cleaned up all examples so that they work with the
current nominal-setting.
theory lam_substs
imports "../nominal"
begin
atom_decl name
nominal_datatype lam = Var "name"
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
section {* strong induction principles for lam *}
lemma lam_induct_aux:
fixes P :: "lam \<Rightarrow> 'a \<Rightarrow> bool"
and f :: "'a \<Rightarrow> name set"
assumes fs: "\<And>x. finite (f x)"
and h1: "\<And>x a. P (Var a) x"
and h2: "\<And>x t1 t2. (\<forall>z. P t1 z) \<Longrightarrow> (\<forall>z. P t2 z) \<Longrightarrow> P (App t1 t2) x"
and h3: "\<And>x a t. a\<notin>f x \<Longrightarrow> (\<forall>z. P t z) \<Longrightarrow> P (Lam [a].t) x"
shows "\<forall>(pi::name prm) x. P (pi\<bullet>t) x"
proof (induct rule: lam.induct_weak)
case (Lam a t)
assume i1: "\<forall>(pi::name prm) x. P (pi\<bullet>t) x"
show ?case
proof (intro strip, simp add: abs_perm)
fix x::"'a" and pi::"name prm"
have f: "\<exists>c::name. c\<sharp>(f x,pi\<bullet>a,pi\<bullet>t)"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1
at_fin_set_supp[OF at_name_inst, OF fs] fs)
then obtain c::"name"
where f1: "c\<noteq>(pi\<bullet>a)" and f2: "c\<sharp>(f x)" and f3: "c\<sharp>(pi\<bullet>t)"
by (force simp add: fresh_prod at_fresh[OF at_name_inst])
have g: "Lam [c].([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t)) = Lam [(pi\<bullet>a)].(pi\<bullet>t)" using f1 f3
by (simp add: lam.inject alpha)
from i1 have "\<forall>x. P (([(c,pi\<bullet>a)]@pi)\<bullet>t) x" by force
hence i1b: "\<forall>x. P ([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t)) x" by (simp add: pt_name2[symmetric])
with h3 f2 have "P (Lam [c].([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t))) x"
by (auto simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs])
with g show "P (Lam [(pi\<bullet>a)].(pi\<bullet>t)) x" by simp
qed
qed (auto intro: h1 h2)
lemma lam_induct'[case_names Fin Var App Lam]:
fixes P :: "lam \<Rightarrow> 'a \<Rightarrow> bool"
and x :: "'a"
and t :: "lam"
and f :: "'a \<Rightarrow> name set"
assumes fs: "\<And>x. finite (f x)"
and h1: "\<And>x a. P (Var a) x"
and h2: "\<And>x t1 t2. (\<forall>z. P t1 z)\<Longrightarrow>(\<forall>z. P t2 z)\<Longrightarrow>P (App t1 t2) x"
and h3: "\<And>x a t. a\<notin>f x \<Longrightarrow> (\<forall>z. P t z)\<Longrightarrow> P (Lam [a].t) x"
shows "P t x"
proof -
from fs h1 h2 h3 have "\<forall>(pi::name prm) x. P (pi\<bullet>t) x" by (rule lam_induct_aux, auto)
hence "P (([]::name prm)\<bullet>t) x" by blast
thus "P t x" by simp
qed
lemma lam_induct[case_names Var App Lam]:
fixes P :: "lam \<Rightarrow> ('a::fs_name) \<Rightarrow> bool"
and x :: "'a::fs_name"
and t :: "lam"
assumes h1: "\<And>x a. P (Var a) x"
and h2: "\<And>x t1 t2. (\<forall>z. P t1 z)\<Longrightarrow>(\<forall>z. P t2 z)\<Longrightarrow>P (App t1 t2) x"
and h3: "\<And>x a t. a\<sharp>x \<Longrightarrow> (\<forall>z. P t z)\<Longrightarrow> P (Lam [a].t) x"
shows "P t x"
by (rule lam_induct'[of "\<lambda>x. ((supp x)::name set)" "P"],
simp_all add: fs_name1 fresh_def[symmetric], auto intro: h1 h2 h3)
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)"
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,pi2\<bullet>a')]))) \<and>
a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,pi2\<bullet>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',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,pi2\<bullet>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 (pi1@[(a,pi2\<bullet>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 (pi1@[(a,pi2\<bullet>a')])))"
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,pi2\<bullet>a')])))"
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 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
lemma f3_fresh_fun_supports:
fixes f3::"('a::pt_name) f3_ty"
and a ::"name"
and pi1::"name prm"
and y ::"name prm \<Rightarrow> 'a::pt_name"
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 "((supp (f3,a,y))::name set) supports (\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')]))))"
proof (auto simp add: "op supports_def" perm_fun_def expand_fun_eq fresh_def[symmetric])
fix b::"name" and c::"name" and pi::"name prm"
assume b1: "b\<sharp>(f3,a,y)"
and b2: "c\<sharp>(f3,a,y)"
from b1 b2 have b3: "[(b,c)]\<bullet>f3=f3" and t4: "[(b,c)]\<bullet>a=a" and t5: "[(b,c)]\<bullet>y=y"
by (simp_all add: pt_fresh_fresh[OF pt_name_inst, OF at_name_inst] fresh_prod)
let ?g = "\<lambda>a'. f3 a' (y (([(b,c)]\<bullet>pi)@[(a,a')]))"
and ?h = "\<lambda>a'. f3 a' (y (pi@[(a,a')]))"
have a0: "finite ((supp (f3,a,[(b,c)]\<bullet>pi,y))::name set)" using a b
by (simp add: supp_prod fs_name1)
have a1: "((supp (f3,a,[(b,c)]\<bullet>pi,y))::name set) supports ?g" by (supports_simp add: perm_append)
hence a2: "finite ((supp ?g)::name set)" using a0 by (rule supports_finite)
have a3: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')"
by (rule f3_freshness_conditions_simple[OF a, OF b, OF c])
have "((supp (f3,a,y))::name set) \<subseteq> (supp (f3,a,[(b,c)]\<bullet>pi,y))" by (force simp add: supp_prod)
have a4: "[(b,c)]\<bullet>?g = ?h" using b1 b2
by (simp add: fresh_prod, perm_simp add: perm_append)
have "[(b,c)]\<bullet>(fresh_fun ?g) = fresh_fun ([(b,c)]\<bullet>?g)"
by (simp add: fresh_fun_equiv[OF pt_name_inst, OF at_name_inst, OF a2, OF a3])
also have "\<dots> = fresh_fun ?h" using a4 by simp
finally show "[(b,c)]\<bullet>(fresh_fun ?g) = fresh_fun ?h" .
qed
lemma f3_fresh_fun_supp_finite:
fixes f3::"('a::pt_name) f3_ty"
and a ::"name"
and pi1::"name prm"
and y ::"name prm \<Rightarrow> 'a::pt_name"
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 "finite ((supp (\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')])))))::name set)"
proof -
have "((supp (f3,a,y))::name set) supports (\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')]))))"
using a b c by (rule f3_fresh_fun_supports)
moreover
have "finite ((supp (f3,a,y))::name set)" using a b by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
(*
types 'a recT = "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> ((lam\<times>name prm\<times>('a::pt_name)) set)"
consts
rec :: "('a::pt_name) recT"
inductive "rec f1 f2 f3"
intros
r1: "(Var a,pi,f1 (pi\<bullet>a))\<in>rec f1 f2 f3"
r2: "\<lbrakk>finite ((supp y1)::name set);(t1,pi,y1)\<in>rec f1 f2 f3;
finite ((supp y2)::name set);(t2,pi,y2)\<in>rec f1 f2 f3\<rbrakk>
\<Longrightarrow> (App t1 t2,pi,f2 y1 y2)\<in>rec f1 f2 f3"
r3: "\<lbrakk>finite ((supp y)::name set);\<forall>a'. ((t,pi@[(a,a')],y)\<in>rec f1 f2 f3)\<rbrakk>
\<Longrightarrow> (Lam [a].t,pi,fresh_fun (\<lambda>a'. f3 a' y))\<in>rec f1 f2 f3"
*)
(*
types 'a recT = "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> (lam\<times>(name prm \<Rightarrow> ('a::pt_name))) set"
consts
rec :: "('a::pt_name) recT"
inductive "rec f1 f2 f3"
intros
r1: "(Var a,\<lambda>pi. f1 (pi\<bullet>a))\<in>rec f1 f2 f3"
r2: "\<lbrakk>finite ((supp y1)::name set);(t1,y1)\<in>rec f1 f2 f3;
finite ((supp y2)::name set);(t2,y2)\<in>rec f1 f2 f3\<rbrakk>
\<Longrightarrow> (App t1 t2,\<lambda>pi. f2 (y1 pi) (y2 pi))\<in>rec f1 f2 f3"
r3: "\<lbrakk>finite ((supp y)::name set);(t,y)\<in>rec f1 f2 f3\<rbrakk>
\<Longrightarrow> (Lam [a].t,fresh_fun (\<lambda>a' pi. f3 a' (y (pi@[(a,a')]))))\<in>rec f1 f2 f3"
*)
term lam_Rep.Var
term lam_Rep.Lam
consts nthe :: "'a nOption \<Rightarrow> 'a"
primrec
"nthe (nSome x) = x"
types 'a recT = "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> (lam\<times>(name prm \<Rightarrow> ('a::pt_name))) set"
(*
consts fn :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam_Rep \<Rightarrow> ('a::pt_name)"
primrec
"fn f1 f2 f3 (lam_Rep.Var a) = f1 a"
"fn f1 f2 f3 (lam_Rep.App t1 t2) = f2 (fn f1 f2 f3 t1) (fn f1 f2 f3 t2)"
"fn f1 f2 f3 (lam_Rep.Lam f) = fresh_fun (\<lambda>a'. f3 a' (fn f1 f2 f3 (fn' (f a'))))"
*)
consts
rec :: "('a::pt_name) recT"
inductive "rec f1 f2 f3"
intros
r1: "(Var a,\<lambda>pi. f1 (pi\<bullet>a))\<in>rec f1 f2 f3"
r2: "\<lbrakk>finite ((supp y1)::name set);(t1,y1)\<in>rec f1 f2 f3;
finite ((supp y2)::name set);(t2,y2)\<in>rec f1 f2 f3\<rbrakk>
\<Longrightarrow> (App t1 t2,\<lambda>pi. f2 (y1 pi) (y2 pi))\<in>rec f1 f2 f3"
r3: "\<lbrakk>finite ((supp y)::name set);(t,y)\<in>rec f1 f2 f3\<rbrakk>
\<Longrightarrow> (Lam [a].t,\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')]))))\<in>rec f1 f2 f3"
constdefs
rfun' :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> name prm \<Rightarrow> ('a::pt_name)"
"rfun' f1 f2 f3 t \<equiv> (THE y. (t,y)\<in>rec f1 f2 f3)"
rfun :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> ('a::pt_name)"
"rfun f1 f2 f3 t \<equiv> rfun' f1 f2 f3 t ([]::name prm)"
lemma rec_prm_eq[rule_format]:
fixes f1 ::"('a::pt_name) f1_ty"
and f2 ::"('a::pt_name) f2_ty"
and f3 ::"('a::pt_name) f3_ty"
and t ::"lam"
and y ::"name prm \<Rightarrow> ('a::pt_name)"
shows "(t,y)\<in>rec f1 f2 f3 \<Longrightarrow> (\<forall>(pi1::name prm) (pi2::name prm). pi1 \<sim> pi2 \<longrightarrow> (y pi1 = y pi2))"
apply(erule rec.induct)
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 spec)
apply(drule_tac x="pi2@[(a,x)]" in spec)
apply(force simp add: prm_eq_def pt2[OF pt_name_inst])
done
(* silly helper lemma *)
lemma rec_trans:
fixes f1 ::"('a::pt_name) f1_ty"
and f2 ::"('a::pt_name) f2_ty"
and f3 ::"('a::pt_name) f3_ty"
and t ::"lam"
and y ::"name prm \<Rightarrow> ('a::pt_name)"
assumes a: "(t,y)\<in>rec f1 f2 f3"
and b: "y=y'"
shows "(t,y')\<in>rec f1 f2 f3"
using a b by simp
lemma rec_prm_equiv:
fixes f1 ::"('a::pt_name) f1_ty"
and f2 ::"('a::pt_name) f2_ty"
and f3 ::"('a::pt_name) f3_ty"
and t ::"lam"
and y ::"name prm \<Rightarrow> ('a::pt_name)"
and 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)"
and a: "(t,y)\<in>rec f1 f2 f3"
shows "(pi\<bullet>t,pi\<bullet>y)\<in>rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)"
using a
proof (induct)
case (r1 c)
let ?g ="pi\<bullet>(\<lambda>(pi'::name prm). f1 (pi'\<bullet>c))"
and ?g'="\<lambda>(pi'::name prm). (pi\<bullet>f1) (pi'\<bullet>(pi\<bullet>c))"
have "?g'=?g"
proof (auto simp only: expand_fun_eq perm_fun_def)
fix pi'::"name prm"
let ?h = "((rev pi)\<bullet>(pi'\<bullet>(pi\<bullet>c)))"
and ?h'= "(((rev pi)\<bullet>pi')\<bullet>c)"
have "?h' = ((rev pi)\<bullet>pi')\<bullet>((rev pi)\<bullet>(pi\<bullet>c))"
by (simp add: pt_rev_pi[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = ?h"
by (simp add: pt_perm_compose[OF pt_name_inst, OF at_name_inst,symmetric])
finally show "pi\<bullet>(f1 ?h) = pi\<bullet>(f1 ?h')" by simp
qed
thus ?case by (force intro: rec_trans rec.r1)
next
case (r2 t1 t2 y1 y2)
assume a1: "finite ((supp y1)::name set)"
and a2: "finite ((supp y2)::name set)"
and a3: "(pi\<bullet>t1,pi\<bullet>y1) \<in> rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)"
and a4: "(pi\<bullet>t2,pi\<bullet>y2) \<in> rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)"
from a1 a2 have a1': "finite ((supp (pi\<bullet>y1))::name set)" and a2': "finite ((supp (pi\<bullet>y2))::name set)"
by (simp_all add: pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst])
let ?g ="pi\<bullet>(\<lambda>(pi'::name prm). f2 (y1 pi') (y2 pi'))"
and ?g'="\<lambda>(pi'::name prm). (pi\<bullet>f2) ((pi\<bullet>y1) pi') ((pi\<bullet>y2) pi')"
have "?g'=?g" by (simp add: expand_fun_eq perm_fun_def pt_rev_pi[OF pt_name_inst, OF at_name_inst])
thus ?case using a1' a2' a3 a4 by (force intro: rec.r2 rec_trans)
next
case (r3 a t y)
assume a1: "finite ((supp y)::name set)"
and a2: "(pi\<bullet>t,pi\<bullet>y) \<in> rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)"
from a1 have a1': "finite ((supp (pi\<bullet>y))::name set)"
by (simp add: pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst])
let ?g ="pi\<bullet>(\<lambda>(pi'::name prm). fresh_fun (\<lambda>a'. f3 a' (y (pi'@[(a,a')]))))"
and ?g'="(\<lambda>(pi'::name prm). fresh_fun (\<lambda>a'. (pi\<bullet>f3) a' ((pi\<bullet>y) (pi'@[((pi\<bullet>a),a')]))))"
have "?g'=?g"
proof (auto simp add: expand_fun_eq perm_fun_def pt_rev_pi[OF pt_name_inst, OF at_name_inst]
perm_append)
fix pi'::"name prm"
let ?h = "\<lambda>a'. pi\<bullet>(f3 ((rev pi)\<bullet>a') (y (((rev pi)\<bullet>pi')@[(a,(rev pi)\<bullet>a')])))"
and ?h' = "\<lambda>a'. f3 a' (y (((rev pi)\<bullet>pi')@[(a,a')]))"
have fs_f3: "finite ((supp f3)::name set)" using f by (simp add: supp_prod)
have fs_h': "finite ((supp ?h')::name set)"
proof -
have "((supp (f3,a,(rev pi)\<bullet>pi',y))::name set) supports ?h'"
by (supports_simp add: perm_append)
moreover
have "finite ((supp (f3,a,(rev pi)\<bullet>pi',y))::name set)" using a1 fs_f3
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
have fcond: "\<exists>(a''::name). a''\<sharp>?h' \<and> a''\<sharp>(?h' a'')"
by (rule f3_freshness_conditions_simple[OF fs_f3, OF a1, OF c])
have "fresh_fun ?h = fresh_fun (pi\<bullet>?h')"
by (simp add: perm_fun_def pt_rev_pi[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = pi\<bullet>(fresh_fun ?h')"
by (simp add: fresh_fun_equiv[OF pt_name_inst, OF at_name_inst, OF fs_h', OF fcond])
finally show "fresh_fun ?h = pi\<bullet>(fresh_fun ?h')" .
qed
thus ?case using a1' a2 by (force intro: rec.r3 rec_trans)
qed
lemma rec_perm:
fixes f1 ::"('a::pt_name) f1_ty"
and f2 ::"('a::pt_name) f2_ty"
and f3 ::"('a::pt_name) f3_ty"
and pi1::"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)"
and a: "(t,y)\<in>rec f1 f2 f3"
shows "(pi1\<bullet>t, \<lambda>pi2. y (pi2@pi1))\<in>rec f1 f2 f3"
proof -
have lem: "\<forall>(y::name prm\<Rightarrow>('a::pt_name))(pi::name prm).
finite ((supp y)::name set) \<longrightarrow> finite ((supp (\<lambda>pi'. y (pi'@pi)))::name set)"
proof (intro strip)
fix y::"name prm\<Rightarrow>('a::pt_name)" and pi::"name prm"
assume "finite ((supp y)::name set)"
hence "finite ((supp (y,pi))::name set)" by (simp add: supp_prod fs_name1)
moreover
have "((supp (y,pi))::name set) supports (\<lambda>pi'. y (pi'@pi))"
by (supports_simp add: perm_append)
ultimately show "finite ((supp (\<lambda>pi'. y (pi'@pi)))::name set)" by (simp add: supports_finite)
qed
from a
show ?thesis
proof (induct)
case (r1 c)
show ?case
by (auto simp add: pt2[OF pt_name_inst], rule rec.r1)
next
case (r2 t1 t2 y1 y2)
thus ?case using lem by (auto intro!: rec.r2)
next
case (r3 c t y)
assume a0: "(t,y)\<in>rec f1 f2 f3"
and a1: "finite ((supp y)::name set)"
and a2: "(pi1\<bullet>t,\<lambda>pi2. y (pi2@pi1))\<in>rec f1 f2 f3"
from f have f': "finite ((supp f3)::name set)" by (simp add: supp_prod)
show ?case
proof(simp)
have a3: "finite ((supp (\<lambda>pi2. y (pi2@pi1)))::name set)" using lem a1 by force
let ?g' = "\<lambda>(pi::name prm). fresh_fun (\<lambda>a'. f3 a' ((\<lambda>pi2. y (pi2@pi1)) (pi@[(pi1\<bullet>c,a')])))"
and ?g = "\<lambda>(pi::name prm). fresh_fun (\<lambda>a'. f3 a' (y (pi@[(pi1\<bullet>c,a')]@pi1)))"
and ?h = "\<lambda>(pi::name prm). fresh_fun (\<lambda>a'. f3 a' (y (pi@pi1@[(c,a')])))"
have eq1: "?g = ?g'" by simp
have eq2: "?g'= ?h"
proof (auto simp only: expand_fun_eq)
fix pi::"name prm"
let ?q = "\<lambda>a'. f3 a' (y ((pi@[(pi1\<bullet>c,a')])@pi1))"
and ?q' = "\<lambda>a'. f3 a' (y (((pi@pi1)@[(c,(rev pi1)\<bullet>a')])))"
and ?r = "\<lambda>a'. f3 a' (y ((pi@pi1)@[(c,a')]))"
and ?r' = "\<lambda>a'. f3 a' (y (pi@(pi1@[(c,a')])))"
have eq3a: "?r = ?r'" by simp
have eq3: "?q = ?q'"
proof (auto simp only: expand_fun_eq)
fix a'::"name"
have "(y ((pi@[(pi1\<bullet>c,a')])@pi1)) = (y (((pi@pi1)@[(c,(rev pi1)\<bullet>a')])))"
proof -
have "((pi@[(pi1\<bullet>c,a')])@pi1) \<sim> ((pi@pi1)@[(c,(rev pi1)\<bullet>a')])"
by (force simp add: prm_eq_def at_append[OF at_name_inst]
at_calc[OF at_name_inst] at_bij[OF at_name_inst]
at_pi_rev[OF at_name_inst] at_rev_pi[OF at_name_inst])
with a0 show ?thesis by (rule rec_prm_eq)
qed
thus "f3 a' (y ((pi@[(pi1\<bullet>c,a')])@pi1)) = f3 a' (y (((pi@pi1)@[(c,(rev pi1)\<bullet>a')])))" by simp
qed
have fs_a: "finite ((supp ?q')::name set)"
proof -
have "((supp (f3,c,(pi@pi1),(rev pi1),y))::name set) supports ?q'"
by (supports_simp add: perm_append fresh_list_append fresh_list_rev)
moreover
have "finite ((supp (f3,c,(pi@pi1),(rev pi1),y))::name set)" using f' a1
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
have fs_b: "finite ((supp ?r)::name set)"
proof -
have "((supp (f3,c,(pi@pi1),y))::name set) supports ?r"
by (supports_simp add: perm_append fresh_list_append)
moreover
have "finite ((supp (f3,c,(pi@pi1),y))::name set)" using f' a1
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
have c1: "\<exists>(a''::name). a''\<sharp>?q' \<and> a''\<sharp>(?q' a'')"
by (rule f3_freshness_conditions[OF f', OF a1, OF c])
have c2: "\<exists>(a''::name). a''\<sharp>?r \<and> a''\<sharp>(?r a'')"
by (rule f3_freshness_conditions_simple[OF f', OF a1, OF c])
have c3: "\<exists>(d::name). d\<sharp>(?q',?r,(rev pi1))"
by (rule at_exists_fresh[OF at_name_inst],
simp only: finite_Un supp_prod fs_a fs_b fs_name1, simp)
then obtain d::"name" where d1: "d\<sharp>?q'" and d2: "d\<sharp>?r" and d3: "d\<sharp>(rev pi1)"
by (auto simp only: fresh_prod)
have eq4: "(rev pi1)\<bullet>d = d" using d3 by (simp add: at_prm_fresh[OF at_name_inst])
have "fresh_fun ?q = fresh_fun ?q'" using eq3 by simp
also have "\<dots> = ?q' d" using fs_a c1 d1
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = ?r d" using fs_b c2 d2 eq4
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = fresh_fun ?r" using fs_b c2 d2
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
finally show "fresh_fun ?q = fresh_fun ?r'" by simp
qed
from a3 a2 have "(Lam [(pi1\<bullet>c)].(pi1\<bullet>t), ?g')\<in>rec f1 f2 f3" by (rule rec.r3)
thus "(Lam [(pi1\<bullet>c)].(pi1\<bullet>t), ?h)\<in>rec f1 f2 f3" using eq2 by simp
qed
qed
qed
lemma rec_perm_rev:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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>rec f1 f2 f3"
shows "(t, \<lambda>pi2. y (pi2@(rev pi)))\<in>rec f1 f2 f3"
proof -
from a have "((rev pi)\<bullet>(pi\<bullet>t),\<lambda>pi2. y (pi2@(rev pi)))\<in>rec f1 f2 f3"
by (rule rec_perm[OF f, OF c])
thus ?thesis by (simp add: pt_rev_pi[OF pt_name_inst, OF at_name_inst])
qed
lemma rec_unique:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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>rec f1 f2 f3"
shows "\<forall>(y'::name prm\<Rightarrow>('a::pt_name))(pi::name prm).
(t,y')\<in>rec f1 f2 f3 \<longrightarrow> y pi = y' pi"
using a
proof (induct)
case (r1 c)
show ?case
apply(auto)
apply(ind_cases "(Var c, y') \<in> rec f1 f2 f3")
apply(simp_all add: lam.distinct lam.inject)
done
next
case (r2 t1 t2 y1 y2)
thus ?case
apply(clarify)
apply(ind_cases "(App t1 t2, y') \<in> rec f1 f2 f3")
apply(simp_all (no_asm_use) only: lam.distinct lam.inject)
apply(clarify)
apply(drule_tac x="y1" in spec)
apply(drule_tac x="y2" in spec)
apply(auto)
done
next
case (r3 c t y)
assume i1: "finite ((supp y)::name set)"
and i2: "(t,y)\<in>rec f1 f2 f3"
and i3: "\<forall>(y'::name prm\<Rightarrow>('a::pt_name))(pi::name prm).
(t,y')\<in>rec f1 f2 f3 \<longrightarrow> y pi = y' pi"
from f have f': "finite ((supp f3)::name set)" by (simp add: supp_prod)
show ?case
proof (auto)
fix y'::"name prm\<Rightarrow>('a::pt_name)" and pi::"name prm"
assume i4: "(Lam [c].t, y') \<in> rec f1 f2 f3"
from i4 show "fresh_fun (\<lambda>a'. f3 a' (y (pi@[(c,a')]))) = y' pi"
proof (cases, simp_all add: lam.distinct lam.inject, auto)
fix a::"name" and t'::"lam" and y''::"name prm\<Rightarrow>('a::pt_name)"
assume i5: "[c].t = [a].t'"
and i6: "(t',y'')\<in>rec f1 f2 f3"
and i6':"finite ((supp y'')::name set)"
let ?g = "\<lambda>a'. f3 a' (y (pi@[(c,a')]))"
and ?h = "\<lambda>a'. f3 a' (y'' (pi@[(a,a')]))"
show "fresh_fun ?g = fresh_fun ?h" using i5
proof (cases "a=c")
case True
assume i7: "a=c"
with i5 have i8: "t=t'" by (simp add: alpha)
show "fresh_fun ?g = fresh_fun ?h" using i3 i6 i7 i8 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 r1: "finite ((supp ?g)::name set)"
proof -
have "((supp (f3,c,pi,y))::name set) supports ?g"
by (supports_simp add: perm_append)
moreover
have "finite ((supp (f3,c,pi,y))::name set)" using f' i1
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
have r2: "finite ((supp ?h)::name set)"
proof -
have "((supp (f3,a,pi,y''))::name set) supports ?h"
by (supports_simp add: perm_append)
moreover
have "finite ((supp (f3,a,pi,y''))::name set)" using f' i6'
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
have c1: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')"
by (rule f3_freshness_conditions_simple[OF f', OF i1, OF c])
have c2: "\<exists>(a''::name). a''\<sharp>?h \<and> a''\<sharp>(?h a'')"
by (rule f3_freshness_conditions_simple[OF f', OF i6', OF c])
have "\<exists>(d::name). d\<sharp>(?g,?h,t,a,c)"
by (rule at_exists_fresh[OF at_name_inst],
simp only: finite_Un supp_prod r1 r2 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,y'')\<in>rec f1 f2 f3" using g1 i10 by (simp only: pt_name2)
hence "(t, \<lambda>pi2. y'' (pi2@(rev ([(a,c)]@[(a,d)]))))\<in>rec f1 f2 f3"
by (simp only: rec_perm_rev[OF f, OF c])
hence g2: "(t, \<lambda>pi2. y'' (pi2@([(a,d)]@[(a,c)])))\<in>rec 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)]) \<sim> (pi@[(a,d)])"
by (force simp add: prm_eq_def at_calc[OF at_name_inst] at_append[OF at_name_inst])
have "fresh_fun ?g = ?g d" using r1 c1 f1
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = f3 d ((\<lambda>pi2. y'' (pi2@([(a,d)]@[(a,c)]))) (pi@[(c,d)]))" using i3 g2 by simp
also have "\<dots> = f3 d (y'' (pi@[(c,d)]@[(a,d)]@[(a,c)]))" by simp
also have "\<dots> = f3 d (y'' (pi@[(a,d)]))" using i6 g3 by (simp add: rec_prm_eq)
also have "\<dots> = fresh_fun ?h" using r2 c2 f2
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst])
finally show "fresh_fun ?g = fresh_fun ?h" .
qed
qed
qed
qed
lemma rec_total_aux:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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::name prm\<Rightarrow>('a::pt_name)). ((t,y)\<in>rec f1 f2 f3) \<and> (finite ((supp y)::name set))"
proof (induct t rule: lam.induct_weak)
case (Var c)
have a1: "(Var c,\<lambda>(pi::name prm). f1 (pi\<bullet>c))\<in>rec f1 f2 f3" by (rule rec.r1)
have a2: "finite ((supp (\<lambda>(pi::name prm). f1 (pi\<bullet>c)))::name set)"
proof -
have "((supp (f1,c))::name set) supports (\<lambda>(pi::name prm). f1 (pi\<bullet>c))" by (supports_simp)
moreover have "finite ((supp (f1,c))::name set)" using f by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
from a1 a2 show ?case by force
next
case (App t1 t2)
assume "\<exists>y1. (t1,y1)\<in>rec f1 f2 f3 \<and> finite ((supp y1)::name set)"
then obtain y1::"name prm \<Rightarrow> ('a::pt_name)"
where a11: "(t1,y1)\<in>rec f1 f2 f3" and a12: "finite ((supp y1)::name set)" by force
assume "\<exists>y2. (t2,y2)\<in>rec f1 f2 f3 \<and> finite ((supp y2)::name set)"
then obtain y2::"name prm \<Rightarrow> ('a::pt_name)"
where a21: "(t2,y2)\<in>rec f1 f2 f3" and a22: "finite ((supp y2)::name set)" by force
have a1: "(App t1 t2,\<lambda>(pi::name prm). f2 (y1 pi) (y2 pi))\<in>rec f1 f2 f3"
using a12 a11 a22 a21 by (rule rec.r2)
have a2: "finite ((supp (\<lambda>(pi::name prm). f2 (y1 pi) (y2 pi)))::name set)"
proof -
have "((supp (f2,y1,y2))::name set) supports (\<lambda>(pi::name prm). f2 (y1 pi) (y2 pi))"
by (supports_simp)
moreover have "finite ((supp (f2,y1,y2))::name set)" using f a21 a22
by (simp add: supp_prod fs_name1)
ultimately show ?thesis by (rule supports_finite)
qed
from a1 a2 show ?case by force
next
case (Lam a t)
assume "\<exists>y. (t,y)\<in>rec f1 f2 f3 \<and> finite ((supp y)::name set)"
then obtain y::"name prm \<Rightarrow> ('a::pt_name)"
where a11: "(t,y)\<in>rec f1 f2 f3" and a12: "finite ((supp y)::name set)" by force
from f have f': "finite ((supp f3)::name set)" by (simp add: supp_prod)
have a1: "(Lam [a].t,\<lambda>(pi::name prm). fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')]))))\<in>rec f1 f2 f3"
using a12 a11 by (rule rec.r3)
have a2: "finite ((supp (\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (y (pi@[(a,a')])))))::name set)"
using f' a12 c by (rule f3_fresh_fun_supp_finite)
from a1 a2 show ?case by force
qed
lemma rec_total:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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::name prm\<Rightarrow>('a::pt_name)). ((t,y)\<in>rec f1 f2 f3)"
proof -
have "\<exists>y. ((t,y)\<in>rec f1 f2 f3) \<and> (finite ((supp y)::name set))"
by (rule rec_total_aux[OF f, OF c])
thus ?thesis by force
qed
lemma rec_function:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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>rec f1 f2 f3"
proof
case goal1
show ?case by (rule rec_total[OF f, OF c])
next
case (goal2 y1 y2)
assume a1: "(t,y1)\<in>rec f1 f2 f3" and a2: "(t,y2)\<in>rec f1 f2 f3"
hence "\<forall>pi. y1 pi = y2 pi" using rec_unique[OF f, OF c] by force
thus ?case by (force simp add: expand_fun_eq)
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 rfun'_equiv:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
and pi::"name prm"
and t ::"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)"
shows "pi\<bullet>(rfun' f1 f2 f3 t) = rfun' (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
apply(auto simp add: rfun'_def)
apply(subgoal_tac "\<exists>y. (t,y)\<in>rec f1 f2 f3 \<and> finite ((supp y)::name set)")
apply(auto)
apply(rule sym)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule rec_function[OF f, OF c])
apply(rule the1_equality)
apply(rule rec_function)
apply(subgoal_tac "finite ((supp (f1,f2,f3))::name set)")
apply(simp add: supp_prod)
apply(simp add: pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst])
apply(rule f)
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: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
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: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
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(simp add: pt_pi_rev[OF pt_name_inst, OF at_name_inst])
apply(simp add: pt_fun_app_eq[OF pt_name_inst, OF at_name_inst])
apply(rule c)
apply(rule rec_prm_equiv)
apply(rule f, rule c, assumption)
apply(rule rec_total_aux)
apply(rule f)
apply(rule c)
done
lemma rfun'_supports:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "((supp (f1,f2,f3,t))::name set) supports (rfun' f1 f2 f3 t)"
proof (auto simp add: "op supports_def" fresh_def[symmetric])
fix a::"name" and b::"name"
assume a1: "a\<sharp>(f1,f2,f3,t)"
and a2: "b\<sharp>(f1,f2,f3,t)"
from a1 a2 have "[(a,b)]\<bullet>f1=f1" and "[(a,b)]\<bullet>f2=f2" and "[(a,b)]\<bullet>f3=f3" and "[(a,b)]\<bullet>t=t"
by (simp_all add: pt_fresh_fresh[OF pt_name_inst, OF at_name_inst] fresh_prod)
thus "[(a,b)]\<bullet>(rfun' f1 f2 f3 t) = rfun' f1 f2 f3 t"
by (simp add: rfun'_equiv[OF f, OF c])
qed
lemma rfun'_finite_supp:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 (rfun' f1 f2 f3 t))::name set)"
apply(rule supports_finite)
apply(rule rfun'_supports[OF f, OF c])
apply(subgoal_tac "finite ((supp (f1,f2,f3))::name set)")
apply(simp add: supp_prod fs_name1)
apply(rule f)
done
lemma rfun'_prm:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
and pi1::"name prm"
and pi2::"name prm"
and t ::"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)"
shows "rfun' f1 f2 f3 (pi1\<bullet>t) pi2 = rfun' f1 f2 f3 t (pi2@pi1)"
apply(auto simp add: rfun'_def)
apply(subgoal_tac "\<exists>y. (t,y)\<in>rec f1 f2 f3 \<and> finite ((supp y)::name set)")
apply(auto)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule rec_function[OF f, OF c])
apply(rule_tac a="\<lambda>pi2. y (pi2@pi1)" in theI2')
apply(rule rec_perm)
apply(rule f, rule c)
apply(assumption)
apply(rule rec_function)
apply(rule f)
apply(rule c)
apply(simp)
apply(rule rec_total_aux)
apply(rule f)
apply(rule c)
done
lemma rfun_Var:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "rfun f1 f2 f3 (Var c) = (f1 c)"
apply(auto simp add: rfun_def rfun'_def)
apply(subgoal_tac "(THE y. (Var c, y) \<in> rec f1 f2 f3) = (\<lambda>(pi::name prm). f1 (pi\<bullet>c))")(*A*)
apply(simp)
apply(rule the1_equality)
apply(rule rec_function)
apply(rule f)
apply(rule c)
apply(rule rec.r1)
done
lemma rfun_App:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "rfun f1 f2 f3 (App t1 t2) = (f2 (rfun f1 f2 f3 t1) (rfun f1 f2 f3 t2))"
apply(auto simp add: rfun_def rfun'_def)
apply(subgoal_tac "(THE y. (App t1 t2, y)\<in>rec f1 f2 f3) =
(\<lambda>(pi::name prm). f2 ((rfun' f1 f2 f3 t1) pi) ((rfun' f1 f2 f3 t2) pi))")(*A*)
apply(simp add: rfun'_def)
apply(rule the1_equality)
apply(rule rec_function[OF f, OF c])
apply(rule rec.r2)
apply(rule rfun'_finite_supp[OF f, OF c])
apply(subgoal_tac "\<exists>y. (t1,y)\<in>rec f1 f2 f3")
apply(erule exE, simp add: rfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule rec_function[OF f, OF c])
apply(assumption)
apply(rule rec_total[OF f, OF c])
apply(rule rfun'_finite_supp[OF f, OF c])
apply(subgoal_tac "\<exists>y. (t2,y)\<in>rec f1 f2 f3")
apply(auto simp add: rfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule rec_function[OF f, OF c])
apply(assumption)
apply(rule rec_total[OF f, OF c])
done
lemma rfun_Lam_aux1:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "rfun f1 f2 f3 (Lam [a].t) = fresh_fun (\<lambda>a'. f3 a' (rfun' f1 f2 f3 t ([]@[(a,a')])))"
apply(simp add: rfun_def rfun'_def)
apply(subgoal_tac "(THE y. (Lam [a].t, y)\<in>rec f1 f2 f3) =
(\<lambda>(pi::name prm). fresh_fun(\<lambda>a'. f3 a' ((rfun' f1 f2 f3 t) (pi@[(a,a')]))))")(*A*)
apply(simp add: rfun'_def[symmetric])
apply(rule the1_equality)
apply(rule rec_function[OF f, OF c])
apply(rule rec.r3)
apply(rule rfun'_finite_supp[OF f, OF c])
apply(subgoal_tac "\<exists>y. (t,y)\<in>rec f1 f2 f3")
apply(erule exE, simp add: rfun'_def)
apply(rule_tac a="y" in theI2')
apply(assumption)
apply(rule rec_function[OF f, OF c])
apply(assumption)
apply(rule rec_total[OF f, OF c])
done
lemma rfun_Lam_aux2:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "rfun f1 f2 f3 (Lam [b].([(a,b)]\<bullet>t)) = f3 b (rfun f1 f2 f3 ([(a,b)]\<bullet>t))"
proof -
from f have f': "finite ((supp f3)::name set)" by (simp add: supp_prod)
have eq1: "rfun f1 f2 f3 (Lam [b].([(a,b)]\<bullet>t)) = rfun f1 f2 f3 (Lam [a].t)"
proof -
have "Lam [a].t = Lam [b].([(a,b)]\<bullet>t)" using a
by (force simp add: lam.inject alpha fresh_prod at_fresh[OF at_name_inst]
pt_swap_bij[OF pt_name_inst, OF at_name_inst]
pt_fresh_left[OF pt_name_inst, OF at_name_inst] at_calc[OF at_name_inst])
thus ?thesis by simp
qed
let ?g = "(\<lambda>a'. f3 a' (rfun' f1 f2 f3 t ([]@[(a,a')])))"
have a0: "((supp (f3,a,([]::name prm),rfun' f1 f2 f3 t))::name set) supports ?g"
by (supports_simp add: perm_append)
have a1: "finite ((supp (f3,a,([]::name prm),rfun' f1 f2 f3 t))::name set)"
using f' by (simp add: supp_prod fs_name1 rfun'_finite_supp[OF f, OF c])
from a0 a1 have a2: "finite ((supp ?g)::name set)"
by (rule supports_finite)
have c0: "finite ((supp (rfun' f1 f2 f3 t))::name set)"
by (rule rfun'_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 (rfun' f1 f2 f3 t)"
by (simp add: rfun'_supports[OF f, OF c])
hence c3: "b\<sharp>(rfun' 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
proof(rule supports_fresh[OF a0, OF a1], simp add: fresh_def[symmetric])
show "b\<sharp>(f3,a,([]::name prm),rfun' f1 f2 f3 t)" using a c3
by (simp add: fresh_prod fresh_list_nil)
qed
qed
(* main proof *)
have "rfun f1 f2 f3 (Lam [b].([(a,b)]\<bullet>t)) = rfun f1 f2 f3 (Lam [a].t)" by (simp add: eq1)
also have "\<dots> = fresh_fun ?g" by (rule rfun_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 (rfun f1 f2 f3 ([(a,b)]\<bullet>t))"
by (simp add: rfun_def rfun'_prm[OF f, OF c])
finally show "rfun f1 f2 f3 (Lam [b].([(a,b)]\<bullet>t)) = f3 b (rfun f1 f2 f3 ([(a,b)]\<bullet>t))" .
qed
lemma rfun_Lam:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 "rfun f1 f2 f3 (Lam [b].t) = f3 b (rfun 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 "rfun f1 f2 f3 (Lam [b].t) = rfun f1 f2 f3 (Lam [b].(([(a,b)])\<bullet>([(a,b)]\<bullet>t)))"
by (simp add: pt_swap_bij[OF pt_name_inst, OF at_name_inst])
also have "\<dots> = f3 b (rfun f1 f2 f3 (([(a,b)])\<bullet>([(a,b)]\<bullet>t)))"
proof(rule rfun_Lam_aux2[OF f, OF c])
have "b\<sharp>([(a,b)]\<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,[(a,b)]\<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 (rfun f1 f2 f3 t)" by (simp add: pt_swap_bij[OF pt_name_inst, OF at_name_inst])
finally show ?thesis .
qed
lemma rec_unique:
fixes fun::"lam \<Rightarrow> ('a::pt_name)"
and f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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 a1: "\<forall>c. fun (Var c) = f1 c"
and a2: "\<forall>t1 t2. fun (App t1 t2) = f2 (fun t1) (fun t2)"
and a3: "\<forall>a t. a\<sharp>(f1,f2,f3) \<longrightarrow> fun (Lam [a].t) = f3 a (fun t)"
shows "fun t = rfun f1 f2 f3 t"
(*apply(nominal_induct t rule: lam_induct')*)
apply (rule lam_induct'[of "\<lambda>_. ((supp (f1,f2,f3))::name set)" "\<lambda>t _. fun t = rfun f1 f2 f3 t"])
(* finite support *)
apply(rule f)
(* VAR *)
apply(simp add: a1 rfun_Var[OF f, OF c, symmetric])
(* APP *)
apply(simp add: a2 rfun_App[OF f, OF c, symmetric])
(* LAM *)
apply(fold fresh_def)
apply(simp add: a3 rfun_Lam[OF f, OF c, symmetric])
done
lemma rec_function:
fixes f1::"('a::pt_name) f1_ty"
and f2::"('a::pt_name) f2_ty"
and f3::"('a::pt_name) f3_ty"
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>!(fun::lam \<Rightarrow> ('a::pt_name)).
(\<forall>c. fun (Var c) = f1 c) \<and>
(\<forall>t1 t2. fun (App t1 t2) = f2 (fun t1) (fun t2)) \<and>
(\<forall>a t. a\<sharp>(f1,f2,f3) \<longrightarrow> fun (Lam [a].t) = f3 a (fun t)))"
apply(rule_tac a="\<lambda>t. rfun f1 f2 f3 t" in ex1I)
(* existence *)
apply(simp add: rfun_Var[OF f, OF c])
apply(simp add: rfun_App[OF f, OF c])
apply(simp add: rfun_Lam[OF f, OF c])
(* uniqueness *)
apply(auto simp add: expand_fun_eq)
apply(rule rec_unique[OF f, OF c])
apply(force)+
done
(* "real" recursion *)
types 'a f1_ty' = "name\<Rightarrow>('a::pt_name)"
'a f2_ty' = "lam\<Rightarrow>lam\<Rightarrow>'a\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
'a f3_ty' = "lam\<Rightarrow>name\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
constdefs
rfun_gen' :: "'a f1_ty' \<Rightarrow> 'a f2_ty' \<Rightarrow> 'a f3_ty' \<Rightarrow> lam \<Rightarrow> (lam\<times>'a::pt_name)"
"rfun_gen' f1 f2 f3 t \<equiv> (rfun
(\<lambda>(a::name). (Var a,f1 a))
(\<lambda>r1 r2. (App (fst r1) (fst r2), f2 (fst r1) (fst r2) (snd r1) (snd r2)))
(\<lambda>(a::name) r. (Lam [a].(fst r),f3 (fst r) a (snd r)))
t)"
rfun_gen :: "'a f1_ty' \<Rightarrow> 'a f2_ty' \<Rightarrow> 'a f3_ty' \<Rightarrow> lam \<Rightarrow> 'a::pt_name"
"rfun_gen f1 f2 f3 t \<equiv> snd(rfun_gen' f1 f2 f3 t)"
lemma f1'_supports:
fixes f1::"('a::pt_name) f1_ty'"
shows "((supp f1)::name set) supports (\<lambda>(a::name). (Var a,f1 a))"
by (supports_simp)
lemma f2'_supports:
fixes f2::"('a::pt_name) f2_ty'"
shows "((supp f2)::name set) supports
(\<lambda>r1 r2. (App (fst r1) (fst r2), f2 (fst r1) (fst r2) (snd r1) (snd r2)))"
by (supports_simp add: perm_fst perm_snd)
lemma f3'_supports:
fixes f3::"('a::pt_name) f3_ty'"
shows "((supp f3)::name set) supports (\<lambda>(a::name) r. (Lam [a].(fst r),f3 (fst r) a (snd r)))"
by (supports_simp add: perm_fst perm_snd)
lemma fcb':
fixes f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
shows "\<exists>a. a \<sharp> (\<lambda>a r. (Lam [a].fst r, f3 (fst r) a (snd r))) \<and>
(\<forall>y. a \<sharp> (Lam [a].fst y, f3 (fst y) a (snd y)))"
using c f
apply(auto)
apply(rule_tac x="a" in exI)
apply(auto simp add: abs_fresh fresh_prod)
apply(rule supports_fresh)
apply(rule f3'_supports)
apply(simp add: supp_prod)
apply(simp add: fresh_def)
done
lemma fsupp':
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
shows "finite((supp
(\<lambda>a. (Var a, f1 a),
\<lambda>r1 r2.
(App (fst r1) (fst r2),
f2 (fst r1) (fst r2) (snd r1) (snd r2)),
\<lambda>a r. (Lam [a].fst r, f3 (fst r) a (snd r))))::name set)"
using f
apply(auto simp add: supp_prod)
apply(rule supports_finite)
apply(rule f1'_supports)
apply(assumption)
apply(rule supports_finite)
apply(rule f2'_supports)
apply(assumption)
apply(rule supports_finite)
apply(rule f3'_supports)
apply(assumption)
done
lemma rfun_gen'_fst:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "(\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y))"
shows "fst (rfun_gen' f1 f2 f3 t) = t"
apply (rule lam_induct'[of "\<lambda>_. ((supp (f1,f2,f3))::name set)" "\<lambda>t _. fst (rfun_gen' f1 f2 f3 t) = t"])
apply(simp add: f)
apply(unfold rfun_gen'_def)
apply(simp only: rfun_Var[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
apply(simp)
apply(simp only: rfun_App[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
apply(simp)
apply(fold fresh_def)
apply(auto)
apply(rule trans)
apply(rule_tac f="fst" in arg_cong)
apply(rule rfun_Lam[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
apply(auto simp add: fresh_prod)
apply(rule supports_fresh)
apply(rule f1'_supports)
apply(subgoal_tac "finite ((supp (f1,f2,f3))::name set)")
apply(simp add: supp_prod)
apply(rule f)
apply(simp add: fresh_def)
apply(rule supports_fresh)
apply(rule f2'_supports)
apply(subgoal_tac "finite ((supp (f1,f2,f3))::name set)")
apply(simp add: supp_prod)
apply(rule f)
apply(simp add: fresh_def)
apply(rule supports_fresh)
apply(rule f3'_supports)
apply(subgoal_tac "finite ((supp (f1,f2,f3))::name set)")
apply(simp add: supp_prod)
apply(rule f)
apply(simp add: fresh_def)
done
lemma rfun_gen'_Var:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
shows "rfun_gen' f1 f2 f3 (Var c) = (Var c, f1 c)"
apply(simp add: rfun_gen'_def)
apply(simp add: rfun_Var[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
done
lemma rfun_gen'_App:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
shows "rfun_gen' f1 f2 f3 (App t1 t2) =
(App t1 t2, f2 t1 t2 (rfun_gen f1 f2 f3 t1) (rfun_gen f1 f2 f3 t2))"
apply(simp add: rfun_gen'_def)
apply(rule trans)
apply(rule rfun_App[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
apply(fold rfun_gen'_def)
apply(simp_all add: rfun_gen'_fst[OF f, OF c])
apply(simp_all add: rfun_gen_def)
done
lemma rfun_gen'_Lam:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
and a: "b\<sharp>(f1,f2,f3)"
shows "rfun_gen' f1 f2 f3 (Lam [b].t) = (Lam [b].t, f3 t b (rfun_gen f1 f2 f3 t))"
using a f
apply(simp add: rfun_gen'_def)
apply(rule trans)
apply(rule rfun_Lam[OF fsupp'[OF f],OF fcb'[OF f, OF c]])
apply(auto simp add: fresh_prod)
apply(rule supports_fresh)
apply(rule f1'_supports)
apply(simp add: supp_prod)
apply(simp add: fresh_def)
apply(rule supports_fresh)
apply(rule f2'_supports)
apply(simp add: supp_prod)
apply(simp add: fresh_def)
apply(rule supports_fresh)
apply(rule f3'_supports)
apply(simp add: supp_prod)
apply(simp add: fresh_def)
apply(fold rfun_gen'_def)
apply(simp_all add: rfun_gen'_fst[OF f, OF c])
apply(simp_all add: rfun_gen_def)
done
lemma rfun_gen_Var:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
shows "rfun_gen f1 f2 f3 (Var c) = f1 c"
apply(unfold rfun_gen_def)
apply(simp add: rfun_gen'_Var[OF f, OF c])
done
lemma rfun_gen_App:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
shows "rfun_gen f1 f2 f3 (App t1 t2) = f2 t1 t2 (rfun_gen f1 f2 f3 t1) (rfun_gen f1 f2 f3 t2)"
apply(unfold rfun_gen_def)
apply(simp add: rfun_gen'_App[OF f, OF c])
apply(simp add: rfun_gen_def)
done
lemma rfun_gen_Lam:
fixes f1::"('a::pt_name) f1_ty'"
and f2::"('a::pt_name) f2_ty'"
and f3::"('a::pt_name) f3_ty'"
assumes f: "finite ((supp (f1,f2,f3))::name set)"
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name) t. a\<sharp>f3 t a y)"
and a: "b\<sharp>(f1,f2,f3)"
shows "rfun_gen f1 f2 f3 (Lam [b].t) = f3 t b (rfun_gen f1 f2 f3 t)"
using a
apply(unfold rfun_gen_def)
apply(simp add: rfun_gen'_Lam[OF f, OF c])
apply(simp add: rfun_gen_def)
done
instance nat :: pt_name
apply(intro_classes)
apply(simp_all add: perm_nat_def)
done
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> rfun 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 supp_depth_Lam)
apply(auto simp add: depth_Lam_def)
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: rfun_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: rfun_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 rfun_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
(* 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> rfun (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: rfun_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: rfun_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: rfun_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
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]
lemma subst_eqvt[simp]:
fixes pi:: "name prm"
and t1:: "lam"
and t2:: "lam"
and a :: "name"
shows "pi\<bullet>(t1[a::=t2]) = (pi\<bullet>t1)[(pi\<bullet>a)::=(pi\<bullet>t2)]"
apply(nominal_induct t1 rule: lam_induct)
apply(auto)
apply(auto simp add: perm_bij fresh_prod fresh_atm)
apply(subgoal_tac "(aa\<bullet>ab)\<sharp>(aa\<bullet>a,aa\<bullet>b)")(*A*)
apply(simp)
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 rule: lam_induct)
apply(auto simp add: lam.supp supp_atm fresh_prod abs_supp)
apply(blast)
apply(blast)
done
(* parallel substitution *)
consts
"domain" :: "(name\<times>lam) list \<Rightarrow> (name list)"
primrec
"domain [] = []"
"domain (x#\<theta>) = (fst x)#(domain \<theta>)"
consts
"apply_sss" :: "(name\<times>lam) list \<Rightarrow> name \<Rightarrow> lam" (" _ < _ >" [80,80] 80)
primrec
"(x#\<theta>)<a> = (if (a = fst x) then (snd x) else \<theta><a>)"
lemma apply_sss_eqvt[rule_format]:
fixes pi::"name prm"
shows "a\<in>set (domain \<theta>) \<longrightarrow> pi\<bullet>(\<theta><a>) = (pi\<bullet>\<theta>)<pi\<bullet>a>"
apply(induct_tac \<theta>)
apply(auto)
apply(simp add: pt_bij[OF pt_name_inst, OF at_name_inst])
done
lemma domain_eqvt[rule_format]:
fixes pi::"name prm"
shows "((pi\<bullet>a)\<in>set (domain \<theta>)) = (a\<in>set (domain ((rev pi)\<bullet>\<theta>)))"
apply(induct_tac \<theta>)
apply(auto)
apply(simp_all add: pt_rev_pi[OF pt_name_inst, OF at_name_inst])
apply(simp_all add: pt_pi_rev[OF pt_name_inst, OF at_name_inst])
done
constdefs
psubst_Var :: "(name\<times>lam) list \<Rightarrow> name \<Rightarrow> lam"
"psubst_Var \<theta> \<equiv> \<lambda>a. (if a\<in>set (domain \<theta>) then \<theta><a> else (Var a))"
psubst_App :: "(name\<times>lam) list \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam"
"psubst_App \<theta> \<equiv> \<lambda>r1 r2. App r1 r2"
psubst_Lam :: "(name\<times>lam) list \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"
"psubst_Lam \<theta> \<equiv> \<lambda>a r. Lam [a].r"
psubst_lam :: "(name\<times>lam) list \<Rightarrow> lam \<Rightarrow> lam"
"psubst_lam \<theta> \<equiv> rfun (psubst_Var \<theta>) (psubst_App \<theta>) (psubst_Lam \<theta>)"
lemma supports_psubst_Var:
shows "((supp \<theta>)::name set) supports (psubst_Var \<theta>)"
by (supports_simp add: psubst_Var_def apply_sss_eqvt domain_eqvt)
lemma supports_psubst_App:
shows "((supp \<theta>)::name set) supports psubst_App \<theta>"
by (supports_simp add: psubst_App_def)
lemma supports_psubst_Lam:
shows "((supp \<theta>)::name set) supports psubst_Lam \<theta>"
by (supports_simp add: psubst_Lam_def)
lemma fin_supp_psubst:
shows "finite ((supp (psubst_Var \<theta>,psubst_App \<theta>,psubst_Lam \<theta>))::name set)"
apply(auto simp add: supp_prod)
apply(rule supports_finite)
apply(rule supports_psubst_Var)
apply(simp add: fs_name1)
apply(rule supports_finite)
apply(rule supports_psubst_App)
apply(simp add: fs_name1)
apply(rule supports_finite)
apply(rule supports_psubst_Lam)
apply(simp add: fs_name1)
done
lemma fresh_psubst_Lam:
shows "\<exists>(a::name). a\<sharp>(psubst_Lam \<theta>)\<and> (\<forall>y. a\<sharp>(psubst_Lam \<theta>) a y)"
apply(subgoal_tac "\<exists>(c::name). c\<sharp>\<theta>")
apply(auto)
apply(rule_tac x="c" in exI)
apply(auto)
apply(rule supports_fresh)
apply(rule supports_psubst_Lam)
apply(simp_all add: fresh_def[symmetric] fs_name1)
apply(simp add: psubst_Lam_def)
apply(simp add: abs_fresh)
apply(rule at_exists_fresh[OF at_name_inst])
apply(simp add: fs_name1)
done
lemma psubst_Var:
shows "psubst_lam \<theta> (Var a) = (if a\<in>set (domain \<theta>) then \<theta><a> else (Var a))"
apply(simp add: psubst_lam_def)
apply(auto simp add: rfun_Var[OF fin_supp_psubst, OF fresh_psubst_Lam])
apply(auto simp add: psubst_Var_def)
done
lemma psubst_App:
shows "psubst_lam \<theta> (App t1 t2) = App (psubst_lam \<theta> t1) (psubst_lam \<theta> t2)"
apply(simp add: psubst_lam_def)
apply(auto simp add: rfun_App[OF fin_supp_psubst, OF fresh_psubst_Lam])
apply(auto simp add: psubst_App_def)
done
lemma psubst_Lam:
assumes a: "a\<sharp>\<theta>"
shows "psubst_lam \<theta> (Lam [a].t1) = Lam [a].(psubst_lam \<theta> t1)"
using a
apply(simp add: psubst_lam_def)
apply(subgoal_tac "a\<sharp>(psubst_Var \<theta>,psubst_App \<theta>,psubst_Lam \<theta>)")
apply(auto simp add: rfun_Lam[OF fin_supp_psubst, OF fresh_psubst_Lam])
apply(simp (no_asm) add: psubst_Lam_def)
apply(auto simp add: fresh_prod)
apply(rule supports_fresh)
apply(rule supports_psubst_Var)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
apply(rule supports_fresh)
apply(rule supports_psubst_App)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
apply(rule supports_fresh)
apply(rule supports_psubst_Lam)
apply(simp_all add: fs_name1 fresh_def[symmetric] fresh_prod)
done
declare psubst_Var[simp]
declare psubst_App[simp]
declare psubst_Lam[simp]
consts
psubst_syn :: "lam \<Rightarrow> (name\<times>lam) list \<Rightarrow> lam" ("_[<_>]" [100,100] 900)
translations
"t[<\<theta>>]" \<rightleftharpoons> "psubst_lam \<theta> t"
end