--- a/src/ZF/WF.thy Sun Mar 04 23:20:43 2012 +0100
+++ b/src/ZF/WF.thy Tue Mar 06 15:15:49 2012 +0000
@@ -21,16 +21,16 @@
definition
wf :: "i=>o" where
(*r is a well-founded relation*)
- "wf(r) == ALL Z. Z=0 | (EX x:Z. ALL y. <y,x>:r --> ~ y:Z)"
+ "wf(r) == \<forall>Z. Z=0 | (\<exists>x\<in>Z. \<forall>y. <y,x>:r \<longrightarrow> ~ y:Z)"
definition
wf_on :: "[i,i]=>o" ("wf[_]'(_')") where
(*r is well-founded on A*)
- "wf_on(A,r) == wf(r Int A*A)"
+ "wf_on(A,r) == wf(r \<inter> A*A)"
definition
is_recfun :: "[i, i, [i,i]=>i, i] =>o" where
- "is_recfun(r,a,H,f) == (f = (lam x: r-``{a}. H(x, restrict(f, r-``{x}))))"
+ "is_recfun(r,a,H,f) == (f = (\<lambda>x\<in>r-``{a}. H(x, restrict(f, r-``{x}))))"
definition
the_recfun :: "[i, i, [i,i]=>i] =>i" where
@@ -47,7 +47,7 @@
definition
wfrec_on :: "[i, i, i, [i,i]=>i] =>i" ("wfrec[_]'(_,_,_')") where
- "wfrec[A](r,a,H) == wfrec(r Int A*A, a, H)"
+ "wfrec[A](r,a,H) == wfrec(r \<inter> A*A, a, H)"
subsection{*Well-Founded Relations*}
@@ -55,9 +55,9 @@
subsubsection{*Equivalences between @{term wf} and @{term wf_on}*}
lemma wf_imp_wf_on: "wf(r) ==> wf[A](r)"
-by (unfold wf_def wf_on_def, force)
+by (unfold wf_def wf_on_def, force)
-lemma wf_on_imp_wf: "[|wf[A](r); r <= A*A|] ==> wf(r)";
+lemma wf_on_imp_wf: "[|wf[A](r); r \<subseteq> A*A|] ==> wf(r)";
by (simp add: wf_on_def subset_Int_iff)
lemma wf_on_field_imp_wf: "wf[field(r)](r) ==> wf(r)"
@@ -80,7 +80,7 @@
text{*If every non-empty subset of @{term A} has an @{term r}-minimal element
then we have @{term "wf[A](r)"}.*}
lemma wf_onI:
- assumes prem: "!!Z u. [| Z<=A; u:Z; ALL x:Z. EX y:Z. <y,x>:r |] ==> False"
+ assumes prem: "!!Z u. [| Z<=A; u:Z; \<forall>x\<in>Z. \<exists>y\<in>Z. <y,x>:r |] ==> False"
shows "wf[A](r)"
apply (unfold wf_on_def wf_def)
apply (rule equals0I [THEN disjCI, THEN allI])
@@ -89,14 +89,14 @@
text{*If @{term r} allows well-founded induction over @{term A}
then we have @{term "wf[A](r)"}. Premise is equivalent to
- @{prop "!!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B"} *}
+ @{prop "!!B. \<forall>x\<in>A. (\<forall>y. <y,x>: r \<longrightarrow> y:B) \<longrightarrow> x:B ==> A<=B"} *}
lemma wf_onI2:
- assumes prem: "!!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A |]
+ assumes prem: "!!y B. [| \<forall>x\<in>A. (\<forall>y\<in>A. <y,x>:r \<longrightarrow> y:B) \<longrightarrow> x:B; y:A |]
==> y:B"
shows "wf[A](r)"
apply (rule wf_onI)
apply (rule_tac c=u in prem [THEN DiffE])
- prefer 3 apply blast
+ prefer 3 apply blast
apply fast+
done
@@ -107,11 +107,11 @@
@{term "P(z)"} does not hold...*}
lemma wf_induct [induct set: wf]:
"[| wf(r);
- !!x.[| ALL y. <y,x>: r --> P(y) |] ==> P(x) |]
+ !!x.[| \<forall>y. <y,x>: r \<longrightarrow> P(y) |] ==> P(x) |]
==> P(a)"
-apply (unfold wf_def)
-apply (erule_tac x = "{z : domain(r). ~ P(z)}" in allE)
-apply blast
+apply (unfold wf_def)
+apply (erule_tac x = "{z \<in> domain(r). ~ P(z)}" in allE)
+apply blast
done
lemmas wf_induct_rule = wf_induct [rule_format, induct set: wf]
@@ -119,20 +119,20 @@
text{*The form of this rule is designed to match @{text wfI}*}
lemma wf_induct2:
"[| wf(r); a:A; field(r)<=A;
- !!x.[| x: A; ALL y. <y,x>: r --> P(y) |] ==> P(x) |]
+ !!x.[| x: A; \<forall>y. <y,x>: r \<longrightarrow> P(y) |] ==> P(x) |]
==> P(a)"
apply (erule_tac P="a:A" in rev_mp)
-apply (erule_tac a=a in wf_induct, blast)
+apply (erule_tac a=a in wf_induct, blast)
done
-lemma field_Int_square: "field(r Int A*A) <= A"
+lemma field_Int_square: "field(r \<inter> A*A) \<subseteq> A"
by blast
lemma wf_on_induct [consumes 2, induct set: wf_on]:
"[| wf[A](r); a:A;
- !!x.[| x: A; ALL y:A. <y,x>: r --> P(y) |] ==> P(x)
+ !!x.[| x: A; \<forall>y\<in>A. <y,x>: r \<longrightarrow> P(y) |] ==> P(x)
|] ==> P(a)"
-apply (unfold wf_on_def)
+apply (unfold wf_on_def)
apply (erule wf_induct2, assumption)
apply (rule field_Int_square, blast)
done
@@ -141,71 +141,71 @@
wf_on_induct [rule_format, consumes 2, induct set: wf_on]
-text{*If @{term r} allows well-founded induction
+text{*If @{term r} allows well-founded induction
then we have @{term "wf(r)"}.*}
lemma wfI:
"[| field(r)<=A;
- !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A|]
+ !!y B. [| \<forall>x\<in>A. (\<forall>y\<in>A. <y,x>:r \<longrightarrow> y:B) \<longrightarrow> x:B; y:A|]
==> y:B |]
==> wf(r)"
apply (rule wf_on_subset_A [THEN wf_on_field_imp_wf])
apply (rule wf_onI2)
- prefer 2 apply blast
-apply blast
+ prefer 2 apply blast
+apply blast
done
subsection{*Basic Properties of Well-Founded Relations*}
-lemma wf_not_refl: "wf(r) ==> <a,a> ~: r"
+lemma wf_not_refl: "wf(r) ==> <a,a> \<notin> r"
by (erule_tac a=a in wf_induct, blast)
-lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. <a,x>:r --> <x,a> ~: r"
+lemma wf_not_sym [rule_format]: "wf(r) ==> \<forall>x. <a,x>:r \<longrightarrow> <x,a> \<notin> r"
by (erule_tac a=a in wf_induct, blast)
-(* [| wf(r); <a,x> : r; ~P ==> <x,a> : r |] ==> P *)
+(* @{term"[| wf(r); <a,x> \<in> r; ~P ==> <x,a> \<in> r |] ==> P"} *)
lemmas wf_asym = wf_not_sym [THEN swap]
-lemma wf_on_not_refl: "[| wf[A](r); a: A |] ==> <a,a> ~: r"
+lemma wf_on_not_refl: "[| wf[A](r); a: A |] ==> <a,a> \<notin> r"
by (erule_tac a=a in wf_on_induct, assumption, blast)
lemma wf_on_not_sym [rule_format]:
- "[| wf[A](r); a:A |] ==> ALL b:A. <a,b>:r --> <b,a>~:r"
+ "[| wf[A](r); a:A |] ==> \<forall>b\<in>A. <a,b>:r \<longrightarrow> <b,a>\<notin>r"
apply (erule_tac a=a in wf_on_induct, assumption, blast)
done
lemma wf_on_asym:
- "[| wf[A](r); ~Z ==> <a,b> : r;
- <b,a> ~: r ==> Z; ~Z ==> a : A; ~Z ==> b : A |] ==> Z"
-by (blast dest: wf_on_not_sym)
+ "[| wf[A](r); ~Z ==> <a,b> \<in> r;
+ <b,a> \<notin> r ==> Z; ~Z ==> a \<in> A; ~Z ==> b \<in> A |] ==> Z"
+by (blast dest: wf_on_not_sym)
(*Needed to prove well_ordI. Could also reason that wf[A](r) means
- wf(r Int A*A); thus wf( (r Int A*A)^+ ) and use wf_not_refl *)
+ wf(r \<inter> A*A); thus wf( (r \<inter> A*A)^+ ) and use wf_not_refl *)
lemma wf_on_chain3:
"[| wf[A](r); <a,b>:r; <b,c>:r; <c,a>:r; a:A; b:A; c:A |] ==> P"
-apply (subgoal_tac "ALL y:A. ALL z:A. <a,y>:r --> <y,z>:r --> <z,a>:r --> P",
- blast)
+apply (subgoal_tac "\<forall>y\<in>A. \<forall>z\<in>A. <a,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <z,a>:r \<longrightarrow> P",
+ blast)
apply (erule_tac a=a in wf_on_induct, assumption, blast)
done
-text{*transitive closure of a WF relation is WF provided
+text{*transitive closure of a WF relation is WF provided
@{term A} is downward closed*}
lemma wf_on_trancl:
- "[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)"
+ "[| wf[A](r); r-``A \<subseteq> A |] ==> wf[A](r^+)"
apply (rule wf_onI2)
apply (frule bspec [THEN mp], assumption+)
apply (erule_tac a = y in wf_on_induct, assumption)
-apply (blast elim: tranclE, blast)
+apply (blast elim: tranclE, blast)
done
lemma wf_trancl: "wf(r) ==> wf(r^+)"
apply (simp add: wf_iff_wf_on_field)
-apply (rule wf_on_subset_A)
+apply (rule wf_on_subset_A)
apply (erule wf_on_trancl)
- apply blast
+ apply blast
apply (rule trancl_type [THEN field_rel_subset])
done
@@ -228,15 +228,15 @@
lemma apply_recfun:
"[| is_recfun(r,a,H,f); <x,a>:r |] ==> f`x = H(x, restrict(f,r-``{x}))"
-apply (unfold is_recfun_def)
+apply (unfold is_recfun_def)
txt{*replace f only on the left-hand side*}
apply (erule_tac P = "%x.?t(x) = ?u" in ssubst)
-apply (simp add: underI)
+apply (simp add: underI)
done
lemma is_recfun_equal [rule_format]:
"[| wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,b,H,g) |]
- ==> <x,a>:r --> <x,b>:r --> f`x=g`x"
+ ==> <x,a>:r \<longrightarrow> <x,b>:r \<longrightarrow> f`x=g`x"
apply (frule_tac f = f in is_recfun_type)
apply (frule_tac f = g in is_recfun_type)
apply (simp add: is_recfun_def)
@@ -245,7 +245,7 @@
apply (elim ssubst)
apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
apply (rule_tac t = "%z. H (?x,z) " in subst_context)
-apply (subgoal_tac "ALL y : r-``{x}. ALL z. <y,z>:f <-> <y,z>:g")
+apply (subgoal_tac "\<forall>y\<in>r-``{x}. \<forall>z. <y,z>:f <-> <y,z>:g")
apply (blast dest: transD)
apply (simp add: apply_iff)
apply (blast dest: transD intro: sym)
@@ -283,13 +283,13 @@
lemma unfold_the_recfun:
"[| wf(r); trans(r) |] ==> is_recfun(r, a, H, the_recfun(r,a,H))"
apply (rule_tac a=a in wf_induct, assumption)
-apply (rename_tac a1)
-apply (rule_tac f = "lam y: r-``{a1}. wftrec (r,y,H)" in is_the_recfun)
+apply (rename_tac a1)
+apply (rule_tac f = "\<lambda>y\<in>r-``{a1}. wftrec (r,y,H)" in is_the_recfun)
apply typecheck
apply (unfold is_recfun_def wftrec_def)
--{*Applying the substitution: must keep the quantified assumption!*}
-apply (rule lam_cong [OF refl])
-apply (drule underD)
+apply (rule lam_cong [OF refl])
+apply (drule underD)
apply (fold is_recfun_def)
apply (rule_tac t = "%z. H(?x,z)" in subst_context)
apply (rule fun_extension)
@@ -298,9 +298,9 @@
apply blast
apply (blast dest: transD)
apply (frule spec [THEN mp], assumption)
-apply (subgoal_tac "<xa,a1> : r")
+apply (subgoal_tac "<xa,a1> \<in> r")
apply (drule_tac x1 = xa in spec [THEN mp], assumption)
-apply (simp add: vimage_singleton_iff
+apply (simp add: vimage_singleton_iff
apply_recfun is_recfun_cut)
apply (blast dest: transD)
done
@@ -316,7 +316,7 @@
(*NOT SUITABLE FOR REWRITING: it is recursive!*)
lemma wftrec:
"[| wf(r); trans(r) |] ==>
- wftrec(r,a,H) = H(a, lam x: r-``{a}. wftrec(r,x,H))"
+ wftrec(r,a,H) = H(a, \<lambda>x\<in>r-``{a}. wftrec(r,x,H))"
apply (unfold wftrec_def)
apply (subst unfold_the_recfun [unfolded is_recfun_def])
apply (simp_all add: vimage_singleton_iff [THEN iff_sym] the_recfun_cut)
@@ -327,8 +327,8 @@
(*NOT SUITABLE FOR REWRITING: it is recursive!*)
lemma wfrec:
- "wf(r) ==> wfrec(r,a,H) = H(a, lam x:r-``{a}. wfrec(r,x,H))"
-apply (unfold wfrec_def)
+ "wf(r) ==> wfrec(r,a,H) = H(a, \<lambda>x\<in>r-``{a}. wfrec(r,x,H))"
+apply (unfold wfrec_def)
apply (erule wf_trancl [THEN wftrec, THEN ssubst])
apply (rule trans_trancl)
apply (rule vimage_pair_mono [THEN restrict_lam_eq, THEN subst_context])
@@ -339,24 +339,24 @@
(*This form avoids giant explosions in proofs. NOTE USE OF == *)
lemma def_wfrec:
"[| !!x. h(x)==wfrec(r,x,H); wf(r) |] ==>
- h(a) = H(a, lam x: r-``{a}. h(x))"
+ h(a) = H(a, \<lambda>x\<in>r-``{a}. h(x))"
apply simp
-apply (elim wfrec)
+apply (elim wfrec)
done
lemma wfrec_type:
"[| wf(r); a:A; field(r)<=A;
- !!x u. [| x: A; u: Pi(r-``{x}, B) |] ==> H(x,u) : B(x)
- |] ==> wfrec(r,a,H) : B(a)"
+ !!x u. [| x: A; u: Pi(r-``{x}, B) |] ==> H(x,u) \<in> B(x)
+ |] ==> wfrec(r,a,H) \<in> B(a)"
apply (rule_tac a = a in wf_induct2, assumption+)
apply (subst wfrec, assumption)
-apply (simp add: lam_type underD)
+apply (simp add: lam_type underD)
done
lemma wfrec_on:
"[| wf[A](r); a: A |] ==>
- wfrec[A](r,a,H) = H(a, lam x: (r-``{a}) Int A. wfrec[A](r,x,H))"
+ wfrec[A](r,a,H) = H(a, \<lambda>x\<in>(r-``{a}) \<inter> A. wfrec[A](r,x,H))"
apply (unfold wf_on_def wfrec_on_def)
apply (erule wfrec [THEN trans])
apply (simp add: vimage_Int_square cons_subset_iff)
@@ -364,7 +364,7 @@
text{*Minimal-element characterization of well-foundedness*}
lemma wf_eq_minimal:
- "wf(r) <-> (ALL Q x. x:Q --> (EX z:Q. ALL y. <y,z>:r --> y~:Q))"
+ "wf(r) <-> (\<forall>Q x. x:Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. <y,z>:r \<longrightarrow> y\<notin>Q))"
by (unfold wf_def, blast)
end