diff --git a/src/HOL/Nominal/Examples/Class1.thy b/src/HOL/Nominal/Examples/Class1.thy
--- a/src/HOL/Nominal/Examples/Class1.thy
+++ b/src/HOL/Nominal/Examples/Class1.thy
@@ -1,8113 +1,7934 @@
 theory Class1
-imports "HOL-Nominal.Nominal" 
+imports "HOL-Nominal.Nominal"
 begin
 
 section \<open>Term-Calculus from Urban's PhD\<close>
 
 atom_decl name coname
 
 text \<open>types\<close>
 
 no_notation not  ("NOT")
 
 nominal_datatype ty =
     PR "string"
   | NOT  "ty"
   | AND  "ty" "ty"   ("_ AND _" [100,100] 100)
   | OR   "ty" "ty"   ("_ OR _" [100,100] 100)
   | IMP  "ty" "ty"   ("_ IMP _" [100,100] 100)
 
 instantiation ty :: size
 begin
 
 nominal_primrec size_ty
 where
   "size (PR s)     = (1::nat)"
 | "size (NOT T)     = 1 + size T"
 | "size (T1 AND T2) = 1 + size T1 + size T2"
 | "size (T1 OR T2)  = 1 + size T1 + size T2"
 | "size (T1 IMP T2) = 1 + size T1 + size T2"
 by (rule TrueI)+
 
 instance ..
 
 end
 
 lemma ty_cases:
   fixes T::ty
   shows "(\<exists>s. T=PR s) \<or> (\<exists>T'. T=NOT T') \<or> (\<exists>S U. T=S OR U) \<or> (\<exists>S U. T=S AND U) \<or> (\<exists>S U. T=S IMP U)"
 by (induct T rule:ty.induct) (auto)
 
 lemma fresh_ty:
   fixes a::"coname"
   and   x::"name"
   and   T::"ty"
   shows "a\<sharp>T" and "x\<sharp>T"
 by (nominal_induct T rule: ty.strong_induct)
-   (auto simp add: fresh_string)
-
-text \<open>terms\<close>
+   (auto simp: fresh_string)
+
+text \<open>terms\<close>                               
 
 nominal_datatype trm = 
     Ax   "name" "coname"
   | Cut  "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm"            ("Cut <_>._ (_)._" [100,100,100,100] 100)
   | NotR "\<guillemotleft>name\<guillemotright>trm" "coname"                 ("NotR (_)._ _" [100,100,100] 100)
   | NotL "\<guillemotleft>coname\<guillemotright>trm" "name"                 ("NotL <_>._ _" [100,100,100] 100)
   | AndR "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>coname\<guillemotright>trm" "coname" ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)
   | AndL1 "\<guillemotleft>name\<guillemotright>trm" "name"                  ("AndL1 (_)._ _" [100,100,100] 100)
   | AndL2 "\<guillemotleft>name\<guillemotright>trm" "name"                  ("AndL2 (_)._ _" [100,100,100] 100)
   | OrR1 "\<guillemotleft>coname\<guillemotright>trm" "coname"               ("OrR1 <_>._ _" [100,100,100] 100)
   | OrR2 "\<guillemotleft>coname\<guillemotright>trm" "coname"               ("OrR2 <_>._ _" [100,100,100] 100)
   | OrL "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name"        ("OrL (_)._ (_)._ _" [100,100,100,100,100] 100)
   | 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)
 
 text \<open>named terms\<close>
 
 nominal_datatype ntrm = Na "\<guillemotleft>name\<guillemotright>trm" ("((_):_)" [100,100] 100)
 
 text \<open>conamed terms\<close>
 
 nominal_datatype ctrm = Co "\<guillemotleft>coname\<guillemotright>trm" ("(<_>:_)" [100,100] 100)
 
 text \<open>renaming functions\<close>
 
 nominal_primrec (freshness_context: "(d::coname,e::coname)") 
   crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) 
 where
   "(Ax x a)[d\<turnstile>c>e] = (if a=d then Ax x e else Ax x a)" 
 | "\<lbrakk>a\<sharp>(d,e,N);x\<sharp>M\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N)[d\<turnstile>c>e] = Cut <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e])" 
 | "(NotR (x).M a)[d\<turnstile>c>e] = (if a=d then NotR (x).(M[d\<turnstile>c>e]) e else NotR (x).(M[d\<turnstile>c>e]) a)" 
 | "a\<sharp>(d,e) \<Longrightarrow> (NotL <a>.M x)[d\<turnstile>c>e] = (NotL <a>.(M[d\<turnstile>c>e]) x)" 
 | "\<lbrakk>a\<sharp>(d,e,N,c);b\<sharp>(d,e,M,c);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[d\<turnstile>c>e] = 
           (if c=d then AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d \<turnstile>c>e]) e else AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d\<turnstile>c>e]) c)" 
 | "x\<sharp>y \<Longrightarrow> (AndL1 (x).M y)[d\<turnstile>c>e] = AndL1 (x).(M[d\<turnstile>c>e]) y"
 | "x\<sharp>y \<Longrightarrow> (AndL2 (x).M y)[d\<turnstile>c>e] = AndL2 (x).(M[d\<turnstile>c>e]) y"
 | "a\<sharp>(d,e,b) \<Longrightarrow> (OrR1 <a>.M b)[d\<turnstile>c>e] = (if b=d then OrR1 <a>.(M[d\<turnstile>c>e]) e else OrR1 <a>.(M[d\<turnstile>c>e]) b)"
 | "a\<sharp>(d,e,b) \<Longrightarrow> (OrR2 <a>.M b)[d\<turnstile>c>e] = (if b=d then OrR2 <a>.(M[d\<turnstile>c>e]) e else OrR2 <a>.(M[d\<turnstile>c>e]) b)"
 | "\<lbrakk>x\<sharp>(N,z);y\<sharp>(M,z);y\<noteq>x\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N z)[d\<turnstile>c>e] = OrL (x).(M[d\<turnstile>c>e]) (y).(N[d\<turnstile>c>e]) z"
 | "a\<sharp>(d,e,b) \<Longrightarrow> (ImpR (x).<a>.M b)[d\<turnstile>c>e] = 
        (if b=d then ImpR (x).<a>.(M[d\<turnstile>c>e]) e else ImpR (x).<a>.(M[d\<turnstile>c>e]) b)"
 | "\<lbrakk>a\<sharp>(d,e,N);x\<sharp>(M,y)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y)[d\<turnstile>c>e] = ImpL <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e]) y"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh abs_supp fin_supp)+
-apply(fresh_guess)+
-done
+  by(finite_guess | simp add: abs_fresh abs_supp fin_supp | fresh_guess)+
 
 nominal_primrec (freshness_context: "(u::name,v::name)") 
   nrename :: "trm \<Rightarrow> name \<Rightarrow> name \<Rightarrow> trm"      ("_[_\<turnstile>n>_]" [100,100,100] 100) 
 where
   "(Ax x a)[u\<turnstile>n>v] = (if x=u then Ax v a else Ax x a)" 
 | "\<lbrakk>a\<sharp>N;x\<sharp>(u,v,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" 
 | "x\<sharp>(u,v) \<Longrightarrow> (NotR (x).M a)[u\<turnstile>n>v] = NotR (x).(M[u\<turnstile>n>v]) a" 
 | "(NotL <a>.M x)[u\<turnstile>n>v] = (if x=u then NotL <a>.(M[u\<turnstile>n>v]) v else NotL <a>.(M[u\<turnstile>n>v]) x)" 
 | "\<lbrakk>a\<sharp>(N,c);b\<sharp>(M,c);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[u\<turnstile>n>v] = AndR <a>.(M[u\<turnstile>n>v]) <b>.(N[u\<turnstile>n>v]) c" 
 | "x\<sharp>(u,v,y) \<Longrightarrow> (AndL1 (x).M y)[u\<turnstile>n>v] = (if y=u then AndL1 (x).(M[u\<turnstile>n>v]) v else AndL1 (x).(M[u\<turnstile>n>v]) y)"
 | "x\<sharp>(u,v,y) \<Longrightarrow> (AndL2 (x).M y)[u\<turnstile>n>v] = (if y=u then AndL2 (x).(M[u\<turnstile>n>v]) v else AndL2 (x).(M[u\<turnstile>n>v]) y)"
 | "a\<sharp>b \<Longrightarrow> (OrR1 <a>.M b)[u\<turnstile>n>v] = OrR1 <a>.(M[u\<turnstile>n>v]) b"
 | "a\<sharp>b \<Longrightarrow> (OrR2 <a>.M b)[u\<turnstile>n>v] = OrR2 <a>.(M[u\<turnstile>n>v]) b"
 | "\<lbrakk>x\<sharp>(u,v,N,z);y\<sharp>(u,v,M,z);y\<noteq>x\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N z)[u\<turnstile>n>v] = 
         (if z=u then OrL (x).(M[u\<turnstile>n>v]) (y).(N[u\<turnstile>n>v]) v else OrL (x).(M[u\<turnstile>n>v]) (y).(N[u\<turnstile>n>v]) z)"
 | "\<lbrakk>a\<sharp>b; x\<sharp>(u,v)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b)[u\<turnstile>n>v] = ImpR (x).<a>.(M[u\<turnstile>n>v]) b"
 | "\<lbrakk>a\<sharp>N;x\<sharp>(u,v,M,y)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y)[u\<turnstile>n>v] = 
         (if y=u then ImpL <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v]) v else ImpL <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v]) y)"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1)+
-apply(fresh_guess)+
-done
+  by(finite_guess | simp add: abs_fresh abs_supp fin_supp | fresh_guess)+
 
 lemmas eq_bij = pt_bij[OF pt_name_inst, OF at_name_inst] pt_bij[OF pt_coname_inst, OF at_coname_inst]
 
 lemma crename_name_eqvt[eqvt]:
   fixes pi::"name prm"
   shows "pi\<bullet>(M[d\<turnstile>c>e]) = (pi\<bullet>M)[(pi\<bullet>d)\<turnstile>c>(pi\<bullet>e)]"
-apply(nominal_induct M avoiding: d e rule: trm.strong_induct)
-apply(auto simp add: fresh_bij eq_bij)
-done
+  by (nominal_induct M avoiding: d e rule: trm.strong_induct) (auto simp: fresh_bij eq_bij)
 
 lemma crename_coname_eqvt[eqvt]:
   fixes pi::"coname prm"
   shows "pi\<bullet>(M[d\<turnstile>c>e]) = (pi\<bullet>M)[(pi\<bullet>d)\<turnstile>c>(pi\<bullet>e)]"
-apply(nominal_induct M avoiding: d e rule: trm.strong_induct)
-apply(auto simp add: fresh_bij eq_bij)
-done
+  by (nominal_induct M avoiding: d e rule: trm.strong_induct)(auto simp: fresh_bij eq_bij)
 
 lemma nrename_name_eqvt[eqvt]:
   fixes pi::"name prm"
   shows "pi\<bullet>(M[x\<turnstile>n>y]) = (pi\<bullet>M)[(pi\<bullet>x)\<turnstile>n>(pi\<bullet>y)]"
-apply(nominal_induct M avoiding: x y rule: trm.strong_induct)
-apply(auto simp add: fresh_bij eq_bij)
-done
+  by(nominal_induct M avoiding: x y rule: trm.strong_induct) (auto simp: fresh_bij eq_bij)
 
 lemma nrename_coname_eqvt[eqvt]:
   fixes pi::"coname prm"
   shows "pi\<bullet>(M[x\<turnstile>n>y]) = (pi\<bullet>M)[(pi\<bullet>x)\<turnstile>n>(pi\<bullet>y)]"
-apply(nominal_induct M avoiding: x y rule: trm.strong_induct)
-apply(auto simp add: fresh_bij eq_bij)
-done
+  by(nominal_induct M avoiding: x y rule: trm.strong_induct)(auto simp: fresh_bij eq_bij)
 
 lemmas rename_eqvts = crename_name_eqvt crename_coname_eqvt
                       nrename_name_eqvt nrename_coname_eqvt
 lemma nrename_fresh:
   assumes a: "x\<sharp>M"
   shows "M[x\<turnstile>n>y] = M"
 using a
 by (nominal_induct M avoiding: x y rule: trm.strong_induct)
-   (auto simp add: trm.inject fresh_atm abs_fresh fin_supp abs_supp)
+   (auto simp: trm.inject fresh_atm abs_fresh fin_supp abs_supp)
 
 lemma crename_fresh:
   assumes a: "a\<sharp>M"
   shows "M[a\<turnstile>c>b] = M"
 using a
 by (nominal_induct M avoiding: a b rule: trm.strong_induct)
-   (auto simp add: trm.inject fresh_atm abs_fresh)
+   (auto simp: trm.inject fresh_atm abs_fresh)
 
 lemma nrename_nfresh:
   fixes x::"name"
   shows "x\<sharp>y\<Longrightarrow>x\<sharp>M[x\<turnstile>n>y]"
 by (nominal_induct M avoiding: x y rule: trm.strong_induct)
-   (auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 
  lemma crename_nfresh:
   fixes x::"name"
   shows "x\<sharp>M\<Longrightarrow>x\<sharp>M[a\<turnstile>c>b]"
 by (nominal_induct M avoiding: a b rule: trm.strong_induct)
-   (auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 
 lemma crename_cfresh:
   fixes a::"coname"
   shows "a\<sharp>b\<Longrightarrow>a\<sharp>M[a\<turnstile>c>b]"
 by (nominal_induct M avoiding: a b rule: trm.strong_induct)
-   (auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 
 lemma nrename_cfresh:
   fixes c::"coname"
   shows "c\<sharp>M\<Longrightarrow>c\<sharp>M[x\<turnstile>n>y]"
 by (nominal_induct M avoiding: x y rule: trm.strong_induct)
-   (auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 
 lemma nrename_nfresh':
   fixes x::"name"
   shows "x\<sharp>(M,z,y)\<Longrightarrow>x\<sharp>M[z\<turnstile>n>y]"
 by (nominal_induct M avoiding: x z y rule: trm.strong_induct)
-   (auto simp add: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)
 
 lemma crename_cfresh':
   fixes a::"coname"
   shows "a\<sharp>(M,b,c)\<Longrightarrow>a\<sharp>M[b\<turnstile>c>c]"
 by (nominal_induct M avoiding: a b c rule: trm.strong_induct)
-   (auto simp add: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)
+   (auto simp: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)
 
 lemma nrename_rename:
   assumes a: "x'\<sharp>M"
   shows "([(x',x)]\<bullet>M)[x'\<turnstile>n>y]= M[x\<turnstile>n>y]"
 using a
 apply(nominal_induct M avoiding: x x' y rule: trm.strong_induct)
-apply(auto simp add: abs_fresh fresh_bij fresh_atm fresh_prod fresh_right calc_atm abs_supp fin_supp)
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)
+apply(auto simp: abs_fresh fresh_bij fresh_atm fresh_prod fresh_right calc_atm abs_supp fin_supp)
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)
 done
 
 lemma crename_rename:
   assumes a: "a'\<sharp>M"
   shows "([(a',a)]\<bullet>M)[a'\<turnstile>c>b]= M[a\<turnstile>c>b]"
 using a
 apply(nominal_induct M avoiding: a a' b rule: trm.strong_induct)
-apply(auto simp add: abs_fresh fresh_bij fresh_atm fresh_prod fresh_right calc_atm abs_supp fin_supp)
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)
+apply(auto simp: abs_fresh fresh_bij fresh_atm fresh_prod fresh_right calc_atm abs_supp fin_supp)
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)
 done
 
 lemmas rename_fresh = nrename_fresh crename_fresh 
                       nrename_nfresh crename_nfresh crename_cfresh nrename_cfresh
                       nrename_nfresh' crename_cfresh'
                       nrename_rename crename_rename
 
 lemma better_nrename_Cut:
   assumes a: "x\<sharp>(u,v)"
   shows "(Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])"
 proof -
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,a,x,u,v)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,x,u,v)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   have "(Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))[u\<turnstile>n>v] = 
                         Cut <a'>.(([(a',a)]\<bullet>M)[u\<turnstile>n>v]) (x').(([(x',x)]\<bullet>N)[u\<turnstile>n>v])"
     using fs1 fs2
-    apply -
-    apply(rule nrename.simps)
-    apply(simp add: fresh_left calc_atm)
-    apply(simp add: fresh_left calc_atm)
-    done
+    by (intro nrename.simps; simp add: fresh_left calc_atm)
   also have "\<dots> = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" using fs1 fs2 a
-    apply -
-    apply(simp add: trm.inject alpha fresh_atm fresh_prod rename_eqvts)
-    apply(simp add: calc_atm)
-    apply(simp add: rename_fresh fresh_atm)
-    done
+    by(simp add: trm.inject alpha fresh_prod rename_eqvts calc_atm rename_fresh fresh_atm)
   finally show "(Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" using eq1
     by simp
 qed
 
 lemma better_crename_Cut:
   assumes a: "a\<sharp>(b,c)"
   shows "(Cut <a>.M (x).N)[b\<turnstile>c>c] = Cut <a>.(M[b\<turnstile>c>c]) (x).(N[b\<turnstile>c>c])"
 proof -
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,a,x,b,c)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,x,b,c)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   have "(Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))[b\<turnstile>c>c] = 
                         Cut <a'>.(([(a',a)]\<bullet>M)[b\<turnstile>c>c]) (x').(([(x',x)]\<bullet>N)[b\<turnstile>c>c])"
     using fs1 fs2
-    apply -
-    apply(rule crename.simps)
-    apply(simp add: fresh_left calc_atm)
-    apply(simp add: fresh_left calc_atm)
-    done
+    by (intro crename.simps; simp add: fresh_left calc_atm)
   also have "\<dots> = Cut <a>.(M[b\<turnstile>c>c]) (x).(N[b\<turnstile>c>c])" using fs1 fs2 a
-    apply -
-    apply(simp add: trm.inject alpha fresh_atm fresh_prod rename_eqvts)
-    apply(simp add: calc_atm)
-    apply(simp add: rename_fresh fresh_atm)
-    done
+    by(simp add: trm.inject alpha calc_atm rename_fresh fresh_atm fresh_prod rename_eqvts)
   finally show "(Cut <a>.M (x).N)[b\<turnstile>c>c] = Cut <a>.(M[b\<turnstile>c>c]) (x).(N[b\<turnstile>c>c])" using eq1
     by simp
 qed
 
 lemma crename_id:
   shows "M[a\<turnstile>c>a] = M"
 by (nominal_induct M avoiding: a rule: trm.strong_induct) (auto)
 
 lemma nrename_id:
   shows "M[x\<turnstile>n>x] = M"
 by (nominal_induct M avoiding: x rule: trm.strong_induct) (auto)
 
 lemma nrename_swap:
   shows "x\<sharp>M \<Longrightarrow> [(x,y)]\<bullet>M = M[y\<turnstile>n>x]"
 by (nominal_induct M avoiding: x y rule: trm.strong_induct) 
    (simp_all add: calc_atm fresh_atm trm.inject alpha abs_fresh abs_supp fin_supp)
 
 lemma crename_swap:
   shows "a\<sharp>M \<Longrightarrow> [(a,b)]\<bullet>M = M[b\<turnstile>c>a]"
 by (nominal_induct M avoiding: a b rule: trm.strong_induct) 
    (simp_all add: calc_atm fresh_atm trm.inject alpha abs_fresh abs_supp fin_supp)
 
 lemma crename_ax:
   assumes a: "M[a\<turnstile>c>b] = Ax x c" "c\<noteq>a" "c\<noteq>b"
   shows "M = Ax x c"
 using a
 apply(nominal_induct M avoiding: a b x c rule: trm.strong_induct)
 apply(simp_all add: trm.inject split: if_splits)
 done
 
 lemma nrename_ax:
   assumes a: "M[x\<turnstile>n>y] = Ax z a" "z\<noteq>x" "z\<noteq>y"
   shows "M = Ax z a"
 using a
 apply(nominal_induct M avoiding: x y z a rule: trm.strong_induct)
 apply(simp_all add: trm.inject split: if_splits)
 done
 
 text \<open>substitution functions\<close>
 
 lemma fresh_perm_coname:
   fixes c::"coname"
   and   pi::"coname prm"
   and   M::"trm"
   assumes a: "c\<sharp>pi" "c\<sharp>M"
   shows "c\<sharp>(pi\<bullet>M)"
-using a
-apply -
-apply(simp add: fresh_left)
-apply(simp add: at_prm_fresh[OF at_coname_inst] fresh_list_rev)
-done
+  by (simp add: assms fresh_perm_app)
 
 lemma fresh_perm_name:
   fixes x::"name"
   and   pi::"name prm"
   and   M::"trm"
   assumes a: "x\<sharp>pi" "x\<sharp>M"
   shows "x\<sharp>(pi\<bullet>M)"
-using a
-apply -
-apply(simp add: fresh_left)
-apply(simp add: at_prm_fresh[OF at_name_inst] fresh_list_rev)
-done
+  by (simp add: assms fresh_perm_app)
 
 lemma fresh_fun_simp_NotL:
   assumes a: "x'\<sharp>P" "x'\<sharp>M"
   shows "fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x') = Cut <c>.P (x').NotL <a>.M x'"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_name_inst)
-apply(rule at_name_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,a,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app)
+  show "pt (TYPE(trm)::trm itself) (TYPE(name)::name itself)"
+    by(rule pt_name_inst)
+  show "at (TYPE(name)::name itself)"
+    by(rule at_name_inst)
+  show "finite (supp (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')::name set)"
+    using a by(finite_guess)
+  obtain n::name where n: "n\<sharp>(c,P,a,M)"
+    by (metis assms fresh_atm(3) fresh_prod)
+  with assms
+  show "\<exists>b. b \<sharp> (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x', Cut <c>.P (b).NotL <a>.M b)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+  show "x' \<sharp> (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')"
+    using assms by(fresh_guess)
+qed
 
 lemma fresh_fun_NotL[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')=
              fresh_fun (pi1\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))"
   and   "pi2\<bullet>fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')=
              fresh_fun (pi2\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))"
-apply -
-apply(perm_simp)
-apply(generate_fresh "name")
-apply(auto simp add: fresh_prod)
-apply(simp add: fresh_fun_simp_NotL)
-apply(rule sym)
-apply(rule trans)
-apply(rule fresh_fun_simp_NotL)
-apply(rule fresh_perm_name)
-apply(assumption)
-apply(assumption)
-apply(rule fresh_perm_name)
-apply(assumption)
-apply(assumption)
-apply(simp add: at_prm_fresh[OF at_name_inst] swap_simps)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_NotL calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(generate_fresh "name")
+   apply(auto simp: fresh_prod fresh_fun_simp_NotL)
+   apply(rule sym)
+   apply(rule trans)
+    apply(rule fresh_fun_simp_NotL)
+     apply(blast intro: fresh_perm_name)
+    apply(metis fresh_perm_name)
+   apply(simp add: at_prm_fresh[OF at_name_inst] swap_simps)
+
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)")
+   apply (metis fresh_fun_simp_NotL fresh_prodD swap_simps(8) trm.perm(14) trm.perm(16))
+  by (meson exists_fresh(1) fs_name1)
 
 lemma fresh_fun_simp_AndL1:
   assumes a: "z'\<sharp>P" "z'\<sharp>M" "z'\<sharp>x"
-  shows "fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z') = Cut <c>.P (z').AndL1 (x).M z'"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_name_inst)
-apply(rule at_name_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,x,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+  shows "fresh_fun (\<lambda>z'. Cut <c>.P(z').AndL1 (x).M z') = Cut <c>.P (z').AndL1 (x).M z'"
+proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
+  obtain n::name where "n\<sharp>(c,P,x,M)"
+    by (meson exists_fresh(1) fs_name1)
+  then show "\<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P(z').AndL1 x. M z', Cut <c>.P(a).AndL1 x. M a)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+next
+  show "z' \<sharp> (\<lambda>z'. Cut <c>.P(z').AndL1 x. M z')"
+    using a by(fresh_guess)
+qed finite_guess
 
 lemma fresh_fun_AndL1[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')=
              fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))"
   and   "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')=
              fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))"
-apply -
 apply(perm_simp)
 apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>x,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndL1 at_prm_fresh[OF at_name_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
+apply(force simp add: fresh_prod fresh_fun_simp_AndL1 at_prm_fresh[OF at_name_inst] swap_simps)
+  apply (meson exists_fresh(1) fs_name1)
 apply(perm_simp)
 apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>x,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndL1 calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+apply(force simp add: fresh_prod fresh_fun_simp_AndL1 calc_atm)
+  by (meson exists_fresh'(1) fs_name1)
 
 lemma fresh_fun_simp_AndL2:
   assumes a: "z'\<sharp>P" "z'\<sharp>M" "z'\<sharp>x"
   shows "fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z') = Cut <c>.P (z').AndL2 (x).M z'"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_name_inst)
-apply(rule at_name_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,x,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
+  obtain n::name where "n\<sharp>(c,P,x,M)"
+    by (meson exists_fresh(1) fs_name1)
+  then show "\<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P(z').AndL2 x. M z', Cut <c>.P(a).AndL2 x. M a)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+next
+  show "z' \<sharp> (\<lambda>z'. Cut <c>.P(z').AndL2 x. M z')"
+    using a by(fresh_guess)           
+qed finite_guess
 
 lemma fresh_fun_AndL2[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')=
              fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))"
   and   "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')=
              fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>x,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndL2 at_prm_fresh[OF at_name_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>x,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndL2 calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>x,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_AndL2 at_prm_fresh[OF at_name_inst] swap_simps)
+   apply (meson exists_fresh(1) fs_name1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>x,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_AndL2 calc_atm)
+  by (meson exists_fresh(1) fs_name1)
 
 lemma fresh_fun_simp_OrL:
   assumes a: "z'\<sharp>P" "z'\<sharp>M" "z'\<sharp>N" "z'\<sharp>u" "z'\<sharp>x"
-  shows "fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (x).M (u).N z') = Cut <c>.P (z').OrL (x).M (u).N z'"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_name_inst)
-apply(rule at_name_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,x,M,u,N)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+  shows "fresh_fun (\<lambda>z'. Cut <c>.P(z').OrL(x).M(u).N z') = Cut <c>.P (z').OrL (x).M (u).N z'"
+proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
+  obtain n::name where "n\<sharp>(c,P,x,M,u,N)"
+    by (meson exists_fresh(1) fs_name1)
+  then show "\<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P(z').OrL(x).M(u).N(z'), Cut <c>.P(a).OrL(x).M(u).N(a))"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+next
+  show "z' \<sharp> (\<lambda>z'. Cut <c>.P(z').OrL(x).M(u).N(z'))"
+    using a by(fresh_guess) 
+qed finite_guess
 
 lemma fresh_fun_OrL[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
-  shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (x).M (u).N z')=
-             fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').OrL (x).M (u).N z'))"
-  and   "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (x).M (u).N z')=
-             fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').OrL (x).M (u).N z'))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,u,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>x,pi1\<bullet>u,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrL at_prm_fresh[OF at_name_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,u,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>x,pi2\<bullet>u,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrL calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+  shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P(z').OrL (x).M(u).N z')=
+             fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').OrL(x).M (u).N z'))"
+  and   "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P(z').OrL (x).M(u).N z')=
+             fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P(z').OrL(x).M(u).N z'))"
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,u,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>x,pi1\<bullet>u,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_OrL at_prm_fresh[OF at_name_inst] swap_simps)
+   apply (meson exists_fresh(1) fs_name1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,u,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>x,pi2\<bullet>u,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_OrL calc_atm)
+  by (meson exists_fresh(1) fs_name1)
 
 lemma fresh_fun_simp_ImpL:
   assumes a: "z'\<sharp>P" "z'\<sharp>M" "z'\<sharp>N" "z'\<sharp>x"
   shows "fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z') = Cut <c>.P (z').ImpL <a>.M (x).N z'"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_name_inst)
-apply(rule at_name_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,x,M,N)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
+  obtain n::name where "n\<sharp>(c,P,x,M,N)"
+    by (meson exists_fresh(1) fs_name1)
+  then show "\<exists>aa. aa \<sharp> (\<lambda>z'. Cut <c>.P(z').ImpL <a>.M(x).N z', Cut <c>.P(aa).ImpL <a>.M(x).N aa)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+next
+  show "z' \<sharp> (\<lambda>z'. Cut <c>.P(z').ImpL <a>.M(x).N z')"
+    using a by(fresh_guess) 
+qed finite_guess
 
 lemma fresh_fun_ImpL[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')=
              fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))"
   and   "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')=
              fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>x,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_ImpL at_prm_fresh[OF at_name_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>x,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_ImpL calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>x,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_ImpL at_prm_fresh[OF at_name_inst] swap_simps)
+   apply (meson exists_fresh(1) fs_name1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>x,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_ImpL calc_atm)
+  by (meson exists_fresh(1) fs_name1)
 
 lemma fresh_fun_simp_NotR:
   assumes a: "a'\<sharp>P" "a'\<sharp>M"
   shows "fresh_fun (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P) = Cut <a'>.(NotR (y).M a') (x).P"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_coname_inst)
-apply(rule at_coname_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(x,P,y,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
+  obtain n::coname where "n\<sharp>(x,P,y,M)"
+    by (metis assms(1) assms(2) fresh_atm(4) fresh_prod)
+  then show "\<exists>a. a \<sharp> (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P, Cut <a>.NotR(y).M(a) (x).P)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+qed (use a in \<open>fresh_guess|finite_guess\<close>)+
 
 lemma fresh_fun_NotR[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P)=
              fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))"
   and   "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P)=
              fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi1\<bullet>P,pi1\<bullet>M,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_NotR calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_NotR at_prm_fresh[OF at_coname_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi1\<bullet>P,pi1\<bullet>M,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_NotR calc_atm)
+   apply (meson exists_fresh(2) fs_coname1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)")
+   apply(force simp: fresh_prod fresh_fun_simp_NotR at_prm_fresh[OF at_coname_inst] swap_simps)
+  by (meson exists_fresh(2) fs_coname1)
 
 lemma fresh_fun_simp_AndR:
   assumes a: "a'\<sharp>P" "a'\<sharp>M" "a'\<sharp>N" "a'\<sharp>b" "a'\<sharp>c"
   shows "fresh_fun (\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P) = Cut <a'>.(AndR <b>.M <c>.N a') (x).P"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_coname_inst)
-apply(rule at_coname_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(x,P,b,M,c,N)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
+  obtain n::coname where "n\<sharp>(x,P,b,M,c,N)"
+    by (meson exists_fresh(2) fs_coname1)
+  then show "\<exists>a. a \<sharp> (\<lambda>a'. Cut <a'>.AndR <b>.M <c>.N(a') (x).P, Cut <a>.AndR <b>.M <c>.N(a) (x).P)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+qed (use a in \<open>fresh_guess|finite_guess\<close>)+
 
 lemma fresh_fun_AndR[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P)=
              fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P))"
   and   "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P)=
              fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>b,pi1\<bullet>c,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndR calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>b,pi2\<bullet>c,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_AndR at_prm_fresh[OF at_coname_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>b,pi1\<bullet>c,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_AndR calc_atm)
+   apply (meson exists_fresh(2) fs_coname1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>b,pi2\<bullet>c,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_AndR at_prm_fresh[OF at_coname_inst] swap_simps)
+  by (meson exists_fresh(2) fs_coname1)
 
 lemma fresh_fun_simp_OrR1:
   assumes a: "a'\<sharp>P" "a'\<sharp>M" "a'\<sharp>b" 
   shows "fresh_fun (\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P) = Cut <a'>.(OrR1 <b>.M a') (x).P"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_coname_inst)
-apply(rule at_coname_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(x,P,b,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
+  obtain n::coname where "n\<sharp>(x,P,b,M)"
+    by (meson exists_fresh(2) fs_coname1)
+  then show "\<exists>a. a \<sharp> (\<lambda>a'. Cut <a'>.OrR1 <b>.M(a') (x).P, Cut <a>.OrR1 <b>.M(a) (x).P)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+qed (use a in \<open>fresh_guess|finite_guess\<close>)+
 
 lemma fresh_fun_OrR1[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P)=
              fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 <b>.M  a') (x).P))"
   and   "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P)=
              fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrR1 calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrR1 at_prm_fresh[OF at_coname_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_OrR1 calc_atm)
+   apply (meson exists_fresh(2) fs_coname1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_OrR1 at_prm_fresh[OF at_coname_inst] swap_simps)
+  by (meson exists_fresh(2) fs_coname1)
 
 lemma fresh_fun_simp_OrR2:
   assumes a: "a'\<sharp>P" "a'\<sharp>M" "a'\<sharp>b" 
   shows "fresh_fun (\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P) = Cut <a'>.(OrR2 <b>.M a') (x).P"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_coname_inst)
-apply(rule at_coname_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(x,P,b,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
+  obtain n::coname where "n\<sharp>(x,P,b,M)"
+    by (meson exists_fresh(2) fs_coname1)
+  then show "\<exists>a. a \<sharp> (\<lambda>a'. Cut <a'>.OrR2 <b>.M(a') (x).P, Cut <a>.OrR2 <b>.M(a) (x).P)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+qed (use a in \<open>fresh_guess|finite_guess\<close>)+
 
 lemma fresh_fun_OrR2[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P)=
              fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 <b>.M  a') (x).P))"
   and   "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P)=
              fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrR2 calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_OrR2 at_prm_fresh[OF at_coname_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_OrR2 calc_atm)
+   apply (meson exists_fresh(2) fs_coname1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_OrR2 at_prm_fresh[OF at_coname_inst] swap_simps)
+  by (meson exists_fresh(2) fs_coname1)
 
 lemma fresh_fun_simp_ImpR:
   assumes a: "a'\<sharp>P" "a'\<sharp>M" "a'\<sharp>b" 
   shows "fresh_fun (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P) = Cut <a'>.(ImpR (y).<b>.M a') (x).P"
-using a
-apply -
-apply(rule fresh_fun_app)
-apply(rule pt_coname_inst)
-apply(rule at_coname_inst)
-apply(finite_guess)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(x,P,y,b,M)")
-apply(erule exE)
-apply(rule_tac x="n" in exI)
-apply(simp add: fresh_prod abs_fresh)
-apply(fresh_guess)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(fresh_guess)
-done
+proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
+  obtain n::coname where "n\<sharp>(x,P,y,b,M)"
+    by (meson exists_fresh(2) fs_coname1)
+  then show "\<exists>a. a \<sharp> (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P, Cut <a>.(ImpR (y).<b>.M a) (x).P)"
+    apply(intro exI [where x="n"])
+    apply(simp add: fresh_prod abs_fresh)
+    apply(fresh_guess)
+    done
+qed (use a in \<open>fresh_guess|finite_guess\<close>)+
 
 lemma fresh_fun_ImpR[eqvt_force]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P)=
              fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M  a') (x).P))"
   and   "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P)=
              fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P))"
-apply -
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_ImpR calc_atm)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-apply(perm_simp)
-apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
-apply(simp add: fresh_prod)
-apply(auto)
-apply(simp add: fresh_fun_simp_ImpR at_prm_fresh[OF at_coname_inst] swap_simps)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+   apply(perm_simp)
+   apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)")
+    apply(force simp add: fresh_prod fresh_fun_simp_ImpR calc_atm)
+   apply (meson exists_fresh(2) fs_coname1)
+  apply(perm_simp)
+  apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)")
+   apply(force simp add: fresh_prod fresh_fun_simp_ImpR at_prm_fresh[OF at_coname_inst] swap_simps)
+  by (meson exists_fresh(2) fs_coname1)
 
 nominal_primrec (freshness_context: "(y::name,c::coname,P::trm)")
   substn :: "trm \<Rightarrow> name   \<Rightarrow> coname \<Rightarrow> trm \<Rightarrow> trm" ("_{_:=<_>._}" [100,100,100,100] 100) 
 where
   "(Ax x a){y:=<c>.P} = (if x=y then Cut <c>.P (y).Ax y a else Ax x a)" 
 | "\<lbrakk>a\<sharp>(c,P,N);x\<sharp>(y,P,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N){y:=<c>.P} = 
   (if M=Ax y a then Cut <c>.P (x).(N{y:=<c>.P}) else Cut <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}))" 
 | "x\<sharp>(y,P) \<Longrightarrow> (NotR (x).M a){y:=<c>.P} = NotR (x).(M{y:=<c>.P}) a" 
 | "a\<sharp>(c,P) \<Longrightarrow> (NotL <a>.M x){y:=<c>.P} = 
   (if x=y then fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.(M{y:=<c>.P}) x') else NotL <a>.(M{y:=<c>.P}) x)"
 | "\<lbrakk>a\<sharp>(c,P,N,d);b\<sharp>(c,P,M,d);b\<noteq>a\<rbrakk> \<Longrightarrow> 
   (AndR <a>.M <b>.N d){y:=<c>.P} = AndR <a>.(M{y:=<c>.P}) <b>.(N{y:=<c>.P}) d" 
 | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL1 (x).M z){y:=<c>.P} = 
   (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).(M{y:=<c>.P}) z') 
    else AndL1 (x).(M{y:=<c>.P}) z)"
 | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL2 (x).M z){y:=<c>.P} = 
   (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).(M{y:=<c>.P}) z') 
    else AndL2 (x).(M{y:=<c>.P}) z)"
 | "a\<sharp>(c,P,b) \<Longrightarrow> (OrR1 <a>.M b){y:=<c>.P} = OrR1 <a>.(M{y:=<c>.P}) b"
 | "a\<sharp>(c,P,b) \<Longrightarrow> (OrR2 <a>.M b){y:=<c>.P} = OrR2 <a>.(M{y:=<c>.P}) b"
 | "\<lbrakk>x\<sharp>(y,N,P,z);u\<sharp>(y,M,P,z);x\<noteq>u\<rbrakk> \<Longrightarrow> (OrL (x).M (u).N z){y:=<c>.P} = 
   (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (x).(M{y:=<c>.P}) (u).(N{y:=<c>.P}) z') 
    else OrL (x).(M{y:=<c>.P}) (u).(N{y:=<c>.P}) z)"
 | "\<lbrakk>a\<sharp>(b,c,P); x\<sharp>(y,P)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b){y:=<c>.P} = ImpR (x).<a>.(M{y:=<c>.P}) b"
 | "\<lbrakk>a\<sharp>(N,c,P);x\<sharp>(y,P,M,z)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N z){y:=<c>.P} = 
   (if y=z then fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}) z') 
    else ImpL <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}) z)"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
+apply(finite_guess | simp add: abs_fresh abs_supp fin_supp | fresh_guess | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_NotL abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_AndL1 abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_AndL2 abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_OrL abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_OrL abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_ImpL abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
+
+apply(simp add: abs_fresh abs_supp fin_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_ImpL abs_fresh fresh_atm)
+apply (meson exists_fresh(1) fs_name1)
 apply(fresh_guess)+
 done
 
 nominal_primrec (freshness_context: "(d::name,z::coname,P::trm)")
   substc :: "trm \<Rightarrow> coname \<Rightarrow> name   \<Rightarrow> trm \<Rightarrow> trm" ("_{_:=(_)._}" [100,100,100,100] 100)
 where
   "(Ax x a){d:=(z).P} = (if d=a then Cut <a>.(Ax x a) (z).P else Ax x a)" 
 | "\<lbrakk>a\<sharp>(d,P,N);x\<sharp>(z,P,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N){d:=(z).P} = 
   (if N=Ax x d then Cut <a>.(M{d:=(z).P}) (z).P else Cut <a>.(M{d:=(z).P}) (x).(N{d:=(z).P}))" 
 | "x\<sharp>(z,P) \<Longrightarrow> (NotR (x).M a){d:=(z).P} = 
   (if d=a then fresh_fun (\<lambda>a'. Cut <a'>.NotR (x).(M{d:=(z).P}) a' (z).P) else NotR (x).(M{d:=(z).P}) a)" 
 | "a\<sharp>(d,P) \<Longrightarrow> (NotL <a>.M x){d:=(z).P} = NotL <a>.(M{d:=(z).P}) x" 
 | "\<lbrakk>a\<sharp>(P,c,N,d);b\<sharp>(P,c,M,d);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c){d:=(z).P} = 
   (if d=c then fresh_fun (\<lambda>a'. Cut <a'>.(AndR <a>.(M{d:=(z).P}) <b>.(N{d:=(z).P}) a') (z).P)
    else AndR <a>.(M{d:=(z).P}) <b>.(N{d:=(z).P}) c)" 
 | "x\<sharp>(y,z,P) \<Longrightarrow> (AndL1 (x).M y){d:=(z).P} = AndL1 (x).(M{d:=(z).P}) y"
 | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL2 (x).M y){d:=(z).P} = AndL2 (x).(M{d:=(z).P}) y"
 | "a\<sharp>(d,P,b) \<Longrightarrow> (OrR1 <a>.M b){d:=(z).P} = 
   (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <a>.(M{d:=(z).P}) a' (z).P) else OrR1 <a>.(M{d:=(z).P}) b)"
 | "a\<sharp>(d,P,b) \<Longrightarrow> (OrR2 <a>.M b){d:=(z).P} = 
   (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <a>.(M{d:=(z).P}) a' (z).P) else OrR2 <a>.(M{d:=(z).P}) b)"
 | "\<lbrakk>x\<sharp>(N,z,P,u);y\<sharp>(M,z,P,u);x\<noteq>y\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N u){d:=(z).P} = 
   OrL (x).(M{d:=(z).P}) (y).(N{d:=(z).P}) u" 
 | "\<lbrakk>a\<sharp>(b,d,P); x\<sharp>(z,P)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b){d:=(z).P} = 
   (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.ImpR (x).<a>.(M{d:=(z).P}) a' (z).P) 
    else ImpR (x).<a>.(M{d:=(z).P}) b)"
 | "\<lbrakk>a\<sharp>(N,d,P);x\<sharp>(y,z,P,M)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y){d:=(z).P} = 
   ImpL <a>.(M{d:=(z).P}) (x).(N{d:=(z).P}) y"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm abs_supp)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(rule impI)
-apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)", erule exE, simp add: fresh_prod)
-apply(erule conjE)+
-apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm)
-apply(rule exists_fresh', simp add: fin_supp)
-apply(simp add: abs_fresh abs_supp)+
-apply(fresh_guess add: abs_fresh fresh_prod)+
+apply(finite_guess | simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_NotR abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_AndR abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)")
+apply(force simp add: fresh_prod fresh_fun_simp_AndR abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_OrR1 abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_OrR2 abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_ImpR abs_fresh fresh_atm abs_supp)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh abs_supp | rule strip)+
+apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)")
+apply(force simp add: fresh_prod fresh_fun_simp_ImpR abs_fresh fresh_atm)
+apply(meson exists_fresh' fin_supp)
+
+apply(simp add: abs_fresh | fresh_guess add: abs_fresh)+
 done
 
+
 lemma csubst_eqvt[eqvt]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>(M{c:=(x).N}) = (pi1\<bullet>M){(pi1\<bullet>c):=(pi1\<bullet>x).(pi1\<bullet>N)}"
   and   "pi2\<bullet>(M{c:=(x).N}) = (pi2\<bullet>M){(pi2\<bullet>c):=(pi2\<bullet>x).(pi2\<bullet>N)}"
 apply(nominal_induct M avoiding: c x N rule: trm.strong_induct)
-apply(auto simp add: eq_bij fresh_bij eqvts)
+apply(auto simp: eq_bij fresh_bij eqvts)
 apply(perm_simp)+
 done
 
 lemma nsubst_eqvt[eqvt]:
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "pi1\<bullet>(M{x:=<c>.N}) = (pi1\<bullet>M){(pi1\<bullet>x):=<(pi1\<bullet>c)>.(pi1\<bullet>N)}"
   and   "pi2\<bullet>(M{x:=<c>.N}) = (pi2\<bullet>M){(pi2\<bullet>x):=<(pi2\<bullet>c)>.(pi2\<bullet>N)}"
 apply(nominal_induct M avoiding: c x N rule: trm.strong_induct)
-apply(auto simp add: eq_bij fresh_bij eqvts)
+apply(auto simp: eq_bij fresh_bij eqvts)
 apply(perm_simp)+
 done
 
 lemma supp_subst1:
   shows "supp (M{y:=<c>.P}) \<subseteq> ((supp M) - {y}) \<union> (supp P)"
 apply(nominal_induct M avoiding: y P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 done
 
 lemma supp_subst2:
   shows "supp (M{y:=<c>.P}) \<subseteq> supp (M) \<union> ((supp P) - {c})"
 apply(nominal_induct M avoiding: y P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)+
 done
 
 lemma supp_subst3:
   shows "supp (M{c:=(x).P}) \<subseteq> ((supp M) - {c}) \<union> (supp P)"
 apply(nominal_induct M avoiding: x P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 done
 
 lemma supp_subst4:
   shows "supp (M{c:=(x).P}) \<subseteq> (supp M) \<union> ((supp P) - {x})"
 apply(nominal_induct M avoiding: x P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 done
 
 lemma supp_subst5:
   shows "(supp M - {y}) \<subseteq> supp (M{y:=<c>.P})"
 apply(nominal_induct M avoiding: y P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 done
 
 lemma supp_subst6:
   shows "(supp M) \<subseteq> ((supp (M{y:=<c>.P}))::coname set)"
 apply(nominal_induct M avoiding: y P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm)
 apply(blast)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(blast)
 done
 
 lemma supp_subst7:
   shows "(supp M - {c}) \<subseteq>  supp (M{c:=(x).P})"
 apply(nominal_induct M avoiding: x P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)
 done
 
 lemma supp_subst8:
   shows "(supp M) \<subseteq> ((supp (M{c:=(x).P}))::name set)"
 apply(nominal_induct M avoiding: x P c rule: trm.strong_induct)
 apply(auto)
-apply(auto simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
+apply(auto simp: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(simp add: fresh_def abs_supp trm.supp supp_atm fin_supp)
 apply(blast)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(blast)+
 done
 
 lemmas subst_supp = supp_subst1 supp_subst2 supp_subst3 supp_subst4
                     supp_subst5 supp_subst6 supp_subst7 supp_subst8
 lemma subst_fresh:
   fixes x::"name"
   and   c::"coname"
   shows "x\<sharp>P \<Longrightarrow> x\<sharp>M{x:=<c>.P}"   
   and   "b\<sharp>P \<Longrightarrow> b\<sharp>M{b:=(y).P}"
   and   "x\<sharp>(M,P) \<Longrightarrow> x\<sharp>M{y:=<c>.P}"
   and   "x\<sharp>M \<Longrightarrow> x\<sharp>M{c:=(x).P}"
   and   "x\<sharp>(M,P) \<Longrightarrow> x\<sharp>M{c:=(y).P}"
   and   "b\<sharp>(M,P) \<Longrightarrow> b\<sharp>M{c:=(y).P}"
   and   "b\<sharp>M \<Longrightarrow> b\<sharp>M{y:=<b>.P}"
   and   "b\<sharp>(M,P) \<Longrightarrow> b\<sharp>M{y:=<c>.P}"
 apply -
 apply(insert subst_supp)
 apply(simp_all add: fresh_def supp_prod)
 apply(blast)+ 
 done
 
 lemma forget:
   shows "x\<sharp>M \<Longrightarrow> M{x:=<c>.P} = M"
   and   "c\<sharp>M \<Longrightarrow> M{c:=(x).P} = M"
 apply(nominal_induct M avoiding: x c P rule: trm.strong_induct)
-apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+apply(auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 done
 
 lemma substc_rename1:
   assumes a: "c\<sharp>(M,a)"
   shows "M{a:=(x).N} = ([(c,a)]\<bullet>M){c:=(x).N}"
 using a
 proof(nominal_induct M avoiding: c a x N rule: trm.strong_induct)
   case (Ax z d)
-  then show ?case by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha)
+  then show ?case by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha)
 next
   case NotL
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (NotR y M d)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (AndR c1 M c2 M' c3)
   then show ?case
-    apply(auto simp add: fresh_prod calc_atm fresh_atm abs_fresh fresh_left)
+    apply(auto simp: fresh_prod calc_atm fresh_atm abs_fresh fresh_left)
     apply (metis (erased, opaque_lifting))
     by metis
 next
   case AndL1
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case AndL2
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (OrR1 d M e)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (OrR2 d M e)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next 
   case ImpL
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
        metis
 next
   case (ImpR y d M e)
   then show ?case
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case (Cut d M y M')
   then show ?case
     by(simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
       (metis crename.simps(1) crename_id crename_rename)
 qed
 
 lemma substc_rename2:
   assumes a: "y\<sharp>(N,x)"
   shows "M{a:=(x).N} = M{a:=(y).([(y,x)]\<bullet>N)}"
 using a
 proof(nominal_induct M avoiding: a x y N rule: trm.strong_induct)
   case (Ax z d)
   then show ?case 
-    by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
+    by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
 next
   case NotL
   then show ?case 
-    by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
+    by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
 next
   case (NotR y M d)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{d:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotR)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (AndR c1 M c2 M' c3)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac 
        "\<exists>a'::coname. a'\<sharp>(N,M{c3:=(y).([(y,x)]\<bullet>N)},M'{c3:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,c1,c2,c3)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndR)
-    apply (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh subst_fresh perm_swap fresh_left)
+    apply (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh subst_fresh perm_swap fresh_left)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case AndL1
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case AndL2
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case (OrR1 d M e)
   then show ?case 
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR1)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrR2 d M e)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR2)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next 
   case ImpL
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case (ImpR y d M e)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpR)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (Cut d M y M')
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
 qed
 
 lemma substn_rename3:
   assumes a: "y\<sharp>(M,x)"
   shows "M{x:=<a>.N} = ([(y,x)]\<bullet>M){y:=<a>.N}"
 using a
 proof(nominal_induct M avoiding: a x y N rule: trm.strong_induct)
   case (Ax z d)
-  then show ?case by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha)
+  then show ?case by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha)
 next
   case NotR
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (NotL d M z)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (AndR c1 M c2 M' c3)
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case OrR1
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case OrR2
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (AndL1 u M v)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (AndL2 u M v)
   then show ?case 
-    by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
+    by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod)
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
     by(simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
       (metis (poly_guards_query))
 next 
   case ImpR
   then show ?case
-  by(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_left abs_supp fin_supp fresh_prod)
+  by(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_left abs_supp fin_supp fresh_prod)
 next
   case (ImpL d M v M' u)
   then show ?case
     by(simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
        (metis (poly_guards_query))
 next
   case (Cut d M y M')
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
     apply(simp add: calc_atm)
     apply metis
     done
 qed
 
 lemma substn_rename4:
   assumes a: "c\<sharp>(N,a)"
   shows "M{x:=<a>.N} = M{x:=<c>.([(c,a)]\<bullet>N)}"
 using a
 proof(nominal_induct M avoiding: x c a N rule: trm.strong_induct)
   case (Ax z d)
   then show ?case 
-    by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
+    by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
 next
   case NotR
   then show ?case 
-    by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
+    by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
 next
   case (NotL d M y)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac 
        "\<exists>a'::name. a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},M'{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,x1,x2,x3)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL)
-    apply (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh subst_fresh perm_swap fresh_left)
+    apply (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh subst_fresh perm_swap fresh_left)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case OrR1
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case OrR2
-  then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+  then show ?case by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case (AndL1 u M v)
   then show ?case 
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,u,v)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndL2 u M v)
   then show ?case 
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,u,v)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndR c1 M c2 M' c3)
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next 
   case ImpR
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 next
   case (ImpL d M y M' u)
   then show ?case
-    apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+    apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
     apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{u:=<c>.([(c,a)]\<bullet>N)},M'{u:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,y,u)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL)
     apply(simp add: trm.inject alpha perm_swap fresh_left calc_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (Cut d M y M')
   then show ?case
-    by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
+    by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
 qed
 
 lemma subst_rename5:
   assumes a: "c'\<sharp>(c,N)" "x'\<sharp>(x,M)"
   shows "M{x:=<c>.N} = ([(x',x)]\<bullet>M){x':=<c'>.([(c',c)]\<bullet>N)}"
 proof -
   have "M{x:=<c>.N} = ([(x',x)]\<bullet>M){x':=<c>.N}" using a by (simp add: substn_rename3)
   also have "\<dots> = ([(x',x)]\<bullet>M){x':=<c'>.([(c',c)]\<bullet>N)}" using a by (simp add: substn_rename4)
   finally show ?thesis by simp
 qed
 
 lemma subst_rename6:
   assumes a: "c'\<sharp>(c,M)" "x'\<sharp>(x,N)"
   shows "M{c:=(x).N} = ([(c',c)]\<bullet>M){c':=(x').([(x',x)]\<bullet>N)}"
 proof -
   have "M{c:=(x).N} = ([(c',c)]\<bullet>M){c':=(x).N}" using a by (simp add: substc_rename1)
   also have "\<dots> = ([(c',c)]\<bullet>M){c':=(x').([(x',x)]\<bullet>N)}" using a by (simp add: substc_rename2)
   finally show ?thesis by simp
 qed
 
 lemmas subst_rename = substc_rename1 substc_rename2 substn_rename3 substn_rename4 subst_rename5 subst_rename6
 
 lemma better_Cut_substn[simp]:
   assumes a: "a\<sharp>[c].P" "x\<sharp>(y,P)"
   shows "(Cut <a>.M (x).N){y:=<c>.P} = 
   (if M=Ax y a then Cut <c>.P (x).(N{y:=<c>.P}) else Cut <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}))"
 proof -   
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,c,P,x,y)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,c,P,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   have eq2: "(M=Ax y a) = (([(a',a)]\<bullet>M)=Ax y a')"
-    apply(auto simp add: calc_atm)
+    apply(auto simp: calc_atm)
     apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
     apply(simp add: calc_atm)
     done
   have "(Cut <a>.M (x).N){y:=<c>.P} = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)){y:=<c>.P}" 
     using eq1 by simp
   also have "\<dots> = (if ([(a',a)]\<bullet>M)=Ax y a' then Cut <c>.P (x').(([(x',x)]\<bullet>N){y:=<c>.P}) 
                               else Cut <a'>.(([(a',a)]\<bullet>M){y:=<c>.P}) (x').(([(x',x)]\<bullet>N){y:=<c>.P}))" 
-    using fs1 fs2 by (auto simp add: fresh_prod fresh_left calc_atm fresh_atm)
+    using fs1 fs2 by (auto simp: fresh_prod fresh_left calc_atm fresh_atm)
   also have "\<dots> =(if M=Ax y a then Cut <c>.P (x).(N{y:=<c>.P}) else Cut <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}))"
     using fs1 fs2 a
     apply -
     apply(simp only: eq2[symmetric])
-    apply(auto simp add: trm.inject)
+    apply(auto simp: trm.inject)
     apply(simp_all add: alpha fresh_atm fresh_prod subst_fresh)
     apply(simp_all add: eqvts perm_fresh_fresh calc_atm)
     apply(auto)
     apply(rule subst_rename)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: abs_fresh)
     apply(simp add: perm_fresh_fresh)
     done
   finally show ?thesis by simp
 qed
     
 lemma better_Cut_substc[simp]:
   assumes a: "a\<sharp>(c,P)" "x\<sharp>[y].P"
   shows "(Cut <a>.M (x).N){c:=(y).P} = 
   (if N=Ax x c then Cut <a>.(M{c:=(y).P}) (y).P else Cut <a>.(M{c:=(y).P}) (x).(N{c:=(y).P}))" 
 proof -   
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,c,P,x,y)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,c,P,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   have eq2: "(N=Ax x c) = (([(x',x)]\<bullet>N)=Ax x' c)"
-    apply(auto simp add: calc_atm)
+    apply(auto simp: calc_atm)
     apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
     apply(simp add: calc_atm)
     done
   have "(Cut <a>.M (x).N){c:=(y).P} = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)){c:=(y).P}" 
     using eq1 by simp
   also have "\<dots> = (if ([(x',x)]\<bullet>N)=Ax x' c then  Cut <a'>.(([(a',a)]\<bullet>M){c:=(y).P}) (y).P
                               else Cut <a'>.(([(a',a)]\<bullet>M){c:=(y).P}) (x').(([(x',x)]\<bullet>N){c:=(y).P}))" 
     using fs1 fs2  by (simp add: fresh_prod fresh_left calc_atm fresh_atm trm.inject)
   also have "\<dots> =(if N=Ax x c then Cut <a>.(M{c:=(y).P}) (y).P else Cut <a>.(M{c:=(y).P}) (x).(N{c:=(y).P}))"
     using fs1 fs2 a
     apply -
     apply(simp only: eq2[symmetric])
-    apply(auto simp add: trm.inject)
+    apply(auto simp: trm.inject)
     apply(simp_all add: alpha fresh_atm fresh_prod subst_fresh)
     apply(simp_all add: eqvts perm_fresh_fresh calc_atm)
     apply(auto)
     apply(rule subst_rename)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: abs_fresh)
     apply(simp add: perm_fresh_fresh)
     done
   finally show ?thesis by simp
 qed
 
 lemma better_Cut_substn':
   assumes a: "a\<sharp>[c].P" "y\<sharp>(N,x)" "M\<noteq>Ax y a"
   shows "(Cut <a>.M (x).N){y:=<c>.P} = Cut <a>.(M{y:=<c>.P}) (x).N"
 using a
 apply -
 apply(generate_fresh "name")
 apply(subgoal_tac "Cut <a>.M (x).N = Cut <a>.M (ca).([(ca,x)]\<bullet>N)")
 apply(simp)
 apply(subgoal_tac"y\<sharp>([(ca,x)]\<bullet>N)")
 apply(simp add: forget)
 apply(simp add: trm.inject)
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)[1]
 apply(simp add: trm.inject)
 apply(rule sym)
 apply(simp add: alpha fresh_prod fresh_atm)
 done
 
 lemma better_NotR_substc:
   assumes a: "d\<sharp>M"
   shows "(NotR (x).M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.NotR (x).M a' (z).P)"
 using a
 apply -
 apply(generate_fresh "name")
 apply(subgoal_tac "NotR (x).M d = NotR (c).([(c,x)]\<bullet>M) d")
-apply(auto simp add: fresh_left calc_atm forget)
+apply(auto simp: fresh_left calc_atm forget)
 apply(generate_fresh "coname")
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(perm_simp add: trm.inject alpha fresh_prod fresh_atm fresh_left, auto)
 done
 
 lemma better_NotL_substn:
   assumes a: "y\<sharp>M"
   shows "(NotL <a>.M y){y:=<c>.P} = fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(subgoal_tac "NotL <a>.M y = NotL <ca>.([(ca,a)]\<bullet>M) y")
-apply(auto simp add: fresh_left calc_atm forget)
+apply(auto simp: fresh_left calc_atm forget)
 apply(generate_fresh "name")
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(perm_simp add: trm.inject alpha fresh_prod fresh_atm fresh_left, auto)
 done
 
 lemma better_AndL1_substn:
   assumes a: "y\<sharp>[x].M"
   shows "(AndL1 (x).M y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')"
 using a
 apply -
 apply(generate_fresh "name")
 apply(subgoal_tac "AndL1 (x).M y = AndL1 (ca).([(ca,x)]\<bullet>M) y")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(generate_fresh "name")
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_AndL2_substn:
   assumes a: "y\<sharp>[x].M"
   shows "(AndL2 (x).M y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')"
 using a
 apply -
 apply(generate_fresh "name")
 apply(subgoal_tac "AndL2 (x).M y = AndL2 (ca).([(ca,x)]\<bullet>M) y")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(generate_fresh "name")
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_AndR_substc:
   assumes a: "c\<sharp>([a].M,[b].N)"
   shows "(AndR <a>.M <b>.N c){c:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.(AndR <a>.M <b>.N a') (z).P)"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(generate_fresh "coname")
 apply(subgoal_tac "AndR <a>.M <b>.N c = AndR <ca>.([(ca,a)]\<bullet>M) <caa>.([(caa,b)]\<bullet>N) c")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule trans)
 apply(rule substc.simps)
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
-apply(auto simp add: fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_prod fresh_atm)[1]
 apply(simp)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule conjI)
 apply(rule forget)
-apply(auto simp add: fresh_left calc_atm abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh)[1]
 apply(rule forget)
-apply(auto simp add: fresh_left calc_atm abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh)[1]
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_OrL_substn:
   assumes a: "x\<sharp>([y].M,[z].N)"
   shows "(OrL (y).M (z).N x){x:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (y).M (z).N z')"
 using a
 apply -
 apply(generate_fresh "name")
 apply(generate_fresh "name")
 apply(subgoal_tac "OrL (y).M (z).N x = OrL (ca).([(ca,y)]\<bullet>M) (caa).([(caa,z)]\<bullet>N) x")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule trans)
 apply(rule substn.simps)
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
-apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
-apply(auto simp add: fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_left calc_atm fresh_prod fresh_atm)[1]
+apply(auto simp: fresh_prod fresh_atm)[1]
 apply(simp)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule conjI)
 apply(rule forget)
-apply(auto simp add: fresh_left calc_atm abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh)[1]
 apply(rule forget)
-apply(auto simp add: fresh_left calc_atm abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh)[1]
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_OrR1_substc:
   assumes a: "d\<sharp>[a].M"
   shows "(OrR1 <a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <a>.M a' (z).P)"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(subgoal_tac "OrR1 <a>.M d = OrR1 <c>.([(c,a)]\<bullet>M) d")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_OrR2_substc:
   assumes a: "d\<sharp>[a].M"
   shows "(OrR2 <a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <a>.M a' (z).P)"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(subgoal_tac "OrR2 <a>.M d = OrR2 <c>.([(c,a)]\<bullet>M) d")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm)
 apply(auto)
 done
 
 lemma better_ImpR_substc:
   assumes a: "d\<sharp>[a].M"
   shows "(ImpR (x).<a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.ImpR (x).<a>.M a' (z).P)"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(generate_fresh "name")
 apply(subgoal_tac "ImpR (x).<a>.M d = ImpR (ca).<c>.([(c,a)]\<bullet>[(ca,x)]\<bullet>M) d")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm abs_fresh fresh_left calc_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm fresh_left calc_atm abs_fresh)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(rule sym)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm abs_fresh abs_perm)
 done
 
 lemma better_ImpL_substn:
   assumes a: "y\<sharp>(M,[x].N)"
   shows "(ImpL <a>.M (x).N y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')"
 using a
 apply -
 apply(generate_fresh "coname")
 apply(generate_fresh "name")
 apply(subgoal_tac "ImpL <a>.M (x).N y = ImpL <ca>.([(ca,a)]\<bullet>M) (caa).([(caa,x)]\<bullet>N) y")
-apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1]
+apply(auto simp: fresh_left calc_atm forget abs_fresh)[1]
 apply(rule_tac f="fresh_fun" in arg_cong)
 apply(simp add:  fun_eq_iff)
 apply(rule allI)
 apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm abs_fresh fresh_left calc_atm)
 apply(rule forget)
 apply(simp add: fresh_left calc_atm)
 apply(auto)[1]
 apply(rule sym)
 apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm abs_fresh abs_perm)
 done
 
 lemma freshn_after_substc:
   fixes x::"name"
   assumes a: "x\<sharp>M{c:=(y).P}"
   shows "x\<sharp>M"
 using a supp_subst8
 apply(simp add: fresh_def)
 apply(blast)
 done
 
 lemma freshn_after_substn:
   fixes x::"name"
   assumes a: "x\<sharp>M{y:=<c>.P}" "x\<noteq>y"
   shows "x\<sharp>M"
 using a
 using a supp_subst5
 apply(simp add: fresh_def)
 apply(blast)
 done
 
 lemma freshc_after_substc:
   fixes a::"coname"
   assumes a: "a\<sharp>M{c:=(y).P}" "a\<noteq>c"
   shows "a\<sharp>M"
 using a supp_subst7
 apply(simp add: fresh_def)
 apply(blast)
 done
 
 lemma freshc_after_substn:
   fixes a::"coname"
   assumes a: "a\<sharp>M{y:=<c>.P}" 
   shows "a\<sharp>M"
 using a supp_subst6
 apply(simp add: fresh_def)
 apply(blast)
 done
 
 lemma substn_crename_comm:
   assumes a: "c\<noteq>a" "c\<noteq>b"
   shows "M{x:=<c>.P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{x:=<c>.(P[a\<turnstile>c>b])}"
 using a
 apply(nominal_induct M avoiding: x c P a b rule: trm.strong_induct)
-apply(auto simp add: subst_fresh rename_fresh trm.inject)
+apply(auto simp: subst_fresh rename_fresh trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,x,c)")
 apply(erule exE)
 apply(subgoal_tac "Cut <c>.P (x).Ax x a = Cut <c>.P (x').Ax x' a")
 apply(simp)
 apply(rule trans)
 apply(rule crename.simps)
 apply(simp add: fresh_prod fresh_atm)
 apply(simp)
 apply(simp add: trm.inject)
 apply(simp add: alpha trm.inject calc_atm fresh_atm)
 apply(simp add: trm.inject)
 apply(simp add: alpha trm.inject calc_atm fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(simp add: crename_id)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
-apply(auto simp add: fresh_atm)[1]
+apply(auto simp: fresh_atm)[1]
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm)
-apply(auto simp add: fresh_atm)[1]
+apply(auto simp: fresh_atm)[1]
 apply(drule crename_ax)
 apply(simp add: fresh_atm)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],x,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],name1,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],name1,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},trm2{x:=<c>.P},P,P[a\<turnstile>c>b],name1,name2,
                                   trm1[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,P[a\<turnstile>c>b],name1,
                                   trm1[a\<turnstile>c>b]{name2:=<c>.P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{name2:=<c>.P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 done
 
 lemma substc_crename_comm:
   assumes a: "c\<noteq>a" "c\<noteq>b"
   shows "M{c:=(x).P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{c:=(x).(P[a\<turnstile>c>b])}"
 using a
 apply(nominal_induct M avoiding: x c P a b rule: trm.strong_induct)
-apply(auto simp add: subst_fresh rename_fresh trm.inject)
+apply(auto simp: subst_fresh rename_fresh trm.inject)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(drule crename_ax)
 apply(simp add: fresh_atm)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(a,b,trm{coname:=(x).P},P,P[a\<turnstile>c>b],x,trm[a\<turnstile>c>b]{coname:=(x).P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,coname2,a,b,trm1{coname3:=(x).P},trm2{coname3:=(x).P},
                    P,P[a\<turnstile>c>b],x,trm1[a\<turnstile>c>b]{coname3:=(x).P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{coname3:=(x).P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b,
                          trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b,
                          trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b,
                          trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(rule trans)
 apply(rule better_crename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 done
 
 lemma substn_nrename_comm:
   assumes a: "x\<noteq>y" "x\<noteq>z"
   shows "M{x:=<c>.P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{x:=<c>.(P[y\<turnstile>n>z])}"
 using a
 apply(nominal_induct M avoiding: x c P y z rule: trm.strong_induct)
-apply(auto simp add: subst_fresh rename_fresh trm.inject)
+apply(auto simp: subst_fresh rename_fresh trm.inject)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_prod fresh_atm)
 apply(simp add: trm.inject)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(drule nrename_ax)
 apply(simp add: fresh_atm)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,z,trm{x:=<c>.P},P,P[y\<turnstile>n>z],x,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[y\<turnstile>n>z],name1,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]},y,z)")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,z,trm{x:=<c>.P},P,P[y\<turnstile>n>z],name1,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},trm2{x:=<c>.P},P,P[y\<turnstile>n>z],name1,name2,y,z,
                                   trm1[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,P[y\<turnstile>n>z],y,z,name1,
                                   trm1[y\<turnstile>n>z]{name2:=<c>.P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{name2:=<c>.P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 done
 
 lemma substc_nrename_comm:
   assumes a: "x\<noteq>y" "x\<noteq>z"
   shows "M{c:=(x).P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{c:=(x).(P[y\<turnstile>n>z])}"
 using a
 apply(nominal_induct M avoiding: x c P y z rule: trm.strong_induct)
-apply(auto simp add: subst_fresh rename_fresh trm.inject)
+apply(auto simp: subst_fresh rename_fresh trm.inject)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(drule nrename_ax)
 apply(simp add: fresh_atm)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(drule nrename_ax)
 apply(simp add: fresh_atm)
 apply(simp add: fresh_atm)
 apply(simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(y,z,trm{coname:=(x).P},P,P[y\<turnstile>n>z],x,trm[y\<turnstile>n>z]{coname:=(x).P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,coname2,y,z,trm1{coname3:=(x).P},trm2{coname3:=(x).P},
                    P,P[y\<turnstile>n>z],x,trm1[y\<turnstile>n>z]{coname3:=(x).P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{coname3:=(x).P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh subst_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z,
                          trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z,
                          trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z,
                          trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(rule trans)
 apply(rule better_nrename_Cut)
 apply(simp add: fresh_atm fresh_prod)
 apply(simp add: rename_fresh fresh_atm)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 done
 
 lemma substn_crename_comm':
   assumes a: "a\<noteq>c" "a\<sharp>P"
   shows "M{x:=<c>.P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{x:=<c>.P}"
 using a
 proof -
   assume a1: "a\<noteq>c"
   assume a2: "a\<sharp>P"
   obtain c'::"coname" where fs2: "c'\<sharp>(c,P,a,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq: "M{x:=<c>.P} = M{x:=<c'>.([(c',c)]\<bullet>P)}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
    have eq': "M[a\<turnstile>c>b]{x:=<c>.P} = M[a\<turnstile>c>b]{x:=<c'>.([(c',c)]\<bullet>P)}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
   have eq2: "([(c',c)]\<bullet>P)[a\<turnstile>c>b] = ([(c',c)]\<bullet>P)" using fs2 a2 a1
     apply -
     apply(rule rename_fresh)
     apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
     done
   have "M{x:=<c>.P}[a\<turnstile>c>b] = M{x:=<c'>.([(c',c)]\<bullet>P)}[a\<turnstile>c>b]" using eq by simp
   also have "\<dots> = M[a\<turnstile>c>b]{x:=<c'>.(([(c',c)]\<bullet>P)[a\<turnstile>c>b])}"
     using fs2
     apply -
     apply(rule substn_crename_comm)
     apply(simp_all add: fresh_prod fresh_atm)
     done
   also have "\<dots> = M[a\<turnstile>c>b]{x:=<c'>.(([(c',c)]\<bullet>P))}" using eq2 by simp
   also have "\<dots> = M[a\<turnstile>c>b]{x:=<c>.P}" using eq' by simp 
   finally show ?thesis by simp
 qed
 
 lemma substc_crename_comm':
   assumes a: "c\<noteq>a" "c\<noteq>b" "a\<sharp>P"
   shows "M{c:=(x).P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{c:=(x).P}"
 using a
 proof -
   assume a1: "c\<noteq>a"
   assume a1': "c\<noteq>b"
   assume a2: "a\<sharp>P"
   obtain c'::"coname" where fs2: "c'\<sharp>(c,M,a,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   have eq: "M{c:=(x).P} = ([(c',c)]\<bullet>M){c':=(x).P}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
    have eq': "([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P} = M[a\<turnstile>c>b]{c:=(x).P}"
     using fs2
     apply -
     apply(rule subst_rename[symmetric])
     apply(simp add: rename_fresh)
     done
   have eq2: "([(c',c)]\<bullet>M)[a\<turnstile>c>b] = ([(c',c)]\<bullet>(M[a\<turnstile>c>b]))" using fs2 a2 a1 a1'
     apply -
     apply(simp add: rename_eqvts)
     apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
     done
   have "M{c:=(x).P}[a\<turnstile>c>b] = ([(c',c)]\<bullet>M){c':=(x).P}[a\<turnstile>c>b]" using eq by simp
   also have "\<dots> = ([(c',c)]\<bullet>M)[a\<turnstile>c>b]{c':=(x).P[a\<turnstile>c>b]}"
     using fs2
     apply -
     apply(rule substc_crename_comm)
     apply(simp_all add: fresh_prod fresh_atm)
     done
   also have "\<dots> = ([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P[a\<turnstile>c>b]}" using eq2 by simp
   also have "\<dots> = ([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P}" using a2 by (simp add: rename_fresh)
   also have "\<dots> = M[a\<turnstile>c>b]{c:=(x).P}" using eq' by simp
   finally show ?thesis by simp
 qed
 
 lemma substn_nrename_comm':
   assumes a: "x\<noteq>y" "x\<noteq>z" "y\<sharp>P"
   shows "M{x:=<c>.P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{x:=<c>.P}"
 using a
 proof -
   assume a1: "x\<noteq>y"
   assume a1': "x\<noteq>z"
   assume a2: "y\<sharp>P"
   obtain x'::"name" where fs2: "x'\<sharp>(x,M,y,z)" by (rule exists_fresh(1), rule fin_supp, blast)
   have eq: "M{x:=<c>.P} = ([(x',x)]\<bullet>M){x':=<c>.P}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
    have eq': "([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P} = M[y\<turnstile>n>z]{x:=<c>.P}"
     using fs2
     apply -
     apply(rule subst_rename[symmetric])
     apply(simp add: rename_fresh)
     done
   have eq2: "([(x',x)]\<bullet>M)[y\<turnstile>n>z] = ([(x',x)]\<bullet>(M[y\<turnstile>n>z]))" using fs2 a2 a1 a1'
     apply -
     apply(simp add: rename_eqvts)
     apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
     done
   have "M{x:=<c>.P}[y\<turnstile>n>z] = ([(x',x)]\<bullet>M){x':=<c>.P}[y\<turnstile>n>z]" using eq by simp
   also have "\<dots> = ([(x',x)]\<bullet>M)[y\<turnstile>n>z]{x':=<c>.P[y\<turnstile>n>z]}"
     using fs2
     apply -
     apply(rule substn_nrename_comm)
     apply(simp_all add: fresh_prod fresh_atm)
     done
   also have "\<dots> = ([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P[y\<turnstile>n>z]}" using eq2 by simp
   also have "\<dots> = ([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P}" using a2 by (simp add: rename_fresh)
   also have "\<dots> = M[y\<turnstile>n>z]{x:=<c>.P}" using eq' by simp
   finally show ?thesis by simp
 qed
 
 lemma substc_nrename_comm':
   assumes a: "x\<noteq>y" "y\<sharp>P"
   shows "M{c:=(x).P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{c:=(x).P}"
 using a
 proof -
   assume a1: "x\<noteq>y"
   assume a2: "y\<sharp>P"
   obtain x'::"name" where fs2: "x'\<sharp>(x,P,y,z)" by (rule exists_fresh(1), rule fin_supp, blast)
   have eq: "M{c:=(x).P} = M{c:=(x').([(x',x)]\<bullet>P)}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
    have eq': "M[y\<turnstile>n>z]{c:=(x).P} = M[y\<turnstile>n>z]{c:=(x').([(x',x)]\<bullet>P)}"
     using fs2
     apply -
     apply(rule subst_rename)
     apply(simp)
     done
   have eq2: "([(x',x)]\<bullet>P)[y\<turnstile>n>z] = ([(x',x)]\<bullet>P)" using fs2 a2 a1
     apply -
     apply(rule rename_fresh)
     apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
     done
   have "M{c:=(x).P}[y\<turnstile>n>z] = M{c:=(x').([(x',x)]\<bullet>P)}[y\<turnstile>n>z]" using eq by simp
   also have "\<dots> = M[y\<turnstile>n>z]{c:=(x').(([(x',x)]\<bullet>P)[y\<turnstile>n>z])}"
     using fs2
     apply -
     apply(rule substc_nrename_comm)
     apply(simp_all add: fresh_prod fresh_atm)
     done
   also have "\<dots> = M[y\<turnstile>n>z]{c:=(x').(([(x',x)]\<bullet>P))}" using eq2 by simp
   also have "\<dots> = M[y\<turnstile>n>z]{c:=(x).P}" using eq' by simp 
   finally show ?thesis by simp
 qed
 
 lemmas subst_comm = substn_crename_comm substc_crename_comm
                     substn_nrename_comm substc_nrename_comm
 lemmas subst_comm' = substn_crename_comm' substc_crename_comm'
                      substn_nrename_comm' substc_nrename_comm'
 
 text \<open>typing contexts\<close>
 
 type_synonym ctxtn = "(name\<times>ty) list"
 type_synonym ctxtc = "(coname\<times>ty) list"
 
 inductive
   validc :: "ctxtc \<Rightarrow> bool"
 where
   vc1[intro]: "validc []"
 | vc2[intro]: "\<lbrakk>a\<sharp>\<Delta>; validc \<Delta>\<rbrakk> \<Longrightarrow> validc ((a,T)#\<Delta>)"
 
 equivariance validc
 
 inductive
   validn :: "ctxtn \<Rightarrow> bool"
 where
   vn1[intro]: "validn []"
 | vn2[intro]: "\<lbrakk>x\<sharp>\<Gamma>; validn \<Gamma>\<rbrakk> \<Longrightarrow> validn ((x,T)#\<Gamma>)"
 
 equivariance validn
 
 lemma fresh_ctxt:
   fixes a::"coname"
   and   x::"name"
   and   \<Gamma>::"ctxtn"
   and   \<Delta>::"ctxtc"
   shows "a\<sharp>\<Gamma>" and "x\<sharp>\<Delta>"
 proof -
-  show "a\<sharp>\<Gamma>" by (induct \<Gamma>) (auto simp add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty)
-next
-  show "x\<sharp>\<Delta>" by (induct \<Delta>) (auto simp add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty)
+  show "a\<sharp>\<Gamma>" by (induct \<Gamma>) (auto simp: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty)
+next
+  show "x\<sharp>\<Delta>" by (induct \<Delta>) (auto simp: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty)
 qed
 
 text \<open>cut-reductions\<close>
 
 declare abs_perm[eqvt]
 
 inductive
   fin :: "trm \<Rightarrow> name \<Rightarrow> bool"
 where
   [intro]: "fin (Ax x a) x"
 | [intro]: "x\<sharp>M \<Longrightarrow> fin (NotL <a>.M x) x"
 | [intro]: "y\<sharp>[x].M \<Longrightarrow> fin (AndL1 (x).M y) y"
 | [intro]: "y\<sharp>[x].M \<Longrightarrow> fin (AndL2 (x).M y) y"
 | [intro]: "\<lbrakk>z\<sharp>[x].M;z\<sharp>[y].N\<rbrakk> \<Longrightarrow> fin (OrL (x).M (y).N z) z"
 | [intro]: "\<lbrakk>y\<sharp>M;y\<sharp>[x].N\<rbrakk> \<Longrightarrow> fin (ImpL <a>.M (x).N y) y"
 
 equivariance fin
 
 lemma fin_Ax_elim:
   assumes a: "fin (Ax x a) y"
   shows "x=y"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fin_NotL_elim:
   assumes a: "fin (NotL <a>.M x) y"
   shows "x=y \<and> x\<sharp>M"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 apply(subgoal_tac "y\<sharp>[aa].Ma")
 apply(drule sym)
 apply(simp_all add: abs_fresh)
 done
 
 lemma fin_AndL1_elim:
   assumes a: "fin (AndL1 (x).M y) z"
   shows "z=y \<and> z\<sharp>[x].M"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fin_AndL2_elim:
   assumes a: "fin (AndL2 (x).M y) z"
   shows "z=y \<and> z\<sharp>[x].M"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fin_OrL_elim:
   assumes a: "fin (OrL (x).M (y).N u) z"
   shows "z=u \<and> z\<sharp>[x].M \<and> z\<sharp>[y].N"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fin_ImpL_elim:
   assumes a: "fin (ImpL <a>.M (x).N z) y"
   shows "z=y \<and> z\<sharp>M \<and> z\<sharp>[x].N"
 using a
 apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 apply(subgoal_tac "y\<sharp>[aa].Ma")
 apply(drule sym)
 apply(simp_all add: abs_fresh)
 apply (metis abs_fresh(5))
 done
  
 
 lemma fin_rest_elims:
   shows "fin (Cut <a>.M (x).N) y \<Longrightarrow> False"
   and   "fin (NotR (x).M c) y \<Longrightarrow> False"
   and   "fin (AndR <a>.M <b>.N c) y \<Longrightarrow> False"
   and   "fin (OrR1 <a>.M b) y \<Longrightarrow> False"
   and   "fin (OrR2 <a>.M b) y \<Longrightarrow> False"
   and   "fin (ImpR (x).<a>.M b) y \<Longrightarrow> False"
 by (erule fin.cases, simp_all add: trm.inject)+
 
 lemmas fin_elims = fin_Ax_elim fin_NotL_elim fin_AndL1_elim fin_AndL2_elim fin_OrL_elim 
                    fin_ImpL_elim fin_rest_elims
 
 lemma fin_rename:
   shows "fin M x \<Longrightarrow> fin ([(x',x)]\<bullet>M) x'"
 by (induct rule: fin.induct)
-   (auto simp add: calc_atm simp add: fresh_left abs_fresh)
+   (auto simp: calc_atm simp add: fresh_left abs_fresh)
 
 lemma not_fin_subst1:
   assumes a: "\<not>(fin M x)" 
   shows "\<not>(fin (M{c:=(y).P}) x)"
 using a [[simproc del: defined_all]]
 apply(nominal_induct M avoiding: x c y P rule: trm.strong_induct)
 apply(auto)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname:=(y).P},P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(drule fin_elims)
 apply(auto)[1]
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,coname3,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,coname3,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(drule fin_AndL1_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substc)
 apply(subgoal_tac "name2\<sharp>[name1]. trm")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(drule fin_AndL2_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substc)
 apply(subgoal_tac "name2\<sharp>[name1].trm")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(drule fin_OrL_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substc)+
 apply(subgoal_tac "name3\<sharp>[name1].trm1 \<and> name3\<sharp>[name2].trm2")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(drule fin_ImpL_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substc)+
 apply(subgoal_tac "x\<sharp>[name1].trm2")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 done
 
 lemma not_fin_subst2:
   assumes a: "\<not>(fin M x)" 
   shows "\<not>(fin (M{y:=<c>.P}) x)"
 using a [[simproc del: defined_all]]
 apply(nominal_induct M avoiding: x c y P rule: trm.strong_induct)
 apply(auto)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fin_NotL_elim)
 apply(auto)[1]
 apply(drule freshn_after_substn)
 apply(simp)
 apply(simp add: fin.intros)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,name1,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fin_AndL1_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substn)
 apply(simp)
 apply(subgoal_tac "name2\<sharp>[name1]. trm")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,name1,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fin_AndL2_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substn)
 apply(simp)
 apply(subgoal_tac "name2\<sharp>[name1].trm")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm1{y:=<c>.P},trm2{y:=<c>.P},name1,name2,P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fin_OrL_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substn)
 apply(simp)
 apply(drule freshn_after_substn)
 apply(simp)
 apply(subgoal_tac "name3\<sharp>[name1].trm1 \<and> name3\<sharp>[name2].trm2")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},name1,P,x)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fin_ImpL_elim)
-apply(auto simp add: abs_fresh)[1]
+apply(auto simp: abs_fresh)[1]
 apply(drule freshn_after_substn)
 apply(simp)
 apply(drule freshn_after_substn)
 apply(simp)
 apply(subgoal_tac "x\<sharp>[name1].trm2")
 apply(simp add: fin.intros)
 apply(simp add: abs_fresh)
 done
 
 lemma fin_subst1:
   assumes a: "fin M x" "x\<noteq>y" "x\<sharp>P"
   shows "fin (M{y:=<c>.P}) x"
 using a
 apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct)
 apply(auto dest!: fin_elims simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh)
 apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh)
 done
 
 lemma fin_subst2:
   assumes a: "fin M y" "x\<noteq>y" "y\<sharp>P" "M\<noteq>Ax y c" 
   shows "fin (M{c:=(x).P}) y"
 using a
 apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct)
 apply(drule fin_elims)
 apply(simp add: trm.inject)
 apply(rule fin.intros)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(rule fin.intros)
 apply(rule subst_fresh)
 apply(simp)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(rule fin.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fin_elims, simp)
 apply(rule fin.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(rule fin.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(rule fin.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 done
 
 lemma fin_substn_nrename:
   assumes a: "fin M x" "x\<noteq>y" "x\<sharp>P"
   shows "M[x\<turnstile>n>y]{y:=<c>.P} = Cut <c>.P (x).(M{y:=<c>.P})"
 using a [[simproc del: defined_all]]
 apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct)
 apply(drule fin_Ax_elim)
 apply(simp)
 apply(simp add: trm.inject)
 apply(simp add: alpha calc_atm fresh_atm)
 apply(simp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_NotL_elim)
 apply(simp)
 apply(subgoal_tac "\<exists>z::name. z\<sharp>(trm,y,x,P,trm[x\<turnstile>n>y]{y:=<c>.P})")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(rule subst_fresh)
 apply(simp)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_AndL1_elim)
 apply(simp)
 apply(subgoal_tac "\<exists>z::name. z\<sharp>(name2,name1,P,trm[name2\<turnstile>n>y]{y:=<c>.P},y,P,trm)")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(rule subst_fresh)
 apply(simp)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(drule fin_AndL2_elim)
 apply(simp)
 apply(subgoal_tac "\<exists>z::name. z\<sharp>(name2,name1,P,trm[name2\<turnstile>n>y]{y:=<c>.P},y,P,trm)")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(rule subst_fresh)
 apply(simp)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_OrL_elim)
 apply(simp add: abs_fresh)
 apply(simp add: subst_fresh rename_fresh)
 apply(subgoal_tac "\<exists>z::name. z\<sharp>(name3,name2,name1,P,trm1[name3\<turnstile>n>y]{y:=<c>.P},trm2[name3\<turnstile>n>y]{y:=<c>.P},y,P,trm1,trm2)")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 apply(drule fin_rest_elims)
 apply(simp)
 apply(drule fin_ImpL_elim)
 apply(simp add: abs_fresh)
 apply(simp add: subst_fresh rename_fresh)
 apply(subgoal_tac "\<exists>z::name. z\<sharp>(name1,x,P,trm1[x\<turnstile>n>y]{y:=<c>.P},trm2[x\<turnstile>n>y]{y:=<c>.P},y,P,trm1,trm2)")
 apply(erule exE, simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(simp add: nsubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: nrename_fresh)
 apply(rule exists_fresh')
 apply(rule fin_supp)
 done
 
 inductive
   fic :: "trm \<Rightarrow> coname \<Rightarrow> bool"
 where
   [intro]: "fic (Ax x a) a"
 | [intro]: "a\<sharp>M \<Longrightarrow> fic (NotR (x).M a) a"
 | [intro]: "\<lbrakk>c\<sharp>[a].M;c\<sharp>[b].N\<rbrakk> \<Longrightarrow> fic (AndR <a>.M <b>.N c) c"
 | [intro]: "b\<sharp>[a].M \<Longrightarrow> fic (OrR1 <a>.M b) b"
 | [intro]: "b\<sharp>[a].M \<Longrightarrow> fic (OrR2 <a>.M b) b"
 | [intro]: "\<lbrakk>b\<sharp>[a].M\<rbrakk> \<Longrightarrow> fic (ImpR (x).<a>.M b) b"
 
 equivariance fic
 
 lemma fic_Ax_elim:
   assumes a: "fic (Ax x a) b"
   shows "a=b"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fic_NotR_elim:
   assumes a: "fic (NotR (x).M a) b"
   shows "a=b \<and> b\<sharp>M"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 apply(subgoal_tac "b\<sharp>[xa].Ma")
 apply(drule sym)
 apply(simp_all add: abs_fresh)
 done
 
 lemma fic_OrR1_elim:
   assumes a: "fic (OrR1 <a>.M b) c"
   shows "b=c \<and> c\<sharp>[a].M"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fic_OrR2_elim:
   assumes a: "fic (OrR2 <a>.M b) c"
   shows "b=c \<and> c\<sharp>[a].M"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fic_AndR_elim:
   assumes a: "fic (AndR <a>.M <b>.N c) d"
   shows "c=d \<and> d\<sharp>[a].M \<and> d\<sharp>[b].N"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 done
 
 lemma fic_ImpR_elim:
   assumes a: "fic (ImpR (x).<a>.M b) c"
   shows "b=c \<and> b\<sharp>[a].M"
 using a
 apply(erule_tac fic.cases)
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 apply(subgoal_tac "c\<sharp>[xa].[aa].Ma")
 apply(drule sym)
 apply(simp_all add: abs_fresh)
 done
 
 lemma fic_rest_elims:
   shows "fic (Cut <a>.M (x).N) d \<Longrightarrow> False"
   and   "fic (NotL <a>.M x) d \<Longrightarrow> False"
   and   "fic (OrL (x).M (y).N z) d \<Longrightarrow> False"
   and   "fic (AndL1 (x).M y) d \<Longrightarrow> False"
   and   "fic (AndL2 (x).M y) d \<Longrightarrow> False"
   and   "fic (ImpL <a>.M (x).N y) d \<Longrightarrow> False"
 by (erule fic.cases, simp_all add: trm.inject)+
 
 lemmas fic_elims = fic_Ax_elim fic_NotR_elim fic_OrR1_elim fic_OrR2_elim fic_AndR_elim 
                    fic_ImpR_elim fic_rest_elims
 
 lemma fic_rename:
   shows "fic M a \<Longrightarrow> fic ([(a',a)]\<bullet>M) a'"
 by (induct rule: fic.induct)
-   (auto simp add: calc_atm simp add: fresh_left abs_fresh)
+   (auto simp: calc_atm simp add: fresh_left abs_fresh)
 
 lemma not_fic_subst1:
   assumes a: "\<not>(fic M a)" 
   shows "\<not>(fic (M{y:=<c>.P}) a)"
 using a
 apply(nominal_induct M avoiding: a c y P rule: trm.strong_induct)
 apply(auto)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims)
 apply(auto)[1]
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},trm2{y:=<c>.P},P,name1,name2,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,name1,name2,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 done
 
 lemma not_fic_subst2:
   assumes a: "\<not>(fic M a)" 
   shows "\<not>(fic (M{c:=(y).P}) a)"
 using a
 apply(nominal_induct M avoiding: a c y P rule: trm.strong_induct)
 apply(auto)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname:=(y).P},P,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fic_elims, simp)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fic_elims, simp)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substc)
 apply(simp)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims, simp)
 done
 
 lemma fic_subst1:
   assumes a: "fic M a" "a\<noteq>b" "a\<sharp>P"
   shows "fic (M{b:=(x).P}) a"
 using a
 apply(nominal_induct M avoiding: x b a P rule: trm.strong_induct)
 apply(drule fic_elims)
 apply(simp add: fic.intros)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(rule subst_fresh)
 apply(simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 done
 
 lemma fic_subst2:
   assumes a: "fic M a" "c\<noteq>a" "a\<sharp>P" "M\<noteq>Ax x a" 
   shows "fic (M{x:=<c>.P}) a"
 using a
 apply(nominal_induct M avoiding: x a c P rule: trm.strong_induct)
 apply(drule fic_elims)
 apply(simp add: trm.inject)
 apply(rule fic.intros)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(rule subst_fresh)
 apply(simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(rule fic.intros)
 apply(simp add: abs_fresh fresh_atm)
 apply(rule subst_fresh)
 apply(auto)[1]
 apply(drule fic_elims, simp)
 done
 
 lemma fic_substc_crename:
   assumes a: "fic M a" "a\<noteq>b" "a\<sharp>P"
   shows "M[a\<turnstile>c>b]{b:=(y).P} = Cut <a>.(M{b:=(y).P}) (y).P"
 using a
 apply(nominal_induct M avoiding: a b  y P rule: trm.strong_induct)
 apply(drule fic_Ax_elim)
 apply(simp)
 apply(simp add: trm.inject)
 apply(simp add: alpha calc_atm fresh_atm trm.inject)
 apply(simp)
 apply(drule fic_rest_elims)
 apply(simp)
 apply(drule fic_NotR_elim)
 apply(simp)
 apply(generate_fresh "coname")
 apply(fresh_fun_simp)
 apply(simp add: trm.inject alpha fresh_atm fresh_prod fresh_atm calc_atm abs_fresh)
 apply(rule conjI)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: crename_fresh)
 apply(rule subst_fresh)
 apply(simp)
 apply(drule fic_rest_elims)
 apply(simp)
 apply(drule fic_AndR_elim)
 apply(simp add: abs_fresh fresh_atm subst_fresh rename_fresh)
 apply(generate_fresh "coname")
 apply(fresh_fun_simp)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod)
 apply(rule conjI)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: subst_fresh)
 apply(drule fic_rest_elims)
 apply(simp)
 apply(drule fic_rest_elims)
 apply(simp)
 apply(drule fic_OrR1_elim)
 apply(simp)
 apply(generate_fresh "coname")
 apply(fresh_fun_simp)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: subst_fresh rename_fresh)
 apply(drule fic_OrR2_elim)
 apply(simp add: abs_fresh fresh_atm)
 apply(generate_fresh "coname")
 apply(fresh_fun_simp)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: subst_fresh rename_fresh)
 apply(drule fic_rest_elims)
 apply(simp)
 apply(drule fic_ImpR_elim)
 apply(simp add: abs_fresh fresh_atm)
 apply(generate_fresh "coname")
 apply(fresh_fun_simp)
 apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod)
 apply(simp add: csubst_eqvt calc_atm)
 apply(simp add: perm_fresh_fresh)
 apply(simp add: subst_fresh rename_fresh)
 apply(drule fic_rest_elims)
 apply(simp)
 done
 
 inductive
   l_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^sub>l _" [100,100] 100)
 where
   LAxR:  "\<lbrakk>x\<sharp>M; a\<sharp>b; fic M a\<rbrakk> \<Longrightarrow> Cut <a>.M (x).(Ax x b) \<longrightarrow>\<^sub>l M[a\<turnstile>c>b]"
 | LAxL:  "\<lbrakk>a\<sharp>M; x\<sharp>y; fin M x\<rbrakk> \<Longrightarrow> Cut <a>.(Ax y a) (x).M \<longrightarrow>\<^sub>l M[x\<turnstile>n>y]"
 | LNot:  "\<lbrakk>y\<sharp>(M,N); x\<sharp>(N,y); a\<sharp>(M,N,b); b\<sharp>M; y\<noteq>x; b\<noteq>a\<rbrakk> \<Longrightarrow>
           Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) \<longrightarrow>\<^sub>l Cut <b>.N (x).M" 
 | LAnd1: "\<lbrakk>b\<sharp>([a1].M1,[a2].M2,N,a1,a2); y\<sharp>([x].N,M1,M2,x); x\<sharp>(M1,M2); a1\<sharp>(M2,N); a2\<sharp>(M1,N); a1\<noteq>a2\<rbrakk> \<Longrightarrow>
           Cut <b>.(AndR <a1>.M1 <a2>.M2 b) (y).(AndL1 (x).N y) \<longrightarrow>\<^sub>l Cut <a1>.M1 (x).N"
 | LAnd2: "\<lbrakk>b\<sharp>([a1].M1,[a2].M2,N,a1,a2); y\<sharp>([x].N,M1,M2,x); x\<sharp>(M1,M2); a1\<sharp>(M2,N); a2\<sharp>(M1,N); a1\<noteq>a2\<rbrakk> \<Longrightarrow>
           Cut <b>.(AndR <a1>.M1 <a2>.M2 b) (y).(AndL2 (x).N y) \<longrightarrow>\<^sub>l Cut <a2>.M2 (x).N"
 | LOr1:  "\<lbrakk>b\<sharp>([a].M,N1,N2,a); y\<sharp>([x1].N1,[x2].N2,M,x1,x2); x1\<sharp>(M,N2); x2\<sharp>(M,N1); a\<sharp>(N1,N2); x1\<noteq>x2\<rbrakk> \<Longrightarrow>
           Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^sub>l Cut <a>.M (x1).N1"
 | LOr2:  "\<lbrakk>b\<sharp>([a].M,N1,N2,a); y\<sharp>([x1].N1,[x2].N2,M,x1,x2); x1\<sharp>(M,N2); x2\<sharp>(M,N1); a\<sharp>(N1,N2); x1\<noteq>x2\<rbrakk> \<Longrightarrow>
           Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^sub>l Cut <a>.M (x2).N2"
 | LImp:  "\<lbrakk>z\<sharp>(N,[y].P,[x].M,y,x); b\<sharp>([a].M,[c].N,P,c,a); x\<sharp>(N,[y].P,y); 
           c\<sharp>(P,[a].M,b,a); a\<sharp>([c].N,P); y\<sharp>(N,[x].M)\<rbrakk> \<Longrightarrow>
           Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z) \<longrightarrow>\<^sub>l Cut <a>.(Cut <c>.N (x).M) (y).P"
 
 equivariance l_redu
 
 lemma l_redu_eqvt':
   fixes pi1::"name prm"
   and   pi2::"coname prm"
   shows "(pi1\<bullet>M) \<longrightarrow>\<^sub>l (pi1\<bullet>M') \<Longrightarrow> M \<longrightarrow>\<^sub>l M'"
   and   "(pi2\<bullet>M) \<longrightarrow>\<^sub>l (pi2\<bullet>M') \<Longrightarrow> M \<longrightarrow>\<^sub>l M'"
 apply -
 apply(drule_tac pi="rev pi1" in l_redu.eqvt(1))
 apply(perm_simp)
 apply(drule_tac pi="rev pi2" in l_redu.eqvt(2))
 apply(perm_simp)
 done
 
 nominal_inductive l_redu
   apply(simp_all add: abs_fresh fresh_atm rename_fresh fresh_prod abs_supp fin_supp)
   apply(force)+
   done
 
 lemma fresh_l_redu:
   fixes x::"name"
   and   a::"coname"
   shows "M \<longrightarrow>\<^sub>l M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'"
   and   "M \<longrightarrow>\<^sub>l M' \<Longrightarrow> a\<sharp>M \<Longrightarrow> a\<sharp>M'"
 apply -
 apply(induct rule: l_redu.induct)
-apply(auto simp add: abs_fresh rename_fresh)
+apply(auto simp: abs_fresh rename_fresh)
 apply(case_tac "xa=x")
 apply(simp add: rename_fresh)
 apply(simp add: rename_fresh fresh_atm)
 apply(simp add: fresh_prod abs_fresh abs_supp fin_supp)+
 apply(induct rule: l_redu.induct)
-apply(auto simp add: abs_fresh rename_fresh)
+apply(auto simp: abs_fresh rename_fresh)
 apply(case_tac "aa=a")
 apply(simp add: rename_fresh)
 apply(simp add: rename_fresh fresh_atm)
 apply(simp add: fresh_prod abs_fresh abs_supp fin_supp)+
 done
 
 lemma better_LAxR_intro[intro]:
   shows "fic M a \<Longrightarrow> Cut <a>.M (x).(Ax x b) \<longrightarrow>\<^sub>l M[a\<turnstile>c>b]"
 proof -
   assume fin: "fic M a"
   obtain x'::"name" where fs1: "x'\<sharp>(M,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(a,M,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.M (x).(Ax x b) =  Cut <a'>.([(a',a)]\<bullet>M) (x').(Ax x' b)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>l ([(a',a)]\<bullet>M)[a'\<turnstile>c>b]" using fs1 fs2 fin
     by (auto intro: l_redu.intros simp add: fresh_left calc_atm fic_rename)
   also have "\<dots> = M[a\<turnstile>c>b]" using fs1 fs2 by (simp add: crename_rename)
   finally show ?thesis by simp
 qed
     
 lemma better_LAxL_intro[intro]:
   shows "fin M x \<Longrightarrow> Cut <a>.(Ax y a) (x).M \<longrightarrow>\<^sub>l M[x\<turnstile>n>y]"
 proof -
   assume fin: "fin M x"
   obtain x'::"name" where fs1: "x'\<sharp>(y,M,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(a,M)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.(Ax y a) (x).M = Cut <a'>.(Ax y a') (x').([(x',x)]\<bullet>M)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>l ([(x',x)]\<bullet>M)[x'\<turnstile>n>y]" using fs1 fs2 fin
     by (auto intro: l_redu.intros simp add: fresh_left calc_atm fin_rename)
   also have "\<dots> = M[x\<turnstile>n>y]" using fs1 fs2 by (simp add: nrename_rename)
   finally show ?thesis by simp
 qed
 
 lemma better_LNot_intro[intro]:
   shows "\<lbrakk>y\<sharp>N; a\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) \<longrightarrow>\<^sub>l Cut <b>.N (x).M"
 proof -
   assume fs: "y\<sharp>N" "a\<sharp>M"
   obtain x'::"name" where f1: "x'\<sharp>(y,N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain y'::"name" where f2: "y'\<sharp>(y,N,M,x,x')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f3: "a'\<sharp>(a,M,N,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b'::"coname" where f4: "b'\<sharp>(a,M,N,b,a')" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) 
                       = Cut <a'>.(NotR (x).([(a',a)]\<bullet>M) a') (y').(NotL <b>.([(y',y)]\<bullet>N) y')"
     using f1 f2 f3 f4 
-    by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm abs_fresh)
+    by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm abs_fresh)
   also have "\<dots> = Cut <a'>.(NotR (x).M a') (y').(NotL <b>.N y')"
     using f1 f2 f3 f4 fs by (perm_simp)
   also have "\<dots> = Cut <a'>.(NotR (x').([(x',x)]\<bullet>M) a') (y').(NotL <b'>.([(b',b)]\<bullet>N) y')"
     using f1 f2 f3 f4 
-    by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <b'>.([(b',b)]\<bullet>N) (x').([(x',x)]\<bullet>M)"
     using f1 f2 f3 f4 fs
     by (auto intro:  l_redu.intros simp add: fresh_prod fresh_left calc_atm fresh_atm)
   also have "\<dots> = Cut <b>.N (x).M"
-    using f1 f2 f3 f4 by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using f1 f2 f3 f4 by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   finally show ?thesis by simp
 qed 
 
 lemma better_LAnd1_intro[intro]:
   shows "\<lbrakk>a\<sharp>([b1].M1,[b2].M2); y\<sharp>[x].N\<rbrakk> 
          \<Longrightarrow> Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL1 (x).N y) \<longrightarrow>\<^sub>l Cut <b1>.M1 (x).N"
 proof -
   assume fs: "a\<sharp>([b1].M1,[b2].M2)" "y\<sharp>[x].N"
   obtain x'::"name" where f1: "x'\<sharp>(y,N,M1,M2,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain y'::"name" where f2: "y'\<sharp>(y,N,M1,M2,x,x')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f3: "a'\<sharp>(a,M1,M2,N,b1,b2)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b1'::"coname" where f4:"b1'\<sharp>(a,M1,M2,N,b1,b2,a')" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b2'::"coname" where f5:"b2'\<sharp>(a,M1,M2,N,b1,b2,a',b1')" by (rule exists_fresh(2),rule fin_supp, blast)
   have "Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL1 (x).N y)
                       = Cut <a'>.(AndR <b1>.M1 <b2>.M2 a') (y').(AndL1 (x).N y')"
     using f1 f2 f3 f4 fs
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
-    apply(auto simp add: perm_fresh_fresh)
+    apply(auto simp: perm_fresh_fresh)
     done
   also have "\<dots> = Cut <a'>.(AndR <b1'>.([(b1',b1)]\<bullet>M1) <b2'>.([(b2',b2)]\<bullet>M2) a') 
                                                                (y').(AndL1 (x').([(x',x)]\<bullet>N) y')"
     using f1 f2 f3 f4 f5 fs 
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
     done
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <b1'>.([(b1',b1)]\<bullet>M1) (x').([(x',x)]\<bullet>N)"
     using f1 f2 f3 f4 f5 fs
     apply -
     apply(rule l_redu.intros)
-    apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
+    apply(auto simp: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
     done
   also have "\<dots> = Cut <b1>.M1 (x).N"
-    using f1 f2 f3 f4 f5 fs by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using f1 f2 f3 f4 f5 fs by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   finally show ?thesis by simp
 qed 
 
 lemma better_LAnd2_intro[intro]:
   shows "\<lbrakk>a\<sharp>([b1].M1,[b2].M2); y\<sharp>[x].N\<rbrakk> 
          \<Longrightarrow> Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL2 (x).N y) \<longrightarrow>\<^sub>l Cut <b2>.M2 (x).N"
 proof -
   assume fs: "a\<sharp>([b1].M1,[b2].M2)" "y\<sharp>[x].N"
   obtain x'::"name" where f1: "x'\<sharp>(y,N,M1,M2,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain y'::"name" where f2: "y'\<sharp>(y,N,M1,M2,x,x')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f3: "a'\<sharp>(a,M1,M2,N,b1,b2)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b1'::"coname" where f4:"b1'\<sharp>(a,M1,M2,N,b1,b2,a')" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b2'::"coname" where f5:"b2'\<sharp>(a,M1,M2,N,b1,b2,a',b1')" by (rule exists_fresh(2),rule fin_supp, blast)
   have "Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL2 (x).N y)
                       = Cut <a'>.(AndR <b1>.M1 <b2>.M2 a') (y').(AndL2 (x).N y')"
     using f1 f2 f3 f4 fs
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
-    apply(auto simp add: perm_fresh_fresh)
+    apply(auto simp: perm_fresh_fresh)
     done
   also have "\<dots> = Cut <a'>.(AndR <b1'>.([(b1',b1)]\<bullet>M1) <b2'>.([(b2',b2)]\<bullet>M2) a') 
                                                                (y').(AndL2 (x').([(x',x)]\<bullet>N) y')"
     using f1 f2 f3 f4 f5 fs 
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
     done
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <b2'>.([(b2',b2)]\<bullet>M2) (x').([(x',x)]\<bullet>N)"
     using f1 f2 f3 f4 f5 fs
     apply -
     apply(rule l_redu.intros)
-    apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
+    apply(auto simp: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
     done
   also have "\<dots> = Cut <b2>.M2 (x).N"
-    using f1 f2 f3 f4 f5 fs by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using f1 f2 f3 f4 f5 fs by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   finally show ?thesis by simp
 qed
 
 lemma better_LOr1_intro[intro]:
   shows "\<lbrakk>y\<sharp>([x1].N1,[x2].N2); b\<sharp>[a].M\<rbrakk> 
          \<Longrightarrow> Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^sub>l Cut <a>.M (x1).N1"
 proof -
   assume fs: "y\<sharp>([x1].N1,[x2].N2)" "b\<sharp>[a].M"
   obtain y'::"name" where f1: "y'\<sharp>(y,M,N1,N2,x1,x2)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain x1'::"name" where f2: "x1'\<sharp>(y,M,N1,N2,x1,x2,y')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain x2'::"name" where f3: "x2'\<sharp>(y,M,N1,N2,x1,x2,y',x1')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f4: "a'\<sharp>(a,N1,N2,M,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b'::"coname" where f5: "b'\<sharp>(a,N1,N2,M,b,a')" by (rule exists_fresh(2),rule fin_supp, blast)
   have "Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y)
                       = Cut <b'>.(OrR1 <a>.M b') (y').(OrL (x1).N1 (x2).N2 y')"
     using f1 f2 f3 f4 f5 fs
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
-    apply(auto simp add: perm_fresh_fresh)
+    apply(auto simp: perm_fresh_fresh)
     done
   also have "\<dots> = Cut <b'>.(OrR1 <a'>.([(a',a)]\<bullet>M) b') 
               (y').(OrL (x1').([(x1',x1)]\<bullet>N1) (x2').([(x2',x2)]\<bullet>N2) y')"   
     using f1 f2 f3 f4 f5 fs 
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
     done
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <a'>.([(a',a)]\<bullet>M) (x1').([(x1',x1)]\<bullet>N1)"
     using f1 f2 f3 f4 f5 fs
     apply -
     apply(rule l_redu.intros)
-    apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
+    apply(auto simp: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
     done
   also have "\<dots> = Cut <a>.M (x1).N1"
-    using f1 f2 f3 f4 f5 fs by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using f1 f2 f3 f4 f5 fs by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   finally show ?thesis by simp
 qed
 
 lemma better_LOr2_intro[intro]:
   shows "\<lbrakk>y\<sharp>([x1].N1,[x2].N2); b\<sharp>[a].M\<rbrakk> 
          \<Longrightarrow> Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^sub>l Cut <a>.M (x2).N2"
 proof -
   assume fs: "y\<sharp>([x1].N1,[x2].N2)" "b\<sharp>[a].M"
   obtain y'::"name" where f1: "y'\<sharp>(y,M,N1,N2,x1,x2)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain x1'::"name" where f2: "x1'\<sharp>(y,M,N1,N2,x1,x2,y')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain x2'::"name" where f3: "x2'\<sharp>(y,M,N1,N2,x1,x2,y',x1')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f4: "a'\<sharp>(a,N1,N2,M,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b'::"coname" where f5: "b'\<sharp>(a,N1,N2,M,b,a')" by (rule exists_fresh(2),rule fin_supp, blast)
   have "Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y)
                       = Cut <b'>.(OrR2 <a>.M b') (y').(OrL (x1).N1 (x2).N2 y')"
     using f1 f2 f3 f4 f5 fs
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
-    apply(auto simp add: perm_fresh_fresh)
+    apply(auto simp: perm_fresh_fresh)
     done
   also have "\<dots> = Cut <b'>.(OrR2 <a'>.([(a',a)]\<bullet>M) b') 
               (y').(OrL (x1').([(x1',x1)]\<bullet>N1) (x2').([(x2',x2)]\<bullet>N2) y')"   
     using f1 f2 f3 f4 f5 fs 
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
     done
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <a'>.([(a',a)]\<bullet>M) (x2').([(x2',x2)]\<bullet>N2)"
     using f1 f2 f3 f4 f5 fs
     apply -
     apply(rule l_redu.intros)
-    apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
+    apply(auto simp: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
     done
   also have "\<dots> = Cut <a>.M (x2).N2"
-    using f1 f2 f3 f4 f5 fs by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using f1 f2 f3 f4 f5 fs by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   finally show ?thesis by simp
 qed 
 
 lemma better_LImp_intro[intro]:
   shows "\<lbrakk>z\<sharp>(N,[y].P); b\<sharp>[a].M; a\<sharp>N\<rbrakk> 
          \<Longrightarrow> Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z) \<longrightarrow>\<^sub>l Cut <a>.(Cut <c>.N (x).M) (y).P"
 proof -
   assume fs: "z\<sharp>(N,[y].P)" "b\<sharp>[a].M" "a\<sharp>N"
   obtain y'::"name" where f1: "y'\<sharp>(y,M,N,P,z,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain x'::"name" where f2: "x'\<sharp>(y,M,N,P,z,x,y')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain z'::"name" where f3: "z'\<sharp>(y,M,N,P,z,x,y',x')" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where f4: "a'\<sharp>(a,N,P,M,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain b'::"coname" where f5: "b'\<sharp>(a,N,P,M,b,c,a')" by (rule exists_fresh(2),rule fin_supp, blast)
   obtain c'::"coname" where f6: "c'\<sharp>(a,N,P,M,b,c,a',b')" by (rule exists_fresh(2),rule fin_supp, blast)
   have " Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z)
                       =  Cut <b'>.(ImpR (x).<a>.M b') (z').(ImpL <c>.N (y).P z')"
     using f1 f2 f3 f4 f5 fs
     apply(rule_tac sym)
     apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh)
-    apply(auto simp add: perm_fresh_fresh)
+    apply(auto simp: perm_fresh_fresh)
     done
   also have "\<dots> = Cut <b'>.(ImpR (x').<a'>.([(a',a)]\<bullet>([(x',x)]\<bullet>M)) b') 
                            (z').(ImpL <c'>.([(c',c)]\<bullet>N) (y').([(y',y)]\<bullet>P) z')"
     using f1 f2 f3 f4 f5 f6 fs 
     apply(rule_tac sym)
     apply(simp add: trm.inject)
     apply(simp add: alpha)
     apply(rule conjI)
     apply(simp add: trm.inject)
     apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh)
     apply(simp add: trm.inject)
     apply(simp add: alpha)
     apply(rule conjI)
     apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh)
     apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh)
     done
   also have "\<dots> \<longrightarrow>\<^sub>l Cut <a'>.(Cut <c'>.([(c',c)]\<bullet>N) (x').([(a',a)]\<bullet>[(x',x)]\<bullet>M)) (y').([(y',y)]\<bullet>P)"
     using f1 f2 f3 f4 f5 f6 fs
     apply -
     apply(rule l_redu.intros)
-    apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
+    apply(auto simp: abs_fresh fresh_prod fresh_left calc_atm fresh_atm)
     done
   also have "\<dots> = Cut <a>.(Cut <c>.N (x).M) (y).P"
     using f1 f2 f3 f4 f5 f6 fs 
     apply(simp add: trm.inject)
     apply(rule conjI)
     apply(simp add: alpha)
     apply(rule disjI2)
     apply(simp add: trm.inject)
     apply(rule conjI)
     apply(simp add: fresh_prod fresh_atm)
     apply(rule conjI)
     apply(perm_simp add: calc_atm)
-    apply(auto simp add: fresh_prod fresh_atm)[1]
+    apply(auto simp: fresh_prod fresh_atm)[1]
     apply(perm_simp add: alpha)
     apply(perm_simp add: alpha)
     apply(perm_simp add: alpha)
     apply(rule conjI)
     apply(perm_simp add: calc_atm)
     apply(rule_tac pi="[(a',a)]" in pt_bij4[OF pt_coname_inst, OF at_coname_inst])
     apply(perm_simp add: abs_perm calc_atm)
     apply(perm_simp add: alpha fresh_prod fresh_atm)
     apply(simp add: abs_fresh)
     apply(perm_simp add: alpha fresh_prod fresh_atm)
     done
   finally show ?thesis by simp
 qed 
 
 lemma alpha_coname:
   fixes M::"trm"
   and   a::"coname"
   assumes a: "[a].M = [b].N" "c\<sharp>(a,b,M,N)"
   shows "M = [(a,c)]\<bullet>[(b,c)]\<bullet>N"
 using a
-apply(auto simp add: alpha_fresh fresh_prod fresh_atm)
+apply(auto simp: alpha_fresh fresh_prod fresh_atm)
 apply(drule sym)
 apply(perm_simp)
 done 
 
 lemma alpha_name:
   fixes M::"trm"
   and   x::"name"
   assumes a: "[x].M = [y].N" "z\<sharp>(x,y,M,N)"
   shows "M = [(x,z)]\<bullet>[(y,z)]\<bullet>N"
 using a
-apply(auto simp add: alpha_fresh fresh_prod fresh_atm)
+apply(auto simp: alpha_fresh fresh_prod fresh_atm)
 apply(drule sym)
 apply(perm_simp)
 done 
 
 lemma alpha_name_coname:
   fixes M::"trm"
   and   x::"name"
   and   a::"coname"
   assumes a: "[x].[b].M = [y].[c].N" "z\<sharp>(x,y,M,N)" "a\<sharp>(b,c,M,N)"
   shows "M = [(x,z)]\<bullet>[(b,a)]\<bullet>[(c,a)]\<bullet>[(y,z)]\<bullet>N"
 using a
-apply(auto simp add: alpha_fresh fresh_prod fresh_atm 
+apply(auto simp: alpha_fresh fresh_prod fresh_atm 
                      abs_supp fin_supp abs_fresh abs_perm fresh_left calc_atm)
 apply(drule sym)
 apply(simp)
 apply(perm_simp)
 done 
 
 lemma Cut_l_redu_elim:
   assumes a: "Cut <a>.M (x).N \<longrightarrow>\<^sub>l R"
   shows "(\<exists>b. R = M[a\<turnstile>c>b]) \<or> (\<exists>y. R = N[x\<turnstile>n>y]) \<or>
   (\<exists>y M' b N'. M = NotR (y).M' a \<and> N = NotL <b>.N' x \<and> R = Cut <b>.N' (y).M' \<and> fic M a \<and> fin N x) \<or>
   (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL1 (y).N' x \<and> R = Cut <b>.M1 (y).N' 
                                                                             \<and> fic M a \<and> fin N x) \<or>
   (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL2 (y).N' x \<and> R = Cut <c>.M2 (y).N' 
                                                                             \<and> fic M a \<and> fin N x) \<or>
   (\<exists>b N' z M1 y M2. M = OrR1 <b>.N' a \<and> N = OrL (z).M1 (y).M2 x \<and> R = Cut <b>.N' (z).M1 
                                                                             \<and> fic M a \<and> fin N x) \<or>
   (\<exists>b N' z M1 y M2. M = OrR2 <b>.N' a \<and> N = OrL (z).M1 (y).M2 x \<and> R = Cut <b>.N' (y).M2 
                                                                             \<and> fic M a \<and> fin N x) \<or>
   (\<exists>z b M' c N1 y N2. M = ImpR (z).<b>.M' a \<and> N = ImpL <c>.N1 (y).N2 x \<and> 
             R = Cut <b>.(Cut <c>.N1 (z).M') (y).N2 \<and> b\<sharp>(c,N1) \<and> fic M a \<and> fin N x)"
 using a
 apply(erule_tac l_redu.cases)
 apply(rule disjI1)
 (* ax case *)
 apply(simp add: trm.inject)
 apply(rule_tac x="b" in exI)
 apply(erule conjE)
 apply(simp add: alpha)
 apply(erule disjE)
 apply(simp)
 apply(simp)
 apply(simp add: rename_fresh)
 apply(rule disjI2)
 apply(rule disjI1)
 (* ax case *)
 apply(simp add: trm.inject)
 apply(rule_tac x="y" in exI)
 apply(erule conjE)
 apply(thin_tac "[a].M = [aa].Ax y aa")
 apply(simp add: alpha)
 apply(erule disjE)
 apply(simp)
 apply(simp)
 apply(simp add: rename_fresh)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI1)
 (* not case *)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="c" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp add: calc_atm)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: calc_atm abs_fresh fresh_prod fresh_atm fresh_left)
 apply(auto)[1]
 apply(drule_tac z="ca" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp add: calc_atm)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule refl)
-apply(auto simp add: calc_atm abs_fresh fresh_left)[1]
+apply(auto simp: calc_atm abs_fresh fresh_left)[1]
 apply(case_tac "y=x")
 apply(perm_simp)
 apply(perm_simp)
 apply(case_tac "aa=a")
 apply(perm_simp)
 apply(perm_simp)
 (* and1 case *)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI1)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="c" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule exI)+
 apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="ca" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh split: if_splits)[1]
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 (* and2 case *)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI1)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="c" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="ca" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh split: if_splits)[1]
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 (* or1 case *)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI1)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="c" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="ca" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule exI)+
 apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule exI)+
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 (* or2 case *)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI1)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="c" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="ca" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 (* imp-case *)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(rule disjI2)
 apply(simp add: trm.inject)
 apply(erule conjE)+
 apply(generate_fresh "coname")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac c="ca" in  alpha_coname)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="a" and t="[(a,ca)]\<bullet>[(b,ca)]\<bullet>b" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cb" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(z,cb)]\<bullet>z" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 apply(generate_fresh "name")
 apply(simp add: abs_fresh fresh_prod fresh_atm)
 apply(auto)[1]
 apply(drule_tac z="cc" in  alpha_name)
 apply(simp add: fresh_prod fresh_atm abs_fresh)
 apply(simp)
 apply(rule exI)+
 apply(rule conjI)
 apply(rule_tac s="x" and t="[(x,cc)]\<bullet>[(z,cc)]\<bullet>z" in subst)
 apply(simp add: calc_atm)
 apply(rule refl)
-apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
+apply(auto simp: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1]
 apply(perm_simp)+
 done
 
 inductive
   c_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^sub>c _" [100,100] 100)
 where
   left[intro]:  "\<lbrakk>\<not>fic M a; a\<sharp>N; x\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>c M{a:=(x).N}"
 | right[intro]: "\<lbrakk>\<not>fin N x; a\<sharp>N; x\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>c N{x:=<a>.M}"
 
 equivariance c_redu
 
 nominal_inductive c_redu
  by (simp_all add: abs_fresh subst_fresh)
 
 lemma better_left[intro]:
   shows "\<not>fic M a \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>c M{a:=(x).N}"
 proof -
   assume not_fic: "\<not>fic M a"
   obtain x'::"name" where fs1: "x'\<sharp>(N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(a,M,N)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.M (x).N =  Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>c ([(a',a)]\<bullet>M){a':=(x').([(x',x)]\<bullet>N)}" using fs1 fs2 not_fic
     apply -
     apply(rule left)
     apply(clarify)
     apply(drule_tac a'="a" in fic_rename)
     apply(simp add: perm_swap)
     apply(simp add: fresh_left calc_atm)+
     done
   also have "\<dots> = M{a:=(x).N}" using fs1 fs2
     by (simp add: subst_rename[symmetric] fresh_atm fresh_prod fresh_left calc_atm)
   finally show ?thesis by simp
 qed
 
 lemma better_right[intro]:
   shows "\<not>fin N x \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>c N{x:=<a>.M}"
 proof -
   assume not_fin: "\<not>fin N x"
   obtain x'::"name" where fs1: "x'\<sharp>(N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(a,M,N)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.M (x).N =  Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>c ([(x',x)]\<bullet>N){x':=<a'>.([(a',a)]\<bullet>M)}" using fs1 fs2 not_fin
     apply -
     apply(rule right)
     apply(clarify)
     apply(drule_tac x'="x" in fin_rename)
     apply(simp add: perm_swap)
     apply(simp add: fresh_left calc_atm)+
     done
   also have "\<dots> = N{x:=<a>.M}" using fs1 fs2
     by (simp add: subst_rename[symmetric] fresh_atm fresh_prod fresh_left calc_atm)
   finally show ?thesis by simp
 qed
 
 lemma fresh_c_redu:
   fixes x::"name"
   and   c::"coname"
   shows "M \<longrightarrow>\<^sub>c M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'"
   and   "M \<longrightarrow>\<^sub>c M' \<Longrightarrow> c\<sharp>M \<Longrightarrow> c\<sharp>M'"
 apply -
 apply(induct rule: c_redu.induct)
-apply(auto simp add: abs_fresh rename_fresh subst_fresh)
+apply(auto simp: abs_fresh rename_fresh subst_fresh)
 apply(induct rule: c_redu.induct)
-apply(auto simp add: abs_fresh rename_fresh subst_fresh)
+apply(auto simp: abs_fresh rename_fresh subst_fresh)
 done
 
 inductive
   a_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^sub>a _" [100,100] 100)
 where
   al_redu[intro]: "M\<longrightarrow>\<^sub>l M' \<Longrightarrow> M \<longrightarrow>\<^sub>a M'"
 | ac_redu[intro]: "M\<longrightarrow>\<^sub>c M' \<Longrightarrow> M \<longrightarrow>\<^sub>a M'"
 | a_Cut_l: "\<lbrakk>a\<sharp>N; x\<sharp>M; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a Cut <a>.M' (x).N"
 | a_Cut_r: "\<lbrakk>a\<sharp>N; x\<sharp>M; N\<longrightarrow>\<^sub>a N'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a Cut <a>.M (x).N'"
 | a_NotL[intro]: "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> NotL <a>.M x \<longrightarrow>\<^sub>a NotL <a>.M' x"
 | a_NotR[intro]: "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> NotR (x).M a \<longrightarrow>\<^sub>a NotR (x).M' a"
 | a_AndR_l: "\<lbrakk>a\<sharp>(N,c); b\<sharp>(M,c); b\<noteq>a; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a AndR <a>.M' <b>.N c"
 | a_AndR_r: "\<lbrakk>a\<sharp>(N,c); b\<sharp>(M,c); b\<noteq>a; N\<longrightarrow>\<^sub>a N'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a AndR <a>.M <b>.N' c"
 | a_AndL1: "\<lbrakk>x\<sharp>y; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^sub>a AndL1 (x).M' y"
 | a_AndL2: "\<lbrakk>x\<sharp>y; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^sub>a AndL2 (x).M' y"
 | a_OrL_l: "\<lbrakk>x\<sharp>(N,z); y\<sharp>(M,z); y\<noteq>x; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a OrL (x).M' (y).N z"
 | a_OrL_r: "\<lbrakk>x\<sharp>(N,z); y\<sharp>(M,z); y\<noteq>x; N\<longrightarrow>\<^sub>a N'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a OrL (x).M (y).N' z"
 | a_OrR1: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^sub>a OrR1 <a>.M' b"
 | a_OrR2: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^sub>a OrR2 <a>.M' b"
 | a_ImpL_l: "\<lbrakk>a\<sharp>N; x\<sharp>(M,y); M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^sub>a ImpL <a>.M' (x).N y"
 | a_ImpL_r: "\<lbrakk>a\<sharp>N; x\<sharp>(M,y); N\<longrightarrow>\<^sub>a N'\<rbrakk> \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^sub>a ImpL <a>.M (x).N' y"
 | a_ImpR: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^sub>a M'\<rbrakk> \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^sub>a ImpR (x).<a>.M' b"
 
 lemma fresh_a_redu:
   fixes x::"name"
   and   c::"coname"
   shows "M \<longrightarrow>\<^sub>a M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'"
   and   "M \<longrightarrow>\<^sub>a M' \<Longrightarrow> c\<sharp>M \<Longrightarrow> c\<sharp>M'"
 apply -
 apply(induct rule: a_redu.induct)
 apply(simp add: fresh_l_redu)
 apply(simp add: fresh_c_redu)
-apply(auto simp add: abs_fresh abs_supp fin_supp)
+apply(auto simp: abs_fresh abs_supp fin_supp)
 apply(induct rule: a_redu.induct)
 apply(simp add: fresh_l_redu)
 apply(simp add: fresh_c_redu)
-apply(auto simp add: abs_fresh abs_supp fin_supp)
+apply(auto simp: abs_fresh abs_supp fin_supp)
 done
 
 equivariance a_redu
 
 nominal_inductive a_redu
   by (simp_all add: abs_fresh fresh_atm fresh_prod abs_supp fin_supp fresh_a_redu)
 
 lemma better_CutL_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a Cut <a>.M' (x).N"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.M (x).N =  Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  Cut <a'>.([(a',a)]\<bullet>M') (x').([(x',x)]\<bullet>N)" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt)
   also have "\<dots> = Cut <a>.M' (x).N" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_CutR_intro[intro]:
   shows "N\<longrightarrow>\<^sub>a N' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a Cut <a>.M (x).N'"
 proof -
   assume red: "N\<longrightarrow>\<^sub>a N'"
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "Cut <a>.M (x).N =  Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N')" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt)
   also have "\<dots> = Cut <a>.M (x).N'" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
     
 lemma better_AndRL_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a AndR <a>.M' <b>.N c"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain b'::"coname" where fs1: "b'\<sharp>(M,N,a,b,c)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,b,c,b')" by (rule exists_fresh(2), rule fin_supp, blast)
   have "AndR <a>.M <b>.N c =  AndR <a'>.([(a',a)]\<bullet>M) <b'>.([(b',b)]\<bullet>N) c"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  AndR <a'>.([(a',a)]\<bullet>M') <b'>.([(b',b)]\<bullet>N) c" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = AndR <a>.M' <b>.N c" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_AndRR_intro[intro]:
   shows "N\<longrightarrow>\<^sub>a N' \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a AndR <a>.M <b>.N' c"
 proof -
   assume red: "N\<longrightarrow>\<^sub>a N'"
   obtain b'::"coname" where fs1: "b'\<sharp>(M,N,a,b,c)" by (rule exists_fresh(2), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,b,c,b')" by (rule exists_fresh(2), rule fin_supp, blast)
   have "AndR <a>.M <b>.N c =  AndR <a'>.([(a',a)]\<bullet>M) <b'>.([(b',b)]\<bullet>N) c"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  AndR <a'>.([(a',a)]\<bullet>M) <b'>.([(b',b)]\<bullet>N') c" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = AndR <a>.M <b>.N' c" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_AndL1_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^sub>a AndL1 (x).M' y"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain x'::"name" where fs1: "x'\<sharp>(M,y,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   have "AndL1 (x).M y = AndL1 (x').([(x',x)]\<bullet>M) y"
-    using fs1 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a AndL1 (x').([(x',x)]\<bullet>M') y" using fs1 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = AndL1 (x).M' y" 
-    using fs1 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_AndL2_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^sub>a AndL2 (x).M' y"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain x'::"name" where fs1: "x'\<sharp>(M,y,x)" by (rule exists_fresh(1), rule fin_supp, blast)
   have "AndL2 (x).M y = AndL2 (x').([(x',x)]\<bullet>M) y"
-    using fs1 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a AndL2 (x').([(x',x)]\<bullet>M') y" using fs1 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = AndL2 (x).M' y" 
-    using fs1 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_OrLL_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a OrL (x).M' (y).N z"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y,z)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain y'::"name" where fs2: "y'\<sharp>(M,N,x,y,z,x')" by (rule exists_fresh(1), rule fin_supp, blast)
   have "OrL (x).M (y).N z =  OrL (x').([(x',x)]\<bullet>M) (y').([(y',y)]\<bullet>N) z"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a OrL (x').([(x',x)]\<bullet>M') (y').([(y',y)]\<bullet>N) z" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = OrL (x).M' (y).N z" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_OrLR_intro[intro]:
   shows "N\<longrightarrow>\<^sub>a N' \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a OrL (x).M (y).N' z"
 proof -
   assume red: "N\<longrightarrow>\<^sub>a N'"
   obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y,z)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain y'::"name" where fs2: "y'\<sharp>(M,N,x,y,z,x')" by (rule exists_fresh(1), rule fin_supp, blast)
   have "OrL (x).M (y).N z =  OrL (x').([(x',x)]\<bullet>M) (y').([(y',y)]\<bullet>N) z"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a OrL (x').([(x',x)]\<bullet>M) (y').([(y',y)]\<bullet>N') z" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = OrL (x).M (y).N' z" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_OrR1_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^sub>a OrR1 <a>.M' b"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain a'::"coname" where fs1: "a'\<sharp>(M,b,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "OrR1 <a>.M b = OrR1 <a'>.([(a',a)]\<bullet>M) b"
-    using fs1 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a OrR1 <a'>.([(a',a)]\<bullet>M') b" using fs1 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = OrR1 <a>.M' b" 
-    using fs1 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_OrR2_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^sub>a OrR2 <a>.M' b"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain a'::"coname" where fs1: "a'\<sharp>(M,b,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "OrR2 <a>.M b = OrR2 <a'>.([(a',a)]\<bullet>M) b"
-    using fs1 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a OrR2 <a'>.([(a',a)]\<bullet>M') b" using fs1 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = OrR2 <a>.M' b" 
-    using fs1 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_ImpLL_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^sub>a ImpL <a>.M' (x).N y"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,x,y)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "ImpL <a>.M (x).N y =  ImpL <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N) y"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  ImpL <a'>.([(a',a)]\<bullet>M') (x').([(x',x)]\<bullet>N) y" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = ImpL <a>.M' (x).N y" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_ImpLR_intro[intro]:
   shows "N\<longrightarrow>\<^sub>a N' \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^sub>a ImpL <a>.M (x).N' y"
 proof -
   assume red: "N\<longrightarrow>\<^sub>a N'"
   obtain x'::"name"   where fs1: "x'\<sharp>(M,N,x,y)" by (rule exists_fresh(1), rule fin_supp, blast)
   obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "ImpL <a>.M (x).N y =  ImpL <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N) y"
-    using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a  ImpL <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N') y" using fs1 fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = ImpL <a>.M (x).N' y" 
-    using fs1 fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs1 fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 lemma better_ImpR_intro[intro]:
   shows "M\<longrightarrow>\<^sub>a M' \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^sub>a ImpR (x).<a>.M' b"
 proof -
   assume red: "M\<longrightarrow>\<^sub>a M'"
   obtain a'::"coname" where fs2: "a'\<sharp>(M,a,b)" by (rule exists_fresh(2), rule fin_supp, blast)
   have "ImpR (x).<a>.M b = ImpR (x).<a'>.([(a',a)]\<bullet>M) b"
-    using fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm)
+    using fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a ImpR (x).<a'>.([(a',a)]\<bullet>M') b" using fs2 red
     by (auto intro: a_redu.intros simp add: fresh_left calc_atm a_redu.eqvt fresh_atm fresh_prod)
   also have "\<dots> = ImpR (x).<a>.M' b" 
-    using fs2 red by (auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
+    using fs2 red by (auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm fresh_a_redu)
   finally show ?thesis by simp
 qed
 
 text \<open>axioms do not reduce\<close>
 
 lemma ax_do_not_l_reduce:
   shows "Ax x a \<longrightarrow>\<^sub>l M \<Longrightarrow> False"
 by (erule_tac l_redu.cases) (simp_all add: trm.inject)
  
 lemma ax_do_not_c_reduce:
   shows "Ax x a \<longrightarrow>\<^sub>c M \<Longrightarrow> False"
 by (erule_tac c_redu.cases) (simp_all add: trm.inject)
 
 lemma ax_do_not_a_reduce:
   shows "Ax x a \<longrightarrow>\<^sub>a M \<Longrightarrow> False"
 apply(erule_tac a_redu.cases) 
-apply(auto simp add: trm.inject)
+apply(auto simp: trm.inject)
 apply(drule ax_do_not_l_reduce)
 apply(simp)
 apply(drule ax_do_not_c_reduce)
 apply(simp)
 done
 
 lemma a_redu_NotL_elim:
   assumes a: "NotL <a>.M x \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = NotL <a>.M' x \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 2)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_NotR_elim:
   assumes a: "NotR (x).M a \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = NotR (x).M' a \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 2)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_AndR_elim:
   assumes a: "AndR <a>.M <b>.N c\<longrightarrow>\<^sub>a R"
   shows "(\<exists>M'. R = AndR <a>.M' <b>.N c \<and> M\<longrightarrow>\<^sub>aM') \<or> (\<exists>N'. R = AndR <a>.M <b>.N' c \<and> N\<longrightarrow>\<^sub>aN')"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rotate_tac 6)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rotate_tac 6)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(b,baa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 done
 
 lemma a_redu_AndL1_elim:
   assumes a: "AndL1 (x).M y \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = AndL1 (x).M' y \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 3)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_AndL2_elim:
   assumes a: "AndL2 (x).M y \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = AndL2 (x).M' y \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 3)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_OrL_elim:
   assumes a: "OrL (x).M (y).N z\<longrightarrow>\<^sub>a R"
   shows "(\<exists>M'. R = OrL (x).M' (y).N z \<and> M\<longrightarrow>\<^sub>aM') \<or> (\<exists>N'. R = OrL (x).M (y).N' z \<and> N\<longrightarrow>\<^sub>aN')"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rotate_tac 6)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rotate_tac 6)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,yaa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 done
 
 lemma a_redu_OrR1_elim:
   assumes a: "OrR1 <a>.M b \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = OrR1 <a>.M' b \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 3)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_OrR2_elim:
   assumes a: "OrR2 <a>.M b \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = OrR2 <a>.M' b \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 3)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt)
+apply(auto simp: alpha a_redu.eqvt)
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)
 apply(simp add: perm_swap)
 done
 
 lemma a_redu_ImpL_elim:
   assumes a: "ImpL <a>.M (y).N z\<longrightarrow>\<^sub>a R"
   shows "(\<exists>M'. R = ImpL <a>.M' (y).N z \<and> M\<longrightarrow>\<^sub>aM') \<or> (\<exists>N'. R = ImpL <a>.M (y).N' z \<and> N\<longrightarrow>\<^sub>aN')"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rotate_tac 5)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rotate_tac 5)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(rule disjI1)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule disjI2)
-apply(auto simp add: alpha a_redu.eqvt)[1]
+apply(auto simp: alpha a_redu.eqvt)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) 
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
-apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
-apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
+apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI)
+apply(auto simp: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1]
 done
 
 lemma a_redu_ImpR_elim:
   assumes a: "ImpR (x).<a>.M b \<longrightarrow>\<^sub>a R"
   shows "\<exists>M'. R = ImpR (x).<a>.M' b \<and> M\<longrightarrow>\<^sub>aM'"
 using a [[simproc del: defined_all]]
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
 apply(auto)
 apply(rotate_tac 3)
 apply(erule_tac a_redu.cases, simp_all add: trm.inject)
 apply(erule_tac l_redu.cases, simp_all add: trm.inject)
 apply(erule_tac c_redu.cases, simp_all add: trm.inject)
-apply(auto simp add: alpha a_redu.eqvt abs_perm abs_fresh)
-apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
-apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: alpha a_redu.eqvt abs_perm abs_fresh)
 apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(a,aa)]\<bullet>[(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule sym)
 apply(rule trans)
 apply(rule perm_compose)
 apply(simp add: calc_atm perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>[(x,xa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule sym)
 apply(rule trans)
 apply(rule perm_compose)
 apply(simp add: calc_atm perm_swap)
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule_tac x="([(a,aa)]\<bullet>[(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule sym)
 apply(rule trans)
 apply(rule perm_compose)
 apply(simp add: calc_atm perm_swap)
 apply(rule_tac x="([(a,aaa)]\<bullet>[(x,xaa)]\<bullet>M'a)" in exI)
-apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
+apply(auto simp: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap)
 apply(rule sym)
 apply(rule trans)
 apply(rule perm_compose)
 apply(simp add: calc_atm perm_swap)
 done
 
 text \<open>Transitive Closure\<close>
 
 abbreviation
  a_star_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^sub>a* _" [100,100] 100)
 where
   "M \<longrightarrow>\<^sub>a* M' \<equiv> (a_redu)\<^sup>*\<^sup>* M M'"
 
 lemma a_starI:
   assumes a: "M \<longrightarrow>\<^sub>a M'"
   shows "M \<longrightarrow>\<^sub>a* M'"
 using a by blast
 
 lemma a_starE:
   assumes a: "M \<longrightarrow>\<^sub>a* M'"
   shows "M = M' \<or> (\<exists>N. M \<longrightarrow>\<^sub>a N \<and> N \<longrightarrow>\<^sub>a* M')"
 using a 
 by (induct) (auto)
 
 lemma a_star_refl:
   shows "M \<longrightarrow>\<^sub>a* M"
   by blast
 
 lemma a_star_trans[trans]:
   assumes a1: "M1\<longrightarrow>\<^sub>a* M2"
   and     a2: "M2\<longrightarrow>\<^sub>a* M3"
   shows "M1 \<longrightarrow>\<^sub>a* M3"
 using a2 a1
 by (induct) (auto)
 
 text \<open>congruence rules for \<open>\<longrightarrow>\<^sub>a*\<close>\<close>
 
 lemma ax_do_not_a_star_reduce:
   shows "Ax x a \<longrightarrow>\<^sub>a* M \<Longrightarrow> M = Ax x a"
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule  ax_do_not_a_reduce)
 apply(simp)
 done
 
 
 lemma a_star_CutL:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a* Cut <a>.M' (x).N"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_CutR:
     "N \<longrightarrow>\<^sub>a* N'\<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a* Cut <a>.M (x).N'"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_Cut:
     "\<lbrakk>M \<longrightarrow>\<^sub>a* M'; N \<longrightarrow>\<^sub>a* N'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^sub>a* Cut <a>.M' (x).N'"
 by (blast intro!: a_star_CutL a_star_CutR intro: rtranclp_trans)
 
 lemma a_star_NotR:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> NotR (x).M a \<longrightarrow>\<^sub>a* NotR (x).M' a"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_NotL:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> NotL <a>.M x \<longrightarrow>\<^sub>a* NotL <a>.M' x"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_AndRL:
     "M \<longrightarrow>\<^sub>a* M'\<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a* AndR <a>.M' <b>.N c"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_AndRR:
     "N \<longrightarrow>\<^sub>a* N'\<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a* AndR <a>.M <b>.N' c"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_AndR:
     "\<lbrakk>M \<longrightarrow>\<^sub>a* M'; N \<longrightarrow>\<^sub>a* N'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a* AndR <a>.M' <b>.N' c"
 by (blast intro!: a_star_AndRL a_star_AndRR intro: rtranclp_trans)
 
 lemma a_star_AndL1:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^sub>a* AndL1 (x).M' y"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_AndL2:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^sub>a* AndL2 (x).M' y"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_OrLL:
     "M \<longrightarrow>\<^sub>a* M'\<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a* OrL (x).M' (y).N z"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_OrLR:
     "N \<longrightarrow>\<^sub>a* N'\<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a* OrL (x).M (y).N' z"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_OrL:
     "\<lbrakk>M \<longrightarrow>\<^sub>a* M'; N \<longrightarrow>\<^sub>a* N'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^sub>a* OrL (x).M' (y).N' z"
 by (blast intro!: a_star_OrLL a_star_OrLR intro: rtranclp_trans)
 
 lemma a_star_OrR1:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^sub>a* OrR1 <a>.M' b"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_OrR2:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^sub>a* OrR2 <a>.M' b"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_ImpLL:
     "M \<longrightarrow>\<^sub>a* M'\<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^sub>a* ImpL <a>.M' (y).N z"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_ImpLR:
     "N \<longrightarrow>\<^sub>a* N'\<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^sub>a* ImpL <a>.M (y).N' z"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemma a_star_ImpL:
     "\<lbrakk>M \<longrightarrow>\<^sub>a* M'; N \<longrightarrow>\<^sub>a* N'\<rbrakk> \<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^sub>a* ImpL <a>.M' (y).N' z"
 by (blast intro!: a_star_ImpLL a_star_ImpLR intro: rtranclp_trans)
 
 lemma a_star_ImpR:
     "M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^sub>a* ImpR (x).<a>.M' b"
 by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
 
 lemmas a_star_congs = a_star_Cut a_star_NotR a_star_NotL a_star_AndR a_star_AndL1 a_star_AndL2
                       a_star_OrL a_star_OrR1 a_star_OrR2 a_star_ImpL a_star_ImpR
 
 lemma a_star_redu_NotL_elim:
   assumes a: "NotL <a>.M x \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = NotL <a>.M' x \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_NotL_elim)
 apply(auto)
 done
 
 lemma a_star_redu_NotR_elim:
   assumes a: "NotR (x).M a \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = NotR (x).M' a \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_NotR_elim)
 apply(auto)
 done
 
 lemma a_star_redu_AndR_elim:
   assumes a: "AndR <a>.M <b>.N c\<longrightarrow>\<^sub>a* R"
   shows "(\<exists>M' N'. R = AndR <a>.M' <b>.N' c \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N')"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_AndR_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_AndL1_elim:
   assumes a: "AndL1 (x).M y \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = AndL1 (x).M' y \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_AndL1_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_AndL2_elim:
   assumes a: "AndL2 (x).M y \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = AndL2 (x).M' y \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_AndL2_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_OrL_elim:
   assumes a: "OrL (x).M (y).N z \<longrightarrow>\<^sub>a* R"
   shows "(\<exists>M' N'. R = OrL (x).M' (y).N' z \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N')"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_OrL_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_OrR1_elim:
   assumes a: "OrR1 <a>.M y \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = OrR1 <a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_OrR1_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_OrR2_elim:
   assumes a: "OrR2 <a>.M y \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = OrR2 <a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_OrR2_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_ImpR_elim:
   assumes a: "ImpR (x).<a>.M y \<longrightarrow>\<^sub>a* R"
   shows "\<exists>M'. R = ImpR (x).<a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_ImpR_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 lemma a_star_redu_ImpL_elim:
   assumes a: "ImpL <a>.M (y).N z \<longrightarrow>\<^sub>a* R"
   shows "(\<exists>M' N'. R = ImpL <a>.M' (y).N' z \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N')"
 using a
 apply(induct set: rtranclp)
 apply(auto)
 apply(drule a_redu_ImpL_elim)
-apply(auto simp add: alpha trm.inject)
+apply(auto simp: alpha trm.inject)
 done
 
 text \<open>Substitution\<close>
 
 lemma subst_not_fin1:
   shows "\<not>fin(M{x:=<c>.P}) x"
 apply(nominal_induct M avoiding: x c P rule: trm.strong_induct)
 apply(auto)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},P,name1,trm2{x:=<c>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(erule fin.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(erule fin.cases, simp_all add: trm.inject)
 done
 
 lemma subst_not_fin2:
   assumes a: "\<not>fin M y"
   shows "\<not>fin(M{c:=(x).P}) y" 
 using a
 apply(nominal_induct M avoiding: x c P y rule: trm.strong_induct)
 apply(auto)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(drule fin_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm1{coname3:=(x).P},P,coname1,trm2{coname3:=(x).P},coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(drule fin_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros abs_fresh)
 apply(drule fin_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros abs_fresh)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(drule fin_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(drule fin_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshn_after_substc)
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros abs_fresh)
 apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(drule fin_elims, simp)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(drule fin_elims, simp)
 apply(drule fin_elims, simp)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshn_after_substc)
 apply(drule freshn_after_substc)
 apply(simp add: fin.intros abs_fresh)
 done
 
 lemma subst_not_fic1:
   shows "\<not>fic (M{a:=(x).P}) a"
 apply(nominal_induct M avoiding: a x P rule: trm.strong_induct)
 apply(auto)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname:=(x).P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR)
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(erule fic.cases, simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)
 done
 
 lemma subst_not_fic2:
   assumes a: "\<not>fic M a"
   shows "\<not>fic(M{x:=<b>.P}) a" 
 using a
 apply(nominal_induct M avoiding: x a P b rule: trm.strong_induct)
 apply(auto)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(auto)[1]
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.P},P,name1,trm2{x:=<b>.P},name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 apply(drule fic_elims, simp)
 apply(simp add: abs_fresh fresh_atm)
 apply(drule freshc_after_substn)
 apply(simp add: fic.intros abs_fresh)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.P},trm2{name2:=<b>.P},P,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL)
 apply(drule fic_elims, simp)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(drule fic_elims, simp)
 done
 
 text \<open>Reductions\<close>
 
 lemma fin_l_reduce:
   assumes  a: "fin M x"
   and      b: "M \<longrightarrow>\<^sub>l M'"
   shows "fin M' x"
 using b a
 apply(induct)
 apply(erule fin.cases)
 apply(simp_all add: trm.inject)
 apply(rotate_tac 3)
 apply(erule fin.cases)
 apply(simp_all add: trm.inject)
 apply(erule fin.cases, simp_all add: trm.inject)+
 done
 
 lemma fin_c_reduce:
   assumes  a: "fin M x"
   and      b: "M \<longrightarrow>\<^sub>c M'"
   shows "fin M' x"
 using b a
 apply(induct)
 apply(erule fin.cases, simp_all add: trm.inject)+
 done
 
 lemma fin_a_reduce:
   assumes  a: "fin M x"
   and      b: "M \<longrightarrow>\<^sub>a M'"
   shows "fin M' x"
 using a b
 apply(induct)
 apply(drule ax_do_not_a_reduce)
 apply(simp)
 apply(drule a_redu_NotL_elim)
 apply(auto)
 apply(rule fin.intros)
 apply(simp add: fresh_a_redu)
 apply(drule a_redu_AndL1_elim)
 apply(auto)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_AndL2_elim)
 apply(auto)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_OrL_elim)
 apply(auto)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_ImpL_elim)
 apply(auto)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(rule fin.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 done
 
 lemma fin_a_star_reduce:
   assumes  a: "fin M x"
   and      b: "M \<longrightarrow>\<^sub>a* M'"
   shows "fin M' x"
 using b a
 apply(induct set: rtranclp)
-apply(auto simp add: fin_a_reduce)
+apply(auto simp: fin_a_reduce)
 done
 
 lemma fic_l_reduce:
   assumes  a: "fic M x"
   and      b: "M \<longrightarrow>\<^sub>l M'"
   shows "fic M' x"
 using b a
 apply(induct)
 apply(erule fic.cases)
 apply(simp_all add: trm.inject)
 apply(rotate_tac 3)
 apply(erule fic.cases)
 apply(simp_all add: trm.inject)
 apply(erule fic.cases, simp_all add: trm.inject)+
 done
 
 lemma fic_c_reduce:
   assumes a: "fic M x"
   and     b: "M \<longrightarrow>\<^sub>c M'"
   shows "fic M' x"
 using b a
 apply(induct)
 apply(erule fic.cases, simp_all add: trm.inject)+
 done
 
 lemma fic_a_reduce:
   assumes a: "fic M x"
   and     b: "M \<longrightarrow>\<^sub>a M'"
   shows "fic M' x"
 using a b
 apply(induct)
 apply(drule ax_do_not_a_reduce)
 apply(simp)
 apply(drule a_redu_NotR_elim)
 apply(auto)
 apply(rule fic.intros)
 apply(simp add: fresh_a_redu)
 apply(drule a_redu_AndR_elim)
 apply(auto)
 apply(rule fic.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(rule fic.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_OrR1_elim)
 apply(auto)
 apply(rule fic.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_OrR2_elim)
 apply(auto)
 apply(rule fic.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 apply(drule a_redu_ImpR_elim)
 apply(auto)
 apply(rule fic.intros)
 apply(force simp add: abs_fresh fresh_a_redu)
 done
 
 lemma fic_a_star_reduce:
   assumes  a: "fic M x"
   and      b: "M \<longrightarrow>\<^sub>a* M'"
   shows "fic M' x"
 using b a
 apply(induct set: rtranclp)
-apply(auto simp add: fic_a_reduce)
+apply(auto simp: fic_a_reduce)
 done
 
 text \<open>substitution properties\<close>
 
 lemma subst_with_ax1:
   shows "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]"
 proof(nominal_induct M avoiding: x a y rule: trm.strong_induct)
   case (Ax z b x a y)
   show "(Ax z b){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Ax z b)[x\<turnstile>n>y]"
   proof (cases "z=x")
     case True
     assume eq: "z=x"
     have "(Ax z b){x:=<a>.Ax y a} = Cut <a>.Ax y a (x).Ax x b" using eq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* (Ax x b)[x\<turnstile>n>y]" by blast
     finally show "Ax z b{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Ax z b)[x\<turnstile>n>y]" using eq by simp
   next
     case False
     assume neq: "z\<noteq>x"
     then show "(Ax z b){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Ax z b)[x\<turnstile>n>y]" using neq by simp
   qed
 next
   case (Cut b M z N x a y)
   have fs: "b\<sharp>x" "b\<sharp>a" "b\<sharp>y" "b\<sharp>N" "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "z\<sharp>M" by fact+
   have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* N[x\<turnstile>n>y]" by fact
   show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]"
   proof (cases "M = Ax x b")
     case True
     assume eq: "M = Ax x b"
     have "(Cut <b>.M (z).N){x:=<a>.Ax y a} = Cut <a>.Ax y a (z).(N{x:=<a>.Ax y a})" using fs eq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.Ax y a (z).(N[x\<turnstile>n>y])" using ih2 a_star_congs by blast
     also have "\<dots> = Cut <b>.(M[x\<turnstile>n>y]) (z).(N[x\<turnstile>n>y])" using eq
       by (simp add: trm.inject alpha calc_atm fresh_atm)
     finally show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]" using fs by simp
   next
     case False
     assume neq: "M \<noteq> Ax x b"
     have "(Cut <b>.M (z).N){x:=<a>.Ax y a} = Cut <b>.(M{x:=<a>.Ax y a}) (z).(N{x:=<a>.Ax y a})" 
       using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* Cut <b>.(M[x\<turnstile>n>y]) (z).(N[x\<turnstile>n>y])" using ih1 ih2 a_star_congs by blast
     finally show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]" using fs by simp
   qed
 next
   case (NotR z M b x a y)
   have fs: "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "z\<sharp>b" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have "(NotR (z).M b){x:=<a>.Ax y a} = NotR (z).(M{x:=<a>.Ax y a}) b" using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* NotR (z).(M[x\<turnstile>n>y]) b" using ih by (auto intro: a_star_congs)
   finally show "(NotR (z).M b){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (NotR (z).M b)[x\<turnstile>n>y]" using fs by simp
 next
   case (NotL b M z x a y)  
   have fs: "b\<sharp>x" "b\<sharp>a" "b\<sharp>y" "b\<sharp>z" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (NotL <b>.M z)[x\<turnstile>n>y]"
   proof(cases "z=x")
     case True
     assume eq: "z=x"
     obtain x'::"name" where new: "x'\<sharp>(Ax y a,M{x:=<a>.Ax y a})" by (rule exists_fresh(1)[OF fs_name1])
     have "(NotL <b>.M z){x:=<a>.Ax y a} = 
                         fresh_fun (\<lambda>x'. Cut <a>.Ax y a (x').NotL <b>.(M{x:=<a>.Ax y a}) x')"
       using eq fs by simp
     also have "\<dots> = Cut <a>.Ax y a (x').NotL <b>.(M{x:=<a>.Ax y a}) x'" 
       using new by (simp add: fresh_fun_simp_NotL fresh_prod)
     also have "\<dots> \<longrightarrow>\<^sub>a* (NotL <b>.(M{x:=<a>.Ax y a}) x')[x'\<turnstile>n>y]"
       using new 
       apply(rule_tac a_starI)
       apply(rule al_redu)
       apply(rule better_LAxL_intro)
       apply(auto)
       done
     also have "\<dots> = NotL <b>.(M{x:=<a>.Ax y a}) y" using new by (simp add: nrename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* NotL <b>.(M[x\<turnstile>n>y]) y" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (NotL <b>.M z)[x\<turnstile>n>y]" using eq by simp
     finally show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (NotL <b>.M z)[x\<turnstile>n>y]" by simp
   next
     case False
     assume neq: "z\<noteq>x"
     have "(NotL <b>.M z){x:=<a>.Ax y a} = NotL <b>.(M{x:=<a>.Ax y a}) z" using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* NotL <b>.(M[x\<turnstile>n>y]) z" using ih by (auto intro: a_star_congs)
     finally show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (NotL <b>.M z)[x\<turnstile>n>y]" using neq by simp
   qed
 next
   case (AndR c M d N e x a y)
   have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "d\<sharp>x" "d\<sharp>a" "d\<sharp>y" "d\<noteq>c" "c\<sharp>N" "c\<sharp>e" "d\<sharp>M" "d\<sharp>e" by fact+
   have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* N[x\<turnstile>n>y]" by fact
   have "(AndR <c>.M <d>.N e){x:=<a>.Ax y a} = AndR <c>.(M{x:=<a>.Ax y a}) <d>.(N{x:=<a>.Ax y a}) e"
     using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* AndR <c>.(M[x\<turnstile>n>y]) <d>.(N[x\<turnstile>n>y]) e" using ih1 ih2 by (auto intro: a_star_congs)
   finally show "(AndR <c>.M <d>.N e){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndR <c>.M <d>.N e)[x\<turnstile>n>y]"
     using fs by simp
 next
   case (AndL1 u M v x a y)
   have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "u\<sharp>v" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   show "(AndL1 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL1 (u).M v)[x\<turnstile>n>y]"
   proof(cases "v=x")
     case True
     assume eq: "v=x"
     obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},u)" by (rule exists_fresh(1)[OF fs_name1])
     have "(AndL1 (u).M v){x:=<a>.Ax y a} = 
                         fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').AndL1 (u).(M{x:=<a>.Ax y a}) v')"
       using eq fs by simp
     also have "\<dots> = Cut <a>.Ax y a (v').AndL1 (u).(M{x:=<a>.Ax y a}) v'" 
       using new by (simp add: fresh_fun_simp_AndL1 fresh_prod)
     also have "\<dots> \<longrightarrow>\<^sub>a* (AndL1 (u).(M{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxL_intro)
       apply(rule fin.intros)
       apply(simp add: abs_fresh)
       done
     also have "\<dots> = AndL1 (u).(M{x:=<a>.Ax y a}) y" using fs new
-      by (auto simp add: fresh_prod fresh_atm nrename_fresh)
+      by (auto simp: fresh_prod fresh_atm nrename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* AndL1 (u).(M[x\<turnstile>n>y]) y" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (AndL1 (u).M v)[x\<turnstile>n>y]" using eq fs by simp
     finally show "(AndL1 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL1 (u).M v)[x\<turnstile>n>y]" by simp
   next
     case False
     assume neq: "v\<noteq>x"
     have "(AndL1 (u).M v){x:=<a>.Ax y a} = AndL1 (u).(M{x:=<a>.Ax y a}) v" using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* AndL1 (u).(M[x\<turnstile>n>y]) v" using ih by (auto intro: a_star_congs)
     finally show "(AndL1 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL1 (u).M v)[x\<turnstile>n>y]" using fs neq by simp
   qed
 next
   case (AndL2 u M v x a y)
   have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "u\<sharp>v" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   show "(AndL2 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL2 (u).M v)[x\<turnstile>n>y]"
   proof(cases "v=x")
     case True
     assume eq: "v=x"
     obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},u)" by (rule exists_fresh(1)[OF fs_name1])
     have "(AndL2 (u).M v){x:=<a>.Ax y a} = 
                         fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').AndL2 (u).(M{x:=<a>.Ax y a}) v')"
       using eq fs by simp
     also have "\<dots> = Cut <a>.Ax y a (v').AndL2 (u).(M{x:=<a>.Ax y a}) v'" 
       using new by (simp add: fresh_fun_simp_AndL2 fresh_prod)
     also have "\<dots> \<longrightarrow>\<^sub>a* (AndL2 (u).(M{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxL_intro)
       apply(rule fin.intros)
       apply(simp add: abs_fresh)
       done
     also have "\<dots> = AndL2 (u).(M{x:=<a>.Ax y a}) y" using fs new
-      by (auto simp add: fresh_prod fresh_atm nrename_fresh)
+      by (auto simp: fresh_prod fresh_atm nrename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* AndL2 (u).(M[x\<turnstile>n>y]) y" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (AndL2 (u).M v)[x\<turnstile>n>y]" using eq fs by simp
     finally show "(AndL2 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL2 (u).M v)[x\<turnstile>n>y]" by simp
   next
     case False
     assume neq: "v\<noteq>x"
     have "(AndL2 (u).M v){x:=<a>.Ax y a} = AndL2 (u).(M{x:=<a>.Ax y a}) v" using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* AndL2 (u).(M[x\<turnstile>n>y]) v" using ih by (auto intro: a_star_congs)
     finally show "(AndL2 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (AndL2 (u).M v)[x\<turnstile>n>y]" using fs neq by simp
   qed
 next
   case (OrR1 c M d  x a y)
   have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "c\<sharp>d" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have "(OrR1 <c>.M d){x:=<a>.Ax y a} = OrR1 <c>.(M{x:=<a>.Ax y a}) d" using fs by (simp add: fresh_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a* OrR1 <c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs)
   finally show "(OrR1 <c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (OrR1 <c>.M d)[x\<turnstile>n>y]" using fs by simp
 next
   case (OrR2 c M d  x a y)
   have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "c\<sharp>d" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have "(OrR2 <c>.M d){x:=<a>.Ax y a} = OrR2 <c>.(M{x:=<a>.Ax y a}) d" using fs by (simp add: fresh_atm)
   also have "\<dots> \<longrightarrow>\<^sub>a* OrR2 <c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs)
   finally show "(OrR2 <c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (OrR2 <c>.M d)[x\<turnstile>n>y]" using fs by simp
 next
   case (OrL u M v N z x a y)
   have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "v\<sharp>x" "v\<sharp>a" "v\<sharp>y" "v\<noteq>u" "u\<sharp>N" "u\<sharp>z" "v\<sharp>M" "v\<sharp>z" by fact+
   have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* N[x\<turnstile>n>y]" by fact
   show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]"
   proof(cases "z=x")
     case True
     assume eq: "z=x"
     obtain z'::"name" where new: "z'\<sharp>(Ax y a,M{x:=<a>.Ax y a},N{x:=<a>.Ax y a},u,v,y,a)" 
       by (rule exists_fresh(1)[OF fs_name1])
     have "(OrL (u).M (v).N z){x:=<a>.Ax y a} = 
                  fresh_fun (\<lambda>z'. Cut <a>.Ax y a (z').OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z')"
       using eq fs by simp
     also have "\<dots> = Cut <a>.Ax y a (z').OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z'" 
       using new by (simp add: fresh_fun_simp_OrL)
     also have "\<dots> \<longrightarrow>\<^sub>a* (OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z')[z'\<turnstile>n>y]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxL_intro)
       apply(rule fin.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) y" using fs new
-      by (auto simp add: fresh_prod fresh_atm nrename_fresh subst_fresh)
+      by (auto simp: fresh_prod fresh_atm nrename_fresh subst_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* OrL (u).(M[x\<turnstile>n>y]) (v).(N[x\<turnstile>n>y]) y" 
       using ih1 ih2 by (auto intro: a_star_congs)
     also have "\<dots> = (OrL (u).M (v).N z)[x\<turnstile>n>y]" using eq fs by simp
     finally show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]" by simp
   next
     case False
     assume neq: "z\<noteq>x"
     have "(OrL (u).M (v).N z){x:=<a>.Ax y a} = OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z" 
       using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* OrL (u).(M[x\<turnstile>n>y]) (v).(N[x\<turnstile>n>y]) z" 
       using ih1 ih2 by (auto intro: a_star_congs)
     finally show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]" using fs neq by simp
   qed
 next
   case (ImpR z c M d x a y)
   have fs: "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "z\<sharp>d" "c\<sharp>d" by fact+
   have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have "(ImpR (z).<c>.M d){x:=<a>.Ax y a} = ImpR (z).<c>.(M{x:=<a>.Ax y a}) d" using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* ImpR (z).<c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs)
   finally show "(ImpR (z).<c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (ImpR (z).<c>.M d)[x\<turnstile>n>y]" using fs by simp
 next
   case (ImpL c M u N v x a y)
   have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "c\<sharp>N" "c\<sharp>v" "u\<sharp>M" "u\<sharp>v" by fact+
   have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]" by fact
   have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* N[x\<turnstile>n>y]" by fact
   show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]"
   proof(cases "v=x")
     case True
     assume eq: "v=x"
     obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},N{x:=<a>.Ax y a},y,a,u)" 
       by (rule exists_fresh(1)[OF fs_name1])
     have "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} = 
                  fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v')"
       using eq fs by simp 
     also have "\<dots> = Cut <a>.Ax y a (v').ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v'" 
       using new by (simp add: fresh_fun_simp_ImpL)
     also have "\<dots> \<longrightarrow>\<^sub>a* (ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxL_intro)
       apply(rule fin.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) y" using fs new
-      by (auto simp add: fresh_prod subst_fresh fresh_atm trm.inject alpha rename_fresh)
+      by (auto simp: fresh_prod subst_fresh fresh_atm trm.inject alpha rename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* ImpL <c>.(M[x\<turnstile>n>y]) (u).(N[x\<turnstile>n>y]) y" 
       using ih1 ih2 by (auto intro: a_star_congs)
     also have "\<dots> = (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" using eq fs by simp
     finally show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" using fs by simp
   next
     case False
     assume neq: "v\<noteq>x"
     have "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} = ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v" 
       using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* ImpL <c>.(M[x\<turnstile>n>y]) (u).(N[x\<turnstile>n>y]) v" 
       using ih1 ih2 by (auto intro: a_star_congs)
     finally show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" 
       using fs neq by simp
   qed
 qed
 
 lemma subst_with_ax2:
   shows "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]"
 proof(nominal_induct M avoiding: b a x rule: trm.strong_induct)
   case (Ax z c b a x)
   show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Ax z c)[b\<turnstile>c>a]"
   proof (cases "c=b")
     case True
     assume eq: "c=b"
     have "(Ax z c){b:=(x).Ax x a} = Cut <b>.Ax z c (x).Ax x a" using eq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* (Ax z c)[b\<turnstile>c>a]" using eq by blast
     finally show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Ax z c)[b\<turnstile>c>a]" by simp
   next
     case False
     assume neq: "c\<noteq>b"
     then show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Ax z c)[b\<turnstile>c>a]" by simp
   qed
 next
   case (Cut c M z N b a x)
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>N" "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "z\<sharp>M" by fact+
   have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* N[b\<turnstile>c>a]" by fact
   show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]"
   proof (cases "N = Ax z b")
     case True
     assume eq: "N = Ax z b"
     have "(Cut <c>.M (z).N){b:=(x).Ax x a} = Cut <c>.(M{b:=(x).Ax x a}) (x).Ax x a" using eq fs by simp 
     also have "\<dots> \<longrightarrow>\<^sub>a* Cut <c>.(M[b\<turnstile>c>a]) (x).Ax x a" using ih1 a_star_congs by blast
     also have "\<dots> = Cut <c>.(M[b\<turnstile>c>a]) (z).(N[b\<turnstile>c>a])" using eq fs
       by (simp add: trm.inject alpha calc_atm fresh_atm)
     finally show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]" using fs by simp
   next
     case False
     assume neq: "N \<noteq> Ax z b"
     have "(Cut <c>.M (z).N){b:=(x).Ax x a} = Cut <c>.(M{b:=(x).Ax x a}) (z).(N{b:=(x).Ax x a})" 
       using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* Cut <c>.(M[b\<turnstile>c>a]) (z).(N[b\<turnstile>c>a])" using ih1 ih2 a_star_congs by blast
     finally show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]" using fs by simp
   qed
 next
   case (NotR z M c b a x)
   have fs: "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "z\<sharp>c" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (NotR (z).M c)[b\<turnstile>c>a]"
   proof (cases "c=b")
     case True
     assume eq: "c=b"
     obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a})" by (rule exists_fresh(2)[OF fs_coname1])
     have "(NotR (z).M c){b:=(x).Ax x a} = 
                         fresh_fun (\<lambda>a'. Cut <a'>.NotR (z).M{b:=(x).Ax x a} a' (x).Ax x a)" 
       using eq fs by simp
     also have "\<dots> = Cut <a'>.NotR (z).M{b:=(x).Ax x a} a' (x).Ax x a"
       using new by (simp add: fresh_fun_simp_NotR fresh_prod)
     also have "\<dots> \<longrightarrow>\<^sub>a* (NotR (z).(M{b:=(x).Ax x a}) a')[a'\<turnstile>c>a]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxR_intro)
       apply(rule fic.intros)
       apply(simp)
       done
     also have "\<dots> = NotR (z).(M{b:=(x).Ax x a}) a" using new by (simp add: crename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* NotR (z).(M[b\<turnstile>c>a]) a" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (NotR (z).M c)[b\<turnstile>c>a]" using eq by simp
     finally show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (NotR (z).M c)[b\<turnstile>c>a]" by simp
   next
     case False
     assume neq: "c\<noteq>b"
     have "(NotR (z).M c){b:=(x).Ax x a} = NotR (z).(M{b:=(x).Ax x a}) c" using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* NotR (z).(M[b\<turnstile>c>a]) c" using ih by (auto intro: a_star_congs)
     finally show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (NotR (z).M c)[b\<turnstile>c>a]" using neq by simp
   qed
 next
   case (NotL c M z b a x)  
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>z" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have "(NotL <c>.M z){b:=(x).Ax x a} = NotL <c>.(M{b:=(x).Ax x a}) z" using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* NotL <c>.(M[b\<turnstile>c>a]) z" using ih by (auto intro: a_star_congs)
   finally show "(NotL <c>.M z){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (NotL <c>.M z)[b\<turnstile>c>a]" using fs by simp
 next
   case (AndR c M d N e b a x)
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "d\<sharp>b" "d\<sharp>a" "d\<sharp>x" "d\<noteq>c" "c\<sharp>N" "c\<sharp>e" "d\<sharp>M" "d\<sharp>e" by fact+
   have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* N[b\<turnstile>c>a]" by fact
   show "(AndR <c>.M <d>.N e){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]"
   proof(cases "e=b")
     case True
     assume eq: "e=b"
     obtain e'::"coname" where new: "e'\<sharp>(Ax x a,M{b:=(x).Ax x a},N{b:=(x).Ax x a},c,d)" 
       by (rule exists_fresh(2)[OF fs_coname1])
     have "(AndR <c>.M <d>.N e){b:=(x).Ax x a} = 
                fresh_fun (\<lambda>e'. Cut <e'>.AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e' (x).Ax x a)"
       using eq fs by simp
     also have "\<dots> = Cut <e'>.AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e' (x).Ax x a"
       using new by (simp add: fresh_fun_simp_AndR fresh_prod)
     also have "\<dots> \<longrightarrow>\<^sub>a* (AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e')[e'\<turnstile>c>a]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxR_intro)
       apply(rule fic.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) a" using fs new
-      by (auto simp add: fresh_prod fresh_atm subst_fresh crename_fresh)
+      by (auto simp: fresh_prod fresh_atm subst_fresh crename_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* AndR <c>.(M[b\<turnstile>c>a]) <d>.(N[b\<turnstile>c>a]) a" using ih1 ih2 by (auto intro: a_star_congs)
     also have "\<dots> = (AndR <c>.M <d>.N e)[b\<turnstile>c>a]" using eq fs by simp
     finally show "(AndR <c>.M <d>.N e){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]" by simp
   next
     case False
     assume neq: "e\<noteq>b"
     have "(AndR <c>.M <d>.N e){b:=(x).Ax x a} = AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e"
       using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* AndR <c>.(M[b\<turnstile>c>a]) <d>.(N[b\<turnstile>c>a]) e" using ih1 ih2 by (auto intro: a_star_congs)
     finally show "(AndR <c>.M <d>.N e){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]"
       using fs neq by simp
   qed
 next
   case (AndL1 u M v b a x)
   have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "u\<sharp>v" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have "(AndL1 (u).M v){b:=(x).Ax x a} = AndL1 (u).(M{b:=(x).Ax x a}) v" using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* AndL1 (u).(M[b\<turnstile>c>a]) v" using ih by (auto intro: a_star_congs)
   finally show "(AndL1 (u).M v){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (AndL1 (u).M v)[b\<turnstile>c>a]" using fs by simp
 next
   case (AndL2 u M v b a x)
   have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "u\<sharp>v" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have "(AndL2 (u).M v){b:=(x).Ax x a} = AndL2 (u).(M{b:=(x).Ax x a}) v" using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* AndL2 (u).(M[b\<turnstile>c>a]) v" using ih by (auto intro: a_star_congs)
   finally show "(AndL2 (u).M v){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (AndL2 (u).M v)[b\<turnstile>c>a]" using fs by simp
 next
   case (OrR1 c M d  b a x)
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>d" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   show "(OrR1 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]"
   proof(cases "d=b")
     case True
     assume eq: "d=b"
     obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},c,x,a)" 
       by (rule exists_fresh(2)[OF fs_coname1])
     have "(OrR1 <c>.M d){b:=(x).Ax x a} = 
              fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by (simp)
     also have "\<dots> = Cut <a'>.OrR1 <c>.M{b:=(x).Ax x a} a' (x).Ax x a"
       using new by (simp add: fresh_fun_simp_OrR1)
     also have "\<dots> \<longrightarrow>\<^sub>a* (OrR1 <c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxR_intro)
       apply(rule fic.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = OrR1 <c>.M{b:=(x).Ax x a} a" using fs new
-      by (auto simp add: fresh_prod fresh_atm crename_fresh subst_fresh)
+      by (auto simp: fresh_prod fresh_atm crename_fresh subst_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* OrR1 <c>.(M[b\<turnstile>c>a]) a" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (OrR1 <c>.M d)[b\<turnstile>c>a]" using eq fs by simp
     finally show "(OrR1 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]" by simp
   next
     case False
     assume neq: "d\<noteq>b"
     have "(OrR1 <c>.M d){b:=(x).Ax x a} = OrR1 <c>.(M{b:=(x).Ax x a}) d" using fs neq by (simp)
     also have "\<dots> \<longrightarrow>\<^sub>a* OrR1 <c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs)
     finally show "(OrR1 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]" using fs neq by simp
   qed
 next
   case (OrR2 c M d  b a x)
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>d" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   show "(OrR2 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]"
   proof(cases "d=b")
     case True
     assume eq: "d=b"
     obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},c,x,a)" 
       by (rule exists_fresh(2)[OF fs_coname1])
     have "(OrR2 <c>.M d){b:=(x).Ax x a} = 
              fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by (simp)
     also have "\<dots> = Cut <a'>.OrR2 <c>.M{b:=(x).Ax x a} a' (x).Ax x a"
       using new by (simp add: fresh_fun_simp_OrR2)
     also have "\<dots> \<longrightarrow>\<^sub>a* (OrR2 <c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxR_intro)
       apply(rule fic.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = OrR2 <c>.M{b:=(x).Ax x a} a" using fs new
-      by (auto simp add: fresh_prod fresh_atm crename_fresh subst_fresh)
+      by (auto simp: fresh_prod fresh_atm crename_fresh subst_fresh)
     also have "\<dots> \<longrightarrow>\<^sub>a* OrR2 <c>.(M[b\<turnstile>c>a]) a" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (OrR2 <c>.M d)[b\<turnstile>c>a]" using eq fs by simp
     finally show "(OrR2 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]" by simp
   next
     case False
     assume neq: "d\<noteq>b"
     have "(OrR2 <c>.M d){b:=(x).Ax x a} = OrR2 <c>.(M{b:=(x).Ax x a}) d" using fs neq by (simp)
     also have "\<dots> \<longrightarrow>\<^sub>a* OrR2 <c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs)
     finally show "(OrR2 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]" using fs neq by simp
   qed
 next
   case (OrL u M v N z b a x)
   have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "v\<sharp>b" "v\<sharp>a" "v\<sharp>x" "v\<noteq>u" "u\<sharp>N" "u\<sharp>z" "v\<sharp>M" "v\<sharp>z" by fact+
   have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* N[b\<turnstile>c>a]" by fact
   have "(OrL (u).M (v).N z){b:=(x).Ax x a} = OrL (u).(M{b:=(x).Ax x a}) (v).(N{b:=(x).Ax x a}) z" 
     using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* OrL (u).(M[b\<turnstile>c>a]) (v).(N[b\<turnstile>c>a]) z" using ih1 ih2 by (auto intro: a_star_congs)
   finally show "(OrL (u).M (v).N z){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (OrL (u).M (v).N z)[b\<turnstile>c>a]" using fs by simp
 next
   case (ImpR z c M d b a x)
   have fs: "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "z\<sharp>d" "c\<sharp>d" by fact+
   have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   show "(ImpR (z).<c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]"
   proof(cases "b=d")
     case True
     assume eq: "b=d"
     obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},x,a,c)" 
       by (rule exists_fresh(2)[OF fs_coname1])
     have "(ImpR (z).<c>.M d){b:=(x).Ax x a} =
                 fresh_fun (\<lambda>a'. Cut <a'>.ImpR z.<c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by simp
     also have "\<dots> = Cut <a'>.ImpR z.<c>.M{b:=(x).Ax x a} a' (x).Ax x a" 
       using new by (simp add: fresh_fun_simp_ImpR)
     also have "\<dots> \<longrightarrow>\<^sub>a* (ImpR z.<c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]"
       using new 
       apply(rule_tac a_starI)
       apply(rule a_redu.intros)
       apply(rule better_LAxR_intro)
       apply(rule fic.intros)
       apply(simp_all add: abs_fresh)
       done
     also have "\<dots> = ImpR z.<c>.M{b:=(x).Ax x a} a" using fs new
-      by (auto simp add: fresh_prod crename_fresh subst_fresh fresh_atm)
+      by (auto simp: fresh_prod crename_fresh subst_fresh fresh_atm)
     also have "\<dots> \<longrightarrow>\<^sub>a* ImpR z.<c>.(M[b\<turnstile>c>a]) a" using ih by (auto intro: a_star_congs)
     also have "\<dots> = (ImpR z.<c>.M b)[b\<turnstile>c>a]" using eq fs by simp
     finally show "(ImpR (z).<c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]" using eq by simp
   next
     case False
     assume neq: "b\<noteq>d"
     have "(ImpR (z).<c>.M d){b:=(x).Ax x a} = ImpR (z).<c>.(M{b:=(x).Ax x a}) d" using fs neq by simp
     also have "\<dots> \<longrightarrow>\<^sub>a* ImpR (z).<c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs)
     finally show "(ImpR (z).<c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]" using neq fs by simp
   qed
 next
   case (ImpL c M u N v b a x)
   have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "c\<sharp>N" "c\<sharp>v" "u\<sharp>M" "u\<sharp>v" by fact+
   have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]" by fact
   have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* N[b\<turnstile>c>a]" by fact
   have "(ImpL <c>.M (u).N v){b:=(x).Ax x a} = ImpL <c>.(M{b:=(x).Ax x a}) (u).(N{b:=(x).Ax x a}) v" 
     using fs by simp
   also have "\<dots> \<longrightarrow>\<^sub>a* ImpL <c>.(M[b\<turnstile>c>a]) (u).(N[b\<turnstile>c>a]) v" 
     using ih1 ih2 by (auto intro: a_star_congs)
   finally show "(ImpL <c>.M (u).N v){b:=(x).Ax x a} \<longrightarrow>\<^sub>a* (ImpL <c>.M (u).N v)[b\<turnstile>c>a]" 
     using fs by simp
 qed
 
 text \<open>substitution lemmas\<close>
 
 lemma not_Ax1:
   shows "\<not>(b\<sharp>M) \<Longrightarrow> M{b:=(y).Q} \<noteq> Ax x a"
 apply(nominal_induct M avoiding: b y Q x a rule: trm.strong_induct)
-apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+apply(auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(y).Q},Q)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(y).Q},Q)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm)
 apply(rule exists_fresh'(2)[OF fs_coname1])
 done
 
 lemma not_Ax2:
   shows "\<not>(x\<sharp>M) \<Longrightarrow> M{x:=<b>.Q} \<noteq> Ax y a"
 apply(nominal_induct M avoiding: b y Q x a rule: trm.strong_induct)
-apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp)
+apply(auto simp: fresh_atm abs_fresh abs_supp fin_supp)
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)")
 apply(erule exE)
 apply(simp add: fresh_prod)
 apply(erule conjE)+
 apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
 apply(rule exists_fresh'(1)[OF fs_name1])
 done
 
 lemma interesting_subst1:
   assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" 
   shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<c>.Ax y c}{y:=<c>.P}"
 using a
 proof(nominal_induct N avoiding: x y c P rule: trm.strong_induct)
   case Ax
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject)
 next 
   case (Cut d M u M' x' y' c P)
   with assms show ?case
     apply(simp)
     apply(auto)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(auto)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(auto)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(rule impI)
     apply(simp add: trm.inject alpha forget)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(auto)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(auto)
     apply(case_tac "y'\<sharp>M")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(auto)
     apply(case_tac "x'\<sharp>M")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     done
 next
   case NotR
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget)
+    by (auto simp: abs_fresh fresh_atm forget)
 next
   case (NotL d M u)
   then show ?case
-    apply (auto simp add: abs_fresh fresh_atm forget)
+    apply (auto simp: abs_fresh fresh_atm forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},y,x)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},Ax y c,y,x)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_NotL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(simp add: trm.inject alpha forget subst_fresh)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: abs_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndR d1 M d2 M' d3)
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (AndL1 u M d)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_AndL1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndL2 u M d)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_AndL2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case OrR1
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case OrR2
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},
                                         M'{y:=<c>.P},M'{x:=<c>.Ax y c}{y:=<c>.P},x1,x2,x3,y,x)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(force)
     apply(simp)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},
                                         M'{x:=<c>.Ax y c},M'{x:=<c>.Ax y c}{y:=<c>.P},x1,x2,x3,y,x)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_OrL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(force)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case ImpR
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (ImpL a M x1 M' x2)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x2:=<c>.P},M{x:=<c>.Ax x2 c}{x2:=<c>.P},
                                         M'{x2:=<c>.P},M'{x:=<c>.Ax x2 c}{x2:=<c>.P},x1,y,x)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(force)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x2:=<c>.Ax y c},M{x2:=<c>.Ax y c}{y:=<c>.P},
                                         M'{x2:=<c>.Ax y c},M'{x2:=<c>.Ax y c}{y:=<c>.P},x1,x2,y,x)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_ImpL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 qed 
 
 lemma interesting_subst1':
   assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" 
   shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<a>.Ax y a}{y:=<c>.P}"
 proof -
   show ?thesis
   proof (cases "c=a")
     case True then show ?thesis using a by (simp add: interesting_subst1)
   next
     case False then show ?thesis using a
       apply - 
       apply(subgoal_tac "N{x:=<a>.Ax y a} = N{x:=<c>.([(c,a)]\<bullet>Ax y a)}") 
       apply(simp add: interesting_subst1 calc_atm)
       apply(rule subst_rename)
       apply(simp add: fresh_prod fresh_atm)
       done
   qed
 qed
 
 lemma interesting_subst2:
   assumes a: "a\<noteq>b" "a\<sharp>P" "b\<sharp>P" 
   shows "N{a:=(y).P}{b:=(y).P} = N{b:=(y).Ax y a}{a:=(y).P}"
 using a
 proof(nominal_induct N avoiding: a b y P rule: trm.strong_induct)
   case Ax
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject)
 next 
   case (Cut d M u M' x' y' c P)
   with assms show ?case
     apply(simp)
-    apply(auto simp add: trm.inject)
+    apply(auto simp: trm.inject)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp)
     apply(simp add: abs_fresh)
     apply(simp add: forget)
     apply(auto)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh) 
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(auto)[1]
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(rule impI)
     apply(simp add: fresh_atm trm.inject alpha forget)
     apply(case_tac "x'\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(auto)
     apply(case_tac "y'\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     done
 next
   case NotL
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget)
+    by (auto simp: abs_fresh fresh_atm forget)
 next
   case (NotR u M d)
   then show ?case
-    apply (auto simp add: abs_fresh fresh_atm forget)
+    apply (auto simp: abs_fresh fresh_atm forget)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{d:=(y).P},M{b:=(y).Ax y d}{d:=(y).P},u,y)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotR)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,M{d:=(y).Ax y a},M{d:=(y).Ax y a}{a:=(y).P},Ax y a,y,d)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_NotR)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget subst_fresh)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (AndR d1 M d2 M' d3)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,M{d3:=(y).P},M{b:=(y).Ax y d3}{d3:=(y).P},
                                         M'{d3:=(y).P},M'{b:=(y).Ax y d3}{d3:=(y).P},d1,d2,d3,b,y)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndR)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh fresh_atm)
     apply(simp add: abs_fresh fresh_atm)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(force)
     apply(simp)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,Ax y a,M{d3:=(y).Ax y a},M{d3:=(y).Ax y a}{a:=(y).P},
                                         M'{d3:=(y).Ax y a},M'{d3:=(y).Ax y a}{a:=(y).P},d1,d2,d3,y,b)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_AndR)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(force)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (AndL1 u M d)
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (AndL2 u M d)
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (OrR1 d M e)
   then show ?case
-    apply (auto simp add: abs_fresh fresh_atm forget)
+    apply (auto simp: abs_fresh fresh_atm forget)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{e:=(y).P},M{b:=(y).Ax y e}{e:=(y).P},d,e)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR1)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,Ax y a,M{e:=(y).Ax y a},M{e:=(y).Ax y a}{a:=(y).P},d,e)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_OrR1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget subst_fresh)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrR2 d M e)
   then show ?case
-    apply (auto simp add: abs_fresh fresh_atm forget)
+    apply (auto simp: abs_fresh fresh_atm forget)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{e:=(y).P},M{b:=(y).Ax y e}{e:=(y).P},d,e)")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR2)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,Ax y a,M{e:=(y).Ax y a},M{e:=(y).Ax y a}{a:=(y).P},d,e)")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_OrR2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget subst_fresh)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrL x1 M x2 M' x3)
   then show ?case
-    by(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case ImpL
   then show ?case
-    by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    by (auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
 next
   case (ImpR u e M d)
   then show ?case
-    apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh)
+    apply(auto simp: abs_fresh fresh_atm forget trm.inject subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,e,d,P,M{d:=(y).P},M{b:=(y).Ax y d}{d:=(y).P})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpR)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(e,d,P,Ax y a,M{d:=(y).Ax y a},M{d:=(y).Ax y a}{a:=(y).P})")
     apply(erule exE, simp only: fresh_prod)
     apply(erule conjE)+
     apply(simp only: fresh_fun_simp_ImpR)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 qed 
 
 lemma interesting_subst2':
   assumes a: "a\<noteq>b" "a\<sharp>P" "b\<sharp>P" 
   shows "N{a:=(y).P}{b:=(y).P} = N{b:=(z).Ax z a}{a:=(y).P}"
 proof -
   show ?thesis
   proof (cases "z=y")
     case True then show ?thesis using a by (simp add: interesting_subst2)
   next
     case False then show ?thesis using a
       apply - 
       apply(subgoal_tac "N{b:=(z).Ax z a} = N{b:=(y).([(y,z)]\<bullet>Ax z a)}") 
       apply(simp add: interesting_subst2 calc_atm)
       apply(rule subst_rename)
       apply(simp add: fresh_prod fresh_atm)
       done
   qed
 qed
 
 lemma subst_subst1:
   assumes a: "a\<sharp>(Q,b)" "x\<sharp>(y,P,Q)" "b\<sharp>Q" "y\<sharp>P" 
   shows "M{x:=<a>.P}{b:=(y).Q} = M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}"
 using a
 proof(nominal_induct M avoiding: x a P b y Q rule: trm.strong_induct)
   case (Ax z c)
   have fs: "a\<sharp>(Q,b)" "x\<sharp>(y,P,Q)" "b\<sharp>Q" "y\<sharp>P" by fact+
   { assume asm: "z=x \<and> c=b"
     have "(Ax x b){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (x).Ax x b){b:=(y).Q}" using fs by simp
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).Q"
       using fs by (simp_all add: fresh_prod fresh_atm)
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" using fs by (simp add: forget)
     also have "\<dots> = (Cut <b>.Ax x b (y).Q){x:=<a>.(P{b:=(y).Q})}"
-      using fs asm by (auto simp add: fresh_prod fresh_atm subst_fresh)
+      using fs asm by (auto simp: fresh_prod fresh_atm subst_fresh)
     also have "\<dots> = (Ax x b){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using fs by simp
     finally have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" 
       using asm by simp
   }
   moreover
   { assume asm: "z\<noteq>x \<and> c=b"
     have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}" using asm by simp
     also have "\<dots> = Cut <b>.Ax z c (y).Q" using fs asm by simp
     also have "\<dots> = Cut <b>.(Ax z c{x:=<a>.(P{b:=(y).Q})}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" 
       using fs asm by (simp add: forget)
     also have "\<dots> = (Cut <b>.Ax z c (y).Q){x:=<a>.(P{b:=(y).Q})}" using asm fs
-      by (auto simp add: trm.inject subst_fresh fresh_prod fresh_atm abs_fresh)
+      by (auto simp: trm.inject subst_fresh fresh_prod fresh_atm abs_fresh)
     also have "\<dots> = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm fs by simp
     finally have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" by simp
   }
   moreover
   { assume asm: "z=x \<and> c\<noteq>b"
     have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (x).Ax z c){b:=(y).Q}" using fs asm by simp
-    also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (x).Ax z c" using fs asm by (auto simp add: trm.inject abs_fresh)
+    also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (x).Ax z c" using fs asm by (auto simp: trm.inject abs_fresh)
     also have "\<dots> = (Ax z c){x:=<a>.(P{b:=(y).Q})}" using fs asm by simp
     also have "\<dots> = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm by auto
     finally have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" by simp
   }
   moreover
   { assume asm: "z\<noteq>x \<and> c\<noteq>b"
     have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm by auto
   }
   ultimately show ?case by blast
 next
   case (Cut c M z N)
   { assume asm: "M = Ax x c \<and> N = Ax z b"
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (z).(N{x:=<a>.P})){b:=(y).Q}" 
       using Cut asm by simp
     also have "\<dots> = (Cut <a>.P (z).N){b:=(y).Q}" using Cut asm by (simp add: fresh_atm)
-    also have "\<dots> = (Cut <a>.(P{b:=(y).Q}) (y).Q)" using Cut asm by (auto simp add: fresh_prod fresh_atm)
+    also have "\<dots> = (Cut <a>.(P{b:=(y).Q}) (y).Q)" using Cut asm by (auto simp: fresh_prod fresh_atm)
     finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <a>.(P{b:=(y).Q}) (y).Q)" by simp
     have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = (Cut <c>.M (y).Q){x:=<a>.(P{b:=(y).Q})}"
       using Cut asm by (simp add: fresh_atm)
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" using Cut asm
-      by (auto simp add: fresh_prod fresh_atm subst_fresh)
+      by (auto simp: fresh_prod fresh_atm subst_fresh)
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).Q" using Cut asm by (simp add: forget)
     finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = Cut <a>.(P{b:=(y).Q}) (y).Q"
       by simp
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" 
       using eq1 eq2 by simp
   }
   moreover
   { assume asm: "M \<noteq> Ax x c \<and> N = Ax z b"
     have neq: "M{b:=(y).Q} \<noteq> Ax x c"
     proof (cases "b\<sharp>M")
       case True then show ?thesis using asm by (simp add: forget)
     next
       case False then show ?thesis by (simp add: not_Ax1)
     qed
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.(M{x:=<a>.P}) (z).(N{x:=<a>.P})){b:=(y).Q}"
       using Cut asm by simp
     also have "\<dots> = (Cut <c>.(M{x:=<a>.P}) (z).N){b:=(y).Q}" using Cut asm by (simp add: fresh_atm)
     also have "\<dots> = Cut <c>.(M{x:=<a>.P}{b:=(y).Q}) (y).Q" using Cut asm by (simp add: abs_fresh)
     also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}}) (y).Q" using Cut asm by simp
     finally 
     have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = Cut <c>.(M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}}) (y).Q" 
       by simp
     have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = 
                (Cut <c>.(M{b:=(y).Q}) (y).Q){x:=<a>.(P{b:=(y).Q})}" using Cut asm by simp
     also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (y).(Q{x:=<a>.(P{b:=(y).Q})})"
-      using Cut asm neq by (auto simp add: fresh_prod fresh_atm subst_fresh abs_fresh)
+      using Cut asm neq by (auto simp: fresh_prod fresh_atm subst_fresh abs_fresh)
     also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (y).Q" using Cut asm by (simp add: forget)
     finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} 
                                        = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (y).Q" by simp
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" 
       using eq1 eq2 by simp
   }
   moreover 
   { assume asm: "M = Ax x c \<and> N \<noteq> Ax z b"
     have neq: "N{x:=<a>.P} \<noteq> Ax z b"
     proof (cases "x\<sharp>N")
       case True then show ?thesis using asm by (simp add: forget)
     next
       case False then show ?thesis by (simp add: not_Ax2)
     qed
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (z).(N{x:=<a>.P})){b:=(y).Q}"
       using Cut asm by simp
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{x:=<a>.P}{b:=(y).Q})" using Cut asm neq 
       by (simp add: abs_fresh)
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" using Cut asm by simp
     finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} 
                     = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp
     have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} 
                     = (Cut <c>.(M{b:=(y).Q}) (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}"
       using Cut asm by auto
     also have "\<dots> = (Cut <c>.M (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}"
-      using Cut asm by (auto simp add: fresh_atm)
+      using Cut asm by (auto simp: fresh_atm)
     also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" 
       using Cut asm by (simp add: fresh_prod fresh_atm subst_fresh)
     finally 
     have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} 
          = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}"
       using eq1 eq2 by simp
   }
   moreover
   { assume asm: "M \<noteq> Ax x c \<and> N \<noteq> Ax z b"
     have neq1: "N{x:=<a>.P} \<noteq> Ax z b"
     proof (cases "x\<sharp>N")
       case True then show ?thesis using asm by (simp add: forget)
     next
       case False then show ?thesis by (simp add: not_Ax2)
     qed
     have neq2: "M{b:=(y).Q} \<noteq> Ax x c"
     proof (cases "b\<sharp>M")
       case True then show ?thesis using asm by (simp add: forget)
     next
       case False then show ?thesis by (simp add: not_Ax1)
     qed
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.(M{x:=<a>.P}) (z).(N{x:=<a>.P})){b:=(y).Q}"
       using Cut asm by simp
     also have "\<dots> = Cut <c>.(M{x:=<a>.P}{b:=(y).Q}) (z).(N{x:=<a>.P}{b:=(y).Q})" using Cut asm neq1
       by (simp add: abs_fresh)
     also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})"
       using Cut asm by simp
     finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q}
              = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp
     have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = 
                 (Cut <c>.(M{b:=(y).Q}) (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}" using Cut asm neq1 by simp
     also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})"
       using Cut asm neq2 by (simp add: fresh_prod fresh_atm subst_fresh)
     finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = 
            Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp
     have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}"
       using eq1 eq2 by simp
   }
   ultimately show ?case by blast
 next
   case (NotR z M c)
   then show ?case
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(M{c:=(y).Q},M{c:=(y).Q}{x:=<a>.P{c:=(y).Q}},Q,a,P,c,y)")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: fresh_prod fresh_atm subst_fresh abs_fresh)
     apply(simp add: fresh_prod fresh_atm subst_fresh abs_fresh)
     apply(simp add: forget)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (NotL c M z)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},y,Q)")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndR c1 M c2 N c3)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c3:=(y).Q},M{c3:=(y).Q}{x:=<a>.P{c3:=(y).Q}},c2,c3,a,
                                      P{c3:=(y).Q},N{c3:=(y).Q},N{c3:=(y).Q}{x:=<a>.P{c3:=(y).Q}},c1)")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp_all add: fresh_atm abs_fresh subst_fresh)
     apply(simp add: forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (AndL1 z1 M z2)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},z1,y,Q,M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}})")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndL2 z1 M z2)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},z1,y,Q,M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}})")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (OrL z1 M z2 N z3)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},z2,z3,a,y,Q,
                                      P{b:=(y).Q},N{x:=<a>.P},N{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},z1)")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp_all add: fresh_atm subst_fresh)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (OrR1 c1 M c2)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c2:=(y).Q},a,P{c2:=(y).Q},c1,
                                                      M{c2:=(y).Q}{x:=<a>.P{c2:=(y).Q}})")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm)
     apply(simp_all add: fresh_atm subst_fresh abs_fresh)
     apply(simp add: forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrR2 c1 M c2)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c2:=(y).Q},a,P{c2:=(y).Q},c1,
                                                      M{c2:=(y).Q}{x:=<a>.P{c2:=(y).Q}})")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm)
     apply(simp_all add: fresh_atm subst_fresh abs_fresh)
     apply(simp add: forget)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (ImpR z c M d)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{d:=(y).Q},a,P{d:=(y).Q},c,
                                                      M{d:=(y).Q}{x:=<a>.P{d:=(y).Q}})")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm)
     apply(simp_all add: fresh_atm subst_fresh forget abs_fresh)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (ImpL c M z N u)
   then show ?case  
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)
     apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,P{b:=(y).Q},M{u:=<a>.P},N{u:=<a>.P},y,Q,
                         M{b:=(y).Q}{u:=<a>.P{b:=(y).Q}},N{b:=(y).Q}{u:=<a>.P{b:=(y).Q}},z)")
     apply(erule exE)
     apply(simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp_all add: fresh_atm subst_fresh forget)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 qed
 
 lemma subst_subst2:
   assumes a: "a\<sharp>(b,P,N)" "x\<sharp>(y,P,M)" "b\<sharp>(M,N)" "y\<sharp>P"
   shows "M{a:=(x).N}{y:=<b>.P} = M{y:=<b>.P}{a:=(x).N{y:=<b>.P}}"
 using a
 proof(nominal_induct M avoiding: a x N y b P rule: trm.strong_induct)
   case (Ax z c)
   then show ?case
-    by (auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    by (auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
 next
   case (Cut d M' u M'')
   then show ?case
     apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh)
     apply(auto)
     apply(simp add: fresh_atm)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: forget)
     apply(simp add: fresh_atm)
     apply(case_tac "a\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_prod fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(auto)[1]
     apply(case_tac "y\<sharp>M''")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     apply(simp add: forget)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh)
     apply(auto)[1]
     apply(case_tac "y\<sharp>M''")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     apply(case_tac "a\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh)
     apply(simp add: subst_fresh abs_fresh)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh fresh_atm)
     apply(simp add: subst_fresh abs_fresh)
     apply(auto)[1]
     apply(case_tac "y\<sharp>M''")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     done
 next
   case (NotR z M' d) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(y,P,N,N{y:=<b>.P},M'{d:=(x).N},M'{y:=<b>.P}{d:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotR)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (NotL d M' z) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(z,y,P,N,N{y:=<b>.P},M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndR d M' e M'' f) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,b,d,e,N,N{y:=<b>.P},M'{f:=(x).N},M''{f:=(x).N},
                   M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}},M''{y:=<b>.P}{f:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndR)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(simp)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (AndL1 z M' u) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,b,z,u,x,N,M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndL2 z M' u) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,b,z,u,x,N,M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (OrL u M' v M'' w) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,b,u,w,v,N,N{y:=<b>.P},M'{y:=<b>.P},M''{y:=<b>.P},
                   M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}},M''{y:=<b>.P}{a:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(simp)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (OrR1 e M' f) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P},
                                         M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR1)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (OrR2 e M' f) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P},
                                         M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrR2)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     done
 next
   case (ImpR x e M' f) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P},
                                         M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpR)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(simp add: fresh_atm trm.inject alpha abs_fresh fin_supp abs_supp)
     apply(rule exists_fresh'(2)[OF fs_coname1])
     apply(simp add: fresh_atm trm.inject alpha abs_fresh fin_supp abs_supp)
     done
 next
   case (ImpL e M' v M'' w) 
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject)
     apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,b,e,w,v,N,N{y:=<b>.P},M'{w:=<b>.P},M''{w:=<b>.P},
                   M'{w:=<b>.P}{a:=(x).N{w:=<b>.P}},M''{w:=<b>.P}{a:=(x).N{w:=<b>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 qed
 
 lemma subst_subst3:
   assumes a: "a\<sharp>(P,N,c)" "c\<sharp>(M,N)" "x\<sharp>(y,P,M)" "y\<sharp>(P,x)" "M\<noteq>Ax y a"
   shows "N{x:=<a>.M}{y:=<c>.P} = N{y:=<c>.P}{x:=<a>.(M{y:=<c>.P})}"
 using a
 proof(nominal_induct N avoiding: x y a c M P rule: trm.strong_induct)
   case (Ax z c)
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (Cut d M' u M'')
   then show ?case
     apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh)
     apply(auto)
     apply(simp add: fresh_atm)
     apply(simp add: trm.inject)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(subgoal_tac "P \<noteq> Ax x c")
     apply(simp)
     apply(simp add: forget)
     apply(clarify)
     apply(simp add: fresh_atm)
     apply(case_tac "x\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(auto)
     apply(case_tac "y\<sharp>M'")
     apply(simp add: forget)
     apply(simp add: not_Ax2)
     done
 next
   case NotR
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (NotL d M' u)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_NotL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_atm subst_fresh fresh_prod)
     apply(subgoal_tac "P \<noteq> Ax x c")
     apply(simp)
     apply(simp add: forget trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(clarify)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case AndR
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (AndL1 u M' v)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(u,y,v,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,u,v,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_atm subst_fresh fresh_prod)
     apply(subgoal_tac "P \<noteq> Ax x c")
     apply(simp)
     apply(simp add: forget trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(clarify)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case (AndL2 u M' v)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(u,y,v,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,u,v,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_AndL2)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_atm subst_fresh fresh_prod)
     apply(subgoal_tac "P \<noteq> Ax x c")
     apply(simp)
     apply(simp add: forget trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(clarify)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case OrR1
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case OrR2
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (OrL x1 M' x2 M'' x3)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}},
                                       x1,x2,x3,M''{x:=<a>.M},M''{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}},
                                       x1,x2,x3,M''{y:=<c>.P},M''{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_OrL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_atm subst_fresh fresh_prod)
     apply(simp add: fresh_prod fresh_atm)
     apply(auto)
     apply(simp add: fresh_atm)
     apply(simp add: forget trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 next
   case ImpR
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (ImpL d M' x1 M'' x2)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x2:=<a>.M},M'{y:=<c>.P}{x2:=<a>.M{y:=<c>.P}},
                                       x1,x2,M''{x2:=<a>.M},M''{y:=<c>.P}{x2:=<a>.M{y:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{x2:=<c>.P},M'{x2:=<c>.P}{x:=<a>.M{x2:=<c>.P}},
                                       x1,x2,M''{x2:=<c>.P},M''{x2:=<c>.P}{x:=<a>.M{x2:=<c>.P}})")
     apply(erule exE, simp add: fresh_prod)
     apply(erule conjE)+
     apply(simp add: fresh_fun_simp_ImpL)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substn)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp add: fresh_atm subst_fresh fresh_prod)
     apply(simp add: fresh_prod fresh_atm)
     apply(auto)
     apply(simp add: fresh_atm)
     apply(simp add: forget trm.inject alpha)
     apply(rule trans)
     apply(rule substn.simps)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm subst_fresh)
     apply(simp add: fresh_atm)
     apply(rule exists_fresh'(1)[OF fs_name1])
     done
 qed
 
 lemma subst_subst4:
   assumes a: "x\<sharp>(P,N,y)" "y\<sharp>(M,N)" "a\<sharp>(c,P,M)" "c\<sharp>(P,a)" "M\<noteq>Ax x c"
   shows "N{a:=(x).M}{c:=(y).P} = N{c:=(y).P}{a:=(x).(M{c:=(y).P})}"
 using a
 proof(nominal_induct N avoiding: x y a c M P rule: trm.strong_induct)
   case (Ax z c)
   then show ?case
-    by (auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by (auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (Cut d M' u M'')
   then show ?case
     apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh)
     apply(auto)
     apply(simp add: fresh_atm)
     apply(simp add: trm.inject)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: abs_fresh subst_fresh fresh_atm)
     apply(simp add: fresh_prod subst_fresh abs_fresh fresh_atm)
     apply(subgoal_tac "P \<noteq> Ax y a")
     apply(simp)
     apply(simp add: forget)
     apply(clarify)
     apply(simp add: fresh_atm)
     apply(case_tac "a\<sharp>M''")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh)
     apply(simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: fresh_prod subst_fresh fresh_atm)
     apply(simp add: abs_fresh subst_fresh)
     apply(auto)
     apply(case_tac "c\<sharp>M''")
     apply(simp add: forget)
     apply(simp add: not_Ax1)
     done
 next
   case NotL
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (NotR u M' d)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(simp add: abs_fresh subst_fresh)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_prod fresh_atm)
-    apply(auto simp add: fresh_atm fresh_prod)[1]
+    apply(auto simp: fresh_atm fresh_prod)[1]
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: fresh_prod fresh_atm subst_fresh)
     apply(simp add: abs_fresh subst_fresh)
-    apply(auto simp add: fresh_atm)
+    apply(auto simp: fresh_atm)
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
     apply(simp add: fresh_atm subst_fresh)
-    apply(auto simp add: fresh_prod fresh_atm) 
+    apply(auto simp: fresh_prod fresh_atm) 
     done
 next
   case AndL1
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case AndL2
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
 next
   case (AndR d M e M' f)
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
-    apply(generate_fresh "coname")
-    apply(fresh_fun_simp)
-    apply(fresh_fun_simp)
-    apply(simp add: abs_fresh subst_fresh)
-    apply(rule trans)
-    apply(rule better_Cut_substc)
-    apply(simp)
-    apply(simp add: abs_fresh)
-    apply(simp)
-    apply(simp add: trm.inject alpha)
-    apply(rule trans)
-    apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(simp)
-    apply(auto simp add: fresh_atm fresh_prod)[1]
-    apply(generate_fresh "coname")
-    apply(fresh_fun_simp)
-    apply(fresh_fun_simp)
-    apply(rule sym)
-    apply(rule trans)
-    apply(rule better_Cut_substc)
-    apply(simp add: subst_fresh fresh_atm fresh_prod)
-    apply(simp add: abs_fresh subst_fresh)
-    apply(auto simp add: fresh_atm)[1]
-    apply(simp add: trm.inject alpha forget)
-    apply(rule trans)
-    apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(simp)
-    apply(auto simp add: fresh_atm fresh_prod)[1]
-    done
-next
-  case OrL
-  then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
-next
-  case (OrR1 d M' e)
-  then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(simp add: abs_fresh subst_fresh)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(simp)
+    apply(auto simp: fresh_atm fresh_prod)[1]
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh fresh_atm fresh_prod)
     apply(simp add: abs_fresh subst_fresh)
-    apply(auto simp add: fresh_atm)[1]
+    apply(auto simp: fresh_atm)[1]
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(simp)
+    apply(auto simp: fresh_atm fresh_prod)[1]
     done
 next
-  case (OrR2 d M' e)
+  case OrL
   then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+next
+  case (OrR1 d M' e)
+  then show ?case
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(simp add: abs_fresh subst_fresh)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh fresh_atm fresh_prod)
     apply(simp add: abs_fresh subst_fresh)
-    apply(auto simp add: fresh_atm)[1]
+    apply(auto simp: fresh_atm)[1]
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
     done
 next
-  case ImpL
+  case (OrR2 d M' e)
   then show ?case
-    by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
-next
-  case (ImpR u d M' e)
-  then show ?case
-    apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(simp add: abs_fresh subst_fresh)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp)
     apply(simp add: abs_fresh)
     apply(simp)
     apply(simp add: trm.inject alpha)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
     apply(generate_fresh "coname")
     apply(fresh_fun_simp)
     apply(fresh_fun_simp)
     apply(rule sym)
     apply(rule trans)
     apply(rule better_Cut_substc)
     apply(simp add: subst_fresh fresh_atm fresh_prod)
     apply(simp add: abs_fresh subst_fresh)
-    apply(auto simp add: fresh_atm)[1]
+    apply(auto simp: fresh_atm)[1]
     apply(simp add: trm.inject alpha forget)
     apply(rule trans)
     apply(rule substc.simps)
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
-    apply(auto simp add: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    done
+next
+  case ImpL
+  then show ?case
+    by(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+next
+  case (ImpR u d M' e)
+  then show ?case
+    apply(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
+    apply(generate_fresh "coname")
+    apply(fresh_fun_simp)
+    apply(fresh_fun_simp)
+    apply(simp add: abs_fresh subst_fresh)
+    apply(rule trans)
+    apply(rule better_Cut_substc)
+    apply(simp)
+    apply(simp add: abs_fresh)
+    apply(simp)
+    apply(simp add: trm.inject alpha)
+    apply(rule trans)
+    apply(rule substc.simps)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
+    apply(generate_fresh "coname")
+    apply(fresh_fun_simp)
+    apply(fresh_fun_simp)
+    apply(rule sym)
+    apply(rule trans)
+    apply(rule better_Cut_substc)
+    apply(simp add: subst_fresh fresh_atm fresh_prod)
+    apply(simp add: abs_fresh subst_fresh)
+    apply(auto simp: fresh_atm)[1]
+    apply(simp add: trm.inject alpha forget)
+    apply(rule trans)
+    apply(rule substc.simps)
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
+    apply(auto simp: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1]
     done
 qed
 
 end