(* $Id$ *)
theory Iteration
imports "../Nominal"
begin
atom_decl name
nominal_datatype lam = Var "name"
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
types 'a f1_ty = "name\<Rightarrow>('a::pt_name)"
'a f2_ty = "'a\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
'a f3_ty = "name\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
consts
it :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> (lam \<times> 'a::pt_name) set"
inductive "it f1 f2 f3"
intros
it1: "(Var a, f1 a) \<in> it f1 f2 f3"
it2: "\<lbrakk>(t1,r1) \<in> it f1 f2 f3; (t2,r2) \<in> it f1 f2 f3\<rbrakk> \<Longrightarrow> (App t1 t2, f2 r1 r2) \<in> it f1 f2 f3"
it3: "\<lbrakk>a\<sharp>(f1,f2,f3); (t,r) \<in> it f1 f2 f3\<rbrakk> \<Longrightarrow> (Lam [a].t,f3 a r) \<in> it f1 f2 f3"
lemma it_equiv:
fixes pi::"name prm"
assumes a: "(t,r) \<in> it f1 f2 f3"
shows "(pi\<bullet>t,pi\<bullet>r) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)"
using a
apply(induct)
apply(perm_simp | auto intro!: it.intros simp add: fresh_right)+
done
lemma it_fin_supp:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and a: "(t,r) \<in> it f1 f2 f3"
shows "finite ((supp r)::name set)"
using a f
apply(induct)
apply(finite_guess, simp add: supp_prod fs_name1)+
done
lemma it_total:
assumes a: "finite ((supp (f1,f2,f3))::name set)"
and b: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "\<exists>r. (t,r)\<in>it f1 f2 f3"
apply(rule_tac lam.induct'[of "\<lambda>_. (supp (f1,f2,f3))" "\<lambda>z. \<lambda>t. \<exists>r. (t,r)\<in>it f1 f2 f3", simplified])
apply(fold fresh_def)
apply(auto intro: it.intros a)
done
lemma it_unique:
assumes a: "finite ((supp (f1,f2,f3))::name set)"
and b: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and c1: "(t,r)\<in>it f1 f2 f3"
and c2: "(t,r')\<in>it f1 f2 f3"
shows "r=r'"
using c1 c2
proof (induct arbitrary: r')
case it1
then show ?case by cases (simp_all add: lam.inject)
next
case (it2 r1 r2 t1 t2)
have ih1: "\<And>r'. (t1,r') \<in> it f1 f2 f3 \<Longrightarrow> r1 = r'" by fact
have ih2: "\<And>r'. (t2,r') \<in> it f1 f2 f3 \<Longrightarrow> r2 = r'" by fact
have "(App t1 t2, r') \<in>it f1 f2 f3" by fact
then show ?case
proof cases
case it2
then show ?thesis using ih1 ih2 by (simp add: lam.inject)
qed (simp_all (no_asm_use))
next
case (it3 a1 r1 t1)
have f1: "a1\<sharp>(f1,f2,f3)" by fact
have ih: "\<And>r'. (t1,r') \<in> it f1 f2 f3 \<Longrightarrow> r1 = r'" by fact
have it1: "(t1,r1) \<in> it f1 f2 f3" by fact
have "(Lam [a1].t1, r') \<in> it f1 f2 f3" by fact
then show ?case
proof cases
case (it3 a2 r2 t2)
then have f2: "a2\<sharp>(f1,f2,f3)"
and it2: "(t2,r2) \<in> it f1 f2 f3"
and eq1: "[a1].t1 = [a2].t2" and eq2: "r' = f3 a2 r2" by (simp_all add: lam.inject)
have "\<exists>(c::name). c\<sharp>(f1,f2,f3,a1,a2,t1,t2,r1,r2)" using a it1 it2
by (auto intro!: at_exists_fresh[OF at_name_inst] simp add: supp_prod fs_name1 it_fin_supp[OF a])
then obtain c where fresh: "c\<sharp>f1" "c\<sharp>f2" "c\<sharp>f3" "c\<noteq>a1" "c\<noteq>a2" "c\<sharp>t1" "c\<sharp>t2" "c\<sharp>r1" "c\<sharp>r2"
by (force simp add: fresh_prod fresh_atm)
have eq3: "[(a1,c)]\<bullet>t1 = [(a2,c)]\<bullet>t2" using eq1 fresh
apply(auto simp add: alpha)
apply(rule trans)
apply(rule perm_compose)
apply(simp add: calc_atm perm_fresh_fresh)
apply(rule pt_name3, rule at_ds5[OF at_name_inst])
done
have eq4: "[(a1,c)]\<bullet>r1 = [(a2,c)]\<bullet>r2" using eq3 it2 f1 f2 fresh
apply(drule_tac sym)
apply(rule_tac pt_bij2[OF pt_name_inst, OF at_name_inst])
apply(rule ih)
apply(drule_tac pi="[(a2,c)]" in it_equiv)
apply(perm_simp only: fresh_prod)
apply(drule_tac pi="[(a1,c)]" in it_equiv)
apply(perm_simp)
done
have fs1: "a1\<sharp>f3 a1 r1" using b f1
apply(auto)
apply(rule_tac pi="[(a1,a)]" in pt_fresh_bij2[OF pt_name_inst, OF at_name_inst])
apply(perm_simp add: calc_atm fresh_prod)
done
have fs2: "a2\<sharp>f3 a2 r2" using b f2
apply(auto)
apply(rule_tac pi="[(a2,a)]" in pt_fresh_bij2[OF pt_name_inst, OF at_name_inst])
apply(perm_simp add: calc_atm fresh_prod)
done
have fs3: "c\<sharp>f3 a1 r1" using fresh it1 a
by (fresh_guess add: supp_prod fs_name1 it_fin_supp[OF a] fresh_atm)
have fs4: "c\<sharp>f3 a2 r2" using fresh it2 a
by (fresh_guess add: supp_prod fs_name1 it_fin_supp[OF a] fresh_atm)
have "f3 a1 r1 = [(a1,c)]\<bullet>(f3 a1 r1)" using fs1 fs3 by perm_simp
also have "\<dots> = f3 c ([(a1,c)]\<bullet>r1)" using f1 fresh by (perm_simp add: calc_atm fresh_prod)
also have "\<dots> = f3 c ([(a2,c)]\<bullet>r2)" using eq4 by simp
also have "\<dots> = [(a2,c)]\<bullet>(f3 a2 r2)" using f2 fresh by (perm_simp add: calc_atm fresh_prod)
also have "\<dots> = f3 a2 r2" using fs2 fs4 by perm_simp
finally have eq4: "f3 a1 r1 = f3 a2 r2" by simp
then show ?thesis using eq2 by simp
qed (simp_all (no_asm_use))
qed
lemma it_function:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)"
shows "\<exists>!r. (t,r) \<in> it f1 f2 f3"
proof (rule ex_ex1I, rule it_total[OF f, OF c])
case (goal1 r1 r2)
have a1: "(t,r1) \<in> it f1 f2 f3" and a2: "(t,r2) \<in> it f1 f2 f3" by fact
thus "r1 = r2" using it_unique[OF f, OF c] by simp
qed
constdefs
itfun :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> ('a::pt_name)"
"itfun f1 f2 f3 t \<equiv> (THE r. (t,r) \<in> it f1 f2 f3)"
lemma itfun_eqvt:
fixes pi::"name prm"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)"
shows "pi\<bullet>(itfun f1 f2 f3 t) = itfun (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
proof -
have f_pi: "finite ((supp (pi\<bullet>f1,pi\<bullet>f2,pi\<bullet>f3))::name set)" using f
by (simp add: supp_prod pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst])
have fs_pi: "\<exists>(a::name). a\<sharp>(pi\<bullet>f3) \<and> (\<forall>(r::'a::pt_name). a\<sharp>(pi\<bullet>f3) a r)"
proof -
from c obtain a where fs1: "a\<sharp>f3" and fs2: "\<forall>(r::'a::pt_name). a\<sharp>f3 a r" by force
have "(pi\<bullet>a)\<sharp>(pi\<bullet>f3)" using fs1 by (simp add: fresh_bij)
moreover
have "\<forall>(r::'a::pt_name). (pi\<bullet>a)\<sharp>((pi\<bullet>f3) (pi\<bullet>a) r)" using fs2 by (perm_simp add: fresh_right)
ultimately show "\<exists>(a::name). a\<sharp>(pi\<bullet>f3) \<and> (\<forall>(r::'a::pt_name). a\<sharp>(pi\<bullet>f3) a r)" by blast
qed
show ?thesis
apply(rule sym)
apply(auto simp add: itfun_def)
apply(rule the1_equality[OF it_function, OF f_pi, OF fs_pi])
apply(rule it_equiv)
apply(rule theI'[OF it_function,OF f, OF c])
done
qed
lemma itfun_Var:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)"
shows "itfun f1 f2 f3 (Var c) = (f1 c)"
using f c by (auto intro!: the1_equality it_function it.intros simp add: itfun_def)
lemma itfun_App:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)"
shows "itfun f1 f2 f3 (App t1 t2) = (f2 (itfun f1 f2 f3 t1) (itfun f1 f2 f3 t2))"
by (auto intro!: the1_equality it_function[OF f, OF c] it.intros
intro: theI'[OF it_function, OF f, OF c] simp add: itfun_def)
lemma itfun_Lam:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)"
and a: "a\<sharp>(f1,f2,f3)"
shows "itfun f1 f2 f3 (Lam [a].t) = f3 a (itfun f1 f2 f3 t)"
using a
by (auto intro!: the1_equality it_function[OF f, OF c] it.intros
intro: theI'[OF it_function, OF f, OF c] simp add: itfun_def)
end