--- a/src/ZF/OrderArith.thy Thu Mar 15 15:54:22 2012 +0000
+++ b/src/ZF/OrderArith.thy Thu Mar 15 16:35:02 2012 +0000
@@ -10,24 +10,24 @@
definition
(*disjoint sum of two relations; underlies ordinal addition*)
radd :: "[i,i,i,i]=>i" where
- "radd(A,r,B,s) ==
- {z: (A+B) * (A+B).
- (\<exists>x y. z = <Inl(x), Inr(y)>) |
- (\<exists>x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) |
+ "radd(A,r,B,s) ==
+ {z: (A+B) * (A+B).
+ (\<exists>x y. z = <Inl(x), Inr(y)>) |
+ (\<exists>x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) |
(\<exists>y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
definition
(*lexicographic product of two relations; underlies ordinal multiplication*)
rmult :: "[i,i,i,i]=>i" where
- "rmult(A,r,B,s) ==
- {z: (A*B) * (A*B).
- \<exists>x' y' x y. z = <<x',y'>, <x,y>> &
+ "rmult(A,r,B,s) ==
+ {z: (A*B) * (A*B).
+ \<exists>x' y' x y. z = <<x',y'>, <x,y>> &
(<x',x>: r | (x'=x & <y',y>: s))}"
definition
(*inverse image of a relation*)
rvimage :: "[i,i,i]=>i" where
- "rvimage(A,f,r) == {z: A*A. \<exists>x y. z = <x,y> & <f`x,f`y>: r}"
+ "rvimage(A,f,r) == {z \<in> A*A. \<exists>x y. z = <x,y> & <f`x,f`y>: r}"
definition
measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i" where
@@ -38,33 +38,33 @@
subsubsection{*Rewrite rules. Can be used to obtain introduction rules*}
-lemma radd_Inl_Inr_iff [iff]:
- "<Inl(a), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow> a:A & b:B"
+lemma radd_Inl_Inr_iff [iff]:
+ "<Inl(a), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow> a \<in> A & b \<in> B"
by (unfold radd_def, blast)
-lemma radd_Inl_iff [iff]:
- "<Inl(a'), Inl(a)> \<in> radd(A,r,B,s) \<longleftrightarrow> a':A & a:A & <a',a>:r"
+lemma radd_Inl_iff [iff]:
+ "<Inl(a'), Inl(a)> \<in> radd(A,r,B,s) \<longleftrightarrow> a':A & a \<in> A & <a',a>:r"
by (unfold radd_def, blast)
-lemma radd_Inr_iff [iff]:
- "<Inr(b'), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow> b':B & b:B & <b',b>:s"
+lemma radd_Inr_iff [iff]:
+ "<Inr(b'), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow> b':B & b \<in> B & <b',b>:s"
by (unfold radd_def, blast)
-lemma radd_Inr_Inl_iff [simp]:
+lemma radd_Inr_Inl_iff [simp]:
"<Inr(b), Inl(a)> \<in> radd(A,r,B,s) \<longleftrightarrow> False"
by (unfold radd_def, blast)
-declare radd_Inr_Inl_iff [THEN iffD1, dest!]
+declare radd_Inr_Inl_iff [THEN iffD1, dest!]
subsubsection{*Elimination Rule*}
lemma raddE:
- "[| <p',p> \<in> radd(A,r,B,s);
- !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;
- !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;
- !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q
+ "[| <p',p> \<in> radd(A,r,B,s);
+ !!x y. [| p'=Inl(x); x \<in> A; p=Inr(y); y \<in> B |] ==> Q;
+ !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x \<in> A |] ==> Q;
+ !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y \<in> B |] ==> Q
|] ==> Q"
-by (unfold radd_def, blast)
+by (unfold radd_def, blast)
subsubsection{*Type checking*}
@@ -77,9 +77,9 @@
subsubsection{*Linearity*}
-lemma linear_radd:
+lemma linear_radd:
"[| linear(A,r); linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
-by (unfold linear_def, blast)
+by (unfold linear_def, blast)
subsubsection{*Well-foundedness*}
@@ -92,17 +92,17 @@
apply (erule_tac V = "y \<in> A + B" in thin_rl)
apply (rule_tac ballI)
apply (erule_tac r = r and a = x in wf_on_induct, assumption)
- apply blast
+ apply blast
txt{*Returning to main part of proof*}
apply safe
apply blast
-apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast)
+apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast)
done
lemma wf_radd: "[| wf(r); wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
apply (simp add: wf_iff_wf_on_field)
apply (rule wf_on_subset_A [OF _ field_radd])
-apply (blast intro: wf_on_radd)
+apply (blast intro: wf_on_radd)
done
lemma well_ord_radd:
@@ -115,17 +115,17 @@
subsubsection{*An @{term ord_iso} congruence law*}
lemma sum_bij:
- "[| f: bij(A,C); g: bij(B,D) |]
+ "[| f \<in> bij(A,C); g \<in> bij(B,D) |]
==> (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) \<in> bij(A+B, C+D)"
-apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))"
+apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))"
in lam_bijective)
-apply (typecheck add: bij_is_inj inj_is_fun)
-apply (auto simp add: left_inverse_bij right_inverse_bij)
+apply (typecheck add: bij_is_inj inj_is_fun)
+apply (auto simp add: left_inverse_bij right_inverse_bij)
done
-lemma sum_ord_iso_cong:
- "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==>
- (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
+lemma sum_ord_iso_cong:
+ "[| f \<in> ord_iso(A,r,A',r'); g \<in> ord_iso(B,s,B',s') |] ==>
+ (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
\<in> ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
apply (unfold ord_iso_def)
apply (safe intro!: sum_bij)
@@ -133,27 +133,27 @@
apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
done
-(*Could we prove an ord_iso result? Perhaps
+(*Could we prove an ord_iso result? Perhaps
ord_iso(A+B, radd(A,r,B,s), A \<union> B, r \<union> s) *)
-lemma sum_disjoint_bij: "A \<inter> B = 0 ==>
+lemma sum_disjoint_bij: "A \<inter> B = 0 ==>
(\<lambda>z\<in>A+B. case(%x. x, %y. y, z)) \<in> bij(A+B, A \<union> B)"
-apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
+apply (rule_tac d = "%z. if z \<in> A then Inl (z) else Inr (z) " in lam_bijective)
apply auto
done
subsubsection{*Associativity*}
lemma sum_assoc_bij:
- "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
+ "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
\<in> bij((A+B)+C, A+(B+C))"
-apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
+apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
in lam_bijective)
apply auto
done
lemma sum_assoc_ord_iso:
- "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
- \<in> ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
+ "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
+ \<in> ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
by (rule sum_assoc_bij [THEN ord_isoI], auto)
@@ -162,19 +162,19 @@
subsubsection{*Rewrite rule. Can be used to obtain introduction rules*}
-lemma rmult_iff [iff]:
- "<<a',b'>, <a,b>> \<in> rmult(A,r,B,s) \<longleftrightarrow>
- (<a',a>: r & a':A & a:A & b': B & b: B) |
- (<b',b>: s & a'=a & a:A & b': B & b: B)"
+lemma rmult_iff [iff]:
+ "<<a',b'>, <a,b>> \<in> rmult(A,r,B,s) \<longleftrightarrow>
+ (<a',a>: r & a':A & a \<in> A & b': B & b \<in> B) |
+ (<b',b>: s & a'=a & a \<in> A & b': B & b \<in> B)"
by (unfold rmult_def, blast)
-lemma rmultE:
- "[| <<a',b'>, <a,b>> \<in> rmult(A,r,B,s);
- [| <a',a>: r; a':A; a:A; b':B; b:B |] ==> Q;
- [| <b',b>: s; a:A; a'=a; b':B; b:B |] ==> Q
+lemma rmultE:
+ "[| <<a',b'>, <a,b>> \<in> rmult(A,r,B,s);
+ [| <a',a>: r; a':A; a \<in> A; b':B; b \<in> B |] ==> Q;
+ [| <b',b>: s; a \<in> A; a'=a; b':B; b \<in> B |] ==> Q
|] ==> Q"
-by blast
+by blast
subsubsection{*Type checking*}
@@ -187,7 +187,7 @@
lemma linear_rmult:
"[| linear(A,r); linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
-by (simp add: linear_def, blast)
+by (simp add: linear_def, blast)
subsubsection{*Well-foundedness*}
@@ -206,7 +206,7 @@
lemma wf_rmult: "[| wf(r); wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
apply (simp add: wf_iff_wf_on_field)
apply (rule wf_on_subset_A [OF _ field_rmult])
-apply (blast intro: wf_on_rmult)
+apply (blast intro: wf_on_rmult)
done
lemma well_ord_rmult:
@@ -220,17 +220,17 @@
subsubsection{*An @{term ord_iso} congruence law*}
lemma prod_bij:
- "[| f: bij(A,C); g: bij(B,D) |]
+ "[| f \<in> bij(A,C); g \<in> bij(B,D) |]
==> (lam <x,y>:A*B. <f`x, g`y>) \<in> bij(A*B, C*D)"
-apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
+apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
in lam_bijective)
-apply (typecheck add: bij_is_inj inj_is_fun)
-apply (auto simp add: left_inverse_bij right_inverse_bij)
+apply (typecheck add: bij_is_inj inj_is_fun)
+apply (auto simp add: left_inverse_bij right_inverse_bij)
done
-lemma prod_ord_iso_cong:
- "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |]
- ==> (lam <x,y>:A*B. <f`x, g`y>)
+lemma prod_ord_iso_cong:
+ "[| f \<in> ord_iso(A,r,A',r'); g \<in> ord_iso(B,s,B',s') |]
+ ==> (lam <x,y>:A*B. <f`x, g`y>)
\<in> ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
apply (unfold ord_iso_def)
apply (safe intro!: prod_bij)
@@ -243,7 +243,7 @@
(*Used??*)
lemma singleton_prod_ord_iso:
- "well_ord({x},xr) ==>
+ "well_ord({x},xr) ==>
(\<lambda>z\<in>A. <x,z>) \<in> ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
apply (rule singleton_prod_bij [THEN ord_isoI])
apply (simp (no_asm_simp))
@@ -253,8 +253,8 @@
(*Here we build a complicated function term, then simplify it using
case_cong, id_conv, comp_lam, case_case.*)
lemma prod_sum_singleton_bij:
- "a\<notin>C ==>
- (\<lambda>x\<in>C*B + D. case(%x. x, %y.<a,y>, x))
+ "a\<notin>C ==>
+ (\<lambda>x\<in>C*B + D. case(%x. x, %y.<a,y>, x))
\<in> bij(C*B + D, C*B \<union> {a}*D)"
apply (rule subst_elem)
apply (rule id_bij [THEN sum_bij, THEN comp_bij])
@@ -267,10 +267,10 @@
done
lemma prod_sum_singleton_ord_iso:
- "[| a:A; well_ord(A,r) |] ==>
- (\<lambda>x\<in>pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
- \<in> ord_iso(pred(A,a,r)*B + pred(B,b,s),
- radd(A*B, rmult(A,r,B,s), B, s),
+ "[| a \<in> A; well_ord(A,r) |] ==>
+ (\<lambda>x\<in>pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
+ \<in> ord_iso(pred(A,a,r)*B + pred(B,b,s),
+ radd(A*B, rmult(A,r,B,s), B, s),
pred(A,a,r)*B \<union> {a}*pred(B,b,s), rmult(A,r,B,s))"
apply (rule prod_sum_singleton_bij [THEN ord_isoI])
apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
@@ -280,14 +280,14 @@
subsubsection{*Distributive law*}
lemma sum_prod_distrib_bij:
- "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
+ "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
\<in> bij((A+B)*C, (A*C)+(B*C))"
-by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) "
+by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) "
in lam_bijective, auto)
lemma sum_prod_distrib_ord_iso:
- "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
- \<in> ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
+ "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
+ \<in> ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
(A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
@@ -298,8 +298,8 @@
by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
lemma prod_assoc_ord_iso:
- "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
- \<in> ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
+ "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
+ \<in> ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
by (rule prod_assoc_bij [THEN ord_isoI], auto)
@@ -307,7 +307,7 @@
subsubsection{*Rewrite rule*}
-lemma rvimage_iff: "<a,b> \<in> rvimage(A,f,r) \<longleftrightarrow> <f`a,f`b>: r & a:A & b:A"
+lemma rvimage_iff: "<a,b> \<in> rvimage(A,f,r) \<longleftrightarrow> <f`a,f`b>: r & a \<in> A & b \<in> A"
by (unfold rvimage_def, blast)
subsubsection{*Type checking*}
@@ -323,20 +323,20 @@
subsubsection{*Partial Ordering Properties*}
-lemma irrefl_rvimage:
- "[| f: inj(A,B); irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
+lemma irrefl_rvimage:
+ "[| f \<in> inj(A,B); irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
apply (unfold irrefl_def rvimage_def)
apply (blast intro: inj_is_fun [THEN apply_type])
done
-lemma trans_on_rvimage:
- "[| f: inj(A,B); trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
+lemma trans_on_rvimage:
+ "[| f \<in> inj(A,B); trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
apply (unfold trans_on_def rvimage_def)
apply (blast intro: inj_is_fun [THEN apply_type])
done
-lemma part_ord_rvimage:
- "[| f: inj(A,B); part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
+lemma part_ord_rvimage:
+ "[| f \<in> inj(A,B); part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
apply (unfold part_ord_def)
apply (blast intro!: irrefl_rvimage trans_on_rvimage)
done
@@ -344,13 +344,13 @@
subsubsection{*Linearity*}
lemma linear_rvimage:
- "[| f: inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
-apply (simp add: inj_def linear_def rvimage_iff)
-apply (blast intro: apply_funtype)
+ "[| f \<in> inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
+apply (simp add: inj_def linear_def rvimage_iff)
+apply (blast intro: apply_funtype)
done
-lemma tot_ord_rvimage:
- "[| f: inj(A,B); tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
+lemma tot_ord_rvimage:
+ "[| f \<in> inj(A,B); tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
apply (unfold tot_ord_def)
apply (blast intro!: part_ord_rvimage linear_rvimage)
done
@@ -361,19 +361,19 @@
lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
apply clarify
-apply (subgoal_tac "\<exists>w. w \<in> {w: {f`x. x:Q}. \<exists>x. x: Q & (f`x = w) }")
+apply (subgoal_tac "\<exists>w. w \<in> {w: {f`x. x \<in> Q}. \<exists>x. x \<in> Q & (f`x = w) }")
apply (erule allE)
apply (erule impE)
apply assumption
apply blast
-apply blast
+apply blast
done
text{*But note that the combination of @{text wf_imp_wf_on} and
@{text wf_rvimage} gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}*}
-lemma wf_on_rvimage: "[| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
+lemma wf_on_rvimage: "[| f \<in> A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
apply (rule wf_onI2)
-apply (subgoal_tac "\<forall>z\<in>A. f`z=f`y \<longrightarrow> z: Ba")
+apply (subgoal_tac "\<forall>z\<in>A. f`z=f`y \<longrightarrow> z \<in> Ba")
apply blast
apply (erule_tac a = "f`y" in wf_on_induct)
apply (blast intro!: apply_funtype)
@@ -382,21 +382,21 @@
(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
lemma well_ord_rvimage:
- "[| f: inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
+ "[| f \<in> inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
apply (rule well_ordI)
apply (unfold well_ord_def tot_ord_def)
apply (blast intro!: wf_on_rvimage inj_is_fun)
apply (blast intro!: linear_rvimage)
done
-lemma ord_iso_rvimage:
- "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
+lemma ord_iso_rvimage:
+ "f \<in> bij(A,B) ==> f \<in> ord_iso(A, rvimage(A,f,s), B, s)"
apply (unfold ord_iso_def)
apply (simp add: rvimage_iff)
done
-lemma ord_iso_rvimage_eq:
- "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r \<inter> A*A"
+lemma ord_iso_rvimage_eq:
+ "f \<in> ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r \<inter> A*A"
by (unfold ord_iso_def rvimage_def, blast)
@@ -463,14 +463,14 @@
text{*Could also be used to prove @{text wf_radd}*}
lemma wf_Un:
"[| range(r) \<inter> domain(s) = 0; wf(r); wf(s) |] ==> wf(r \<union> s)"
-apply (simp add: wf_def, clarify)
-apply (rule equalityI)
- prefer 2 apply blast
-apply clarify
+apply (simp add: wf_def, clarify)
+apply (rule equalityI)
+ prefer 2 apply blast
+apply clarify
apply (drule_tac x=Z in spec)
apply (drule_tac x="Z \<inter> domain(s)" in spec)
-apply simp
-apply (blast intro: elim: equalityE)
+apply simp
+apply (blast intro: elim: equalityE)
done
subsubsection{*The Empty Relation*}
@@ -496,29 +496,29 @@
lemma wf_measure [iff]: "wf(measure(A,f))"
by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
-lemma measure_iff [iff]: "<x,y> \<in> measure(A,f) \<longleftrightarrow> x:A & y:A & f(x)<f(y)"
+lemma measure_iff [iff]: "<x,y> \<in> measure(A,f) \<longleftrightarrow> x \<in> A & y \<in> A & f(x)<f(y)"
by (simp (no_asm) add: measure_def)
-lemma linear_measure:
+lemma linear_measure:
assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
and inj: "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
shows "linear(A, measure(A,f))"
-apply (auto simp add: linear_def)
-apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt)
- apply (simp_all add: Ordf)
-apply (blast intro: inj)
+apply (auto simp add: linear_def)
+apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt)
+ apply (simp_all add: Ordf)
+apply (blast intro: inj)
done
lemma wf_on_measure: "wf[B](measure(A,f))"
by (rule wf_imp_wf_on [OF wf_measure])
-lemma well_ord_measure:
+lemma well_ord_measure:
assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
and inj: "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
shows "well_ord(A, measure(A,f))"
apply (rule well_ordI)
-apply (rule wf_on_measure)
-apply (blast intro: linear_measure Ordf inj)
+apply (rule wf_on_measure)
+apply (blast intro: linear_measure Ordf inj)
done
lemma measure_type: "measure(A,f) \<subseteq> A*A"
@@ -529,7 +529,7 @@
lemma wf_on_Union:
assumes wfA: "wf[A](r)"
and wfB: "!!a. a\<in>A ==> wf[B(a)](s)"
- and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|]
+ and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|]
==> (\<exists>a'\<in>A. <a',a> \<in> r & u \<in> B(a')) | u \<in> B(a)"
shows "wf[\<Union>a\<in>A. B(a)](s)"
apply (rule wf_onI2)
@@ -538,25 +538,25 @@
apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
apply (rule ballI)
apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
-apply (rename_tac u)
-apply (drule_tac x=u in bspec, blast)
+apply (rename_tac u)
+apply (drule_tac x=u in bspec, blast)
apply (erule mp, clarify)
-apply (frule ok, assumption+, blast)
+apply (frule ok, assumption+, blast)
done
subsubsection{*Bijections involving Powersets*}
lemma Pow_sum_bij:
- "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)
+ "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)
\<in> bij(Pow(A+B), Pow(A)*Pow(B))"
-apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} \<union> {Inr (y). y \<in> Y}"
+apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} \<union> {Inr (y). y \<in> Y}"
in lam_bijective)
apply force+
done
text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *}
lemma Pow_Sigma_bij:
- "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})
+ "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})
\<in> bij(Pow(Sigma(A,B)), \<Pi> x \<in> A. Pow(B(x)))"
apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
apply (blast intro: lam_type)