(* Title: ZF/Constructible/Separation.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
*)
section\<open>Early Instances of Separation and Strong Replacement\<close>
theory Separation imports L_axioms WF_absolute begin
text\<open>This theory proves all instances needed for locale \<open>M_basic\<close>\<close>
text\<open>Helps us solve for de Bruijn indices!\<close>
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\<in>A. P(x) & Q(x)} \<in> 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 \<in> Lset(j). x \<in> z & P(x)} \<in> 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, \x. P(x))"
apply (unfold separation_def, clarify)
apply (rule_tac x="{x\z. P(x)}" in rexI)
apply simp_all
done
text\<open>Reduces the original comprehension to the reflected one\<close>
lemma reflection_imp_L_separation:
"[| \x\Lset(j). P(x) \ Q(x);
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
Ord(j); z \<in> Lset(j)|] ==> L({x \<in> 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\<open>Encapsulates the standard proof script for proving instances of
Separation.\<close>
lemma gen_separation:
assumes reflection: "REFLECTS [P,Q]"
and Lu: "L(u)"
and collI: "!!j. u \ Lset(j)
\<Longrightarrow> Collect(Lset(j), Q(j)) \<in> 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\<open>As above, but typically \<^term>\<open>u\<close> is a finite enumeration such as
\<^term>\<open>{a,b}\<close>; thus the new subgoal gets the assumption
\<^term>\<open>{a,b} \<subseteq> Lset(i)\<close>, which is logically equivalent to
\<^term>\<open>a \<in> Lset(i)\<close> and \<^term>\<open>b \<in> Lset(i)\<close>.\<close>
lemma gen_separation_multi:
assumes reflection: "REFLECTS [P,Q]"
and Lu: "L(u)"
and collI: "!!j. u \ Lset(j)
\<Longrightarrow> Collect(Lset(j), Q(j)) \<in> 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\<open>Separation for Intersection\<close>
lemma Inter_Reflects:
"REFLECTS[\x. \y[L]. y\A \ x \ y,
\<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A \<longrightarrow> x \<in> y]"
by (intro FOL_reflections)
lemma Inter_separation:
"L(A) ==> separation(L, \x. \y[L]. y\A \ x\y)"
apply (rule gen_separation [OF Inter_Reflects], simp)
apply (rule DPow_LsetI)
txt\<open>I leave this one example of a manual proof. The tedium of manually
instantiating \<^term>\<open>i\<close>, \<^term>\<open>j\<close> and \<^term>\<open>env\<close> is obvious.\<close>
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\<open>Separation for Set Difference\<close>
lemma Diff_Reflects:
"REFLECTS[\x. x \ B, \i x. x \ B]"
by (intro FOL_reflections)
lemma Diff_separation:
"L(B) ==> separation(L, \x. 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\<open>Separation for Cartesian Product\<close>
lemma cartprod_Reflects:
"REFLECTS[\z. \x[L]. x\A & (\y[L]. y\B & pair(L,x,y,z)),
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
pair(##Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)
lemma cartprod_separation:
"[| L(A); L(B) |]
==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>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\<open>Separation for Image\<close>
lemma image_Reflects:
"REFLECTS[\y. \p[L]. p\r & (\x[L]. x\A & pair(L,x,y,p)),
\<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(##Lset(i),x,y,p))]"
by (intro FOL_reflections function_reflections)
lemma image_separation:
"[| L(A); L(r) |]
==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>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\<open>Separation for Converse\<close>
lemma converse_Reflects:
"REFLECTS[\z. \p[L]. p\r & (\x[L]. \y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>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,
\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>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\<open>Separation for Restriction\<close>
lemma restrict_Reflects:
"REFLECTS[\z. \x[L]. x\A & (\y[L]. pair(L,x,y,z)),
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(##Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)
lemma restrict_separation:
"L(A) ==> separation(L, \z. \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\<open>Separation for Composition\<close>
lemma comp_Reflects:
"REFLECTS[\xz. \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\<in>s & yz\<in>r,
\<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) &
pair(##Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
by (intro FOL_reflections function_reflections)
lemma comp_separation:
"[| L(r); L(s) |]
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
xy\<in>s & yz\<in>r)"
apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto)
txt\<open>Subgoals after applying general ``separation'' rule:
@{subgoals[display,indent=0,margin=65]}\<close>
apply (rule_tac env="[r,s]" in DPow_LsetI)
txt\<open>Subgoals ready for automatic synthesis of a formula:
@{subgoals[display,indent=0,margin=65]}\<close>
apply (rule sep_rules | simp)+
done
subsection\<open>Separation for Predecessors in an Order\<close>
lemma pred_Reflects:
"REFLECTS[\y. \p[L]. p\r & pair(L,y,x,p),
\<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(##Lset(i),y,x,p)]"
by (intro FOL_reflections function_reflections)
lemma pred_separation:
"[| L(r); L(x) |] ==> separation(L, \y. \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\<open>Separation for the Membership Relation\<close>
lemma Memrel_Reflects:
"REFLECTS[\z. \x[L]. \y[L]. pair(L,x,y,z) & x \ y,
\<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(##Lset(i),x,y,z) & x \<in> y]"
by (intro FOL_reflections function_reflections)
lemma Memrel_separation:
"separation(L, \z. \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\<open>Replacement for FunSpace\<close>
lemma funspace_succ_Reflects:
"REFLECTS[\z. \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)),
\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
\<exists>nb \<in> Lset(i). \<exists>cnbf \<in> 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, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>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\<open>Separation for a Theorem about \<^term>\<open>is_recfun\<close>\<close>
lemma is_recfun_reflects:
"REFLECTS[\x. \xa[L]. \xb[L].
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
fx \<noteq> gx),
\<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
pair(##Lset(i),x,a,xa) & xa \<in> r & pair(##Lset(i),x,b,xb) & xb \<in> r &
(\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(##Lset(i),f,x,fx) &
fun_apply(##Lset(i),g,x,gx) & fx \<noteq> gx)]"
by (intro FOL_reflections function_reflections fun_plus_reflections)
lemma is_recfun_separation:
\<comment> \<open>for well-founded recursion\<close>
"[| L(r); L(f); L(g); L(a); L(b) |]
==> separation(L,
\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
fx \<noteq> 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\<open>Instantiating the locale \<open>M_basic\<close>\<close>
text\<open>Separation (and Strong Replacement) for basic set-theoretic constructions
such as intersection, Cartesian Product and image.\<close>
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
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