(* $Id$ *)
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> (name prm \<Rightarrow> 'a::pt_name)) set"
inductive "it f1 f2 f3"
intros
it1: "(Var a,\<lambda>pi. f1(pi\<bullet>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,\<lambda>pi. f2 (r1 pi) (r2 pi)) \<in> it f1 f2 f3"
it3: "(t,r) \<in> it f1 f2 f3 \<Longrightarrow>
(Lam [a].t,\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (r (pi@[(a,a')])))) \<in> it f1 f2 f3"
constdefs
itfun' :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> name prm \<Rightarrow> ('a::pt_name)"
"itfun' f1 f2 f3 t \<equiv> (THE y. (t,y) \<in> it f1 f2 f3)"
itfun :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> ('a::pt_name)"
"itfun f1 f2 f3 t \<equiv> itfun' f1 f2 f3 t ([]::name prm)"
lemma it_total:
shows "\<exists>r. (t,r) \<in> it f1 f2 f3"
apply(induct t rule: lam.induct_weak)
apply(auto intro: it.intros)
done
lemma it_prm_eq:
assumes a: "(t,y) \<in> it f1 f2 f3"
and b: "pi1 \<triangleq> pi2"
shows "y pi1 = y pi2"
using a b
apply(induct fixing: pi1 pi2)
apply(auto simp add: pt3[OF pt_name_inst])
apply(rule_tac f="fresh_fun" in arg_cong)
apply(auto simp add: expand_fun_eq)
apply(drule_tac x="pi1@[(a,x)]" in meta_spec)
apply(drule_tac x="pi2@[(a,x)]" in meta_spec)
apply(force simp add: prm_eq_def pt2[OF pt_name_inst])
done
lemma f3_freshness_conditions_simple:
fixes f3::"('a::pt_name) f3_ty"
and y ::"name prm \<Rightarrow> 'a::pt_name"
and a ::"name"
and pi::"name prm"
assumes a: "finite ((supp f3)::name set)"
and b: "finite ((supp y)::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "\<exists>(a''::name). a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')]))) \<and> a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')]))) a''"
proof -
from c obtain a' where d0: "a'\<sharp>f3" and d1: "\<forall>(y::'a::pt_name). a'\<sharp>f3 a' y" by force
have "\<exists>(a''::name). a''\<sharp>(f3,a,a',pi,y)"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 a b)
then obtain a'' where d2: "a''\<sharp>f3" and d3: "a''\<noteq>a'" and d3b: "a''\<sharp>(f3,a,pi,y)"
by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
have d3c: "a''\<notin>((supp (f3,a,pi,y))::name set)" using d3b by (simp add: fresh_def)
have d4: "a''\<sharp>f3 a'' (y (pi@[(a,a'')]))"
proof -
have d5: "[(a'',a')]\<bullet>f3 = f3"
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF d2, OF d0])
from d1 have "\<forall>(y::'a::pt_name). ([(a'',a')]\<bullet>a')\<sharp>([(a'',a')]\<bullet>(f3 a' y))"
by (simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
hence "\<forall>(y::'a::pt_name). a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>y))" using d3 d5
by (simp add: at_calc[OF at_name_inst] pt_fun_app_eq[OF pt_name_inst, OF at_name_inst])
hence "a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>((rev [(a'',a')])\<bullet>(y (pi@[(a,a'')])))))" by force
thus ?thesis by (simp only: pt_pi_rev[OF pt_name_inst, OF at_name_inst])
qed
have d6: "a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')])))"
proof -
from a b have d7: "finite ((supp (f3,a,pi,y))::name set)" by (simp add: supp_prod fs_name1)
have "((supp (f3,a,pi,y))::name set) supports (\<lambda>a'. f3 a' (y (pi@[(a,a')])))"
by (supports_simp add: perm_append)
with d7 d3c show ?thesis by (simp add: supports_fresh)
qed
from d6 d4 show ?thesis by force
qed
text {* FIXME: this lemma should be thrown out somehow *}
lemma f3_freshness_conditions:
fixes f3::"('a::pt_name) f3_ty"
and y ::"name prm \<Rightarrow> 'a::pt_name"
and a ::"name"
and pi1::"name prm"
and pi2::"name prm"
assumes a: "finite ((supp f3)::name set)"
and b: "finite ((supp y)::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "\<exists>(a''::name). a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2))) \<and>
a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2))) a''"
proof -
from c obtain a' where d0: "a'\<sharp>f3" and d1: "\<forall>(y::'a::pt_name). a'\<sharp>f3 a' y" by force
have "\<exists>(a''::name). a''\<sharp>(f3,a,a',pi1,pi2,y)"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 a b)
then obtain a'' where d2: "a''\<sharp>f3" and d3: "a''\<noteq>a'" and d3b: "a''\<sharp>(f3,a,pi1,pi2,y)"
by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
have d3c: "a''\<notin>((supp (f3,a,pi1,pi2,y))::name set)" using d3b by (simp add: fresh_def)
have d4: "a''\<sharp>f3 a'' (y (pi1@[(a,a'')]@pi2))"
proof -
have d5: "[(a'',a')]\<bullet>f3 = f3"
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF d2, OF d0])
from d1 have "\<forall>(y::'a::pt_name). ([(a'',a')]\<bullet>a')\<sharp>([(a'',a')]\<bullet>(f3 a' y))"
by (simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
hence "\<forall>(y::'a::pt_name). a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>y))" using d3 d5
by (simp add: at_calc[OF at_name_inst] pt_fun_app_eq[OF pt_name_inst, OF at_name_inst])
hence "a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>((rev [(a'',a')])\<bullet>(y (pi1@[(a,a'')]@pi2)))))" by force
thus ?thesis by (simp only: pt_pi_rev[OF pt_name_inst, OF at_name_inst])
qed
have d6: "a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2)))"
proof -
from a b have d7: "finite ((supp (f3,a,pi1,pi2,y))::name set)" by (simp add: supp_prod fs_name1)
have e: "((supp (f3,a,pi1,pi2,y))::name set) supports (\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2)))"
by (supports_simp add: perm_append)
from e d7 d3c show ?thesis by (rule supports_fresh)
qed
from d6 d4 show ?thesis by force
qed
lemma it_fin_supp:
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: "(t,r) \<in> it f1 f2 f3"
shows "finite ((supp r)::name set)"
using a
proof (induct)
case it1 thus ?case using f by (finite_guess add: supp_prod fs_name1)
next
case it2 thus ?case using f by (finite_guess add: supp_prod fs_name1)
next
case (it3 c r t)
have ih: "finite ((supp r)::name set)" by fact
let ?g' = "\<lambda>pi a'. f3 a' (r (pi@[(c,a')]))" --"helper function"
--"two facts used by fresh_fun_equiv"
have fact1: "\<forall>pi. finite ((supp (?g' pi))::name set)" using f ih
by (rule_tac allI, finite_guess add: perm_append supp_prod fs_name1)
have fact2: "\<forall>pi. \<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')"
proof
fix pi::"name prm"
show "\<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" using f c ih
by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod)
qed
show ?case using fact1 fact2 ih f
by (finite_guess add: fresh_fun_equiv[OF pt_name_inst, OF at_name_inst]
perm_append supp_prod fs_name1)
qed
lemma it_trans:
shows "\<lbrakk>(t,r)\<in>rec f1 f2 f3; r=r'\<rbrakk> \<Longrightarrow> (t,r')\<in>rec f1 f2 f3" by simp
lemma it_perm:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and a: "(t,y) \<in> it f1 f2 f3"
shows "(pi1\<bullet>t, \<lambda>pi2. y (pi2@pi1)) \<in> it f1 f2 f3"
using a
proof (induct)
case (it1 c)
show ?case by (auto simp add: pt2[OF pt_name_inst], rule it.intros)
next
case (it2 t1 t2 r1 r2)
thus ?case by (auto intro: it.intros)
next
case (it3 c r t)
let ?g = "\<lambda>pi' a'. f3 a' (r (pi'@[(pi1\<bullet>c,a')]@pi1))"
and ?h = "\<lambda>pi' a'. f3 a' (r ((pi'@pi1)@[(c,a')]))"
have ih': "(t,r) \<in> it f1 f2 f3" by fact
hence fin_r: "finite ((supp r)::name set)" using f c by (auto intro: it_fin_supp)
have ih: "(pi1\<bullet>t,\<lambda>pi2. r (pi2@pi1)) \<in> it f1 f2 f3" by fact
hence "(Lam [(pi1\<bullet>c)].(pi1\<bullet>t),\<lambda>pi'. fresh_fun (?g pi')) \<in> it f1 f2 f3"
by (auto intro: it_trans it.intros)
moreover
have "\<forall>pi'. (fresh_fun (?g pi')) = (fresh_fun (?h pi'))"
proof
fix pi'::"name prm"
have fin_g: "finite ((supp (?g pi'))::name set)"
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1)
have fr_g: "\<exists>(a::name). (a\<sharp>(?g pi')\<and> a\<sharp>(?g pi' a))"
using f c fin_r by (rule_tac f3_freshness_conditions, simp_all add: supp_prod)
have fin_h: "finite ((supp (?h pi'))::name set)"
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1)
have fr_h: "\<exists>(a::name). (a\<sharp>(?h pi')\<and> a\<sharp>(?h pi' a))"
using f c fin_r by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod)
show "fresh_fun (?g pi') = fresh_fun (?h pi')"
proof -
have "\<exists>(d::name). d\<sharp>(?g pi', ?h pi', pi1)" using fin_g fin_h
by (rule_tac at_exists_fresh[OF at_name_inst], simp only: supp_prod finite_Un fs_name1, simp)
then obtain d::"name" where f1: "d\<sharp>?g pi'" and f2: "d\<sharp>?h pi'" and f3: "d\<sharp>(rev pi1)"
by (auto simp only: fresh_prod fresh_list_rev)
have "?g pi' d = ?h pi' d"
proof -
have "r (pi'@[(pi1\<bullet>c,d)]@pi1) = r ((pi'@pi1)@[(c,d)])" using f3 ih'
by (auto intro!: it_prm_eq at_prm_eq_append[OF at_name_inst]
simp only: append_assoc at_ds10[OF at_name_inst])
then show ?thesis by simp
qed
then show "fresh_fun (?g pi') = fresh_fun (?h pi')"
using f1 fin_g fr_g f2 fin_h fr_h
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
qed
qed
hence "(\<lambda>pi'. (fresh_fun (?g pi'))) = (\<lambda>pi'. (fresh_fun (?h pi')))" by (simp add: expand_fun_eq)
ultimately show ?case by simp
qed
lemma it_perm_rev:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and a: "(pi\<bullet>t,y) \<in> it f1 f2 f3"
shows "(t, \<lambda>pi2. y (pi2@(rev pi))) \<in> it f1 f2 f3"
proof -
from a have "((rev pi)\<bullet>(pi\<bullet>t),\<lambda>pi2. y (pi2@(rev pi))) \<in> it f1 f2 f3"
by (rule it_perm[OF f, OF c])
thus ?thesis by perm_simp
qed
lemma it_unique:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and a: "(t,y) \<in> it f1 f2 f3"
and b: "(t,y') \<in> it f1 f2 f3"
shows "y pi = y' pi"
using a b
proof (induct fixing: y' pi)
case (it1 c)
thus ?case by (cases, simp_all add: lam.inject)
next
case (it2 r1 r2 t1 t2)
with prems(9) show ?case
by (cases, simp_all (no_asm_use) add: lam.inject, force)
next
case (it3 c r t r')
have "(t,r) \<in> it f1 f2 f3" by fact
hence fin_r: "finite ((supp r)::name set)" using f c by (auto intro: it_fin_supp)
have ih: "\<And>r' pi. (t,r') \<in> it f1 f2 f3 \<Longrightarrow> r pi = r' pi" by fact
have "(Lam [c].t, r') \<in> it f1 f2 f3" by fact
then show "fresh_fun (\<lambda>a'. f3 a' (r (pi@[(c,a')]))) = r' pi"
proof (cases, simp_all add: lam.inject, auto)
fix a::"name" and t'::"lam" and r''::"name prm\<Rightarrow>'a::pt_name"
assume i5: "[c].t = [a].t'"
and i6: "(t',r'') \<in> it f1 f2 f3"
hence fin_r'': "finite ((supp r'')::name set)" using f c by (auto intro: it_fin_supp)
let ?g = "\<lambda>a'. f3 a' (r (pi@[(c,a')]))"
and ?h = "\<lambda>a'. f3 a' (r'' (pi@[(a,a')]))"
show "fresh_fun ?g = fresh_fun ?h" using i5
proof (cases "a=c")
case True
have i7: "a=c" by fact
with i5 have i8: "t=t'" by (simp add: alpha)
show "fresh_fun ?g = fresh_fun ?h" using i6 i7 i8 ih by simp
next
case False
assume i9: "a\<noteq>c"
with i5[symmetric] have i10: "t'=[(a,c)]\<bullet>t" and i11: "a\<sharp>t" by (simp_all add: alpha)
have fin_g: "finite ((supp ?g)::name set)"
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1)
have fin_h: "finite ((supp ?h)::name set)"
using f fin_r'' by (finite_guess add: perm_append supp_prod fs_name1)
have fr_g: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')"
using f c fin_r by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod)
have fr_h: "\<exists>(a''::name). a''\<sharp>?h \<and> a''\<sharp>(?h a'')"
using f c fin_r'' by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod)
have "\<exists>(d::name). d\<sharp>(?g,?h,t,a,c)" using fin_g fin_h
by (rule_tac at_exists_fresh[OF at_name_inst], simp only: finite_Un supp_prod fs_name1, simp)
then obtain d::"name"
where f1: "d\<sharp>?g" and f2: "d\<sharp>?h" and f3: "d\<sharp>t" and f4: "d\<noteq>a" and f5: "d\<noteq>c"
by (force simp add: fresh_prod at_fresh[OF at_name_inst] at_fresh[OF at_name_inst])
have g1: "[(a,d)]\<bullet>t = t"
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF i11, OF f3])
from i6 have "(([(a,c)]@[(a,d)])\<bullet>t,r'') \<in> it f1 f2 f3" using g1 i10 by (simp only: pt_name2)
hence "(t, \<lambda>pi2. r'' (pi2@(rev ([(a,c)]@[(a,d)])))) \<in> it f1 f2 f3"
by (simp only: it_perm_rev[OF f, OF c])
hence g2: "(t, \<lambda>pi2. r'' (pi2@([(a,d)]@[(a,c)]))) \<in> it f1 f2 f3" by simp
have "distinct [a,c,d]" using i9 f4 f5 by force
hence g3: "(pi@[(c,d)]@[(a,d)]@[(a,c)]) \<triangleq> (pi@[(a,d)])"
by (rule_tac at_prm_eq_append[OF at_name_inst],
force simp add: prm_eq_def at_calc[OF at_name_inst])
have "fresh_fun ?g = ?g d" using fin_g fr_g f1
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = f3 d ((\<lambda>pi2. r'' (pi2@([(a,d)]@[(a,c)]))) (pi@[(c,d)]))" using ih g2 by simp
also have "\<dots> = f3 d (r'' (pi@[(c,d)]@[(a,d)]@[(a,c)]))" by simp
also have "\<dots> = f3 d (r'' (pi@[(a,d)]))" using i6 g3 by (simp add: it_prm_eq)
also have "\<dots> = fresh_fun ?h" using fin_h fr_h f2
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
finally show "fresh_fun ?g = fresh_fun ?h" by simp
qed
qed
qed
lemma it_function:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "\<exists>!y. (t,y) \<in> it f1 f2 f3"
proof (rule ex_ex1I)
case goal1
show ?case by (rule it_total)
next
case (goal2 y1 y2)
have a1: "(t,y1) \<in> it f1 f2 f3" and a2: "(t,y2) \<in> it f1 f2 f3" by fact
hence "\<forall>pi. y1 pi = y2 pi" using it_unique[OF f, OF c] by force
thus ?case by (simp add: expand_fun_eq)
qed
lemma it_eqvt:
fixes pi::"name prm"
assumes a: "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 b: "(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 b
proof(induct)
case it1 show ?case by (perm_simp add: it.intros)
next
case it2 thus ?case by (perm_simp add: it.intros)
next
case (it3 c r t) (* lam case *)
let ?g = "pi\<bullet>(\<lambda>pi'. fresh_fun (\<lambda>a'. f3 a' (r (pi'@[(c,a')]))))"
and ?h = "\<lambda>pi'. fresh_fun (\<lambda>a'. (pi\<bullet>f3) a' ((pi\<bullet>r) (pi'@[((pi\<bullet>c),a')])))"
have "(t,r) \<in> it f1 f2 f3" by fact
hence fin_r: "finite ((supp r)::name set)" using a c by (auto intro: it_fin_supp)
have ih: "(pi\<bullet>t,pi\<bullet>r) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by fact
hence "(Lam [(pi\<bullet>c)].(pi\<bullet>t),?h) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by (simp add: it.intros)
moreover
have "?g = ?h"
proof -
let ?g' = "\<lambda>pi a'. f3 a' (r (pi@[(c,a')]))"
have fact1: "\<forall>pi. finite ((supp (?g' pi))::name set)" using fin_r a
by (rule_tac allI, finite_guess add: perm_append supp_prod fs_name1)
have fact2: "\<forall>pi. \<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')"
proof
fix pi::"name prm"
show "\<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" using a c fin_r
by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod)
qed
show ?thesis using fact1 fact2
by (perm_simp add: fresh_fun_equiv[OF pt_name_inst, OF at_name_inst] perm_append)
qed
ultimately show "(pi\<bullet>Lam [c].t,?g) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by simp
qed
lemma theI2':
assumes a1: "P a"
and a2: "\<exists>!x. P x"
and a3: "P a \<Longrightarrow> Q a"
shows "Q (THE x. P x)"
apply(rule theI2)
apply(rule a1)
apply(subgoal_tac "\<exists>!x. P x")
apply(auto intro: a1 simp add: Ex1_def)
apply(fold Ex1_def)
apply(rule a2)
apply(subgoal_tac "x=a")
apply(simp)
apply(rule a3)
apply(assumption)
apply(subgoal_tac "\<exists>!x. P x")
apply(auto intro: a1 simp add: Ex1_def)
apply(fold Ex1_def)
apply(rule a2)
done
lemma itfun'_equiv:
fixes pi::"name prm"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "pi\<bullet>(itfun' f1 f2 f3 t) = itfun' (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
using f
apply(auto simp add: itfun'_def)
apply(subgoal_tac "\<exists>y. (t,y) \<in> it f1 f2 f3")(*A*)
apply(auto)
apply(rule sym)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(rule the1_equality)
apply(rule it_function)
apply(simp add: supp_prod)
apply(simp add: pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst])
apply(subgoal_tac "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)")
apply(auto)
apply(rule_tac x="pi\<bullet>a" in exI)
apply(auto)
apply(simp add: fresh_eqvt)
apply(drule_tac x="(rev pi)\<bullet>x" in spec)
apply(subgoal_tac "(pi\<bullet>f3) (pi\<bullet>a) x = pi\<bullet>(f3 a ((rev pi)\<bullet>x))")
apply(simp add: fresh_eqvt)
apply(subgoal_tac "pi\<bullet>(f3 a ((rev pi)\<bullet>x)) = (pi\<bullet>f3) (pi\<bullet>a) (pi\<bullet>((rev pi)\<bullet>x))")
apply(simp)
apply(perm_simp)
apply(perm_simp)
apply(rule c)
apply(rule it_eqvt)
apply(rule f, rule c, assumption)
(*A*)
apply(rule it_total)
done
lemma itfun'_equiv_aux:
fixes pi::"name prm"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "pi\<bullet>(itfun' f1 f2 f3 t pi') = itfun' (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t) (pi\<bullet>pi')" (is "?LHS=?RHS")
proof -
have "?LHS = (pi\<bullet>itfun' f1 f2 f3 t) (pi\<bullet>pi')" by (simp add: perm_app)
also have "\<dots> = ?RHS" by (simp add: itfun'_equiv[OF f, OF c])
finally show "?LHS = ?RHS" by simp
qed
lemma itfun'_finite_supp:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "finite ((supp (itfun' f1 f2 f3 t))::name set)"
using f by (finite_guess add: itfun'_equiv[OF f, OF c] supp_prod fs_name1)
lemma itfun'_prm:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "itfun' f1 f2 f3 (pi1\<bullet>t) pi2 = itfun' f1 f2 f3 t (pi2@pi1)"
apply(auto simp add: itfun'_def)
apply(subgoal_tac "\<exists>y. (t,y) \<in> it f1 f2 f3")(*A*)
apply(auto)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(rule_tac a="\<lambda>pi2. y (pi2@pi1)" in theI2')
apply(rule it_perm)
apply(rule f, rule c)
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(simp)
(*A*)
apply(rule it_total)
done
lemma itfun_Var:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "itfun f1 f2 f3 (Var c) = (f1 c)"
apply(auto simp add: itfun_def itfun'_def)
apply(subgoal_tac "(THE y. (Var c, y) \<in> it f1 f2 f3) = (\<lambda>(pi::name prm). f1 (pi\<bullet>c))")(*A*)
apply(simp)
apply(rule the1_equality)
apply(rule it_function[OF f, OF c])
apply(rule it.intros)
done
lemma itfun_App:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "itfun f1 f2 f3 (App t1 t2) = (f2 (itfun f1 f2 f3 t1) (itfun f1 f2 f3 t2))"
apply(auto simp add: itfun_def itfun'_def)
apply(subgoal_tac "(THE y. (App t1 t2, y) \<in> it f1 f2 f3) =
(\<lambda>(pi::name prm). f2 ((itfun' f1 f2 f3 t1) pi) ((itfun' f1 f2 f3 t2) pi))")(*A*)
apply(simp add: itfun'_def)
apply(rule the1_equality)
apply(rule it_function[OF f, OF c])
apply(rule it.intros)
apply(subgoal_tac "\<exists>y. (t1,y) \<in> it f1 f2 f3")(*A*)
apply(erule exE, simp add: itfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(assumption)
(*A*)
apply(rule it_total)
apply(subgoal_tac "\<exists>y. (t2,y) \<in> it f1 f2 f3")(*B*)
apply(auto simp add: itfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(assumption)
(*B*)
apply(rule it_total)
done
lemma itfun_Lam_aux1:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
shows "itfun f1 f2 f3 (Lam [a].t) = fresh_fun (\<lambda>a'. f3 a' (itfun' f1 f2 f3 t ([]@[(a,a')])))"
apply(simp add: itfun_def itfun'_def)
apply(subgoal_tac "(THE y. (Lam [a].t, y) \<in> it f1 f2 f3) =
(\<lambda>(pi::name prm). fresh_fun(\<lambda>a'. f3 a' ((itfun' f1 f2 f3 t) (pi@[(a,a')]))))")(*A*)
apply(simp add: itfun'_def[symmetric])
apply(rule the1_equality)
apply(rule it_function[OF f, OF c])
apply(rule it.intros)
apply(subgoal_tac "\<exists>y. (t,y) \<in> it f1 f2 f3")(*B*)
apply(erule exE, simp add: itfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule it_function[OF f, OF c])
apply(assumption)
(*B*)
apply(rule it_total)
done
lemma itfun_Lam_aux2:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and a: "b\<sharp>(a,t,f1,f2,f3)"
shows "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))"
proof -
from f have f': "finite ((supp f3)::name set)" by (simp add: supp_prod)
have eq1: "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = itfun f1 f2 f3 (Lam [a].t)"
proof -
have "Lam [b].([(b,a)]\<bullet>t) = Lam [a].t" using a
by (simp add: lam.inject alpha fresh_prod fresh_atm)
thus ?thesis by simp
qed
let ?g = "(\<lambda>a'. f3 a' (itfun' f1 f2 f3 t ([]@[(a,a')])))"
(* FIXME: cleanup*)
have a0: "((supp (f1,f3,f2,t,a))::name set) supports ?g"
by (supports_simp add: itfun'_equiv_aux[OF f, OF c] perm_append)
have a1: "finite ((supp (f1,f3,f2,t,a))::name set)" using f
by (simp add: supp_prod fs_name1)
have a2: "finite ((supp ?g)::name set)"
using f by (finite_guess add: itfun'_equiv_aux[OF f, OF c] supp_prod fs_name1)
have c0: "finite ((supp (itfun' f1 f2 f3 t))::name set)"
by (rule itfun'_finite_supp[OF f, OF c])
have c1: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')"
by (rule f3_freshness_conditions_simple[OF f', OF c0, OF c])
have c2: "b\<sharp>?g"
proof -
have fs_g: "finite ((supp (f1,f2,f3,t))::name set)" using f
by (simp add: supp_prod fs_name1)
have "((supp (f1,f2,f3,t))::name set) supports (itfun' f1 f2 f3 t)"
by (supports_simp add: itfun'_equiv[OF f, OF c])
hence c3: "b\<sharp>(itfun' f1 f2 f3 t)" using fs_g
proof(rule supports_fresh, simp add: fresh_def[symmetric])
show "b\<sharp>(f1,f2,f3,t)" using a by (simp add: fresh_prod)
qed
show ?thesis using a
by (rule_tac supports_fresh[OF a0, OF a1], simp add: fresh_def[symmetric], simp add: fresh_prod)
qed
(* main proof *)
have "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = itfun f1 f2 f3 (Lam [a].t)" by (simp add: eq1)
also have "\<dots> = fresh_fun ?g" by (rule itfun_Lam_aux1[OF f, OF c])
also have "\<dots> = ?g b" using c2
by (rule fresh_fun_app[OF pt_name_inst, OF at_name_inst, OF a2, OF c1])
also have "\<dots> = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))"
by (simp add: itfun_def itfun'_prm[OF f, OF c])
finally show "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))" by simp
qed
lemma itfun_Lam:
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)"
and a: "b\<sharp>(f1,f2,f3)"
shows "itfun f1 f2 f3 (Lam [b].t) = f3 b (itfun f1 f2 f3 t)"
proof -
have "\<exists>(a::name). a\<sharp>(b,t)"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1)
then obtain a::"name" where a1: "a\<sharp>b" and a2: "a\<sharp>t" by (force simp add: fresh_prod)
have "itfun f1 f2 f3 (Lam [b].t) = itfun f1 f2 f3 (Lam [b].(([(b,a)])\<bullet>([(b,a)]\<bullet>t)))"
by (simp add: pt_swap_bij[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = f3 b (itfun f1 f2 f3 (([(a,b)])\<bullet>([(b,a)]\<bullet>t)))"
proof(rule itfun_Lam_aux2[OF f, OF c])
have "b\<sharp>([(b,a)]\<bullet>t)" using a1 a2
by (simp add: pt_fresh_left[OF pt_name_inst, OF at_name_inst] at_calc[OF at_name_inst]
at_fresh[OF at_name_inst])
thus "b\<sharp>(a,[(b,a)]\<bullet>t,f1,f2,f3)" using a a1 by (force simp add: fresh_prod at_fresh[OF at_name_inst])
qed
also have "\<dots> = f3 b (itfun f1 f2 f3 t)" by (simp add: pt_swap_bij'[OF pt_name_inst, OF at_name_inst])
finally show ?thesis .
qed
constdefs
depth_Var :: "name \<Rightarrow> nat"
"depth_Var \<equiv> \<lambda>(a::name). 1"
depth_App :: "nat \<Rightarrow> nat \<Rightarrow> nat"
"depth_App \<equiv> \<lambda>n1 n2. (max n1 n2)+1"
depth_Lam :: "name \<Rightarrow> nat \<Rightarrow> nat"
"depth_Lam \<equiv> \<lambda>(a::name) n. n+1"
depth_lam :: "lam \<Rightarrow> nat"
"depth_lam \<equiv> itfun depth_Var depth_App depth_Lam"
lemma supp_depth_Var:
shows "((supp depth_Var)::name set)={}"
apply(simp add: depth_Var_def)
apply(simp add: supp_def)
apply(simp add: perm_fun_def)
apply(simp add: perm_nat_def)
done
lemma supp_depth_App:
shows "((supp depth_App)::name set)={}"
apply(simp add: depth_App_def)
apply(simp add: supp_def)
apply(simp add: perm_fun_def)
apply(simp add: perm_nat_def)
done
lemma supp_depth_Lam:
shows "((supp depth_Lam)::name set)={}"
apply(simp add: depth_Lam_def)
apply(simp add: supp_def)
apply(simp add: perm_fun_def)
apply(simp add: perm_nat_def)
done
lemma fin_supp_depth:
shows "finite ((supp (depth_Var,depth_App,depth_Lam))::name set)"
using supp_depth_Var supp_depth_Lam supp_depth_App
by (simp add: supp_prod)
lemma fresh_depth_Lam:
shows "\<exists>(a::name). a\<sharp>depth_Lam \<and> (\<forall>n. a\<sharp>depth_Lam a n)"
apply(rule exI)
apply(rule conjI)
apply(simp add: fresh_def)
apply(force simp add: supp_depth_Lam)
apply(unfold fresh_def)
apply(simp add: supp_def)
apply(simp add: perm_nat_def)
done
lemma depth_Var:
shows "depth_lam (Var c) = 1"
apply(simp add: depth_lam_def)
apply(simp add: itfun_Var[OF fin_supp_depth, OF fresh_depth_Lam])
apply(simp add: depth_Var_def)
done
lemma depth_App:
shows "depth_lam (App t1 t2) = (max (depth_lam t1) (depth_lam t2))+1"
apply(simp add: depth_lam_def)
apply(simp add: itfun_App[OF fin_supp_depth, OF fresh_depth_Lam])
apply(simp add: depth_App_def)
done
lemma depth_Lam:
shows "depth_lam (Lam [a].t) = (depth_lam t)+1"
apply(simp add: depth_lam_def)
apply(rule trans)
apply(rule itfun_Lam[OF fin_supp_depth, OF fresh_depth_Lam])
apply(simp add: fresh_def supp_prod supp_depth_Var supp_depth_App supp_depth_Lam)
apply(simp add: depth_Lam_def)
done
text {* substitution *}
constdefs
subst_Var :: "name \<Rightarrow>lam \<Rightarrow> name \<Rightarrow> lam"
"subst_Var b t \<equiv> \<lambda>a. (if a=b then t else (Var a))"
subst_App :: "name \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam"
"subst_App b t \<equiv> \<lambda>r1 r2. App r1 r2"
subst_Lam :: "name \<Rightarrow> lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"
"subst_Lam b t \<equiv> \<lambda>a r. Lam [a].r"
subst_lam :: "name \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam"
"subst_lam b t \<equiv> itfun (subst_Var b t) (subst_App b t) (subst_Lam b t)"
lemma supports_subst_Var:
shows "((supp (b,t))::name set) supports (subst_Var b t)"
apply(supports_simp add: subst_Var_def)
apply(rule impI)
apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
apply(simp add: pt_fresh_fresh[OF pt_name_inst, OF at_name_inst])
done
lemma supports_subst_App:
shows "((supp (b,t))::name set) supports subst_App b t"
by (supports_simp add: subst_App_def)
lemma supports_subst_Lam:
shows "((supp (b,t))::name set) supports subst_Lam b t"
by (supports_simp add: subst_Lam_def)
lemma fin_supp_subst:
shows "finite ((supp (subst_Var b t,subst_App b t,subst_Lam b t))::name set)"
apply(auto simp add: supp_prod)
apply(rule supports_finite)
apply(rule supports_subst_Var)
apply(simp add: fs_name1)
apply(rule supports_finite)
apply(rule supports_subst_App)
apply(simp add: fs_name1)
apply(rule supports_finite)
apply(rule supports_subst_Lam)
apply(simp add: fs_name1)
done
lemma fresh_subst_Lam:
shows "\<exists>(a::name). a\<sharp>(subst_Lam b t)\<and> (\<forall>y. a\<sharp>(subst_Lam b t) a y)"
apply(subgoal_tac "\<exists>(c::name). c\<sharp>(b,t)")
apply(auto)
apply(rule_tac x="c" in exI)
apply(auto)
apply(rule supports_fresh)
apply(rule supports_subst_Lam)
apply(simp_all add: fresh_def[symmetric] fs_name1)
apply(simp add: subst_Lam_def)
apply(simp add: abs_fresh)
apply(rule at_exists_fresh[OF at_name_inst])
apply(simp add: fs_name1)
done
lemma subst_Var:
shows "subst_lam b t (Var a) = (if a=b then t else (Var a))"
apply(simp add: subst_lam_def)
apply(auto simp add: itfun_Var[OF fin_supp_subst, OF fresh_subst_Lam])
apply(auto simp add: subst_Var_def)
done
lemma subst_App:
shows "subst_lam b t (App t1 t2) = App (subst_lam b t t1) (subst_lam b t t2)"
apply(simp add: subst_lam_def)
apply(auto simp add: itfun_App[OF fin_supp_subst, OF fresh_subst_Lam])
apply(auto simp add: subst_App_def)
done
lemma subst_Lam:
assumes a: "a\<sharp>(b,t)"
shows "subst_lam b t (Lam [a].t1) = Lam [a].(subst_lam b t t1)"
using a
apply(simp add: subst_lam_def)
apply(subgoal_tac "a\<sharp>(subst_Var b t,subst_App b t,subst_Lam b t)")
apply(auto simp add: itfun_Lam[OF fin_supp_subst, OF fresh_subst_Lam])
apply(simp (no_asm) add: subst_Lam_def)
apply(auto simp add: fresh_prod)
apply(rule supports_fresh)
apply(rule supports_subst_Var)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
apply(rule supports_fresh)
apply(rule supports_subst_App)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
apply(rule supports_fresh)
apply(rule supports_subst_Lam)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
done
lemma subst_Lam':
assumes a: "a\<noteq>b"
and b: "a\<sharp>t"
shows "subst_lam b t (Lam [a].t1) = Lam [a].(subst_lam b t t1)"
apply(rule subst_Lam)
apply(simp add: fresh_prod a b fresh_atm)
done
lemma subst_Lam'':
assumes a: "a\<sharp>b"
and b: "a\<sharp>t"
shows "subst_lam b t (Lam [a].t1) = Lam [a].(subst_lam b t t1)"
apply(rule subst_Lam)
apply(simp add: fresh_prod a b)
done
consts
subst_syn :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 900)
translations
"t1[a::=t2]" \<rightleftharpoons> "subst_lam a t2 t1"
declare subst_Var[simp]
declare subst_App[simp]
declare subst_Lam[simp]
declare subst_Lam'[simp]
declare subst_Lam''[simp]
lemma subst_eqvt[simp]:
fixes pi:: "name prm"
and t1:: "lam"
and t2:: "lam"
and a :: "name"
shows "pi\<bullet>(t1[b::=t2]) = (pi\<bullet>t1)[(pi\<bullet>b)::=(pi\<bullet>t2)]"
apply(nominal_induct t1 avoiding: b t2 rule: lam.induct)
apply(auto simp add: perm_bij fresh_prod fresh_atm)
apply(subgoal_tac "(pi\<bullet>name)\<sharp>(pi\<bullet>b,pi\<bullet>t2)")(*A*)
apply(simp)
(*A*)
apply(simp add: perm_bij fresh_prod fresh_atm pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
done
lemma subst_supp: "supp(t1[a::=t2])\<subseteq>(((supp(t1)-{a})\<union>supp(t2))::name set)"
apply(nominal_induct t1 avoiding: a t2 rule: lam.induct)
apply(auto simp add: lam.supp supp_atm fresh_prod abs_supp)
apply(blast)
apply(blast)
done
end