theory class
imports "../nominal"
begin
atom_decl name coname
section {* Term-Calculus from my PHD *}
nominal_datatype trm = Ax "name" "coname"
| ImpR "\<guillemotleft>name\<guillemotright>(\<guillemotleft>coname\<guillemotright>trm)" "coname" ("ImpR [_].[_]._ _" [100,100,100,100] 100)
| ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name"("ImpL [_]._ [_]._ _" [100,100,100,100,100] 100)
| Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Cut [_]._ [_]._" [100,100,100,100] 100)
lemma trm_induct_aux:
fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool"
and f1 :: "'a \<Rightarrow> name set"
and f2 :: "'a \<Rightarrow> coname set"
assumes fs1: "\<And>x. finite (f1 x)"
and fs2: "\<And>x. finite (f2 x)"
and h1: "\<And>k x a. P (Ax x a) k"
and h2: "\<And>k x a t b. x\<notin>f1 k \<Longrightarrow> a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k"
and h3: "\<And>k a t1 x t2 y. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k"
and h4: "\<And>k a t1 x t2. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (Cut [a].t1 [x].t2) k"
shows "\<forall>(pi1::name prm) (pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k"
proof (induct rule: trm.induct_weak)
case (goal1 a)
show ?case using h1 by simp
next
case (goal2 x a t b)
assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k"
show ?case
proof (intro strip, simp add: abs_perm perm_dj)
fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<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 fs1] fs1)
then obtain u::"name"
where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t))"
by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t))"
by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1
at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
then obtain c::"coname"
where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t))"
by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
have g: "ImpR [u].[c].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) (pi2\<bullet>b)
=ImpR [(pi1\<bullet>x)].[(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t)) (pi2\<bullet>b)" using f1 f3 e1 e3
apply (auto simp add: ImpR_inject alpha abs_fresh abs_perm perm_dj,
simp add: dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
apply(simp add: pt_fresh_left_ineq[OF pt_name_inst, OF pt_name_inst,
OF at_name_inst, OF cp_name_coname_inst] perm_dj)
done
from i1 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t)) k" by force
hence i1b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) k"
by (simp add: pt_name2[symmetric] pt_coname2[symmetric])
with h2 f2 e2 have "P (ImpR [u].[c].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) (pi2\<bullet>b)) k"
by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
at_fin_set_supp[OF at_coname_inst, OF fs2])
with g show "P (ImpR [(pi1\<bullet>x)].[(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t)) (pi2\<bullet>b)) k" by simp
qed
next
case (goal3 a t1 x t2 y)
assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t1)) k"
and i2: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t2)) k"
show ?case
proof (intro strip, simp add: abs_perm)
fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<bullet>t2))"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1
at_fin_set_supp[OF at_name_inst, OF fs1] fs1)
then obtain u::"name"
where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t2))"
by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t1))"
by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1
at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
then obtain c::"coname"
where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t1))"
by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
have g: "ImpL [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) (pi1\<bullet>y)
=ImpL [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2)) (pi1\<bullet>y)" using f1 f3 e1 e3
by (simp add: ImpL_inject alpha abs_fresh abs_perm perm_dj)
from i2 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(pi2\<bullet>t2)) k" by force
hence i2b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) k"
by (simp add: pt_name2[symmetric])
from i1 have "\<forall>k. P (pi1\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t1)) k" by force
hence i1b: "\<forall>k. P ([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) k"
by (simp add: pt_coname2[symmetric] dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
from h3 f2 e2 i1b i2b
have "P (ImpL [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) (pi1\<bullet>y)) k"
by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
at_fin_set_supp[OF at_coname_inst, OF fs2])
with g show "P (ImpL [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2)) (pi1\<bullet>y)) k" by simp
qed
next
case (goal4 a t1 x t2)
assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t1)) k"
and i2: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t2)) k"
show ?case
proof (intro strip, simp add: abs_perm)
fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<bullet>t2))"
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1
at_fin_set_supp[OF at_name_inst, OF fs1] fs1)
then obtain u::"name"
where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t2))"
by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t1))"
by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1
at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
then obtain c::"coname"
where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t1))"
by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
have g: "Cut [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2)))
=Cut [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2))" using f1 f3 e1 e3
by (simp add: Cut_inject alpha abs_fresh abs_perm perm_dj)
from i2 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(pi2\<bullet>t2)) k" by force
hence i2b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) k"
by (simp add: pt_name2[symmetric])
from i1 have "\<forall>k. P (pi1\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t1)) k" by force
hence i1b: "\<forall>k. P ([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) k"
by (simp add: pt_coname2[symmetric] dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
from h3 f2 e2 i1b i2b
have "P (Cut [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2)))) k"
by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
at_fin_set_supp[OF at_coname_inst, OF fs2])
with g show "P (Cut [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2))) k" by simp
qed
qed
lemma trm_induct'[case_names Ax ImpR ImpL Cut]:
fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool"
and f1 :: "'a \<Rightarrow> name set"
and f2 :: "'a \<Rightarrow> coname set"
assumes fs1: "\<And>x. finite (f1 x)"
and fs2: "\<And>x. finite (f2 x)"
and h1: "\<And>k x a. P (Ax x a) k"
and h2: "\<And>k x a t b. x\<notin>f1 k \<Longrightarrow> a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k"
and h3: "\<And>k a t1 x t2 y. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k"
and h4: "\<And>k a t1 x t2. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (Cut [a].t1 [x].t2) k"
shows "P t k"
proof -
have "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k"
using fs1 fs2 h1 h2 h3 h4 by (rule trm_induct_aux, auto)
hence "P (([]::name prm)\<bullet>(([]::coname prm)\<bullet>t)) k" by blast
thus "P t k" by simp
qed
lemma trm_induct[case_names Ax ImpR ImpL Cut]:
fixes P :: "trm \<Rightarrow> ('a::{fs_name,fs_coname}) \<Rightarrow> bool"
assumes h1: "\<And>k x a. P (Ax x a) k"
and h2: "\<And>k x a t b. x\<sharp>k \<Longrightarrow> a\<sharp>k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k"
and h3: "\<And>k a t1 x t2 y. a\<sharp>k \<Longrightarrow> x\<sharp>k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k"
and h4: "\<And>k a t1 x t2. a\<sharp>k \<Longrightarrow> x\<sharp>k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l)
\<Longrightarrow> P (Cut [a].t1 [x].t2) k"
shows "P t k"
by (rule trm_induct'[of "\<lambda>x. ((supp x)::name set)" "\<lambda>x. ((supp x)::coname set)" "P"],
simp_all add: fs_name1 fs_coname1 fresh_def[symmetric], auto intro: h1 h2 h3 h4)