--- a/src/HOL/Nominal/Examples/Class1.thy Thu May 23 11:18:46 2024 +0200
+++ b/src/HOL/Nominal/Examples/Class1.thy Thu May 23 20:22:52 2024 +0200
@@ -1,5 +1,5 @@
theory Class1
-imports "HOL-Nominal.Nominal"
+imports "HOL-Nominal.Nominal"
begin
section \<open>Term-Calculus from Urban's PhD\<close>
@@ -44,9 +44,9 @@
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"
@@ -89,11 +89,7 @@
| "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)
@@ -112,41 +108,29 @@
| "\<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
@@ -155,68 +139,64 @@
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)
-done
+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'\<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)
-done
+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
@@ -230,21 +210,13 @@
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
@@ -256,21 +228,13 @@
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
@@ -286,28 +250,28 @@
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)
+ (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)
+
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)
+ (auto simp: abs_fresh abs_supp fin_supp fresh_left calc_atm fresh_atm)
+
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
+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\<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
+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>
@@ -317,11 +281,7 @@
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"
@@ -329,30 +289,27 @@
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 have "n \<sharp> (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')"
+ by (fresh_guess)
+ then show "\<exists>b. b \<sharp> (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x', Cut <c>.P (b).NotL <a>.M b)"
+ by (metis abs_fresh(1) abs_fresh(5) fresh_prod n trm.fresh(3))
+ 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"
@@ -361,48 +318,30 @@
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 (metis fresh_bij(1) fresh_fun_simp_NotL name_prm_coname_def)
+ 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"
@@ -411,41 +350,30 @@
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"
@@ -454,84 +382,62 @@
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"
@@ -540,41 +446,27 @@
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"
@@ -583,41 +475,27 @@
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"
@@ -626,41 +504,27 @@
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"
@@ -669,41 +533,27 @@
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"
@@ -712,41 +562,27 @@
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"
@@ -755,22 +591,14 @@
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)
@@ -798,50 +626,40 @@
| "\<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
@@ -870,801 +688,377 @@
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(perm_simp)+
-done
+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\<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(perm_simp)+
-done
+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:=<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(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
-
+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:=<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(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
+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}) \<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(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
+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'\<sharp>(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'\<sharp>(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'\<sharp>(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'\<sharp>(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}) \<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(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
+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'\<sharp>(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'\<sharp>(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'\<sharp>(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'\<sharp>(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}) \<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(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
+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) \<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(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
+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}) \<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(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
-
+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) \<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(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
-
+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:
@@ -1678,17 +1072,14 @@
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
+ using subst_supp
+ by(fastforce simp add: fresh_def supp_prod)+
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:
@@ -1696,198 +1087,81 @@
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)
-next
- case NotL
- then show ?case by (auto simp add: 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)
-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 (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)
-next
- case AndL2
- then show ?case by (auto simp add: 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)
-next
- case (OrR2 d M e)
- then show ?case
- by(auto simp add: 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)
+ 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 add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+ by (auto simp: calc_atm 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)
-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
+qed (auto simp: calc_atm alpha fresh_atm abs_fresh fresh_prod fresh_left trm.inject)
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)
-next
- case NotL
- then show ?case
- by (auto simp add: 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(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
+ obtain a::coname where "a\<sharp>(N,M{d:=(y).([(y,x)]\<bullet>N)},[(y,x)]\<bullet>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)
- then show ?case
- apply(auto simp add: 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(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)
-next
- case AndL2
- then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+ 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)
- then show ?case
- apply(auto simp add: 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
+ 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)
- then show ?case
- apply(auto simp add: 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)
-next
- case ImpL
- then show ?case
- by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+ 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)
- then show ?case
- apply(auto simp add: 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)
-qed
+ 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\<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)
-next
- case NotR
- then show ?case by (auto simp add: 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)
-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)
-next
- case OrR1
- then show ?case by (auto simp add: 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)
-next
- case (AndL1 u M v)
- then show ?case
- by(auto simp add: 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)
-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)
+ 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
- by(simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
- (metis (poly_guards_query))
+ 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 add: 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
+ 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\<sharp>(N,a)"
@@ -1896,86 +1170,51 @@
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(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
+ 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 show ?case
- apply(auto simp add: 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(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)
-next
- case OrR2
- then show ?case by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+ 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 show ?case
- apply(auto simp add: 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
+ 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 show ?case
- apply(auto simp add: 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)
-next
- case ImpR
- then show ?case
- by (auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left)
+ 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 show ?case
- apply(auto simp add: 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
+ 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 add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left perm_swap)
-qed
+ 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'\<sharp>(c,N)" "x'\<sharp>(x,M)"
@@ -2005,30 +1244,20 @@
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(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
- apply(simp add: calc_atm)
- done
+ by (metis perm_swap(4) swap_simps(4) swap_simps(8) trm.perm(13))
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(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
+ 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(2) fresh_atm(2) fresh_prod perm_fresh_fresh(2) substn_rename4)
finally show ?thesis by simp
qed
@@ -2040,12 +1269,9 @@
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(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
- apply(simp add: calc_atm)
- done
+ by (metis perm_dj(1) perm_swap(1) swap_simps(1) trm.perm(1))
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
@@ -2053,52 +1279,42 @@
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(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
+ 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: "a\<sharp>[c].P" "y\<sharp>(N,x)" "M\<noteq>Ax y a"
+ assumes "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(simp add: trm.inject)
-apply(rule sym)
-apply(simp add: alpha fresh_prod fresh_atm)
-done
+proof -
+ obtain d::name where d: "d \<sharp> (M, N, P, a, c, x, y)"
+ by (meson exists_fresh(1) fs_name1)
+ then have *: "y\<sharp>([(d,x)]\<bullet>N)"
+ by (metis assms(2) fresh_atm(1) fresh_list_cons fresh_list_nil fresh_perm_name fresh_prod)
+ with d have "Cut <a>.M (x).N = Cut <a>.M (d).([(d,x)]\<bullet>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\<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(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
+proof -
+ obtain c::name where c: "c \<sharp> (M, P, d, x, z)"
+ by (meson exists_fresh(1) fs_name1)
+ obtain e::coname where e: "e \<sharp> (M, P, d, x, z, c)"
+ by (meson exists_fresh'(2) fs_coname1)
+ with c have "NotR (x).M d = NotR (c).([(c,x)]\<bullet>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\<sharp>M"
@@ -2107,7 +1323,7 @@
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)
@@ -2123,7 +1339,7 @@
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)
@@ -2148,7 +1364,7 @@
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)
@@ -2174,12 +1390,12 @@
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)
@@ -2187,9 +1403,9 @@
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
@@ -2202,12 +1418,12 @@
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)
@@ -2215,9 +1431,9 @@
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
@@ -2229,7 +1445,7 @@
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)
@@ -2253,7 +1469,7 @@
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)
@@ -2278,7 +1494,7 @@
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)
@@ -2303,7 +1519,7 @@
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)
@@ -2317,47 +1533,34 @@
lemma freshn_after_substc:
fixes x::"name"
- assumes a: "x\<sharp>M{c:=(y).P}"
+ assumes "x\<sharp>M{c:=(y).P}"
shows "x\<sharp>M"
-using a supp_subst8
-apply(simp add: fresh_def)
-apply(blast)
-done
+ by (meson assms fresh_def subsetD supp_subst8)
lemma freshn_after_substn:
fixes x::"name"
- assumes a: "x\<sharp>M{y:=<c>.P}" "x\<noteq>y"
+ assumes "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
+ by (meson DiffI assms fresh_def singleton_iff subset_eq supp_subst5)
lemma freshc_after_substc:
fixes a::"coname"
- assumes a: "a\<sharp>M{c:=(y).P}" "a\<noteq>c"
+ assumes "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
+ by (meson Diff_iff assms fresh_def in_mono singleton_iff supp_subst7)
lemma freshc_after_substn:
fixes a::"coname"
- assumes a: "a\<sharp>M{y:=<c>.P}"
+ assumes "a\<sharp>M{y:=<c>.P}"
shows "a\<sharp>M"
-using a supp_subst6
-apply(simp add: fresh_def)
-apply(blast)
-done
+ by (meson assms fresh_def subset_iff supp_subst6)
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")
@@ -2381,11 +1584,11 @@
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)
@@ -2449,7 +1652,7 @@
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)
@@ -2523,7 +1726,7 @@
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)
@@ -2595,7 +1798,7 @@
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)
@@ -2681,152 +1884,82 @@
done
lemma substn_crename_comm':
- assumes a: "a\<noteq>c" "a\<sharp>P"
+ assumes "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
+ using fs2 substn_rename4 by force
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
+ using fs2 by (simp add: substn_rename4)
+ have eq2: "([(c',c)]\<bullet>P)[a\<turnstile>c>b] = ([(c',c)]\<bullet>P)" using fs2 assms
+ by (metis crename_fresh fresh_atm(2) fresh_aux(2) fresh_prod)
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
+ by (simp add: fresh_atm(2) fresh_prod substn_crename_comm)
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"
+ assumes "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
+ using fs2 by (simp add: substc_rename1)
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
+ using fs2 by (metis crename_cfresh' fresh_prod substc_rename1)
+ have eq2: "([(c',c)]\<bullet>M)[a\<turnstile>c>b] = ([(c',c)]\<bullet>(M[a\<turnstile>c>b]))" using fs2 assms
+ by (simp add: crename_coname_eqvt fresh_atm(2) fresh_prod swap_simps(6))
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
+ using fs2 assms
+ by (metis crename_fresh eq eq' eq2 substc_crename_comm)
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> = ([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P}"
+ using assms 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"
+ assumes "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
+ using fs2 by (simp add: substn_rename3)
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
+ using fs2 by (metis fresh_prod nrename_nfresh' substn_rename3)
+ have eq2: "([(x',x)]\<bullet>M)[y\<turnstile>n>z] = ([(x',x)]\<bullet>(M[y\<turnstile>n>z]))"
+ using fs2 by (simp add: assms fresh_atm(1) fresh_prod nrename_name_eqvt swap_simps(5))
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
+ using fs2 by (metis assms eq eq' eq2 nrename_fresh substn_nrename_comm)
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> = ([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P}" using assms 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"
+ assumes "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
+ using substc_rename2 by auto
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
+ using fs2 by (simp add: substc_rename2)
+ have eq2: "([(x',x)]\<bullet>P)[y\<turnstile>n>z] = ([(x',x)]\<bullet>P)"
+ using fs2 by (metis assms(2) fresh_atm(1) fresh_aux(1) fresh_prod nrename_fresh)
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
+ using fs2 by (simp add: fresh_atm(1) fresh_prod substc_nrename_comm)
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
@@ -2865,9 +1998,9 @@
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)
+ 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 add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty)
+ 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>
@@ -2887,59 +2020,36 @@
equivariance fin
lemma fin_Ax_elim:
- assumes a: "fin (Ax x a) y"
+ assumes "fin (Ax x a) y"
shows "x=y"
-using a
-apply(erule_tac fin.cases)
-apply(auto simp add: trm.inject)
-done
+ using assms fin.simps trm.inject(1) by auto
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(subgoal_tac "y\<sharp>[aa].Ma")
-apply(drule sym)
-apply(simp_all add: abs_fresh)
-done
+ using assms
+ by (cases rule: fin.cases; simp add: trm.inject; metis abs_fresh(5))
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)
-done
+ using assms by (cases rule: fin.cases; simp add: trm.inject)
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)
-done
+ using assms by (cases rule: fin.cases; simp add: trm.inject)
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)
-done
+ using assms by (cases rule: fin.cases; simp add: trm.inject)
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(subgoal_tac "y\<sharp>[aa].Ma")
-apply(drule sym)
-apply(simp_all add: abs_fresh)
-apply (metis abs_fresh(5))
-done
+ using assms
+ by (cases rule: fin.cases; simp add: trm.inject; metis abs_fresh(5))
lemma fin_rest_elims:
@@ -2957,7 +2067,7 @@
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)"
@@ -2995,13 +2105,13 @@
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)
@@ -3023,7 +2133,7 @@
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)
@@ -3037,7 +2147,7 @@
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)
@@ -3075,7 +2185,7 @@
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")
@@ -3089,7 +2199,7 @@
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")
@@ -3105,7 +2215,7 @@
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)
@@ -3122,7 +2232,7 @@
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)
@@ -3315,7 +2425,7 @@
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:
@@ -3323,7 +2433,7 @@
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)
@@ -3334,7 +2444,7 @@
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:
@@ -3342,7 +2452,7 @@
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:
@@ -3350,7 +2460,7 @@
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:
@@ -3358,7 +2468,7 @@
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)
@@ -3379,7 +2489,7 @@
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)"
@@ -3734,13 +2844,13 @@
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)
@@ -3754,7 +2864,7 @@
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)
@@ -3768,7 +2878,7 @@
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)
@@ -3786,17 +2896,17 @@
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
@@ -3815,7 +2925,7 @@
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')"
@@ -3827,10 +2937,10 @@
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
@@ -3849,7 +2959,7 @@
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')"
@@ -3861,10 +2971,10 @@
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
@@ -3883,7 +2993,7 @@
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')"
@@ -3895,10 +3005,10 @@
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
@@ -3917,7 +3027,7 @@
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')"
@@ -3929,10 +3039,10 @@
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
@@ -3952,7 +3062,7 @@
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')"
@@ -3973,7 +3083,7 @@
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
@@ -3986,7 +3096,7 @@
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)
@@ -4007,7 +3117,7 @@
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
@@ -4018,7 +3128,7 @@
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
@@ -4030,7 +3140,7 @@
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)
@@ -4099,7 +3209,7 @@
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)
@@ -4136,7 +3246,7 @@
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]
@@ -4148,7 +3258,7 @@
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)
@@ -4161,7 +3271,7 @@
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)
@@ -4174,7 +3284,7 @@
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)
@@ -4206,7 +3316,7 @@
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]
@@ -4218,7 +3328,7 @@
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)
@@ -4231,7 +3341,7 @@
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)
@@ -4244,7 +3354,7 @@
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)
@@ -4278,7 +3388,7 @@
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)
@@ -4292,7 +3402,7 @@
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)
@@ -4326,7 +3436,7 @@
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)
@@ -4339,7 +3449,7 @@
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)
@@ -4373,7 +3483,7 @@
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)
@@ -4386,7 +3496,7 @@
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
@@ -4408,7 +3518,7 @@
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)
@@ -4429,7 +3539,7 @@
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)
@@ -4450,9 +3560,9 @@
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
@@ -4485,11 +3595,11 @@
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
@@ -4504,11 +3614,11 @@
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
@@ -4519,11 +3629,11 @@
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
@@ -4534,11 +3644,11 @@
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
@@ -4549,11 +3659,11 @@
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
@@ -4563,11 +3673,11 @@
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
@@ -4577,11 +3687,11 @@
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
@@ -4592,11 +3702,11 @@
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
@@ -4607,11 +3717,11 @@
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
@@ -4621,11 +3731,11 @@
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
@@ -4635,11 +3745,11 @@
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
@@ -4650,11 +3760,11 @@
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
@@ -4665,11 +3775,11 @@
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
@@ -4679,11 +3789,11 @@
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
@@ -4700,7 +3810,7 @@
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)
@@ -4719,12 +3829,12 @@
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
@@ -4740,12 +3850,12 @@
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
@@ -4761,81 +3871,81 @@
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(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(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,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 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 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 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 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(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(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,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 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 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 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 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(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 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]
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:
@@ -4850,12 +3960,12 @@
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
@@ -4871,12 +3981,12 @@
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
@@ -4892,81 +4002,81 @@
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(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(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,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 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 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 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 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(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(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,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 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 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 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 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(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 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]
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:
@@ -4981,12 +4091,12 @@
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
@@ -5002,12 +4112,12 @@
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
@@ -5023,81 +4133,81 @@
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(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(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,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 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 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 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 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(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 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(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(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(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,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 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 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 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 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(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 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]
done
lemma a_redu_ImpR_elim:
@@ -5112,43 +4222,43 @@
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(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 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="([(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)
@@ -5301,7 +4411,7 @@
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:
@@ -5311,7 +4421,7 @@
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:
@@ -5321,7 +4431,7 @@
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:
@@ -5331,7 +4441,7 @@
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:
@@ -5341,7 +4451,7 @@
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:
@@ -5351,7 +4461,7 @@
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:
@@ -5361,7 +4471,7 @@
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:
@@ -5371,7 +4481,7 @@
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>
@@ -5701,7 +4811,7 @@
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:
@@ -5767,7 +4877,7 @@
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>
@@ -5882,7 +4992,7 @@
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
@@ -5916,7 +5026,7 @@
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
@@ -5966,7 +5076,7 @@
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
@@ -6012,7 +5122,7 @@
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
@@ -6132,7 +5242,7 @@
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
@@ -6182,7 +5292,7 @@
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
@@ -6216,7 +5326,7 @@
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
@@ -6259,7 +5369,7 @@
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
@@ -6288,7 +5398,7 @@
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)
@@ -6360,7 +5470,7 @@
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)
@@ -6442,7 +5552,7 @@
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
@@ -6529,11 +5639,11 @@
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)+
@@ -6543,7 +5653,7 @@
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)")
@@ -6566,11 +5676,11 @@
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)+
@@ -6580,7 +5690,7 @@
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)")
@@ -6593,13 +5703,13 @@
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)+
@@ -6609,7 +5719,7 @@
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)")
@@ -6622,21 +5732,21 @@
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)
@@ -6647,7 +5757,7 @@
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)
@@ -6674,17 +5784,17 @@
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)
@@ -6695,7 +5805,7 @@
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)
@@ -6720,7 +5830,7 @@
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
@@ -6750,12 +5860,12 @@
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)
@@ -6815,11 +5925,11 @@
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)+
@@ -6829,7 +5939,7 @@
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)")
@@ -6852,7 +5962,7 @@
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)
@@ -6863,7 +5973,7 @@
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)
@@ -6890,21 +6000,21 @@
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)+
@@ -6914,7 +6024,7 @@
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)")
@@ -6937,7 +6047,7 @@
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)+
@@ -6947,7 +6057,7 @@
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)")
@@ -6970,15 +6080,15 @@
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)+
@@ -6988,7 +6098,7 @@
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})")
@@ -7007,7 +6117,7 @@
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
@@ -7043,7 +6153,7 @@
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
@@ -7055,14 +6165,14 @@
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
@@ -7078,12 +6188,12 @@
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
@@ -7109,7 +6219,7 @@
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
@@ -7135,7 +6245,7 @@
= (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
@@ -7179,7 +6289,7 @@
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)
@@ -7197,7 +6307,7 @@
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)
@@ -7208,7 +6318,7 @@
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)
@@ -7225,7 +6335,7 @@
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)
@@ -7236,7 +6346,7 @@
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)
@@ -7247,7 +6357,7 @@
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)
@@ -7263,7 +6373,7 @@
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)
@@ -7277,7 +6387,7 @@
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)
@@ -7291,7 +6401,7 @@
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)
@@ -7304,7 +6414,7 @@
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)
@@ -7326,7 +6436,7 @@
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
@@ -7385,7 +6495,7 @@
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)+
@@ -7405,7 +6515,7 @@
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)+
@@ -7427,7 +6537,7 @@
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)
@@ -7450,7 +6560,7 @@
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)+
@@ -7471,7 +6581,7 @@
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)+
@@ -7492,7 +6602,7 @@
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)
@@ -7516,7 +6626,7 @@
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)
@@ -7537,7 +6647,7 @@
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)
@@ -7558,7 +6668,7 @@
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)
@@ -7581,7 +6691,7 @@
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)
@@ -7610,7 +6720,7 @@
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
@@ -7650,11 +6760,11 @@
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)+
@@ -7693,11 +6803,11 @@
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)+
@@ -7736,7 +6846,7 @@
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)+
@@ -7775,15 +6885,15 @@
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)
@@ -7827,11 +6937,11 @@
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)
@@ -7879,7 +6989,7 @@
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
@@ -7919,11 +7029,11 @@
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)
@@ -7937,7 +7047,7 @@
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)
@@ -7946,25 +7056,25 @@
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(auto simp: subst_fresh abs_fresh fresh_atm fresh_prod forget)
apply(generate_fresh "coname")
apply(fresh_fun_simp)
apply(fresh_fun_simp)
@@ -7977,10 +7087,10 @@
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 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)
@@ -7989,23 +7099,23 @@
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 add: fresh_atm fresh_prod)[1]
+ apply(auto simp: 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)
+ by(auto simp: 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)
@@ -8018,8 +7128,8 @@
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)
@@ -8028,17 +7138,17 @@
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 (OrR2 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)
@@ -8051,8 +7161,8 @@
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)
@@ -8061,21 +7171,21 @@
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
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 (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)
@@ -8088,9 +7198,9 @@
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(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)
@@ -8099,14 +7209,14 @@
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]
+ 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