src/ZF/OrderArith.thy
author wenzelm
Sat Nov 04 19:17:19 2017 +0100 (21 months ago)
changeset 67006 b1278ed3cd46
parent 61980 6b780867d426
child 67443 3abf6a722518
permissions -rw-r--r--
prefer main entry points of HOL;
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(*  Title:      ZF/OrderArith.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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section\<open>Combining Orderings: Foundations of Ordinal Arithmetic\<close>
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theory OrderArith imports Order Sum Ordinal begin
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definition
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  (*disjoint sum of two relations; underlies ordinal addition*)
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  radd    :: "[i,i,i,i]=>i"  where
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    "radd(A,r,B,s) ==
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                {z: (A+B) * (A+B).
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                    (\<exists>x y. z = <Inl(x), Inr(y)>)   |
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                    (\<exists>x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |
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                    (\<exists>y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
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definition
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  (*lexicographic product of two relations; underlies ordinal multiplication*)
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  rmult   :: "[i,i,i,i]=>i"  where
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    "rmult(A,r,B,s) ==
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                {z: (A*B) * (A*B).
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                    \<exists>x' y' x y. z = <<x',y'>, <x,y>> &
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                       (<x',x>: r | (x'=x & <y',y>: s))}"
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definition
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  (*inverse image of a relation*)
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  rvimage :: "[i,i,i]=>i"  where
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    "rvimage(A,f,r) == {z \<in> A*A. \<exists>x y. z = <x,y> & <f`x,f`y>: r}"
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definition
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  measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"  where
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    "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
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subsection\<open>Addition of Relations -- Disjoint Sum\<close>
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subsubsection\<open>Rewrite rules.  Can be used to obtain introduction rules\<close>
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lemma radd_Inl_Inr_iff [iff]:
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    "<Inl(a), Inr(b)> \<in> radd(A,r,B,s)  \<longleftrightarrow>  a \<in> A & b \<in> B"
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by (unfold radd_def, blast)
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lemma radd_Inl_iff [iff]:
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    "<Inl(a'), Inl(a)> \<in> radd(A,r,B,s)  \<longleftrightarrow>  a':A & a \<in> A & <a',a>:r"
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by (unfold radd_def, blast)
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lemma radd_Inr_iff [iff]:
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    "<Inr(b'), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow>  b':B & b \<in> B & <b',b>:s"
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by (unfold radd_def, blast)
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lemma radd_Inr_Inl_iff [simp]:
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    "<Inr(b), Inl(a)> \<in> radd(A,r,B,s) \<longleftrightarrow> False"
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by (unfold radd_def, blast)
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declare radd_Inr_Inl_iff [THEN iffD1, dest!]
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subsubsection\<open>Elimination Rule\<close>
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lemma raddE:
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    "[| <p',p> \<in> radd(A,r,B,s);
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        !!x y. [| p'=Inl(x); x \<in> A; p=Inr(y); y \<in> B |] ==> Q;
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        !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x \<in> A |] ==> Q;
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        !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y \<in> B |] ==> Q
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     |] ==> Q"
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by (unfold radd_def, blast)
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subsubsection\<open>Type checking\<close>
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lemma radd_type: "radd(A,r,B,s) \<subseteq> (A+B) * (A+B)"
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apply (unfold radd_def)
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apply (rule Collect_subset)
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done
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lemmas field_radd = radd_type [THEN field_rel_subset]
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subsubsection\<open>Linearity\<close>
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lemma linear_radd:
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    "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
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by (unfold linear_def, blast)
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subsubsection\<open>Well-foundedness\<close>
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lemma wf_on_radd: "[| wf[A](r);  wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
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apply (rule wf_onI2)
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apply (subgoal_tac "\<forall>x\<in>A. Inl (x) \<in> Ba")
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 \<comment>\<open>Proving the lemma, which is needed twice!\<close>
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 prefer 2
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 apply (erule_tac V = "y \<in> A + B" in thin_rl)
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 apply (rule_tac ballI)
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 apply (erule_tac r = r and a = x in wf_on_induct, assumption)
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 apply blast
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txt\<open>Returning to main part of proof\<close>
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apply safe
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apply blast
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apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast)
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done
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lemma wf_radd: "[| wf(r);  wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
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apply (simp add: wf_iff_wf_on_field)
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apply (rule wf_on_subset_A [OF _ field_radd])
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apply (blast intro: wf_on_radd)
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done
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lemma well_ord_radd:
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     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
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apply (rule well_ordI)
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apply (simp add: well_ord_def wf_on_radd)
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apply (simp add: well_ord_def tot_ord_def linear_radd)
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done
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subsubsection\<open>An @{term ord_iso} congruence law\<close>
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lemma sum_bij:
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     "[| f \<in> bij(A,C);  g \<in> bij(B,D) |]
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      ==> (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) \<in> bij(A+B, C+D)"
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apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))"
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       in lam_bijective)
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apply (typecheck add: bij_is_inj inj_is_fun)
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apply (auto simp add: left_inverse_bij right_inverse_bij)
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done
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lemma sum_ord_iso_cong:
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    "[| f \<in> ord_iso(A,r,A',r');  g \<in> ord_iso(B,s,B',s') |] ==>
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            (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
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            \<in> ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
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apply (unfold ord_iso_def)
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apply (safe intro!: sum_bij)
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(*Do the beta-reductions now*)
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apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
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done
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(*Could we prove an ord_iso result?  Perhaps
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     ord_iso(A+B, radd(A,r,B,s), A \<union> B, r \<union> s) *)
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lemma sum_disjoint_bij: "A \<inter> B = 0 ==>
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            (\<lambda>z\<in>A+B. case(%x. x, %y. y, z)) \<in> bij(A+B, A \<union> B)"
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apply (rule_tac d = "%z. if z \<in> A then Inl (z) else Inr (z) " in lam_bijective)
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apply auto
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done
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subsubsection\<open>Associativity\<close>
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lemma sum_assoc_bij:
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     "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
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      \<in> bij((A+B)+C, A+(B+C))"
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apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
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       in lam_bijective)
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apply auto
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done
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lemma sum_assoc_ord_iso:
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     "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
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      \<in> ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
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                A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
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by (rule sum_assoc_bij [THEN ord_isoI], auto)
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subsection\<open>Multiplication of Relations -- Lexicographic Product\<close>
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subsubsection\<open>Rewrite rule.  Can be used to obtain introduction rules\<close>
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lemma  rmult_iff [iff]:
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    "<<a',b'>, <a,b>> \<in> rmult(A,r,B,s) \<longleftrightarrow>
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            (<a',a>: r  & a':A & a \<in> A & b': B & b \<in> B) |
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            (<b',b>: s  & a'=a & a \<in> A & b': B & b \<in> B)"
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by (unfold rmult_def, blast)
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lemma rmultE:
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    "[| <<a',b'>, <a,b>> \<in> rmult(A,r,B,s);
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        [| <a',a>: r;  a':A;  a \<in> A;  b':B;  b \<in> B |] ==> Q;
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        [| <b',b>: s;  a \<in> A;  a'=a;  b':B;  b \<in> B |] ==> Q
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     |] ==> Q"
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by blast
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subsubsection\<open>Type checking\<close>
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lemma rmult_type: "rmult(A,r,B,s) \<subseteq> (A*B) * (A*B)"
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by (unfold rmult_def, rule Collect_subset)
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lemmas field_rmult = rmult_type [THEN field_rel_subset]
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subsubsection\<open>Linearity\<close>
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lemma linear_rmult:
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    "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
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by (simp add: linear_def, blast)
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subsubsection\<open>Well-foundedness\<close>
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lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
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apply (rule wf_onI2)
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apply (erule SigmaE)
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apply (erule ssubst)
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apply (subgoal_tac "\<forall>b\<in>B. <x,b>: Ba", blast)
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apply (erule_tac a = x in wf_on_induct, assumption)
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apply (rule ballI)
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apply (erule_tac a = b in wf_on_induct, assumption)
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apply (best elim!: rmultE bspec [THEN mp])
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done
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lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
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apply (simp add: wf_iff_wf_on_field)
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apply (rule wf_on_subset_A [OF _ field_rmult])
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apply (blast intro: wf_on_rmult)
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done
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lemma well_ord_rmult:
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     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
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apply (rule well_ordI)
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apply (simp add: well_ord_def wf_on_rmult)
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apply (simp add: well_ord_def tot_ord_def linear_rmult)
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done
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subsubsection\<open>An @{term ord_iso} congruence law\<close>
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lemma prod_bij:
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     "[| f \<in> bij(A,C);  g \<in> bij(B,D) |]
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      ==> (lam <x,y>:A*B. <f`x, g`y>) \<in> bij(A*B, C*D)"
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apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
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       in lam_bijective)
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apply (typecheck add: bij_is_inj inj_is_fun)
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apply (auto simp add: left_inverse_bij right_inverse_bij)
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done
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lemma prod_ord_iso_cong:
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    "[| f \<in> ord_iso(A,r,A',r');  g \<in> ord_iso(B,s,B',s') |]
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     ==> (lam <x,y>:A*B. <f`x, g`y>)
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         \<in> ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
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apply (unfold ord_iso_def)
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apply (safe intro!: prod_bij)
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apply (simp_all add: bij_is_fun [THEN apply_type])
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apply (blast intro: bij_is_inj [THEN inj_apply_equality])
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done
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lemma singleton_prod_bij: "(\<lambda>z\<in>A. <x,z>) \<in> bij(A, {x}*A)"
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by (rule_tac d = snd in lam_bijective, auto)
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(*Used??*)
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lemma singleton_prod_ord_iso:
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     "well_ord({x},xr) ==>
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          (\<lambda>z\<in>A. <x,z>) \<in> ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
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apply (rule singleton_prod_bij [THEN ord_isoI])
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apply (simp (no_asm_simp))
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apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
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done
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(*Here we build a complicated function term, then simplify it using
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  case_cong, id_conv, comp_lam, case_case.*)
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lemma prod_sum_singleton_bij:
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     "a\<notin>C ==>
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       (\<lambda>x\<in>C*B + D. case(%x. x, %y.<a,y>, x))
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       \<in> bij(C*B + D, C*B \<union> {a}*D)"
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apply (rule subst_elem)
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apply (rule id_bij [THEN sum_bij, THEN comp_bij])
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apply (rule singleton_prod_bij)
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apply (rule sum_disjoint_bij, blast)
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apply (simp (no_asm_simp) cong add: case_cong)
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apply (rule comp_lam [THEN trans, symmetric])
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apply (fast elim!: case_type)
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apply (simp (no_asm_simp) add: case_case)
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done
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lemma prod_sum_singleton_ord_iso:
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 "[| a \<in> A;  well_ord(A,r) |] ==>
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    (\<lambda>x\<in>pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
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    \<in> ord_iso(pred(A,a,r)*B + pred(B,b,s),
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                  radd(A*B, rmult(A,r,B,s), B, s),
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              pred(A,a,r)*B \<union> {a}*pred(B,b,s), rmult(A,r,B,s))"
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apply (rule prod_sum_singleton_bij [THEN ord_isoI])
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apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
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apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
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done
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subsubsection\<open>Distributive law\<close>
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lemma sum_prod_distrib_bij:
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     "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
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      \<in> bij((A+B)*C, (A*C)+(B*C))"
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by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) "
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    in lam_bijective, auto)
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   288
lemma sum_prod_distrib_ord_iso:
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 "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
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   290
  \<in> ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
paulson@13140
   291
            (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
paulson@13356
   292
by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
paulson@13140
   293
wenzelm@60770
   294
subsubsection\<open>Associativity\<close>
paulson@13140
   295
paulson@13140
   296
lemma prod_assoc_bij:
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   297
     "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) \<in> bij((A*B)*C, A*(B*C))"
paulson@13356
   298
by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
paulson@13140
   299
paulson@13140
   300
lemma prod_assoc_ord_iso:
paulson@46953
   301
 "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
paulson@46953
   302
  \<in> ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
paulson@13140
   303
            A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
paulson@13356
   304
by (rule prod_assoc_bij [THEN ord_isoI], auto)
paulson@13140
   305
wenzelm@60770
   306
subsection\<open>Inverse Image of a Relation\<close>
paulson@13140
   307
wenzelm@60770
   308
subsubsection\<open>Rewrite rule\<close>
paulson@13140
   309
paulson@46953
   310
lemma rvimage_iff: "<a,b> \<in> rvimage(A,f,r)  \<longleftrightarrow>  <f`a,f`b>: r & a \<in> A & b \<in> A"
paulson@13269
   311
by (unfold rvimage_def, blast)
paulson@13140
   312
wenzelm@60770
   313
subsubsection\<open>Type checking\<close>
paulson@13140
   314
paulson@46820
   315
lemma rvimage_type: "rvimage(A,f,r) \<subseteq> A*A"
paulson@13784
   316
by (unfold rvimage_def, rule Collect_subset)
paulson@13140
   317
paulson@13140
   318
lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
paulson@13140
   319
paulson@13140
   320
lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
paulson@13269
   321
by (unfold rvimage_def, blast)
paulson@13140
   322
paulson@13140
   323
wenzelm@60770
   324
subsubsection\<open>Partial Ordering Properties\<close>
paulson@13140
   325
paulson@46953
   326
lemma irrefl_rvimage:
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   327
    "[| f \<in> inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
paulson@13140
   328
apply (unfold irrefl_def rvimage_def)
paulson@13140
   329
apply (blast intro: inj_is_fun [THEN apply_type])
paulson@13140
   330
done
paulson@13140
   331
paulson@46953
   332
lemma trans_on_rvimage:
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   333
    "[| f \<in> inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
paulson@13140
   334
apply (unfold trans_on_def rvimage_def)
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   335
apply (blast intro: inj_is_fun [THEN apply_type])
paulson@13140
   336
done
paulson@13140
   337
paulson@46953
   338
lemma part_ord_rvimage:
paulson@46953
   339
    "[| f \<in> inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
paulson@13140
   340
apply (unfold part_ord_def)
paulson@13140
   341
apply (blast intro!: irrefl_rvimage trans_on_rvimage)
paulson@13140
   342
done
paulson@13140
   343
wenzelm@60770
   344
subsubsection\<open>Linearity\<close>
paulson@13140
   345
paulson@13140
   346
lemma linear_rvimage:
paulson@46953
   347
    "[| f \<in> inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
paulson@46953
   348
apply (simp add: inj_def linear_def rvimage_iff)
paulson@46953
   349
apply (blast intro: apply_funtype)
paulson@13140
   350
done
paulson@13140
   351
paulson@46953
   352
lemma tot_ord_rvimage:
paulson@46953
   353
    "[| f \<in> inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
paulson@13140
   354
apply (unfold tot_ord_def)
paulson@13140
   355
apply (blast intro!: part_ord_rvimage linear_rvimage)
paulson@13140
   356
done
paulson@13140
   357
paulson@13140
   358
wenzelm@60770
   359
subsubsection\<open>Well-foundedness\<close>
paulson@13140
   360
paulson@13140
   361
lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
paulson@13140
   362
apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
paulson@13140
   363
apply clarify
paulson@46953
   364
apply (subgoal_tac "\<exists>w. w \<in> {w: {f`x. x \<in> Q}. \<exists>x. x \<in> Q & (f`x = w) }")
paulson@13140
   365
 apply (erule allE)
paulson@13140
   366
 apply (erule impE)
paulson@13269
   367
 apply assumption
paulson@13140
   368
 apply blast
paulson@46953
   369
apply blast
paulson@13140
   370
done
paulson@13140
   371
wenzelm@61798
   372
text\<open>But note that the combination of \<open>wf_imp_wf_on\<close> and
wenzelm@61798
   373
 \<open>wf_rvimage\<close> gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}\<close>
paulson@46953
   374
lemma wf_on_rvimage: "[| f \<in> A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
paulson@13140
   375
apply (rule wf_onI2)
paulson@46953
   376
apply (subgoal_tac "\<forall>z\<in>A. f`z=f`y \<longrightarrow> z \<in> Ba")
paulson@13140
   377
 apply blast
paulson@13140
   378
apply (erule_tac a = "f`y" in wf_on_induct)
paulson@13140
   379
 apply (blast intro!: apply_funtype)
paulson@13140
   380
apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
paulson@13140
   381
done
paulson@13140
   382
paulson@13140
   383
(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
paulson@13140
   384
lemma well_ord_rvimage:
paulson@46953
   385
     "[| f \<in> inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
paulson@13140
   386
apply (rule well_ordI)
paulson@13140
   387
apply (unfold well_ord_def tot_ord_def)
paulson@13140
   388
apply (blast intro!: wf_on_rvimage inj_is_fun)
paulson@13140
   389
apply (blast intro!: linear_rvimage)
paulson@13140
   390
done
paulson@13140
   391
paulson@46953
   392
lemma ord_iso_rvimage:
paulson@46953
   393
    "f \<in> bij(A,B) ==> f \<in> ord_iso(A, rvimage(A,f,s), B, s)"
paulson@13140
   394
apply (unfold ord_iso_def)
paulson@13140
   395
apply (simp add: rvimage_iff)
paulson@13140
   396
done
paulson@13140
   397
paulson@46953
   398
lemma ord_iso_rvimage_eq:
paulson@46953
   399
    "f \<in> ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r \<inter> A*A"
paulson@13356
   400
by (unfold ord_iso_def rvimage_def, blast)
paulson@13140
   401
paulson@13140
   402
wenzelm@60770
   403
subsection\<open>Every well-founded relation is a subset of some inverse image of
wenzelm@60770
   404
      an ordinal\<close>
paulson@13634
   405
paulson@13634
   406
lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
paulson@13634
   407
by (blast intro: wf_rvimage wf_Memrel)
paulson@13634
   408
paulson@13634
   409
wenzelm@24893
   410
definition
wenzelm@24893
   411
  wfrank :: "[i,i]=>i"  where
paulson@13634
   412
    "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
paulson@13634
   413
wenzelm@24893
   414
definition
wenzelm@24893
   415
  wftype :: "i=>i"  where
paulson@13634
   416
    "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
paulson@13634
   417
paulson@13634
   418
lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
paulson@13634
   419
by (subst wfrank_def [THEN def_wfrec], simp_all)
paulson@13634
   420
paulson@13634
   421
lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
paulson@13634
   422
apply (rule_tac a=a in wf_induct, assumption)
paulson@13634
   423
apply (subst wfrank, assumption)
paulson@13634
   424
apply (rule Ord_succ [THEN Ord_UN], blast)
paulson@13634
   425
done
paulson@13634
   426
paulson@13634
   427
lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
paulson@13634
   428
apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
paulson@13634
   429
apply (rule UN_I [THEN ltI])
paulson@13634
   430
apply (simp add: Ord_wfrank vimage_iff)+
paulson@13634
   431
done
paulson@13634
   432
paulson@13634
   433
lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
paulson@13634
   434
by (simp add: wftype_def Ord_wfrank)
paulson@13634
   435
paulson@13634
   436
lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
paulson@13634
   437
apply (simp add: wftype_def)
paulson@13634
   438
apply (blast intro: wfrank_lt [THEN ltD])
paulson@13634
   439
done
paulson@13634
   440
paulson@13634
   441
paulson@13634
   442
lemma wf_imp_subset_rvimage:
paulson@46820
   443
     "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i))"
paulson@13634
   444
apply (rule_tac x="wftype(r)" in exI)
paulson@13634
   445
apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
paulson@13634
   446
apply (simp add: Ord_wftype, clarify)
paulson@13634
   447
apply (frule subsetD, assumption, clarify)
paulson@13634
   448
apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
paulson@13634
   449
apply (blast intro: wftypeI)
paulson@13634
   450
done
paulson@13634
   451
paulson@13634
   452
theorem wf_iff_subset_rvimage:
paulson@46821
   453
  "relation(r) ==> wf(r) \<longleftrightarrow> (\<exists>i f A. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i)))"
paulson@13634
   454
by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
paulson@13634
   455
          intro: wf_rvimage_Ord [THEN wf_subset])
paulson@13634
   456
paulson@13634
   457
wenzelm@60770
   458
subsection\<open>Other Results\<close>
paulson@13544
   459
paulson@46820
   460
lemma wf_times: "A \<inter> B = 0 ==> wf(A*B)"
paulson@13544
   461
by (simp add: wf_def, blast)
paulson@13544
   462
wenzelm@61798
   463
text\<open>Could also be used to prove \<open>wf_radd\<close>\<close>
paulson@13544
   464
lemma wf_Un:
paulson@46820
   465
     "[| range(r) \<inter> domain(s) = 0; wf(r);  wf(s) |] ==> wf(r \<union> s)"
paulson@46953
   466
apply (simp add: wf_def, clarify)
paulson@46953
   467
apply (rule equalityI)
paulson@46953
   468
 prefer 2 apply blast
paulson@46953
   469
apply clarify
paulson@13544
   470
apply (drule_tac x=Z in spec)
paulson@46820
   471
apply (drule_tac x="Z \<inter> domain(s)" in spec)
paulson@46953
   472
apply simp
paulson@46953
   473
apply (blast intro: elim: equalityE)
paulson@13544
   474
done
paulson@13544
   475
wenzelm@60770
   476
subsubsection\<open>The Empty Relation\<close>
paulson@13544
   477
paulson@13544
   478
lemma wf0: "wf(0)"
paulson@13544
   479
by (simp add: wf_def, blast)
paulson@13544
   480
paulson@13544
   481
lemma linear0: "linear(0,0)"
paulson@13544
   482
by (simp add: linear_def)
paulson@13544
   483
paulson@13544
   484
lemma well_ord0: "well_ord(0,0)"
paulson@13544
   485
by (blast intro: wf_imp_wf_on well_ordI wf0 linear0)
paulson@13512
   486
wenzelm@60770
   487
subsubsection\<open>The "measure" relation is useful with wfrec\<close>
paulson@13140
   488
paulson@13140
   489
lemma measure_eq_rvimage_Memrel:
paulson@13140
   490
     "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
paulson@13140
   491
apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
paulson@13269
   492
apply (rule equalityI, auto)
paulson@13140
   493
apply (auto intro: Ord_in_Ord simp add: lt_def)
paulson@13140
   494
done
paulson@13140
   495
paulson@13140
   496
lemma wf_measure [iff]: "wf(measure(A,f))"
paulson@13356
   497
by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
paulson@13140
   498
paulson@46953
   499
lemma measure_iff [iff]: "<x,y> \<in> measure(A,f) \<longleftrightarrow> x \<in> A & y \<in> A & f(x)<f(y)"
paulson@13356
   500
by (simp (no_asm) add: measure_def)
paulson@13140
   501
paulson@46953
   502
lemma linear_measure:
paulson@13544
   503
 assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
paulson@13544
   504
     and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
paulson@13544
   505
 shows "linear(A, measure(A,f))"
paulson@46953
   506
apply (auto simp add: linear_def)
paulson@46953
   507
apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt)
paulson@46953
   508
    apply (simp_all add: Ordf)
paulson@46953
   509
apply (blast intro: inj)
paulson@13544
   510
done
paulson@13544
   511
paulson@13544
   512
lemma wf_on_measure: "wf[B](measure(A,f))"
paulson@13544
   513
by (rule wf_imp_wf_on [OF wf_measure])
paulson@13544
   514
paulson@46953
   515
lemma well_ord_measure:
paulson@13544
   516
 assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
paulson@13544
   517
     and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
paulson@13544
   518
 shows "well_ord(A, measure(A,f))"
paulson@13544
   519
apply (rule well_ordI)
paulson@46953
   520
apply (rule wf_on_measure)
paulson@46953
   521
apply (blast intro: linear_measure Ordf inj)
paulson@13544
   522
done
paulson@13544
   523
paulson@46820
   524
lemma measure_type: "measure(A,f) \<subseteq> A*A"
paulson@13544
   525
by (auto simp add: measure_def)
paulson@13544
   526
wenzelm@60770
   527
subsubsection\<open>Well-foundedness of Unions\<close>
paulson@13512
   528
paulson@13512
   529
lemma wf_on_Union:
paulson@13512
   530
 assumes wfA: "wf[A](r)"
paulson@13512
   531
     and wfB: "!!a. a\<in>A ==> wf[B(a)](s)"
paulson@46953
   532
     and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|]
paulson@13512
   533
                       ==> (\<exists>a'\<in>A. <a',a> \<in> r & u \<in> B(a')) | u \<in> B(a)"
paulson@13512
   534
 shows "wf[\<Union>a\<in>A. B(a)](s)"
paulson@13512
   535
apply (rule wf_onI2)
paulson@13512
   536
apply (erule UN_E)
paulson@13512
   537
apply (subgoal_tac "\<forall>z \<in> B(a). z \<in> Ba", blast)
paulson@13512
   538
apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
paulson@13512
   539
apply (rule ballI)
paulson@13512
   540
apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
paulson@46953
   541
apply (rename_tac u)
paulson@46953
   542
apply (drule_tac x=u in bspec, blast)
paulson@13512
   543
apply (erule mp, clarify)
paulson@46953
   544
apply (frule ok, assumption+, blast)
paulson@13512
   545
done
paulson@13512
   546
wenzelm@60770
   547
subsubsection\<open>Bijections involving Powersets\<close>
paulson@14120
   548
paulson@14120
   549
lemma Pow_sum_bij:
paulson@46953
   550
    "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)
paulson@14120
   551
     \<in> bij(Pow(A+B), Pow(A)*Pow(B))"
paulson@46953
   552
apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} \<union> {Inr (y). y \<in> Y}"
paulson@14120
   553
       in lam_bijective)
paulson@14120
   554
apply force+
paulson@14120
   555
done
paulson@14120
   556
wenzelm@60770
   557
text\<open>As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"}\<close>
paulson@14120
   558
lemma Pow_Sigma_bij:
paulson@46953
   559
    "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})
wenzelm@61980
   560
     \<in> bij(Pow(Sigma(A,B)), \<Prod>x \<in> A. Pow(B(x)))"
paulson@14120
   561
apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
paulson@14120
   562
apply (blast intro: lam_type)
paulson@14120
   563
apply (blast dest: apply_type, simp_all)
paulson@14120
   564
apply fast (*strange, but blast can't do it*)
paulson@14120
   565
apply (rule fun_extension, auto)
paulson@14120
   566
by blast
paulson@14120
   567
lcp@437
   568
end