Class1.thy Interaktion und
PortierbarkeitIsabelle

theory Class1
imports "HOL-Nominal.Nominal"
begin

section \<open>Term-Calculus from Urban's PhD\<close>

atom_decl name coname

text \<open>types\<close>

unbundle no bit_operations_syntax

nominal_datatype ty =
 PR "string"
 | NOT "ty"
 | AND "ty" "ty" (\<open>_ AND _\<close> [100,100] 100)
 | OR "ty" "ty" (\<open>_ OR _\<close> [100,100] 100)
 | IMP "ty" "ty" (\<open>_ IMP _\<close> [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 "(\s. T=PR s) \ (\T'. T=NOT T') \ (\S U. T=S OR U) \ (\S U. T=S AND U) \ (\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\T" and "x\T"
by (nominal_induct T rule: ty.strong_induct)
 (auto simp: fresh_string)

text \<open>terms\<close>

nominal_datatype trm =
 Ax "name" "coname"
 | Cut "\coname\trm" "\name\trm" (\Cut <_>._ ('(_'))._\ [0,0,0,100] 101)
 | NotR "\name\trm" "coname" (\NotR ('(_'))._ _\ [0,100,100] 101)
 | NotL "\coname\trm" "name" (\NotL <_>._ _\ [0,100,100] 101)
 | AndR "\coname\trm" "\coname\trm" "coname" (\AndR <_>._ <_>._ _\ [0,100,0,100,100] 101)
 | AndL1 "\name\trm" "name" (\AndL1 (_)._ _\ [100,100,100] 101)
 | AndL2 "\name\trm" "name" (\AndL2 (_)._ _\ [100,100,100] 101)
 | OrR1 "\coname\trm" "coname" (\OrR1 <_>._ _\ [100,100,100] 101)
 | OrR2 "\coname\trm" "coname" (\OrR2 <_>._ _\ [100,100,100] 101)
 | OrL "\name\trm" "\name\trm" "name" (\OrL (_)._ (_)._ _\ [100,100,100,100,100] 101)
 | ImpR "\name\(\coname\trm)" "coname" (\ImpR (_).<_>._ _\ [100,100,100,100] 101)
 | ImpL "\coname\trm" "\name\trm" "name" (\ImpL <_>._ (_)._ _\ [100,100,100,100,100] 101)

text \<open>named terms\<close>

nominal_datatype ntrm = Na "\name\trm" (\((_):_)\ [100,100] 100)

text \<open>conamed terms\<close>

nominal_datatype ctrm = Co "\coname\trm" (\(<_>:_)\ [100,100] 100)

text \<open>renaming functions\<close>

nominal_primrec (freshness_context: "(d::coname,e::coname)")
 crename :: "trm \ coname \ coname \ trm" (\_[_\c>_]\ [100,0,0] 100)
where
 "(Ax x a)[d\c>e] = (if a=d then Ax x e else Ax x a)"
| "\a\(d,e,N);x\M\ \ (Cut .M (x).N)[d\c>e] = Cut .(M[d\c>e]) (x).(N[d\c>e])"
| "(NotR (x).M a)[d\c>e] = (if a=d then NotR (x).(M[d\c>e]) e else NotR (x).(M[d\c>e]) a)"
| "a\(d,e) \ (NotL .M x)[d\c>e] = (NotL .(M[d\c>e]) x)"
| "\a\(d,e,N,c);b\(d,e,M,c);b\a\ \ (AndR .M .N c)[d\c>e] =
 (if c=d then AndR <a>.(M[d\<turnstile>c>e]) .(N[d \<turnstile>c>e]) e else AndR <a>.(M[d\<turnstile>c>e]) .(N[d\<turnstile>c>e]) c)"
| "x\y \ (AndL1 (x).M y)[d\c>e] = AndL1 (x).(M[d\c>e]) y"
| "x\y \ (AndL2 (x).M y)[d\c>e] = AndL2 (x).(M[d\c>e]) y"
| "a\(d,e,b) \ (OrR1 .M b)[d\c>e] = (if b=d then OrR1 .(M[d\c>e]) e else OrR1 .(M[d\c>e]) b)"
| "a\(d,e,b) \ (OrR2 .M b)[d\c>e] = (if b=d then OrR2 .(M[d\c>e]) e else OrR2 .(M[d\c>e]) b)"
| "\x\(N,z);y\(M,z);y\x\ \ (OrL (x).M (y).N z)[d\c>e] = OrL (x).(M[d\c>e]) (y).(N[d\c>e]) z"
| "a\(d,e,b) \ (ImpR (x)..M b)[d\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)"
| "\a\(d,e,N);x\(M,y)\ \ (ImpL .M (x).N y)[d\c>e] = ImpL .(M[d\c>e]) (x).(N[d\c>e]) y"
 by(finite_guess | simp add: abs_fresh abs_supp fin_supp | fresh_guess)+

nominal_primrec (freshness_context: "(u::name,v::name)")
 nrename :: "trm \ name \ name \ trm" (\_[_\n>_]\ [100,0,0] 100)
where
 "(Ax x a)[u\n>v] = (if x=u then Ax v a else Ax x a)"
| "\a\N;x\(u,v,M)\ \ (Cut .M (x).N)[u\n>v] = Cut .(M[u\n>v]) (x).(N[u\n>v])"
| "x\(u,v) \ (NotR (x).M a)[u\n>v] = NotR (x).(M[u\n>v]) a"
| "(NotL .M x)[u\n>v] = (if x=u then NotL .(M[u\n>v]) v else NotL .(M[u\n>v]) x)"
| "\a\(N,c);b\(M,c);b\a\ \ (AndR .M .N c)[u\n>v] = AndR .(M[u\n>v]) .(N[u\n>v]) c"
| "x\(u,v,y) \ (AndL1 (x).M y)[u\n>v] = (if y=u then AndL1 (x).(M[u\n>v]) v else AndL1 (x).(M[u\n>v]) y)"
| "x\(u,v,y) \ (AndL2 (x).M y)[u\n>v] = (if y=u then AndL2 (x).(M[u\n>v]) v else AndL2 (x).(M[u\n>v]) y)"
| "a\b \ (OrR1 .M b)[u\n>v] = OrR1 .(M[u\n>v]) b"
| "a\b \ (OrR2 .M b)[u\n>v] = OrR2 .(M[u\n>v]) b"
| "\x\(u,v,N,z);y\(u,v,M,z);y\x\ \ (OrL (x).M (y).N z)[u\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)"
| "\a\b; x\(u,v)\ \ (ImpR (x)..M b)[u\n>v] = ImpR (x)..(M[u\n>v]) b"
| "\a\N;x\(u,v,M,y)\ \ (ImpL .M (x).N y)[u\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)"
 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\(M[d\c>e]) = (pi\M)[(pi\d)\c>(pi\e)]"
 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\(M[d\c>e]) = (pi\M)[(pi\d)\c>(pi\e)]"
 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\(M[x\n>y]) = (pi\M)[(pi\x)\n>(pi\y)]"
 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\(M[x\n>y]) = (pi\M)[(pi\x)\n>(pi\y)]"
 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\M"
 shows "M[x\n>y] = M"
using a
by (nominal_induct M avoiding: x y rule: trm.strong_induct)
 (auto simp: trm.inject fresh_atm abs_fresh fin_supp abs_supp)

lemma crename_fresh:
 assumes a: "a\M"
 shows "M[a\c>b] = M"
using a
by (nominal_induct M avoiding: a b rule: trm.strong_induct)
 (auto simp: trm.inject fresh_atm abs_fresh)

lemma nrename_nfresh:
 fixes x::"name"
 shows "x\y\x\M[x\n>y]"
by (nominal_induct M avoiding: x y rule: trm.strong_induct)
 (auto simp: fresh_atm abs_fresh abs_supp fin_supp)

lemma crename_nfresh:
 fixes x::"name"
 shows "x\M\x\M[a\c>b]"
by (nominal_induct M avoiding: a b rule: trm.strong_induct)
 (auto simp: fresh_atm abs_fresh abs_supp fin_supp)

lemma crename_cfresh:
 fixes a::"coname"
 shows "a\b\a\M[a\c>b]"
by (nominal_induct M avoiding: a b rule: trm.strong_induct)
 (auto simp: fresh_atm abs_fresh abs_supp fin_supp)

lemma nrename_cfresh:
 fixes c::"coname"
 shows "c\M\c\M[x\n>y]"
by (nominal_induct M avoiding: x y rule: trm.strong_induct)
 (auto simp: fresh_atm abs_fresh abs_supp fin_supp)

lemma nrename_nfresh':
 fixes x::"name"
 shows "x\(M,z,y)\x\M[z\n>y]"
by (nominal_induct M avoiding: x z y rule: trm.strong_induct)
 (auto simp: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)

lemma crename_cfresh':
 fixes a::"coname"
 shows "a\(M,b,c)\a\M[b\c>c]"
by (nominal_induct M avoiding: a b c rule: trm.strong_induct)
 (auto simp: fresh_prod fresh_atm abs_fresh abs_supp fin_supp)

lemma nrename_rename:
 assumes a: "x'\M"
 shows "([(x',x)]\M)[x'\n>y]= M[x\n>y]"
using a
proof (nominal_induct M avoiding: x x' y rule: trm.strong_induct)
qed (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)

lemma crename_rename:
 assumes a: "a'\M"
 shows "([(a',a)]\M)[a'\c>b]= M[a\c>b]"
using a
proof (nominal_induct M avoiding: a a' b rule: trm.strong_induct)
qed (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)

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\(u,v)"
 shows "(Cut .M (x).N)[u\n>v] = Cut .(M[u\n>v]) (x).(N[u\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 .M (x).N) = (Cut .([(a',a)]\M) (x').([(x',x)]\N))"
 using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
 have "(Cut .([(a',a)]\M) (x').([(x',x)]\N))[u\n>v] =
 Cut <a'>.(([(a',a)]\<bullet>M)[u\<turnstile>n>v]) (x').(([(x',x)]\<bullet>N)[u\<turnstile>n>v])"
 using fs1 fs2
 by (intro nrename.simps; simp add: fresh_left calc_atm)
 also have "\ = Cut .(M[u\n>v]) (x).(N[u\n>v])" using fs1 fs2 a
 by(simp add: trm.inject alpha fresh_prod rename_eqvts calc_atm rename_fresh fresh_atm)
 finally show "(Cut .M (x).N)[u\n>v] = Cut .(M[u\n>v]) (x).(N[u\n>v])" using eq1
 by simp
qed

lemma better_crename_Cut:
 assumes a: "a\(b,c)"
 shows "(Cut .M (x).N)[b\c>c] = Cut .(M[b\c>c]) (x).(N[b\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 .M (x).N) = (Cut .([(a',a)]\M) (x').([(x',x)]\N))"
 using fs1 fs2 by (rule_tac sym, auto simp: trm.inject alpha fresh_atm fresh_prod calc_atm)
 have "(Cut .([(a',a)]\M) (x').([(x',x)]\N))[b\c>c] =
 Cut <a'>.(([(a',a)]\<bullet>M)[b\<turnstile>c>c]) (x').(([(x',x)]\<bullet>N)[b\<turnstile>c>c])"
 using fs1 fs2
 by (intro crename.simps; simp add: fresh_left calc_atm)
 also have "\ = Cut .(M[b\c>c]) (x).(N[b\c>c])" using fs1 fs2 a
 by(simp add: trm.inject alpha calc_atm rename_fresh fresh_atm fresh_prod rename_eqvts)
 finally show "(Cut .M (x).N)[b\c>c] = Cut .(M[b\c>c]) (x).(N[b\c>c])" using eq1
 by simp
qed

lemma crename_id:
 shows "M[a\c>a] = M"
by (nominal_induct M avoiding: a rule: trm.strong_induct) (auto)

lemma nrename_id:
 shows "M[x\n>x] = M"
by (nominal_induct M avoiding: x rule: trm.strong_induct) (auto)

lemma nrename_swap:
 shows "x\M \ [(x,y)]\M = M[y\n>x]"
by (nominal_induct M avoiding: x y rule: trm.strong_induct)
 (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)

lemma crename_swap:
 shows "a\M \ [(a,b)]\M = M[b\c>a]"
by (nominal_induct M avoiding: a b rule: trm.strong_induct)
 (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)

lemma crename_ax:
 assumes a: "M[a\c>b] = Ax x c" "c\a" "c\b"
 shows "M = Ax x c"
using a
proof (nominal_induct M avoiding: a b x c rule: trm.strong_induct)
qed (simp_all add: trm.inject split: if_splits)

lemma nrename_ax:
 assumes a: "M[x\n>y] = Ax z a" "z\x" "z\y"
 shows "M = Ax z a"
using a
proof (nominal_induct M avoiding: x y z a rule: trm.strong_induct)
qed (simp_all add: trm.inject split: if_splits)

text \<open>substitution functions\<close>

lemma fresh_perm_coname:
 fixes c::"coname"
 and pi::"coname prm"
 and M::"trm"
 assumes a: "c\pi" "c\M"
 shows "c\(pi\M)"
 by (simp add: assms fresh_perm_app)

lemma fresh_perm_name:
 fixes x::"name"
 and pi::"name prm"
 and M::"trm"
 assumes a: "x\pi" "x\M"
 shows "x\(pi\M)"
 by (simp add: assms fresh_perm_app)

lemma fresh_fun_simp_NotL:
 assumes a: "x'\P" "x'\M"
 shows "fresh_fun (\x'. Cut .P (x').NotL .M x') = Cut .P (x').NotL .M x'"
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 (\x'. Cut .P (x').NotL .M x')::name set)"
 using a by(finite_guess)
 obtain n::name where n: "n\(c,P,a,M)"
 by (metis assms fresh_atm(3) fresh_prod)
 with assms have "n \ (\x'. Cut .P (x').NotL .M x')"
 by (fresh_guess)
 then show "\b. b \ (\x'. Cut .P (x').NotL .M x', Cut .P (b).NotL .M b)"
 by (metis abs_fresh(1) abs_fresh(5) fresh_prod n trm.fresh(3))
 show "x' \ (\x'. Cut .P (x').NotL .M x')"
 using assms by(fresh_guess)
qed

lemma fresh_fun_NotL[eqvt_force]:
 fixes pi1::"name prm"
 and pi2::"coname prm"
 shows "pi1\fresh_fun (\x'. Cut .P (x').NotL .M x')=
 fresh_fun (pi1\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))"
 and "pi2\fresh_fun (\x'. Cut .P (x').NotL .M x')=
 fresh_fun (pi2\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))"
 apply(perm_simp)
 apply(generate_fresh "name")
 apply(auto simp: fresh_prod fresh_fun_simp_NotL)
 apply (metis fresh_bij(1) fresh_fun_simp_NotL name_prm_coname_def)
 apply(perm_simp)
 apply(subgoal_tac "\n::name. n\(P,M,pi2\P,pi2\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'\P" "z'\M" "z'\x"
 shows "fresh_fun (\z'. Cut .P(z').AndL1 (x).M z') = Cut .P (z').AndL1 (x).M z'"
proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
 obtain n::name where "n\(c,P,x,M)"
 by (meson exists_fresh(1) fs_name1)
 then show "\a. a \ (\z'. Cut .P(z').AndL1 x. M z', Cut .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' \ (\z'. Cut .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\fresh_fun (\z'. Cut .P (z').AndL1 (x).M z')=
 fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))"
 and "pi2\fresh_fun (\z'. Cut .P (z').AndL1 (x).M z')=
 fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))"
apply(perm_simp)
apply(subgoal_tac "\n::name. n\(P,M,x,pi1\P,pi1\M,pi1\x,pi1)")
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 "\n::name. n\(P,M,x,pi2\P,pi2\M,pi2\x,pi2)")
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'\P" "z'\M" "z'\x"
 shows "fresh_fun (\z'. Cut .P (z').AndL2 (x).M z') = Cut .P (z').AndL2 (x).M z'"
proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
 obtain n::name where "n\(c,P,x,M)"
 by (meson exists_fresh(1) fs_name1)
 then show "\a. a \ (\z'. Cut .P(z').AndL2 x. M z', Cut .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' \ (\z'. Cut .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\fresh_fun (\z'. Cut .P (z').AndL2 (x).M z')=
 fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))"
 and "pi2\fresh_fun (\z'. Cut .P (z').AndL2 (x).M z')=
 fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))"
 apply(perm_simp)
 apply(subgoal_tac "\n::name. n\(P,M,x,pi1\P,pi1\M,pi1\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 "\n::name. n\(P,M,x,pi2\P,pi2\M,pi2\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'\P" "z'\M" "z'\N" "z'\u" "z'\x"
 shows "fresh_fun (\z'. Cut .P(z').OrL(x).M(u).N z') = Cut .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\(c,P,x,M,u,N)"
 by (meson exists_fresh(1) fs_name1)
 then show "\a. a \ (\z'. Cut .P(z').OrL(x).M(u).N(z'), Cut .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' \ (\z'. Cut .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\fresh_fun (\z'. Cut .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\fresh_fun (\z'. Cut .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 "\n::name. n\(P,M,N,x,u,pi1\P,pi1\M,pi1\N,pi1\x,pi1\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 "\n::name. n\(P,M,N,x,u,pi2\P,pi2\M,pi2\N,pi2\x,pi2\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'\P" "z'\M" "z'\N" "z'\x"
 shows "fresh_fun (\z'. Cut .P (z').ImpL .M (x).N z') = Cut .P (z').ImpL .M (x).N z'"
proof (rule fresh_fun_app [OF pt_name_inst at_name_inst])
 obtain n::name where "n\(c,P,x,M,N)"
 by (meson exists_fresh(1) fs_name1)
 then show "\aa. aa \ (\z'. Cut .P(z').ImpL .M(x).N z', Cut .P(aa).ImpL .M(x).N aa)"
 apply(intro exI [where x="n"])
 apply(simp add: fresh_prod abs_fresh)
 apply(fresh_guess)
 done
next
 show "z' \ (\z'. Cut .P(z').ImpL .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\fresh_fun (\z'. Cut .P (z').ImpL .M (x).N z')=
 fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))"
 and "pi2\fresh_fun (\z'. Cut .P (z').ImpL .M (x).N z')=
 fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))"
 apply(perm_simp)
 apply(subgoal_tac "\n::name. n\(P,M,N,x,pi1\P,pi1\M,pi1\N,pi1\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 "\n::name. n\(P,M,N,x,pi2\P,pi2\M,pi2\N,pi2\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'\P" "a'\M"
 shows "fresh_fun (\a'. Cut .(NotR (y).M a') (x).P) = Cut .(NotR (y).M a') (x).P"
proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
 obtain n::coname where "n\(x,P,y,M)"
 by (metis assms(1) assms(2) fresh_atm(4) fresh_prod)
 then show "\a. a \ (\a'. Cut .(NotR (y).M a') (x).P, Cut .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\fresh_fun (\a'. Cut .(NotR (y).M a') (x).P)=
 fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))"
 and "pi2\fresh_fun (\a'. Cut .(NotR (y).M a') (x).P)=
 fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))"
 apply(perm_simp)
 apply(subgoal_tac "\n::coname. n\(P,M,pi1\P,pi1\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 "\n::coname. n\(P,M,pi2\P,pi2\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'\P" "a'\M" "a'\N" "a'\b" "a'\c"
 shows "fresh_fun (\a'. Cut .(AndR .M .N a') (x).P) = Cut .(AndR .M .N a') (x).P"
proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
 obtain n::coname where "n\(x,P,b,M,c,N)"
 by (meson exists_fresh(2) fs_coname1)
 then show "\a. a \ (\a'. Cut .AndR .M .N(a') (x).P, Cut .AndR .M .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\fresh_fun (\a'. Cut .(AndR .M .N a') (x).P)=
 fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(AndR .M <c>.N a') (x).P))"
 and "pi2\fresh_fun (\a'. Cut .(AndR .M .N a') (x).P)=
 fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(AndR .M <c>.N a') (x).P))"
 apply(perm_simp)
 apply(subgoal_tac "\n::coname. n\(P,M,N,b,c,pi1\P,pi1\M,pi1\N,pi1\b,pi1\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 "\n::coname. n\(P,M,N,b,c,pi2\P,pi2\M,pi2\N,pi2\b,pi2\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'\P" "a'\M" "a'\b"
 shows "fresh_fun (\a'. Cut .(OrR1 .M a') (x).P) = Cut .(OrR1 .M a') (x).P"
proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
 obtain n::coname where "n\(x,P,b,M)"
 by (meson exists_fresh(2) fs_coname1)
 then show "\a. a \ (\a'. Cut .OrR1 .M(a') (x).P, Cut .OrR1 .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\fresh_fun (\a'. Cut .(OrR1 .M a') (x).P) =
 fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 .M a') (x).P))" (is "?t1")
 and "pi2\fresh_fun (\a'. Cut .(OrR1 .M a') (x).P) =
 fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 .M a') (x).P))" (is "?t2")
proof -
 obtain n::coname where "n\(P,M,b,pi1\P,pi1\M,pi1\b,pi1)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t1
 by perm_simp (force simp add: fresh_prod fresh_fun_simp_OrR1 calc_atm)
 obtain n::coname where "n\(P,M,b,pi2\P,pi2\M,pi2\b,pi2)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t2
 by perm_simp
 (force simp add: fresh_prod fresh_fun_simp_OrR1 at_prm_fresh[OF at_coname_inst] swap_simps)
qed

lemma fresh_fun_simp_OrR2:
 assumes "a'\P" "a'\M" "a'\b"
 shows "fresh_fun (\a'. Cut .(OrR2 .M a') (x).P) = Cut .(OrR2 .M a') (x).P"
proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
 obtain n::coname where "n\(x,P,b,M)"
 by (meson exists_fresh(2) fs_coname1)
 then show "\a. a \ (\a'. Cut .OrR2 .M(a') (x).P, Cut .OrR2 .M(a) (x).P)"
 apply(intro exI [where x="n"])
 apply(simp add: fresh_prod abs_fresh)
 apply(fresh_guess)
 done
qed (use assms in \<open>fresh_guess|finite_guess\<close>)+

lemma fresh_fun_OrR2[eqvt_force]:
 fixes pi1::"name prm"
 and pi2::"coname prm"
 shows "pi1\fresh_fun (\a'. Cut .(OrR2 .M a') (x).P) =
 fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 .M a') (x).P))" (is "?t1")
 and "pi2\fresh_fun (\a'. Cut .(OrR2 .M a') (x).P) =
 fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 .M a') (x).P))" (is "?t2")
proof -
 obtain n::coname where "n\(P,M,b,pi1\P,pi1\M,pi1\b,pi1)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t1
 by perm_simp (force simp add: fresh_prod fresh_fun_simp_OrR2 calc_atm)
 obtain n::coname where "n\(P,M,b,pi2\P,pi2\M,pi2\b,pi2)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t2
 by perm_simp
 (force simp add: fresh_prod fresh_fun_simp_OrR2 at_prm_fresh[OF at_coname_inst] swap_simps)
qed

lemma fresh_fun_simp_ImpR:
 assumes a: "a'\P" "a'\M" "a'\b"
 shows "fresh_fun (\a'. Cut .(ImpR (y)..M a') (x).P) = Cut .(ImpR (y)..M a') (x).P"
proof (rule fresh_fun_app [OF pt_coname_inst at_coname_inst])
 obtain n::coname where "n\(x,P,y,b,M)"
 by (meson exists_fresh(2) fs_coname1)
 then show "\a. a \ (\a'. Cut .(ImpR (y)..M a') (x).P, Cut .(ImpR (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_ImpR[eqvt_force]:
 fixes pi1::"name prm"
 and pi2::"coname prm"
 shows "pi1\fresh_fun (\a'. Cut .(ImpR (y)..M a') (x).P) =
 fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y)..M a') (x).P))" (is "?t1")
 and "pi2\fresh_fun (\a'. Cut .(ImpR (y)..M a') (x).P)=
 fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y)..M a') (x).P))" (is "?t2")
proof -
 obtain n::coname where "n\(P,M,b,pi1\P,pi1\M,pi1\b,pi1)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t1
 by perm_simp (force simp add: fresh_prod fresh_fun_simp_ImpR calc_atm)
 obtain n::coname where "n\(P,M,b,pi2\P,pi2\M,pi2\b,pi2)"
 by (meson exists_fresh(2) fs_coname1)
 then show ?t2
 by perm_simp
 (force simp add: fresh_prod fresh_fun_simp_ImpR at_prm_fresh[OF at_coname_inst] swap_simps)
qed

nominal_primrec (freshness_context: "(y::name,c::coname,P::trm)")
 substn :: "trm \ name \ coname \ trm \ trm" (\_{_:=<_>._}\ [100,100,100,100] 100)
where
 "(Ax x a){y:=.P} = (if x=y then Cut .P (y).Ax y a else Ax x a)"
| "\a\(c,P,N);x\(y,P,M)\ \ (Cut .M (x).N){y:=.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\(y,P) \ (NotR (x).M a){y:=.P} = NotR (x).(M{y:=.P}) a"
| "a\(c,P) \ (NotL .M x){y:=.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)"
| "\a\(c,P,N,d);b\(c,P,M,d);b\a\ \
 (AndR <a>.M .N d){y:=<c>.P} = AndR <a>.(M{y:=<c>.P}) .(N{y:=<c>.P}) d"
| "x\(y,P,z) \ (AndL1 (x).M z){y:=.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\(y,P,z) \ (AndL2 (x).M z){y:=.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\(c,P,b) \ (OrR1 .M b){y:=.P} = OrR1 .(M{y:=.P}) b"
| "a\(c,P,b) \ (OrR2 .M b){y:=.P} = OrR2 .(M{y:=.P}) b"
| "\x\(y,N,P,z);u\(y,M,P,z);x\u\ \ (OrL (x).M (u).N z){y:=.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)"
| "\a\(b,c,P); x\(y,P)\ \ (ImpR (x)..M b){y:=.P} = ImpR (x)..(M{y:=.P}) b"
| "\a\(N,c,P);x\(y,P,M,z)\ \ (ImpL .M (x).N z){y:=.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 | simp add: abs_fresh abs_supp fin_supp | fresh_guess | rule strip)+
apply(subgoal_tac "\x::name. x\(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 "\x::name. x\(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 "\x::name. x\(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 "\x::name. x\(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 "\x::name. x\(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 "\x::name. x\(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 "\x::name. x\(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 \ coname \ name \ trm \ trm" (\_{_:=('(_'))._}\ [100,0,0,100] 100)
where
 "(Ax x a){d:=(z).P} = (if d=a then Cut .(Ax x a) (z).P else Ax x a)"
| "\a\(d,P,N);x\(z,P,M)\ \ (Cut .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\(z,P) \ (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\(d,P) \ (NotL .M x){d:=(z).P} = NotL .(M{d:=(z).P}) x"
| "\a\(P,c,N,d);b\(P,c,M,d);b\a\ \ (AndR .M .N c){d:=(z).P} =
 (if d=c then fresh_fun (\<lambda>a'. Cut <a'>.(AndR <a>.(M{d:=(z).P}) .(N{d:=(z).P}) a') (z).P)
 else AndR <a>.(M{d:=(z).P}) .(N{d:=(z).P}) c)"
| "x\(y,z,P) \ (AndL1 (x).M y){d:=(z).P} = AndL1 (x).(M{d:=(z).P}) y"
| "x\(y,P,z) \ (AndL2 (x).M y){d:=(z).P} = AndL2 (x).(M{d:=(z).P}) y"
| "a\(d,P,b) \ (OrR1 .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\(d,P,b) \ (OrR2 .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)"
| "\x\(N,z,P,u);y\(M,z,P,u);x\y\ \ (OrL (x).M (y).N u){d:=(z).P} =
 OrL (x).(M{d:=(z).P}) (y).(N{d:=(z).P}) u"
| "\a\(b,d,P); x\(z,P)\ \ (ImpR (x)..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)"
| "\a\(N,d,P);x\(y,z,P,M)\ \ (ImpL .M (x).N y){d:=(z).P} =
 ImpL <a>.(M{d:=(z).P}) (x).(N{d:=(z).P}) y"
apply(finite_guess | simp add: abs_fresh abs_supp fs_name1 fs_coname1 | rule strip)+
apply(subgoal_tac "\x::coname. x\(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 "\x::coname. x\(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 "\x::coname. x\(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 "\x::coname. x\(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 "\x::coname. x\(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 "\x::coname. x\(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 "\x::coname. x\(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\(M{c:=(x).N}) = (pi1\M){(pi1\c):=(pi1\x).(pi1\N)}"
 and "pi2\(M{c:=(x).N}) = (pi2\M){(pi2\c):=(pi2\x).(pi2\N)}"
by (nominal_induct M avoiding: c x N rule: trm.strong_induct)
 (auto simp: eq_bij fresh_bij eqvts; perm_simp)+

lemma nsubst_eqvt[eqvt]:
 fixes pi1::"name prm"
 and pi2::"coname prm"
 shows "pi1\(M{x:=.N}) = (pi1\M){(pi1\x):=<(pi1\c)>.(pi1\N)}"
 and "pi2\(M{x:=.N}) = (pi2\M){(pi2\x):=<(pi2\c)>.(pi2\N)}"
by (nominal_induct M avoiding: c x N rule: trm.strong_induct)
 (auto simp: eq_bij fresh_bij eqvts; perm_simp)+

lemma supp_subst1:
 shows "supp (M{y:=.P}) \ ((supp M) - {y}) \ (supp P)"
proof (nominal_induct M avoiding: y P c rule: trm.strong_induct)
 case (NotL coname trm name)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P)"
 by (meson exists_fresh(1) fs_name1)
 with NotL
 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotL; blast)
next
 case (AndL1 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL1 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL1 fresh_atm; blast)
next
 case (AndL2 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL2 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL2 fresh_atm; blast)
next
 case (OrL name1 trm1 name2 trm2 name3)
 obtain x'::name where "x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)"
 by (meson exists_fresh(1) fs_name1)
 with OrL show ?case
 by (auto simp: fs_name1 fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrL fresh_atm; blast)
next
 case (ImpL coname trm1 name1 trm2 name2)
 obtain x'::name where "x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})"
 by (meson exists_fresh(1) fs_name1)
 with ImpL show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpL fresh_atm; blast)
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

text \<open>Identical to the previous proof\<close>
lemma supp_subst2:
 shows "supp (M{y:=.P}) \ supp (M) \ ((supp P) - {c})"
proof (nominal_induct M avoiding: y P c rule: trm.strong_induct)
 case (NotL coname trm name)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P)"
 by (meson exists_fresh(1) fs_name1)
 with NotL
 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotL; blast)
next
 case (AndL1 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL1 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL1 fresh_atm; blast)
next
 case (AndL2 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL2 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL2 fresh_atm; blast)
next
 case (OrL name1 trm1 name2 trm2 name3)
 obtain x'::name where "x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)"
 by (meson exists_fresh(1) fs_name1)
 with OrL show ?case
 by (auto simp: fs_name1 fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrL fresh_atm; blast)
next
 case (ImpL coname trm1 name1 trm2 name2)
 obtain x'::name where "x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})"
 by (meson exists_fresh(1) fs_name1)
 with ImpL show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpL fresh_atm; blast)
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst3:
 shows "supp (M{c:=(x).P}) \ ((supp M) - {c}) \ (supp P)"
proof (nominal_induct M avoiding: x P c rule: trm.strong_induct)
 case (NotR name trm coname)
 obtain x'::coname where "x'\<sharp>(trm{coname:=(x).P},P)"
 by (meson exists_fresh'(2) fs_coname1)
 with NotR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotR; blast)
next
 case (AndR coname1 trm1 coname2 trm2 coname3)
 obtain x'::coname where x': "x'\(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)"
 by (meson exists_fresh'(2) fs_coname1)
 with AndR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndR; blast)
next
 case (OrR1 coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR1 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR1; blast)
next
 case (OrR2 coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR2 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR2; blast)
next
 case (ImpR name coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with ImpR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpR; blast)
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst4:
 shows "supp (M{c:=(x).P}) \ (supp M) \ ((supp P) - {x})"
proof (nominal_induct M avoiding: x P c rule: trm.strong_induct)
 case (NotR name trm coname)
 obtain x'::coname where "x'\<sharp>(trm{coname:=(x).P},P)"
 by (meson exists_fresh'(2) fs_coname1)
 with NotR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotR; blast)
next
 case (AndR coname1 trm1 coname2 trm2 coname3)
 obtain x'::coname where x': "x'\(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)"
 by (meson exists_fresh'(2) fs_coname1)
 with AndR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndR; blast)
next
 case (OrR1 coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR1 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR1; blast)
next
 case (OrR2 coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR2 show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR2; blast)
next
 case (ImpR name coname1 trm coname2)
 obtain x'::coname where x': "x'\(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with ImpR show ?case
 by (auto simp: fresh_prod abs_supp supp_atm trm.supp fs_name1 fresh_fun_simp_ImpR; blast)
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst5:
 shows "(supp M - {y}) \ supp (M{y:=.P})"
proof (nominal_induct M avoiding: y P c rule: trm.strong_induct)
 case (NotL coname trm name)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P)"
 by (meson exists_fresh(1) fs_name1)
 with NotL
 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotL)
 apply (auto simp: fresh_def)
 done
next
 case (AndL1 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL1 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL1)
 apply (auto simp: fresh_def)
 done
next
 case (AndL2 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL2 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL2)
 apply (auto simp: fresh_def)
 done
next
 case (OrL name1 trm1 name2 trm2 name3)
 obtain x'::name where "x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)"
 by (meson exists_fresh(1) fs_name1)
 with OrL show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrL)
 apply (fastforce simp: fresh_def)+
 done
next
 case (ImpL coname trm1 name1 trm2 name2)
 obtain x'::name where "x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})"
 by (meson exists_fresh(1) fs_name1)
 with ImpL show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpL)
 apply (fastforce simp: fresh_def)+
 done
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst6:
 shows "(supp M) \ ((supp (M{y:=.P}))::coname set)"
proof (nominal_induct M avoiding: y P c rule: trm.strong_induct)
 case (NotL coname trm name)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P)"
 by (meson exists_fresh(1) fs_name1)
 with NotL
 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotL)
 apply (auto simp: fresh_def)
 done
next
 case (AndL1 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL1 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL1)
 apply (auto simp: fresh_def)
 done
next
 case (AndL2 name1 trm name2)
 obtain x'::name where "x'\<sharp>(trm{y:=<c>.P},P,name1)"
 by (meson exists_fresh(1) fs_name1)
 with AndL2 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndL2)
 apply (auto simp: fresh_def)
 done
next
 case (OrL name1 trm1 name2 trm2 name3)
 obtain x'::name where "x'\<sharp>(trm1{y:=<c>.P},P,name1,trm2{y:=<c>.P},name2)"
 by (meson exists_fresh(1) fs_name1)
 with OrL show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrL)
 apply (fastforce simp: fresh_def)+
 done
next
 case (ImpL coname trm1 name1 trm2 name2)
 obtain x'::name where "x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})"
 by (meson exists_fresh(1) fs_name1)
 with ImpL show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpL)
 apply (fastforce simp: fresh_def)+
 done
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst7:
 shows "(supp M - {c}) \ supp (M{c:=(x).P})"
proof (nominal_induct M avoiding: x P c rule: trm.strong_induct)
 case (NotR name trm coname)
 obtain x'::coname where "x'\<sharp>(trm{coname:=(x).P},P)"
 by (meson exists_fresh'(2) fs_coname1)
 with NotR show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotR)
 apply (auto simp: fresh_def)
 done
next
 case (AndR coname1 trm1 coname2 trm2 coname3)
 obtain x'::coname where "x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)"
 by (meson exists_fresh'(2) fs_coname1)
 with AndR show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndR)
 apply (fastforce simp: fresh_def)+
 done
next
 case (OrR1 coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR1 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR1)
 apply (auto simp: fresh_def)
 done
next
 case (OrR2 coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR2 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR2)
 apply (auto simp: fresh_def)
 done
next
 case (ImpR name coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with ImpR show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_ImpR)
 apply (auto simp: fresh_def)
 done
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

lemma supp_subst8:
 shows "(supp M) \ ((supp (M{c:=(x).P}))::name set)"
proof (nominal_induct M avoiding: x P c rule: trm.strong_induct)
 case (NotR name trm coname)
 obtain x'::coname where "x'\<sharp>(trm{coname:=(x).P},P)"
 by (meson exists_fresh'(2) fs_coname1)
 with NotR show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_NotR)
 apply (auto simp: fresh_def)
 done
next
 case (AndR coname1 trm1 coname2 trm2 coname3)
 obtain x'::coname where "x'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)"
 by (meson exists_fresh'(2) fs_coname1)
 with AndR show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_AndR)
 apply (fastforce simp: fresh_def)+
 done
next
 case (OrR1 coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR1 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR1)
 apply (auto simp: fresh_def)
 done
next
 case (OrR2 coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with OrR2 show ?case
 apply (auto simp: fresh_prod abs_supp supp_atm trm.supp fresh_fun_simp_OrR2)
 apply (auto simp: fresh_def)
 done
next
 case (ImpR name coname1 trm coname2)
 obtain x'::coname where "x'\<sharp>(trm{coname2:=(x).P},P,coname1)"
 by (meson exists_fresh'(2) fs_coname1)
 with ImpR show ?case
 by (force simp: fresh_prod abs_supp fs_name1 supp_atm trm.supp fresh_fun_simp_ImpR)
qed (simp add: abs_supp supp_atm trm.supp fs_name1; blast)+

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\P \ x\M{x:=.P}"
 and "b\P \ b\M{b:=(y).P}"
 and "x\(M,P) \ x\M{y:=.P}"
 and "x\M \ x\M{c:=(x).P}"
 and "x\(M,P) \ x\M{c:=(y).P}"
 and "b\(M,P) \ b\M{c:=(y).P}"
 and "b\M \ b\M{y:=.P}"
 and "b\(M,P) \ b\M{y:=.P}"
 using subst_supp
 by(fastforce simp add: fresh_def supp_prod)+

lemma forget:
 shows "x\M \ M{x:=.P} = M"
 and "c\M \ M{c:=(x).P} = M"
 by (nominal_induct M avoiding: x c P rule: trm.strong_induct)
 (auto simp: fresh_atm abs_fresh abs_supp fin_supp)

lemma substc_rename1:
 assumes a: "c\(M,a)"
 shows "M{a:=(x).N} = ([(c,a)]\M){c:=(x).N}"
using a
proof(nominal_induct M avoiding: c a x N rule: trm.strong_induct)
 case (AndR c1 M c2 M' c3)
 then show ?case
 apply(auto simp: fresh_prod calc_atm fresh_atm abs_fresh fresh_left)
 apply (metis (no_types, lifting))+
 done
next
 case ImpL
 then show ?case
 by (auto simp: calc_atm alpha fresh_atm abs_fresh fresh_prod fresh_left)
 metis
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 (auto simp: calc_atm alpha fresh_atm abs_fresh fresh_prod fresh_left trm.inject)

lemma substc_rename2:
 assumes a: "y\(N,x)"
 shows "M{a:=(x).N} = M{a:=(y).([(y,x)]\N)}"
using a
proof(nominal_induct M avoiding: a x y N rule: trm.strong_induct)
 case (NotR y M d)
 obtain a::coname where "a\(N,M{d:=(y).([(y,x)]\N)},[(y,x)]\N)"
 by (meson exists_fresh(2) fs_coname1)
 with NotR show ?case
 apply(auto simp: calc_atm alpha fresh_atm fresh_prod fresh_left)
 by (metis (no_types, opaque_lifting) alpha(1) trm.inject(2))
next
 case (AndR c1 M c2 M' c3)
 obtain a'::coname where "a'\<sharp>(N,M{c3:=(y).([(y,x)]\<bullet>N)},M'{c3:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,c1,c2,c3)"
 by (meson exists_fresh(2) fs_coname1)
 with AndR show ?case
 apply(auto simp: calc_atm alpha fresh_atm fresh_prod fresh_left)
 by (metis (no_types, opaque_lifting) alpha(1) trm.inject(2))
next
 case (OrR1 d M e)
 obtain a'::coname where "a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)"
 by (meson exists_fresh(2) fs_coname1)
 with OrR1 show ?case
 by (auto simp: perm_swap calc_atm trm.inject alpha fresh_atm fresh_prod fresh_left fresh_fun_simp_OrR1)
next
 case (OrR2 d M e)
 obtain a'::coname where "a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)"
 by (meson exists_fresh(2) fs_coname1)
 with OrR2 show ?case
 by (auto simp: perm_swap calc_atm trm.inject alpha fresh_atm fresh_prod fresh_left fresh_fun_simp_OrR2)
next
 case (ImpR y d M e)
 obtain a'::coname where "a'\<sharp>(N,M{e:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>N,d,e)"
 by (meson exists_fresh(2) fs_coname1)
 with ImpR show ?case
 by (auto simp: perm_swap calc_atm trm.inject alpha fresh_atm fresh_prod fresh_left fresh_fun_simp_ImpR)
qed (auto simp: calc_atm trm.inject alpha fresh_atm fresh_prod fresh_left perm_swap)

lemma substn_rename3:
 assumes a: "y\(M,x)"
 shows "M{x:=.N} = ([(y,x)]\M){y:=.N}"
using a
proof(nominal_induct M avoiding: a x y N rule: trm.strong_induct)
 case (OrL x1 M x2 M' x3)
 then show ?case
 apply(auto simp add: calc_atm fresh_atm abs_fresh fresh_prod fresh_left)
 by (metis (mono_tags))+
next
 case (ImpL d M v M' u)
 then show ?case
 apply(auto simp add: calc_atm fresh_atm abs_fresh fresh_prod fresh_left)
 by (metis (mono_tags))+
next
 case (Cut d M y M')
 then show ?case
 apply(auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
 by (metis nrename.simps(1) nrename_id nrename_rename)+
qed (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_left abs_supp fin_supp fresh_prod)

lemma substn_rename4:
 assumes a: "c\(N,a)"
 shows "M{x:=.N} = M{x:=.([(c,a)]\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: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
next
 case NotR
 then show ?case
 by (auto simp: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left)
next
 case (NotL d M y)
 then obtain a'::name where "a'\<sharp>(N, M{x:=<c>.([(c,a)]\<bullet>N)}, [(c,a)]\<bullet>N)"
 by (meson exists_fresh(1) fs_name1)
 with NotL show ?case
 apply(auto simp: calc_atm trm.inject fresh_atm fresh_prod fresh_left)
 by (metis (no_types, opaque_lifting) alpha(2) trm.inject(2))
next
 case (OrL x1 M x2 M' x3)
 then obtain a'::name where "a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},M'{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,x1,x2,x3)"
 by (meson exists_fresh(1) fs_name1)
 with OrL show ?case
 apply(auto simp: calc_atm trm.inject fresh_atm fresh_prod fresh_left)
 by (metis (no_types) alpha'(2) trm.inject(2))
next
 case (AndL1 u M v)
 then obtain a'::name where "a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,u,v)"
 by (meson exists_fresh(1) fs_name1)
 with AndL1 show ?case
 apply(auto simp: calc_atm trm.inject fresh_atm fresh_prod fresh_left)
 by (metis (mono_tags, opaque_lifting) alpha'(2) trm.inject(2))
next
 case (AndL2 u M v)
 then obtain a'::name where "a'\<sharp>(N,M{x:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,u,v)"
 by (meson exists_fresh(1) fs_name1)
 with AndL2 show ?case
 apply(auto simp: calc_atm trm.inject fresh_atm fresh_prod fresh_left)
 by (metis (mono_tags, opaque_lifting) alpha'(2) trm.inject(2))
next
 case (ImpL d M y M' u)
 then obtain a'::name where "a'\<sharp>(N,M{u:=<c>.([(c,a)]\<bullet>N)},M'{u:=<c>.([(c,a)]\<bullet>N)},[(c,a)]\<bullet>N,y,u)"
 by (meson exists_fresh(1) fs_name1)
 with ImpL show ?case
 apply(auto simp: calc_atm trm.inject fresh_atm fresh_prod fresh_left)
 by (metis (no_types) alpha'(2) trm.inject(2))
next
 case (Cut d M y M')
 then show ?case
 by (auto simp: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
qed (auto simp: calc_atm fresh_atm fresh_left)

lemma subst_rename5:
 assumes a: "c'\(c,N)" "x'\(x,M)"
 shows "M{x:=.N} = ([(x',x)]\M){x':=.([(c',c)]\N)}"
proof -
 have "M{x:=.N} = ([(x',x)]\M){x':=.N}" using a by (simp add: substn_rename3)
 also have "\ = ([(x',x)]\M){x':=.([(c',c)]\N)}" using a by (simp add: substn_rename4)
 finally show ?thesis by simp
qed

lemma subst_rename6:
 assumes a: "c'\(c,M)" "x'\(x,N)"
 shows "M{c:=(x).N} = ([(c',c)]\M){c':=(x').([(x',x)]\N)}"
proof -
 have "M{c:=(x).N} = ([(c',c)]\M){c':=(x).N}" using a by (simp add: substc_rename1)
 also have "\ = ([(c',c)]\M){c':=(x').([(x',x)]\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\[c].P" "x\(y,P)"
 shows "(Cut .M (x).N){y:=.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 .M (x).N) = (Cut .([(a',a)]\M) (x').([(x',x)]\N))"
 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)]\M)=Ax y a')"
 by (metis perm_swap(4) swap_simps(4) swap_simps(8) trm.perm(13))
 have "(Cut .M (x).N){y:=.P} = (Cut .([(a',a)]\M) (x').([(x',x)]\N)){y:=.P}"
 using eq1 by simp
 also have "\ = (if ([(a',a)]\M)=Ax y a' then Cut .P (x').(([(x',x)]\N){y:=.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: fresh_prod fresh_left calc_atm fresh_atm)
 also have "\ =(if M=Ax y a then Cut .P (x).(N{y:=.P}) else Cut .(M{y:=.P}) (x).(N{y:=.P}))"
 using fs1 fs2 a
 unfolding eq2[symmetric]
 apply(auto simp: trm.inject)
 apply(simp_all add: alpha fresh_atm fresh_prod subst_fresh eqvts perm_fresh_fresh calc_atm)
 apply (simp add: fresh_atm(2) substn_rename4)
 by (metis abs_fresh(2) fresh_atm(2) fresh_prod perm_fresh_fresh(2) substn_rename4)
 finally show ?thesis by simp
qed

lemma better_Cut_substc[simp]:
 assumes a: "a\(c,P)" "x\[y].P"
 shows "(Cut .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 .M (x).N) = (Cut .([(a',a)]\M) (x').([(x',x)]\N))"
 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)]\N)=Ax x' c)"
 by (metis perm_dj(1) perm_swap(1) swap_simps(1) trm.perm(1))
 have "(Cut .M (x).N){c:=(y).P} = (Cut .([(a',a)]\M) (x').([(x',x)]\N)){c:=(y).P}"
 using eq1 by simp
 also have "\ = (if ([(x',x)]\N)=Ax x' c then Cut .(([(a',a)]\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 "\ =(if N=Ax x c then Cut .(M{c:=(y).P}) (y).P else Cut .(M{c:=(y).P}) (x).(N{c:=(y).P}))"
 using fs1 fs2 a
 unfolding eq2[symmetric]
 apply(auto simp: trm.inject)
 apply(simp_all add: alpha fresh_atm fresh_prod subst_fresh eqvts perm_fresh_fresh calc_atm)
 by (metis abs_fresh(1) fresh_atm(1) fresh_prod perm_fresh_fresh(1) substc_rename2)
 finally show ?thesis by simp
qed

lemma better_Cut_substn':
 assumes "a\[c].P" "y\(N,x)" "M\Ax y a"
 shows "(Cut .M (x).N){y:=.P} = Cut .(M{y:=.P}) (x).N"
proof -
 obtain d::name where d: "d \ (M, N, P, a, c, x, y)"
 by (meson exists_fresh(1) fs_name1)
 then have *: "y\([(d,x)]\N)"
 by (metis assms(2) fresh_atm(1) fresh_list_cons fresh_list_nil fresh_perm_name fresh_prod)
 with d have "Cut .M (x).N = Cut .M (d).([(d,x)]\N)"
 by (metis (no_types, lifting) alpha(1) fresh_prodD perm_fresh_fresh(1) trm.inject(2))
 with * d assms show ?thesis
 apply(simp add: fresh_prod)
 by (smt (verit, ccfv_SIG) forget(1) trm.inject(2))
qed

lemma better_NotR_substc:
 assumes a: "d\M"
 shows "(NotR (x).M d){d:=(z).P} = fresh_fun (\a'. Cut .NotR (x).M a' (z).P)"
proof -
 obtain c::name where c: "c \ (M, P, d, x, z)"
 by (meson exists_fresh(1) fs_name1)
 obtain e::coname where e: "e \ (M, P, d, x, z, c)"
 by (meson exists_fresh'(2) fs_coname1)
 with c have "NotR (x).M d = NotR (c).([(c,x)]\M) d"
 by (metis alpha'(1) fresh_prodD(1) nrename_id nrename_swap trm.inject(3))
 with c e assms show ?thesis
 apply(simp add: fresh_prod)
 by (metis forget(2) fresh_perm_app(3) trm.inject(3))
qed

lemma better_NotL_substn:
 assumes a: "y\M"
 shows "(NotL .M y){y:=.P} = fresh_fun (\x'. Cut .P (x').NotL .M x')"
proof (generate_fresh "coname")
 fix ca :: coname
 assume d: "ca \ (M, P, a, c, y)"
 then have "NotL .M y = NotL .([(ca,a)]\M) y"
 by (metis alpha(2) fresh_prod perm_fresh_fresh(2) trm.inject(4))
 with a d show ?thesis
 apply(simp add: fresh_left calc_atm forget)
 apply (metis trm.inject(4))
 done
qed

lemma better_AndL1_substn:
 assumes a: "y\[x].M"
 shows "(AndL1 (x).M y){y:=.P} = fresh_fun (\z'. Cut .P (z').AndL1 (x).M z')"
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