(*  Title:      ZF/Constructible/WFrec.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Relativized Well-Founded Recursion›

theory WFrec imports Wellorderings begin


subsection‹General Lemmas›

(*Many of these might be useful in WF.thy*)

lemma apply_recfun2:
    "[is_recfun(r,a,H,f); ⟨x,i⟩:f] ==> i = H(x, restrict(f,r-``{x}))"
apply (frule apply_recfun) 
 apply (blast dest: is_recfun_type fun_is_rel) 
apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
done

text‹Expresses ‹is_recfun› as a recursion equation›
lemma is_recfun_iff_equation:
     "is_recfun(r,a,H,f) ⟷
           f ∈ r -`` {a} → range(f) ∧
           (∀x ∈ r-``{a}. f`x = H(x, restrict(f, r-``{x})))"  
apply (rule iffI) 
 apply (simp add: is_recfun_type apply_recfun Ball_def vimage_singleton_iff, 
        clarify)  
apply (simp add: is_recfun_def) 
apply (rule fun_extension) 
  apply assumption
 apply (fast intro: lam_type, simp) 
done

lemma is_recfun_imp_in_r: "[is_recfun(r,a,H,f); ⟨x,i⟩ ∈ f] ==> ⟨x, a⟩ ∈ r"
by (blast dest: is_recfun_type fun_is_rel)

lemma trans_Int_eq:
      "[trans(r); ⟨y,x⟩ ∈ r] ==> r -`` {x} ∩ r -`` {y} = r -`` {y}"
by (blast intro: transD) 

lemma is_recfun_restrict_idem:
     "is_recfun(r,a,H,f) ==> restrict(f, r -`` {a}) = f"
apply (drule is_recfun_type)
apply (auto simp add: Pi_iff subset_Sigma_imp_relation restrict_idem)  
done

lemma is_recfun_cong_lemma:
  "[is_recfun(r,a,H,f); r = r'; a = a'; f = f';
      ∧x g. [<x,a'> ∈ r'; relation(g); domain(g) ⊆ r' -``{x}]
             ==> H(x,g) = H'(x,g)]
   ==> is_recfun(r',a',H',f')"
apply (simp add: is_recfun_def) 
apply (erule trans) 
apply (rule lam_cong) 
apply (simp_all add: vimage_singleton_iff Int_lower2)  
done

text‹For ‹is_recfun› we need only pay attention to functions
 whose domains are initial segments of term‹r›.›
lemma is_recfun_cong:
  "[r = r'; a = a'; f = f';
      ∧x g. [<x,a'> ∈ r'; relation(g); domain(g) ⊆ r' -``{x}]
             ==> H(x,g) = H'(x,g)]
   ==> is_recfun(r,a,H,f) ⟷ is_recfun(r',a',H',f')"
apply (rule iffI)
txt‹Messy: fast and blast don't work for some reason›
apply (erule is_recfun_cong_lemma, auto) 
apply (erule is_recfun_cong_lemma)
apply (blast intro: sym)+
done

subsection‹Reworking of the Recursion Theory Within term‹M››

lemma (in M_basic) is_recfun_separation':
    "[f ∈ r -`` {a} → range(f); g ∈ r -`` {b} → range(g);
        M(r); M(f); M(g); M(a); M(b)]
     ==> separation(M, λx. ¬ (⟨x, a⟩ ∈ r ⟶ ⟨x, b⟩ ∈ r ⟶ f ` x = g ` x))"
apply (insert is_recfun_separation [of r f g a b]) 
apply (simp add: vimage_singleton_iff)
done

text‹Stated using term‹trans(r)› rather than
 term‹transitive_rel(M,A,r)› because the latter rewrites to
 the former anyway, by ‹transitive_rel_abs›.
 As always, theorems should be expressed in simplified form.
 The last three M-premises are redundant because of term‹M(r)›,
 but without them we'd have to undertake
 more work to set up the induction formula.›
lemma (in M_basic) is_recfun_equal [rule_format]: 
    "[is_recfun(r,a,H,f); is_recfun(r,b,H,g);
       wellfounded(M,r); trans(r);
       M(f); M(g); M(r); M(x); M(a); M(b)]
     ==> ⟨x,a⟩ ∈ r ⟶ ⟨x,b⟩ ∈ r ⟶ f`x=g`x"
apply (frule_tac f=f in is_recfun_type) 
apply (frule_tac f=g in is_recfun_type) 
apply (simp add: is_recfun_def)
apply (erule_tac a=x in wellfounded_induct, assumption+)
txt‹Separation to justify the induction›
 apply (blast intro: is_recfun_separation') 
txt‹Now the inductive argument itself›
apply clarify 
apply (erule ssubst)+
apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
apply (rename_tac x1)
apply (rule_tac t="λz. H(x1,z)" in subst_context) 
apply (subgoal_tac "∀y ∈ r-``{x1}. ∀z. ⟨y,z⟩∈f ⟷ ⟨y,z⟩∈g")
 apply (blast intro: transD) 
apply (simp add: apply_iff) 
apply (blast intro: transD sym) 
done

lemma (in M_basic) is_recfun_cut: 
    "[is_recfun(r,a,H,f); is_recfun(r,b,H,g);
       wellfounded(M,r); trans(r);
       M(f); M(g); M(r); ⟨b,a⟩ ∈ r]
      ==> restrict(f, r-``{b}) = g"
apply (frule_tac f=f in is_recfun_type) 
apply (rule fun_extension) 
apply (blast intro: transD restrict_type2) 
apply (erule is_recfun_type, simp) 
apply (blast intro: is_recfun_equal transD dest: transM) 
done

lemma (in M_basic) is_recfun_functional:
     "[is_recfun(r,a,H,f); is_recfun(r,a,H,g);
       wellfounded(M,r); trans(r); M(f); M(g); M(r)] ==> f=g"
apply (rule fun_extension)
apply (erule is_recfun_type)+
apply (blast intro!: is_recfun_equal dest: transM) 
done 

text‹Tells us that ‹is_recfun› can (in principle) be relativized.›
lemma (in M_basic) is_recfun_relativize:
  "[M(r); M(f); ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))]
   ==> is_recfun(r,a,H,f) ⟷
       (∀z[M]. z ∈ f ⟷
        (∃x[M]. ⟨x,a⟩ ∈ r ∧ z = <x, H(x, restrict(f, r-``{x}))>))"
apply (simp add: is_recfun_def lam_def)
apply (safe intro!: equalityI) 
   apply (drule equalityD1 [THEN subsetD], assumption) 
   apply (blast dest: pair_components_in_M) 
  apply (blast elim!: equalityE dest: pair_components_in_M)
 apply (frule transM, assumption) 
 apply simp  
 apply blast
apply (subgoal_tac "is_function(M,f)")
 txt‹We use term‹is_function› rather than term‹function› because
 the subgoal's easier to prove with relativized quantifiers!›
 prefer 2 apply (simp add: is_function_def) 
apply (frule pair_components_in_M, assumption) 
apply (simp add: is_recfun_imp_function function_restrictI) 
done

lemma (in M_basic) is_recfun_restrict:
     "[wellfounded(M,r); trans(r); is_recfun(r,x,H,f); ⟨y,x⟩ ∈ r;
       M(r); M(f);
       ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))]
       ==> is_recfun(r, y, H, restrict(f, r -`` {y}))"
apply (frule pair_components_in_M, assumption, clarify) 
apply (simp (no_asm_simp) add: is_recfun_relativize restrict_iff
           trans_Int_eq)
apply safe
  apply (simp_all add: vimage_singleton_iff is_recfun_type [THEN apply_iff]) 
  apply (frule_tac x=xa in pair_components_in_M, assumption)
  apply (frule_tac x=xa in apply_recfun, blast intro: transD)  
  apply (simp add: is_recfun_type [THEN apply_iff] 
                   is_recfun_imp_function function_restrictI)
apply (blast intro: apply_recfun dest: transD)
done
 
lemma (in M_basic) restrict_Y_lemma:
   "[wellfounded(M,r); trans(r); M(r);
       ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g)); M(Y);
       ∀b[M].
           b ∈ Y ⟷
           (∃x[M]. ⟨x,a1⟩ ∈ r ∧
            (∃y[M]. b = ⟨x,y⟩ ∧ (∃g[M]. is_recfun(r,x,H,g) ∧ y = H(x,g))));
          ⟨x,a1⟩ ∈ r; is_recfun(r,x,H,f); M(f)]
       ==> restrict(Y, r -`` {x}) = f"
apply (subgoal_tac "∀y ∈ r-``{x}. ∀z. ⟨y,z⟩:Y ⟷ ⟨y,z⟩:f") 
 apply (simp (no_asm_simp) add: restrict_def) 
 apply (thin_tac "rall(M,P)" for P)+  ― ‹essential for efficiency›
 apply (frule is_recfun_type [THEN fun_is_rel], blast)
apply (frule pair_components_in_M, assumption, clarify) 
apply (rule iffI)
 apply (frule_tac y="⟨y,z⟩" in transM, assumption)
 apply (clarsimp simp add: vimage_singleton_iff is_recfun_type [THEN apply_iff]
                           apply_recfun is_recfun_cut) 
txt‹Opposite inclusion: something in f, show in Y›
apply (frule_tac y="⟨y,z⟩" in transM, assumption)  
apply (simp add: vimage_singleton_iff) 
apply (rule conjI) 
 apply (blast dest: transD) 
apply (rule_tac x="restrict(f, r -`` {y})" in rexI) 
apply (simp_all add: is_recfun_restrict
                     apply_recfun is_recfun_type [THEN apply_iff]) 
done

text‹For typical applications of Replacement for recursive definitions›
lemma (in M_basic) univalent_is_recfun:
     "[wellfounded(M,r); trans(r); M(r)]
      ==> univalent (M, A, λx p.
              ∃y[M]. p = ⟨x,y⟩ ∧ (∃f[M]. is_recfun(r,x,H,f) ∧ y = H(x,f)))"
apply (simp add: univalent_def) 
apply (blast dest: is_recfun_functional) 
done


text‹Proof of the inductive step for ‹exists_is_recfun›, since
 we must prove two versions.›
lemma (in M_basic) exists_is_recfun_indstep:
    "[∀y. ⟨y, a1⟩ ∈ r ⟶ (∃f[M]. is_recfun(r, y, H, f));
       wellfounded(M,r); trans(r); M(r); M(a1);
       strong_replacement(M, λx z.
              ∃y[M]. ∃g[M]. pair(M,x,y,z) ∧ is_recfun(r,x,H,g) ∧ y = H(x,g));
       ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))]
      ==> ∃f[M]. is_recfun(r,a1,H,f)"
apply (drule_tac A="r-``{a1}" in strong_replacementD)
  apply blast 
 txt‹Discharge the "univalent" obligation of Replacement›
 apply (simp add: univalent_is_recfun) 
txt‹Show that the constructed object satisfies ‹is_recfun››
apply clarify 
apply (rule_tac x=Y in rexI)  
txt‹Unfold only the top-level occurrence of term‹is_recfun››
apply (simp (no_asm_simp) add: is_recfun_relativize [of concl: _ a1])
txt‹The big iff-formula defining term‹Y› is now redundant›
apply safe 
 apply (simp add: vimage_singleton_iff restrict_Y_lemma [of r H _ a1]) 
txt‹one more case›
apply (simp (no_asm_simp) add: Bex_def vimage_singleton_iff)
apply (drule_tac x1=x in spec [THEN mp], assumption, clarify) 
apply (rename_tac f) 
apply (rule_tac x=f in rexI) 
apply (simp_all add: restrict_Y_lemma [of r H])
txt‹FIXME: should not be needed!›
apply (subst restrict_Y_lemma [of r H])
apply (simp add: vimage_singleton_iff)+
apply blast+
done

text‹Relativized version, when we have the (currently weaker) premise
 term‹wellfounded(M,r)››
lemma (in M_basic) wellfounded_exists_is_recfun:
    "[wellfounded(M,r); trans(r);
       separation(M, λx. ¬ (∃f[M]. is_recfun(r, x, H, f)));
       strong_replacement(M, λx z.
          ∃y[M]. ∃g[M]. pair(M,x,y,z) ∧ is_recfun(r,x,H,g) ∧ y = H(x,g));
       M(r); M(a);
       ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))]
      ==> ∃f[M]. is_recfun(r,a,H,f)"
apply (rule wellfounded_induct, assumption+, clarify)
apply (rule exists_is_recfun_indstep, assumption+)
done

lemma (in M_basic) wf_exists_is_recfun [rule_format]:
    "[wf(r); trans(r); M(r);
       strong_replacement(M, λx z.
         ∃y[M]. ∃g[M]. pair(M,x,y,z) ∧ is_recfun(r,x,H,g) ∧ y = H(x,g));
       ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))]
      ==> M(a) ⟶ (∃f[M]. is_recfun(r,a,H,f))"
apply (rule wf_induct, assumption+)
apply (frule wf_imp_relativized)
apply (intro impI)
apply (rule exists_is_recfun_indstep) 
      apply (blast dest: transM del: rev_rallE, assumption+)
done


subsection‹Relativization of the ZF Predicate term‹is_recfun››

definition
  M_is_recfun :: "[i→o, [i,i,i]→o, i, i, i] → o" where
   "M_is_recfun(M,MH,r,a,f) ≡
     ∀z[M]. z ∈ f ⟷
            (∃x[M]. ∃y[M]. ∃xa[M]. ∃sx[M]. ∃r_sx[M]. ∃f_r_sx[M].
               pair(M,x,y,z) ∧ pair(M,x,a,xa) ∧ upair(M,x,x,sx) ∧
               pre_image(M,r,sx,r_sx) ∧ restriction(M,f,r_sx,f_r_sx) ∧
               xa ∈ r ∧ MH(x, f_r_sx, y))"

definition
  is_wfrec :: "[i→o, [i,i,i]→o, i, i, i] → o" where
   "is_wfrec(M,MH,r,a,z) ≡
      ∃f[M]. M_is_recfun(M,MH,r,a,f) ∧ MH(a,f,z)"

definition
  wfrec_replacement :: "[i→o, [i,i,i]→o, i] → o" where
   "wfrec_replacement(M,MH,r) ≡
        strong_replacement(M,
             λx z. ∃y[M]. pair(M,x,y,z) ∧ is_wfrec(M,MH,r,x,y))"

lemma (in M_basic) is_recfun_abs:
     "[∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g)); M(r); M(a); M(f);
         relation2(M,MH,H)]
      ==> M_is_recfun(M,MH,r,a,f) ⟷ is_recfun(r,a,H,f)"
apply (simp add: M_is_recfun_def relation2_def is_recfun_relativize)
apply (rule rall_cong)
apply (blast dest: transM)
done

lemma M_is_recfun_cong [cong]:
     "[r = r'; a = a'; f = f';
       ∧x g y. [M(x); M(g); M(y)] ==> MH(x,g,y) ⟷ MH'(x,g,y)]
      ==> M_is_recfun(M,MH,r,a,f) ⟷ M_is_recfun(M,MH',r',a',f')"
by (simp add: M_is_recfun_def) 

lemma (in M_basic) is_wfrec_abs:
     "[∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g));
         relation2(M,MH,H); M(r); M(a); M(z)]
      ==> is_wfrec(M,MH,r,a,z) ⟷
          (∃g[M]. is_recfun(r,a,H,g) ∧ z = H(a,g))"
by (simp add: is_wfrec_def relation2_def is_recfun_abs)

text‹Relating term‹wfrec_replacement› to native constructs›
lemma (in M_basic) wfrec_replacement':
  "[wfrec_replacement(M,MH,r);
     ∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g));
     relation2(M,MH,H); M(r)]
   ==> strong_replacement(M, λx z. ∃y[M].
                pair(M,x,y,z) ∧ (∃g[M]. is_recfun(r,x,H,g) ∧ y = H(x,g)))"
by (simp add: wfrec_replacement_def is_wfrec_abs) 

lemma wfrec_replacement_cong [cong]:
     "[∧x y z. [M(x); M(y); M(z)] ==> MH(x,y,z) ⟷ MH'(x,y,z);
         r=r']
      ==> wfrec_replacement(M, λx y. MH(x,y), r) ⟷
          wfrec_replacement(M, λx y. MH'(x,y), r')" 
by (simp add: is_wfrec_def wfrec_replacement_def) 


end