src/ZF/Constructible/WFrec.thy
author wenzelm
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(*  Title:      ZF/Constructible/WFrec.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header{*Relativized Well-Founded Recursion*}
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theory WFrec imports Wellorderings begin
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subsection{*General Lemmas*}
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(*Many of these might be useful in WF.thy*)
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lemma apply_recfun2:
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    "[| is_recfun(r,a,H,f); <x,i>:f |] ==> i = H(x, restrict(f,r-``{x}))"
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apply (frule apply_recfun) 
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 apply (blast dest: is_recfun_type fun_is_rel) 
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apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
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done
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text{*Expresses @{text is_recfun} as a recursion equation*}
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lemma is_recfun_iff_equation:
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     "is_recfun(r,a,H,f) \<longleftrightarrow>
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           f \<in> r -`` {a} \<rightarrow> range(f) &
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           (\<forall>x \<in> r-``{a}. f`x = H(x, restrict(f, r-``{x})))"  
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apply (rule iffI) 
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 apply (simp add: is_recfun_type apply_recfun Ball_def vimage_singleton_iff, 
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        clarify)  
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apply (simp add: is_recfun_def) 
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apply (rule fun_extension) 
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  apply assumption
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 apply (fast intro: lam_type, simp) 
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done
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lemma is_recfun_imp_in_r: "[|is_recfun(r,a,H,f); \<langle>x,i\<rangle> \<in> f|] ==> \<langle>x, a\<rangle> \<in> r"
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by (blast dest: is_recfun_type fun_is_rel)
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lemma trans_Int_eq:
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      "[| trans(r); <y,x> \<in> r |] ==> r -`` {x} \<inter> r -`` {y} = r -`` {y}"
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by (blast intro: transD) 
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lemma is_recfun_restrict_idem:
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     "is_recfun(r,a,H,f) ==> restrict(f, r -`` {a}) = f"
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apply (drule is_recfun_type)
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apply (auto simp add: Pi_iff subset_Sigma_imp_relation restrict_idem)  
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done
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lemma is_recfun_cong_lemma:
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  "[| is_recfun(r,a,H,f); r = r'; a = a'; f = f'; 
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      !!x g. [| <x,a'> \<in> r'; relation(g); domain(g) \<subseteq> r' -``{x} |] 
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             ==> H(x,g) = H'(x,g) |]
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   ==> is_recfun(r',a',H',f')"
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apply (simp add: is_recfun_def) 
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apply (erule trans) 
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apply (rule lam_cong) 
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apply (simp_all add: vimage_singleton_iff Int_lower2)  
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done
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text{*For @{text is_recfun} we need only pay attention to functions
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      whose domains are initial segments of @{term r}.*}
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lemma is_recfun_cong:
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  "[| r = r'; a = a'; f = f'; 
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      !!x g. [| <x,a'> \<in> r'; relation(g); domain(g) \<subseteq> r' -``{x} |] 
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             ==> H(x,g) = H'(x,g) |]
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   ==> is_recfun(r,a,H,f) \<longleftrightarrow> is_recfun(r',a',H',f')"
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apply (rule iffI)
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txt{*Messy: fast and blast don't work for some reason*}
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apply (erule is_recfun_cong_lemma, auto) 
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apply (erule is_recfun_cong_lemma)
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apply (blast intro: sym)+
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done
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subsection{*Reworking of the Recursion Theory Within @{term M}*}
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lemma (in M_basic) is_recfun_separation':
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    "[| f \<in> r -`` {a} \<rightarrow> range(f); g \<in> r -`` {b} \<rightarrow> range(g);
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        M(r); M(f); M(g); M(a); M(b) |] 
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     ==> separation(M, \<lambda>x. \<not> (\<langle>x, a\<rangle> \<in> r \<longrightarrow> \<langle>x, b\<rangle> \<in> r \<longrightarrow> f ` x = g ` x))"
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apply (insert is_recfun_separation [of r f g a b]) 
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apply (simp add: vimage_singleton_iff)
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done
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text{*Stated using @{term "trans(r)"} rather than
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      @{term "transitive_rel(M,A,r)"} because the latter rewrites to
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      the former anyway, by @{text transitive_rel_abs}.
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      As always, theorems should be expressed in simplified form.
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      The last three M-premises are redundant because of @{term "M(r)"}, 
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      but without them we'd have to undertake
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      more work to set up the induction formula.*}
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lemma (in M_basic) is_recfun_equal [rule_format]: 
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    "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
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       wellfounded(M,r);  trans(r);
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       M(f); M(g); M(r); M(x); M(a); M(b) |] 
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     ==> <x,a> \<in> r \<longrightarrow> <x,b> \<in> r \<longrightarrow> f`x=g`x"
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apply (frule_tac f=f in is_recfun_type) 
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apply (frule_tac f=g in is_recfun_type) 
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apply (simp add: is_recfun_def)
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apply (erule_tac a=x in wellfounded_induct, assumption+)
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txt{*Separation to justify the induction*}
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 apply (blast intro: is_recfun_separation') 
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txt{*Now the inductive argument itself*}
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apply clarify 
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apply (erule ssubst)+
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apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
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apply (rename_tac x1)
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apply (rule_tac t="%z. H(x1,z)" in subst_context) 
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apply (subgoal_tac "\<forall>y \<in> r-``{x1}. \<forall>z. <y,z>\<in>f \<longleftrightarrow> <y,z>\<in>g")
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 apply (blast intro: transD) 
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apply (simp add: apply_iff) 
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apply (blast intro: transD sym) 
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done
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lemma (in M_basic) is_recfun_cut: 
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    "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
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       wellfounded(M,r); trans(r); 
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       M(f); M(g); M(r); <b,a> \<in> r |]   
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      ==> restrict(f, r-``{b}) = g"
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apply (frule_tac f=f in is_recfun_type) 
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apply (rule fun_extension) 
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apply (blast intro: transD restrict_type2) 
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apply (erule is_recfun_type, simp) 
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apply (blast intro: is_recfun_equal transD dest: transM) 
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done
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lemma (in M_basic) is_recfun_functional:
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     "[|is_recfun(r,a,H,f);  is_recfun(r,a,H,g);  
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       wellfounded(M,r); trans(r); M(f); M(g); M(r) |] ==> f=g"
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apply (rule fun_extension)
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apply (erule is_recfun_type)+
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apply (blast intro!: is_recfun_equal dest: transM) 
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done 
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text{*Tells us that @{text is_recfun} can (in principle) be relativized.*}
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lemma (in M_basic) is_recfun_relativize:
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  "[| M(r); M(f); \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |] 
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   ==> is_recfun(r,a,H,f) \<longleftrightarrow>
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       (\<forall>z[M]. z \<in> f \<longleftrightarrow> 
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        (\<exists>x[M]. <x,a> \<in> r & z = <x, H(x, restrict(f, r-``{x}))>))";
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apply (simp add: is_recfun_def lam_def)
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apply (safe intro!: equalityI) 
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   apply (drule equalityD1 [THEN subsetD], assumption) 
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   apply (blast dest: pair_components_in_M) 
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  apply (blast elim!: equalityE dest: pair_components_in_M)
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 apply (frule transM, assumption) 
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 apply simp  
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 apply blast
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apply (subgoal_tac "is_function(M,f)")
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 txt{*We use @{term "is_function"} rather than @{term "function"} because
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      the subgoal's easier to prove with relativized quantifiers!*}
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 prefer 2 apply (simp add: is_function_def) 
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apply (frule pair_components_in_M, assumption) 
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apply (simp add: is_recfun_imp_function function_restrictI) 
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done
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lemma (in M_basic) is_recfun_restrict:
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     "[| wellfounded(M,r); trans(r); is_recfun(r,x,H,f); \<langle>y,x\<rangle> \<in> r; 
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       M(r); M(f); 
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       \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]
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       ==> is_recfun(r, y, H, restrict(f, r -`` {y}))"
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apply (frule pair_components_in_M, assumption, clarify) 
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apply (simp (no_asm_simp) add: is_recfun_relativize restrict_iff
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           trans_Int_eq)
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apply safe
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  apply (simp_all add: vimage_singleton_iff is_recfun_type [THEN apply_iff]) 
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  apply (frule_tac x=xa in pair_components_in_M, assumption)
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  apply (frule_tac x=xa in apply_recfun, blast intro: transD)  
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  apply (simp add: is_recfun_type [THEN apply_iff] 
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                   is_recfun_imp_function function_restrictI)
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apply (blast intro: apply_recfun dest: transD)
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done
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lemma (in M_basic) restrict_Y_lemma:
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   "[| wellfounded(M,r); trans(r); M(r);
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       \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g));  M(Y);
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       \<forall>b[M]. 
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           b \<in> Y \<longleftrightarrow>
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           (\<exists>x[M]. <x,a1> \<in> r &
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            (\<exists>y[M]. b = \<langle>x,y\<rangle> & (\<exists>g[M]. is_recfun(r,x,H,g) \<and> y = H(x,g))));
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          \<langle>x,a1\<rangle> \<in> r; is_recfun(r,x,H,f); M(f) |]
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       ==> restrict(Y, r -`` {x}) = f"
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apply (subgoal_tac "\<forall>y \<in> r-``{x}. \<forall>z. <y,z>:Y \<longleftrightarrow> <y,z>:f") 
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 apply (simp (no_asm_simp) add: restrict_def) 
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 apply (thin_tac "rall(M,?P)")+  --{*essential for efficiency*}
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 apply (frule is_recfun_type [THEN fun_is_rel], blast)
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apply (frule pair_components_in_M, assumption, clarify) 
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apply (rule iffI)
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 apply (frule_tac y="<y,z>" in transM, assumption)
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 apply (clarsimp simp add: vimage_singleton_iff is_recfun_type [THEN apply_iff]
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                           apply_recfun is_recfun_cut) 
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txt{*Opposite inclusion: something in f, show in Y*}
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apply (frule_tac y="<y,z>" in transM, assumption)  
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apply (simp add: vimage_singleton_iff) 
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apply (rule conjI) 
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 apply (blast dest: transD) 
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apply (rule_tac x="restrict(f, r -`` {y})" in rexI) 
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apply (simp_all add: is_recfun_restrict
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                     apply_recfun is_recfun_type [THEN apply_iff]) 
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done
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text{*For typical applications of Replacement for recursive definitions*}
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lemma (in M_basic) univalent_is_recfun:
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     "[|wellfounded(M,r); trans(r); M(r)|]
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      ==> univalent (M, A, \<lambda>x p. 
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              \<exists>y[M]. p = \<langle>x,y\<rangle> & (\<exists>f[M]. is_recfun(r,x,H,f) & y = H(x,f)))"
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apply (simp add: univalent_def) 
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apply (blast dest: is_recfun_functional) 
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done
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text{*Proof of the inductive step for @{text exists_is_recfun}, since
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      we must prove two versions.*}
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lemma (in M_basic) exists_is_recfun_indstep:
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    "[|\<forall>y. \<langle>y, a1\<rangle> \<in> r \<longrightarrow> (\<exists>f[M]. is_recfun(r, y, H, f)); 
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       wellfounded(M,r); trans(r); M(r); M(a1);
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       strong_replacement(M, \<lambda>x z. 
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              \<exists>y[M]. \<exists>g[M]. pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
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       \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]   
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      ==> \<exists>f[M]. is_recfun(r,a1,H,f)"
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apply (drule_tac A="r-``{a1}" in strong_replacementD)
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  apply blast 
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 txt{*Discharge the "univalent" obligation of Replacement*}
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 apply (simp add: univalent_is_recfun) 
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txt{*Show that the constructed object satisfies @{text is_recfun}*} 
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apply clarify 
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apply (rule_tac x=Y in rexI)  
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txt{*Unfold only the top-level occurrence of @{term is_recfun}*}
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apply (simp (no_asm_simp) add: is_recfun_relativize [of concl: _ a1])
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txt{*The big iff-formula defining @{term Y} is now redundant*}
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apply safe 
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 apply (simp add: vimage_singleton_iff restrict_Y_lemma [of r H _ a1]) 
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txt{*one more case*}
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apply (simp (no_asm_simp) add: Bex_def vimage_singleton_iff)
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apply (drule_tac x1=x in spec [THEN mp], assumption, clarify) 
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apply (rename_tac f) 
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apply (rule_tac x=f in rexI) 
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apply (simp_all add: restrict_Y_lemma [of r H])
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txt{*FIXME: should not be needed!*}
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apply (subst restrict_Y_lemma [of r H])
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apply (simp add: vimage_singleton_iff)+
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apply blast+
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done
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text{*Relativized version, when we have the (currently weaker) premise
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      @{term "wellfounded(M,r)"}*}
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lemma (in M_basic) wellfounded_exists_is_recfun:
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    "[|wellfounded(M,r);  trans(r);  
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       separation(M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r, x, H, f)));
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       strong_replacement(M, \<lambda>x z. 
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          \<exists>y[M]. \<exists>g[M]. pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
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       M(r);  M(a);  
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       \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]   
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      ==> \<exists>f[M]. is_recfun(r,a,H,f)"
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apply (rule wellfounded_induct, assumption+, clarify)
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apply (rule exists_is_recfun_indstep, assumption+)
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done
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lemma (in M_basic) wf_exists_is_recfun [rule_format]:
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    "[|wf(r);  trans(r);  M(r);  
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       strong_replacement(M, \<lambda>x z. 
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         \<exists>y[M]. \<exists>g[M]. pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
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       \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]   
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      ==> M(a) \<longrightarrow> (\<exists>f[M]. is_recfun(r,a,H,f))"
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apply (rule wf_induct, assumption+)
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apply (frule wf_imp_relativized)
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apply (intro impI)
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apply (rule exists_is_recfun_indstep) 
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      apply (blast dest: transM del: rev_rallE, assumption+)
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done
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subsection{*Relativization of the ZF Predicate @{term is_recfun}*}
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definition
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  M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o" where
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   "M_is_recfun(M,MH,r,a,f) == 
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     \<forall>z[M]. z \<in> f \<longleftrightarrow> 
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            (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M]. 
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               pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
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               pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
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               xa \<in> r & MH(x, f_r_sx, y))"
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definition
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  is_wfrec :: "[i=>o, [i,i,i]=>o, i, i, i] => o" where
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   "is_wfrec(M,MH,r,a,z) == 
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      \<exists>f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)"
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definition
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  wfrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o" where
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   "wfrec_replacement(M,MH,r) ==
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        strong_replacement(M, 
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             \<lambda>x z. \<exists>y[M]. pair(M,x,y,z) & is_wfrec(M,MH,r,x,y))"
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lemma (in M_basic) is_recfun_abs:
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     "[| \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g));  M(r); M(a); M(f); 
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         relation2(M,MH,H) |] 
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      ==> M_is_recfun(M,MH,r,a,f) \<longleftrightarrow> is_recfun(r,a,H,f)"
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apply (simp add: M_is_recfun_def relation2_def is_recfun_relativize)
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apply (rule rall_cong)
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apply (blast dest: transM)
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done
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lemma M_is_recfun_cong [cong]:
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     "[| r = r'; a = a'; f = f'; 
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       !!x g y. [| M(x); M(g); M(y) |] ==> MH(x,g,y) \<longleftrightarrow> MH'(x,g,y) |]
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      ==> M_is_recfun(M,MH,r,a,f) \<longleftrightarrow> M_is_recfun(M,MH',r',a',f')"
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by (simp add: M_is_recfun_def) 
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lemma (in M_basic) is_wfrec_abs:
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     "[| \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)); 
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         relation2(M,MH,H);  M(r); M(a); M(z) |]
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      ==> is_wfrec(M,MH,r,a,z) \<longleftrightarrow> 
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          (\<exists>g[M]. is_recfun(r,a,H,g) & z = H(a,g))"
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by (simp add: is_wfrec_def relation2_def is_recfun_abs)
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text{*Relating @{term wfrec_replacement} to native constructs*}
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lemma (in M_basic) wfrec_replacement':
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  "[|wfrec_replacement(M,MH,r);
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     \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)); 
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     relation2(M,MH,H);  M(r)|] 
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   ==> strong_replacement(M, \<lambda>x z. \<exists>y[M]. 
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                pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)))"
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by (simp add: wfrec_replacement_def is_wfrec_abs) 
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lemma wfrec_replacement_cong [cong]:
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     "[| !!x y z. [| M(x); M(y); M(z) |] ==> MH(x,y,z) \<longleftrightarrow> MH'(x,y,z);
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         r=r' |] 
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      ==> wfrec_replacement(M, %x y. MH(x,y), r) \<longleftrightarrow> 
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          wfrec_replacement(M, %x y. MH'(x,y), r')" 
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by (simp add: is_wfrec_def wfrec_replacement_def) 
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end
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