp
hide_const (open) real_of_int
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Code_Target_Int.thy
--- a/src/HOL/Library/Code_Target_Int.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Code_Target_Int.thy Tue Nov 19 17:07:52 2013 +0100
@@ -30,7 +30,7 @@
by transfer simp
lemma [code_abbrev]:
- "int_of_integer (neg_numeral k) = Int.Neg k"
+ "int_of_integer (- numeral k) = Int.Neg k"
by transfer simp
lemma [code, symmetric, code_post]:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Extended.thy
--- a/src/HOL/Library/Extended.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Extended.thy Tue Nov 19 17:07:52 2013 +0100
@@ -161,8 +161,8 @@
apply (simp only: numeral_inc one_extended_def plus_extended.simps(1)[symmetric])
done
-lemma Fin_neg_numeral: "Fin(neg_numeral w) = - numeral w"
-by (simp only: Fin_numeral minus_numeral[symmetric] uminus_extended.simps[symmetric])
+lemma Fin_neg_numeral: "Fin (- numeral w) = - numeral w"
+by (simp only: Fin_numeral uminus_extended.simps[symmetric])
instantiation extended :: (lattice)bounded_lattice
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Float.thy
--- a/src/HOL/Library/Float.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Float.thy Tue Nov 19 17:07:52 2013 +0100
@@ -45,7 +45,7 @@
lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp
-lemma neg_numeral_float[simp]: "neg_numeral i \<in> float" by (intro floatI[of "neg_numeral i" 0]) simp
+lemma neg_numeral_float[simp]: "- numeral i \<in> float" by (intro floatI[of "- numeral i" 0]) simp
lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp
lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
@@ -53,7 +53,7 @@
lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
-lemma two_powr_neg_numeral_float[simp]: "2 powr neg_numeral i \<in> float" by (intro floatI[of 1 "neg_numeral i"]) simp
+lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float" by (intro floatI[of 1 "- numeral i"]) simp
lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp
@@ -121,11 +121,11 @@
qed
lemma div_neg_numeral_Bit0_float[simp]:
- assumes x: "x / numeral n \<in> float" shows "x / (neg_numeral (Num.Bit0 n)) \<in> float"
+ assumes x: "x / numeral n \<in> float" shows "x / (- numeral (Num.Bit0 n)) \<in> float"
proof -
have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp
- also have "- (x / numeral (Num.Bit0 n)) = x / neg_numeral (Num.Bit0 n)"
- unfolding neg_numeral_def by (simp del: minus_numeral)
+ also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)"
+ by simp
finally show ?thesis .
qed
@@ -197,7 +197,7 @@
then show "\<exists>c. a < c \<and> c < b"
apply (intro exI[of _ "(a + b) * Float 1 -1"])
apply transfer
- apply (simp add: powr_neg_numeral)
+ apply (simp add: powr_minus)
done
qed
@@ -226,16 +226,16 @@
"fun_rel (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp
-lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
- by (simp add: minus_numeral[symmetric] del: minus_numeral)
+lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x"
+ by simp
lemma transfer_neg_numeral [transfer_rule]:
- "fun_rel (op =) pcr_float (neg_numeral :: _ \<Rightarrow> real) (neg_numeral :: _ \<Rightarrow> float)"
+ "fun_rel (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"
unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp
lemma
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
- and float_of_neg_numeral[simp]: "neg_numeral k = float_of (neg_numeral k)"
+ and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
unfolding real_of_float_eq by simp_all
subsection {* Represent floats as unique mantissa and exponent *}
@@ -439,7 +439,7 @@
by transfer simp
hide_fact (open) compute_float_numeral
-lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k"
+lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
by transfer simp
hide_fact (open) compute_float_neg_numeral
@@ -960,7 +960,7 @@
also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
apply (rule mult_strict_right_mono) by (insert assms) auto
also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
- using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_neg_numeral)
+ using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_minus)
also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Formal_Power_Series.thy
--- a/src/HOL/Library/Formal_Power_Series.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Formal_Power_Series.thy Tue Nov 19 17:07:52 2013 +0100
@@ -384,8 +384,8 @@
by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1
fps_const_add [symmetric])
-lemma neg_numeral_fps_const: "neg_numeral k = fps_const (neg_numeral k)"
- by (simp only: neg_numeral_def numeral_fps_const fps_const_neg)
+lemma neg_numeral_fps_const: "- numeral k = fps_const (- numeral k)"
+ by (simp only: numeral_fps_const fps_const_neg)
subsection{* The eXtractor series X*}
@@ -1202,7 +1202,7 @@
have eq: "(1 + X) * ?r = 1"
unfolding minus_one_power_iff
by (auto simp add: field_simps fps_eq_iff)
- show ?thesis by (auto simp add: eq intro: fps_inverse_unique simp del: minus_one)
+ show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
qed
@@ -1245,7 +1245,7 @@
lemma numeral_compose[simp]: "(numeral k::('a::{comm_ring_1}) fps) oo b = numeral k"
unfolding numeral_fps_const by simp
-lemma neg_numeral_compose[simp]: "(neg_numeral k::('a::{comm_ring_1}) fps) oo b = neg_numeral k"
+lemma neg_numeral_compose[simp]: "(- numeral k::('a::{comm_ring_1}) fps) oo b = - numeral k"
unfolding neg_numeral_fps_const by simp
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
@@ -2363,7 +2363,7 @@
next
case (Suc n1)
have "?i $ n = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
- by (simp add: fps_compose_nth Suc startsby_zero_power_nth_same[OF a0] del: power_Suc)
+ by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
also have "\<dots> = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
(X$ Suc n1 - setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
using a0 a1 Suc by (simp add: fps_inv_def)
@@ -2404,7 +2404,7 @@
next
case (Suc n1)
have "?i $ n = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
- by (simp add: fps_compose_nth Suc startsby_zero_power_nth_same[OF a0] del: power_Suc)
+ by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
(b$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
using a0 a1 Suc by (simp add: fps_ginv_def)
@@ -2564,9 +2564,9 @@
lemma fps_compose_mult_distrib:
- assumes c0: "c$0 = (0::'a::idom)"
- shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r")
- apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
+ assumes c0: "c $ 0 = (0::'a::idom)"
+ shows "(a * b) oo c = (a oo c) * (b oo c)"
+ apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
apply (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
done
@@ -2941,7 +2941,7 @@
(is "inverse ?l = ?r")
proof -
have th: "?l * ?r = 1"
- by (auto simp add: field_simps fps_eq_iff minus_one_power_iff simp del: minus_one)
+ by (auto simp add: field_simps fps_eq_iff minus_one_power_iff)
have th': "?l $ 0 \<noteq> 0" by (simp add: )
from fps_inverse_unique[OF th' th] show ?thesis .
qed
@@ -3165,7 +3165,7 @@
have th: "?r$0 \<noteq> 0" by simp
have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
by (simp add: fps_inverse_deriv[OF th] fps_divide_def
- power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg minus_one)
+ power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg)
have eq: "inverse ?r $ 0 = 1"
by (simp add: fps_inverse_def)
from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
@@ -3276,7 +3276,7 @@
have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
unfolding m1nk
unfolding m h pochhammer_Suc_setprod
- apply (simp add: field_simps del: fact_Suc minus_one)
+ apply (simp add: field_simps del: fact_Suc)
unfolding fact_altdef_nat id_def
unfolding of_nat_setprod
unfolding setprod_timesf[symmetric]
@@ -3593,7 +3593,7 @@
unfolding even_mult_two_ex by blast
have "?l $n = ?r$n"
- by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus)
+ by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
}
moreover
{
@@ -3602,7 +3602,7 @@
unfolding odd_nat_equiv_def2 by (auto simp add: mult_2)
have "?l $n = ?r$n"
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
- power_mult power_minus)
+ power_mult power_minus [of "c ^ 2"])
}
ultimately have "?l $n = ?r$n" by blast
} then show ?thesis by (simp add: fps_eq_iff)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Fundamental_Theorem_Algebra.thy
--- a/src/HOL/Library/Fundamental_Theorem_Algebra.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy Tue Nov 19 17:07:52 2013 +0100
@@ -207,12 +207,14 @@
from unimodular_reduce_norm[OF th0] o
have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
- apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp del: minus_one add: minus_one [symmetric])
+ apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
apply (rule_tac x="- ii" in exI, simp add: m power_mult)
apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
- apply (rule_tac x="ii" in exI, simp add: m power_mult)
+ apply (auto simp add: m power_mult)
+ apply (rule_tac x="ii" in exI)
+ apply (auto simp add: m power_mult)
done
then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
let ?w = "v / complex_of_real (root n (cmod b))"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Order_Relation.thy
--- a/src/HOL/Library/Order_Relation.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Order_Relation.thy Tue Nov 19 17:07:52 2013 +0100
@@ -93,7 +93,7 @@
using mono_Field[of "r - Id" r] Diff_subset[of r Id]
proof(auto)
have "r \<noteq> {}" using NID by fast
- then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by fast
+ then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by auto
hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)
(* *)
fix a assume *: "a \<in> Field r"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Order_Union.thy
--- a/src/HOL/Library/Order_Union.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Order_Union.thy Tue Nov 19 17:07:52 2013 +0100
@@ -7,7 +7,7 @@
header {* Order Union *}
theory Order_Union
-imports "~~/src/HOL/Cardinals/Wellfounded_More_Base"
+imports "~~/src/HOL/Cardinals/Wellfounded_More_FP"
begin
definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "Osum" 60) where
@@ -31,7 +31,7 @@
assume Case1: "B \<noteq> {}"
hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
- using WF unfolding wf_eq_minimal2 by blast
+ using WF unfolding wf_eq_minimal2 by metis
hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
(* *)
have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
@@ -59,7 +59,7 @@
assume Case2: "B = {}"
hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
- using WF' unfolding wf_eq_minimal2 by blast
+ using WF' unfolding wf_eq_minimal2 by metis
hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
(* *)
have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
@@ -299,7 +299,7 @@
using assms Total_Id_Field by blast
hence ?thesis unfolding Osum_def by auto
}
- ultimately show ?thesis using * unfolding Osum_def by blast
+ ultimately show ?thesis using * unfolding Osum_def by fast
qed
}
thus ?thesis by(auto simp add: Osum_def)
@@ -308,12 +308,7 @@
lemma wf_Int_Times:
assumes "A Int B = {}"
shows "wf(A \<times> B)"
-proof(unfold wf_def, auto)
- fix P x
- assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
- moreover have "\<forall>y \<in> A. P y" using assms * by blast
- ultimately show "P x" using * by (case_tac "x \<in> B", auto)
-qed
+unfolding wf_def using assms by blast
lemma Osum_wf_Id:
assumes TOT: "Total r" and TOT': "Total r'" and
@@ -343,7 +338,7 @@
using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
by (auto simp add: Un_commute)
}
- ultimately have ?thesis by (auto simp add: wf_subset)
+ ultimately have ?thesis by (metis wf_subset)
}
moreover
{assume Case22: "r' \<le> Id"
@@ -356,7 +351,7 @@
using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
by (auto simp add: Un_commute)
}
- ultimately have ?thesis by (auto simp add: wf_subset)
+ ultimately have ?thesis by (metis wf_subset)
}
ultimately show ?thesis by blast
qed
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Polynomial.thy
--- a/src/HOL/Library/Polynomial.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Polynomial.thy Tue Nov 19 17:07:52 2013 +0100
@@ -1575,12 +1575,12 @@
lemma poly_div_minus_left [simp]:
fixes x y :: "'a::field poly"
shows "(- x) div y = - (x div y)"
- using div_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
+ using div_smult_left [of "- 1::'a"] by simp
lemma poly_mod_minus_left [simp]:
fixes x y :: "'a::field poly"
shows "(- x) mod y = - (x mod y)"
- using mod_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
+ using mod_smult_left [of "- 1::'a"] by simp
lemma pdivmod_rel_smult_right:
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
@@ -1597,13 +1597,12 @@
lemma poly_div_minus_right [simp]:
fixes x y :: "'a::field poly"
shows "x div (- y) = - (x div y)"
- using div_smult_right [of "- 1::'a"]
- by (simp add: nonzero_inverse_minus_eq del: minus_one) (* FIXME *)
+ using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)
lemma poly_mod_minus_right [simp]:
fixes x y :: "'a::field poly"
shows "x mod (- y) = x mod y"
- using mod_smult_right [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
+ using mod_smult_right [of "- 1::'a"] by simp
lemma pdivmod_rel_mult:
"\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Predicate_Compile_Alternative_Defs.thy
--- a/src/HOL/Library/Predicate_Compile_Alternative_Defs.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Predicate_Compile_Alternative_Defs.thy Tue Nov 19 17:07:52 2013 +0100
@@ -45,8 +45,8 @@
section {* Setup for Numerals *}
-setup {* Predicate_Compile_Data.ignore_consts [@{const_name numeral}, @{const_name neg_numeral}] *}
-setup {* Predicate_Compile_Data.keep_functions [@{const_name numeral}, @{const_name neg_numeral}] *}
+setup {* Predicate_Compile_Data.ignore_consts [@{const_name numeral}] *}
+setup {* Predicate_Compile_Data.keep_functions [@{const_name numeral}] *}
setup {* Predicate_Compile_Data.ignore_consts [@{const_name div}, @{const_name mod}, @{const_name times}] *}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Sublist.thy
--- a/src/HOL/Library/Sublist.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Sublist.thy Tue Nov 19 17:07:52 2013 +0100
@@ -107,16 +107,22 @@
lemma append_one_prefixeq:
"prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys"
- unfolding prefixeq_def
- by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
- eq_Nil_appendI nth_drop')
+ proof (unfold prefixeq_def)
+ assume a1: "\<exists>zs. ys = xs @ zs"
+ then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
+ assume a2: "length xs < length ys"
+ have f1: "\<And>v. ([]\<Colon>'a list) @ v = v" using append_Nil2 by simp
+ have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force
+ hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl)
+ thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
+ qed
theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys"
by (auto simp add: prefixeq_def)
lemma prefixeq_same_cases:
"prefixeq (xs\<^sub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^sub>2 ys \<Longrightarrow> prefixeq xs\<^sub>1 xs\<^sub>2 \<or> prefixeq xs\<^sub>2 xs\<^sub>1"
- unfolding prefixeq_def by (metis append_eq_append_conv2)
+ unfolding prefixeq_def by (force simp: append_eq_append_conv2)
lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
by (auto simp add: prefixeq_def)
@@ -224,9 +230,9 @@
then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
next
fix c cs assume ys': "ys' = c # cs"
- then show ?thesis
- by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI
- same_prefixeq_prefixeq snoc.prems ys)
+ have "x \<noteq> c" using snoc.prems ys ys' by fastforce
+ thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs"
+ using ys ys' by blast
qed
next
assume "prefix ys xs"
@@ -464,7 +470,7 @@
then show ?case by (metis append_Cons)
next
case (list_hembeq_Cons2 x y xs ys)
- then show ?case by (cases xs) (auto, blast+)
+ then show ?case by blast
qed
lemma list_hembeq_appendD:
@@ -475,9 +481,14 @@
case Nil then show ?case by auto
next
case (Cons x xs)
- then obtain us v vs where "zs = us @ v # vs"
- and "P\<^sup>=\<^sup>= x v" and "list_hembeq P (xs @ ys) vs" by (auto dest: list_hembeq_ConsD)
- with Cons show ?case by (metis append_Cons append_assoc list_hembeq_Cons2 list_hembeq_append2)
+ then obtain us v vs where
+ zs: "zs = us @ v # vs" and p: "P\<^sup>=\<^sup>= x v" and lh: "list_hembeq P (xs @ ys) vs"
+ by (auto dest: list_hembeq_ConsD)
+ obtain sk\<^sub>0 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" and sk\<^sub>1 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+ sk: "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_hembeq P (xs @ x\<^sub>0) x\<^sub>1 \<or> sk\<^sub>0 x\<^sub>0 x\<^sub>1 @ sk\<^sub>1 x\<^sub>0 x\<^sub>1 = x\<^sub>1 \<and> list_hembeq P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_hembeq P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
+ using Cons(1) by (metis (no_types))
+ hence "\<forall>x\<^sub>2. list_hembeq P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto
+ thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
qed
lemma list_hembeq_suffix:
@@ -550,7 +561,7 @@
by (simp_all)
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
- by (induct xs) (auto dest: list_hembeq_ConsD)
+ by (induct xs, simp, blast dest: list_hembeq_ConsD)
lemma sublisteq_Cons2':
assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
@@ -579,11 +590,11 @@
from list_hembeq_Nil2 [OF this] show ?case by simp
next
case list_hembeq_Cons2
- then show ?case by simp
+ thus ?case by simp
next
case list_hembeq_Cons
- then show ?case
- by (metis sublisteq_Cons' list_hembeq_length Suc_length_conv Suc_n_not_le_n)
+ hence False using sublisteq_Cons' by fastforce
+ thus ?case ..
qed
lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
@@ -650,7 +661,7 @@
lemma sublisteq_filter [simp]:
assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
- using assms by (induct) auto
+ using assms by induct auto
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
proof
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Sum_of_Squares/sum_of_squares.ML
--- a/src/HOL/Library/Sum_of_Squares/sum_of_squares.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Sum_of_Squares/sum_of_squares.ML Tue Nov 19 17:07:52 2013 +0100
@@ -875,7 +875,6 @@
@{term "0::real"}, @{term "1::real"},
@{term "numeral :: num => nat"},
@{term "numeral :: num => real"},
- @{term "neg_numeral :: num => real"},
@{term "Num.Bit0"}, @{term "Num.Bit1"}, @{term "Num.One"}];
fun check_sos kcts ct =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Wfrec.thy
--- a/src/HOL/Library/Wfrec.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Wfrec.thy Tue Nov 19 17:07:52 2013 +0100
@@ -48,7 +48,7 @@
apply (fast dest!: theI')
apply (erule wfrec_rel.cases, simp)
apply (erule allE, erule allE, erule allE, erule mp)
-apply (fast intro: the_equality [symmetric])
+apply (blast intro: the_equality [symmetric])
done
lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Library/Zorn.thy
--- a/src/HOL/Library/Zorn.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Library/Zorn.thy Tue Nov 19 17:07:52 2013 +0100
@@ -71,7 +71,7 @@
lemma suc_not_equals:
"chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
- by (auto simp: suc_def) (metis less_irrefl not_maxchain_Some)
+ by (auto simp: suc_def) (metis (no_types) less_irrefl not_maxchain_Some)
lemma subset_suc:
assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y"
@@ -258,7 +258,7 @@
shows "chain X"
using assms
proof (induct)
- case (suc X) then show ?case by (simp add: suc_def) (metis not_maxchain_Some)
+ case (suc X) then show ?case by (simp add: suc_def) (metis (no_types) not_maxchain_Some)
next
case (Union X)
then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier)
@@ -378,7 +378,7 @@
using `subset.maxchain A M` by (auto simp: subset.maxchain_def)
qed
qed
- ultimately show ?thesis by blast
+ ultimately show ?thesis by metis
qed
text{*Alternative version of Zorn's lemma for the subset relation.*}
@@ -423,7 +423,7 @@
unfolding Chains_def by blast
lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
- by (auto simp add: chain_subset_def subset.chain_def)
+ unfolding chain_subset_def subset.chain_def by fast
lemma chains_alt_def: "chains A = {C. subset.chain A C}"
by (simp add: chains_def chain_subset_alt_def subset.chain_def)
@@ -487,7 +487,7 @@
fix a B assume aB: "B \<in> C" "a \<in> B"
with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
thus "(a, u) \<in> r" using uA and aB and `Preorder r`
- by (auto simp add: preorder_on_def refl_on_def) (metis transD)
+ unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
qed
then have "\<exists>u\<in>Field r. ?P u" using `u \<in> Field r` by blast
}
@@ -524,8 +524,7 @@
lemma trans_init_seg_of:
"r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
- by (simp (no_asm_use) add: init_seg_of_def)
- (metis UnCI Un_absorb2 subset_trans)
+ by (simp (no_asm_use) add: init_seg_of_def) blast
lemma antisym_init_seg_of:
"r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
@@ -539,14 +538,13 @@
"chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans (\<Union>R)"
apply (auto simp add: chain_subset_def)
apply (simp (no_asm_use) add: trans_def)
-apply (metis subsetD)
-done
+by (metis subsetD)
lemma chain_subset_antisym_Union:
"chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym (\<Union>R)"
-apply (auto simp add: chain_subset_def antisym_def)
-apply (metis subsetD)
-done
+unfolding chain_subset_def antisym_def
+apply simp
+by (metis (no_types) subsetD)
lemma chain_subset_Total_Union:
assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
@@ -558,11 +556,11 @@
thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
proof
assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A
- by (simp add: total_on_def) (metis mono_Field subsetD)
+ by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
thus ?thesis using `s \<in> R` by blast
next
assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A
- by (simp add: total_on_def) (metis mono_Field subsetD)
+ by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
thus ?thesis using `r \<in> R` by blast
qed
qed
@@ -604,7 +602,7 @@
def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
- by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+ unfolding init_seg_of_def chain_subset_def Chains_def by blast
have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
by (simp add: Chains_def I_def) blast
have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def)
@@ -619,7 +617,7 @@
have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
- have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by (auto simp: refl_on_def)
+ have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` unfolding refl_on_def by fastforce
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
moreover have "antisym (\<Union>R)"
@@ -630,7 +628,7 @@
proof -
have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
with `\<forall>r\<in>R. wf (r - Id)` and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
- show ?thesis by (simp (no_asm_simp)) blast
+ show ?thesis by fastforce
qed
ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
@@ -643,7 +641,7 @@
--{*Zorn's Lemma yields a maximal well-order m:*}
then obtain m::"'a rel" where "Well_order m" and
max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
- using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
+ using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
--{*Now show by contradiction that m covers the whole type:*}
{ fix x::'a assume "x \<notin> Field m"
--{*We assume that x is not covered and extend m at the top with x*}
@@ -666,7 +664,7 @@
have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
using `Well_order m` by (simp_all add: order_on_defs)
--{*We show that the extension is a well-order*}
- have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
+ have "Refl ?m" using `Refl m` Fm unfolding refl_on_def by blast
moreover have "trans ?m" using `trans m` and `x \<notin> Field m`
unfolding trans_def Field_def by blast
moreover have "antisym ?m" using `antisym m` and `x \<notin> Field m`
@@ -678,7 +676,7 @@
by (auto simp add: wf_eq_minimal Field_def) metis
thus ?thesis using `wf (m - Id)` and `x \<notin> Field m`
wf_subset [OF `wf ?s` Diff_subset]
- by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
+ unfolding Un_Diff Field_def by (auto intro: wf_Un)
qed
ultimately have "Well_order ?m" by (simp add: order_on_defs)
--{*We show that the extension is above m*}
@@ -709,7 +707,7 @@
moreover have "Total ?r" using `Total r` by (simp add:total_on_def 1 univ)
moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf (r - Id)`]) blast
ultimately have "Well_order ?r" by (simp add: order_on_defs)
- with 1 show ?thesis by metis
+ with 1 show ?thesis by auto
qed
subsection {* Extending Well-founded Relations to Well-Orders *}
@@ -732,7 +730,7 @@
lemma downset_onD:
"downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"
- by (auto simp: downset_on_def)
+ unfolding downset_on_def by blast
text {*Extensions of relations w.r.t.\ a given set.*}
definition extension_on where
@@ -755,7 +753,8 @@
assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
shows "extension_on (Field (\<Union>R)) (\<Union>R) p"
using assms
- by (simp add: chain_subset_def extension_on_def) (metis mono_Field set_mp)
+ by (simp add: chain_subset_def extension_on_def)
+ (metis (no_types) mono_Field set_mp)
lemma downset_on_empty [simp]: "downset_on {} p"
by (auto simp: downset_on_def)
@@ -789,7 +788,7 @@
"\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and
"\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
- have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by (auto simp: refl_on_def)
+ have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` unfolding refl_on_def by fastforce
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
moreover have "antisym (\<Union>R)"
@@ -800,7 +799,7 @@
proof -
have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
with `\<forall>r\<in>R. wf (r - Id)` wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
- show ?thesis by (simp (no_asm_simp)) blast
+ show ?thesis by fastforce
qed
ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)
moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris
@@ -853,8 +852,9 @@
using `extension_on (Field m) m p` `downset_on (Field m) p`
by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
moreover have "downset_on (Field ?m) p"
+ apply (subst Fm)
using `downset_on (Field m) p` and min
- by (subst Fm, simp add: downset_on_def Field_def) (metis Domain_iff)
+ unfolding downset_on_def Field_def by blast
moreover have "(m, ?m) \<in> I"
using `Well_order m` and `Well_order ?m` and
`downset_on (Field m) p` and `downset_on (Field ?m) p` and
@@ -867,7 +867,7 @@
qed
have "p \<subseteq> m"
using `Field p \<subseteq> Field m` and `extension_on (Field m) m p`
- by (force simp: Field_def extension_on_def)
+ unfolding Field_def extension_on_def by auto fast
with `Well_order m` show ?thesis by blast
qed
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/List.thy
--- a/src/HOL/List.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/List.thy Tue Nov 19 17:07:52 2013 +0100
@@ -3075,9 +3075,9 @@
lemmas upto_rec_numeral [simp] =
upto.simps[of "numeral m" "numeral n"]
- upto.simps[of "numeral m" "neg_numeral n"]
- upto.simps[of "neg_numeral m" "numeral n"]
- upto.simps[of "neg_numeral m" "neg_numeral n"] for m n
+ upto.simps[of "numeral m" "- numeral n"]
+ upto.simps[of "- numeral m" "numeral n"]
+ upto.simps[of "- numeral m" "- numeral n"] for m n
lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
by(simp add: upto.simps)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Matrix_LP/ComputeFloat.thy
--- a/src/HOL/Matrix_LP/ComputeFloat.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Matrix_LP/ComputeFloat.thy Tue Nov 19 17:07:52 2013 +0100
@@ -79,8 +79,8 @@
lemma real_is_int_numeral[simp]: "real_is_int (numeral x)"
by (auto simp: real_is_int_def intro!: exI[of _ "numeral x"])
-lemma real_is_int_neg_numeral[simp]: "real_is_int (neg_numeral x)"
- by (auto simp: real_is_int_def intro!: exI[of _ "neg_numeral x"])
+lemma real_is_int_neg_numeral[simp]: "real_is_int (- numeral x)"
+ by (auto simp: real_is_int_def intro!: exI[of _ "- numeral x"])
lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
by (simp add: int_of_real_def)
@@ -96,7 +96,7 @@
by (intro some_equality)
(auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
-lemma int_of_real_neg_numeral[simp]: "int_of_real (neg_numeral b) = neg_numeral b"
+lemma int_of_real_neg_numeral[simp]: "int_of_real (- numeral b) = - numeral b"
unfolding int_of_real_def
by (intro some_equality)
(auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Tue Nov 19 17:07:52 2013 +0100
@@ -228,7 +228,10 @@
then show ?case by vector
qed
-lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1"
+lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
+ by vector
+
+lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
by vector
instance vec :: (semiring_char_0, finite) semiring_char_0
@@ -244,8 +247,8 @@
lemma numeral_index [simp]: "numeral w $ i = numeral w"
by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
-lemma neg_numeral_index [simp]: "neg_numeral w $ i = neg_numeral w"
- by (simp only: neg_numeral_def vector_uminus_component numeral_index)
+lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
+ by (simp only: vector_uminus_component numeral_index)
instance vec :: (comm_ring_1, finite) comm_ring_1 ..
instance vec :: (ring_char_0, finite) ring_char_0 ..
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Tue Nov 19 17:07:52 2013 +0100
@@ -337,10 +337,10 @@
by (simp add: bilinear_def linear_iff)
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
- by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
+ by (drule bilinear_lmul [of _ "- 1"]) simp
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
- by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
+ by (drule bilinear_rmul [of _ _ "- 1"]) simp
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
using add_imp_eq[of x y 0] by auto
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Nov 19 17:07:52 2013 +0100
@@ -5163,9 +5163,8 @@
lemma open_negations:
fixes s :: "'a::real_normed_vector set"
- shows "open s \<Longrightarrow> open ((\<lambda> x. -x) ` s)"
- unfolding scaleR_minus1_left [symmetric]
- by (rule open_scaling, auto)
+ shows "open s \<Longrightarrow> open ((\<lambda>x. - x) ` s)"
+ using open_scaling [of "- 1" s] by simp
lemma open_translation:
fixes s :: "'a::real_normed_vector set"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/NSA/HTranscendental.thy
--- a/src/HOL/NSA/HTranscendental.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/NSA/HTranscendental.thy Tue Nov 19 17:07:52 2013 +0100
@@ -258,7 +258,7 @@
simp add: mult_assoc)
apply (rule approx_add_right_cancel [where d="-1"])
apply (rule approx_sym [THEN [2] approx_trans2])
-apply (auto simp add: mem_infmal_iff minus_one [symmetric] simp del: minus_one)
+apply (auto simp add: mem_infmal_iff)
done
lemma STAR_exp_epsilon [simp]: "( *f* exp) epsilon @= 1"
@@ -450,7 +450,7 @@
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
simp add: mult_assoc)
apply (rule approx_add_right_cancel [where d = "-1"])
-apply (simp add: minus_one [symmetric] del: minus_one)
+apply simp
done
lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/NSA/HyperDef.thy
--- a/src/HOL/NSA/HyperDef.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/NSA/HyperDef.thy Tue Nov 19 17:07:52 2013 +0100
@@ -425,7 +425,7 @@
declare power_hypreal_of_real_numeral [of _ "numeral w", simp] for w
lemma power_hypreal_of_real_neg_numeral:
- "(neg_numeral v :: hypreal) ^ n = hypreal_of_real ((neg_numeral v) ^ n)"
+ "(- numeral v :: hypreal) ^ n = hypreal_of_real ((- numeral v) ^ n)"
by simp
declare power_hypreal_of_real_neg_numeral [of _ "numeral w", simp] for w
(*
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/NSA/NSA.thy
--- a/src/HOL/NSA/NSA.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/NSA/NSA.thy Tue Nov 19 17:07:52 2013 +0100
@@ -654,7 +654,7 @@
(*reorientation simplification procedure: reorients (polymorphic)
0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
simproc_setup approx_reorient_simproc
- ("0 @= x" | "1 @= y" | "numeral w @= z" | "neg_numeral w @= r") =
+ ("0 @= x" | "1 @= y" | "numeral w @= z" | "- 1 @= y" | "- numeral w @= r") =
{*
let val rule = @{thm approx_reorient} RS eq_reflection
fun proc phi ss ct = case term_of ct of
@@ -1849,8 +1849,12 @@
lemma st_numeral [simp]: "st (numeral w) = numeral w"
by (rule Reals_numeral [THEN st_SReal_eq])
-lemma st_neg_numeral [simp]: "st (neg_numeral w) = neg_numeral w"
-by (rule Reals_neg_numeral [THEN st_SReal_eq])
+lemma st_neg_numeral [simp]: "st (- numeral w) = - numeral w"
+proof -
+ from Reals_numeral have "numeral w \<in> \<real>" .
+ then have "- numeral w \<in> \<real>" by simp
+ with st_SReal_eq show ?thesis .
+qed
lemma st_0 [simp]: "st 0 = 0"
by (simp add: st_SReal_eq)
@@ -1858,6 +1862,9 @@
lemma st_1 [simp]: "st 1 = 1"
by (simp add: st_SReal_eq)
+lemma st_neg_1 [simp]: "st (- 1) = - 1"
+by (simp add: st_SReal_eq)
+
lemma st_minus: "x \<in> HFinite \<Longrightarrow> st (- x) = - st x"
by (simp add: st_unique st_SReal st_approx_self approx_minus)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/NSA/NSComplex.thy
--- a/src/HOL/NSA/NSComplex.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/NSA/NSComplex.thy Tue Nov 19 17:07:52 2013 +0100
@@ -635,7 +635,7 @@
by transfer (rule of_real_numeral [symmetric])
lemma hcomplex_hypreal_neg_numeral:
- "hcomplex_of_complex (neg_numeral w) = hcomplex_of_hypreal(neg_numeral w)"
+ "hcomplex_of_complex (- numeral w) = hcomplex_of_hypreal(- numeral w)"
by transfer (rule of_real_neg_numeral [symmetric])
lemma hcomplex_numeral_hcnj [simp]:
@@ -647,7 +647,7 @@
by transfer (rule norm_numeral)
lemma hcomplex_neg_numeral_hcmod [simp]:
- "hcmod(neg_numeral v :: hcomplex) = (numeral v :: hypreal)"
+ "hcmod(- numeral v :: hcomplex) = (numeral v :: hypreal)"
by transfer (rule norm_neg_numeral)
lemma hcomplex_numeral_hRe [simp]:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/NSA/StarDef.thy
--- a/src/HOL/NSA/StarDef.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/NSA/StarDef.thy Tue Nov 19 17:07:52 2013 +0100
@@ -968,13 +968,13 @@
by transfer (rule refl)
lemma star_neg_numeral_def [transfer_unfold]:
- "neg_numeral k = star_of (neg_numeral k)"
-by (simp only: neg_numeral_def star_of_minus star_of_numeral)
+ "- numeral k = star_of (- numeral k)"
+by (simp only: star_of_minus star_of_numeral)
-lemma Standard_neg_numeral [simp]: "neg_numeral k \<in> Standard"
-by (simp add: star_neg_numeral_def)
+lemma Standard_neg_numeral [simp]: "- numeral k \<in> Standard"
+ using star_neg_numeral_def [of k] by simp
-lemma star_of_neg_numeral [simp]: "star_of (neg_numeral k) = neg_numeral k"
+lemma star_of_neg_numeral [simp]: "star_of (- numeral k) = - numeral k"
by transfer (rule refl)
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)"
@@ -987,12 +987,12 @@
star_of_less [of _ "numeral k", simplified star_of_numeral]
star_of_le [of _ "numeral k", simplified star_of_numeral]
star_of_eq [of _ "numeral k", simplified star_of_numeral]
- star_of_less [of "neg_numeral k", simplified star_of_numeral]
- star_of_le [of "neg_numeral k", simplified star_of_numeral]
- star_of_eq [of "neg_numeral k", simplified star_of_numeral]
- star_of_less [of _ "neg_numeral k", simplified star_of_numeral]
- star_of_le [of _ "neg_numeral k", simplified star_of_numeral]
- star_of_eq [of _ "neg_numeral k", simplified star_of_numeral] for k
+ star_of_less [of "- numeral k", simplified star_of_numeral]
+ star_of_le [of "- numeral k", simplified star_of_numeral]
+ star_of_eq [of "- numeral k", simplified star_of_numeral]
+ star_of_less [of _ "- numeral k", simplified star_of_numeral]
+ star_of_le [of _ "- numeral k", simplified star_of_numeral]
+ star_of_eq [of _ "- numeral k", simplified star_of_numeral] for k
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard"
by (simp add: star_of_nat_def)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Nominal/Nominal.thy
--- a/src/HOL/Nominal/Nominal.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Nominal/Nominal.thy Tue Nov 19 17:07:52 2013 +0100
@@ -3517,7 +3517,7 @@
by (simp add: perm_int_def perm_int_def)
lemma neg_numeral_int_eqvt:
- shows "pi\<bullet>((neg_numeral n)::int) = neg_numeral n"
+ shows "pi\<bullet>((- numeral n)::int) = - numeral n"
by (simp add: perm_int_def perm_int_def)
lemma max_int_eqvt:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Num.thy
--- a/src/HOL/Num.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Num.thy Tue Nov 19 17:07:52 2013 +0100
@@ -275,16 +275,6 @@
end
-text {* Negative numerals. *}
-
-class neg_numeral = numeral + group_add
-begin
-
-definition neg_numeral :: "num \<Rightarrow> 'a" where
- "neg_numeral k = - numeral k"
-
-end
-
text {* Numeral syntax. *}
syntax
@@ -299,8 +289,8 @@
| (n, 0) => Syntax.const @{const_name Bit0} $ num_of_int n
| (n, 1) => Syntax.const @{const_name Bit1} $ num_of_int n)
else raise Match
- val pos = Syntax.const @{const_name numeral}
- val neg = Syntax.const @{const_name neg_numeral}
+ val numeral = Syntax.const @{const_name numeral}
+ val uminus = Syntax.const @{const_name uminus}
val one = Syntax.const @{const_name Groups.one}
val zero = Syntax.const @{const_name Groups.zero}
fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
@@ -311,8 +301,9 @@
in
if value = 0 then zero else
if value > 0
- then pos $ num_of_int value
- else neg $ num_of_int (~value)
+ then numeral $ num_of_int value
+ else if value = ~1 then uminus $ one
+ else uminus $ (numeral $ num_of_int (~value))
end
| numeral_tr ts = raise TERM ("numeral_tr", ts);
in [("_Numeral", K numeral_tr)] end
@@ -323,12 +314,12 @@
fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n
| dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1
| dest_num (Const (@{const_syntax One}, _)) = 1;
- fun num_tr' sign ctxt T [n] =
+ fun num_tr' ctxt T [n] =
let
val k = dest_num n;
val t' =
Syntax.const @{syntax_const "_Numeral"} $
- Syntax.free (sign ^ string_of_int k);
+ Syntax.free (string_of_int k);
in
(case T of
Type (@{type_name fun}, [_, T']) =>
@@ -339,8 +330,7 @@
| _ => if T = dummyT then t' else raise Match)
end;
in
- [(@{const_syntax numeral}, num_tr' ""),
- (@{const_syntax neg_numeral}, num_tr' "-")]
+ [(@{const_syntax numeral}, num_tr')]
end
*}
@@ -383,9 +373,13 @@
Structures with negation: class @{text neg_numeral}
*}
-context neg_numeral
+class neg_numeral = numeral + group_add
begin
+lemma uminus_numeral_One:
+ "- Numeral1 = - 1"
+ by (simp add: numeral_One)
+
text {* Numerals form an abelian subgroup. *}
inductive is_num :: "'a \<Rightarrow> bool" where
@@ -403,7 +397,7 @@
apply simp
apply (rule_tac a=x in add_left_imp_eq)
apply (rule_tac a=x in add_right_imp_eq)
- apply (simp add: add_assoc minus_add_cancel)
+ apply (simp add: add_assoc)
apply (simp add: add_assoc [symmetric], simp add: add_assoc)
apply (rule_tac a=x in add_left_imp_eq)
apply (rule_tac a=x in add_right_imp_eq)
@@ -431,77 +425,85 @@
by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
lemma dbl_simps [simp]:
- "dbl (neg_numeral k) = neg_numeral (Bit0 k)"
+ "dbl (- numeral k) = - dbl (numeral k)"
"dbl 0 = 0"
"dbl 1 = 2"
+ "dbl (- 1) = - 2"
"dbl (numeral k) = numeral (Bit0 k)"
- by (simp_all add: dbl_def neg_numeral_def numeral.simps minus_add)
+ by (simp_all add: dbl_def numeral.simps minus_add)
lemma dbl_inc_simps [simp]:
- "dbl_inc (neg_numeral k) = neg_numeral (BitM k)"
+ "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
"dbl_inc 0 = 1"
"dbl_inc 1 = 3"
+ "dbl_inc (- 1) = - 1"
"dbl_inc (numeral k) = numeral (Bit1 k)"
- by (simp_all add: dbl_inc_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
+ by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
lemma dbl_dec_simps [simp]:
- "dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"
- "dbl_dec 0 = -1"
+ "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
+ "dbl_dec 0 = - 1"
"dbl_dec 1 = 1"
+ "dbl_dec (- 1) = - 3"
"dbl_dec (numeral k) = numeral (BitM k)"
- by (simp_all add: dbl_dec_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize)
+ by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
lemma sub_num_simps [simp]:
"sub One One = 0"
- "sub One (Bit0 l) = neg_numeral (BitM l)"
- "sub One (Bit1 l) = neg_numeral (Bit0 l)"
+ "sub One (Bit0 l) = - numeral (BitM l)"
+ "sub One (Bit1 l) = - numeral (Bit0 l)"
"sub (Bit0 k) One = numeral (BitM k)"
"sub (Bit1 k) One = numeral (Bit0 k)"
"sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
"sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
"sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
"sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
- by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def neg_numeral_def numeral.simps
+ by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_simps:
- "numeral m + neg_numeral n = sub m n"
- "neg_numeral m + numeral n = sub n m"
- "neg_numeral m + neg_numeral n = neg_numeral (m + n)"
- by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize
+ "numeral m + - numeral n = sub m n"
+ "- numeral m + numeral n = sub n m"
+ "- numeral m + - numeral n = - (numeral m + numeral n)"
+ by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_special:
- "1 + neg_numeral m = sub One m"
- "neg_numeral m + 1 = sub One m"
- by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize)
+ "1 + - numeral m = sub One m"
+ "- numeral m + 1 = sub One m"
+ "numeral m + - 1 = sub m One"
+ "- 1 + numeral n = sub n One"
+ "- 1 + - numeral n = - numeral (inc n)"
+ "- numeral m + - 1 = - numeral (inc m)"
+ "1 + - 1 = 0"
+ "- 1 + 1 = 0"
+ "- 1 + - 1 = - 2"
+ by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
+ del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_simps:
"numeral m - numeral n = sub m n"
- "numeral m - neg_numeral n = numeral (m + n)"
- "neg_numeral m - numeral n = neg_numeral (m + n)"
- "neg_numeral m - neg_numeral n = sub n m"
- by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps is_num_normalize
+ "numeral m - - numeral n = numeral (m + n)"
+ "- numeral m - numeral n = - numeral (m + n)"
+ "- numeral m - - numeral n = sub n m"
+ by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_special:
"1 - numeral n = sub One n"
- "1 - neg_numeral n = numeral (One + n)"
"numeral m - 1 = sub m One"
- "neg_numeral m - 1 = neg_numeral (m + One)"
- by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps add: is_num_normalize)
-
-lemma minus_one: "- 1 = -1"
- unfolding neg_numeral_def numeral.simps ..
-
-lemma minus_numeral: "- numeral n = neg_numeral n"
- unfolding neg_numeral_def ..
-
-lemma minus_neg_numeral: "- neg_numeral n = numeral n"
- unfolding neg_numeral_def by simp
-
-lemmas minus_numeral_simps [simp] =
- minus_one minus_numeral minus_neg_numeral
+ "1 - - numeral n = numeral (One + n)"
+ "- numeral m - 1 = - numeral (m + One)"
+ "- 1 - numeral n = - numeral (inc n)"
+ "numeral m - - 1 = numeral (inc m)"
+ "- 1 - - numeral n = sub n One"
+ "- numeral m - - 1 = sub One m"
+ "1 - 1 = 0"
+ "- 1 - 1 = - 2"
+ "1 - - 1 = 2"
+ "- 1 - - 1 = 0"
+ by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
+ del: add_uminus_conv_diff add: diff_conv_add_uminus)
end
@@ -675,17 +677,17 @@
subclass neg_numeral ..
lemma mult_neg_numeral_simps:
- "neg_numeral m * neg_numeral n = numeral (m * n)"
- "neg_numeral m * numeral n = neg_numeral (m * n)"
- "numeral m * neg_numeral n = neg_numeral (m * n)"
- unfolding neg_numeral_def mult_minus_left mult_minus_right
+ "- numeral m * - numeral n = numeral (m * n)"
+ "- numeral m * numeral n = - numeral (m * n)"
+ "numeral m * - numeral n = - numeral (m * n)"
+ unfolding mult_minus_left mult_minus_right
by (simp_all only: minus_minus numeral_mult)
-lemma mult_minus1 [simp]: "-1 * z = - z"
- unfolding neg_numeral_def numeral.simps mult_minus_left by simp
+lemma mult_minus1 [simp]: "- 1 * z = - z"
+ unfolding numeral.simps mult_minus_left by simp
-lemma mult_minus1_right [simp]: "z * -1 = - z"
- unfolding neg_numeral_def numeral.simps mult_minus_right by simp
+lemma mult_minus1_right [simp]: "z * - 1 = - z"
+ unfolding numeral.simps mult_minus_right by simp
end
@@ -708,9 +710,15 @@
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
by (simp add: numeral_One)
+lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
+ by (simp add: iszero_def)
+
+lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
+ by (simp add: numeral_One)
+
lemma iszero_neg_numeral [simp]:
- "iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"
- unfolding iszero_def neg_numeral_def
+ "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
+ unfolding iszero_def
by (rule neg_equal_0_iff_equal)
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
@@ -730,17 +738,17 @@
lemma eq_numeral_iff_iszero:
"numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
- "numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"
- "neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
- "neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"
+ "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
+ "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
+ "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
"numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
"1 = numeral y \<longleftrightarrow> iszero (sub One y)"
- "neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
- "1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"
+ "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
+ "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
"numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
"0 = numeral y \<longleftrightarrow> iszero (numeral y)"
- "neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
- "0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"
+ "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
+ "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
by simp_all
@@ -756,33 +764,69 @@
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
by (simp add: iszero_def)
-lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"
- by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)
+lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
+ by simp
-lemma numeral_neq_neg_numeral: "numeral m \<noteq> neg_numeral n"
- unfolding neg_numeral_def eq_neg_iff_add_eq_0
+lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
+ unfolding eq_neg_iff_add_eq_0
by (simp add: numeral_plus_numeral)
-lemma neg_numeral_neq_numeral: "neg_numeral m \<noteq> numeral n"
+lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
by (rule numeral_neq_neg_numeral [symmetric])
-lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"
- unfolding neg_numeral_def neg_0_equal_iff_equal by simp
+lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
+ unfolding neg_0_equal_iff_equal by simp
-lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"
- unfolding neg_numeral_def neg_equal_0_iff_equal by simp
+lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
+ unfolding neg_equal_0_iff_equal by simp
-lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"
+lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
-lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"
+lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
+lemma neg_one_neq_numeral:
+ "- 1 \<noteq> numeral n"
+ using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
+
+lemma numeral_neq_neg_one:
+ "numeral n \<noteq> - 1"
+ using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
+
+lemma neg_one_eq_numeral_iff:
+ "- 1 = - numeral n \<longleftrightarrow> n = One"
+ using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
+
+lemma numeral_eq_neg_one_iff:
+ "- numeral n = - 1 \<longleftrightarrow> n = One"
+ using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
+
+lemma neg_one_neq_zero:
+ "- 1 \<noteq> 0"
+ by simp
+
+lemma zero_neq_neg_one:
+ "0 \<noteq> - 1"
+ by simp
+
+lemma neg_one_neq_one:
+ "- 1 \<noteq> 1"
+ using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
+
+lemma one_neq_neg_one:
+ "1 \<noteq> - 1"
+ using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
+
lemmas eq_neg_numeral_simps [simp] =
neg_numeral_eq_iff
numeral_neq_neg_numeral neg_numeral_neq_numeral
one_neq_neg_numeral neg_numeral_neq_one
zero_neq_neg_numeral neg_numeral_neq_zero
+ neg_one_neq_numeral numeral_neq_neg_one
+ neg_one_eq_numeral_iff numeral_eq_neg_one_iff
+ neg_one_neq_zero zero_neq_neg_one
+ neg_one_neq_one one_neq_neg_one
end
@@ -795,48 +839,72 @@
subclass ring_char_0 ..
-lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"
- by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)
+lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
+ by (simp only: neg_le_iff_le numeral_le_iff)
-lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"
- by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)
+lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
+ by (simp only: neg_less_iff_less numeral_less_iff)
-lemma neg_numeral_less_zero: "neg_numeral n < 0"
- by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)
+lemma neg_numeral_less_zero: "- numeral n < 0"
+ by (simp only: neg_less_0_iff_less zero_less_numeral)
-lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"
- by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)
+lemma neg_numeral_le_zero: "- numeral n \<le> 0"
+ by (simp only: neg_le_0_iff_le zero_le_numeral)
-lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"
+lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
by (simp only: not_less neg_numeral_le_zero)
-lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"
+lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
by (simp only: not_le neg_numeral_less_zero)
-lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"
+lemma neg_numeral_less_numeral: "- numeral m < numeral n"
using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
-lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"
+lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
by (simp only: less_imp_le neg_numeral_less_numeral)
-lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"
+lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
by (simp only: not_less neg_numeral_le_numeral)
-lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"
+lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
by (simp only: not_le neg_numeral_less_numeral)
-lemma neg_numeral_less_one: "neg_numeral m < 1"
+lemma neg_numeral_less_one: "- numeral m < 1"
by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
-lemma neg_numeral_le_one: "neg_numeral m \<le> 1"
+lemma neg_numeral_le_one: "- numeral m \<le> 1"
by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
-lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"
+lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
by (simp only: not_less neg_numeral_le_one)
-lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"
+lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
by (simp only: not_le neg_numeral_less_one)
+lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
+ using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
+
+lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
+ using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
+
+lemma neg_one_less_numeral: "- 1 < numeral m"
+ using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
+
+lemma neg_one_le_numeral: "- 1 \<le> numeral m"
+ using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
+
+lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
+ by (cases m) simp_all
+
+lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
+ by simp
+
+lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
+ by simp
+
+lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
+ by (cases m) simp_all
+
lemma sub_non_negative:
"sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
by (simp only: sub_def le_diff_eq) simp
@@ -858,18 +926,40 @@
neg_numeral_le_numeral not_numeral_le_neg_numeral
neg_numeral_le_zero not_zero_le_neg_numeral
neg_numeral_le_one not_one_le_neg_numeral
+ neg_one_le_numeral not_numeral_le_neg_one
+ neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
+
+lemma le_minus_one_simps [simp]:
+ "- 1 \<le> 0"
+ "- 1 \<le> 1"
+ "\<not> 0 \<le> - 1"
+ "\<not> 1 \<le> - 1"
+ by simp_all
lemmas less_neg_numeral_simps [simp] =
neg_numeral_less_iff
neg_numeral_less_numeral not_numeral_less_neg_numeral
neg_numeral_less_zero not_zero_less_neg_numeral
neg_numeral_less_one not_one_less_neg_numeral
+ neg_one_less_numeral not_numeral_less_neg_one
+ neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
+
+lemma less_minus_one_simps [simp]:
+ "- 1 < 0"
+ "- 1 < 1"
+ "\<not> 0 < - 1"
+ "\<not> 1 < - 1"
+ by (simp_all add: less_le)
lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
by simp
-lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"
- by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)
+lemma abs_neg_numeral [simp]: "abs (- numeral n) = numeral n"
+ by (simp only: abs_minus_cancel abs_numeral)
+
+lemma abs_neg_one [simp]:
+ "abs (- 1) = 1"
+ by simp
end
@@ -1032,31 +1122,36 @@
text{*Theorem lists for the cancellation simprocs. The use of a binary
numeral for 1 reduces the number of special cases.*}
-lemmas mult_1s =
- mult_numeral_1 mult_numeral_1_right
- mult_minus1 mult_minus1_right
+lemma mult_1s:
+ fixes a :: "'a::semiring_numeral"
+ and b :: "'b::ring_1"
+ shows "Numeral1 * a = a"
+ "a * Numeral1 = a"
+ "- Numeral1 * b = - b"
+ "b * - Numeral1 = - b"
+ by simp_all
setup {*
Reorient_Proc.add
(fn Const (@{const_name numeral}, _) $ _ => true
- | Const (@{const_name neg_numeral}, _) $ _ => true
+ | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
| _ => false)
*}
simproc_setup reorient_numeral
- ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
+ ("numeral w = x" | "- numeral w = y") = Reorient_Proc.proc
subsubsection {* Simplification of arithmetic operations on integer constants. *}
lemmas arith_special = (* already declared simp above *)
add_numeral_special add_neg_numeral_special
- diff_numeral_special minus_one
+ diff_numeral_special
(* rules already in simpset *)
lemmas arith_extra_simps =
numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
- minus_numeral minus_neg_numeral minus_zero minus_one
+ minus_zero
diff_numeral_simps diff_0 diff_0_right
numeral_times_numeral mult_neg_numeral_simps
mult_zero_left mult_zero_right
@@ -1089,15 +1184,15 @@
lemmas rel_simps =
le_num_simps less_num_simps eq_num_simps
- le_numeral_simps le_neg_numeral_simps le_numeral_extra
- less_numeral_simps less_neg_numeral_simps less_numeral_extra
+ le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
+ less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..
-lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
+lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..
@@ -1133,16 +1228,16 @@
by (simp_all add: add_assoc [symmetric])
lemma add_neg_numeral_left [simp]:
- "numeral v + (neg_numeral w + y) = (sub v w + y)"
- "neg_numeral v + (numeral w + y) = (sub w v + y)"
- "neg_numeral v + (neg_numeral w + y) = (neg_numeral(v + w) + y)"
+ "numeral v + (- numeral w + y) = (sub v w + y)"
+ "- numeral v + (numeral w + y) = (sub w v + y)"
+ "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
by (simp_all add: add_assoc [symmetric])
lemma mult_numeral_left [simp]:
"numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
- "neg_numeral v * (numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
- "numeral v * (neg_numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
- "neg_numeral v * (neg_numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
+ "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
+ "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
+ "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
by (simp_all add: mult_assoc [symmetric])
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Number_Theory/Cong.thy
--- a/src/HOL/Number_Theory/Cong.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Number_Theory/Cong.thy Tue Nov 19 17:07:52 2013 +0100
@@ -323,8 +323,6 @@
\<Longrightarrow> [a = 1] (mod p) \<or> [a = - 1] (mod p)"
apply (simp only: cong_altdef_int)
apply (subst prime_dvd_mult_eq_int [symmetric], assumption)
- (* any way around this? *)
- apply (subgoal_tac "a * a - 1 = (a - 1) * (a - -1)")
apply (auto simp add: field_simps)
done
@@ -665,7 +663,6 @@
apply auto
apply (cases "n \<ge> 0")
apply auto
- apply (subst cong_int_def, auto)
apply (frule cong_solve_int [of a n])
apply auto
done
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Number_Theory/Residues.thy
--- a/src/HOL/Number_Theory/Residues.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Number_Theory/Residues.thy Tue Nov 19 17:07:52 2013 +0100
@@ -455,6 +455,7 @@
apply (subst fact_altdef_int, simp)
apply (subst cong_int_def)
apply simp
+ apply arith
apply (rule residues_prime.wilson_theorem1)
apply (rule residues_prime.intro)
apply auto
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Numeral_Simprocs.thy
--- a/src/HOL/Numeral_Simprocs.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Numeral_Simprocs.thy Tue Nov 19 17:07:52 2013 +0100
@@ -17,7 +17,7 @@
if_False if_True
add_0 add_Suc add_numeral_left
add_neg_numeral_left mult_numeral_left
- numeral_1_eq_1 [symmetric] Suc_eq_plus1
+ numeral_One [symmetric] uminus_numeral_One [symmetric] Suc_eq_plus1
eq_numeral_iff_iszero not_iszero_Numeral1
declare split_div [of _ _ "numeral k", arith_split] for k
@@ -85,18 +85,19 @@
text {* For @{text cancel_factor} *}
-lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
-by auto
+lemmas nat_mult_le_cancel_disj = mult_le_cancel1
-lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
-by auto
+lemmas nat_mult_less_cancel_disj = mult_less_cancel1
-lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
-by auto
+lemma nat_mult_eq_cancel_disj:
+ fixes k m n :: nat
+ shows "k * m = k * n \<longleftrightarrow> k = 0 \<or> m = n"
+ by auto
-lemma nat_mult_div_cancel_disj[simp]:
- "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
-by (simp add: nat_mult_div_cancel1)
+lemma nat_mult_div_cancel_disj [simp]:
+ fixes k m n :: nat
+ shows "(k * m) div (k * n) = (if k = 0 then 0 else m div n)"
+ by (fact div_mult_mult1_if)
ML_file "Tools/numeral_simprocs.ML"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Parity.thy
--- a/src/HOL/Parity.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Parity.thy Tue Nov 19 17:07:52 2013 +0100
@@ -78,7 +78,7 @@
unfolding even_def by simp
(* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
-declare even_def [of "neg_numeral v", simp] for v
+declare even_def [of "- numeral v", simp] for v
lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
unfolding even_nat_def by simp
@@ -190,14 +190,9 @@
by (induct n) simp_all
lemma (in comm_ring_1)
- shows minus_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
- and minus_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
- by (simp_all add: neg_power_if del: minus_one)
-
-lemma (in comm_ring_1)
- shows neg_one_even_power [simp]: "even n \<Longrightarrow> (-1) ^ n = 1"
- and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (-1) ^ n = - 1"
- by (simp_all add: minus_one [symmetric] del: minus_one)
+ shows neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
+ and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
+ by (simp_all add: neg_power_if)
lemma zero_le_even_power: "even n ==>
0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Power.thy
--- a/src/HOL/Power.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Power.thy Tue Nov 19 17:07:52 2013 +0100
@@ -209,14 +209,6 @@
"(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
-lemma power_neg_numeral_Bit0 [simp]:
- "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
- by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
-
-lemma power_neg_numeral_Bit1 [simp]:
- "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
- by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
-
lemma power2_minus [simp]:
"(- a)\<^sup>2 = a\<^sup>2"
by (rule power_minus_Bit0)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/ROOT
--- a/src/HOL/ROOT Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/ROOT Tue Nov 19 17:07:52 2013 +0100
@@ -687,14 +687,14 @@
theories Nominal_Examples
theories [quick_and_dirty] VC_Condition
-session "HOL-Cardinals-Base" in Cardinals = HOL +
+session "HOL-Cardinals-FP" in Cardinals = HOL +
description {*
- Ordinals and Cardinals, Base Theories.
+ Ordinals and Cardinals, Theories Needed for BNF FP Constructions.
*}
options [document = false]
- theories Cardinal_Arithmetic
+ theories Cardinal_Arithmetic_FP
-session "HOL-Cardinals" in Cardinals = "HOL-Cardinals-Base" +
+session "HOL-Cardinals" in Cardinals = "HOL-Cardinals-FP" +
description {*
Ordinals and Cardinals, Full Theories.
*}
@@ -705,16 +705,16 @@
"document/root.tex"
"document/root.bib"
-session "HOL-BNF-LFP" in BNF = "HOL-Cardinals-Base" +
+session "HOL-BNF-FP" in BNF = "HOL-Cardinals-FP" +
description {*
- Bounded Natural Functors for Datatypes.
+ Bounded Natural Functors for (Co)datatypes.
*}
options [document = false]
- theories BNF_LFP
+ theories BNF_LFP BNF_GFP
-session "HOL-BNF" in BNF = "HOL-Cardinals" +
+session "HOL-BNF" in BNF = "HOL-BNF-FP" +
description {*
- Bounded Natural Functors for (Co)datatypes.
+ Bounded Natural Functors for (Co)datatypes, Including More BNFs.
*}
options [document = false]
theories BNF
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Rat.thy
--- a/src/HOL/Rat.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Rat.thy Tue Nov 19 17:07:52 2013 +0100
@@ -215,17 +215,19 @@
"Fract 0 k = 0"
"Fract 1 1 = 1"
"Fract (numeral w) 1 = numeral w"
- "Fract (neg_numeral w) 1 = neg_numeral w"
+ "Fract (- numeral w) 1 = - numeral w"
+ "Fract (- 1) 1 = - 1"
"Fract k 0 = 0"
using Fract_of_int_eq [of "numeral w"]
- using Fract_of_int_eq [of "neg_numeral w"]
+ using Fract_of_int_eq [of "- numeral w"]
by (simp_all add: Zero_rat_def One_rat_def eq_rat)
lemma rat_number_expand:
"0 = Fract 0 1"
"1 = Fract 1 1"
"numeral k = Fract (numeral k) 1"
- "neg_numeral k = Fract (neg_numeral k) 1"
+ "- 1 = Fract (- 1) 1"
+ "- numeral k = Fract (- numeral k) 1"
by (simp_all add: rat_number_collapse)
lemma Rat_cases_nonzero [case_names Fract 0]:
@@ -356,7 +358,8 @@
"quotient_of 0 = (0, 1)"
"quotient_of 1 = (1, 1)"
"quotient_of (numeral k) = (numeral k, 1)"
- "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
+ "quotient_of (- 1) = (- 1, 1)"
+ "quotient_of (- numeral k) = (- numeral k, 1)"
by (simp_all add: rat_number_expand quotient_of_Fract)
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
@@ -620,7 +623,7 @@
#> Lin_Arith.add_simps [@{thm neg_less_iff_less},
@{thm True_implies_equals},
read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm distrib_left},
- read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm distrib_left},
+ read_instantiate @{context} [(("a", 0), "(- numeral ?v)")] @{thm distrib_left},
@{thm divide_1}, @{thm divide_zero_left},
@{thm times_divide_eq_right}, @{thm times_divide_eq_left},
@{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
@@ -664,6 +667,10 @@
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
by transfer simp
+lemma of_rat_neg_one [simp]:
+ "of_rat (- 1) = - 1"
+ by (simp add: of_rat_minus)
+
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
using of_rat_add [of a "- b"] by (simp add: of_rat_minus)
@@ -778,8 +785,8 @@
using of_rat_of_int_eq [of "numeral w"] by simp
lemma of_rat_neg_numeral_eq [simp]:
- "of_rat (neg_numeral w) = neg_numeral w"
-using of_rat_of_int_eq [of "neg_numeral w"] by simp
+ "of_rat (- numeral w) = - numeral w"
+using of_rat_of_int_eq [of "- numeral w"] by simp
lemmas zero_rat = Zero_rat_def
lemmas one_rat = One_rat_def
@@ -820,9 +827,6 @@
lemma Rats_number_of [simp]: "numeral w \<in> Rats"
by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
-lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
-by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
-
lemma Rats_0 [simp]: "0 \<in> Rats"
apply (unfold Rats_def)
apply (rule range_eqI)
@@ -943,7 +947,7 @@
by (simp add: Rat.of_int_def)
lemma [code_unfold]:
- "neg_numeral k = Rat.of_int (neg_numeral k)"
+ "- numeral k = Rat.of_int (- numeral k)"
by (simp add: Rat.of_int_def)
lemma Frct_code_post [code_post]:
@@ -951,13 +955,13 @@
"Frct (a, 0) = 0"
"Frct (1, 1) = 1"
"Frct (numeral k, 1) = numeral k"
- "Frct (neg_numeral k, 1) = neg_numeral k"
+ "Frct (- numeral k, 1) = - numeral k"
"Frct (1, numeral k) = 1 / numeral k"
- "Frct (1, neg_numeral k) = 1 / neg_numeral k"
+ "Frct (1, - numeral k) = 1 / - numeral k"
"Frct (numeral k, numeral l) = numeral k / numeral l"
- "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
- "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
- "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
+ "Frct (numeral k, - numeral l) = numeral k / - numeral l"
+ "Frct (- numeral k, numeral l) = - numeral k / numeral l"
+ "Frct (- numeral k, - numeral l) = - numeral k / - numeral l"
by (simp_all add: Fract_of_int_quotient)
@@ -1156,7 +1160,7 @@
in
if i = 0 then Syntax.const @{const_syntax Groups.zero}
else if i > 0 then Syntax.const @{const_syntax Num.numeral} $ mk i
- else Syntax.const @{const_syntax Num.neg_numeral} $ mk (~i)
+ else Syntax.const @{const_syntax Groups.uminus} $ (Syntax.const @{const_syntax Num.numeral} $ mk (~i))
end;
fun mk_frac str =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Real.thy
--- a/src/HOL/Real.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Real.thy Tue Nov 19 17:07:52 2013 +0100
@@ -1447,13 +1447,13 @@
lemma [code_abbrev]:
"real_of_int (numeral k) = numeral k"
- "real_of_int (neg_numeral k) = neg_numeral k"
+ "real_of_int (- numeral k) = - numeral k"
by simp_all
-text{*Collapse applications of @{term real} to @{term number_of}*}
+text{*Collapse applications of @{const real} to @{const numeral}*}
lemma real_numeral [simp]:
"real (numeral v :: int) = numeral v"
- "real (neg_numeral v :: int) = neg_numeral v"
+ "real (- numeral v :: int) = - numeral v"
by (simp_all add: real_of_int_def)
lemma real_of_nat_numeral [simp]:
@@ -1559,11 +1559,11 @@
unfolding real_of_int_le_iff[symmetric] by simp
lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
- "(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"
+ "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
unfolding real_of_int_le_iff[symmetric] by simp
lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
- "real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"
+ "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
unfolding real_of_int_le_iff[symmetric] by simp
subsection{*Density of the Reals*}
@@ -2051,7 +2051,7 @@
by simp
lemma [code_abbrev]:
- "(of_rat (neg_numeral k) :: real) = neg_numeral k"
+ "(of_rat (- numeral k) :: real) = - numeral k"
by simp
lemma [code_post]:
@@ -2059,14 +2059,14 @@
"(of_rat (r / 0) :: real) = 0"
"(of_rat (1 / 1) :: real) = 1"
"(of_rat (numeral k / 1) :: real) = numeral k"
- "(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"
+ "(of_rat (- numeral k / 1) :: real) = - numeral k"
"(of_rat (1 / numeral k) :: real) = 1 / numeral k"
- "(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"
+ "(of_rat (1 / - numeral k) :: real) = 1 / - numeral k"
"(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
- "(of_rat (numeral k / neg_numeral l) :: real) = numeral k / neg_numeral l"
- "(of_rat (neg_numeral k / numeral l) :: real) = neg_numeral k / numeral l"
- "(of_rat (neg_numeral k / neg_numeral l) :: real) = neg_numeral k / neg_numeral l"
- by (simp_all add: of_rat_divide)
+ "(of_rat (numeral k / - numeral l) :: real) = numeral k / - numeral l"
+ "(of_rat (- numeral k / numeral l) :: real) = - numeral k / numeral l"
+ "(of_rat (- numeral k / - numeral l) :: real) = - numeral k / - numeral l"
+ by (simp_all add: of_rat_divide of_rat_minus)
text {* Operations *}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Real_Vector_Spaces.thy Tue Nov 19 17:07:52 2013 +0100
@@ -307,8 +307,8 @@
lemma of_real_numeral: "of_real (numeral w) = numeral w"
using of_real_of_int_eq [of "numeral w"] by simp
-lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"
-using of_real_of_int_eq [of "neg_numeral w"] by simp
+lemma of_real_neg_numeral: "of_real (- numeral w) = - numeral w"
+using of_real_of_int_eq [of "- numeral w"] by simp
text{*Every real algebra has characteristic zero*}
@@ -341,9 +341,6 @@
lemma Reals_numeral [simp]: "numeral w \<in> Reals"
by (subst of_real_numeral [symmetric], rule Reals_of_real)
-lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals"
-by (subst of_real_neg_numeral [symmetric], rule Reals_of_real)
-
lemma Reals_0 [simp]: "0 \<in> Reals"
apply (unfold Reals_def)
apply (rule range_eqI)
@@ -602,7 +599,7 @@
by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
lemma norm_neg_numeral [simp]:
- "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w"
+ "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
lemma norm_of_int [simp]:
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Rings.thy
--- a/src/HOL/Rings.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Rings.thy Tue Nov 19 17:07:52 2013 +0100
@@ -1058,6 +1058,34 @@
"\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
by(subst abs_dvd_iff[symmetric]) simp
+text {* The following lemmas can be proven in more generale structures, but
+are dangerous as simp rules in absence of @{thm neg_equal_zero},
+@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
+
+lemma equation_minus_iff_1 [simp, no_atp]:
+ "1 = - a \<longleftrightarrow> a = - 1"
+ by (fact equation_minus_iff)
+
+lemma minus_equation_iff_1 [simp, no_atp]:
+ "- a = 1 \<longleftrightarrow> a = - 1"
+ by (subst minus_equation_iff, auto)
+
+lemma le_minus_iff_1 [simp, no_atp]:
+ "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
+ by (fact le_minus_iff)
+
+lemma minus_le_iff_1 [simp, no_atp]:
+ "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
+ by (fact minus_le_iff)
+
+lemma less_minus_iff_1 [simp, no_atp]:
+ "1 < - b \<longleftrightarrow> b < - 1"
+ by (fact less_minus_iff)
+
+lemma minus_less_iff_1 [simp, no_atp]:
+ "- a < 1 \<longleftrightarrow> - 1 < a"
+ by (fact minus_less_iff)
+
end
text {* Simprules for comparisons where common factors can be cancelled. *}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/SMT_Examples/SMT_Tests.thy
--- a/src/HOL/SMT_Examples/SMT_Tests.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/SMT_Examples/SMT_Tests.thy Tue Nov 19 17:07:52 2013 +0100
@@ -374,7 +374,6 @@
lemma
"(0::int) = 0"
- "(0::int) = -0"
"(0::int) = (- 0)"
"(1::int) = 1"
"\<not>(-1 = (1::int))"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/SMT_Examples/SMT_Word_Examples.certs
--- a/src/HOL/SMT_Examples/SMT_Word_Examples.certs Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/SMT_Examples/SMT_Word_Examples.certs Tue Nov 19 17:07:52 2013 +0100
@@ -54,3 +54,5 @@
unsat
e5c27ae0a583eeafeaa4ef3c59b1b4ec53e06b0f 1 0
unsat
+7d3ef49480d3ed3a7e5f2d7a12e7108cf7fc7819 1 0
+unsat
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/TPTP/atp_problem_import.ML
--- a/src/HOL/TPTP/atp_problem_import.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/TPTP/atp_problem_import.ML Tue Nov 19 17:07:52 2013 +0100
@@ -20,7 +20,7 @@
val sledgehammer_tptp_file : theory -> int -> string -> unit
val isabelle_tptp_file : theory -> int -> string -> unit
val isabelle_hot_tptp_file : theory -> int -> string -> unit
- val translate_tptp_file : theory -> string -> string -> string -> unit
+ val translate_tptp_file : theory -> string -> string -> unit
end;
structure ATP_Problem_Import : ATP_PROBLEM_IMPORT =
@@ -301,9 +301,9 @@
(** Translator between TPTP(-like) file formats **)
-fun translate_tptp_file thy format_str in_file_name out_file_name =
+fun translate_tptp_file thy format_str file_name =
let
- val (conjs, (defs, nondefs), ctxt) = read_tptp_file thy I in_file_name
+ val (conjs, (defs, nondefs), ctxt) = read_tptp_file thy I file_name
val conj = make_conj ([], []) (map snd conjs)
val (format, type_enc, lam_trans) =
@@ -327,7 +327,7 @@
val ord_info = K []
val lines = lines_of_atp_problem format ord ord_info atp_problem
in
- File.write_list (exploded_absolute_path out_file_name) lines
+ List.app Output.physical_stdout lines
end
end;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/TPTP/lib/Tools/tptp_translate
--- a/src/HOL/TPTP/lib/Tools/tptp_translate Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/TPTP/lib/Tools/tptp_translate Tue Nov 19 17:07:52 2013 +0100
@@ -9,20 +9,21 @@
function usage() {
echo
- echo "Usage: isabelle $PRG FORMAT IN_FILE OUT_FILE"
+ echo "Usage: isabelle $PRG FORMAT FILE"
echo
echo " Translates TPTP input file to the specified format (\"FOF\", \"TFF0\", \"THF0\", or \"DFG\")."
+ echo " Emits the result to standard output."
echo
exit 1
}
-[ "$#" -ne 3 -o "$1" = "-?" ] && usage
+[ "$#" -ne 2 -o "$1" = "-?" ] && usage
SCRATCH="Scratch_${PRG}_$$_${RANDOM}"
args=("$@")
echo "theory $SCRATCH imports \"$TPTP_HOME/ATP_Problem_Import\" begin \
-ML {* ATP_Problem_Import.translate_tptp_file @{theory} \"${args[0]}\" \"${args[1]}\" \"${args[2]}\" *} end;" \
+ML {* ATP_Problem_Import.translate_tptp_file @{theory} \"${args[0]}\" \"${args[1]}\" *} end;" \
> /tmp/$SCRATCH.thy
"$ISABELLE_PROCESS" -q -e "use_thy \"/tmp/$SCRATCH\"; exit 1;" HOL-TPTP | grep --line-buffered -v "^###\|^PROOF FAILED for depth\|^Failure node\|inferences so far. Searching to depth\|^val \|^Loading theory"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Datatype/datatype_data.ML
--- a/src/HOL/Tools/Datatype/datatype_data.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Datatype/datatype_data.ML Tue Nov 19 17:07:52 2013 +0100
@@ -118,7 +118,7 @@
expands = [],
sel_splits = [],
sel_split_asms = [],
- case_ifs = []};
+ case_eq_ifs = []};
fun register dt_infos =
Data.map (fn {types, constrs, cases} =>
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Nitpick/nitpick_hol.ML
--- a/src/HOL/Tools/Nitpick/nitpick_hol.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Nitpick/nitpick_hol.ML Tue Nov 19 17:07:52 2013 +0100
@@ -1645,10 +1645,10 @@
(hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
def_tables, ground_thm_table, ersatz_table, ...}) =
let
- fun do_numeral depth Ts mult T t0 t1 =
+ fun do_numeral depth Ts mult T some_t0 t1 t2 =
(if is_number_type ctxt T then
let
- val j = mult * HOLogic.dest_num t1
+ val j = mult * HOLogic.dest_num t2
in
if j = 1 then
raise SAME ()
@@ -1667,15 +1667,16 @@
handle TERM _ => raise SAME ()
else
raise SAME ())
- handle SAME () => s_betapply [] (do_term depth Ts t0, do_term depth Ts t1)
+ handle SAME () => (case some_t0 of NONE => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)
+ | SOME t0 => s_betapply [] (do_term depth Ts t0, s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)))
and do_term depth Ts t =
case t of
- (t0 as Const (@{const_name Num.neg_numeral_class.neg_numeral},
- Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
- do_numeral depth Ts ~1 ran_T t0 t1
- | (t0 as Const (@{const_name Num.numeral_class.numeral},
- Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
- do_numeral depth Ts 1 ran_T t0 t1
+ (t0 as Const (@{const_name uminus}, _) $ ((t1 as Const (@{const_name numeral},
+ Type (@{type_name fun}, [_, ran_T]))) $ t2)) =>
+ do_numeral depth Ts ~1 ran_T (SOME t0) t1 t2
+ | (t1 as Const (@{const_name numeral},
+ Type (@{type_name fun}, [_, ran_T]))) $ t2 =>
+ do_numeral depth Ts 1 ran_T NONE t1 t2
| Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
| (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Qelim/cooper.ML
--- a/src/HOL/Tools/Qelim/cooper.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Qelim/cooper.ML Tue Nov 19 17:07:52 2013 +0100
@@ -42,7 +42,6 @@
@{term "nat"}, @{term "int"},
@{term "Num.One"}, @{term "Num.Bit0"}, @{term "Num.Bit1"},
@{term "Num.numeral :: num => int"}, @{term "Num.numeral :: num => nat"},
- @{term "Num.neg_numeral :: num => int"},
@{term "0::int"}, @{term "1::int"}, @{term "0::nat"}, @{term "1::nat"},
@{term "True"}, @{term "False"}];
@@ -610,8 +609,6 @@
| num_of_term vs @{term "1::int"} = Proc.C (Proc.Int_of_integer 1)
| num_of_term vs (t as Const (@{const_name numeral}, _) $ _) =
Proc.C (Proc.Int_of_integer (dest_number t))
- | num_of_term vs (t as Const (@{const_name neg_numeral}, _) $ _) =
- Proc.Neg (Proc.C (Proc.Int_of_integer (dest_number t)))
| num_of_term vs (Const (@{const_name Groups.uminus}, _) $ t') =
Proc.Neg (num_of_term vs t')
| num_of_term vs (Const (@{const_name Groups.plus}, _) $ t1 $ t2) =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/SMT/smt_builtin.ML
--- a/src/HOL/Tools/SMT/smt_builtin.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/SMT/smt_builtin.ML Tue Nov 19 17:07:52 2013 +0100
@@ -144,9 +144,10 @@
(case try HOLogic.dest_number t of
NONE => NONE
| SOME (T, i) =>
- (case lookup_builtin_typ ctxt T of
- SOME (_, Int (_, g)) => g T i |> Option.map (rpair T)
- | _ => NONE))
+ if i < 0 then NONE else
+ (case lookup_builtin_typ ctxt T of
+ SOME (_, Int (_, g)) => g T i |> Option.map (rpair T)
+ | _ => NONE))
val is_builtin_num = is_some oo dest_builtin_num
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/SMT/smt_normalize.ML
--- a/src/HOL/Tools/SMT/smt_normalize.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/SMT/smt_normalize.ML Tue Nov 19 17:07:52 2013 +0100
@@ -526,23 +526,26 @@
local
(*
- rewrite negative numerals into positive numerals,
- rewrite Numeral0 into 0
rewrite Numeral1 into 1
+ rewrite - 0 into 0
*)
- fun is_strange_number ctxt (t as Const (@{const_name neg_numeral}, _) $ _) =
- SMT_Builtin.is_builtin_num ctxt t
- | is_strange_number _ _ = false
+ fun is_irregular_number (Const (@{const_name numeral}, _) $ Const (@{const_name num.One}, _)) =
+ true
+ | is_irregular_number (Const (@{const_name uminus}, _) $ Const (@{const_name Groups.zero}, _)) =
+ true
+ | is_irregular_number _ =
+ false;
- val pos_num_ss =
+ fun is_strange_number ctxt t = is_irregular_number t andalso SMT_Builtin.is_builtin_num ctxt t;
+
+ val proper_num_ss =
simpset_of (put_simpset HOL_ss @{context}
- addsimps [@{thm Num.numeral_One}]
- addsimps [@{thm Num.neg_numeral_def}])
+ addsimps @{thms Num.numeral_One minus_zero})
fun norm_num_conv ctxt =
SMT_Utils.if_conv (is_strange_number ctxt)
- (Simplifier.rewrite (put_simpset pos_num_ss ctxt)) Conv.no_conv
+ (Simplifier.rewrite (put_simpset proper_num_ss ctxt)) Conv.no_conv
in
fun normalize_numerals_conv ctxt =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/SMT/smt_utils.ML
--- a/src/HOL/Tools/SMT/smt_utils.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/SMT/smt_utils.ML Tue Nov 19 17:07:52 2013 +0100
@@ -140,7 +140,6 @@
is_num env t andalso is_num env u
| is_num env (Const (@{const_name Let}, _) $ t $ Abs (_, _, u)) =
is_num (t :: env) u
- | is_num env (Const (@{const_name uminus}, _) $ t) = is_num env t
| is_num env (Bound i) = i < length env andalso is_num env (nth env i)
| is_num _ t = can HOLogic.dest_number t
in is_num [] end
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Sledgehammer/sledgehammer_provers.ML
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_provers.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_provers.ML Tue Nov 19 17:07:52 2013 +0100
@@ -838,9 +838,10 @@
Output.urgent_message "Generating proof text..."
else
()
+ val atp_proof = (fn () => termify_atp_proof ctxt pool lifted sym_tab atp_proof)
val isar_params =
- (debug, verbose, preplay_timeout, isar_compress, isar_try0,
- pool, fact_names, lifted, sym_tab, atp_proof, goal)
+ (debug, verbose, preplay_timeout, isar_compress, isar_try0, fact_names, atp_proof,
+ goal)
val one_line_params =
(preplay, proof_banner mode name, used_facts,
choose_minimize_command ctxt params minimize_command name
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML Tue Nov 19 17:07:52 2013 +0100
@@ -7,14 +7,14 @@
signature SLEDGEHAMMER_RECONSTRUCT =
sig
+ type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
type 'a atp_proof = 'a ATP_Proof.atp_proof
type stature = ATP_Problem_Generate.stature
- type one_line_params = Sledgehammer_Print.one_line_params
+ type one_line_params = Sledgehammer_Reconstructor.one_line_params
type isar_params =
- bool * bool * Time.time option * real * bool * string Symtab.table
- * (string * stature) list vector * (string * term) list * int Symtab.table
- * string atp_proof * thm
+ bool * bool * Time.time option * real * bool * (string * stature) list vector
+ * (unit -> (term, string) atp_step list) * thm
val lam_trans_of_atp_proof : string atp_proof -> string -> string
val is_typed_helper_used_in_atp_proof : string atp_proof -> bool
@@ -24,6 +24,9 @@
val used_facts_in_unsound_atp_proof :
Proof.context -> (string * stature) list vector -> 'a atp_proof ->
string list option
+ val termify_atp_proof :
+ Proof.context -> string Symtab.table -> (string * term) list ->
+ int Symtab.table -> string atp_proof -> (term, string) atp_step list
val isar_proof_text :
Proof.context -> bool option -> isar_params -> one_line_params -> string
val proof_text :
@@ -407,14 +410,19 @@
and chain_proofs proofs = map (chain_proof) proofs
in chain_proof end
+fun termify_atp_proof ctxt pool lifted sym_tab =
+ clean_up_atp_proof_dependencies
+ #> nasty_atp_proof pool
+ #> map_term_names_in_atp_proof repair_name
+ #> map (decode_line ctxt lifted sym_tab)
+ #> repair_waldmeister_endgame
+
type isar_params =
- bool * bool * Time.time option * real * bool * string Symtab.table
- * (string * stature) list vector * (string * term) list * int Symtab.table
- * string atp_proof * thm
+ bool * bool * Time.time option * real * bool * (string * stature) list vector
+ * (unit -> (term, string) atp_step list) * thm
fun isar_proof_text ctxt isar_proofs
- (debug, verbose, preplay_timeout, isar_compress, isar_try0, pool,
- fact_names, lifted, sym_tab, atp_proof, goal)
+ (debug, verbose, preplay_timeout, isar_compress, isar_try0, fact_names, atp_proof, goal)
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
val (params, hyp_ts, concl_t) = strip_subgoal goal subgoal ctxt
@@ -425,6 +433,7 @@
ctxt |> Variable.set_body false
|> Proof_Context.add_fixes fixes)
val one_line_proof = one_line_proof_text 0 one_line_params
+ val atp_proof = atp_proof ()
val type_enc =
if is_typed_helper_used_in_atp_proof atp_proof then full_typesN
else partial_typesN
@@ -435,11 +444,6 @@
let
val atp_proof =
atp_proof
- |> clean_up_atp_proof_dependencies
- |> nasty_atp_proof pool
- |> map_term_names_in_atp_proof repair_name
- |> map (decode_line ctxt lifted sym_tab)
- |> repair_waldmeister_endgame
|> rpair [] |-> fold_rev (add_line fact_names)
|> rpair [] |-> fold_rev add_nontrivial_line
|> rpair (0, [])
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/Sledgehammer/sledgehammer_reconstructor.ML
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_reconstructor.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_reconstructor.ML Tue Nov 19 17:07:52 2013 +0100
@@ -7,7 +7,6 @@
signature SLEDGEHAMMER_RECONSTRUCTOR =
sig
-
type stature = ATP_Problem_Generate.stature
datatype reconstructor =
@@ -25,8 +24,7 @@
play * string * (string * stature) list * minimize_command * int * int
val smtN : string
-
-end
+end;
structure Sledgehammer_Reconstructor : SLEDGEHAMMER_RECONSTRUCTOR =
struct
@@ -49,4 +47,4 @@
val smtN = "smt"
-end
+end;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/ctr_sugar.ML
--- a/src/HOL/Tools/ctr_sugar.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/ctr_sugar.ML Tue Nov 19 17:07:52 2013 +0100
@@ -30,7 +30,7 @@
expands: thm list,
sel_splits: thm list,
sel_split_asms: thm list,
- case_ifs: thm list};
+ case_eq_ifs: thm list};
val morph_ctr_sugar: morphism -> ctr_sugar -> ctr_sugar
val transfer_ctr_sugar: Proof.context -> ctr_sugar -> ctr_sugar
@@ -90,7 +90,7 @@
expands: thm list,
sel_splits: thm list,
sel_split_asms: thm list,
- case_ifs: thm list};
+ case_eq_ifs: thm list};
fun eq_ctr_sugar ({ctrs = ctrs1, casex = case1, discs = discs1, selss = selss1, ...} : ctr_sugar,
{ctrs = ctrs2, casex = case2, discs = discs2, selss = selss2, ...} : ctr_sugar) =
@@ -98,7 +98,7 @@
fun morph_ctr_sugar phi {ctrs, casex, discs, selss, exhaust, nchotomy, injects, distincts,
case_thms, case_cong, weak_case_cong, split, split_asm, disc_thmss, discIs, sel_thmss,
- disc_exhausts, sel_exhausts, collapses, expands, sel_splits, sel_split_asms, case_ifs} =
+ disc_exhausts, sel_exhausts, collapses, expands, sel_splits, sel_split_asms, case_eq_ifs} =
{ctrs = map (Morphism.term phi) ctrs,
casex = Morphism.term phi casex,
discs = map (Morphism.term phi) discs,
@@ -121,7 +121,7 @@
expands = map (Morphism.thm phi) expands,
sel_splits = map (Morphism.thm phi) sel_splits,
sel_split_asms = map (Morphism.thm phi) sel_split_asms,
- case_ifs = map (Morphism.thm phi) case_ifs};
+ case_eq_ifs = map (Morphism.thm phi) case_eq_ifs};
val transfer_ctr_sugar =
morph_ctr_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;
@@ -160,7 +160,7 @@
val caseN = "case";
val case_congN = "case_cong";
-val case_ifN = "case_if";
+val case_eq_ifN = "case_eq_if";
val collapseN = "collapse";
val disc_excludeN = "disc_exclude";
val disc_exhaustN = "disc_exhaust";
@@ -390,7 +390,7 @@
qualify false
(if Binding.is_empty raw_case_binding orelse
Binding.eq_name (raw_case_binding, standard_binding) then
- Binding.suffix_name ("_" ^ caseN) fc_b
+ Binding.prefix_name (caseN ^ "_") fc_b
else
raw_case_binding);
@@ -657,7 +657,7 @@
val (all_sel_thms, sel_thmss, disc_thmss, nontriv_disc_thms, discI_thms, nontriv_discI_thms,
disc_exclude_thms, disc_exhaust_thms, sel_exhaust_thms, all_collapse_thms,
- safe_collapse_thms, expand_thms, sel_split_thms, sel_split_asm_thms, case_if_thms) =
+ safe_collapse_thms, expand_thms, sel_split_thms, sel_split_asm_thms, case_eq_if_thms) =
if no_discs_sels then
([], [], [], [], [], [], [], [], [], [], [], [], [], [], [])
else
@@ -861,12 +861,12 @@
(thm, asm_thm)
end;
- val case_if_thm =
+ val case_eq_if_thm =
let
val goal = mk_Trueprop_eq (ufcase, mk_IfN B udiscs usel_fs);
in
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
- mk_case_if_tac ctxt n uexhaust_thm case_thms disc_thmss' sel_thmss)
+ mk_case_eq_if_tac ctxt n uexhaust_thm case_thms disc_thmss' sel_thmss)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
@@ -874,7 +874,7 @@
(all_sel_thms, sel_thmss, disc_thmss, nontriv_disc_thms, discI_thms,
nontriv_discI_thms, disc_exclude_thms, [disc_exhaust_thm], [sel_exhaust_thm],
all_collapse_thms, safe_collapse_thms, [expand_thm], [sel_split_thm],
- [sel_split_asm_thm], [case_if_thm])
+ [sel_split_asm_thm], [case_eq_if_thm])
end;
val exhaust_case_names_attr = Attrib.internal (K (Rule_Cases.case_names exhaust_cases));
@@ -890,7 +890,7 @@
val notes =
[(caseN, case_thms, code_nitpicksimp_simp_attrs),
(case_congN, [case_cong_thm], []),
- (case_ifN, case_if_thms, []),
+ (case_eq_ifN, case_eq_if_thms, []),
(collapseN, safe_collapse_thms, simp_attrs),
(discN, nontriv_disc_thms, simp_attrs),
(discIN, nontriv_discI_thms, []),
@@ -921,7 +921,7 @@
discIs = discI_thms, sel_thmss = sel_thmss, disc_exhausts = disc_exhaust_thms,
sel_exhausts = sel_exhaust_thms, collapses = all_collapse_thms, expands = expand_thms,
sel_splits = sel_split_thms, sel_split_asms = sel_split_asm_thms,
- case_ifs = case_if_thms};
+ case_eq_ifs = case_eq_if_thms};
in
(ctr_sugar,
lthy
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/ctr_sugar_tactics.ML
--- a/src/HOL/Tools/ctr_sugar_tactics.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/ctr_sugar_tactics.ML Tue Nov 19 17:07:52 2013 +0100
@@ -18,8 +18,8 @@
val mk_alternate_disc_def_tac: Proof.context -> int -> thm -> thm -> thm -> tactic
val mk_case_tac: Proof.context -> int -> int -> thm -> thm list -> thm list list -> tactic
val mk_case_cong_tac: Proof.context -> thm -> thm list -> tactic
- val mk_case_if_tac: Proof.context -> int -> thm -> thm list -> thm list list -> thm list list ->
- tactic
+ val mk_case_eq_if_tac: Proof.context -> int -> thm -> thm list -> thm list list ->
+ thm list list -> tactic
val mk_collapse_tac: Proof.context -> int -> thm -> thm list -> tactic
val mk_disc_exhaust_tac: int -> thm -> thm list -> tactic
val mk_expand_tac: Proof.context -> int -> int list -> thm -> thm -> thm list ->
@@ -28,8 +28,8 @@
val mk_nchotomy_tac: int -> thm -> tactic
val mk_other_half_disc_exclude_tac: thm -> tactic
val mk_sel_exhaust_tac: int -> thm -> thm list -> tactic
- val mk_split_tac: Proof.context -> thm -> thm list -> thm list list -> thm list list -> thm list
- list list -> tactic
+ val mk_split_tac: Proof.context -> thm -> thm list -> thm list list -> thm list list ->
+ thm list list list -> tactic
val mk_split_asm_tac: Proof.context -> thm -> tactic
val mk_unique_disc_def_tac: int -> thm -> tactic
end;
@@ -143,17 +143,17 @@
else mk_case_distinct_ctrs_tac ctxt distincts)) ks distinctss))
end;
-fun mk_case_if_tac ctxt n uexhaust cases discss' selss =
+fun mk_case_cong_tac ctxt uexhaust cases =
+ HEADGOAL (rtac uexhaust THEN'
+ EVERY' (maps (fn casex => [dtac sym, asm_simp_tac (ss_only [casex] ctxt)]) cases));
+
+fun mk_case_eq_if_tac ctxt n uexhaust cases discss' selss =
HEADGOAL (rtac uexhaust THEN'
EVERY' (map3 (fn casex => fn if_discs => fn sels =>
EVERY' [hyp_subst_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt (if_discs @ sels)),
rtac casex])
cases (map2 (seq_conds if_P_or_not_P_OF n) (1 upto n) discss') selss));
-fun mk_case_cong_tac ctxt uexhaust cases =
- HEADGOAL (rtac uexhaust THEN'
- EVERY' (maps (fn casex => [dtac sym, asm_simp_tac (ss_only [casex] ctxt)]) cases));
-
fun mk_split_tac ctxt uexhaust cases selss injectss distinctsss =
HEADGOAL (rtac uexhaust) THEN
ALLGOALS (fn k => (hyp_subst_tac ctxt THEN'
@@ -169,4 +169,4 @@
end;
-structure Ctr_Sugar_General_Tactics: CTR_SUGAR_GENERAL_TACTICS = Ctr_Sugar_Tactics;
+structure Ctr_Sugar_General_Tactics : CTR_SUGAR_GENERAL_TACTICS = Ctr_Sugar_Tactics;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/ctr_sugar_util.ML
--- a/src/HOL/Tools/ctr_sugar_util.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/ctr_sugar_util.ML Tue Nov 19 17:07:52 2013 +0100
@@ -176,9 +176,7 @@
fun rapp u t = betapply (t, u);
fun list_quant_free quant_const =
- fold_rev (fn free => fn P =>
- let val (x, T) = Term.dest_Free free;
- in quant_const T $ Term.absfree (x, T) P end);
+ fold_rev (fn Free (xT as (_, T)) => fn P => quant_const T $ Term.absfree xT P);
val list_all_free = list_quant_free HOLogic.all_const;
val list_exists_free = list_quant_free HOLogic.exists_const;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/hologic.ML
--- a/src/HOL/Tools/hologic.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/hologic.ML Tue Nov 19 17:07:52 2013 +0100
@@ -104,7 +104,6 @@
val mk_numeral: int -> term
val dest_num: term -> int
val numeral_const: typ -> term
- val neg_numeral_const: typ -> term
val add_numerals: term -> (term * typ) list -> (term * typ) list
val mk_number: typ -> int -> term
val dest_number: term -> typ * int
@@ -548,7 +547,6 @@
| dest_num t = raise TERM ("dest_num", [t]);
fun numeral_const T = Const ("Num.numeral_class.numeral", numT --> T);
-fun neg_numeral_const T = Const ("Num.neg_numeral_class.neg_numeral", numT --> T);
fun add_numerals (Const ("Num.numeral_class.numeral", Type (_, [_, T])) $ t) = cons (t, T)
| add_numerals (t $ u) = add_numerals t #> add_numerals u
@@ -559,14 +557,14 @@
| mk_number T 1 = Const ("Groups.one_class.one", T)
| mk_number T i =
if i > 0 then numeral_const T $ mk_numeral i
- else neg_numeral_const T $ mk_numeral (~ i);
+ else Const ("Groups.uminus_class.uminus", T --> T) $ mk_number T (~ i);
fun dest_number (Const ("Groups.zero_class.zero", T)) = (T, 0)
| dest_number (Const ("Groups.one_class.one", T)) = (T, 1)
| dest_number (Const ("Num.numeral_class.numeral", Type ("fun", [_, T])) $ t) =
(T, dest_num t)
- | dest_number (Const ("Num.neg_numeral_class.neg_numeral", Type ("fun", [_, T])) $ t) =
- (T, ~ (dest_num t))
+ | dest_number (Const ("Groups.uminus_class.uminus", Type ("fun", [_, T])) $ t) =
+ apsnd (op ~) (dest_number t)
| dest_number t = raise TERM ("dest_number", [t]);
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/lin_arith.ML
--- a/src/HOL/Tools/lin_arith.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/lin_arith.ML Tue Nov 19 17:07:52 2013 +0100
@@ -183,9 +183,6 @@
| demult (t as Const ("Num.numeral_class.numeral", _) $ n, m) =
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_num n)))
handle TERM _ => (SOME t, m))
- | demult (t as Const ("Num.neg_numeral_class.neg_numeral", _) $ n, m) =
- ((NONE, Rat.mult m (Rat.rat_of_int (~ (HOLogic.dest_num n))))
- handle TERM _ => (SOME t, m))
| demult (t as Const (@{const_name Suc}, _) $ _, m) =
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
handle TERM _ => (SOME t, m))
@@ -212,6 +209,10 @@
pi
| poly (Const (@{const_name Groups.one}, _), m, (p, i)) =
(p, Rat.add i m)
+ | poly (all as Const ("Num.numeral_class.numeral", Type(_,[_,_])) $ t, m, pi as (p, i)) =
+ (let val k = HOLogic.dest_num t
+ in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k))) end
+ handle TERM _ => add_atom all m pi)
| poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
poly (t, m, (p, Rat.add i m))
| poly (all as Const (@{const_name Groups.times}, _) $ _ $ _, m, pi as (p, i)) =
@@ -222,14 +223,6 @@
(case demult inj_consts (all, m) of
(NONE, m') => (p, Rat.add i m')
| (SOME u, m') => add_atom u m' pi)
- | poly (all as Const ("Num.numeral_class.numeral", Type(_,[_,_])) $ t, m, pi as (p, i)) =
- (let val k = HOLogic.dest_num t
- in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k))) end
- handle TERM _ => add_atom all m pi)
- | poly (all as Const ("Num.neg_numeral_class.neg_numeral", Type(_,[_,_])) $ t, m, pi as (p, i)) =
- (let val k = HOLogic.dest_num t
- in (p, Rat.add i (Rat.mult m (Rat.rat_of_int (~ k)))) end
- handle TERM _ => add_atom all m pi)
| poly (all as Const f $ x, m, pi) =
if member (op =) inj_consts f then poly (x, m, pi) else add_atom all m pi
| poly (all, m, pi) =
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/numeral.ML
--- a/src/HOL/Tools/numeral.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/numeral.ML Tue Nov 19 17:07:52 2013 +0100
@@ -45,8 +45,8 @@
val numeral = @{cpat "numeral"};
val numeralT = Thm.ctyp_of @{theory} (Term.range_type (Thm.typ_of (Thm.ctyp_of_term numeral)));
-val neg_numeral = @{cpat "neg_numeral"};
-val neg_numeralT = Thm.ctyp_of @{theory} (Term.range_type (Thm.typ_of (Thm.ctyp_of_term neg_numeral)));
+val uminus = @{cpat "uminus"};
+val uminusT = Thm.ctyp_of @{theory} (Term.range_type (Thm.typ_of (Thm.ctyp_of_term uminus)));
fun instT T V = Thm.instantiate_cterm ([(V, T)], []);
@@ -56,7 +56,7 @@
| mk_cnumber T 1 = instT T oneT one
| mk_cnumber T i =
if i > 0 then Thm.apply (instT T numeralT numeral) (mk_cnumeral i)
- else Thm.apply (instT T neg_numeralT neg_numeral) (mk_cnumeral (~i));
+ else Thm.apply (instT T uminusT uminus) (Thm.apply (instT T numeralT numeral) (mk_cnumeral (~i)));
end;
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Tools/numeral_simprocs.ML
--- a/src/HOL/Tools/numeral_simprocs.ML Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Tools/numeral_simprocs.ML Tue Nov 19 17:07:52 2013 +0100
@@ -56,9 +56,6 @@
val long_mk_sum = Arith_Data.long_mk_sum;
val dest_sum = Arith_Data.dest_sum;
-val mk_diff = HOLogic.mk_binop @{const_name Groups.minus};
-val dest_diff = HOLogic.dest_bin @{const_name Groups.minus} dummyT;
-
val mk_times = HOLogic.mk_binop @{const_name Groups.times};
fun one_of T = Const(@{const_name Groups.one}, T);
@@ -181,7 +178,7 @@
(*Simplify 0+n, n+0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
val add_0s = @{thms add_0s};
-val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};
+val mult_1s = @{thms mult_1s divide_numeral_1 mult_1_left mult_1_right mult_minus1 mult_minus1_right divide_1};
(* For post-simplification of the rhs of simproc-generated rules *)
val post_simps =
@@ -194,9 +191,8 @@
val field_post_simps =
post_simps @ [@{thm divide_zero_left}, @{thm divide_1}]
-(*Simplify inverse Numeral1, a/Numeral1*)
+(*Simplify inverse Numeral1*)
val inverse_1s = [@{thm inverse_numeral_1}];
-val divide_1s = [@{thm divide_numeral_1}];
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the Numeral_Simprocs, such as 3 * (5 * x). *)
@@ -217,7 +213,7 @@
@{thms add_neg_numeral_simps}) simps;
(*To evaluate binary negations of coefficients*)
-val minus_simps = [@{thm minus_zero}, @{thm minus_one}, @{thm minus_numeral}, @{thm minus_neg_numeral}];
+val minus_simps = [@{thm minus_zero}, @{thm minus_minus}];
(*To let us treat subtraction as addition*)
val diff_simps = [@{thm diff_conv_add_uminus}, @{thm minus_add_distrib}, @{thm minus_minus}];
@@ -225,16 +221,13 @@
(*To let us treat division as multiplication*)
val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
-(*push the unary minus down*)
-val minus_mult_eq_1_to_2 = @{lemma "- (a::'a::ring) * b = a * - b" by simp};
-
(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
[@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
- [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
+ [@{thm mult_assoc}, @{thm minus_mult_right}, @{thm minus_mult_commute}];
val norm_ss1 =
simpset_of (put_simpset num_ss @{context}
@@ -247,7 +240,7 @@
val norm_ss3 =
simpset_of (put_simpset num_ss @{context}
- addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac})
+ addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac minus_mult_commute})
structure CancelNumeralsCommon =
struct
@@ -330,7 +323,7 @@
structure FieldCombineNumeralsData =
struct
type coeff = int * int
- val iszero = (fn (p, q) => p = 0)
+ val iszero = (fn (p, _) => p = 0)
val add = add_frac
val mk_sum = long_mk_sum
val dest_sum = dest_sum
@@ -368,7 +361,7 @@
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
- val assoc_ss = simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac})
+ val assoc_ss = simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac minus_mult_commute})
val eq_reflection = eq_reflection
val is_numeral = can HOLogic.dest_number
end;
@@ -388,7 +381,7 @@
val norm_ss2 =
simpset_of (put_simpset HOL_basic_ss @{context} addsimps simps @ mult_minus_simps)
val norm_ss3 =
- simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac})
+ simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac minus_mult_commute})
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
@@ -463,9 +456,9 @@
["((l::'a::field_inverse_zero) * m) / n",
"(l::'a::field_inverse_zero) / (m * n)",
"((numeral v)::'a::field_inverse_zero) / (numeral w)",
- "((numeral v)::'a::field_inverse_zero) / (neg_numeral w)",
- "((neg_numeral v)::'a::field_inverse_zero) / (numeral w)",
- "((neg_numeral v)::'a::field_inverse_zero) / (neg_numeral w)"],
+ "((numeral v)::'a::field_inverse_zero) / (- numeral w)",
+ "((- numeral v)::'a::field_inverse_zero) / (numeral w)",
+ "((- numeral v)::'a::field_inverse_zero) / (- numeral w)"],
DivideCancelNumeralFactor.proc)]
@@ -516,7 +509,7 @@
val find_first = find_first_t []
val trans_tac = trans_tac
val norm_ss =
- simpset_of (put_simpset HOL_basic_ss @{context} addsimps mult_1s @ @{thms mult_ac})
+ simpset_of (put_simpset HOL_basic_ss @{context} addsimps mult_1s @ @{thms mult_ac minus_mult_commute})
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
val simplify_meta_eq = cancel_simplify_meta_eq
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Transcendental.thy Tue Nov 19 17:07:52 2013 +0100
@@ -2000,8 +2000,8 @@
apply (subst powr_add, simp, simp)
done
-lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x^(numeral n)"
- unfolding real_of_nat_numeral[symmetric] by (rule powr_realpow)
+lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
+ unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
apply (case_tac "x = 0", simp, simp)
@@ -2020,11 +2020,17 @@
then show ?thesis by (simp add: assms powr_realpow[symmetric])
qed
-lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
- using powr_realpow[of x "numeral n"] by simp
-
-lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
- using powr_int[of x "neg_numeral n"] by simp
+lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x"
+ using powr_realpow [of x 1] by simp
+
+lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
+ by (fact powr_realpow_numeral)
+
+lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
+ using powr_int [of x "- 1"] by simp
+
+lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
+ using powr_int [of x "- numeral n"] by simp
lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
@@ -4047,7 +4053,7 @@
show "sgn x * pi / 2 - arctan x < pi / 2"
using arctan_bounded [of "- x"] assms
unfolding sgn_real_def arctan_minus
- by auto
+ by (auto simp add: algebra_simps)
show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
unfolding sgn_real_def
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Word/Bit_Int.thy
--- a/src/HOL/Word/Bit_Int.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Word/Bit_Int.thy Tue Nov 19 17:07:52 2013 +0100
@@ -52,10 +52,10 @@
lemma int_not_simps [simp]:
"NOT (0::int) = -1"
"NOT (1::int) = -2"
- "NOT (-1::int) = 0"
- "NOT (numeral w::int) = neg_numeral (w + Num.One)"
- "NOT (neg_numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
- "NOT (neg_numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
+ "NOT (- 1::int) = 0"
+ "NOT (numeral w::int) = - numeral (w + Num.One)"
+ "NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
+ "NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
unfolding int_not_def by simp_all
lemma int_not_not [simp]: "NOT (NOT (x::int)) = x"
@@ -228,11 +228,11 @@
by (metis bin_rl_simp)
lemma bin_rest_neg_numeral_BitM [simp]:
- "bin_rest (neg_numeral (Num.BitM w)) = neg_numeral w"
+ "bin_rest (- numeral (Num.BitM w)) = - numeral w"
by (simp only: BIT_bin_simps [symmetric] bin_rest_BIT)
lemma bin_last_neg_numeral_BitM [simp]:
- "bin_last (neg_numeral (Num.BitM w)) = 1"
+ "bin_last (- numeral (Num.BitM w)) = 1"
by (simp only: BIT_bin_simps [symmetric] bin_last_BIT)
text {* FIXME: The rule sets below are very large (24 rules for each
@@ -243,26 +243,26 @@
"numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 0"
"numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT 0"
"numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 1"
- "numeral (Num.Bit0 x) AND neg_numeral (Num.Bit0 y) = (numeral x AND neg_numeral y) BIT 0"
- "numeral (Num.Bit0 x) AND neg_numeral (Num.Bit1 y) = (numeral x AND neg_numeral (y + Num.One)) BIT 0"
- "numeral (Num.Bit1 x) AND neg_numeral (Num.Bit0 y) = (numeral x AND neg_numeral y) BIT 0"
- "numeral (Num.Bit1 x) AND neg_numeral (Num.Bit1 y) = (numeral x AND neg_numeral (y + Num.One)) BIT 1"
- "neg_numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (neg_numeral x AND numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (neg_numeral x AND numeral y) BIT 0"
- "neg_numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) AND numeral y) BIT 0"
- "neg_numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) AND numeral y) BIT 1"
- "neg_numeral (Num.Bit0 x) AND neg_numeral (Num.Bit0 y) = (neg_numeral x AND neg_numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) AND neg_numeral (Num.Bit1 y) = (neg_numeral x AND neg_numeral (y + Num.One)) BIT 0"
- "neg_numeral (Num.Bit1 x) AND neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) AND neg_numeral y) BIT 0"
- "neg_numeral (Num.Bit1 x) AND neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) AND neg_numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT 0"
+ "numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT 0"
+ "numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT 0"
+ "numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT 1"
+ "- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (- numeral x AND numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (- numeral x AND numeral y) BIT 0"
+ "- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND numeral y) BIT 0"
+ "- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND numeral y) BIT 1"
+ "- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (- numeral x AND - numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (- numeral x AND - numeral (y + Num.One)) BIT 0"
+ "- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND - numeral y) BIT 0"
+ "- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND - numeral (y + Num.One)) BIT 1"
"(1::int) AND numeral (Num.Bit0 y) = 0"
"(1::int) AND numeral (Num.Bit1 y) = 1"
- "(1::int) AND neg_numeral (Num.Bit0 y) = 0"
- "(1::int) AND neg_numeral (Num.Bit1 y) = 1"
+ "(1::int) AND - numeral (Num.Bit0 y) = 0"
+ "(1::int) AND - numeral (Num.Bit1 y) = 1"
"numeral (Num.Bit0 x) AND (1::int) = 0"
"numeral (Num.Bit1 x) AND (1::int) = 1"
- "neg_numeral (Num.Bit0 x) AND (1::int) = 0"
- "neg_numeral (Num.Bit1 x) AND (1::int) = 1"
+ "- numeral (Num.Bit0 x) AND (1::int) = 0"
+ "- numeral (Num.Bit1 x) AND (1::int) = 1"
by (rule bin_rl_eqI, simp, simp)+
lemma int_or_numerals [simp]:
@@ -270,26 +270,26 @@
"numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1"
"numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT 1"
"numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1"
- "numeral (Num.Bit0 x) OR neg_numeral (Num.Bit0 y) = (numeral x OR neg_numeral y) BIT 0"
- "numeral (Num.Bit0 x) OR neg_numeral (Num.Bit1 y) = (numeral x OR neg_numeral (y + Num.One)) BIT 1"
- "numeral (Num.Bit1 x) OR neg_numeral (Num.Bit0 y) = (numeral x OR neg_numeral y) BIT 1"
- "numeral (Num.Bit1 x) OR neg_numeral (Num.Bit1 y) = (numeral x OR neg_numeral (y + Num.One)) BIT 1"
- "neg_numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (neg_numeral x OR numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (neg_numeral x OR numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) OR numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) OR numeral y) BIT 1"
- "neg_numeral (Num.Bit0 x) OR neg_numeral (Num.Bit0 y) = (neg_numeral x OR neg_numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) OR neg_numeral (Num.Bit1 y) = (neg_numeral x OR neg_numeral (y + Num.One)) BIT 1"
- "neg_numeral (Num.Bit1 x) OR neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) OR neg_numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) OR neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) OR neg_numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT 0"
+ "numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT 1"
+ "numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT 1"
+ "- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (- numeral x OR numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (- numeral x OR numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR numeral y) BIT 1"
+ "- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (- numeral x OR - numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (- numeral x OR - numeral (y + Num.One)) BIT 1"
+ "- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR - numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR - numeral (y + Num.One)) BIT 1"
"(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)"
- "(1::int) OR neg_numeral (Num.Bit0 y) = neg_numeral (Num.BitM y)"
- "(1::int) OR neg_numeral (Num.Bit1 y) = neg_numeral (Num.Bit1 y)"
+ "(1::int) OR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
+ "(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)"
"numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)"
"numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)"
- "neg_numeral (Num.Bit0 x) OR (1::int) = neg_numeral (Num.BitM x)"
- "neg_numeral (Num.Bit1 x) OR (1::int) = neg_numeral (Num.Bit1 x)"
+ "- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)"
+ "- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)"
by (rule bin_rl_eqI, simp, simp)+
lemma int_xor_numerals [simp]:
@@ -297,26 +297,26 @@
"numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 1"
"numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT 1"
"numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 0"
- "numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit0 y) = (numeral x XOR neg_numeral y) BIT 0"
- "numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit1 y) = (numeral x XOR neg_numeral (y + Num.One)) BIT 1"
- "numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit0 y) = (numeral x XOR neg_numeral y) BIT 1"
- "numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit1 y) = (numeral x XOR neg_numeral (y + Num.One)) BIT 0"
- "neg_numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (neg_numeral x XOR numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (neg_numeral x XOR numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) XOR numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) XOR numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit0 y) = (neg_numeral x XOR neg_numeral y) BIT 0"
- "neg_numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit1 y) = (neg_numeral x XOR neg_numeral (y + Num.One)) BIT 1"
- "neg_numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) XOR neg_numeral y) BIT 1"
- "neg_numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) XOR neg_numeral (y + Num.One)) BIT 0"
+ "numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT 0"
+ "numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT 1"
+ "numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT 0"
+ "- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (- numeral x XOR numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (- numeral x XOR numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (- numeral x XOR - numeral y) BIT 0"
+ "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (- numeral x XOR - numeral (y + Num.One)) BIT 1"
+ "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR - numeral y) BIT 1"
+ "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR - numeral (y + Num.One)) BIT 0"
"(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)"
- "(1::int) XOR neg_numeral (Num.Bit0 y) = neg_numeral (Num.BitM y)"
- "(1::int) XOR neg_numeral (Num.Bit1 y) = neg_numeral (Num.Bit0 (y + Num.One))"
+ "(1::int) XOR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
+ "(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))"
"numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)"
"numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)"
- "neg_numeral (Num.Bit0 x) XOR (1::int) = neg_numeral (Num.BitM x)"
- "neg_numeral (Num.Bit1 x) XOR (1::int) = neg_numeral (Num.Bit0 (x + Num.One))"
+ "- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)"
+ "- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))"
by (rule bin_rl_eqI, simp, simp)+
subsubsection {* Interactions with arithmetic *}
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Word/Bit_Representation.thy
--- a/src/HOL/Word/Bit_Representation.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Word/Bit_Representation.thy Tue Nov 19 17:07:52 2013 +0100
@@ -61,21 +61,23 @@
lemma BIT_bin_simps [simp]:
"numeral k BIT 0 = numeral (Num.Bit0 k)"
"numeral k BIT 1 = numeral (Num.Bit1 k)"
- "neg_numeral k BIT 0 = neg_numeral (Num.Bit0 k)"
- "neg_numeral k BIT 1 = neg_numeral (Num.BitM k)"
- unfolding neg_numeral_def numeral.simps numeral_BitM
+ "(- numeral k) BIT 0 = - numeral (Num.Bit0 k)"
+ "(- numeral k) BIT 1 = - numeral (Num.BitM k)"
+ unfolding numeral.simps numeral_BitM
unfolding Bit_def bitval_simps
by (simp_all del: arith_simps add_numeral_special diff_numeral_special)
lemma BIT_special_simps [simp]:
- shows "0 BIT 0 = 0" and "0 BIT 1 = 1" and "1 BIT 0 = 2" and "1 BIT 1 = 3"
+ shows "0 BIT 0 = 0" and "0 BIT 1 = 1"
+ and "1 BIT 0 = 2" and "1 BIT 1 = 3"
+ and "(- 1) BIT 0 = - 2" and "(- 1) BIT 1 = - 1"
unfolding Bit_def by simp_all
lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> b = 0"
by (subst BIT_eq_iff [symmetric], simp)
-lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b = 1"
- by (subst BIT_eq_iff [symmetric], simp)
+lemma Bit_eq_m1_iff: "w BIT b = - 1 \<longleftrightarrow> w = - 1 \<and> b = 1"
+ by (cases b) (auto simp add: Bit_def, arith)
lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
by (induct w, simp_all)
@@ -83,8 +85,8 @@
lemma expand_BIT:
"numeral (Num.Bit0 w) = numeral w BIT 0"
"numeral (Num.Bit1 w) = numeral w BIT 1"
- "neg_numeral (Num.Bit0 w) = neg_numeral w BIT 0"
- "neg_numeral (Num.Bit1 w) = neg_numeral (w + Num.One) BIT 1"
+ "- numeral (Num.Bit0 w) = - numeral w BIT 0"
+ "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT 1"
unfolding add_One by (simp_all add: BitM_inc)
lemma bin_last_numeral_simps [simp]:
@@ -94,9 +96,9 @@
"bin_last Numeral1 = 1"
"bin_last (numeral (Num.Bit0 w)) = 0"
"bin_last (numeral (Num.Bit1 w)) = 1"
- "bin_last (neg_numeral (Num.Bit0 w)) = 0"
- "bin_last (neg_numeral (Num.Bit1 w)) = 1"
- unfolding expand_BIT bin_last_BIT by (simp_all add: bin_last_def)
+ "bin_last (- numeral (Num.Bit0 w)) = 0"
+ "bin_last (- numeral (Num.Bit1 w)) = 1"
+ unfolding expand_BIT bin_last_BIT by (simp_all add: bin_last_def zmod_zminus1_eq_if)
lemma bin_rest_numeral_simps [simp]:
"bin_rest 0 = 0"
@@ -105,9 +107,9 @@
"bin_rest Numeral1 = 0"
"bin_rest (numeral (Num.Bit0 w)) = numeral w"
"bin_rest (numeral (Num.Bit1 w)) = numeral w"
- "bin_rest (neg_numeral (Num.Bit0 w)) = neg_numeral w"
- "bin_rest (neg_numeral (Num.Bit1 w)) = neg_numeral (w + Num.One)"
- unfolding expand_BIT bin_rest_BIT by (simp_all add: bin_rest_def)
+ "bin_rest (- numeral (Num.Bit0 w)) = - numeral w"
+ "bin_rest (- numeral (Num.Bit1 w)) = - numeral (w + Num.One)"
+ unfolding expand_BIT bin_rest_BIT by (simp_all add: bin_rest_def zdiv_zminus1_eq_if)
lemma less_Bits:
"(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"
@@ -197,42 +199,45 @@
lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
unfolding bin_rest_def [symmetric] by (rule bin_rest_BIT)
-lemma bin_nth_lem [rule_format]:
- "ALL y. bin_nth x = bin_nth y --> x = y"
- apply (induct x rule: bin_induct)
- apply safe
- apply (erule rev_mp)
- apply (induct_tac y rule: bin_induct)
+lemma bin_nth_eq_iff:
+ "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
+proof -
+ have bin_nth_lem [rule_format]: "ALL y. bin_nth x = bin_nth y --> x = y"
+ apply (induct x rule: bin_induct)
apply safe
- apply (drule_tac x=0 in fun_cong, force)
- apply (erule notE, rule ext,
+ apply (erule rev_mp)
+ apply (induct_tac y rule: bin_induct)
+ apply safe
+ apply (drule_tac x=0 in fun_cong, force)
+ apply (erule notE, rule ext,
drule_tac x="Suc x" in fun_cong, force)
- apply (drule_tac x=0 in fun_cong, force)
- apply (erule rev_mp)
- apply (induct_tac y rule: bin_induct)
- apply safe
+ apply (drule_tac x=0 in fun_cong, force)
+ apply (erule rev_mp)
+ apply (induct_tac y rule: bin_induct)
+ apply safe
+ apply (drule_tac x=0 in fun_cong, force)
+ apply (erule notE, rule ext,
+ drule_tac x="Suc x" in fun_cong, force)
+ apply (metis Bit_eq_m1_iff Z bin_last_BIT)
+ apply (case_tac y rule: bin_exhaust)
+ apply clarify
+ apply (erule allE)
+ apply (erule impE)
+ prefer 2
+ apply (erule conjI)
apply (drule_tac x=0 in fun_cong, force)
- apply (erule notE, rule ext,
- drule_tac x="Suc x" in fun_cong, force)
- apply (drule_tac x=0 in fun_cong, force)
- apply (case_tac y rule: bin_exhaust)
- apply clarify
- apply (erule allE)
- apply (erule impE)
- prefer 2
- apply (erule conjI)
- apply (drule_tac x=0 in fun_cong, force)
- apply (rule ext)
- apply (drule_tac x="Suc ?x" in fun_cong, force)
- done
-
-lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"
+ apply (rule ext)
+ apply (drule_tac x="Suc ?x" in fun_cong, force)
+ done
+ show ?thesis
by (auto elim: bin_nth_lem)
+qed
lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]]
-lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
- by (auto intro!: bin_nth_lem del: equalityI)
+lemma bin_eq_iff:
+ "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
+ using bin_nth_eq_iff by auto
lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"
by (induct n) auto
@@ -276,8 +281,9 @@
lemma bin_sign_simps [simp]:
"bin_sign 0 = 0"
"bin_sign 1 = 0"
+ "bin_sign (- 1) = - 1"
"bin_sign (numeral k) = 0"
- "bin_sign (neg_numeral k) = -1"
+ "bin_sign (- numeral k) = -1"
"bin_sign (w BIT b) = bin_sign w"
unfolding bin_sign_def Bit_def bitval_def
by (simp_all split: bit.split)
@@ -331,18 +337,18 @@
"bintrunc (Suc n) -1 = bintrunc n -1 BIT 1"
"bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT 0"
"bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT 1"
- "bintrunc (Suc n) (neg_numeral (Num.Bit0 w)) =
- bintrunc n (neg_numeral w) BIT 0"
- "bintrunc (Suc n) (neg_numeral (Num.Bit1 w)) =
- bintrunc n (neg_numeral (w + Num.One)) BIT 1"
+ "bintrunc (Suc n) (- numeral (Num.Bit0 w)) =
+ bintrunc n (- numeral w) BIT 0"
+ "bintrunc (Suc n) (- numeral (Num.Bit1 w)) =
+ bintrunc n (- numeral (w + Num.One)) BIT 1"
by simp_all
lemma sbintrunc_0_numeral [simp]:
"sbintrunc 0 1 = -1"
"sbintrunc 0 (numeral (Num.Bit0 w)) = 0"
"sbintrunc 0 (numeral (Num.Bit1 w)) = -1"
- "sbintrunc 0 (neg_numeral (Num.Bit0 w)) = 0"
- "sbintrunc 0 (neg_numeral (Num.Bit1 w)) = -1"
+ "sbintrunc 0 (- numeral (Num.Bit0 w)) = 0"
+ "sbintrunc 0 (- numeral (Num.Bit1 w)) = -1"
by simp_all
lemma sbintrunc_Suc_numeral:
@@ -351,10 +357,10 @@
sbintrunc n (numeral w) BIT 0"
"sbintrunc (Suc n) (numeral (Num.Bit1 w)) =
sbintrunc n (numeral w) BIT 1"
- "sbintrunc (Suc n) (neg_numeral (Num.Bit0 w)) =
- sbintrunc n (neg_numeral w) BIT 0"
- "sbintrunc (Suc n) (neg_numeral (Num.Bit1 w)) =
- sbintrunc n (neg_numeral (w + Num.One)) BIT 1"
+ "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) =
+ sbintrunc n (- numeral w) BIT 0"
+ "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) =
+ sbintrunc n (- numeral (w + Num.One)) BIT 1"
by simp_all
lemma bit_bool:
@@ -580,10 +586,10 @@
bintrunc (pred_numeral k) (numeral w) BIT 0"
"bintrunc (numeral k) (numeral (Num.Bit1 w)) =
bintrunc (pred_numeral k) (numeral w) BIT 1"
- "bintrunc (numeral k) (neg_numeral (Num.Bit0 w)) =
- bintrunc (pred_numeral k) (neg_numeral w) BIT 0"
- "bintrunc (numeral k) (neg_numeral (Num.Bit1 w)) =
- bintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1"
+ "bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
+ bintrunc (pred_numeral k) (- numeral w) BIT 0"
+ "bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
+ bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1"
"bintrunc (numeral k) 1 = 1"
by (simp_all add: bintrunc_numeral)
@@ -592,10 +598,10 @@
sbintrunc (pred_numeral k) (numeral w) BIT 0"
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
sbintrunc (pred_numeral k) (numeral w) BIT 1"
- "sbintrunc (numeral k) (neg_numeral (Num.Bit0 w)) =
- sbintrunc (pred_numeral k) (neg_numeral w) BIT 0"
- "sbintrunc (numeral k) (neg_numeral (Num.Bit1 w)) =
- sbintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1"
+ "sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
+ sbintrunc (pred_numeral k) (- numeral w) BIT 0"
+ "sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
+ sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1"
"sbintrunc (numeral k) 1 = 1"
by (simp_all add: sbintrunc_numeral)
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Word/Word.thy
--- a/src/HOL/Word/Word.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Word/Word.thy Tue Nov 19 17:07:52 2013 +0100
@@ -591,24 +591,24 @@
declare word_numeral_alt [symmetric, code_abbrev]
lemma word_neg_numeral_alt:
- "neg_numeral b = word_of_int (neg_numeral b)"
- by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg)
+ "- numeral b = word_of_int (- numeral b)"
+ by (simp only: word_numeral_alt wi_hom_neg)
declare word_neg_numeral_alt [symmetric, code_abbrev]
lemma word_numeral_transfer [transfer_rule]:
"(fun_rel op = pcr_word) numeral numeral"
- "(fun_rel op = pcr_word) neg_numeral neg_numeral"
- unfolding fun_rel_def word.pcr_cr_eq cr_word_def word_numeral_alt word_neg_numeral_alt
- by simp_all
+ "(fun_rel op = pcr_word) (- numeral) (- numeral)"
+ apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def)
+ using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+
lemma uint_bintrunc [simp]:
"uint (numeral bin :: 'a word) =
bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
unfolding word_numeral_alt by (rule word_ubin.eq_norm)
-lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) =
- bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)"
+lemma uint_bintrunc_neg [simp]: "uint (- numeral bin :: 'a word) =
+ bintrunc (len_of TYPE ('a :: len0)) (- numeral bin)"
by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
lemma sint_sbintrunc [simp]:
@@ -616,8 +616,8 @@
sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
by (simp only: word_numeral_alt word_sbin.eq_norm)
-lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) =
- sbintrunc (len_of TYPE ('a :: len) - 1) (neg_numeral bin)"
+lemma sint_sbintrunc_neg [simp]: "sint (- numeral bin :: 'a word) =
+ sbintrunc (len_of TYPE ('a :: len) - 1) (- numeral bin)"
by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
lemma unat_bintrunc [simp]:
@@ -626,8 +626,8 @@
by (simp only: unat_def uint_bintrunc)
lemma unat_bintrunc_neg [simp]:
- "unat (neg_numeral bin :: 'a :: len0 word) =
- nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))"
+ "unat (- numeral bin :: 'a :: len0 word) =
+ nat (bintrunc (len_of TYPE('a)) (- numeral bin))"
by (simp only: unat_def uint_bintrunc_neg)
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
@@ -678,7 +678,7 @@
by (simp only: int_word_uint)
lemma uint_neg_numeral:
- "uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)"
+ "uint (- numeral b :: 'a :: len0 word) = - numeral b mod 2 ^ len_of TYPE('a)"
unfolding word_neg_numeral_alt
by (simp only: int_word_uint)
@@ -702,13 +702,16 @@
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
unfolding word_1_wi ..
+lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
+ by (simp add: wi_hom_syms)
+
lemma word_of_int_numeral [simp] :
"(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
unfolding word_numeral_alt ..
lemma word_of_int_neg_numeral [simp]:
- "(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)"
- unfolding neg_numeral_def word_numeral_alt wi_hom_syms ..
+ "(word_of_int (- numeral bin) :: 'a :: len0 word) = (- numeral bin)"
+ unfolding word_numeral_alt wi_hom_syms ..
lemma word_int_case_wi:
"word_int_case f (word_of_int i :: 'b word) =
@@ -880,8 +883,8 @@
unfolding word_numeral_alt by (rule to_bl_of_bin)
lemma to_bl_neg_numeral [simp]:
- "to_bl (neg_numeral bin::'a::len0 word) =
- bin_to_bl (len_of TYPE('a)) (neg_numeral bin)"
+ "to_bl (- numeral bin::'a::len0 word) =
+ bin_to_bl (len_of TYPE('a)) (- numeral bin)"
unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
@@ -1156,11 +1159,8 @@
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
-lemma word_1_no: "(1::'a::len0 word) = Numeral1"
- by (simp add: word_numeral_alt)
-
-lemma word_m1_wi: "-1 = word_of_int -1"
- by (rule word_neg_numeral_alt)
+lemma word_m1_wi: "- 1 = word_of_int (- 1)"
+ using word_neg_numeral_alt [of Num.One] by simp
lemma word_0_bl [simp]: "of_bl [] = 0"
unfolding of_bl_def by simp
@@ -1215,9 +1215,9 @@
unfolding scast_def by simp
lemma sint_n1 [simp] : "sint -1 = -1"
- unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
-
-lemma scast_n1 [simp]: "scast -1 = -1"
+ unfolding word_m1_wi word_sbin.eq_norm by simp
+
+lemma scast_n1 [simp]: "scast (- 1) = - 1"
unfolding scast_def by simp
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
@@ -1270,8 +1270,8 @@
lemma succ_pred_no [simp]:
"word_succ (numeral w) = numeral w + 1"
"word_pred (numeral w) = numeral w - 1"
- "word_succ (neg_numeral w) = neg_numeral w + 1"
- "word_pred (neg_numeral w) = neg_numeral w - 1"
+ "word_succ (- numeral w) = - numeral w + 1"
+ "word_pred (- numeral w) = - numeral w - 1"
unfolding word_succ_p1 word_pred_m1 by simp_all
lemma word_sp_01 [simp] :
@@ -2151,19 +2151,19 @@
lemma word_no_log_defs [simp]:
"NOT (numeral a) = word_of_int (NOT (numeral a))"
- "NOT (neg_numeral a) = word_of_int (NOT (neg_numeral a))"
+ "NOT (- numeral a) = word_of_int (NOT (- numeral a))"
"numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
- "numeral a AND neg_numeral b = word_of_int (numeral a AND neg_numeral b)"
- "neg_numeral a AND numeral b = word_of_int (neg_numeral a AND numeral b)"
- "neg_numeral a AND neg_numeral b = word_of_int (neg_numeral a AND neg_numeral b)"
+ "numeral a AND - numeral b = word_of_int (numeral a AND - numeral b)"
+ "- numeral a AND numeral b = word_of_int (- numeral a AND numeral b)"
+ "- numeral a AND - numeral b = word_of_int (- numeral a AND - numeral b)"
"numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
- "numeral a OR neg_numeral b = word_of_int (numeral a OR neg_numeral b)"
- "neg_numeral a OR numeral b = word_of_int (neg_numeral a OR numeral b)"
- "neg_numeral a OR neg_numeral b = word_of_int (neg_numeral a OR neg_numeral b)"
+ "numeral a OR - numeral b = word_of_int (numeral a OR - numeral b)"
+ "- numeral a OR numeral b = word_of_int (- numeral a OR numeral b)"
+ "- numeral a OR - numeral b = word_of_int (- numeral a OR - numeral b)"
"numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
- "numeral a XOR neg_numeral b = word_of_int (numeral a XOR neg_numeral b)"
- "neg_numeral a XOR numeral b = word_of_int (neg_numeral a XOR numeral b)"
- "neg_numeral a XOR neg_numeral b = word_of_int (neg_numeral a XOR neg_numeral b)"
+ "numeral a XOR - numeral b = word_of_int (numeral a XOR - numeral b)"
+ "- numeral a XOR numeral b = word_of_int (- numeral a XOR numeral b)"
+ "- numeral a XOR - numeral b = word_of_int (- numeral a XOR - numeral b)"
by (transfer, rule refl)+
text {* Special cases for when one of the arguments equals 1. *}
@@ -2171,17 +2171,17 @@
lemma word_bitwise_1_simps [simp]:
"NOT (1::'a::len0 word) = -2"
"1 AND numeral b = word_of_int (1 AND numeral b)"
- "1 AND neg_numeral b = word_of_int (1 AND neg_numeral b)"
+ "1 AND - numeral b = word_of_int (1 AND - numeral b)"
"numeral a AND 1 = word_of_int (numeral a AND 1)"
- "neg_numeral a AND 1 = word_of_int (neg_numeral a AND 1)"
+ "- numeral a AND 1 = word_of_int (- numeral a AND 1)"
"1 OR numeral b = word_of_int (1 OR numeral b)"
- "1 OR neg_numeral b = word_of_int (1 OR neg_numeral b)"
+ "1 OR - numeral b = word_of_int (1 OR - numeral b)"
"numeral a OR 1 = word_of_int (numeral a OR 1)"
- "neg_numeral a OR 1 = word_of_int (neg_numeral a OR 1)"
+ "- numeral a OR 1 = word_of_int (- numeral a OR 1)"
"1 XOR numeral b = word_of_int (1 XOR numeral b)"
- "1 XOR neg_numeral b = word_of_int (1 XOR neg_numeral b)"
+ "1 XOR - numeral b = word_of_int (1 XOR - numeral b)"
"numeral a XOR 1 = word_of_int (numeral a XOR 1)"
- "neg_numeral a XOR 1 = word_of_int (neg_numeral a XOR 1)"
+ "- numeral a XOR 1 = word_of_int (- numeral a XOR 1)"
by (transfer, simp)+
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
@@ -2220,8 +2220,8 @@
by transfer (rule refl)
lemma test_bit_neg_numeral [simp]:
- "(neg_numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
- n < len_of TYPE('a) \<and> bin_nth (neg_numeral w) n"
+ "(- numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
+ n < len_of TYPE('a) \<and> bin_nth (- numeral w) n"
by transfer (rule refl)
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
@@ -2398,7 +2398,7 @@
unfolding word_numeral_alt by (rule msb_word_of_int)
lemma word_msb_neg_numeral [simp]:
- "msb (neg_numeral w::'a::len word) = bin_nth (neg_numeral w) (len_of TYPE('a) - 1)"
+ "msb (- numeral w::'a::len word) = bin_nth (- numeral w) (len_of TYPE('a) - 1)"
unfolding word_neg_numeral_alt by (rule msb_word_of_int)
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
@@ -2528,7 +2528,7 @@
unfolding word_lsb_alt test_bit_numeral by simp
lemma word_lsb_neg_numeral [simp]:
- "lsb (neg_numeral bin :: 'a :: len word) = (bin_last (neg_numeral bin) = 1)"
+ "lsb (- numeral bin :: 'a :: len word) = (bin_last (- numeral bin) = 1)"
unfolding word_lsb_alt test_bit_neg_numeral by simp
lemma set_bit_word_of_int:
@@ -2544,8 +2544,8 @@
unfolding word_numeral_alt by (rule set_bit_word_of_int)
lemma word_set_neg_numeral [simp]:
- "set_bit (neg_numeral bin::'a::len0 word) n b =
- word_of_int (bin_sc n (if b then 1 else 0) (neg_numeral bin))"
+ "set_bit (- numeral bin::'a::len0 word) n b =
+ word_of_int (bin_sc n (if b then 1 else 0) (- numeral bin))"
unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
lemma word_set_bit_0 [simp]:
@@ -2612,8 +2612,14 @@
apply clarsimp
apply clarsimp
apply (drule word_gt_0 [THEN iffD1])
- apply (safe intro!: word_eqI bin_nth_lem)
- apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric])
+ apply (safe intro!: word_eqI)
+ apply (auto simp add: nth_2p_bin)
+ apply (erule notE)
+ apply (simp (no_asm_use) add: uint_word_of_int word_size)
+ apply (subst mod_pos_pos_trivial)
+ apply simp
+ apply (rule power_strict_increasing)
+ apply simp_all
done
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n"
@@ -2670,7 +2676,7 @@
unfolding word_numeral_alt shiftl1_wi by simp
lemma shiftl1_neg_numeral [simp]:
- "shiftl1 (neg_numeral w) = neg_numeral (Num.Bit0 w)"
+ "shiftl1 (- numeral w) = - numeral (Num.Bit0 w)"
unfolding word_neg_numeral_alt shiftl1_wi by simp
lemma shiftl1_0 [simp] : "shiftl1 0 = 0"
@@ -4638,9 +4644,6 @@
"1 + n \<noteq> (0::'a::len word) \<Longrightarrow> unat (1 + n) = Suc (unat n)"
by unat_arith
-lemma word_no_1 [simp]: "(Numeral1::'a::len0 word) = 1"
- by (fact word_1_no [symmetric])
-
declare bin_to_bl_def [simp]
ML_file "Tools/word_lib.ML"
diff -r c76dec4df4d7 -r f7fef6b00bfe src/HOL/Word/WordBitwise.thy
--- a/src/HOL/Word/WordBitwise.thy Mon Nov 18 17:15:01 2013 +0100
+++ b/src/HOL/Word/WordBitwise.thy Tue Nov 19 17:07:52 2013 +0100
@@ -461,18 +461,18 @@
= True # rev (bin_to_bl n (numeral nm))"
"rev (bin_to_bl (Suc n) (numeral (num.One)))
= True # replicate n False"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.Bit0 nm)))
- = False # rev (bin_to_bl n (neg_numeral nm))"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.Bit1 nm)))
- = True # rev (bin_to_bl n (neg_numeral (nm + num.One)))"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.One)))
+ "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm)))
+ = False # rev (bin_to_bl n (- numeral nm))"
+ "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm)))
+ = True # rev (bin_to_bl n (- numeral (nm + num.One)))"
+ "rev (bin_to_bl (Suc n) (- numeral (num.One)))
= True # replicate n True"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.Bit0 nm + num.One)))
- = True # rev (bin_to_bl n (neg_numeral (nm + num.One)))"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.Bit1 nm + num.One)))
- = False # rev (bin_to_bl n (neg_numeral (nm + num.One)))"
- "rev (bin_to_bl (Suc n) (neg_numeral (num.One + num.One)))
- = False # rev (bin_to_bl n (neg_numeral num.One))"
+ "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm + num.One)))
+ = True # rev (bin_to_bl n (- numeral (nm + num.One)))"
+ "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm + num.One)))
+ = False # rev (bin_to_bl n (- numeral (nm + num.One)))"
+ "rev (bin_to_bl (Suc n) (- numeral (num.One + num.One)))
+ = False # rev (bin_to_bl n (- numeral num.One))"
apply (simp_all add: bin_to_bl_def)
apply (simp_all only: bin_to_bl_aux_alt)
apply (simp_all)