--- a/src/ZF/Perm.thy Sun Mar 04 23:20:43 2012 +0100
+++ b/src/ZF/Perm.thy Tue Mar 06 15:15:49 2012 +0000
@@ -15,28 +15,28 @@
definition
(*composition of relations and functions; NOT Suppes's relative product*)
comp :: "[i,i]=>i" (infixr "O" 60) where
- "r O s == {xz : domain(s)*range(r) .
- EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r}"
+ "r O s == {xz \<in> domain(s)*range(r) .
+ \<exists>x y z. xz=<x,z> & <x,y>:s & <y,z>:r}"
definition
(*the identity function for A*)
id :: "i=>i" where
- "id(A) == (lam x:A. x)"
+ "id(A) == (\<lambda>x\<in>A. x)"
definition
(*one-to-one functions from A to B*)
inj :: "[i,i]=>i" where
- "inj(A,B) == { f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x}"
+ "inj(A,B) == { f: A->B. \<forall>w\<in>A. \<forall>x\<in>A. f`w=f`x \<longrightarrow> w=x}"
definition
(*onto functions from A to B*)
surj :: "[i,i]=>i" where
- "surj(A,B) == { f: A->B . ALL y:B. EX x:A. f`x=y}"
+ "surj(A,B) == { f: A->B . \<forall>y\<in>B. \<exists>x\<in>A. f`x=y}"
definition
(*one-to-one and onto functions*)
bij :: "[i,i]=>i" where
- "bij(A,B) == inj(A,B) Int surj(A,B)"
+ "bij(A,B) == inj(A,B) \<inter> surj(A,B)"
subsection{*Surjective Function Space*}
@@ -46,7 +46,7 @@
apply (erule CollectD1)
done
-lemma fun_is_surj: "f : Pi(A,B) ==> f: surj(A,range(f))"
+lemma fun_is_surj: "f \<in> Pi(A,B) ==> f: surj(A,range(f))"
apply (unfold surj_def)
apply (blast intro: apply_equality range_of_fun domain_type)
done
@@ -67,13 +67,13 @@
"[| !!x. x:A ==> c(x): B;
!!y. y:B ==> d(y): A;
!!y. y:B ==> c(d(y)) = y
- |] ==> (lam x:A. c(x)) : surj(A,B)"
+ |] ==> (\<lambda>x\<in>A. c(x)) \<in> surj(A,B)"
apply (rule_tac d = d in f_imp_surjective)
apply (simp_all add: lam_type)
done
text{*Cantor's theorem revisited*}
-lemma cantor_surj: "f ~: surj(A,Pow(A))"
+lemma cantor_surj: "f \<notin> surj(A,Pow(A))"
apply (unfold surj_def, safe)
apply (cut_tac cantor)
apply (best del: subsetI)
@@ -99,7 +99,7 @@
text{* A function with a left inverse is an injection *}
-lemma f_imp_injective: "[| f: A->B; ALL x:A. d(f`x)=x |] ==> f: inj(A,B)"
+lemma f_imp_injective: "[| f: A->B; \<forall>x\<in>A. d(f`x)=x |] ==> f: inj(A,B)"
apply (simp (no_asm_simp) add: inj_def)
apply (blast intro: subst_context [THEN box_equals])
done
@@ -107,7 +107,7 @@
lemma lam_injective:
"[| !!x. x:A ==> c(x): B;
!!x. x:A ==> d(c(x)) = x |]
- ==> (lam x:A. c(x)) : inj(A,B)"
+ ==> (\<lambda>x\<in>A. c(x)) \<in> inj(A,B)"
apply (rule_tac d = d in f_imp_injective)
apply (simp_all add: lam_type)
done
@@ -132,13 +132,13 @@
!!y. y:B ==> d(y): A;
!!x. x:A ==> d(c(x)) = x;
!!y. y:B ==> c(d(y)) = y
- |] ==> (lam x:A. c(x)) : bij(A,B)"
+ |] ==> (\<lambda>x\<in>A. c(x)) \<in> bij(A,B)"
apply (unfold bij_def)
apply (blast intro!: lam_injective lam_surjective)
done
-lemma RepFun_bijective: "(ALL y : x. EX! y'. f(y') = f(y))
- ==> (lam z:{f(y). y:x}. THE y. f(y) = z) : bij({f(y). y:x}, x)"
+lemma RepFun_bijective: "(\<forall>y\<in>x. EX! y'. f(y') = f(y))
+ ==> (\<lambda>z\<in>{f(y). y:x}. THE y. f(y) = z) \<in> bij({f(y). y:x}, x)"
apply (rule_tac d = f in lam_bijective)
apply (auto simp add: the_equality2)
done
@@ -146,7 +146,7 @@
subsection{*Identity Function*}
-lemma idI [intro!]: "a:A ==> <a,a> : id(A)"
+lemma idI [intro!]: "a:A ==> <a,a> \<in> id(A)"
apply (unfold id_def)
apply (erule lamI)
done
@@ -154,7 +154,7 @@
lemma idE [elim!]: "[| p: id(A); !!x.[| x:A; p=<x,x> |] ==> P |] ==> P"
by (simp add: id_def lam_def, blast)
-lemma id_type: "id(A) : A->A"
+lemma id_type: "id(A) \<in> A->A"
apply (unfold id_def)
apply (rule lam_type, assumption)
done
@@ -164,7 +164,7 @@
apply (simp (no_asm_simp))
done
-lemma id_mono: "A<=B ==> id(A) <= id(B)"
+lemma id_mono: "A<=B ==> id(A) \<subseteq> id(B)"
apply (unfold id_def)
apply (erule lam_mono)
done
@@ -186,7 +186,7 @@
apply (blast intro: id_inj id_surj)
done
-lemma subset_iff_id: "A <= B <-> id(A) : A->B"
+lemma subset_iff_id: "A \<subseteq> B <-> id(A) \<in> A->B"
apply (unfold id_def)
apply (force intro!: lam_type dest: apply_type)
done
@@ -198,7 +198,7 @@
subsection{*Converse of a Function*}
-lemma inj_converse_fun: "f: inj(A,B) ==> converse(f) : range(f)->A"
+lemma inj_converse_fun: "f: inj(A,B) ==> converse(f) \<in> range(f)->A"
apply (unfold inj_def)
apply (simp (no_asm_simp) add: Pi_iff function_def)
apply (erule CollectE)
@@ -259,19 +259,19 @@
subsection{*Composition of Two Relations*}
-text{*The inductive definition package could derive these theorems for @term{r O s}*}
+text{*The inductive definition package could derive these theorems for @{term"r O s"}*}
-lemma compI [intro]: "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s"
+lemma compI [intro]: "[| <a,b>:s; <b,c>:r |] ==> <a,c> \<in> r O s"
by (unfold comp_def, blast)
lemma compE [elim!]:
- "[| xz : r O s;
+ "[| xz \<in> r O s;
!!x y z. [| xz=<x,z>; <x,y>:s; <y,z>:r |] ==> P |]
==> P"
by (unfold comp_def, blast)
lemma compEpair:
- "[| <a,c> : r O s;
+ "[| <a,c> \<in> r O s;
!!y. [| <a,y>:s; <y,c>:r |] ==> P |]
==> P"
by (erule compE, simp)
@@ -283,35 +283,35 @@
subsection{*Domain and Range -- see Suppes, Section 3.1*}
text{*Boyer et al., Set Theory in First-Order Logic, JAR 2 (1986), 287-327*}
-lemma range_comp: "range(r O s) <= range(r)"
+lemma range_comp: "range(r O s) \<subseteq> range(r)"
by blast
-lemma range_comp_eq: "domain(r) <= range(s) ==> range(r O s) = range(r)"
+lemma range_comp_eq: "domain(r) \<subseteq> range(s) ==> range(r O s) = range(r)"
by (rule range_comp [THEN equalityI], blast)
-lemma domain_comp: "domain(r O s) <= domain(s)"
+lemma domain_comp: "domain(r O s) \<subseteq> domain(s)"
by blast
-lemma domain_comp_eq: "range(s) <= domain(r) ==> domain(r O s) = domain(s)"
+lemma domain_comp_eq: "range(s) \<subseteq> domain(r) ==> domain(r O s) = domain(s)"
by (rule domain_comp [THEN equalityI], blast)
lemma image_comp: "(r O s)``A = r``(s``A)"
by blast
-lemma inj_inj_range: "f: inj(A,B) ==> f : inj(A,range(f))"
+lemma inj_inj_range: "f: inj(A,B) ==> f \<in> inj(A,range(f))"
by (auto simp add: inj_def Pi_iff function_def)
-lemma inj_bij_range: "f: inj(A,B) ==> f : bij(A,range(f))"
+lemma inj_bij_range: "f: inj(A,B) ==> f \<in> bij(A,range(f))"
by (auto simp add: bij_def intro: inj_inj_range inj_is_fun fun_is_surj)
subsection{*Other Results*}
-lemma comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"
+lemma comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') \<subseteq> (r O s)"
by blast
text{*composition preserves relations*}
-lemma comp_rel: "[| s<=A*B; r<=B*C |] ==> (r O s) <= A*C"
+lemma comp_rel: "[| s<=A*B; r<=B*C |] ==> (r O s) \<subseteq> A*C"
by blast
text{*associative law for composition*}
@@ -319,14 +319,14 @@
by blast
(*left identity of composition; provable inclusions are
- id(A) O r <= r
- and [| r<=A*B; B<=C |] ==> r <= id(C) O r *)
+ id(A) O r \<subseteq> r
+ and [| r<=A*B; B<=C |] ==> r \<subseteq> id(C) O r *)
lemma left_comp_id: "r<=A*B ==> id(B) O r = r"
by blast
(*right identity of composition; provable inclusions are
- r O id(A) <= r
- and [| r<=A*B; A<=C |] ==> r <= r O id(C) *)
+ r O id(A) \<subseteq> r
+ and [| r<=A*B; A<=C |] ==> r \<subseteq> r O id(C) *)
lemma right_comp_id: "r<=A*B ==> r O id(A) = r"
by blast
@@ -337,7 +337,7 @@
by (unfold function_def, blast)
text{*Don't think the premises can be weakened much*}
-lemma comp_fun: "[| g: A->B; f: B->C |] ==> (f O g) : A->C"
+lemma comp_fun: "[| g: A->B; f: B->C |] ==> (f O g) \<in> A->C"
apply (auto simp add: Pi_def comp_function Pow_iff comp_rel)
apply (subst range_rel_subset [THEN domain_comp_eq], auto)
done
@@ -353,8 +353,8 @@
text{*Simplifies compositions of lambda-abstractions*}
lemma comp_lam:
"[| !!x. x:A ==> b(x): B |]
- ==> (lam y:B. c(y)) O (lam x:A. b(x)) = (lam x:A. c(b(x)))"
-apply (subgoal_tac "(lam x:A. b(x)) : A -> B")
+ ==> (\<lambda>y\<in>B. c(y)) O (\<lambda>x\<in>A. b(x)) = (\<lambda>x\<in>A. c(b(x)))"
+apply (subgoal_tac "(\<lambda>x\<in>A. b(x)) \<in> A -> B")
apply (rule fun_extension)
apply (blast intro: comp_fun lam_funtype)
apply (rule lam_funtype)
@@ -363,7 +363,7 @@
done
lemma comp_inj:
- "[| g: inj(A,B); f: inj(B,C) |] ==> (f O g) : inj(A,C)"
+ "[| g: inj(A,B); f: inj(B,C) |] ==> (f O g) \<in> inj(A,C)"
apply (frule inj_is_fun [of g])
apply (frule inj_is_fun [of f])
apply (rule_tac d = "%y. converse (g) ` (converse (f) ` y)" in f_imp_injective)
@@ -371,13 +371,13 @@
done
lemma comp_surj:
- "[| g: surj(A,B); f: surj(B,C) |] ==> (f O g) : surj(A,C)"
+ "[| g: surj(A,B); f: surj(B,C) |] ==> (f O g) \<in> surj(A,C)"
apply (unfold surj_def)
apply (blast intro!: comp_fun comp_fun_apply)
done
lemma comp_bij:
- "[| g: bij(A,B); f: bij(B,C) |] ==> (f O g) : bij(A,C)"
+ "[| g: bij(A,B); f: bij(B,C) |] ==> (f O g) \<in> bij(A,C)"
apply (unfold bij_def)
apply (blast intro: comp_inj comp_surj)
done
@@ -420,7 +420,7 @@
subsubsection{*Inverses of Composition*}
text{*left inverse of composition; one inclusion is
- @term{f: A->B ==> id(A) <= converse(f) O f} *}
+ @{term "f: A->B ==> id(A) \<subseteq> converse(f) O f"} *}
lemma left_comp_inverse: "f: inj(A,B) ==> converse(f) O f = id(A)"
apply (unfold inj_def, clarify)
apply (rule equalityI)
@@ -428,7 +428,7 @@
done
text{*right inverse of composition; one inclusion is
- @term{f: A->B ==> f O converse(f) <= id(B)} *}
+ @{term "f: A->B ==> f O converse(f) \<subseteq> id(B)"} *}
lemma right_comp_inverse:
"f: surj(A,B) ==> f O converse(f) = id(B)"
apply (simp add: surj_def, clarify)
@@ -441,7 +441,7 @@
subsubsection{*Proving that a Function is a Bijection*}
lemma comp_eq_id_iff:
- "[| f: A->B; g: B->A |] ==> f O g = id(B) <-> (ALL y:B. f`(g`y)=y)"
+ "[| f: A->B; g: B->A |] ==> f O g = id(B) <-> (\<forall>y\<in>B. f`(g`y)=y)"
apply (unfold id_def, safe)
apply (drule_tac t = "%h. h`y " in subst_context)
apply simp
@@ -451,17 +451,17 @@
done
lemma fg_imp_bijective:
- "[| f: A->B; g: B->A; f O g = id(B); g O f = id(A) |] ==> f : bij(A,B)"
+ "[| f: A->B; g: B->A; f O g = id(B); g O f = id(A) |] ==> f \<in> bij(A,B)"
apply (unfold bij_def)
apply (simp add: comp_eq_id_iff)
apply (blast intro: f_imp_injective f_imp_surjective apply_funtype)
done
-lemma nilpotent_imp_bijective: "[| f: A->A; f O f = id(A) |] ==> f : bij(A,A)"
+lemma nilpotent_imp_bijective: "[| f: A->A; f O f = id(A) |] ==> f \<in> bij(A,A)"
by (blast intro: fg_imp_bijective)
lemma invertible_imp_bijective:
- "[| converse(f): B->A; f: A->B |] ==> f : bij(A,B)"
+ "[| converse(f): B->A; f: A->B |] ==> f \<in> bij(A,B)"
by (simp add: fg_imp_bijective comp_eq_id_iff
left_inverse_lemma right_inverse_lemma)
@@ -471,16 +471,16 @@
text{*Theorem by KG, proof by LCP*}
lemma inj_disjoint_Un:
- "[| f: inj(A,B); g: inj(C,D); B Int D = 0 |]
- ==> (lam a: A Un C. if a:A then f`a else g`a) : inj(A Un C, B Un D)"
+ "[| f: inj(A,B); g: inj(C,D); B \<inter> D = 0 |]
+ ==> (\<lambda>a\<in>A \<union> C. if a:A then f`a else g`a) \<in> inj(A \<union> C, B \<union> D)"
apply (rule_tac d = "%z. if z:B then converse (f) `z else converse (g) `z"
in lam_injective)
apply (auto simp add: inj_is_fun [THEN apply_type])
done
lemma surj_disjoint_Un:
- "[| f: surj(A,B); g: surj(C,D); A Int C = 0 |]
- ==> (f Un g) : surj(A Un C, B Un D)"
+ "[| f: surj(A,B); g: surj(C,D); A \<inter> C = 0 |]
+ ==> (f \<union> g) \<in> surj(A \<union> C, B \<union> D)"
apply (simp add: surj_def fun_disjoint_Un)
apply (blast dest!: domain_of_fun
intro!: fun_disjoint_apply1 fun_disjoint_apply2)
@@ -489,8 +489,8 @@
text{*A simple, high-level proof; the version for injections follows from it,
using @term{f:inj(A,B) <-> f:bij(A,range(f))} *}
lemma bij_disjoint_Un:
- "[| f: bij(A,B); g: bij(C,D); A Int C = 0; B Int D = 0 |]
- ==> (f Un g) : bij(A Un C, B Un D)"
+ "[| f: bij(A,B); g: bij(C,D); A \<inter> C = 0; B \<inter> D = 0 |]
+ ==> (f \<union> g) \<in> bij(A \<union> C, B \<union> D)"
apply (rule invertible_imp_bijective)
apply (subst converse_Un)
apply (auto intro: fun_disjoint_Un bij_is_fun bij_converse_bij)
@@ -505,7 +505,7 @@
apply (blast intro: apply_equality apply_Pair Pi_type)
done
-lemma restrict_image [simp]: "restrict(f,A) `` B = f `` (A Int B)"
+lemma restrict_image [simp]: "restrict(f,A) `` B = f `` (A \<inter> B)"
by (auto simp add: restrict_def)
lemma restrict_inj:
@@ -534,7 +534,7 @@
done
lemma inj_succ_restrict:
- "[| f: inj(succ(m), A) |] ==> restrict(f,m) : inj(m, A-{f`m})"
+ "[| f: inj(succ(m), A) |] ==> restrict(f,m) \<in> inj(m, A-{f`m})"
apply (rule restrict_bij [THEN bij_is_inj, THEN inj_weaken_type], assumption, blast)
apply (unfold inj_def)
apply (fast elim: range_type mem_irrefl dest: apply_equality)
@@ -542,8 +542,8 @@
lemma inj_extend:
- "[| f: inj(A,B); a~:A; b~:B |]
- ==> cons(<a,b>,f) : inj(cons(a,A), cons(b,B))"
+ "[| f: inj(A,B); a\<notin>A; b\<notin>B |]
+ ==> cons(<a,b>,f) \<in> inj(cons(a,A), cons(b,B))"
apply (unfold inj_def)
apply (force intro: apply_type simp add: fun_extend)
done