Theory Separation

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

section‹Early Instances of Separation and Strong Replacement›

theory Separation imports L_axioms WF_absolute begin

text‹This theory proves all instances needed for locale ‹M_basic››

text‹Helps us solve for de Bruijn indices!›
lemma nth_ConsI: "[|nth(n,l) = x; n ∈ nat|] ==> nth(succ(n), Cons(a,l)) = x"
by simp

lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
                   fun_plus_iff_sats

lemma Collect_conj_in_DPow:
     "[| {x∈A. P(x)} ∈ DPow(A);  {x∈A. Q(x)} ∈ DPow(A) |]
      ==> {x∈A. P(x) & Q(x)} ∈ DPow(A)"
by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])

lemma Collect_conj_in_DPow_Lset:
     "[|z ∈ Lset(j); {x ∈ Lset(j). P(x)} ∈ DPow(Lset(j))|]
      ==> {x ∈ Lset(j). x ∈ z & P(x)} ∈ DPow(Lset(j))"
apply (frule mem_Lset_imp_subset_Lset)
apply (simp add: Collect_conj_in_DPow Collect_mem_eq
                 subset_Int_iff2 elem_subset_in_DPow)
done

lemma separation_CollectI:
     "(⋀z. L(z) ==> L({x ∈ z . P(x)})) ==> separation(L, λx. P(x))"
apply (unfold separation_def, clarify)
apply (rule_tac x="{x∈z. P(x)}" in rexI)
apply simp_all
done

text‹Reduces the original comprehension to the reflected one›
lemma reflection_imp_L_separation:
      "[| ∀x∈Lset(j). P(x) ⟷ Q(x);
          {x ∈ Lset(j) . Q(x)} ∈ DPow(Lset(j));
          Ord(j);  z ∈ Lset(j)|] ==> L({x ∈ z . P(x)})"
apply (rule_tac i = "succ(j)" in L_I)
 prefer 2 apply simp
apply (subgoal_tac "{x ∈ z. P(x)} = {x ∈ Lset(j). x ∈ z & (Q(x))}")
 prefer 2
 apply (blast dest: mem_Lset_imp_subset_Lset)
apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
done

text‹Encapsulates the standard proof script for proving instances of 
      Separation.›
lemma gen_separation:
 assumes reflection: "REFLECTS [P,Q]"
     and Lu:         "L(u)"
     and collI: "!!j. u ∈ Lset(j)
                ⟹ Collect(Lset(j), Q(j)) ∈ DPow(Lset(j))"
 shows "separation(L,P)"
apply (rule separation_CollectI)
apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu)
apply (rule ReflectsE [OF reflection], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
  apply (simp_all add: lt_Ord2, clarify)
apply (rule collI, assumption)
done

text‹As above, but typically \<^term>‹u› is a finite enumeration such as
  \<^term>‹{a,b}›; thus the new subgoal gets the assumption
  \<^term>‹{a,b} ⊆ Lset(i)›, which is logically equivalent to 
  \<^term>‹a ∈ Lset(i)› and \<^term>‹b ∈ Lset(i)›.›
lemma gen_separation_multi:
 assumes reflection: "REFLECTS [P,Q]"
     and Lu:         "L(u)"
     and collI: "!!j. u ⊆ Lset(j)
                ⟹ Collect(Lset(j), Q(j)) ∈ DPow(Lset(j))"
 shows "separation(L,P)"
apply (rule gen_separation [OF reflection Lu])
apply (drule mem_Lset_imp_subset_Lset)
apply (erule collI) 
done


subsection‹Separation for Intersection›

lemma Inter_Reflects:
     "REFLECTS[λx. ∀y[L]. y∈A ⟶ x ∈ y,
               λi x. ∀y∈Lset(i). y∈A ⟶ x ∈ y]"
by (intro FOL_reflections)

lemma Inter_separation:
     "L(A) ==> separation(L, λx. ∀y[L]. y∈A ⟶ x∈y)"
apply (rule gen_separation [OF Inter_Reflects], simp)
apply (rule DPow_LsetI)
 txt‹I leave this one example of a manual proof.  The tedium of manually
      instantiating \<^term>‹i›, \<^term>‹j› and \<^term>‹env› is obvious.›
apply (rule ball_iff_sats)
apply (rule imp_iff_sats)
apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
apply (rule_tac i=0 and j=2 in mem_iff_sats)
apply (simp_all add: succ_Un_distrib [symmetric])
done

subsection‹Separation for Set Difference›

lemma Diff_Reflects:
     "REFLECTS[λx. x ∉ B, λi x. x ∉ B]"
by (intro FOL_reflections)  

lemma Diff_separation:
     "L(B) ==> separation(L, λx. x ∉ B)"
apply (rule gen_separation [OF Diff_Reflects], simp)
apply (rule_tac env="[B]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done

subsection‹Separation for Cartesian Product›

lemma cartprod_Reflects:
     "REFLECTS[λz. ∃x[L]. x∈A & (∃y[L]. y∈B & pair(L,x,y,z)),
                λi z. ∃x∈Lset(i). x∈A & (∃y∈Lset(i). y∈B &
                                   pair(##Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)

lemma cartprod_separation:
     "[| L(A); L(B) |]
      ==> separation(L, λz. ∃x[L]. x∈A & (∃y[L]. y∈B & pair(L,x,y,z)))"
apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto)
apply (rule_tac env="[A,B]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done

subsection‹Separation for Image›

lemma image_Reflects:
     "REFLECTS[λy. ∃p[L]. p∈r & (∃x[L]. x∈A & pair(L,x,y,p)),
           λi y. ∃p∈Lset(i). p∈r & (∃x∈Lset(i). x∈A & pair(##Lset(i),x,y,p))]"
by (intro FOL_reflections function_reflections)

lemma image_separation:
     "[| L(A); L(r) |]
      ==> separation(L, λy. ∃p[L]. p∈r & (∃x[L]. x∈A & pair(L,x,y,p)))"
apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto)
apply (rule_tac env="[A,r]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Separation for Converse›

lemma converse_Reflects:
  "REFLECTS[λz. ∃p[L]. p∈r & (∃x[L]. ∃y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
     λi z. ∃p∈Lset(i). p∈r & (∃x∈Lset(i). ∃y∈Lset(i).
                     pair(##Lset(i),x,y,p) & pair(##Lset(i),y,x,z))]"
by (intro FOL_reflections function_reflections)

lemma converse_separation:
     "L(r) ==> separation(L,
         λz. ∃p[L]. p∈r & (∃x[L]. ∃y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
apply (rule gen_separation [OF converse_Reflects], simp)
apply (rule_tac env="[r]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Separation for Restriction›

lemma restrict_Reflects:
     "REFLECTS[λz. ∃x[L]. x∈A & (∃y[L]. pair(L,x,y,z)),
        λi z. ∃x∈Lset(i). x∈A & (∃y∈Lset(i). pair(##Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)

lemma restrict_separation:
   "L(A) ==> separation(L, λz. ∃x[L]. x∈A & (∃y[L]. pair(L,x,y,z)))"
apply (rule gen_separation [OF restrict_Reflects], simp)
apply (rule_tac env="[A]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Separation for Composition›

lemma comp_Reflects:
     "REFLECTS[λxz. ∃x[L]. ∃y[L]. ∃z[L]. ∃xy[L]. ∃yz[L].
                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
                  xy∈s & yz∈r,
        λi xz. ∃x∈Lset(i). ∃y∈Lset(i). ∃z∈Lset(i). ∃xy∈Lset(i). ∃yz∈Lset(i).
                  pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) &
                  pair(##Lset(i),y,z,yz) & xy∈s & yz∈r]"
by (intro FOL_reflections function_reflections)

lemma comp_separation:
     "[| L(r); L(s) |]
      ==> separation(L, λxz. ∃x[L]. ∃y[L]. ∃z[L]. ∃xy[L]. ∃yz[L].
                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
                  xy∈s & yz∈r)"
apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto)
txt‹Subgoals after applying general ``separation'' rule:
     @{subgoals[display,indent=0,margin=65]}›
apply (rule_tac env="[r,s]" in DPow_LsetI)
txt‹Subgoals ready for automatic synthesis of a formula:
     @{subgoals[display,indent=0,margin=65]}›
apply (rule sep_rules | simp)+
done


subsection‹Separation for Predecessors in an Order›

lemma pred_Reflects:
     "REFLECTS[λy. ∃p[L]. p∈r & pair(L,y,x,p),
                    λi y. ∃p ∈ Lset(i). p∈r & pair(##Lset(i),y,x,p)]"
by (intro FOL_reflections function_reflections)

lemma pred_separation:
     "[| L(r); L(x) |] ==> separation(L, λy. ∃p[L]. p∈r & pair(L,y,x,p))"
apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto)
apply (rule_tac env="[r,x]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Separation for the Membership Relation›

lemma Memrel_Reflects:
     "REFLECTS[λz. ∃x[L]. ∃y[L]. pair(L,x,y,z) & x ∈ y,
            λi z. ∃x ∈ Lset(i). ∃y ∈ Lset(i). pair(##Lset(i),x,y,z) & x ∈ y]"
by (intro FOL_reflections function_reflections)

lemma Memrel_separation:
     "separation(L, λz. ∃x[L]. ∃y[L]. pair(L,x,y,z) & x ∈ y)"
apply (rule gen_separation [OF Memrel_Reflects nonempty])
apply (rule_tac env="[]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Replacement for FunSpace›

lemma funspace_succ_Reflects:
 "REFLECTS[λz. ∃p[L]. p∈A & (∃f[L]. ∃b[L]. ∃nb[L]. ∃cnbf[L].
            pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
            upair(L,cnbf,cnbf,z)),
        λi z. ∃p ∈ Lset(i). p∈A & (∃f ∈ Lset(i). ∃b ∈ Lset(i).
              ∃nb ∈ Lset(i). ∃cnbf ∈ Lset(i).
                pair(##Lset(i),f,b,p) & pair(##Lset(i),n,b,nb) &
                is_cons(##Lset(i),nb,f,cnbf) & upair(##Lset(i),cnbf,cnbf,z))]"
by (intro FOL_reflections function_reflections)

lemma funspace_succ_replacement:
     "L(n) ==>
      strong_replacement(L, λp z. ∃f[L]. ∃b[L]. ∃nb[L]. ∃cnbf[L].
                pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
                upair(L,cnbf,cnbf,z))"
apply (rule strong_replacementI)
apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects], 
       auto)
apply (rule_tac env="[n,B]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Separation for a Theorem about \<^term>‹is_recfun››

lemma is_recfun_reflects:
  "REFLECTS[λx. ∃xa[L]. ∃xb[L].
                pair(L,x,a,xa) & xa ∈ r & pair(L,x,b,xb) & xb ∈ r &
                (∃fx[L]. ∃gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
                                   fx ≠ gx),
   λi x. ∃xa ∈ Lset(i). ∃xb ∈ Lset(i).
          pair(##Lset(i),x,a,xa) & xa ∈ r & pair(##Lset(i),x,b,xb) & xb ∈ r &
                (∃fx ∈ Lset(i). ∃gx ∈ Lset(i). fun_apply(##Lset(i),f,x,fx) &
                  fun_apply(##Lset(i),g,x,gx) & fx ≠ gx)]"
by (intro FOL_reflections function_reflections fun_plus_reflections)

lemma is_recfun_separation:
     ― ‹for well-founded recursion›
     "[| L(r); L(f); L(g); L(a); L(b) |]
     ==> separation(L,
            λx. ∃xa[L]. ∃xb[L].
                pair(L,x,a,xa) & xa ∈ r & pair(L,x,b,xb) & xb ∈ r &
                (∃fx[L]. ∃gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
                                   fx ≠ gx))"
apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"], 
            auto)
apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI)
apply (rule sep_rules | simp)+
done


subsection‹Instantiating the locale ‹M_basic››
text‹Separation (and Strong Replacement) for basic set-theoretic constructions
such as intersection, Cartesian Product and image.›

lemma M_basic_axioms_L: "M_basic_axioms(L)"
  apply (rule M_basic_axioms.intro)
       apply (assumption | rule
         Inter_separation Diff_separation cartprod_separation image_separation
         converse_separation restrict_separation
         comp_separation pred_separation Memrel_separation
         funspace_succ_replacement is_recfun_separation power_ax)+
  done

theorem M_basic_L: " M_basic(L)"
by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])

interpretation L: M_basic L by (rule M_basic_L)


end