# Theory Separation

theory Separation
imports L_axioms WF_absolute
```(*  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)+
done

theorem M_basic_L: "PROP 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
```