src/ZF/OrderArith.thy
author wenzelm
Sun Oct 07 21:19:31 2007 +0200 (2007-10-07)
changeset 24893 b8ef7afe3a6b
parent 22710 f44439cdce77
child 35762 af3ff2ba4c54
permissions -rw-r--r--
modernized specifications;
removed legacy ML bindings;
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(*  Title:      ZF/OrderArith.thy
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    ID:         $Id$
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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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header{*Combining Orderings: Foundations of Ordinal Arithmetic*}
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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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                    (EX x y. z = <Inl(x), Inr(y)>)   |   
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                    (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |      
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                    (EX 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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                    EX 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: A*A. EX 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{*Addition of Relations -- Disjoint Sum*}
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subsubsection{*Rewrite rules.  Can be used to obtain introduction rules*}
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lemma radd_Inl_Inr_iff [iff]: 
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    "<Inl(a), Inr(b)> : radd(A,r,B,s)  <->  a:A & b: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)> : radd(A,r,B,s)  <->  a':A & a: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)> : radd(A,r,B,s) <->  b':B & b: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)> : radd(A,r,B,s) <-> 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{*Elimination Rule*}
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lemma raddE:
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    "[| <p',p> : radd(A,r,B,s);                  
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        !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;        
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        !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;  
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        !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q   
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     |] ==> Q"
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by (unfold radd_def, blast) 
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subsubsection{*Type checking*}
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lemma radd_type: "radd(A,r,B,s) <= (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{*Linearity*}
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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{*Well-foundedness*}
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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 "ALL x:A. Inl (x) : Ba")
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 --{*Proving the lemma, which is needed twice!*}
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 prefer 2
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 apply (erule_tac V = "y : 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{*Returning to main part of proof*}
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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{*An @{term ord_iso} congruence law*}
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lemma sum_bij:
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     "[| f: bij(A,C);  g: bij(B,D) |]
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      ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : 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: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |] ==>      
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            (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))             
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            : 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 Un B, r Un s) *)
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lemma sum_disjoint_bij: "A Int B = 0 ==>      
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            (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
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apply (rule_tac d = "%z. if z: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{*Associativity*}
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lemma sum_assoc_bij:
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     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
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      : 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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     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
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      : 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{*Multiplication of Relations -- Lexicographic Product*}
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subsubsection{*Rewrite rule.  Can be used to obtain introduction rules*}
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lemma  rmult_iff [iff]: 
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    "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->        
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            (<a',a>: r  & a':A & a:A & b': B & b: B) |   
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            (<b',b>: s  & a'=a & a:A & b': B & b: B)"
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by (unfold rmult_def, blast)
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lemma rmultE: 
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    "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);               
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        [| <a',a>: r;  a':A;  a:A;  b':B;  b:B |] ==> Q;         
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        [| <b',b>: s;  a:A;  a'=a;  b':B;  b:B |] ==> Q  
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     |] ==> Q"
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by blast 
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subsubsection{*Type checking*}
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lemma rmult_type: "rmult(A,r,B,s) <= (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{*Linearity*}
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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{*Well-foundedness*}
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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 "ALL b: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{*An @{term ord_iso} congruence law*}
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lemma prod_bij:
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     "[| f: bij(A,C);  g: bij(B,D) |] 
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      ==> (lam <x,y>:A*B. <f`x, g`y>) : 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: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |]      
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     ==> (lam <x,y>:A*B. <f`x, g`y>)                                  
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         : 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: "(lam z:A. <x,z>) : 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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          (lam z:A. <x,z>) : 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~:C ==>  
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       (lam x:C*B + D. case(%x. x, %y.<a,y>, x))  
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       : bij(C*B + D, C*B Un {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:A;  well_ord(A,r) |] ==>  
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    (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))  
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    : 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 Un {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{*Distributive law*}
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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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      : 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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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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  : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),  
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            (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
paulson@13356
   294
by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
paulson@13140
   295
paulson@13512
   296
subsubsection{*Associativity*}
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   297
paulson@13140
   298
lemma prod_assoc_bij:
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   299
     "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
paulson@13356
   300
by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
paulson@13140
   301
paulson@13140
   302
lemma prod_assoc_ord_iso:
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   303
 "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)                    
paulson@13140
   304
  : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),   
paulson@13140
   305
            A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
paulson@13356
   306
by (rule prod_assoc_bij [THEN ord_isoI], auto)
paulson@13140
   307
paulson@13356
   308
subsection{*Inverse Image of a Relation*}
paulson@13140
   309
paulson@13512
   310
subsubsection{*Rewrite rule*}
paulson@13140
   311
paulson@13140
   312
lemma rvimage_iff: "<a,b> : rvimage(A,f,r)  <->  <f`a,f`b>: r & a:A & b:A"
paulson@13269
   313
by (unfold rvimage_def, blast)
paulson@13140
   314
paulson@13512
   315
subsubsection{*Type checking*}
paulson@13140
   316
paulson@13140
   317
lemma rvimage_type: "rvimage(A,f,r) <= A*A"
paulson@13784
   318
by (unfold rvimage_def, rule Collect_subset)
paulson@13140
   319
paulson@13140
   320
lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
paulson@13140
   321
paulson@13140
   322
lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
paulson@13269
   323
by (unfold rvimage_def, blast)
paulson@13140
   324
paulson@13140
   325
paulson@13512
   326
subsubsection{*Partial Ordering Properties*}
paulson@13140
   327
paulson@13140
   328
lemma irrefl_rvimage: 
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   329
    "[| f: inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
paulson@13140
   330
apply (unfold irrefl_def rvimage_def)
paulson@13140
   331
apply (blast intro: inj_is_fun [THEN apply_type])
paulson@13140
   332
done
paulson@13140
   333
paulson@13140
   334
lemma trans_on_rvimage: 
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   335
    "[| f: inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
paulson@13140
   336
apply (unfold trans_on_def rvimage_def)
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   337
apply (blast intro: inj_is_fun [THEN apply_type])
paulson@13140
   338
done
paulson@13140
   339
paulson@13140
   340
lemma part_ord_rvimage: 
paulson@13140
   341
    "[| f: inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
paulson@13140
   342
apply (unfold part_ord_def)
paulson@13140
   343
apply (blast intro!: irrefl_rvimage trans_on_rvimage)
paulson@13140
   344
done
paulson@13140
   345
paulson@13512
   346
subsubsection{*Linearity*}
paulson@13140
   347
paulson@13140
   348
lemma linear_rvimage:
paulson@13140
   349
    "[| f: inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
paulson@13140
   350
apply (simp add: inj_def linear_def rvimage_iff) 
paulson@13269
   351
apply (blast intro: apply_funtype) 
paulson@13140
   352
done
paulson@13140
   353
paulson@13140
   354
lemma tot_ord_rvimage: 
paulson@13140
   355
    "[| f: inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
paulson@13140
   356
apply (unfold tot_ord_def)
paulson@13140
   357
apply (blast intro!: part_ord_rvimage linear_rvimage)
paulson@13140
   358
done
paulson@13140
   359
paulson@13140
   360
paulson@13512
   361
subsubsection{*Well-foundedness*}
paulson@13140
   362
paulson@13140
   363
lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
paulson@13140
   364
apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
paulson@13140
   365
apply clarify
paulson@13140
   366
apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
paulson@13140
   367
 apply (erule allE)
paulson@13140
   368
 apply (erule impE)
paulson@13269
   369
 apply assumption
paulson@13140
   370
 apply blast
paulson@13269
   371
apply blast 
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   372
done
paulson@13140
   373
paulson@13544
   374
text{*But note that the combination of @{text wf_imp_wf_on} and
wenzelm@22710
   375
 @{text wf_rvimage} gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}*}
paulson@13140
   376
lemma wf_on_rvimage: "[| f: A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
paulson@13140
   377
apply (rule wf_onI2)
paulson@13140
   378
apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
paulson@13140
   379
 apply blast
paulson@13140
   380
apply (erule_tac a = "f`y" in wf_on_induct)
paulson@13140
   381
 apply (blast intro!: apply_funtype)
paulson@13140
   382
apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
paulson@13140
   383
done
paulson@13140
   384
paulson@13140
   385
(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
paulson@13140
   386
lemma well_ord_rvimage:
paulson@13140
   387
     "[| f: inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
paulson@13140
   388
apply (rule well_ordI)
paulson@13140
   389
apply (unfold well_ord_def tot_ord_def)
paulson@13140
   390
apply (blast intro!: wf_on_rvimage inj_is_fun)
paulson@13140
   391
apply (blast intro!: linear_rvimage)
paulson@13140
   392
done
paulson@13140
   393
paulson@13140
   394
lemma ord_iso_rvimage: 
paulson@13140
   395
    "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
paulson@13140
   396
apply (unfold ord_iso_def)
paulson@13140
   397
apply (simp add: rvimage_iff)
paulson@13140
   398
done
paulson@13140
   399
paulson@13140
   400
lemma ord_iso_rvimage_eq: 
paulson@13140
   401
    "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
paulson@13356
   402
by (unfold ord_iso_def rvimage_def, blast)
paulson@13140
   403
paulson@13140
   404
paulson@13634
   405
subsection{*Every well-founded relation is a subset of some inverse image of
paulson@13634
   406
      an ordinal*}
paulson@13634
   407
paulson@13634
   408
lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
paulson@13634
   409
by (blast intro: wf_rvimage wf_Memrel)
paulson@13634
   410
paulson@13634
   411
wenzelm@24893
   412
definition
wenzelm@24893
   413
  wfrank :: "[i,i]=>i"  where
paulson@13634
   414
    "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
paulson@13634
   415
wenzelm@24893
   416
definition
wenzelm@24893
   417
  wftype :: "i=>i"  where
paulson@13634
   418
    "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
paulson@13634
   419
paulson@13634
   420
lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
paulson@13634
   421
by (subst wfrank_def [THEN def_wfrec], simp_all)
paulson@13634
   422
paulson@13634
   423
lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
paulson@13634
   424
apply (rule_tac a=a in wf_induct, assumption)
paulson@13634
   425
apply (subst wfrank, assumption)
paulson@13634
   426
apply (rule Ord_succ [THEN Ord_UN], blast)
paulson@13634
   427
done
paulson@13634
   428
paulson@13634
   429
lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
paulson@13634
   430
apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
paulson@13634
   431
apply (rule UN_I [THEN ltI])
paulson@13634
   432
apply (simp add: Ord_wfrank vimage_iff)+
paulson@13634
   433
done
paulson@13634
   434
paulson@13634
   435
lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
paulson@13634
   436
by (simp add: wftype_def Ord_wfrank)
paulson@13634
   437
paulson@13634
   438
lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
paulson@13634
   439
apply (simp add: wftype_def)
paulson@13634
   440
apply (blast intro: wfrank_lt [THEN ltD])
paulson@13634
   441
done
paulson@13634
   442
paulson@13634
   443
paulson@13634
   444
lemma wf_imp_subset_rvimage:
paulson@13634
   445
     "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
paulson@13634
   446
apply (rule_tac x="wftype(r)" in exI)
paulson@13634
   447
apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
paulson@13634
   448
apply (simp add: Ord_wftype, clarify)
paulson@13634
   449
apply (frule subsetD, assumption, clarify)
paulson@13634
   450
apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
paulson@13634
   451
apply (blast intro: wftypeI)
paulson@13634
   452
done
paulson@13634
   453
paulson@13634
   454
theorem wf_iff_subset_rvimage:
paulson@13634
   455
  "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
paulson@13634
   456
by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
paulson@13634
   457
          intro: wf_rvimage_Ord [THEN wf_subset])
paulson@13634
   458
paulson@13634
   459
paulson@13544
   460
subsection{*Other Results*}
paulson@13544
   461
paulson@13544
   462
lemma wf_times: "A Int B = 0 ==> wf(A*B)"
paulson@13544
   463
by (simp add: wf_def, blast)
paulson@13544
   464
paulson@13544
   465
text{*Could also be used to prove @{text wf_radd}*}
paulson@13544
   466
lemma wf_Un:
paulson@13544
   467
     "[| range(r) Int domain(s) = 0; wf(r);  wf(s) |] ==> wf(r Un s)"
paulson@13544
   468
apply (simp add: wf_def, clarify) 
paulson@13544
   469
apply (rule equalityI) 
paulson@13544
   470
 prefer 2 apply blast 
paulson@13544
   471
apply clarify 
paulson@13544
   472
apply (drule_tac x=Z in spec)
paulson@13544
   473
apply (drule_tac x="Z Int domain(s)" in spec)
paulson@13544
   474
apply simp 
paulson@13544
   475
apply (blast intro: elim: equalityE) 
paulson@13544
   476
done
paulson@13544
   477
paulson@13544
   478
subsubsection{*The Empty Relation*}
paulson@13544
   479
paulson@13544
   480
lemma wf0: "wf(0)"
paulson@13544
   481
by (simp add: wf_def, blast)
paulson@13544
   482
paulson@13544
   483
lemma linear0: "linear(0,0)"
paulson@13544
   484
by (simp add: linear_def)
paulson@13544
   485
paulson@13544
   486
lemma well_ord0: "well_ord(0,0)"
paulson@13544
   487
by (blast intro: wf_imp_wf_on well_ordI wf0 linear0)
paulson@13512
   488
paulson@13512
   489
subsubsection{*The "measure" relation is useful with wfrec*}
paulson@13140
   490
paulson@13140
   491
lemma measure_eq_rvimage_Memrel:
paulson@13140
   492
     "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
paulson@13140
   493
apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
paulson@13269
   494
apply (rule equalityI, auto)
paulson@13140
   495
apply (auto intro: Ord_in_Ord simp add: lt_def)
paulson@13140
   496
done
paulson@13140
   497
paulson@13140
   498
lemma wf_measure [iff]: "wf(measure(A,f))"
paulson@13356
   499
by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
paulson@13140
   500
paulson@13140
   501
lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
paulson@13356
   502
by (simp (no_asm) add: measure_def)
paulson@13140
   503
paulson@13544
   504
lemma linear_measure: 
paulson@13544
   505
 assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
paulson@13544
   506
     and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
paulson@13544
   507
 shows "linear(A, measure(A,f))"
paulson@13544
   508
apply (auto simp add: linear_def) 
paulson@13544
   509
apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt) 
paulson@13544
   510
    apply (simp_all add: Ordf) 
paulson@13544
   511
apply (blast intro: inj) 
paulson@13544
   512
done
paulson@13544
   513
paulson@13544
   514
lemma wf_on_measure: "wf[B](measure(A,f))"
paulson@13544
   515
by (rule wf_imp_wf_on [OF wf_measure])
paulson@13544
   516
paulson@13544
   517
lemma well_ord_measure: 
paulson@13544
   518
 assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
paulson@13544
   519
     and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
paulson@13544
   520
 shows "well_ord(A, measure(A,f))"
paulson@13544
   521
apply (rule well_ordI)
paulson@13544
   522
apply (rule wf_on_measure) 
paulson@13544
   523
apply (blast intro: linear_measure Ordf inj) 
paulson@13544
   524
done
paulson@13544
   525
paulson@13544
   526
lemma measure_type: "measure(A,f) <= A*A"
paulson@13544
   527
by (auto simp add: measure_def)
paulson@13544
   528
paulson@13512
   529
subsubsection{*Well-foundedness of Unions*}
paulson@13512
   530
paulson@13512
   531
lemma wf_on_Union:
paulson@13512
   532
 assumes wfA: "wf[A](r)"
paulson@13512
   533
     and wfB: "!!a. a\<in>A ==> wf[B(a)](s)"
paulson@13512
   534
     and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|] 
paulson@13512
   535
                       ==> (\<exists>a'\<in>A. <a',a> \<in> r & u \<in> B(a')) | u \<in> B(a)"
paulson@13512
   536
 shows "wf[\<Union>a\<in>A. B(a)](s)"
paulson@13512
   537
apply (rule wf_onI2)
paulson@13512
   538
apply (erule UN_E)
paulson@13512
   539
apply (subgoal_tac "\<forall>z \<in> B(a). z \<in> Ba", blast)
paulson@13512
   540
apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
paulson@13512
   541
apply (rule ballI)
paulson@13512
   542
apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
paulson@13512
   543
apply (rename_tac u) 
paulson@13512
   544
apply (drule_tac x=u in bspec, blast) 
paulson@13512
   545
apply (erule mp, clarify)
paulson@13784
   546
apply (frule ok, assumption+, blast) 
paulson@13512
   547
done
paulson@13512
   548
paulson@14120
   549
subsubsection{*Bijections involving Powersets*}
paulson@14120
   550
paulson@14120
   551
lemma Pow_sum_bij:
paulson@14120
   552
    "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)  
paulson@14120
   553
     \<in> bij(Pow(A+B), Pow(A)*Pow(B))"
paulson@14120
   554
apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} Un {Inr (y). y \<in> Y}" 
paulson@14120
   555
       in lam_bijective)
paulson@14120
   556
apply force+
paulson@14120
   557
done
paulson@14120
   558
paulson@14120
   559
text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *}
paulson@14120
   560
lemma Pow_Sigma_bij:
paulson@14120
   561
    "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})  
skalberg@14171
   562
     \<in> bij(Pow(Sigma(A,B)), \<Pi> x \<in> A. Pow(B(x)))"
paulson@14120
   563
apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
paulson@14120
   564
apply (blast intro: lam_type)
paulson@14120
   565
apply (blast dest: apply_type, simp_all)
paulson@14120
   566
apply fast (*strange, but blast can't do it*)
paulson@14120
   567
apply (rule fun_extension, auto)
paulson@14120
   568
by blast
paulson@14120
   569
lcp@437
   570
end