author | wenzelm |
Thu, 22 Apr 2010 22:01:06 +0200 | |
changeset 36277 | 9be4ab2acc13 |
child 39198 | f967a16dfcdd |
permissions | -rw-r--r-- |
theory Class1 imports "../Nominal" begin section {* Term-Calculus from Urban's PhD *} atom_decl name coname text {* types *} nominal_datatype ty = PR "string" | NOT "ty" | AND "ty" "ty" ("_ AND _" [100,100] 100) | OR "ty" "ty" ("_ OR _" [100,100] 100) | IMP "ty" "ty" ("_ IMP _" [100,100] 100) instantiation ty :: size begin nominal_primrec size_ty where "size (PR s) = (1::nat)" | "size (NOT T) = 1 + size T" | "size (T1 AND T2) = 1 + size T1 + size T2" | "size (T1 OR T2) = 1 + size T1 + size T2" | "size (T1 IMP T2) = 1 + size T1 + size T2" by (rule TrueI)+ instance .. end lemma ty_cases: fixes T::ty shows "(\<exists>s. T=PR s) \<or> (\<exists>T'. T=NOT T') \<or> (\<exists>S U. T=S OR U) \<or> (\<exists>S U. T=S AND U) \<or> (\<exists>S U. T=S IMP U)" by (induct T rule:ty.induct) (auto) lemma fresh_ty: fixes a::"coname" and x::"name" and T::"ty" shows "a\<sharp>T" and "x\<sharp>T" by (nominal_induct T rule: ty.strong_induct) (auto simp add: fresh_string) text {* terms *} nominal_datatype trm = Ax "name" "coname" | Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Cut <_>._ (_)._" [100,100,100,100] 100) | NotR "\<guillemotleft>name\<guillemotright>trm" "coname" ("NotR (_)._ _" [100,100,100] 100) | NotL "\<guillemotleft>coname\<guillemotright>trm" "name" ("NotL <_>._ _" [100,100,100] 100) | AndR "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>coname\<guillemotright>trm" "coname" ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100) | AndL1 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL1 (_)._ _" [100,100,100] 100) | AndL2 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL2 (_)._ _" [100,100,100] 100) | OrR1 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR1 <_>._ _" [100,100,100] 100) | OrR2 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR2 <_>._ _" [100,100,100] 100) | OrL "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("OrL (_)._ (_)._ _" [100,100,100,100,100] 100) | ImpR "\<guillemotleft>name\<guillemotright>(\<guillemotleft>coname\<guillemotright>trm)" "coname" ("ImpR (_).<_>._ _" [100,100,100,100] 100) | ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("ImpL <_>._ (_)._ _" [100,100,100,100,100] 100) text {* named terms *} nominal_datatype ntrm = Na "\<guillemotleft>name\<guillemotright>trm" ("((_):_)" [100,100] 100) text {* conamed terms *} nominal_datatype ctrm = Co "\<guillemotleft>coname\<guillemotright>trm" ("(<_>:_)" [100,100] 100) text {* renaming functions *} nominal_primrec (freshness_context: "(d::coname,e::coname)") crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm" ("_[_\<turnstile>c>_]" [100,100,100] 100) where "(Ax x a)[d\<turnstile>c>e] = (if a=d then Ax x e else Ax x a)" | "\<lbrakk>a\<sharp>(d,e,N);x\<sharp>M\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N)[d\<turnstile>c>e] = Cut <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e])" | "(NotR (x).M a)[d\<turnstile>c>e] = (if a=d then NotR (x).(M[d\<turnstile>c>e]) e else NotR (x).(M[d\<turnstile>c>e]) a)" | "a\<sharp>(d,e) \<Longrightarrow> (NotL <a>.M x)[d\<turnstile>c>e] = (NotL <a>.(M[d\<turnstile>c>e]) x)" | "\<lbrakk>a\<sharp>(d,e,N,c);b\<sharp>(d,e,M,c);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[d\<turnstile>c>e] = (if c=d then AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d \<turnstile>c>e]) e else AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d\<turnstile>c>e]) c)" | "x\<sharp>y \<Longrightarrow> (AndL1 (x).M y)[d\<turnstile>c>e] = AndL1 (x).(M[d\<turnstile>c>e]) y" | "x\<sharp>y \<Longrightarrow> (AndL2 (x).M y)[d\<turnstile>c>e] = AndL2 (x).(M[d\<turnstile>c>e]) y" | "a\<sharp>(d,e,b) \<Longrightarrow> (OrR1 <a>.M b)[d\<turnstile>c>e] = (if b=d then OrR1 <a>.(M[d\<turnstile>c>e]) e else OrR1 <a>.(M[d\<turnstile>c>e]) b)" | "a\<sharp>(d,e,b) \<Longrightarrow> (OrR2 <a>.M b)[d\<turnstile>c>e] = (if b=d then OrR2 <a>.(M[d\<turnstile>c>e]) e else OrR2 <a>.(M[d\<turnstile>c>e]) b)" | "\<lbrakk>x\<sharp>(N,z);y\<sharp>(M,z);y\<noteq>x\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N z)[d\<turnstile>c>e] = OrL (x).(M[d\<turnstile>c>e]) (y).(N[d\<turnstile>c>e]) z" | "a\<sharp>(d,e,b) \<Longrightarrow> (ImpR (x).<a>.M b)[d\<turnstile>c>e] = (if b=d then ImpR (x).<a>.(M[d\<turnstile>c>e]) e else ImpR (x).<a>.(M[d\<turnstile>c>e]) b)" | "\<lbrakk>a\<sharp>(d,e,N);x\<sharp>(M,y)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y)[d\<turnstile>c>e] = ImpL <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e]) y" apply(finite_guess)+ apply(rule TrueI)+ apply(simp add: abs_fresh abs_supp fin_supp)+ apply(fresh_guess)+ done nominal_primrec (freshness_context: "(u::name,v::name)") nrename :: "trm \<Rightarrow> name \<Rightarrow> name \<Rightarrow> trm" ("_[_\<turnstile>n>_]" [100,100,100] 100) where "(Ax x a)[u\<turnstile>n>v] = (if x=u then Ax v a else Ax x a)" | "\<lbrakk>a\<sharp>N;x\<sharp>(u,v,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" | "x\<sharp>(u,v) \<Longrightarrow> (NotR (x).M a)[u\<turnstile>n>v] = NotR (x).(M[u\<turnstile>n>v]) a" | "(NotL <a>.M x)[u\<turnstile>n>v] = (if x=u then NotL <a>.(M[u\<turnstile>n>v]) v else NotL <a>.(M[u\<turnstile>n>v]) x)" | "\<lbrakk>a\<sharp>(N,c);b\<sharp>(M,c);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[u\<turnstile>n>v] = AndR <a>.(M[u\<turnstile>n>v]) <b>.(N[u\<turnstile>n>v]) c" | "x\<sharp>(u,v,y) \<Longrightarrow> (AndL1 (x).M y)[u\<turnstile>n>v] = (if y=u then AndL1 (x).(M[u\<turnstile>n>v]) v else AndL1 (x).(M[u\<turnstile>n>v]) y)" | "x\<sharp>(u,v,y) \<Longrightarrow> (AndL2 (x).M y)[u\<turnstile>n>v] = (if y=u then AndL2 (x).(M[u\<turnstile>n>v]) v else AndL2 (x).(M[u\<turnstile>n>v]) y)" | "a\<sharp>b \<Longrightarrow> (OrR1 <a>.M b)[u\<turnstile>n>v] = OrR1 <a>.(M[u\<turnstile>n>v]) b" | "a\<sharp>b \<Longrightarrow> (OrR2 <a>.M b)[u\<turnstile>n>v] = OrR2 <a>.(M[u\<turnstile>n>v]) b" | "\<lbrakk>x\<sharp>(u,v,N,z);y\<sharp>(u,v,M,z);y\<noteq>x\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N z)[u\<turnstile>n>v] = (if z=u then OrL (x).(M[u\<turnstile>n>v]) (y).(N[u\<turnstile>n>v]) v else OrL (x).(M[u\<turnstile>n>v]) (y).(N[u\<turnstile>n>v]) z)" | "\<lbrakk>a\<sharp>b; x\<sharp>(u,v)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b)[u\<turnstile>n>v] = ImpR (x).<a>.(M[u\<turnstile>n>v]) b" | "\<lbrakk>a\<sharp>N;x\<sharp>(u,v,M,y)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y)[u\<turnstile>n>v] = (if y=u then ImpL <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v]) v else ImpL <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v]) y)" apply(finite_guess)+ apply(rule TrueI)+ apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1)+ apply(fresh_guess)+ done 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 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 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 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 lemmas rename_eqvts = crename_name_eqvt crename_coname_eqvt nrename_name_eqvt nrename_coname_eqvt lemma nrename_fresh: assumes a: "x\<sharp>M" shows "M[x\<turnstile>n>y] = M" using a by (nominal_induct M avoiding: x y rule: trm.strong_induct) (auto simp add: trm.inject fresh_atm abs_fresh fin_supp abs_supp) 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) 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) 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) 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) 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) 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) 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) 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 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 lemmas rename_fresh = nrename_fresh crename_fresh nrename_nfresh crename_nfresh crename_cfresh nrename_cfresh nrename_nfresh' crename_cfresh' nrename_rename crename_rename lemma better_nrename_Cut: assumes a: "x\<sharp>(u,v)" shows "(Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" proof - obtain x'::"name" where fs1: "x'\<sharp>(M,N,a,x,u,v)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,x,u,v)" by (rule exists_fresh(2), rule fin_supp, blast) have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) 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 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 finally show "(Cut <a>.M (x).N)[u\<turnstile>n>v] = Cut <a>.(M[u\<turnstile>n>v]) (x).(N[u\<turnstile>n>v])" using eq1 by simp qed lemma better_crename_Cut: assumes a: "a\<sharp>(b,c)" shows "(Cut <a>.M (x).N)[b\<turnstile>c>c] = Cut <a>.(M[b\<turnstile>c>c]) (x).(N[b\<turnstile>c>c])" proof - obtain x'::"name" where fs1: "x'\<sharp>(M,N,a,x,b,c)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,x,b,c)" by (rule exists_fresh(2), rule fin_supp, blast) have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) 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 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 finally show "(Cut <a>.M (x).N)[b\<turnstile>c>c] = Cut <a>.(M[b\<turnstile>c>c]) (x).(N[b\<turnstile>c>c])" using eq1 by simp qed lemma crename_id: shows "M[a\<turnstile>c>a] = M" by (nominal_induct M avoiding: a rule: trm.strong_induct) (auto) lemma nrename_id: shows "M[x\<turnstile>n>x] = M" by (nominal_induct M avoiding: x rule: trm.strong_induct) (auto) lemma nrename_swap: shows "x\<sharp>M \<Longrightarrow> [(x,y)]\<bullet>M = M[y\<turnstile>n>x]" by (nominal_induct M avoiding: x y rule: trm.strong_induct) (simp_all add: calc_atm fresh_atm trm.inject alpha abs_fresh abs_supp fin_supp) lemma crename_swap: shows "a\<sharp>M \<Longrightarrow> [(a,b)]\<bullet>M = M[b\<turnstile>c>a]" by (nominal_induct M avoiding: a b rule: trm.strong_induct) (simp_all add: calc_atm fresh_atm trm.inject alpha abs_fresh abs_supp fin_supp) lemma crename_ax: assumes a: "M[a\<turnstile>c>b] = Ax x c" "c\<noteq>a" "c\<noteq>b" shows "M = Ax x c" using a apply(nominal_induct M avoiding: a b x c rule: trm.strong_induct) apply(simp_all add: trm.inject split: if_splits) done lemma nrename_ax: assumes a: "M[x\<turnstile>n>y] = Ax z a" "z\<noteq>x" "z\<noteq>y" shows "M = Ax z a" using a apply(nominal_induct M avoiding: x y z a rule: trm.strong_induct) apply(simp_all add: trm.inject split: if_splits) done text {* substitution functions *} lemma fresh_perm_coname: fixes c::"coname" and pi::"coname prm" and M::"trm" assumes a: "c\<sharp>pi" "c\<sharp>M" shows "c\<sharp>(pi\<bullet>M)" using a apply - apply(simp add: fresh_left) apply(simp add: at_prm_fresh[OF at_coname_inst] fresh_list_rev) done lemma fresh_perm_name: fixes x::"name" and pi::"name prm" and M::"trm" assumes a: "x\<sharp>pi" "x\<sharp>M" shows "x\<sharp>(pi\<bullet>M)" using a apply - apply(simp add: fresh_left) apply(simp add: at_prm_fresh[OF at_name_inst] fresh_list_rev) done 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 lemma fresh_fun_NotL[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')= fresh_fun (pi1\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))" and "pi2\<bullet>fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')= fresh_fun (pi2\<bullet>(\<lambda>x'. Cut <c>.P (x').NotL <a>.M x'))" apply - apply(perm_simp) apply(generate_fresh "name") apply(auto simp add: fresh_prod) apply(simp add: fresh_fun_simp_NotL) apply(rule sym) apply(rule trans) apply(rule fresh_fun_simp_NotL) apply(rule fresh_perm_name) apply(assumption) apply(assumption) apply(rule fresh_perm_name) apply(assumption) apply(assumption) apply(simp add: at_prm_fresh[OF at_name_inst] swap_simps) apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_NotL calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_AndL1[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')= fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))" and "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')= fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z'))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>x,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_AndL1 at_prm_fresh[OF at_name_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) apply(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 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 lemma fresh_fun_AndL2[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')= fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))" and "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')= fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z'))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>x,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_AndL2 at_prm_fresh[OF at_name_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>x,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_AndL2 calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 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 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 lemma fresh_fun_ImpL[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')= fresh_fun (pi1\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))" and "pi2\<bullet>fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')= fresh_fun (pi2\<bullet>(\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z'))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>x,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_ImpL at_prm_fresh[OF at_name_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::name. n\<sharp>(P,M,N,x,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>x,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_ImpL calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_NotR[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P)= fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))" and "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P)= fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(NotR (y).M a') (x).P))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi1\<bullet>P,pi1\<bullet>M,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_NotR calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,pi2\<bullet>P,pi2\<bullet>M,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_NotR at_prm_fresh[OF at_coname_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_AndR[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P)= fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P))" and "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P)= fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(AndR <b>.M <c>.N a') (x).P))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>N,pi1\<bullet>b,pi1\<bullet>c,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_AndR calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,N,b,c,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>N,pi2\<bullet>b,pi2\<bullet>c,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_AndR at_prm_fresh[OF at_coname_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_OrR1[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P)= fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P))" and "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P)= fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR1 <b>.M a') (x).P))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_OrR1 calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_OrR1 at_prm_fresh[OF at_coname_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_OrR2[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P)= fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P))" and "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P)= fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(OrR2 <b>.M a') (x).P))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_OrR2 calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_OrR2 at_prm_fresh[OF at_coname_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) done 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 lemma fresh_fun_ImpR[eqvt_force]: fixes pi1::"name prm" and pi2::"coname prm" shows "pi1\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P)= fresh_fun (pi1\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P))" and "pi2\<bullet>fresh_fun (\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P)= fresh_fun (pi2\<bullet>(\<lambda>a'. Cut <a'>.(ImpR (y).<b>.M a') (x).P))" apply - apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi1\<bullet>P,pi1\<bullet>M,pi1\<bullet>b,pi1)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_ImpR calc_atm) apply(rule exists_fresh') apply(simp add: fin_supp) apply(perm_simp) apply(subgoal_tac "\<exists>n::coname. n\<sharp>(P,M,b,pi2\<bullet>P,pi2\<bullet>M,pi2\<bullet>b,pi2)") apply(simp add: fresh_prod) apply(auto) apply(simp add: fresh_fun_simp_ImpR at_prm_fresh[OF at_coname_inst] swap_simps) apply(rule exists_fresh') apply(simp add: fin_supp) done nominal_primrec (freshness_context: "(y::name,c::coname,P::trm)") substn :: "trm \<Rightarrow> name \<Rightarrow> coname \<Rightarrow> trm \<Rightarrow> trm" ("_{_:=<_>._}" [100,100,100,100] 100) where "(Ax x a){y:=<c>.P} = (if x=y then Cut <c>.P (y).Ax y a else Ax x a)" | "\<lbrakk>a\<sharp>(c,P,N);x\<sharp>(y,P,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N){y:=<c>.P} = (if M=Ax y a then Cut <c>.P (x).(N{y:=<c>.P}) else Cut <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}))" | "x\<sharp>(y,P) \<Longrightarrow> (NotR (x).M a){y:=<c>.P} = NotR (x).(M{y:=<c>.P}) a" | "a\<sharp>(c,P) \<Longrightarrow> (NotL <a>.M x){y:=<c>.P} = (if x=y then fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.(M{y:=<c>.P}) x') else NotL <a>.(M{y:=<c>.P}) x)" | "\<lbrakk>a\<sharp>(c,P,N,d);b\<sharp>(c,P,M,d);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N d){y:=<c>.P} = AndR <a>.(M{y:=<c>.P}) <b>.(N{y:=<c>.P}) d" | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL1 (x).M z){y:=<c>.P} = (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).(M{y:=<c>.P}) z') else AndL1 (x).(M{y:=<c>.P}) z)" | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL2 (x).M z){y:=<c>.P} = (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).(M{y:=<c>.P}) z') else AndL2 (x).(M{y:=<c>.P}) z)" | "a\<sharp>(c,P,b) \<Longrightarrow> (OrR1 <a>.M b){y:=<c>.P} = OrR1 <a>.(M{y:=<c>.P}) b" | "a\<sharp>(c,P,b) \<Longrightarrow> (OrR2 <a>.M b){y:=<c>.P} = OrR2 <a>.(M{y:=<c>.P}) b" | "\<lbrakk>x\<sharp>(y,N,P,z);u\<sharp>(y,M,P,z);x\<noteq>u\<rbrakk> \<Longrightarrow> (OrL (x).M (u).N z){y:=<c>.P} = (if z=y then fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (x).(M{y:=<c>.P}) (u).(N{y:=<c>.P}) z') else OrL (x).(M{y:=<c>.P}) (u).(N{y:=<c>.P}) z)" | "\<lbrakk>a\<sharp>(b,c,P); x\<sharp>(y,P)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b){y:=<c>.P} = ImpR (x).<a>.(M{y:=<c>.P}) b" | "\<lbrakk>a\<sharp>(N,c,P);x\<sharp>(y,P,M,z)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N z){y:=<c>.P} = (if y=z then fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}) z') else ImpL <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}) z)" apply(finite_guess)+ apply(rule TrueI)+ apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::name. x\<sharp>(x3,P,y1,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(fresh_guess)+ done nominal_primrec (freshness_context: "(d::name,z::coname,P::trm)") substc :: "trm \<Rightarrow> coname \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_{_:=(_)._}" [100,100,100,100] 100) where "(Ax x a){d:=(z).P} = (if d=a then Cut <a>.(Ax x a) (z).P else Ax x a)" | "\<lbrakk>a\<sharp>(d,P,N);x\<sharp>(z,P,M)\<rbrakk> \<Longrightarrow> (Cut <a>.M (x).N){d:=(z).P} = (if N=Ax x d then Cut <a>.(M{d:=(z).P}) (z).P else Cut <a>.(M{d:=(z).P}) (x).(N{d:=(z).P}))" | "x\<sharp>(z,P) \<Longrightarrow> (NotR (x).M a){d:=(z).P} = (if d=a then fresh_fun (\<lambda>a'. Cut <a'>.NotR (x).(M{d:=(z).P}) a' (z).P) else NotR (x).(M{d:=(z).P}) a)" | "a\<sharp>(d,P) \<Longrightarrow> (NotL <a>.M x){d:=(z).P} = NotL <a>.(M{d:=(z).P}) x" | "\<lbrakk>a\<sharp>(P,c,N,d);b\<sharp>(P,c,M,d);b\<noteq>a\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c){d:=(z).P} = (if d=c then fresh_fun (\<lambda>a'. Cut <a'>.(AndR <a>.(M{d:=(z).P}) <b>.(N{d:=(z).P}) a') (z).P) else AndR <a>.(M{d:=(z).P}) <b>.(N{d:=(z).P}) c)" | "x\<sharp>(y,z,P) \<Longrightarrow> (AndL1 (x).M y){d:=(z).P} = AndL1 (x).(M{d:=(z).P}) y" | "x\<sharp>(y,P,z) \<Longrightarrow> (AndL2 (x).M y){d:=(z).P} = AndL2 (x).(M{d:=(z).P}) y" | "a\<sharp>(d,P,b) \<Longrightarrow> (OrR1 <a>.M b){d:=(z).P} = (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <a>.(M{d:=(z).P}) a' (z).P) else OrR1 <a>.(M{d:=(z).P}) b)" | "a\<sharp>(d,P,b) \<Longrightarrow> (OrR2 <a>.M b){d:=(z).P} = (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <a>.(M{d:=(z).P}) a' (z).P) else OrR2 <a>.(M{d:=(z).P}) b)" | "\<lbrakk>x\<sharp>(N,z,P,u);y\<sharp>(M,z,P,u);x\<noteq>y\<rbrakk> \<Longrightarrow> (OrL (x).M (y).N u){d:=(z).P} = OrL (x).(M{d:=(z).P}) (y).(N{d:=(z).P}) u" | "\<lbrakk>a\<sharp>(b,d,P); x\<sharp>(z,P)\<rbrakk> \<Longrightarrow> (ImpR (x).<a>.M b){d:=(z).P} = (if d=b then fresh_fun (\<lambda>a'. Cut <a'>.ImpR (x).<a>.(M{d:=(z).P}) a' (z).P) else ImpR (x).<a>.(M{d:=(z).P}) b)" | "\<lbrakk>a\<sharp>(N,d,P);x\<sharp>(y,z,P,M)\<rbrakk> \<Longrightarrow> (ImpL <a>.M (x).N y){d:=(z).P} = ImpL <a>.(M{d:=(z).P}) (x).(N{d:=(z).P}) y" apply(finite_guess)+ apply(rule TrueI)+ apply(simp add: abs_fresh abs_supp fs_name1 fs_coname1)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1,x3,y2)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm abs_supp) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(rule impI) apply(subgoal_tac "\<exists>x::coname. x\<sharp>(x1,P,x2,y1)", erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm) apply(rule exists_fresh', simp add: fin_supp) apply(simp add: abs_fresh abs_supp)+ apply(fresh_guess add: abs_fresh fresh_prod)+ 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 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 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 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 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 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 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 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 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 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 lemmas subst_supp = supp_subst1 supp_subst2 supp_subst3 supp_subst4 supp_subst5 supp_subst6 supp_subst7 supp_subst8 lemma subst_fresh: fixes x::"name" and c::"coname" shows "x\<sharp>P \<Longrightarrow> x\<sharp>M{x:=<c>.P}" and "b\<sharp>P \<Longrightarrow> b\<sharp>M{b:=(y).P}" and "x\<sharp>(M,P) \<Longrightarrow> x\<sharp>M{y:=<c>.P}" and "x\<sharp>M \<Longrightarrow> x\<sharp>M{c:=(x).P}" and "x\<sharp>(M,P) \<Longrightarrow> x\<sharp>M{c:=(y).P}" and "b\<sharp>(M,P) \<Longrightarrow> b\<sharp>M{c:=(y).P}" and "b\<sharp>M \<Longrightarrow> b\<sharp>M{y:=<b>.P}" and "b\<sharp>(M,P) \<Longrightarrow> b\<sharp>M{y:=<c>.P}" apply - apply(insert subst_supp) apply(simp_all add: fresh_def supp_prod) apply(blast)+ done lemma forget: shows "x\<sharp>M \<Longrightarrow> M{x:=<c>.P} = M" and "c\<sharp>M \<Longrightarrow> M{c:=(x).P} = M" apply(nominal_induct M avoiding: x c P rule: trm.strong_induct) apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp) done lemma substc_rename1: assumes a: "c\<sharp>(M,a)" shows "M{a:=(x).N} = ([(c,a)]\<bullet>M){c:=(x).N}" using a proof(nominal_induct M avoiding: c a x N rule: trm.strong_induct) case (Ax z d) then show ?case by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha) 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 apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{d:=(x).N},([(c,d)]\<bullet>M){c:=(x).N})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(simp add: trm.inject alpha) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (AndR c1 M c2 M' c3) then show ?case apply(simp) apply(auto) apply(simp add: fresh_prod calc_atm fresh_atm abs_fresh) apply(simp add: fresh_prod calc_atm fresh_atm abs_fresh fresh_left) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{c3:=(x).N}, M'{c3:=(x).N},c1,c2,c3,([(c,c3)]\<bullet>M){c:=(x).N},([(c,c3)]\<bullet>M'){c:=(x).N})") 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) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(simp add: fresh_prod calc_atm fresh_atm abs_fresh fresh_left) apply(simp add: fresh_prod calc_atm fresh_atm abs_fresh fresh_left) apply(auto simp add: trm.inject alpha) done 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 apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{e:=(x).N},([(c,e)]\<bullet>M){c:=(x).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) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (OrR2 d M e) then show ?case apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(N,M{e:=(x).N},([(c,e)]\<bullet>M){c:=(x).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) 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) 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:=(x).N},([(c,e)]\<bullet>M){c:=(x).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) apply(rule exists_fresh'(2)[OF fs_coname1]) done 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_coname_inst, OF at_coname_inst]) apply(simp add: calc_atm) done qed lemma substc_rename2: assumes a: "y\<sharp>(N,x)" shows "M{a:=(x).N} = M{a:=(y).([(y,x)]\<bullet>N)}" using a proof(nominal_induct M avoiding: a x y N rule: trm.strong_induct) case (Ax z d) then show ?case by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left) 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 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) 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 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) 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 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 apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<a>.N},([(y,x)]\<bullet>M){y:=<a>.N})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(simp add: trm.inject alpha) 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 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 apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<a>.N},([(y,x)]\<bullet>M){y:=<a>.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) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndL2 u M v) then show ?case apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<a>.N},([(y,x)]\<bullet>M){y:=<a>.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) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (OrL x1 M x2 M' x3) then show ?case apply(auto simp add: calc_atm trm.inject alpha fresh_atm abs_fresh fresh_prod fresh_left) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(N,M{x:=<a>.N},M'{x:=<a>.N},([(y,x)]\<bullet>M){y:=<a>.N},([(y,x)]\<bullet>M'){y:=<a>.N},x1,x2)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(simp add: trm.inject alpha) apply(rule exists_fresh'(1)[OF fs_name1]) done 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) next case (ImpL d M v 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:=<a>.N},M'{u:=<a>.N},([(y,u)]\<bullet>M){y:=<a>.N},([(y,u)]\<bullet>M'){y:=<a>.N},d,v)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(simp add: trm.inject alpha) apply(rule exists_fresh'(1)[OF fs_name1]) done 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) done qed lemma substn_rename4: assumes a: "c\<sharp>(N,a)" shows "M{x:=<a>.N} = M{x:=<c>.([(c,a)]\<bullet>N)}" using a proof(nominal_induct M avoiding: x c a N rule: trm.strong_induct) case (Ax z d) then show ?case by (auto simp add: fresh_prod fresh_atm calc_atm trm.inject alpha perm_swap fresh_left) next case NotR then show ?case by (auto simp add: 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 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) 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 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) 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 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 lemma subst_rename5: assumes a: "c'\<sharp>(c,N)" "x'\<sharp>(x,M)" shows "M{x:=<c>.N} = ([(x',x)]\<bullet>M){x':=<c'>.([(c',c)]\<bullet>N)}" proof - have "M{x:=<c>.N} = ([(x',x)]\<bullet>M){x':=<c>.N}" using a by (simp add: substn_rename3) also have "\<dots> = ([(x',x)]\<bullet>M){x':=<c'>.([(c',c)]\<bullet>N)}" using a by (simp add: substn_rename4) finally show ?thesis by simp qed lemma subst_rename6: assumes a: "c'\<sharp>(c,M)" "x'\<sharp>(x,N)" shows "M{c:=(x).N} = ([(c',c)]\<bullet>M){c':=(x').([(x',x)]\<bullet>N)}" proof - have "M{c:=(x).N} = ([(c',c)]\<bullet>M){c':=(x).N}" using a by (simp add: substc_rename1) also have "\<dots> = ([(c',c)]\<bullet>M){c':=(x').([(x',x)]\<bullet>N)}" using a by (simp add: substc_rename2) finally show ?thesis by simp qed lemmas subst_rename = substc_rename1 substc_rename2 substn_rename3 substn_rename4 subst_rename5 subst_rename6 lemma better_Cut_substn[simp]: assumes a: "a\<sharp>[c].P" "x\<sharp>(y,P)" shows "(Cut <a>.M (x).N){y:=<c>.P} = (if M=Ax y a then Cut <c>.P (x).(N{y:=<c>.P}) else Cut <a>.(M{y:=<c>.P}) (x).(N{y:=<c>.P}))" proof - obtain x'::"name" where fs1: "x'\<sharp>(M,N,c,P,x,y)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,c,P,a)" by (rule exists_fresh(2), rule fin_supp, blast) have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) 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 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) 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 finally show ?thesis by simp qed lemma better_Cut_substc[simp]: assumes a: "a\<sharp>(c,P)" "x\<sharp>[y].P" shows "(Cut <a>.M (x).N){c:=(y).P} = (if N=Ax x c then Cut <a>.(M{c:=(y).P}) (y).P else Cut <a>.(M{c:=(y).P}) (x).(N{c:=(y).P}))" proof - obtain x'::"name" where fs1: "x'\<sharp>(M,N,c,P,x,y)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,c,P,a)" by (rule exists_fresh(2), rule fin_supp, blast) have eq1: "(Cut <a>.M (x).N) = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N))" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) 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 have "(Cut <a>.M (x).N){c:=(y).P} = (Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)){c:=(y).P}" using eq1 by simp also have "\<dots> = (if ([(x',x)]\<bullet>N)=Ax x' c then Cut <a'>.(([(a',a)]\<bullet>M){c:=(y).P}) (y).P else Cut <a'>.(([(a',a)]\<bullet>M){c:=(y).P}) (x').(([(x',x)]\<bullet>N){c:=(y).P}))" using fs1 fs2 by (simp add: fresh_prod fresh_left calc_atm fresh_atm trm.inject) also have "\<dots> =(if N=Ax x c then Cut <a>.(M{c:=(y).P}) (y).P else Cut <a>.(M{c:=(y).P}) (x).(N{c:=(y).P}))" using fs1 fs2 a apply - apply(simp only: eq2[symmetric]) apply(auto simp add: trm.inject) apply(simp_all add: alpha fresh_atm fresh_prod subst_fresh) apply(simp_all add: eqvts perm_fresh_fresh calc_atm) apply(auto) apply(rule subst_rename) apply(simp add: fresh_prod fresh_atm) apply(simp add: abs_fresh) apply(simp add: perm_fresh_fresh) done finally show ?thesis by simp qed lemma better_Cut_substn': assumes a: "a\<sharp>[c].P" "y\<sharp>(N,x)" "M\<noteq>Ax y a" shows "(Cut <a>.M (x).N){y:=<c>.P} = Cut <a>.(M{y:=<c>.P}) (x).N" using a apply - apply(generate_fresh "name") apply(subgoal_tac "Cut <a>.M (x).N = Cut <a>.M (ca).([(ca,x)]\<bullet>N)") apply(simp) apply(subgoal_tac"y\<sharp>([(ca,x)]\<bullet>N)") apply(simp add: forget) apply(simp add: trm.inject) apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1] apply(simp add: trm.inject) apply(rule sym) apply(simp add: alpha fresh_prod fresh_atm) done lemma better_NotR_substc: assumes a: "d\<sharp>M" shows "(NotR (x).M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.NotR (x).M a' (z).P)" using a apply - apply(generate_fresh "name") apply(subgoal_tac "NotR (x).M d = NotR (c).([(c,x)]\<bullet>M) d") apply(auto simp add: fresh_left calc_atm forget) apply(generate_fresh "coname") apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(perm_simp add: trm.inject alpha fresh_prod fresh_atm fresh_left, auto) done lemma better_NotL_substn: assumes a: "y\<sharp>M" shows "(NotL <a>.M y){y:=<c>.P} = fresh_fun (\<lambda>x'. Cut <c>.P (x').NotL <a>.M x')" using a apply - apply(generate_fresh "coname") apply(subgoal_tac "NotL <a>.M y = NotL <ca>.([(ca,a)]\<bullet>M) y") apply(auto simp add: fresh_left calc_atm forget) apply(generate_fresh "name") apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(perm_simp add: trm.inject alpha fresh_prod fresh_atm fresh_left, auto) done lemma better_AndL1_substn: assumes a: "y\<sharp>[x].M" shows "(AndL1 (x).M y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL1 (x).M z')" using a apply - apply(generate_fresh "name") apply(subgoal_tac "AndL1 (x).M y = AndL1 (ca).([(ca,x)]\<bullet>M) y") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(generate_fresh "name") apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_AndL2_substn: assumes a: "y\<sharp>[x].M" shows "(AndL2 (x).M y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z')" using a apply - apply(generate_fresh "name") apply(subgoal_tac "AndL2 (x).M y = AndL2 (ca).([(ca,x)]\<bullet>M) y") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(generate_fresh "name") apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_AndR_substc: assumes a: "c\<sharp>([a].M,[b].N)" shows "(AndR <a>.M <b>.N c){c:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.(AndR <a>.M <b>.N a') (z).P)" using a apply - apply(generate_fresh "coname") apply(generate_fresh "coname") apply(subgoal_tac "AndR <a>.M <b>.N c = AndR <ca>.([(ca,a)]\<bullet>M) <caa>.([(caa,b)]\<bullet>N) c") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(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(simp) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule conjI) apply(rule forget) apply(auto simp add: fresh_left calc_atm abs_fresh)[1] apply(rule forget) apply(auto simp add: fresh_left calc_atm abs_fresh)[1] apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_OrL_substn: assumes a: "x\<sharp>([y].M,[z].N)" shows "(OrL (y).M (z).N x){x:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').OrL (y).M (z).N z')" using a apply - apply(generate_fresh "name") apply(generate_fresh "name") apply(subgoal_tac "OrL (y).M (z).N x = OrL (ca).([(ca,y)]\<bullet>M) (caa).([(caa,z)]\<bullet>N) x") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(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(simp) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule conjI) apply(rule forget) apply(auto simp add: fresh_left calc_atm abs_fresh)[1] apply(rule forget) apply(auto simp add: fresh_left calc_atm abs_fresh)[1] apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_OrR1_substc: assumes a: "d\<sharp>[a].M" shows "(OrR1 <a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <a>.M a' (z).P)" using a apply - apply(generate_fresh "coname") apply(subgoal_tac "OrR1 <a>.M d = OrR1 <c>.([(c,a)]\<bullet>M) d") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_OrR2_substc: assumes a: "d\<sharp>[a].M" shows "(OrR2 <a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <a>.M a' (z).P)" using a apply - apply(generate_fresh "coname") apply(subgoal_tac "OrR2 <a>.M d = OrR2 <c>.([(c,a)]\<bullet>M) d") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm) apply(auto) done lemma better_ImpR_substc: assumes a: "d\<sharp>[a].M" shows "(ImpR (x).<a>.M d){d:=(z).P} = fresh_fun (\<lambda>a'. Cut <a'>.ImpR (x).<a>.M a' (z).P)" using a apply - apply(generate_fresh "coname") apply(generate_fresh "name") apply(subgoal_tac "ImpR (x).<a>.M d = ImpR (ca).<c>.([(c,a)]\<bullet>[(ca,x)]\<bullet>M) d") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm abs_fresh fresh_left calc_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm fresh_left calc_atm abs_fresh) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(rule sym) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm abs_fresh abs_perm) done lemma better_ImpL_substn: assumes a: "y\<sharp>(M,[x].N)" shows "(ImpL <a>.M (x).N y){y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P (z').ImpL <a>.M (x).N z')" using a apply - apply(generate_fresh "coname") apply(generate_fresh "name") apply(subgoal_tac "ImpL <a>.M (x).N y = ImpL <ca>.([(ca,a)]\<bullet>M) (caa).([(caa,x)]\<bullet>N) y") apply(auto simp add: fresh_left calc_atm forget abs_fresh)[1] apply(rule_tac f="fresh_fun" in arg_cong) apply(simp add: expand_fun_eq) apply(rule allI) apply(simp add: trm.inject alpha fresh_prod fresh_atm abs_perm abs_fresh fresh_left calc_atm) apply(rule forget) apply(simp add: fresh_left calc_atm) apply(auto)[1] apply(rule sym) apply(perm_simp add: trm.inject alpha fresh_left calc_atm fresh_prod fresh_atm abs_fresh abs_perm) done lemma freshn_after_substc: fixes x::"name" assumes a: "x\<sharp>M{c:=(y).P}" shows "x\<sharp>M" using a supp_subst8 apply(simp add: fresh_def) apply(blast) done lemma freshn_after_substn: fixes x::"name" assumes a: "x\<sharp>M{y:=<c>.P}" "x\<noteq>y" shows "x\<sharp>M" using a using a supp_subst5 apply(simp add: fresh_def) apply(blast) done lemma freshc_after_substc: fixes a::"coname" assumes a: "a\<sharp>M{c:=(y).P}" "a\<noteq>c" shows "a\<sharp>M" using a supp_subst7 apply(simp add: fresh_def) apply(blast) done lemma freshc_after_substn: fixes a::"coname" assumes a: "a\<sharp>M{y:=<c>.P}" shows "a\<sharp>M" using a supp_subst6 apply(simp add: fresh_def) apply(blast) done lemma substn_crename_comm: assumes a: "c\<noteq>a" "c\<noteq>b" shows "M{x:=<c>.P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{x:=<c>.(P[a\<turnstile>c>b])}" using a apply(nominal_induct M avoiding: x c P a b rule: trm.strong_induct) apply(auto simp add: subst_fresh rename_fresh trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,x,c)") apply(erule exE) apply(subgoal_tac "Cut <c>.P (x).Ax x a = Cut <c>.P (x').Ax x' a") apply(simp) apply(rule trans) apply(rule crename.simps) apply(simp add: fresh_prod fresh_atm) apply(simp) apply(simp add: trm.inject) apply(simp add: alpha trm.inject calc_atm fresh_atm) apply(simp add: trm.inject) apply(simp add: alpha trm.inject calc_atm fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm) apply(simp) apply(simp add: crename_id) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(auto simp add: fresh_atm)[1] apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm) apply(auto simp add: fresh_atm)[1] apply(drule crename_ax) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(simp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],x,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],name1,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[a\<turnstile>c>b],name1,trm[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},trm2{x:=<c>.P},P,P[a\<turnstile>c>b],name1,name2, trm1[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{x:=<c>.P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,P[a\<turnstile>c>b],name1, trm1[a\<turnstile>c>b]{name2:=<c>.P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{name2:=<c>.P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) done lemma substc_crename_comm: assumes a: "c\<noteq>a" "c\<noteq>b" shows "M{c:=(x).P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{c:=(x).(P[a\<turnstile>c>b])}" using a apply(nominal_induct M avoiding: x c P a b rule: trm.strong_induct) apply(auto simp add: subst_fresh rename_fresh trm.inject) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(drule crename_ax) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(a,b,trm{coname:=(x).P},P,P[a\<turnstile>c>b],x,trm[a\<turnstile>c>b]{coname:=(x).P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,coname2,a,b,trm1{coname3:=(x).P},trm2{coname3:=(x).P}, P,P[a\<turnstile>c>b],x,trm1[a\<turnstile>c>b]{coname3:=(x).P[a\<turnstile>c>b]},trm2[a\<turnstile>c>b]{coname3:=(x).P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b, trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b, trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[a\<turnstile>c>b],a,b, trm[a\<turnstile>c>b]{coname2:=(x).P[a\<turnstile>c>b]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(rule trans) apply(rule better_crename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) done lemma substn_nrename_comm: assumes a: "x\<noteq>y" "x\<noteq>z" shows "M{x:=<c>.P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{x:=<c>.(P[y\<turnstile>n>z])}" using a apply(nominal_induct M avoiding: x c P y z rule: trm.strong_induct) apply(auto simp add: subst_fresh rename_fresh trm.inject) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_prod fresh_atm) apply(simp add: trm.inject) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm) apply(simp) apply(drule nrename_ax) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(simp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,z,trm{x:=<c>.P},P,P[y\<turnstile>n>z],x,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,P[y\<turnstile>n>z],name1,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]},y,z)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,z,trm{x:=<c>.P},P,P[y\<turnstile>n>z],name1,trm[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},trm2{x:=<c>.P},P,P[y\<turnstile>n>z],name1,name2,y,z, trm1[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{x:=<c>.P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,P[y\<turnstile>n>z],y,z,name1, trm1[y\<turnstile>n>z]{name2:=<c>.P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{name2:=<c>.P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) done lemma substc_nrename_comm: assumes a: "x\<noteq>y" "x\<noteq>z" shows "M{c:=(x).P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{c:=(x).(P[y\<turnstile>n>z])}" using a apply(nominal_induct M avoiding: x c P y z rule: trm.strong_induct) apply(auto simp add: subst_fresh rename_fresh trm.inject) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(drule nrename_ax) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(simp) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(drule nrename_ax) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(y,z,trm{coname:=(x).P},P,P[y\<turnstile>n>z],x,trm[y\<turnstile>n>z]{coname:=(x).P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,coname2,y,z,trm1{coname3:=(x).P},trm2{coname3:=(x).P}, P,P[y\<turnstile>n>z],x,trm1[y\<turnstile>n>z]{coname3:=(x).P[y\<turnstile>n>z]},trm2[y\<turnstile>n>z]{coname3:=(x).P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh subst_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z, trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z, trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(coname1,trm{coname2:=(x).P},P,P[y\<turnstile>n>z],y,z, trm[y\<turnstile>n>z]{coname2:=(x).P[y\<turnstile>n>z]})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(rule trans) apply(rule better_nrename_Cut) apply(simp add: fresh_atm fresh_prod) apply(simp add: rename_fresh fresh_atm) apply(rule exists_fresh') apply(rule fin_supp) done lemma substn_crename_comm': assumes a: "a\<noteq>c" "a\<sharp>P" shows "M{x:=<c>.P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{x:=<c>.P}" using a proof - assume a1: "a\<noteq>c" assume a2: "a\<sharp>P" obtain c'::"coname" where fs2: "c'\<sharp>(c,P,a,b)" by (rule exists_fresh(2), rule fin_supp, blast) have eq: "M{x:=<c>.P} = M{x:=<c'>.([(c',c)]\<bullet>P)}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq': "M[a\<turnstile>c>b]{x:=<c>.P} = M[a\<turnstile>c>b]{x:=<c'>.([(c',c)]\<bullet>P)}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq2: "([(c',c)]\<bullet>P)[a\<turnstile>c>b] = ([(c',c)]\<bullet>P)" using fs2 a2 a1 apply - apply(rule rename_fresh) apply(simp add: fresh_left calc_atm fresh_prod fresh_atm) done have "M{x:=<c>.P}[a\<turnstile>c>b] = M{x:=<c'>.([(c',c)]\<bullet>P)}[a\<turnstile>c>b]" using eq by simp also have "\<dots> = M[a\<turnstile>c>b]{x:=<c'>.(([(c',c)]\<bullet>P)[a\<turnstile>c>b])}" using fs2 apply - apply(rule substn_crename_comm) apply(simp_all add: fresh_prod fresh_atm) done also have "\<dots> = M[a\<turnstile>c>b]{x:=<c'>.(([(c',c)]\<bullet>P))}" using eq2 by simp also have "\<dots> = M[a\<turnstile>c>b]{x:=<c>.P}" using eq' by simp finally show ?thesis by simp qed lemma substc_crename_comm': assumes a: "c\<noteq>a" "c\<noteq>b" "a\<sharp>P" shows "M{c:=(x).P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{c:=(x).P}" using a proof - assume a1: "c\<noteq>a" assume a1': "c\<noteq>b" assume a2: "a\<sharp>P" obtain c'::"coname" where fs2: "c'\<sharp>(c,M,a,b)" by (rule exists_fresh(2), rule fin_supp, blast) have eq: "M{c:=(x).P} = ([(c',c)]\<bullet>M){c':=(x).P}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq': "([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P} = M[a\<turnstile>c>b]{c:=(x).P}" using fs2 apply - apply(rule subst_rename[symmetric]) apply(simp add: rename_fresh) done have eq2: "([(c',c)]\<bullet>M)[a\<turnstile>c>b] = ([(c',c)]\<bullet>(M[a\<turnstile>c>b]))" using fs2 a2 a1 a1' apply - apply(simp add: rename_eqvts) apply(simp add: fresh_left calc_atm fresh_prod fresh_atm) done have "M{c:=(x).P}[a\<turnstile>c>b] = ([(c',c)]\<bullet>M){c':=(x).P}[a\<turnstile>c>b]" using eq by simp also have "\<dots> = ([(c',c)]\<bullet>M)[a\<turnstile>c>b]{c':=(x).P[a\<turnstile>c>b]}" using fs2 apply - apply(rule substc_crename_comm) apply(simp_all add: fresh_prod fresh_atm) done also have "\<dots> = ([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P[a\<turnstile>c>b]}" using eq2 by simp also have "\<dots> = ([(c',c)]\<bullet>(M[a\<turnstile>c>b])){c':=(x).P}" using a2 by (simp add: rename_fresh) also have "\<dots> = M[a\<turnstile>c>b]{c:=(x).P}" using eq' by simp finally show ?thesis by simp qed lemma substn_nrename_comm': assumes a: "x\<noteq>y" "x\<noteq>z" "y\<sharp>P" shows "M{x:=<c>.P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{x:=<c>.P}" using a proof - assume a1: "x\<noteq>y" assume a1': "x\<noteq>z" assume a2: "y\<sharp>P" obtain x'::"name" where fs2: "x'\<sharp>(x,M,y,z)" by (rule exists_fresh(1), rule fin_supp, blast) have eq: "M{x:=<c>.P} = ([(x',x)]\<bullet>M){x':=<c>.P}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq': "([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P} = M[y\<turnstile>n>z]{x:=<c>.P}" using fs2 apply - apply(rule subst_rename[symmetric]) apply(simp add: rename_fresh) done have eq2: "([(x',x)]\<bullet>M)[y\<turnstile>n>z] = ([(x',x)]\<bullet>(M[y\<turnstile>n>z]))" using fs2 a2 a1 a1' apply - apply(simp add: rename_eqvts) apply(simp add: fresh_left calc_atm fresh_prod fresh_atm) done have "M{x:=<c>.P}[y\<turnstile>n>z] = ([(x',x)]\<bullet>M){x':=<c>.P}[y\<turnstile>n>z]" using eq by simp also have "\<dots> = ([(x',x)]\<bullet>M)[y\<turnstile>n>z]{x':=<c>.P[y\<turnstile>n>z]}" using fs2 apply - apply(rule substn_nrename_comm) apply(simp_all add: fresh_prod fresh_atm) done also have "\<dots> = ([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P[y\<turnstile>n>z]}" using eq2 by simp also have "\<dots> = ([(x',x)]\<bullet>(M[y\<turnstile>n>z])){x':=<c>.P}" using a2 by (simp add: rename_fresh) also have "\<dots> = M[y\<turnstile>n>z]{x:=<c>.P}" using eq' by simp finally show ?thesis by simp qed lemma substc_nrename_comm': assumes a: "x\<noteq>y" "y\<sharp>P" shows "M{c:=(x).P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{c:=(x).P}" using a proof - assume a1: "x\<noteq>y" assume a2: "y\<sharp>P" obtain x'::"name" where fs2: "x'\<sharp>(x,P,y,z)" by (rule exists_fresh(1), rule fin_supp, blast) have eq: "M{c:=(x).P} = M{c:=(x').([(x',x)]\<bullet>P)}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq': "M[y\<turnstile>n>z]{c:=(x).P} = M[y\<turnstile>n>z]{c:=(x').([(x',x)]\<bullet>P)}" using fs2 apply - apply(rule subst_rename) apply(simp) done have eq2: "([(x',x)]\<bullet>P)[y\<turnstile>n>z] = ([(x',x)]\<bullet>P)" using fs2 a2 a1 apply - apply(rule rename_fresh) apply(simp add: fresh_left calc_atm fresh_prod fresh_atm) done have "M{c:=(x).P}[y\<turnstile>n>z] = M{c:=(x').([(x',x)]\<bullet>P)}[y\<turnstile>n>z]" using eq by simp also have "\<dots> = M[y\<turnstile>n>z]{c:=(x').(([(x',x)]\<bullet>P)[y\<turnstile>n>z])}" using fs2 apply - apply(rule substc_nrename_comm) apply(simp_all add: fresh_prod fresh_atm) done also have "\<dots> = M[y\<turnstile>n>z]{c:=(x').(([(x',x)]\<bullet>P))}" using eq2 by simp also have "\<dots> = M[y\<turnstile>n>z]{c:=(x).P}" using eq' by simp finally show ?thesis by simp qed lemmas subst_comm = substn_crename_comm substc_crename_comm substn_nrename_comm substc_nrename_comm lemmas subst_comm' = substn_crename_comm' substc_crename_comm' substn_nrename_comm' substc_nrename_comm' text {* typing contexts *} types ctxtn = "(name\<times>ty) list" ctxtc = "(coname\<times>ty) list" inductive validc :: "ctxtc \<Rightarrow> bool" where vc1[intro]: "validc []" | vc2[intro]: "\<lbrakk>a\<sharp>\<Delta>; validc \<Delta>\<rbrakk> \<Longrightarrow> validc ((a,T)#\<Delta>)" equivariance validc inductive validn :: "ctxtn \<Rightarrow> bool" where vn1[intro]: "validn []" | vn2[intro]: "\<lbrakk>x\<sharp>\<Gamma>; validn \<Gamma>\<rbrakk> \<Longrightarrow> validn ((x,T)#\<Gamma>)" equivariance validn lemma fresh_ctxt: fixes a::"coname" and x::"name" and \<Gamma>::"ctxtn" and \<Delta>::"ctxtc" shows "a\<sharp>\<Gamma>" and "x\<sharp>\<Delta>" proof - show "a\<sharp>\<Gamma>" by (induct \<Gamma>) (auto simp add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty) next show "x\<sharp>\<Delta>" by (induct \<Delta>) (auto simp add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm fresh_ty) qed text {* cut-reductions *} declare abs_perm[eqvt] inductive fin :: "trm \<Rightarrow> name \<Rightarrow> bool" where [intro]: "fin (Ax x a) x" | [intro]: "x\<sharp>M \<Longrightarrow> fin (NotL <a>.M x) x" | [intro]: "y\<sharp>[x].M \<Longrightarrow> fin (AndL1 (x).M y) y" | [intro]: "y\<sharp>[x].M \<Longrightarrow> fin (AndL2 (x).M y) y" | [intro]: "\<lbrakk>z\<sharp>[x].M;z\<sharp>[y].N\<rbrakk> \<Longrightarrow> fin (OrL (x).M (y).N z) z" | [intro]: "\<lbrakk>y\<sharp>M;y\<sharp>[x].N\<rbrakk> \<Longrightarrow> fin (ImpL <a>.M (x).N y) y" equivariance fin lemma fin_Ax_elim: assumes a: "fin (Ax x a) y" shows "x=y" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) done lemma fin_NotL_elim: assumes a: "fin (NotL <a>.M x) y" shows "x=y \<and> x\<sharp>M" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) apply(subgoal_tac "y\<sharp>[aa].Ma") apply(drule sym) apply(simp_all add: abs_fresh) done lemma fin_AndL1_elim: assumes a: "fin (AndL1 (x).M y) z" shows "z=y \<and> z\<sharp>[x].M" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) done lemma fin_AndL2_elim: assumes a: "fin (AndL2 (x).M y) z" shows "z=y \<and> z\<sharp>[x].M" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) done lemma fin_OrL_elim: assumes a: "fin (OrL (x).M (y).N u) z" shows "z=u \<and> z\<sharp>[x].M \<and> z\<sharp>[y].N" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) done lemma fin_ImpL_elim: assumes a: "fin (ImpL <a>.M (x).N z) y" shows "z=y \<and> z\<sharp>M \<and> z\<sharp>[x].N" using a apply(erule_tac fin.cases) apply(auto simp add: trm.inject) apply(subgoal_tac "y\<sharp>[aa].Ma") apply(drule sym) apply(simp_all add: abs_fresh) done lemma fin_rest_elims: shows "fin (Cut <a>.M (x).N) y \<Longrightarrow> False" and "fin (NotR (x).M c) y \<Longrightarrow> False" and "fin (AndR <a>.M <b>.N c) y \<Longrightarrow> False" and "fin (OrR1 <a>.M b) y \<Longrightarrow> False" and "fin (OrR2 <a>.M b) y \<Longrightarrow> False" and "fin (ImpR (x).<a>.M b) y \<Longrightarrow> False" by (erule fin.cases, simp_all add: trm.inject)+ lemmas fin_elims = fin_Ax_elim fin_NotL_elim fin_AndL1_elim fin_AndL2_elim fin_OrL_elim fin_ImpL_elim fin_rest_elims lemma fin_rename: shows "fin M x \<Longrightarrow> fin ([(x',x)]\<bullet>M) x'" by (induct rule: fin.induct) (auto simp add: calc_atm simp add: fresh_left abs_fresh) lemma not_fin_subst1: assumes a: "\<not>(fin M x)" shows "\<not>(fin (M{c:=(y).P}) x)" using a apply(nominal_induct M avoiding: x c y P rule: trm.strong_induct) apply(auto) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname:=(y).P},P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(drule fin_elims) apply(auto)[1] apply(drule freshn_after_substc) apply(simp add: fin.intros) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,coname3,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,coname3,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fin.cases, simp_all add: trm.inject) apply(drule fin_AndL1_elim) apply(auto simp add: abs_fresh)[1] apply(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(drule freshn_after_substc) apply(subgoal_tac "name2\<sharp>[name1].trm") apply(simp add: fin.intros) apply(simp add: abs_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fin.cases, simp_all add: trm.inject) apply(drule fin_OrL_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substc)+ apply(subgoal_tac "name3\<sharp>[name1].trm1 \<and> name3\<sharp>[name2].trm2") apply(simp add: fin.intros) apply(simp add: abs_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(y).P},coname1,P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fin.cases, simp_all add: trm.inject) apply(drule fin_ImpL_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substc)+ apply(subgoal_tac "x\<sharp>[name1].trm2") apply(simp add: fin.intros) apply(simp add: abs_fresh) done lemma not_fin_subst2: assumes a: "\<not>(fin M x)" shows "\<not>(fin (M{y:=<c>.P}) x)" using a apply(nominal_induct M avoiding: x c y P rule: trm.strong_induct) apply(auto) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fin_NotL_elim) apply(auto)[1] apply(drule freshn_after_substn) apply(simp) apply(simp add: fin.intros) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,name1,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fin_AndL1_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substn) apply(simp) apply(subgoal_tac "name2\<sharp>[name1]. trm") apply(simp add: fin.intros) apply(simp add: abs_fresh) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm{y:=<c>.P},P,name1,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fin_AndL2_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substn) apply(simp) apply(subgoal_tac "name2\<sharp>[name1].trm") apply(simp add: fin.intros) apply(simp add: abs_fresh) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm1{y:=<c>.P},trm2{y:=<c>.P},name1,name2,P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fin_OrL_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substn) apply(simp) apply(drule freshn_after_substn) apply(simp) apply(subgoal_tac "name3\<sharp>[name1].trm1 \<and> name3\<sharp>[name2].trm2") apply(simp add: fin.intros) apply(simp add: abs_fresh) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},name1,P,x)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fin_ImpL_elim) apply(auto simp add: abs_fresh)[1] apply(drule freshn_after_substn) apply(simp) apply(drule freshn_after_substn) apply(simp) apply(subgoal_tac "x\<sharp>[name1].trm2") apply(simp add: fin.intros) apply(simp add: abs_fresh) done lemma fin_subst1: assumes a: "fin M x" "x\<noteq>y" "x\<sharp>P" shows "fin (M{y:=<c>.P}) x" using a apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct) apply(auto dest!: fin_elims simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh) apply(rule fin.intros, simp add: subst_fresh abs_fresh, simp add: subst_fresh abs_fresh) done lemma fin_subst2: assumes a: "fin M y" "x\<noteq>y" "y\<sharp>P" "M\<noteq>Ax y c" shows "fin (M{c:=(x).P}) y" using a apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct) apply(drule fin_elims) apply(simp add: trm.inject) apply(rule fin.intros) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(rule fin.intros) apply(auto)[1] apply(rule subst_fresh) apply(simp) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(rule fin.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fin_elims, simp) apply(rule fin.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(rule fin.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(rule fin.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] done lemma fin_substn_nrename: assumes a: "fin M x" "x\<noteq>y" "x\<sharp>P" shows "M[x\<turnstile>n>y]{y:=<c>.P} = Cut <c>.P (x).(M{y:=<c>.P})" using a apply(nominal_induct M avoiding: x y c P rule: trm.strong_induct) apply(drule fin_Ax_elim) apply(simp) apply(simp add: trm.inject) apply(simp add: alpha calc_atm fresh_atm) apply(simp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_NotL_elim) apply(simp) apply(subgoal_tac "\<exists>z::name. z\<sharp>(trm,y,x,P,trm[x\<turnstile>n>y]{y:=<c>.P})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(rule subst_fresh) apply(simp) apply(rule exists_fresh') apply(rule fin_supp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_AndL1_elim) apply(simp) apply(subgoal_tac "\<exists>z::name. z\<sharp>(name2,name1,P,trm[name2\<turnstile>n>y]{y:=<c>.P},y,P,trm)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(rule subst_fresh) apply(simp) apply(rule exists_fresh') apply(rule fin_supp) apply(drule fin_AndL2_elim) apply(simp) apply(subgoal_tac "\<exists>z::name. z\<sharp>(name2,name1,P,trm[name2\<turnstile>n>y]{y:=<c>.P},y,P,trm)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(rule subst_fresh) apply(simp) apply(rule exists_fresh') apply(rule fin_supp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_OrL_elim) apply(simp add: abs_fresh) apply(simp add: subst_fresh rename_fresh) apply(subgoal_tac "\<exists>z::name. z\<sharp>(name3,name2,name1,P,trm1[name3\<turnstile>n>y]{y:=<c>.P},trm2[name3\<turnstile>n>y]{y:=<c>.P},y,P,trm1,trm2)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(rule exists_fresh') apply(rule fin_supp) apply(drule fin_rest_elims) apply(simp) apply(drule fin_ImpL_elim) apply(simp add: abs_fresh) apply(simp add: subst_fresh rename_fresh) apply(subgoal_tac "\<exists>z::name. z\<sharp>(name1,x,P,trm1[x\<turnstile>n>y]{y:=<c>.P},trm2[x\<turnstile>n>y]{y:=<c>.P},y,P,trm1,trm2)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(simp add: nsubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: nrename_fresh) apply(rule exists_fresh') apply(rule fin_supp) done inductive fic :: "trm \<Rightarrow> coname \<Rightarrow> bool" where [intro]: "fic (Ax x a) a" | [intro]: "a\<sharp>M \<Longrightarrow> fic (NotR (x).M a) a" | [intro]: "\<lbrakk>c\<sharp>[a].M;c\<sharp>[b].N\<rbrakk> \<Longrightarrow> fic (AndR <a>.M <b>.N c) c" | [intro]: "b\<sharp>[a].M \<Longrightarrow> fic (OrR1 <a>.M b) b" | [intro]: "b\<sharp>[a].M \<Longrightarrow> fic (OrR2 <a>.M b) b" | [intro]: "\<lbrakk>b\<sharp>[a].M\<rbrakk> \<Longrightarrow> fic (ImpR (x).<a>.M b) b" equivariance fic lemma fic_Ax_elim: assumes a: "fic (Ax x a) b" shows "a=b" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) done lemma fic_NotR_elim: assumes a: "fic (NotR (x).M a) b" shows "a=b \<and> b\<sharp>M" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) apply(subgoal_tac "b\<sharp>[xa].Ma") apply(drule sym) apply(simp_all add: abs_fresh) done lemma fic_OrR1_elim: assumes a: "fic (OrR1 <a>.M b) c" shows "b=c \<and> c\<sharp>[a].M" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) done lemma fic_OrR2_elim: assumes a: "fic (OrR2 <a>.M b) c" shows "b=c \<and> c\<sharp>[a].M" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) done lemma fic_AndR_elim: assumes a: "fic (AndR <a>.M <b>.N c) d" shows "c=d \<and> d\<sharp>[a].M \<and> d\<sharp>[b].N" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) done lemma fic_ImpR_elim: assumes a: "fic (ImpR (x).<a>.M b) c" shows "b=c \<and> b\<sharp>[a].M" using a apply(erule_tac fic.cases) apply(auto simp add: trm.inject) apply(subgoal_tac "c\<sharp>[xa].[aa].Ma") apply(drule sym) apply(simp_all add: abs_fresh) done lemma fic_rest_elims: shows "fic (Cut <a>.M (x).N) d \<Longrightarrow> False" and "fic (NotL <a>.M x) d \<Longrightarrow> False" and "fic (OrL (x).M (y).N z) d \<Longrightarrow> False" and "fic (AndL1 (x).M y) d \<Longrightarrow> False" and "fic (AndL2 (x).M y) d \<Longrightarrow> False" and "fic (ImpL <a>.M (x).N y) d \<Longrightarrow> False" by (erule fic.cases, simp_all add: trm.inject)+ lemmas fic_elims = fic_Ax_elim fic_NotR_elim fic_OrR1_elim fic_OrR2_elim fic_AndR_elim fic_ImpR_elim fic_rest_elims lemma fic_rename: shows "fic M a \<Longrightarrow> fic ([(a',a)]\<bullet>M) a'" by (induct rule: fic.induct) (auto simp add: calc_atm simp add: fresh_left abs_fresh) lemma not_fic_subst1: assumes a: "\<not>(fic M a)" shows "\<not>(fic (M{y:=<c>.P}) a)" using a apply(nominal_induct M avoiding: a c y P rule: trm.strong_induct) apply(auto) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims) apply(auto)[1] apply(drule freshc_after_substn) apply(simp add: fic.intros) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{y:=<c>.P},P,name1,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{y:=<c>.P},trm2{y:=<c>.P},P,name1,name2,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},trm2{name2:=<c>.P},P,name1,name2,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) done lemma not_fic_subst2: assumes a: "\<not>(fic M a)" shows "\<not>(fic (M{c:=(y).P}) a)" using a apply(nominal_induct M avoiding: a c y P rule: trm.strong_induct) apply(auto) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname:=(y).P},P,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(drule fic_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fic_elims, simp) apply(erule conjE)+ apply(drule freshc_after_substc) apply(simp) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm1{coname3:=(y).P},trm2{coname3:=(y).P},P,coname1,coname2,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(drule fic_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substc) apply(simp) apply(drule freshc_after_substc) apply(simp) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(drule fic_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substc) apply(simp) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(drule fic_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substc) apply(simp) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(y).P},P,coname1,a)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(drule fic_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substc) apply(simp) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims, simp) done lemma fic_subst1: assumes a: "fic M a" "a\<noteq>b" "a\<sharp>P" shows "fic (M{b:=(x).P}) a" using a apply(nominal_induct M avoiding: x b a P rule: trm.strong_induct) apply(drule fic_elims) apply(simp add: fic.intros) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(auto)[1] apply(rule subst_fresh) apply(simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) done lemma fic_subst2: assumes a: "fic M a" "c\<noteq>a" "a\<sharp>P" "M\<noteq>Ax x a" shows "fic (M{x:=<c>.P}) a" using a apply(nominal_induct M avoiding: x a c P rule: trm.strong_induct) apply(drule fic_elims) apply(simp add: trm.inject) apply(rule fic.intros) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(auto)[1] apply(rule subst_fresh) apply(simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(rule fic.intros) apply(simp add: abs_fresh fresh_atm) apply(rule subst_fresh) apply(auto)[1] apply(drule fic_elims, simp) done lemma fic_substc_crename: assumes a: "fic M a" "a\<noteq>b" "a\<sharp>P" shows "M[a\<turnstile>c>b]{b:=(y).P} = Cut <a>.(M{b:=(y).P}) (y).P" using a apply(nominal_induct M avoiding: a b y P rule: trm.strong_induct) apply(drule fic_Ax_elim) apply(simp) apply(simp add: trm.inject) apply(simp add: alpha calc_atm fresh_atm trm.inject) apply(simp) apply(drule fic_rest_elims) apply(simp) apply(drule fic_NotR_elim) apply(simp) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(simp add: trm.inject alpha fresh_atm fresh_prod fresh_atm calc_atm abs_fresh) apply(rule conjI) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: crename_fresh) apply(rule subst_fresh) apply(simp) apply(drule fic_rest_elims) apply(simp) apply(drule fic_AndR_elim) apply(simp add: abs_fresh fresh_atm subst_fresh rename_fresh) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod) apply(rule conjI) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: subst_fresh) apply(drule fic_rest_elims) apply(simp) apply(drule fic_rest_elims) apply(simp) apply(drule fic_OrR1_elim) apply(simp) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: subst_fresh rename_fresh) apply(drule fic_OrR2_elim) apply(simp add: abs_fresh fresh_atm) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: subst_fresh rename_fresh) apply(drule fic_rest_elims) apply(simp) apply(drule fic_ImpR_elim) apply(simp add: abs_fresh fresh_atm) apply(auto)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(simp add: trm.inject alpha fresh_atm calc_atm abs_fresh fresh_prod) apply(simp add: csubst_eqvt calc_atm) apply(simp add: perm_fresh_fresh) apply(simp add: subst_fresh rename_fresh) apply(drule fic_rest_elims) apply(simp) done inductive l_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^isub>l _" [100,100] 100) where LAxR: "\<lbrakk>x\<sharp>M; a\<sharp>b; fic M a\<rbrakk> \<Longrightarrow> Cut <a>.M (x).(Ax x b) \<longrightarrow>\<^isub>l M[a\<turnstile>c>b]" | LAxL: "\<lbrakk>a\<sharp>M; x\<sharp>y; fin M x\<rbrakk> \<Longrightarrow> Cut <a>.(Ax y a) (x).M \<longrightarrow>\<^isub>l M[x\<turnstile>n>y]" | LNot: "\<lbrakk>y\<sharp>(M,N); x\<sharp>(N,y); a\<sharp>(M,N,b); b\<sharp>M; y\<noteq>x; b\<noteq>a\<rbrakk> \<Longrightarrow> Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) \<longrightarrow>\<^isub>l Cut <b>.N (x).M" | LAnd1: "\<lbrakk>b\<sharp>([a1].M1,[a2].M2,N,a1,a2); y\<sharp>([x].N,M1,M2,x); x\<sharp>(M1,M2); a1\<sharp>(M2,N); a2\<sharp>(M1,N); a1\<noteq>a2\<rbrakk> \<Longrightarrow> Cut <b>.(AndR <a1>.M1 <a2>.M2 b) (y).(AndL1 (x).N y) \<longrightarrow>\<^isub>l Cut <a1>.M1 (x).N" | LAnd2: "\<lbrakk>b\<sharp>([a1].M1,[a2].M2,N,a1,a2); y\<sharp>([x].N,M1,M2,x); x\<sharp>(M1,M2); a1\<sharp>(M2,N); a2\<sharp>(M1,N); a1\<noteq>a2\<rbrakk> \<Longrightarrow> Cut <b>.(AndR <a1>.M1 <a2>.M2 b) (y).(AndL2 (x).N y) \<longrightarrow>\<^isub>l Cut <a2>.M2 (x).N" | LOr1: "\<lbrakk>b\<sharp>([a].M,N1,N2,a); y\<sharp>([x1].N1,[x2].N2,M,x1,x2); x1\<sharp>(M,N2); x2\<sharp>(M,N1); a\<sharp>(N1,N2); x1\<noteq>x2\<rbrakk> \<Longrightarrow> Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^isub>l Cut <a>.M (x1).N1" | LOr2: "\<lbrakk>b\<sharp>([a].M,N1,N2,a); y\<sharp>([x1].N1,[x2].N2,M,x1,x2); x1\<sharp>(M,N2); x2\<sharp>(M,N1); a\<sharp>(N1,N2); x1\<noteq>x2\<rbrakk> \<Longrightarrow> Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^isub>l Cut <a>.M (x2).N2" | LImp: "\<lbrakk>z\<sharp>(N,[y].P,[x].M,y,x); b\<sharp>([a].M,[c].N,P,c,a); x\<sharp>(N,[y].P,y); c\<sharp>(P,[a].M,b,a); a\<sharp>([c].N,P); y\<sharp>(N,[x].M)\<rbrakk> \<Longrightarrow> Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z) \<longrightarrow>\<^isub>l Cut <a>.(Cut <c>.N (x).M) (y).P" equivariance l_redu lemma l_redu_eqvt': fixes pi1::"name prm" and pi2::"coname prm" shows "(pi1\<bullet>M) \<longrightarrow>\<^isub>l (pi1\<bullet>M') \<Longrightarrow> M \<longrightarrow>\<^isub>l M'" and "(pi2\<bullet>M) \<longrightarrow>\<^isub>l (pi2\<bullet>M') \<Longrightarrow> M \<longrightarrow>\<^isub>l M'" apply - apply(drule_tac pi="rev pi1" in l_redu.eqvt(1)) apply(perm_simp) apply(drule_tac pi="rev pi2" in l_redu.eqvt(2)) apply(perm_simp) done nominal_inductive l_redu apply(simp_all add: abs_fresh fresh_atm rename_fresh fresh_prod abs_supp fin_supp) apply(force)+ done lemma fresh_l_redu: fixes x::"name" and a::"coname" shows "M \<longrightarrow>\<^isub>l M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'" and "M \<longrightarrow>\<^isub>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(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(case_tac "aa=a") apply(simp add: rename_fresh) apply(simp add: rename_fresh fresh_atm) apply(simp add: fresh_prod abs_fresh abs_supp fin_supp)+ done lemma better_LAxR_intro[intro]: shows "fic M a \<Longrightarrow> Cut <a>.M (x).(Ax x b) \<longrightarrow>\<^isub>l M[a\<turnstile>c>b]" proof - assume fin: "fic M a" obtain x'::"name" where fs1: "x'\<sharp>(M,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(a,M,b)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.M (x).(Ax x b) = Cut <a'>.([(a',a)]\<bullet>M) (x').(Ax x' b)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>l ([(a',a)]\<bullet>M)[a'\<turnstile>c>b]" using fs1 fs2 fin by (auto intro: l_redu.intros simp add: fresh_left calc_atm fic_rename) also have "\<dots> = M[a\<turnstile>c>b]" using fs1 fs2 by (simp add: crename_rename) finally show ?thesis by simp qed lemma better_LAxL_intro[intro]: shows "fin M x \<Longrightarrow> Cut <a>.(Ax y a) (x).M \<longrightarrow>\<^isub>l M[x\<turnstile>n>y]" proof - assume fin: "fin M x" obtain x'::"name" where fs1: "x'\<sharp>(y,M,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(a,M)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.(Ax y a) (x).M = Cut <a'>.(Ax y a') (x').([(x',x)]\<bullet>M)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>l ([(x',x)]\<bullet>M)[x'\<turnstile>n>y]" using fs1 fs2 fin by (auto intro: l_redu.intros simp add: fresh_left calc_atm fin_rename) also have "\<dots> = M[x\<turnstile>n>y]" using fs1 fs2 by (simp add: nrename_rename) finally show ?thesis by simp qed lemma better_LNot_intro[intro]: shows "\<lbrakk>y\<sharp>N; a\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) \<longrightarrow>\<^isub>l Cut <b>.N (x).M" proof - assume fs: "y\<sharp>N" "a\<sharp>M" obtain x'::"name" where f1: "x'\<sharp>(y,N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain y'::"name" where f2: "y'\<sharp>(y,N,M,x,x')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f3: "a'\<sharp>(a,M,N,b)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b'::"coname" where f4: "b'\<sharp>(a,M,N,b,a')" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.(NotR (x).M a) (y).(NotL <b>.N y) = Cut <a'>.(NotR (x).([(a',a)]\<bullet>M) a') (y').(NotL <b>.([(y',y)]\<bullet>N) y')" using f1 f2 f3 f4 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm abs_fresh) 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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_LAnd1_intro[intro]: shows "\<lbrakk>a\<sharp>([b1].M1,[b2].M2); y\<sharp>[x].N\<rbrakk> \<Longrightarrow> Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL1 (x).N y) \<longrightarrow>\<^isub>l Cut <b1>.M1 (x).N" proof - assume fs: "a\<sharp>([b1].M1,[b2].M2)" "y\<sharp>[x].N" obtain x'::"name" where f1: "x'\<sharp>(y,N,M1,M2,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain y'::"name" where f2: "y'\<sharp>(y,N,M1,M2,x,x')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f3: "a'\<sharp>(a,M1,M2,N,b1,b2)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b1'::"coname" where f4:"b1'\<sharp>(a,M1,M2,N,b1,b2,a')" by (rule exists_fresh(2), rule fin_supp, blast) obtain b2'::"coname" where f5:"b2'\<sharp>(a,M1,M2,N,b1,b2,a',b1')" by (rule exists_fresh(2),rule fin_supp, blast) have "Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL1 (x).N y) = Cut <a'>.(AndR <b1>.M1 <b2>.M2 a') (y').(AndL1 (x).N y')" using f1 f2 f3 f4 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) apply(auto simp add: perm_fresh_fresh) done also have "\<dots> = Cut <a'>.(AndR <b1'>.([(b1',b1)]\<bullet>M1) <b2'>.([(b2',b2)]\<bullet>M2) a') (y').(AndL1 (x').([(x',x)]\<bullet>N) y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) done also have "\<dots> \<longrightarrow>\<^isub>l Cut <b1'>.([(b1',b1)]\<bullet>M1) (x').([(x',x)]\<bullet>N)" using f1 f2 f3 f4 f5 fs apply - apply(rule l_redu.intros) apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm) 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) finally show ?thesis by simp qed lemma better_LAnd2_intro[intro]: shows "\<lbrakk>a\<sharp>([b1].M1,[b2].M2); y\<sharp>[x].N\<rbrakk> \<Longrightarrow> Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL2 (x).N y) \<longrightarrow>\<^isub>l Cut <b2>.M2 (x).N" proof - assume fs: "a\<sharp>([b1].M1,[b2].M2)" "y\<sharp>[x].N" obtain x'::"name" where f1: "x'\<sharp>(y,N,M1,M2,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain y'::"name" where f2: "y'\<sharp>(y,N,M1,M2,x,x')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f3: "a'\<sharp>(a,M1,M2,N,b1,b2)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b1'::"coname" where f4:"b1'\<sharp>(a,M1,M2,N,b1,b2,a')" by (rule exists_fresh(2), rule fin_supp, blast) obtain b2'::"coname" where f5:"b2'\<sharp>(a,M1,M2,N,b1,b2,a',b1')" by (rule exists_fresh(2),rule fin_supp, blast) have "Cut <a>.(AndR <b1>.M1 <b2>.M2 a) (y).(AndL2 (x).N y) = Cut <a'>.(AndR <b1>.M1 <b2>.M2 a') (y').(AndL2 (x).N y')" using f1 f2 f3 f4 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) apply(auto simp add: perm_fresh_fresh) done also have "\<dots> = Cut <a'>.(AndR <b1'>.([(b1',b1)]\<bullet>M1) <b2'>.([(b2',b2)]\<bullet>M2) a') (y').(AndL2 (x').([(x',x)]\<bullet>N) y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) done also have "\<dots> \<longrightarrow>\<^isub>l Cut <b2'>.([(b2',b2)]\<bullet>M2) (x').([(x',x)]\<bullet>N)" using f1 f2 f3 f4 f5 fs apply - apply(rule l_redu.intros) apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm) 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) finally show ?thesis by simp qed lemma better_LOr1_intro[intro]: shows "\<lbrakk>y\<sharp>([x1].N1,[x2].N2); b\<sharp>[a].M\<rbrakk> \<Longrightarrow> Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^isub>l Cut <a>.M (x1).N1" proof - assume fs: "y\<sharp>([x1].N1,[x2].N2)" "b\<sharp>[a].M" obtain y'::"name" where f1: "y'\<sharp>(y,M,N1,N2,x1,x2)" by (rule exists_fresh(1), rule fin_supp, blast) obtain x1'::"name" where f2: "x1'\<sharp>(y,M,N1,N2,x1,x2,y')" by (rule exists_fresh(1), rule fin_supp, blast) obtain x2'::"name" where f3: "x2'\<sharp>(y,M,N1,N2,x1,x2,y',x1')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f4: "a'\<sharp>(a,N1,N2,M,b)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b'::"coname" where f5: "b'\<sharp>(a,N1,N2,M,b,a')" by (rule exists_fresh(2),rule fin_supp, blast) have "Cut <b>.(OrR1 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) = Cut <b'>.(OrR1 <a>.M b') (y').(OrL (x1).N1 (x2).N2 y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) apply(auto simp add: perm_fresh_fresh) done also have "\<dots> = Cut <b'>.(OrR1 <a'>.([(a',a)]\<bullet>M) b') (y').(OrL (x1').([(x1',x1)]\<bullet>N1) (x2').([(x2',x2)]\<bullet>N2) y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) done also have "\<dots> \<longrightarrow>\<^isub>l Cut <a'>.([(a',a)]\<bullet>M) (x1').([(x1',x1)]\<bullet>N1)" using f1 f2 f3 f4 f5 fs apply - apply(rule l_redu.intros) apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm) 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) finally show ?thesis by simp qed lemma better_LOr2_intro[intro]: shows "\<lbrakk>y\<sharp>([x1].N1,[x2].N2); b\<sharp>[a].M\<rbrakk> \<Longrightarrow> Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) \<longrightarrow>\<^isub>l Cut <a>.M (x2).N2" proof - assume fs: "y\<sharp>([x1].N1,[x2].N2)" "b\<sharp>[a].M" obtain y'::"name" where f1: "y'\<sharp>(y,M,N1,N2,x1,x2)" by (rule exists_fresh(1), rule fin_supp, blast) obtain x1'::"name" where f2: "x1'\<sharp>(y,M,N1,N2,x1,x2,y')" by (rule exists_fresh(1), rule fin_supp, blast) obtain x2'::"name" where f3: "x2'\<sharp>(y,M,N1,N2,x1,x2,y',x1')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f4: "a'\<sharp>(a,N1,N2,M,b)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b'::"coname" where f5: "b'\<sharp>(a,N1,N2,M,b,a')" by (rule exists_fresh(2),rule fin_supp, blast) have "Cut <b>.(OrR2 <a>.M b) (y).(OrL (x1).N1 (x2).N2 y) = Cut <b'>.(OrR2 <a>.M b') (y').(OrL (x1).N1 (x2).N2 y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) apply(auto simp add: perm_fresh_fresh) done also have "\<dots> = Cut <b'>.(OrR2 <a'>.([(a',a)]\<bullet>M) b') (y').(OrL (x1').([(x1',x1)]\<bullet>N1) (x2').([(x2',x2)]\<bullet>N2) y')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) done also have "\<dots> \<longrightarrow>\<^isub>l Cut <a'>.([(a',a)]\<bullet>M) (x2').([(x2',x2)]\<bullet>N2)" using f1 f2 f3 f4 f5 fs apply - apply(rule l_redu.intros) apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm) 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) finally show ?thesis by simp qed lemma better_LImp_intro[intro]: shows "\<lbrakk>z\<sharp>(N,[y].P); b\<sharp>[a].M; a\<sharp>N\<rbrakk> \<Longrightarrow> Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z) \<longrightarrow>\<^isub>l Cut <a>.(Cut <c>.N (x).M) (y).P" proof - assume fs: "z\<sharp>(N,[y].P)" "b\<sharp>[a].M" "a\<sharp>N" obtain y'::"name" where f1: "y'\<sharp>(y,M,N,P,z,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain x'::"name" where f2: "x'\<sharp>(y,M,N,P,z,x,y')" by (rule exists_fresh(1), rule fin_supp, blast) obtain z'::"name" where f3: "z'\<sharp>(y,M,N,P,z,x,y',x')" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where f4: "a'\<sharp>(a,N,P,M,b)" by (rule exists_fresh(2), rule fin_supp, blast) obtain b'::"coname" where f5: "b'\<sharp>(a,N,P,M,b,c,a')" by (rule exists_fresh(2),rule fin_supp, blast) obtain c'::"coname" where f6: "c'\<sharp>(a,N,P,M,b,c,a',b')" by (rule exists_fresh(2),rule fin_supp, blast) have " Cut <b>.(ImpR (x).<a>.M b) (z).(ImpL <c>.N (y).P z) = Cut <b'>.(ImpR (x).<a>.M b') (z').(ImpL <c>.N (y).P z')" using f1 f2 f3 f4 f5 fs apply(rule_tac sym) apply(perm_simp add: trm.inject alpha calc_atm fresh_prod fresh_left fresh_atm abs_fresh) apply(auto simp add: perm_fresh_fresh) done also have "\<dots> = Cut <b'>.(ImpR (x').<a'>.([(a',a)]\<bullet>([(x',x)]\<bullet>M)) b') (z').(ImpL <c'>.([(c',c)]\<bullet>N) (y').([(y',y)]\<bullet>P) z')" using f1 f2 f3 f4 f5 f6 fs apply(rule_tac sym) apply(simp add: trm.inject) apply(simp add: alpha) apply(rule conjI) apply(simp add: trm.inject) apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh) apply(simp add: trm.inject) apply(simp add: alpha) apply(rule conjI) apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh) apply(simp add: alpha fresh_prod fresh_atm abs_perm calc_atm fresh_left abs_fresh) done also have "\<dots> \<longrightarrow>\<^isub>l Cut <a'>.(Cut <c'>.([(c',c)]\<bullet>N) (x').([(a',a)]\<bullet>[(x',x)]\<bullet>M)) (y').([(y',y)]\<bullet>P)" using f1 f2 f3 f4 f5 f6 fs apply - apply(rule l_redu.intros) apply(auto simp add: abs_fresh fresh_prod fresh_left calc_atm fresh_atm) done also have "\<dots> = Cut <a>.(Cut <c>.N (x).M) (y).P" using f1 f2 f3 f4 f5 f6 fs apply(simp add: trm.inject) apply(rule conjI) apply(simp add: alpha) apply(rule disjI2) apply(simp add: trm.inject) apply(rule conjI) apply(simp add: fresh_prod fresh_atm) apply(rule conjI) apply(perm_simp add: calc_atm) apply(auto simp add: fresh_prod fresh_atm)[1] apply(perm_simp add: alpha) apply(perm_simp add: alpha) apply(perm_simp add: alpha) apply(rule conjI) apply(perm_simp add: calc_atm) apply(rule_tac pi="[(a',a)]" in pt_bij4[OF pt_coname_inst, OF at_coname_inst]) apply(perm_simp add: abs_perm calc_atm) apply(perm_simp add: alpha fresh_prod fresh_atm) apply(simp add: abs_fresh) apply(perm_simp add: alpha fresh_prod fresh_atm) done finally show ?thesis by simp qed lemma alpha_coname: fixes M::"trm" and a::"coname" assumes a: "[a].M = [b].N" "c\<sharp>(a,b,M,N)" shows "M = [(a,c)]\<bullet>[(b,c)]\<bullet>N" using a apply(auto simp add: alpha_fresh fresh_prod fresh_atm) apply(drule sym) apply(perm_simp) done lemma alpha_name: fixes M::"trm" and x::"name" assumes a: "[x].M = [y].N" "z\<sharp>(x,y,M,N)" shows "M = [(x,z)]\<bullet>[(y,z)]\<bullet>N" using a apply(auto simp add: alpha_fresh fresh_prod fresh_atm) apply(drule sym) apply(perm_simp) done lemma alpha_name_coname: fixes M::"trm" and x::"name" and a::"coname" assumes a: "[x].[b].M = [y].[c].N" "z\<sharp>(x,y,M,N)" "a\<sharp>(b,c,M,N)" shows "M = [(x,z)]\<bullet>[(b,a)]\<bullet>[(c,a)]\<bullet>[(y,z)]\<bullet>N" using a apply(auto simp add: alpha_fresh fresh_prod fresh_atm abs_supp fin_supp abs_fresh abs_perm fresh_left calc_atm) apply(drule sym) apply(simp) apply(perm_simp) done lemma Cut_l_redu_elim: assumes a: "Cut <a>.M (x).N \<longrightarrow>\<^isub>l R" shows "(\<exists>b. R = M[a\<turnstile>c>b]) \<or> (\<exists>y. R = N[x\<turnstile>n>y]) \<or> (\<exists>y M' b N'. M = NotR (y).M' a \<and> N = NotL <b>.N' x \<and> R = Cut <b>.N' (y).M' \<and> fic M a \<and> fin N x) \<or> (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL1 (y).N' x \<and> R = Cut <b>.M1 (y).N' \<and> fic M a \<and> fin N x) \<or> (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL2 (y).N' x \<and> R = Cut <c>.M2 (y).N' \<and> fic M a \<and> fin N x) \<or> (\<exists>b N' z M1 y M2. M = OrR1 <b>.N' a \<and> N = OrL (z).M1 (y).M2 x \<and> R = Cut <b>.N' (z).M1 \<and> fic M a \<and> fin N x) \<or> (\<exists>b N' z M1 y M2. M = OrR2 <b>.N' a \<and> N = OrL (z).M1 (y).M2 x \<and> R = Cut <b>.N' (y).M2 \<and> fic M a \<and> fin N x) \<or> (\<exists>z b M' c N1 y N2. M = ImpR (z).<b>.M' a \<and> N = ImpL <c>.N1 (y).N2 x \<and> R = Cut <b>.(Cut <c>.N1 (z).M') (y).N2 \<and> b\<sharp>(c,N1) \<and> fic M a \<and> fin N x)" using a apply(erule_tac l_redu.cases) apply(rule disjI1) (* ax case *) apply(simp add: trm.inject) apply(rule_tac x="b" in exI) apply(erule conjE) apply(simp add: alpha) apply(erule disjE) apply(simp) apply(simp) apply(simp add: rename_fresh) apply(rule disjI2) apply(rule disjI1) (* ax case *) apply(simp add: trm.inject) apply(rule_tac x="y" in exI) apply(erule conjE) apply(thin_tac "[a].M = [aa].Ax y aa") apply(simp add: alpha) apply(erule disjE) apply(simp) apply(simp) apply(simp add: rename_fresh) apply(rule disjI2) apply(rule disjI2) apply(rule disjI1) (* not case *) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="c" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp add: calc_atm) apply(rule exI)+ apply(rule conjI) apply(rule refl) apply(generate_fresh "name") apply(simp add: calc_atm abs_fresh fresh_prod fresh_atm fresh_left) apply(auto)[1] apply(drule_tac z="ca" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp add: calc_atm) apply(rule exI)+ apply(rule conjI) apply(rule refl) apply(auto simp add: calc_atm abs_fresh fresh_left)[1] apply(case_tac "y=x") apply(perm_simp) apply(perm_simp) apply(case_tac "aa=a") apply(perm_simp) apply(perm_simp) (* and1 case *) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI1) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="c" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule exI)+ apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst) apply(simp add: calc_atm) apply(rule refl) apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="ca" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh split: if_splits)[1] apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ (* and2 case *) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI1) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="c" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst) apply(simp add: calc_atm) apply(rule refl) apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="ca" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh split: if_splits)[1] apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ (* or1 case *) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI1) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="c" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst) apply(simp add: calc_atm) apply(rule refl) apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="ca" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule exI)+ apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule exI)+ apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ (* or2 case *) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI1) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="c" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="a" and t="[(a,c)]\<bullet>[(b,c)]\<bullet>b" in subst) apply(simp add: calc_atm) apply(rule refl) apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="ca" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,ca)]\<bullet>[(y,ca)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(y,cb)]\<bullet>y" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ (* imp-case *) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(rule disjI2) apply(simp add: trm.inject) apply(erule conjE)+ apply(generate_fresh "coname") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac c="ca" in alpha_coname) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="a" and t="[(a,ca)]\<bullet>[(b,ca)]\<bullet>b" in subst) apply(simp add: calc_atm) apply(rule refl) apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cb" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cb)]\<bullet>[(z,cb)]\<bullet>z" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ apply(generate_fresh "name") apply(simp add: abs_fresh fresh_prod fresh_atm) apply(auto)[1] apply(drule_tac z="cc" in alpha_name) apply(simp add: fresh_prod fresh_atm abs_fresh) apply(simp) apply(rule exI)+ apply(rule conjI) apply(rule_tac s="x" and t="[(x,cc)]\<bullet>[(z,cc)]\<bullet>z" in subst) apply(simp add: calc_atm) apply(rule refl) apply(auto simp add: fresh_left calc_atm abs_fresh alpha perm_fresh_fresh split: if_splits)[1] apply(perm_simp)+ done inductive c_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^isub>c _" [100,100] 100) where left[intro]: "\<lbrakk>\<not>fic M a; a\<sharp>N; x\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>c M{a:=(x).N}" | right[intro]: "\<lbrakk>\<not>fin N x; a\<sharp>N; x\<sharp>M\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>c N{x:=<a>.M}" equivariance c_redu nominal_inductive c_redu by (simp_all add: abs_fresh subst_fresh) lemma better_left[intro]: shows "\<not>fic M a \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>c M{a:=(x).N}" proof - assume not_fic: "\<not>fic M a" obtain x'::"name" where fs1: "x'\<sharp>(N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(a,M,N)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.M (x).N = Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>c ([(a',a)]\<bullet>M){a':=(x').([(x',x)]\<bullet>N)}" using fs1 fs2 not_fic apply - apply(rule left) apply(clarify) apply(drule_tac a'="a" in fic_rename) apply(simp add: perm_swap) apply(simp add: fresh_left calc_atm)+ done also have "\<dots> = M{a:=(x).N}" using fs1 fs2 by (simp add: subst_rename[symmetric] fresh_atm fresh_prod fresh_left calc_atm) finally show ?thesis by simp qed lemma better_right[intro]: shows "\<not>fin N x \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>c N{x:=<a>.M}" proof - assume not_fin: "\<not>fin N x" obtain x'::"name" where fs1: "x'\<sharp>(N,M,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(a,M,N)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.M (x).N = Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>c ([(x',x)]\<bullet>N){x':=<a'>.([(a',a)]\<bullet>M)}" using fs1 fs2 not_fin apply - apply(rule right) apply(clarify) apply(drule_tac x'="x" in fin_rename) apply(simp add: perm_swap) apply(simp add: fresh_left calc_atm)+ done also have "\<dots> = N{x:=<a>.M}" using fs1 fs2 by (simp add: subst_rename[symmetric] fresh_atm fresh_prod fresh_left calc_atm) finally show ?thesis by simp qed lemma fresh_c_redu: fixes x::"name" and c::"coname" shows "M \<longrightarrow>\<^isub>c M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'" and "M \<longrightarrow>\<^isub>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(induct rule: c_redu.induct) apply(auto simp add: abs_fresh rename_fresh subst_fresh) done inductive a_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^isub>a _" [100,100] 100) where al_redu[intro]: "M\<longrightarrow>\<^isub>l M' \<Longrightarrow> M \<longrightarrow>\<^isub>a M'" | ac_redu[intro]: "M\<longrightarrow>\<^isub>c M' \<Longrightarrow> M \<longrightarrow>\<^isub>a M'" | a_Cut_l: "\<lbrakk>a\<sharp>N; x\<sharp>M; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a Cut <a>.M' (x).N" | a_Cut_r: "\<lbrakk>a\<sharp>N; x\<sharp>M; N\<longrightarrow>\<^isub>a N'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a Cut <a>.M (x).N'" | a_NotL[intro]: "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> NotL <a>.M x \<longrightarrow>\<^isub>a NotL <a>.M' x" | a_NotR[intro]: "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> NotR (x).M a \<longrightarrow>\<^isub>a NotR (x).M' a" | a_AndR_l: "\<lbrakk>a\<sharp>(N,c); b\<sharp>(M,c); b\<noteq>a; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a AndR <a>.M' <b>.N c" | a_AndR_r: "\<lbrakk>a\<sharp>(N,c); b\<sharp>(M,c); b\<noteq>a; N\<longrightarrow>\<^isub>a N'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a AndR <a>.M <b>.N' c" | a_AndL1: "\<lbrakk>x\<sharp>y; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^isub>a AndL1 (x).M' y" | a_AndL2: "\<lbrakk>x\<sharp>y; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^isub>a AndL2 (x).M' y" | a_OrL_l: "\<lbrakk>x\<sharp>(N,z); y\<sharp>(M,z); y\<noteq>x; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a OrL (x).M' (y).N z" | a_OrL_r: "\<lbrakk>x\<sharp>(N,z); y\<sharp>(M,z); y\<noteq>x; N\<longrightarrow>\<^isub>a N'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a OrL (x).M (y).N' z" | a_OrR1: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^isub>a OrR1 <a>.M' b" | a_OrR2: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^isub>a OrR2 <a>.M' b" | a_ImpL_l: "\<lbrakk>a\<sharp>N; x\<sharp>(M,y); M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^isub>a ImpL <a>.M' (x).N y" | a_ImpL_r: "\<lbrakk>a\<sharp>N; x\<sharp>(M,y); N\<longrightarrow>\<^isub>a N'\<rbrakk> \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^isub>a ImpL <a>.M (x).N' y" | a_ImpR: "\<lbrakk>a\<sharp>b; M\<longrightarrow>\<^isub>a M'\<rbrakk> \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^isub>a ImpR (x).<a>.M' b" lemma fresh_a_redu: fixes x::"name" and c::"coname" shows "M \<longrightarrow>\<^isub>a M' \<Longrightarrow> x\<sharp>M \<Longrightarrow> x\<sharp>M'" and "M \<longrightarrow>\<^isub>a M' \<Longrightarrow> c\<sharp>M \<Longrightarrow> c\<sharp>M'" apply - apply(induct rule: a_redu.induct) apply(simp add: fresh_l_redu) apply(simp add: fresh_c_redu) apply(auto simp add: abs_fresh abs_supp fin_supp) apply(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) done equivariance a_redu nominal_inductive a_redu by (simp_all add: abs_fresh fresh_atm fresh_prod abs_supp fin_supp fresh_a_redu) lemma better_CutL_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a Cut <a>.M' (x).N" proof - assume red: "M\<longrightarrow>\<^isub>a M'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.M (x).N = Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_CutR_intro[intro]: shows "N\<longrightarrow>\<^isub>a N' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a Cut <a>.M (x).N'" proof - assume red: "N\<longrightarrow>\<^isub>a N'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast) have "Cut <a>.M (x).N = Cut <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N)" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_AndRL_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a AndR <a>.M' <b>.N c" proof - assume red: "M\<longrightarrow>\<^isub>a M'" obtain b'::"coname" where fs1: "b'\<sharp>(M,N,a,b,c)" by (rule exists_fresh(2), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,b,c,b')" by (rule exists_fresh(2), rule fin_supp, blast) have "AndR <a>.M <b>.N c = AndR <a'>.([(a',a)]\<bullet>M) <b'>.([(b',b)]\<bullet>N) c" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_AndRR_intro[intro]: shows "N\<longrightarrow>\<^isub>a N' \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a AndR <a>.M <b>.N' c" proof - assume red: "N\<longrightarrow>\<^isub>a N'" obtain b'::"coname" where fs1: "b'\<sharp>(M,N,a,b,c)" by (rule exists_fresh(2), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a,b,c,b')" by (rule exists_fresh(2), rule fin_supp, blast) have "AndR <a>.M <b>.N c = AndR <a'>.([(a',a)]\<bullet>M) <b'>.([(b',b)]\<bullet>N) c" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_AndL1_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^isub>a AndL1 (x).M' y" proof - assume red: "M\<longrightarrow>\<^isub>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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_AndL2_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^isub>a AndL2 (x).M' y" proof - assume red: "M\<longrightarrow>\<^isub>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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_OrLL_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a OrL (x).M' (y).N z" proof - assume red: "M\<longrightarrow>\<^isub>a M'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y,z)" by (rule exists_fresh(1), rule fin_supp, blast) obtain y'::"name" where fs2: "y'\<sharp>(M,N,x,y,z,x')" by (rule exists_fresh(1), rule fin_supp, blast) have "OrL (x).M (y).N z = OrL (x').([(x',x)]\<bullet>M) (y').([(y',y)]\<bullet>N) z" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_OrLR_intro[intro]: shows "N\<longrightarrow>\<^isub>a N' \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a OrL (x).M (y).N' z" proof - assume red: "N\<longrightarrow>\<^isub>a N'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y,z)" by (rule exists_fresh(1), rule fin_supp, blast) obtain y'::"name" where fs2: "y'\<sharp>(M,N,x,y,z,x')" by (rule exists_fresh(1), rule fin_supp, blast) have "OrL (x).M (y).N z = OrL (x').([(x',x)]\<bullet>M) (y').([(y',y)]\<bullet>N) z" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_OrR1_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^isub>a OrR1 <a>.M' b" proof - assume red: "M\<longrightarrow>\<^isub>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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_OrR2_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^isub>a OrR2 <a>.M' b" proof - assume red: "M\<longrightarrow>\<^isub>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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_ImpLL_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^isub>a ImpL <a>.M' (x).N y" proof - assume red: "M\<longrightarrow>\<^isub>a M'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast) have "ImpL <a>.M (x).N y = ImpL <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N) y" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_ImpLR_intro[intro]: shows "N\<longrightarrow>\<^isub>a N' \<Longrightarrow> ImpL <a>.M (x).N y \<longrightarrow>\<^isub>a ImpL <a>.M (x).N' y" proof - assume red: "N\<longrightarrow>\<^isub>a N'" obtain x'::"name" where fs1: "x'\<sharp>(M,N,x,y)" by (rule exists_fresh(1), rule fin_supp, blast) obtain a'::"coname" where fs2: "a'\<sharp>(M,N,a)" by (rule exists_fresh(2), rule fin_supp, blast) have "ImpL <a>.M (x).N y = ImpL <a'>.([(a',a)]\<bullet>M) (x').([(x',x)]\<bullet>N) y" using fs1 fs2 by (rule_tac sym, auto simp add: trm.inject alpha fresh_atm fresh_prod calc_atm) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed lemma better_ImpR_intro[intro]: shows "M\<longrightarrow>\<^isub>a M' \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^isub>a ImpR (x).<a>.M' b" proof - assume red: "M\<longrightarrow>\<^isub>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) also have "\<dots> \<longrightarrow>\<^isub>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) finally show ?thesis by simp qed text {* axioms do not reduce *} lemma ax_do_not_l_reduce: shows "Ax x a \<longrightarrow>\<^isub>l M \<Longrightarrow> False" by (erule_tac l_redu.cases) (simp_all add: trm.inject) lemma ax_do_not_c_reduce: shows "Ax x a \<longrightarrow>\<^isub>c M \<Longrightarrow> False" by (erule_tac c_redu.cases) (simp_all add: trm.inject) lemma ax_do_not_a_reduce: shows "Ax x a \<longrightarrow>\<^isub>a M \<Longrightarrow> False" apply(erule_tac a_redu.cases) apply(auto simp add: trm.inject) apply(drule ax_do_not_l_reduce) apply(simp) apply(drule ax_do_not_c_reduce) apply(simp) done lemma a_redu_NotL_elim: assumes a: "NotL <a>.M x \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = NotL <a>.M' x \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 1) 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(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(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(simp add: perm_swap) done lemma a_redu_NotR_elim: assumes a: "NotR (x).M a \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = NotR (x).M' a \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 1) 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(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(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(simp add: perm_swap) done lemma a_redu_AndR_elim: assumes a: "AndR <a>.M <b>.N c\<longrightarrow>\<^isub>a R" shows "(\<exists>M'. R = AndR <a>.M' <b>.N c \<and> M\<longrightarrow>\<^isub>aM') \<or> (\<exists>N'. R = AndR <a>.M <b>.N' c \<and> N\<longrightarrow>\<^isub>aN')" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rotate_tac 6) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rule disjI1) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: 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(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(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(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(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(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(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(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(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(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(b,ba)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(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(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(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(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] done lemma a_redu_AndL1_elim: assumes a: "AndL1 (x).M y \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = AndL1 (x).M' y \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 2) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto simp add: alpha a_redu.eqvt) apply(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(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(simp add: perm_swap) done lemma a_redu_AndL2_elim: assumes a: "AndL2 (x).M y \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = AndL2 (x).M' y \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 2) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto simp add: alpha a_redu.eqvt) apply(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(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(simp add: perm_swap) done lemma a_redu_OrL_elim: assumes a: "OrL (x).M (y).N z\<longrightarrow>\<^isub>a R" shows "(\<exists>M'. R = OrL (x).M' (y).N z \<and> M\<longrightarrow>\<^isub>aM') \<or> (\<exists>N'. R = OrL (x).M (y).N' z \<and> N\<longrightarrow>\<^isub>aN')" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rotate_tac 6) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rule disjI1) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: 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(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(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(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(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(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(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(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(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(rule_tac x="([(x,xa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(x,xaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,ya)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(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(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(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(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] done lemma a_redu_OrR1_elim: assumes a: "OrR1 <a>.M b \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = OrR1 <a>.M' b \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 2) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto simp add: alpha a_redu.eqvt) apply(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(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(simp add: perm_swap) done lemma a_redu_OrR2_elim: assumes a: "OrR2 <a>.M b \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = OrR2 <a>.M' b \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 2) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto simp add: alpha a_redu.eqvt) apply(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(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(simp add: perm_swap) done lemma a_redu_ImpL_elim: assumes a: "ImpL <a>.M (y).N z\<longrightarrow>\<^isub>a R" shows "(\<exists>M'. R = ImpL <a>.M' (y).N z \<and> M\<longrightarrow>\<^isub>aM') \<or> (\<exists>N'. R = ImpL <a>.M (y).N' z \<and> N\<longrightarrow>\<^isub>aN')" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rotate_tac 5) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(rule disjI1) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(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(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(a,aaa)]\<bullet>M')" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule disjI2) apply(auto simp add: alpha a_redu.eqvt)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] apply(rule_tac x="([(y,xa)]\<bullet>N'a)" in exI) apply(auto simp add: perm_swap fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu)[1] done lemma a_redu_ImpR_elim: assumes a: "ImpR (x).<a>.M b \<longrightarrow>\<^isub>a R" shows "\<exists>M'. R = ImpR (x).<a>.M' b \<and> M\<longrightarrow>\<^isub>aM'" using a apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto) apply(rotate_tac 2) apply(erule_tac a_redu.cases, simp_all add: trm.inject) apply(erule_tac l_redu.cases, simp_all add: trm.inject) apply(erule_tac c_redu.cases, simp_all add: trm.inject) apply(auto simp add: alpha a_redu.eqvt abs_perm abs_fresh) apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap) apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap) apply(rule_tac x="([(a,aa)]\<bullet>M'a)" in exI) apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap) apply(rule_tac x="([(a,aaa)]\<bullet>M'a)" in exI) apply(auto simp add: fresh_left alpha a_redu.eqvt calc_atm fresh_a_redu perm_swap) apply(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(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(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(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(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(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(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(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(rule sym) apply(rule trans) apply(rule perm_compose) apply(simp add: calc_atm perm_swap) done text {* Transitive Closure*} abbreviation a_star_redu :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longrightarrow>\<^isub>a* _" [100,100] 100) where "M \<longrightarrow>\<^isub>a* M' \<equiv> (a_redu)^** M M'" lemma a_starI: assumes a: "M \<longrightarrow>\<^isub>a M'" shows "M \<longrightarrow>\<^isub>a* M'" using a by blast lemma a_starE: assumes a: "M \<longrightarrow>\<^isub>a* M'" shows "M = M' \<or> (\<exists>N. M \<longrightarrow>\<^isub>a N \<and> N \<longrightarrow>\<^isub>a* M')" using a by (induct) (auto) lemma a_star_refl: shows "M \<longrightarrow>\<^isub>a* M" by blast lemma a_star_trans[trans]: assumes a1: "M1\<longrightarrow>\<^isub>a* M2" and a2: "M2\<longrightarrow>\<^isub>a* M3" shows "M1 \<longrightarrow>\<^isub>a* M3" using a2 a1 by (induct) (auto) text {* congruence rules for \<longrightarrow>\<^isub>a* *} lemma ax_do_not_a_star_reduce: shows "Ax x a \<longrightarrow>\<^isub>a* M \<Longrightarrow> M = Ax x a" apply(induct set: rtranclp) apply(auto) apply(drule ax_do_not_a_reduce) apply(simp) done lemma a_star_CutL: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a* Cut <a>.M' (x).N" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_CutR: "N \<longrightarrow>\<^isub>a* N'\<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a* Cut <a>.M (x).N'" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_Cut: "\<lbrakk>M \<longrightarrow>\<^isub>a* M'; N \<longrightarrow>\<^isub>a* N'\<rbrakk> \<Longrightarrow> Cut <a>.M (x).N \<longrightarrow>\<^isub>a* Cut <a>.M' (x).N'" by (blast intro!: a_star_CutL a_star_CutR intro: rtranclp_trans) lemma a_star_NotR: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> NotR (x).M a \<longrightarrow>\<^isub>a* NotR (x).M' a" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_NotL: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> NotL <a>.M x \<longrightarrow>\<^isub>a* NotL <a>.M' x" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_AndRL: "M \<longrightarrow>\<^isub>a* M'\<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a* AndR <a>.M' <b>.N c" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_AndRR: "N \<longrightarrow>\<^isub>a* N'\<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a* AndR <a>.M <b>.N' c" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_AndR: "\<lbrakk>M \<longrightarrow>\<^isub>a* M'; N \<longrightarrow>\<^isub>a* N'\<rbrakk> \<Longrightarrow> AndR <a>.M <b>.N c \<longrightarrow>\<^isub>a* AndR <a>.M' <b>.N' c" by (blast intro!: a_star_AndRL a_star_AndRR intro: rtranclp_trans) lemma a_star_AndL1: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> AndL1 (x).M y \<longrightarrow>\<^isub>a* AndL1 (x).M' y" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_AndL2: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> AndL2 (x).M y \<longrightarrow>\<^isub>a* AndL2 (x).M' y" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_OrLL: "M \<longrightarrow>\<^isub>a* M'\<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a* OrL (x).M' (y).N z" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_OrLR: "N \<longrightarrow>\<^isub>a* N'\<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a* OrL (x).M (y).N' z" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_OrL: "\<lbrakk>M \<longrightarrow>\<^isub>a* M'; N \<longrightarrow>\<^isub>a* N'\<rbrakk> \<Longrightarrow> OrL (x).M (y).N z \<longrightarrow>\<^isub>a* OrL (x).M' (y).N' z" by (blast intro!: a_star_OrLL a_star_OrLR intro: rtranclp_trans) lemma a_star_OrR1: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> OrR1 <a>.M b \<longrightarrow>\<^isub>a* OrR1 <a>.M' b" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_OrR2: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> OrR2 <a>.M b \<longrightarrow>\<^isub>a* OrR2 <a>.M' b" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_ImpLL: "M \<longrightarrow>\<^isub>a* M'\<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^isub>a* ImpL <a>.M' (y).N z" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_ImpLR: "N \<longrightarrow>\<^isub>a* N'\<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^isub>a* ImpL <a>.M (y).N' z" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemma a_star_ImpL: "\<lbrakk>M \<longrightarrow>\<^isub>a* M'; N \<longrightarrow>\<^isub>a* N'\<rbrakk> \<Longrightarrow> ImpL <a>.M (y).N z \<longrightarrow>\<^isub>a* ImpL <a>.M' (y).N' z" by (blast intro!: a_star_ImpLL a_star_ImpLR intro: rtranclp_trans) lemma a_star_ImpR: "M \<longrightarrow>\<^isub>a* M' \<Longrightarrow> ImpR (x).<a>.M b \<longrightarrow>\<^isub>a* ImpR (x).<a>.M' b" by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+ lemmas a_star_congs = a_star_Cut a_star_NotR a_star_NotL a_star_AndR a_star_AndL1 a_star_AndL2 a_star_OrL a_star_OrR1 a_star_OrR2 a_star_ImpL a_star_ImpR lemma a_star_redu_NotL_elim: assumes a: "NotL <a>.M x \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = NotL <a>.M' x \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_NotL_elim) apply(auto) done lemma a_star_redu_NotR_elim: assumes a: "NotR (x).M a \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = NotR (x).M' a \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_NotR_elim) apply(auto) done lemma a_star_redu_AndR_elim: assumes a: "AndR <a>.M <b>.N c\<longrightarrow>\<^isub>a* R" shows "(\<exists>M' N'. R = AndR <a>.M' <b>.N' c \<and> M \<longrightarrow>\<^isub>a* M' \<and> N \<longrightarrow>\<^isub>a* N')" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_AndR_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_AndL1_elim: assumes a: "AndL1 (x).M y \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = AndL1 (x).M' y \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_AndL1_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_AndL2_elim: assumes a: "AndL2 (x).M y \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = AndL2 (x).M' y \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_AndL2_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_OrL_elim: assumes a: "OrL (x).M (y).N z \<longrightarrow>\<^isub>a* R" shows "(\<exists>M' N'. R = OrL (x).M' (y).N' z \<and> M \<longrightarrow>\<^isub>a* M' \<and> N \<longrightarrow>\<^isub>a* N')" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_OrL_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_OrR1_elim: assumes a: "OrR1 <a>.M y \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = OrR1 <a>.M' y \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_OrR1_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_OrR2_elim: assumes a: "OrR2 <a>.M y \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = OrR2 <a>.M' y \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_OrR2_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_ImpR_elim: assumes a: "ImpR (x).<a>.M y \<longrightarrow>\<^isub>a* R" shows "\<exists>M'. R = ImpR (x).<a>.M' y \<and> M \<longrightarrow>\<^isub>a* M'" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_ImpR_elim) apply(auto simp add: alpha trm.inject) done lemma a_star_redu_ImpL_elim: assumes a: "ImpL <a>.M (y).N z \<longrightarrow>\<^isub>a* R" shows "(\<exists>M' N'. R = ImpL <a>.M' (y).N' z \<and> M \<longrightarrow>\<^isub>a* M' \<and> N \<longrightarrow>\<^isub>a* N')" using a apply(induct set: rtranclp) apply(auto) apply(drule a_redu_ImpL_elim) apply(auto simp add: alpha trm.inject) done text {* Substitution *} lemma subst_not_fin1: shows "\<not>fin(M{x:=<c>.P}) x" apply(nominal_induct M avoiding: x c P rule: trm.strong_induct) apply(auto) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<c>.P},P,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<c>.P},P,name1,trm2{x:=<c>.P},name2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(erule fin.cases, simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<c>.P},P,name1,trm2{name2:=<c>.P})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(erule fin.cases, simp_all add: trm.inject) apply(rule exists_fresh'(1)[OF fs_name1]) apply(erule fin.cases, simp_all add: trm.inject) done lemma subst_not_fin2: assumes a: "\<not>fin M y" shows "\<not>fin(M{c:=(x).P}) y" using a apply(nominal_induct M avoiding: x c P y rule: trm.strong_induct) apply(auto) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname:=(x).P},P)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(drule fin_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(auto)[1] apply(drule freshn_after_substc) apply(simp add: fin.intros) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm1{coname3:=(x).P},P,coname1,trm2{coname3:=(x).P},coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(drule fin_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshn_after_substc) apply(simp add: fin.intros abs_fresh) apply(drule fin_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshn_after_substc) apply(simp add: fin.intros abs_fresh) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(drule fin_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(drule fin_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshn_after_substc) apply(drule freshn_after_substc) apply(simp add: fin.intros abs_fresh) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(trm{coname2:=(x).P},P,coname1,coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(drule fin_elims, simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(drule fin_elims, simp) apply(drule fin_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshn_after_substc) apply(drule freshn_after_substc) apply(simp add: fin.intros abs_fresh) done lemma subst_not_fic1: shows "\<not>fic (M{a:=(x).P}) a" apply(nominal_induct M avoiding: a x P rule: trm.strong_induct) apply(auto) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname:=(x).P},P)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(erule fic.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm1{coname3:=(x).P},P,trm2{coname3:=(x).P},coname1,coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(erule fic.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(erule fic.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fic.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(erule fic.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(trm{coname2:=(x).P},P,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(erule fic.cases, simp_all add: trm.inject) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(erule fic.cases, simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject) done lemma subst_not_fic2: assumes a: "\<not>fic M a" shows "\<not>fic(M{x:=<b>.P}) a" using a apply(nominal_induct M avoiding: x a P b rule: trm.strong_induct) apply(auto) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(auto)[1] apply(drule freshc_after_substn) apply(simp add: fic.intros) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.P},P,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.P},P,name1,trm2{x:=<b>.P},name2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) apply(drule fic_elims, simp) apply(auto)[1] apply(simp add: abs_fresh fresh_atm) apply(drule freshc_after_substn) apply(simp add: fic.intros abs_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.P},trm2{name2:=<b>.P},P,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(drule fic_elims, simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(drule fic_elims, simp) done text {* Reductions *} lemma fin_l_reduce: assumes a: "fin M x" and b: "M \<longrightarrow>\<^isub>l M'" shows "fin M' x" using b a apply(induct) apply(erule fin.cases) apply(simp_all add: trm.inject) apply(rotate_tac 3) apply(erule fin.cases) apply(simp_all add: trm.inject) apply(erule fin.cases, simp_all add: trm.inject)+ done lemma fin_c_reduce: assumes a: "fin M x" and b: "M \<longrightarrow>\<^isub>c M'" shows "fin M' x" using b a apply(induct) apply(erule fin.cases, simp_all add: trm.inject)+ done lemma fin_a_reduce: assumes a: "fin M x" and b: "M \<longrightarrow>\<^isub>a M'" shows "fin M' x" using a b apply(induct) apply(drule ax_do_not_a_reduce) apply(simp) apply(drule a_redu_NotL_elim) apply(auto) apply(rule fin.intros) apply(simp add: fresh_a_redu) apply(drule a_redu_AndL1_elim) apply(auto) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_AndL2_elim) apply(auto) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_OrL_elim) apply(auto) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_ImpL_elim) apply(auto) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) apply(rule fin.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) done lemma fin_a_star_reduce: assumes a: "fin M x" and b: "M \<longrightarrow>\<^isub>a* M'" shows "fin M' x" using b a apply(induct set: rtranclp) apply(auto simp add: fin_a_reduce) done lemma fic_l_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^isub>l M'" shows "fic M' x" using b a apply(induct) apply(erule fic.cases) apply(simp_all add: trm.inject) apply(rotate_tac 3) apply(erule fic.cases) apply(simp_all add: trm.inject) apply(erule fic.cases, simp_all add: trm.inject)+ done lemma fic_c_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^isub>c M'" shows "fic M' x" using b a apply(induct) apply(erule fic.cases, simp_all add: trm.inject)+ done lemma fic_a_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^isub>a M'" shows "fic M' x" using a b apply(induct) apply(drule ax_do_not_a_reduce) apply(simp) apply(drule a_redu_NotR_elim) apply(auto) apply(rule fic.intros) apply(simp add: fresh_a_redu) apply(drule a_redu_AndR_elim) apply(auto) apply(rule fic.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) apply(rule fic.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_OrR1_elim) apply(auto) apply(rule fic.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_OrR2_elim) apply(auto) apply(rule fic.intros) apply(force simp add: abs_fresh fresh_a_redu) apply(drule a_redu_ImpR_elim) apply(auto) apply(rule fic.intros) apply(force simp add: abs_fresh fresh_a_redu) done lemma fic_a_star_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^isub>a* M'" shows "fic M' x" using b a apply(induct set: rtranclp) apply(auto simp add: fic_a_reduce) done text {* substitution properties *} lemma subst_with_ax1: shows "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" proof(nominal_induct M avoiding: x a y rule: trm.strong_induct) case (Ax z b x a y) show "(Ax z b){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Ax z b)[x\<turnstile>n>y]" proof (cases "z=x") case True assume eq: "z=x" have "(Ax z b){x:=<a>.Ax y a} = Cut <a>.Ax y a (x).Ax x b" using eq by simp also have "\<dots> \<longrightarrow>\<^isub>a* (Ax x b)[x\<turnstile>n>y]" by blast finally show "Ax z b{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Ax z b)[x\<turnstile>n>y]" using eq by simp next case False assume neq: "z\<noteq>x" then show "(Ax z b){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Ax z b)[x\<turnstile>n>y]" using neq by simp qed next case (Cut b M z N x a y) have fs: "b\<sharp>x" "b\<sharp>a" "b\<sharp>y" "b\<sharp>N" "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "z\<sharp>M" by fact+ have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* N[x\<turnstile>n>y]" by fact show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]" proof (cases "M = Ax x b") case True assume eq: "M = Ax x b" have "(Cut <b>.M (z).N){x:=<a>.Ax y a} = Cut <a>.Ax y a (z).(N{x:=<a>.Ax y a})" using fs eq by simp also have "\<dots> \<longrightarrow>\<^isub>a* Cut <a>.Ax y a (z).(N[x\<turnstile>n>y])" using ih2 a_star_congs by blast also have "\<dots> = Cut <b>.(M[x\<turnstile>n>y]) (z).(N[x\<turnstile>n>y])" using eq by (simp add: trm.inject alpha calc_atm fresh_atm) finally show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]" using fs by simp next case False assume neq: "M \<noteq> Ax x b" have "(Cut <b>.M (z).N){x:=<a>.Ax y a} = Cut <b>.(M{x:=<a>.Ax y a}) (z).(N{x:=<a>.Ax y a})" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* Cut <b>.(M[x\<turnstile>n>y]) (z).(N[x\<turnstile>n>y])" using ih1 ih2 a_star_congs by blast finally show "(Cut <b>.M (z).N){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (Cut <b>.M (z).N)[x\<turnstile>n>y]" using fs by simp qed next case (NotR z M b x a y) have fs: "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "z\<sharp>b" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have "(NotR (z).M b){x:=<a>.Ax y a} = NotR (z).(M{x:=<a>.Ax y a}) b" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* NotR (z).(M[x\<turnstile>n>y]) b" using ih by (auto intro: a_star_congs) finally show "(NotR (z).M b){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (NotR (z).M b)[x\<turnstile>n>y]" using fs by simp next case (NotL b M z x a y) have fs: "b\<sharp>x" "b\<sharp>a" "b\<sharp>y" "b\<sharp>z" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (NotL <b>.M z)[x\<turnstile>n>y]" proof(cases "z=x") case True assume eq: "z=x" obtain x'::"name" where new: "x'\<sharp>(Ax y a,M{x:=<a>.Ax y a})" by (rule exists_fresh(1)[OF fs_name1]) have "(NotL <b>.M z){x:=<a>.Ax y a} = fresh_fun (\<lambda>x'. Cut <a>.Ax y a (x').NotL <b>.(M{x:=<a>.Ax y a}) x')" using eq fs by simp also have "\<dots> = Cut <a>.Ax y a (x').NotL <b>.(M{x:=<a>.Ax y a}) x'" using new by (simp add: fresh_fun_simp_NotL fresh_prod) also have "\<dots> \<longrightarrow>\<^isub>a* (NotL <b>.(M{x:=<a>.Ax y a}) x')[x'\<turnstile>n>y]" using new apply(rule_tac a_starI) apply(rule al_redu) apply(rule better_LAxL_intro) apply(auto) done also have "\<dots> = NotL <b>.(M{x:=<a>.Ax y a}) y" using new by (simp add: nrename_fresh) also have "\<dots> \<longrightarrow>\<^isub>a* NotL <b>.(M[x\<turnstile>n>y]) y" using ih by (auto intro: a_star_congs) also have "\<dots> = (NotL <b>.M z)[x\<turnstile>n>y]" using eq by simp finally show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (NotL <b>.M z)[x\<turnstile>n>y]" by simp next case False assume neq: "z\<noteq>x" have "(NotL <b>.M z){x:=<a>.Ax y a} = NotL <b>.(M{x:=<a>.Ax y a}) z" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* NotL <b>.(M[x\<turnstile>n>y]) z" using ih by (auto intro: a_star_congs) finally show "(NotL <b>.M z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (NotL <b>.M z)[x\<turnstile>n>y]" using neq by simp qed next case (AndR c M d N e x a y) have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "d\<sharp>x" "d\<sharp>a" "d\<sharp>y" "d\<noteq>c" "c\<sharp>N" "c\<sharp>e" "d\<sharp>M" "d\<sharp>e" by fact+ have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* N[x\<turnstile>n>y]" by fact have "(AndR <c>.M <d>.N e){x:=<a>.Ax y a} = AndR <c>.(M{x:=<a>.Ax y a}) <d>.(N{x:=<a>.Ax y a}) e" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndR <c>.(M[x\<turnstile>n>y]) <d>.(N[x\<turnstile>n>y]) e" using ih1 ih2 by (auto intro: a_star_congs) finally show "(AndR <c>.M <d>.N e){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (AndR <c>.M <d>.N e)[x\<turnstile>n>y]" using fs by simp next case (AndL1 u M v x a y) have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "u\<sharp>v" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact show "(AndL1 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (AndL1 (u).M v)[x\<turnstile>n>y]" proof(cases "v=x") case True assume eq: "v=x" obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},u)" by (rule exists_fresh(1)[OF fs_name1]) have "(AndL1 (u).M v){x:=<a>.Ax y a} = fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').AndL1 (u).(M{x:=<a>.Ax y a}) v')" using eq fs by simp also have "\<dots> = Cut <a>.Ax y a (v').AndL1 (u).(M{x:=<a>.Ax y a}) v'" using new by (simp add: fresh_fun_simp_AndL1 fresh_prod) also have "\<dots> \<longrightarrow>\<^isub>a* (AndL1 (u).(M{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxL_intro) apply(rule fin.intros) apply(simp add: abs_fresh) done also have "\<dots> = AndL1 (u).(M{x:=<a>.Ax y a}) y" using fs new by (auto simp add: fresh_prod fresh_atm nrename_fresh) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (AndL1 (u).M v)[x\<turnstile>n>y]" by simp next case False assume neq: "v\<noteq>x" have "(AndL1 (u).M v){x:=<a>.Ax y a} = AndL1 (u).(M{x:=<a>.Ax y a}) v" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndL1 (u).(M[x\<turnstile>n>y]) v" using ih by (auto intro: a_star_congs) finally show "(AndL1 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (AndL1 (u).M v)[x\<turnstile>n>y]" using fs neq by simp qed next case (AndL2 u M v x a y) have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "u\<sharp>v" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact show "(AndL2 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (AndL2 (u).M v)[x\<turnstile>n>y]" proof(cases "v=x") case True assume eq: "v=x" obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},u)" by (rule exists_fresh(1)[OF fs_name1]) have "(AndL2 (u).M v){x:=<a>.Ax y a} = fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').AndL2 (u).(M{x:=<a>.Ax y a}) v')" using eq fs by simp also have "\<dots> = Cut <a>.Ax y a (v').AndL2 (u).(M{x:=<a>.Ax y a}) v'" using new by (simp add: fresh_fun_simp_AndL2 fresh_prod) also have "\<dots> \<longrightarrow>\<^isub>a* (AndL2 (u).(M{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxL_intro) apply(rule fin.intros) apply(simp add: abs_fresh) done also have "\<dots> = AndL2 (u).(M{x:=<a>.Ax y a}) y" using fs new by (auto simp add: fresh_prod fresh_atm nrename_fresh) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (AndL2 (u).M v)[x\<turnstile>n>y]" by simp next case False assume neq: "v\<noteq>x" have "(AndL2 (u).M v){x:=<a>.Ax y a} = AndL2 (u).(M{x:=<a>.Ax y a}) v" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndL2 (u).(M[x\<turnstile>n>y]) v" using ih by (auto intro: a_star_congs) finally show "(AndL2 (u).M v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (AndL2 (u).M v)[x\<turnstile>n>y]" using fs neq by simp qed next case (OrR1 c M d x a y) have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "c\<sharp>d" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have "(OrR1 <c>.M d){x:=<a>.Ax y a} = OrR1 <c>.(M{x:=<a>.Ax y a}) d" using fs by (simp add: fresh_atm) also have "\<dots> \<longrightarrow>\<^isub>a* OrR1 <c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs) finally show "(OrR1 <c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (OrR1 <c>.M d)[x\<turnstile>n>y]" using fs by simp next case (OrR2 c M d x a y) have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "c\<sharp>d" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have "(OrR2 <c>.M d){x:=<a>.Ax y a} = OrR2 <c>.(M{x:=<a>.Ax y a}) d" using fs by (simp add: fresh_atm) also have "\<dots> \<longrightarrow>\<^isub>a* OrR2 <c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs) finally show "(OrR2 <c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (OrR2 <c>.M d)[x\<turnstile>n>y]" using fs by simp next case (OrL u M v N z x a y) have fs: "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "v\<sharp>x" "v\<sharp>a" "v\<sharp>y" "v\<noteq>u" "u\<sharp>N" "u\<sharp>z" "v\<sharp>M" "v\<sharp>z" by fact+ have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* N[x\<turnstile>n>y]" by fact show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]" proof(cases "z=x") case True assume eq: "z=x" obtain z'::"name" where new: "z'\<sharp>(Ax y a,M{x:=<a>.Ax y a},N{x:=<a>.Ax y a},u,v,y,a)" by (rule exists_fresh(1)[OF fs_name1]) have "(OrL (u).M (v).N z){x:=<a>.Ax y a} = fresh_fun (\<lambda>z'. Cut <a>.Ax y a (z').OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z')" using eq fs by simp also have "\<dots> = Cut <a>.Ax y a (z').OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z'" using new by (simp add: fresh_fun_simp_OrL) also have "\<dots> \<longrightarrow>\<^isub>a* (OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z')[z'\<turnstile>n>y]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxL_intro) apply(rule fin.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) y" using fs new by (auto simp add: fresh_prod fresh_atm nrename_fresh subst_fresh) also have "\<dots> \<longrightarrow>\<^isub>a* OrL (u).(M[x\<turnstile>n>y]) (v).(N[x\<turnstile>n>y]) y" using ih1 ih2 by (auto intro: a_star_congs) also have "\<dots> = (OrL (u).M (v).N z)[x\<turnstile>n>y]" using eq fs by simp finally show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]" by simp next case False assume neq: "z\<noteq>x" have "(OrL (u).M (v).N z){x:=<a>.Ax y a} = OrL (u).(M{x:=<a>.Ax y a}) (v).(N{x:=<a>.Ax y a}) z" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* OrL (u).(M[x\<turnstile>n>y]) (v).(N[x\<turnstile>n>y]) z" using ih1 ih2 by (auto intro: a_star_congs) finally show "(OrL (u).M (v).N z){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (OrL (u).M (v).N z)[x\<turnstile>n>y]" using fs neq by simp qed next case (ImpR z c M d x a y) have fs: "z\<sharp>x" "z\<sharp>a" "z\<sharp>y" "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "z\<sharp>d" "c\<sharp>d" by fact+ have ih: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have "(ImpR (z).<c>.M d){x:=<a>.Ax y a} = ImpR (z).<c>.(M{x:=<a>.Ax y a}) d" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* ImpR (z).<c>.(M[x\<turnstile>n>y]) d" using ih by (auto intro: a_star_congs) finally show "(ImpR (z).<c>.M d){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (ImpR (z).<c>.M d)[x\<turnstile>n>y]" using fs by simp next case (ImpL c M u N v x a y) have fs: "c\<sharp>x" "c\<sharp>a" "c\<sharp>y" "u\<sharp>x" "u\<sharp>a" "u\<sharp>y" "c\<sharp>N" "c\<sharp>v" "u\<sharp>M" "u\<sharp>v" by fact+ have ih1: "M{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* M[x\<turnstile>n>y]" by fact have ih2: "N{x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* N[x\<turnstile>n>y]" by fact show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" proof(cases "v=x") case True assume eq: "v=x" obtain v'::"name" where new: "v'\<sharp>(Ax y a,M{x:=<a>.Ax y a},N{x:=<a>.Ax y a},y,a,u)" by (rule exists_fresh(1)[OF fs_name1]) have "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} = fresh_fun (\<lambda>v'. Cut <a>.Ax y a (v').ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v')" using eq fs by simp also have "\<dots> = Cut <a>.Ax y a (v').ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v'" using new by (simp add: fresh_fun_simp_ImpL) also have "\<dots> \<longrightarrow>\<^isub>a* (ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v')[v'\<turnstile>n>y]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxL_intro) apply(rule fin.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) y" using fs new by (auto simp add: fresh_prod subst_fresh fresh_atm trm.inject alpha rename_fresh) also have "\<dots> \<longrightarrow>\<^isub>a* ImpL <c>.(M[x\<turnstile>n>y]) (u).(N[x\<turnstile>n>y]) y" using ih1 ih2 by (auto intro: a_star_congs) also have "\<dots> = (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" using eq fs by simp finally show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" using fs by simp next case False assume neq: "v\<noteq>x" have "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} = ImpL <c>.(M{x:=<a>.Ax y a}) (u).(N{x:=<a>.Ax y a}) v" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* ImpL <c>.(M[x\<turnstile>n>y]) (u).(N[x\<turnstile>n>y]) v" using ih1 ih2 by (auto intro: a_star_congs) finally show "(ImpL <c>.M (u).N v){x:=<a>.Ax y a} \<longrightarrow>\<^isub>a* (ImpL <c>.M (u).N v)[x\<turnstile>n>y]" using fs neq by simp qed qed lemma subst_with_ax2: shows "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" proof(nominal_induct M avoiding: b a x rule: trm.strong_induct) case (Ax z c b a x) show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Ax z c)[b\<turnstile>c>a]" proof (cases "c=b") case True assume eq: "c=b" have "(Ax z c){b:=(x).Ax x a} = Cut <b>.Ax z c (x).Ax x a" using eq by simp also have "\<dots> \<longrightarrow>\<^isub>a* (Ax z c)[b\<turnstile>c>a]" using eq by blast finally show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Ax z c)[b\<turnstile>c>a]" by simp next case False assume neq: "c\<noteq>b" then show "(Ax z c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Ax z c)[b\<turnstile>c>a]" by simp qed next case (Cut c M z N b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>N" "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "z\<sharp>M" by fact+ have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* N[b\<turnstile>c>a]" by fact show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]" proof (cases "N = Ax z b") case True assume eq: "N = Ax z b" have "(Cut <c>.M (z).N){b:=(x).Ax x a} = Cut <c>.(M{b:=(x).Ax x a}) (x).Ax x a" using eq fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* Cut <c>.(M[b\<turnstile>c>a]) (x).Ax x a" using ih1 a_star_congs by blast also have "\<dots> = Cut <c>.(M[b\<turnstile>c>a]) (z).(N[b\<turnstile>c>a])" using eq fs by (simp add: trm.inject alpha calc_atm fresh_atm) finally show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]" using fs by simp next case False assume neq: "N \<noteq> Ax z b" have "(Cut <c>.M (z).N){b:=(x).Ax x a} = Cut <c>.(M{b:=(x).Ax x a}) (z).(N{b:=(x).Ax x a})" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* Cut <c>.(M[b\<turnstile>c>a]) (z).(N[b\<turnstile>c>a])" using ih1 ih2 a_star_congs by blast finally show "(Cut <c>.M (z).N){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (Cut <c>.M (z).N)[b\<turnstile>c>a]" using fs by simp qed next case (NotR z M c b a x) have fs: "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "z\<sharp>c" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (NotR (z).M c)[b\<turnstile>c>a]" proof (cases "c=b") case True assume eq: "c=b" obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a})" by (rule exists_fresh(2)[OF fs_coname1]) have "(NotR (z).M c){b:=(x).Ax x a} = fresh_fun (\<lambda>a'. Cut <a'>.NotR (z).M{b:=(x).Ax x a} a' (x).Ax x a)" using eq fs by simp also have "\<dots> = Cut <a'>.NotR (z).M{b:=(x).Ax x a} a' (x).Ax x a" using new by (simp add: fresh_fun_simp_NotR fresh_prod) also have "\<dots> \<longrightarrow>\<^isub>a* (NotR (z).(M{b:=(x).Ax x a}) a')[a'\<turnstile>c>a]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxR_intro) apply(rule fic.intros) apply(simp) done also have "\<dots> = NotR (z).(M{b:=(x).Ax x a}) a" using new by (simp add: crename_fresh) also have "\<dots> \<longrightarrow>\<^isub>a* NotR (z).(M[b\<turnstile>c>a]) a" using ih by (auto intro: a_star_congs) also have "\<dots> = (NotR (z).M c)[b\<turnstile>c>a]" using eq by simp finally show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (NotR (z).M c)[b\<turnstile>c>a]" by simp next case False assume neq: "c\<noteq>b" have "(NotR (z).M c){b:=(x).Ax x a} = NotR (z).(M{b:=(x).Ax x a}) c" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* NotR (z).(M[b\<turnstile>c>a]) c" using ih by (auto intro: a_star_congs) finally show "(NotR (z).M c){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (NotR (z).M c)[b\<turnstile>c>a]" using neq by simp qed next case (NotL c M z b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>z" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have "(NotL <c>.M z){b:=(x).Ax x a} = NotL <c>.(M{b:=(x).Ax x a}) z" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* NotL <c>.(M[b\<turnstile>c>a]) z" using ih by (auto intro: a_star_congs) finally show "(NotL <c>.M z){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (NotL <c>.M z)[b\<turnstile>c>a]" using fs by simp next case (AndR c M d N e b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "d\<sharp>b" "d\<sharp>a" "d\<sharp>x" "d\<noteq>c" "c\<sharp>N" "c\<sharp>e" "d\<sharp>M" "d\<sharp>e" by fact+ have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* N[b\<turnstile>c>a]" by fact show "(AndR <c>.M <d>.N e){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]" proof(cases "e=b") case True assume eq: "e=b" obtain e'::"coname" where new: "e'\<sharp>(Ax x a,M{b:=(x).Ax x a},N{b:=(x).Ax x a},c,d)" by (rule exists_fresh(2)[OF fs_coname1]) have "(AndR <c>.M <d>.N e){b:=(x).Ax x a} = fresh_fun (\<lambda>e'. Cut <e'>.AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e' (x).Ax x a)" using eq fs by simp also have "\<dots> = Cut <e'>.AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e' (x).Ax x a" using new by (simp add: fresh_fun_simp_AndR fresh_prod) also have "\<dots> \<longrightarrow>\<^isub>a* (AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e')[e'\<turnstile>c>a]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxR_intro) apply(rule fic.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) a" using fs new by (auto simp add: fresh_prod fresh_atm subst_fresh crename_fresh) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]" by simp next case False assume neq: "e\<noteq>b" have "(AndR <c>.M <d>.N e){b:=(x).Ax x a} = AndR <c>.(M{b:=(x).Ax x a}) <d>.(N{b:=(x).Ax x a}) e" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndR <c>.(M[b\<turnstile>c>a]) <d>.(N[b\<turnstile>c>a]) e" using ih1 ih2 by (auto intro: a_star_congs) finally show "(AndR <c>.M <d>.N e){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (AndR <c>.M <d>.N e)[b\<turnstile>c>a]" using fs neq by simp qed next case (AndL1 u M v b a x) have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "u\<sharp>v" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have "(AndL1 (u).M v){b:=(x).Ax x a} = AndL1 (u).(M{b:=(x).Ax x a}) v" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndL1 (u).(M[b\<turnstile>c>a]) v" using ih by (auto intro: a_star_congs) finally show "(AndL1 (u).M v){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (AndL1 (u).M v)[b\<turnstile>c>a]" using fs by simp next case (AndL2 u M v b a x) have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "u\<sharp>v" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have "(AndL2 (u).M v){b:=(x).Ax x a} = AndL2 (u).(M{b:=(x).Ax x a}) v" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* AndL2 (u).(M[b\<turnstile>c>a]) v" using ih by (auto intro: a_star_congs) finally show "(AndL2 (u).M v){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (AndL2 (u).M v)[b\<turnstile>c>a]" using fs by simp next case (OrR1 c M d b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>d" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact show "(OrR1 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]" proof(cases "d=b") case True assume eq: "d=b" obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},c,x,a)" by (rule exists_fresh(2)[OF fs_coname1]) have "(OrR1 <c>.M d){b:=(x).Ax x a} = fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by (simp) also have "\<dots> = Cut <a'>.OrR1 <c>.M{b:=(x).Ax x a} a' (x).Ax x a" using new by (simp add: fresh_fun_simp_OrR1) also have "\<dots> \<longrightarrow>\<^isub>a* (OrR1 <c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxR_intro) apply(rule fic.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = OrR1 <c>.M{b:=(x).Ax x a} a" using fs new by (auto simp add: fresh_prod fresh_atm crename_fresh subst_fresh) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]" by simp next case False assume neq: "d\<noteq>b" have "(OrR1 <c>.M d){b:=(x).Ax x a} = OrR1 <c>.(M{b:=(x).Ax x a}) d" using fs neq by (simp) also have "\<dots> \<longrightarrow>\<^isub>a* OrR1 <c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs) finally show "(OrR1 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (OrR1 <c>.M d)[b\<turnstile>c>a]" using fs neq by simp qed next case (OrR2 c M d b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "c\<sharp>d" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact show "(OrR2 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]" proof(cases "d=b") case True assume eq: "d=b" obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},c,x,a)" by (rule exists_fresh(2)[OF fs_coname1]) have "(OrR2 <c>.M d){b:=(x).Ax x a} = fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by (simp) also have "\<dots> = Cut <a'>.OrR2 <c>.M{b:=(x).Ax x a} a' (x).Ax x a" using new by (simp add: fresh_fun_simp_OrR2) also have "\<dots> \<longrightarrow>\<^isub>a* (OrR2 <c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxR_intro) apply(rule fic.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = OrR2 <c>.M{b:=(x).Ax x a} a" using fs new by (auto simp add: fresh_prod fresh_atm crename_fresh subst_fresh) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]" by simp next case False assume neq: "d\<noteq>b" have "(OrR2 <c>.M d){b:=(x).Ax x a} = OrR2 <c>.(M{b:=(x).Ax x a}) d" using fs neq by (simp) also have "\<dots> \<longrightarrow>\<^isub>a* OrR2 <c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs) finally show "(OrR2 <c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (OrR2 <c>.M d)[b\<turnstile>c>a]" using fs neq by simp qed next case (OrL u M v N z b a x) have fs: "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "v\<sharp>b" "v\<sharp>a" "v\<sharp>x" "v\<noteq>u" "u\<sharp>N" "u\<sharp>z" "v\<sharp>M" "v\<sharp>z" by fact+ have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* N[b\<turnstile>c>a]" by fact have "(OrL (u).M (v).N z){b:=(x).Ax x a} = OrL (u).(M{b:=(x).Ax x a}) (v).(N{b:=(x).Ax x a}) z" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* OrL (u).(M[b\<turnstile>c>a]) (v).(N[b\<turnstile>c>a]) z" using ih1 ih2 by (auto intro: a_star_congs) finally show "(OrL (u).M (v).N z){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (OrL (u).M (v).N z)[b\<turnstile>c>a]" using fs by simp next case (ImpR z c M d b a x) have fs: "z\<sharp>b" "z\<sharp>a" "z\<sharp>x" "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "z\<sharp>d" "c\<sharp>d" by fact+ have ih: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact show "(ImpR (z).<c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]" proof(cases "b=d") case True assume eq: "b=d" obtain a'::"coname" where new: "a'\<sharp>(Ax x a,M{b:=(x).Ax x a},x,a,c)" by (rule exists_fresh(2)[OF fs_coname1]) have "(ImpR (z).<c>.M d){b:=(x).Ax x a} = fresh_fun (\<lambda>a'. Cut <a'>.ImpR z.<c>.M{b:=(x).Ax x a} a' (x).Ax x a)" using fs eq by simp also have "\<dots> = Cut <a'>.ImpR z.<c>.M{b:=(x).Ax x a} a' (x).Ax x a" using new by (simp add: fresh_fun_simp_ImpR) also have "\<dots> \<longrightarrow>\<^isub>a* (ImpR z.<c>.M{b:=(x).Ax x a} a')[a'\<turnstile>c>a]" using new apply(rule_tac a_starI) apply(rule a_redu.intros) apply(rule better_LAxR_intro) apply(rule fic.intros) apply(simp_all add: abs_fresh) done also have "\<dots> = ImpR z.<c>.M{b:=(x).Ax x a} a" using fs new by (auto simp add: fresh_prod crename_fresh subst_fresh fresh_atm) also have "\<dots> \<longrightarrow>\<^isub>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>\<^isub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]" using eq by simp next case False assume neq: "b\<noteq>d" have "(ImpR (z).<c>.M d){b:=(x).Ax x a} = ImpR (z).<c>.(M{b:=(x).Ax x a}) d" using fs neq by simp also have "\<dots> \<longrightarrow>\<^isub>a* ImpR (z).<c>.(M[b\<turnstile>c>a]) d" using ih by (auto intro: a_star_congs) finally show "(ImpR (z).<c>.M d){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (ImpR (z).<c>.M d)[b\<turnstile>c>a]" using neq fs by simp qed next case (ImpL c M u N v b a x) have fs: "c\<sharp>b" "c\<sharp>a" "c\<sharp>x" "u\<sharp>b" "u\<sharp>a" "u\<sharp>x" "c\<sharp>N" "c\<sharp>v" "u\<sharp>M" "u\<sharp>v" by fact+ have ih1: "M{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* M[b\<turnstile>c>a]" by fact have ih2: "N{b:=(x).Ax x a} \<longrightarrow>\<^isub>a* N[b\<turnstile>c>a]" by fact have "(ImpL <c>.M (u).N v){b:=(x).Ax x a} = ImpL <c>.(M{b:=(x).Ax x a}) (u).(N{b:=(x).Ax x a}) v" using fs by simp also have "\<dots> \<longrightarrow>\<^isub>a* ImpL <c>.(M[b\<turnstile>c>a]) (u).(N[b\<turnstile>c>a]) v" using ih1 ih2 by (auto intro: a_star_congs) finally show "(ImpL <c>.M (u).N v){b:=(x).Ax x a} \<longrightarrow>\<^isub>a* (ImpL <c>.M (u).N v)[b\<turnstile>c>a]" using fs by simp qed text {* substitution lemmas *} 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(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(y).Q},Q)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname:=(y).Q},Q)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm1{coname3:=(y).Q},Q,trm2{coname3:=(y).Q},coname1,coname2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>x'::coname. x'\<sharp>(trm{coname2:=(y).Q},Q,coname1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR abs_fresh abs_supp fin_supp fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done lemma not_Ax2: shows "\<not>(x\<sharp>M) \<Longrightarrow> M{x:=<b>.Q} \<noteq> Ax y a" apply(nominal_induct M avoiding: b y Q x a rule: trm.strong_induct) apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm{x:=<b>.Q},Q,name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{x:=<b>.Q},Q,trm2{x:=<b>.Q},name1,name2)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(trm1{name2:=<b>.Q},Q,trm2{name2:=<b>.Q},name1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done lemma interesting_subst1: assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<c>.Ax y c}{y:=<c>.P}" using a proof(nominal_induct N avoiding: x y c P rule: trm.strong_induct) case Ax then show ?case by (auto simp add: abs_fresh fresh_atm forget trm.inject) next case (Cut d M u M' x' y' c P) from prems show ?case apply(simp) apply(auto) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(rule impI) apply(simp add: trm.inject alpha forget) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto) apply(case_tac "y'\<sharp>M") apply(simp add: forget) apply(simp add: not_Ax2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto) apply(case_tac "x'\<sharp>M") apply(simp add: forget) apply(simp add: not_Ax2) done next case NotR then show ?case by (auto simp add: abs_fresh fresh_atm forget) next case (NotL d M u) then show ?case apply (auto simp add: abs_fresh fresh_atm forget) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},y,x)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},Ax y c,y,x)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_NotL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(simp add: trm.inject alpha forget subst_fresh) apply(rule trans) apply(rule substn.simps) apply(simp add: abs_fresh fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndR d1 M d2 M' d3) then show ?case by (auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh) next case (AndL1 u M d) then show ?case apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_AndL1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(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(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P},u,y,x)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_AndL2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(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) next case OrR2 then show ?case by (auto simp add: 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(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{y:=<c>.P},M{x:=<c>.Ax y c}{y:=<c>.P}, M'{y:=<c>.P},M'{x:=<c>.Ax y c}{y:=<c>.P},x1,x2,x3,y,x)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(force) apply(simp) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x:=<c>.Ax y c},M{x:=<c>.Ax y c}{y:=<c>.P}, M'{x:=<c>.Ax y c},M'{x:=<c>.Ax y c}{y:=<c>.P},x1,x2,x3,y,x)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_OrL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(force) apply(simp) apply(auto simp add: fresh_atm) apply(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) 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(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x2:=<c>.P},M{x:=<c>.Ax x2 c}{x2:=<c>.P}, M'{x2:=<c>.P},M'{x:=<c>.Ax x2 c}{x2:=<c>.P},x1,y,x)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(force) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,Ax y c,M{x2:=<c>.Ax y c},M{x2:=<c>.Ax y c}{y:=<c>.P}, M'{x2:=<c>.Ax y c},M'{x2:=<c>.Ax y c}{y:=<c>.P},x1,x2,x3,y,x)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_ImpL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh) apply(simp) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp) apply(auto simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done qed lemma interesting_subst1': assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<a>.Ax y a}{y:=<c>.P}" proof - show ?thesis proof (cases "c=a") case True then show ?thesis using a by (simp add: interesting_subst1) next case False then show ?thesis using a apply - apply(subgoal_tac "N{x:=<a>.Ax y a} = N{x:=<c>.([(c,a)]\<bullet>Ax y a)}") apply(simp add: interesting_subst1 calc_atm) apply(rule subst_rename) apply(simp add: fresh_prod fresh_atm) done qed qed lemma interesting_subst2: assumes a: "a\<noteq>b" "a\<sharp>P" "b\<sharp>P" shows "N{a:=(y).P}{b:=(y).P} = N{b:=(y).Ax y a}{a:=(y).P}" using a proof(nominal_induct N avoiding: a b y P rule: trm.strong_induct) case Ax then show ?case by (auto simp add: abs_fresh fresh_atm forget trm.inject) next case (Cut d M u M' x' y' c P) from prems show ?case apply(simp) apply(auto simp add: trm.inject) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp add: forget) apply(auto) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(auto)[1] apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(rule impI) apply(simp add: fresh_atm trm.inject alpha forget) apply(case_tac "x'\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(auto) apply(case_tac "y'\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax1) done next case NotL then show ?case by (auto simp add: abs_fresh fresh_atm forget) next case (NotR u M d) then show ?case apply (auto simp add: abs_fresh fresh_atm forget) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{d:=(y).P},M{b:=(y).Ax y d}{d:=(y).P},u,y)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,M{d:=(y).Ax y a},M{d:=(y).Ax y a}{a:=(y).P},Ax y a,y,d)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_NotR) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget subst_fresh) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (AndR d1 M d2 M' d3) then show ?case apply(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,M{d3:=(y).P},M{b:=(y).Ax y d3}{d3:=(y).P}, M'{d3:=(y).P},M'{b:=(y).Ax y d3}{d3:=(y).P},d1,d2,d3,b,y)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh fresh_atm) apply(simp add: abs_fresh fresh_atm) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(force) apply(simp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,Ax y a,M{d3:=(y).Ax y a},M{d3:=(y).Ax y a}{a:=(y).P}, M'{d3:=(y).Ax y a},M'{d3:=(y).Ax y a}{a:=(y).P},d1,d2,d3,y,b)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_AndR) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(force) apply(simp) apply(auto simp add: fresh_atm) apply(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) next case (AndL2 u M d) then show ?case by (auto simp add: 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(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{e:=(y).P},M{b:=(y).Ax y e}{e:=(y).P},d,e)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,Ax y a,M{e:=(y).Ax y a},M{e:=(y).Ax y a}{a:=(y).P},d,e)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_OrR1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget subst_fresh) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (OrR2 d M e) then show ?case apply (auto simp add: abs_fresh fresh_atm forget) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,M{e:=(y).P},M{b:=(y).Ax y e}{e:=(y).P},d,e)") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,P,Ax y a,M{e:=(y).Ax y a},M{e:=(y).Ax y a}{a:=(y).P},d,e)") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_OrR2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget subst_fresh) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (OrL x1 M x2 M' x3) then show ?case by(auto simp add: abs_fresh fresh_atm forget trm.inject subst_fresh) next case ImpL then show ?case by (auto simp add: 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(subgoal_tac "\<exists>a'::coname. a'\<sharp>(b,e,d,P,M{d:=(y).P},M{b:=(y).Ax y d}{d:=(y).P})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(auto simp add: fresh_atm) apply(simp add: trm.inject alpha forget) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(e,d,P,Ax y a,M{d:=(y).Ax y a},M{d:=(y).Ax y a}{a:=(y).P})") apply(erule exE, simp only: fresh_prod) apply(erule conjE)+ apply(simp only: fresh_fun_simp_ImpR) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp) apply(auto simp add: fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done qed lemma interesting_subst2': assumes a: "a\<noteq>b" "a\<sharp>P" "b\<sharp>P" shows "N{a:=(y).P}{b:=(y).P} = N{b:=(z).Ax z a}{a:=(y).P}" proof - show ?thesis proof (cases "z=y") case True then show ?thesis using a by (simp add: interesting_subst2) next case False then show ?thesis using a apply - apply(subgoal_tac "N{b:=(z).Ax z a} = N{b:=(y).([(y,z)]\<bullet>Ax z a)}") apply(simp add: interesting_subst2 calc_atm) apply(rule subst_rename) apply(simp add: fresh_prod fresh_atm) done qed qed lemma subst_subst1: assumes a: "a\<sharp>(Q,b)" "x\<sharp>(y,P,Q)" "b\<sharp>Q" "y\<sharp>P" shows "M{x:=<a>.P}{b:=(y).Q} = M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using a proof(nominal_induct M avoiding: x a P b y Q rule: trm.strong_induct) case (Ax z c) have fs: "a\<sharp>(Q,b)" "x\<sharp>(y,P,Q)" "b\<sharp>Q" "y\<sharp>P" by fact+ { assume asm: "z=x \<and> c=b" have "(Ax x b){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (x).Ax x b){b:=(y).Q}" using fs by simp also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).Q" using fs by (simp_all add: fresh_prod fresh_atm) also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" using fs by (simp add: forget) also have "\<dots> = (Cut <b>.Ax x b (y).Q){x:=<a>.(P{b:=(y).Q})}" using fs asm by (auto simp add: fresh_prod fresh_atm subst_fresh) also have "\<dots> = (Ax x b){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using fs by simp finally have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm by simp } moreover { assume asm: "z\<noteq>x \<and> c=b" have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}" using asm by simp also have "\<dots> = Cut <b>.Ax z c (y).Q" using fs asm by simp also have "\<dots> = Cut <b>.(Ax z c{x:=<a>.(P{b:=(y).Q})}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" using fs asm by (simp add: forget) also have "\<dots> = (Cut <b>.Ax z c (y).Q){x:=<a>.(P{b:=(y).Q})}" using asm fs by (auto simp add: trm.inject subst_fresh fresh_prod fresh_atm abs_fresh) 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> = (Ax z c){x:=<a>.(P{b:=(y).Q})}" using fs asm by simp also have "\<dots> = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm by auto finally have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" by simp } moreover { assume asm: "z\<noteq>x \<and> c\<noteq>b" have "(Ax z c){x:=<a>.P}{b:=(y).Q} = (Ax z c){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using asm by auto } ultimately show ?case by blast next case (Cut c M z N) { assume asm: "M = Ax x c \<and> N = Ax z b" have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (z).(N{x:=<a>.P})){b:=(y).Q}" using asm prems by simp also have "\<dots> = (Cut <a>.P (z).N){b:=(y).Q}" using asm prems by (simp add: fresh_atm) also have "\<dots> = (Cut <a>.(P{b:=(y).Q}) (y).Q)" using asm prems by (auto simp add: 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 prems asm by (simp add: fresh_atm) also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).(Q{x:=<a>.(P{b:=(y).Q})})" using asm prems by (auto simp add: fresh_prod fresh_atm subst_fresh) also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (y).Q" using asm prems by (simp add: forget) finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = Cut <a>.(P{b:=(y).Q}) (y).Q" by simp have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using eq1 eq2 by simp } moreover { assume asm: "M \<noteq> Ax x c \<and> N = Ax z b" have neq: "M{b:=(y).Q} \<noteq> Ax x c" proof (cases "b\<sharp>M") case True then show ?thesis using asm by (simp add: forget) next case False then show ?thesis by (simp add: not_Ax1) qed have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.(M{x:=<a>.P}) (z).(N{x:=<a>.P})){b:=(y).Q}" using asm prems by simp also have "\<dots> = (Cut <c>.(M{x:=<a>.P}) (z).N){b:=(y).Q}" using asm prems by (simp add: fresh_atm) also have "\<dots> = Cut <c>.(M{x:=<a>.P}{b:=(y).Q}) (y).Q" using asm prems by (simp add: abs_fresh) also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}}) (y).Q" using asm prems by simp finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = Cut <c>.(M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}}) (y).Q" by simp have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = (Cut <c>.(M{b:=(y).Q}) (y).Q){x:=<a>.(P{b:=(y).Q})}" using asm prems 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 asm prems neq by (auto simp add: 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 asm prems by (simp add: forget) finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (y).Q" by simp have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using eq1 eq2 by simp } moreover { assume asm: "M = Ax x c \<and> N \<noteq> Ax z b" have neq: "N{x:=<a>.P} \<noteq> Ax z b" proof (cases "x\<sharp>N") case True then show ?thesis using asm by (simp add: forget) next case False then show ?thesis by (simp add: not_Ax2) qed have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <a>.P (z).(N{x:=<a>.P})){b:=(y).Q}" using asm prems by simp also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{x:=<a>.P}{b:=(y).Q})" using asm prems neq by (simp add: abs_fresh) also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" using asm prems by simp finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = (Cut <c>.(M{b:=(y).Q}) (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}" using asm prems by auto also have "\<dots> = (Cut <c>.M (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}" using asm prems by (auto simp add: fresh_atm) also have "\<dots> = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" using asm prems by (simp add: fresh_prod fresh_atm subst_fresh) finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = Cut <a>.(P{b:=(y).Q}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using eq1 eq2 by simp } moreover { assume asm: "M \<noteq> Ax x c \<and> N \<noteq> Ax z b" have neq1: "N{x:=<a>.P} \<noteq> Ax z b" proof (cases "x\<sharp>N") case True then show ?thesis using asm by (simp add: forget) next case False then show ?thesis by (simp add: not_Ax2) qed have neq2: "M{b:=(y).Q} \<noteq> Ax x c" proof (cases "b\<sharp>M") case True then show ?thesis using asm by (simp add: forget) next case False then show ?thesis by (simp add: not_Ax1) qed have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.(M{x:=<a>.P}) (z).(N{x:=<a>.P})){b:=(y).Q}" using asm prems by simp also have "\<dots> = Cut <c>.(M{x:=<a>.P}{b:=(y).Q}) (z).(N{x:=<a>.P}{b:=(y).Q})" using asm prems neq1 by (simp add: abs_fresh) also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" using asm prems by simp finally have eq1: "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp have "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = (Cut <c>.(M{b:=(y).Q}) (z).(N{b:=(y).Q})){x:=<a>.(P{b:=(y).Q})}" using asm neq1 prems by simp also have "\<dots> = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" using asm neq2 prems by (simp add: fresh_prod fresh_atm subst_fresh) finally have eq2: "(Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})} = Cut <c>.(M{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}) (z).(N{b:=(y).Q}{x:=<a>.(P{b:=(y).Q})})" by simp have "(Cut <c>.M (z).N){x:=<a>.P}{b:=(y).Q} = (Cut <c>.M (z).N){b:=(y).Q}{x:=<a>.(P{b:=(y).Q})}" using eq1 eq2 by simp } ultimately show ?case by blast next case (NotR z M c) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(M{c:=(y).Q},M{c:=(y).Q}{x:=<a>.P{c:=(y).Q}},Q,a,P,c,y)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR abs_fresh fresh_atm) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: fresh_prod fresh_atm subst_fresh abs_fresh) apply(simp add: fresh_prod fresh_atm subst_fresh abs_fresh) apply(simp add: forget) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (NotL c M z) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},y,Q)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndR c1 M c2 N c3) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c3:=(y).Q},M{c3:=(y).Q}{x:=<a>.P{c3:=(y).Q}},c2,c3,a, P{c3:=(y).Q},N{c3:=(y).Q},N{c3:=(y).Q}{x:=<a>.P{c3:=(y).Q}},c1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR abs_fresh fresh_atm) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp_all add: fresh_atm abs_fresh subst_fresh) apply(simp add: forget) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (AndL1 z1 M z2) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},z1,y,Q,M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndL2 z1 M z2) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},P{b:=(y).Q},z1,y,Q,M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2 abs_fresh fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (OrL z1 M z2 N z3) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,M{x:=<a>.P},M{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},z2,z3,a,y,Q, P{b:=(y).Q},N{x:=<a>.P},N{b:=(y).Q}{x:=<a>.P{b:=(y).Q}},z1)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL abs_fresh fresh_atm) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(simp_all add: fresh_atm subst_fresh) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (OrR1 c1 M c2) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c2:=(y).Q},a,P{c2:=(y).Q},c1, M{c2:=(y).Q}{x:=<a>.P{c2:=(y).Q}})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1 abs_fresh fresh_atm) apply(simp_all add: fresh_atm subst_fresh abs_fresh) apply(simp add: forget) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (OrR2 c1 M c2) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{c2:=(y).Q},a,P{c2:=(y).Q},c1, M{c2:=(y).Q}{x:=<a>.P{c2:=(y).Q}})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2 abs_fresh fresh_atm) apply(simp_all add: fresh_atm subst_fresh abs_fresh) apply(simp add: forget) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (ImpR z c M d) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(Q,M{d:=(y).Q},a,P{d:=(y).Q},c, M{d:=(y).Q}{x:=<a>.P{d:=(y).Q}})") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR abs_fresh fresh_atm) apply(simp_all add: fresh_atm subst_fresh forget abs_fresh) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (ImpL c M z N u) then show ?case apply(auto simp add: fresh_prod fresh_atm subst_fresh) apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,P{b:=(y).Q},M{u:=<a>.P},N{u:=<a>.P},y,Q, M{b:=(y).Q}{u:=<a>.P{b:=(y).Q}},N{b:=(y).Q}{u:=<a>.P{b:=(y).Q}},z)") apply(erule exE) apply(simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL abs_fresh fresh_atm) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(simp_all add: fresh_atm subst_fresh forget) apply(rule exists_fresh'(1)[OF fs_name1]) done qed lemma subst_subst2: assumes a: "a\<sharp>(b,P,N)" "x\<sharp>(y,P,M)" "b\<sharp>(M,N)" "y\<sharp>P" shows "M{a:=(x).N}{y:=<b>.P} = M{y:=<b>.P}{a:=(x).N{y:=<b>.P}}" using a proof(nominal_induct M avoiding: a x N y b P rule: trm.strong_induct) case (Ax z c) then show ?case by (auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) next case (Cut d M' u M'') then show ?case apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh) apply(auto) apply(simp add: fresh_atm) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_prod fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: forget) apply(simp add: fresh_atm) apply(case_tac "a\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_prod fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(auto)[1] apply(case_tac "y\<sharp>M''") apply(simp add: forget) apply(simp add: not_Ax2) apply(simp add: forget) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh) apply(auto)[1] apply(case_tac "y\<sharp>M''") apply(simp add: forget) apply(simp add: not_Ax2) apply(case_tac "a\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh) apply(simp add: subst_fresh abs_fresh) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm) apply(simp add: subst_fresh abs_fresh) apply(auto)[1] apply(case_tac "y\<sharp>M''") apply(simp add: forget) apply(simp add: not_Ax2) done next case (NotR z M' d) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(y,P,N,N{y:=<b>.P},M'{d:=(x).N},M'{y:=<b>.P}{d:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotR) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (NotL d M' z) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(z,y,P,N,N{y:=<b>.P},M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm) apply(simp add: fresh_atm subst_fresh) apply(simp) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndR d M' e M'' f) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>a'::coname. a'\<sharp>(P,b,d,e,N,N{y:=<b>.P},M'{f:=(x).N},M''{f:=(x).N}, M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}},M''{y:=<b>.P}{f:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndR) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(simp) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (AndL1 z M' u) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,b,z,u,x,N,M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndL2 z M' u) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(P,b,z,u,x,N,M'{y:=<b>.P},M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (OrL u M' v M'' w) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,b,u,w,v,N,N{y:=<b>.P},M'{y:=<b>.P},M''{y:=<b>.P}, M'{y:=<b>.P}{a:=(x).N{y:=<b>.P}},M''{y:=<b>.P}{a:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(simp) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (OrR1 e M' f) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P}, M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR1) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (OrR2 e M' f) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P}, M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrR2) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp) apply(rule exists_fresh'(2)[OF fs_coname1]) done next case (ImpR x e M' f) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>c'::coname. c'\<sharp>(P,b,e,f,x,N,N{y:=<b>.P}, M'{f:=(x).N},M'{y:=<b>.P}{f:=(x).N{y:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpR) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(simp add: fresh_atm trm.inject alpha abs_fresh fin_supp abs_supp) apply(rule exists_fresh'(2)[OF fs_coname1]) apply(simp add: fresh_atm trm.inject alpha abs_fresh fin_supp abs_supp) done next case (ImpL e M' v M'' w) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget trm.inject) apply(subgoal_tac "\<exists>z'::name. z'\<sharp>(P,b,e,w,v,N,N{y:=<b>.P},M'{w:=<b>.P},M''{w:=<b>.P}, M'{w:=<b>.P}{a:=(x).N{w:=<b>.P}},M''{w:=<b>.P}{a:=(x).N{w:=<b>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh fresh_atm abs_fresh) apply(simp) apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done qed lemma subst_subst3: assumes a: "a\<sharp>(P,N,c)" "c\<sharp>(M,N)" "x\<sharp>(y,P,M)" "y\<sharp>(P,x)" "M\<noteq>Ax y a" shows "N{x:=<a>.M}{y:=<c>.P} = N{y:=<c>.P}{x:=<a>.(M{y:=<c>.P})}" using a proof(nominal_induct N avoiding: x y a c M P rule: trm.strong_induct) case (Ax z c) then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (Cut d M' u M'') then show ?case apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh) apply(auto) apply(simp add: fresh_atm) apply(simp add: trm.inject) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(subgoal_tac "P \<noteq> Ax x c") apply(simp) apply(simp add: forget) apply(clarify) apply(simp add: fresh_atm) apply(case_tac "x\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(auto) apply(case_tac "y\<sharp>M'") apply(simp add: forget) apply(simp add: not_Ax2) done next case NotR then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (NotL d M' u) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_NotL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_atm subst_fresh fresh_prod) apply(subgoal_tac "P \<noteq> Ax x c") apply(simp) apply(simp add: forget trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(clarify) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case AndR then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (AndL1 u M' v) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(u,y,v,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,u,v,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_atm subst_fresh fresh_prod) apply(subgoal_tac "P \<noteq> Ax x c") apply(simp) apply(simp add: forget trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(clarify) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case (AndL2 u M' v) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(u,y,v,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,u,v,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_AndL2) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_atm subst_fresh fresh_prod) apply(subgoal_tac "P \<noteq> Ax x c") apply(simp) apply(simp add: forget trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(clarify) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case OrR1 then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case OrR2 then show ?case by(auto simp add: 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(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x:=<a>.M},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}}, x1,x2,x3,M''{x:=<a>.M},M''{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{y:=<c>.P},M'{y:=<c>.P}{x:=<a>.M{y:=<c>.P}}, x1,x2,x3,M''{y:=<c>.P},M''{y:=<c>.P}{x:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_OrL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_atm subst_fresh fresh_prod) apply(simp add: fresh_prod fresh_atm) apply(auto) apply(simp add: fresh_atm) apply(simp add: forget trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done next case ImpR then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) 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(subgoal_tac "\<exists>x'::name. x'\<sharp>(y,P,M,M{y:=<c>.P},M'{x2:=<a>.M},M'{y:=<c>.P}{x2:=<a>.M{y:=<c>.P}}, x1,x2,M''{x2:=<a>.M},M''{y:=<c>.P}{x2:=<a>.M{y:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) apply(subgoal_tac "\<exists>x'::name. x'\<sharp>(x,y,P,M,M'{x2:=<c>.P},M'{x2:=<c>.P}{x:=<a>.M{x2:=<c>.P}}, x1,x2,M''{x2:=<c>.P},M''{x2:=<c>.P}{x:=<a>.M{x2:=<c>.P}})") apply(erule exE, simp add: fresh_prod) apply(erule conjE)+ apply(simp add: fresh_fun_simp_ImpL) apply(rule sym) apply(rule trans) apply(rule better_Cut_substn) apply(simp add: abs_fresh subst_fresh) apply(simp add: fresh_atm subst_fresh fresh_prod) apply(simp add: fresh_prod fresh_atm) apply(auto) apply(simp add: fresh_atm) apply(simp add: forget trm.inject alpha) apply(rule trans) apply(rule substn.simps) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm subst_fresh) apply(simp add: fresh_atm) apply(rule exists_fresh'(1)[OF fs_name1]) done qed lemma subst_subst4: assumes a: "x\<sharp>(P,N,y)" "y\<sharp>(M,N)" "a\<sharp>(c,P,M)" "c\<sharp>(P,a)" "M\<noteq>Ax x c" shows "N{a:=(x).M}{c:=(y).P} = N{c:=(y).P}{a:=(x).(M{c:=(y).P})}" using a proof(nominal_induct N avoiding: x y a c M P rule: trm.strong_induct) case (Ax z c) then show ?case by (auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (Cut d M' u M'') then show ?case apply(simp add: fresh_atm fresh_prod trm.inject abs_fresh) apply(auto) apply(simp add: fresh_atm) apply(simp add: trm.inject) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: abs_fresh subst_fresh fresh_atm) apply(simp add: fresh_prod subst_fresh abs_fresh fresh_atm) apply(subgoal_tac "P \<noteq> Ax y a") apply(simp) apply(simp add: forget) apply(clarify) apply(simp add: fresh_atm) apply(case_tac "a\<sharp>M''") apply(simp add: forget) apply(simp add: not_Ax1) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh) apply(simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: fresh_prod subst_fresh fresh_atm) apply(simp add: abs_fresh subst_fresh) apply(auto) apply(case_tac "c\<sharp>M''") apply(simp add: forget) apply(simp add: not_Ax1) done next case NotL then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (NotR u M' d) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(simp add: abs_fresh subst_fresh) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(simp add: fresh_prod fresh_atm) apply(auto simp add: fresh_atm fresh_prod)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: fresh_prod fresh_atm subst_fresh) apply(simp add: abs_fresh subst_fresh) apply(auto simp add: 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) done next case AndL1 then show ?case by(auto simp add: 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) next case (AndR d M e M' f) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(simp add: abs_fresh subst_fresh) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(simp) apply(auto simp add: fresh_atm fresh_prod)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm fresh_prod) apply(simp add: abs_fresh subst_fresh) apply(auto simp add: fresh_atm)[1] apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(simp) apply(auto simp add: fresh_atm fresh_prod)[1] done next case OrL then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (OrR1 d M' e) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(simp add: abs_fresh subst_fresh) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm fresh_prod) apply(simp add: abs_fresh subst_fresh) apply(auto simp add: fresh_atm)[1] apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] done next case (OrR2 d M' e) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(simp add: abs_fresh subst_fresh) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm fresh_prod) apply(simp add: abs_fresh subst_fresh) apply(auto simp add: fresh_atm)[1] apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] done next case ImpL then show ?case by(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) next case (ImpR u d M' e) then show ?case apply(auto simp add: subst_fresh abs_fresh fresh_atm fresh_prod forget) apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(simp add: abs_fresh subst_fresh) apply(rule trans) apply(rule better_Cut_substc) apply(simp) apply(simp add: abs_fresh) apply(simp) apply(simp add: trm.inject alpha) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh abs_fresh abs_supp fin_supp)[1] apply(generate_fresh "coname") apply(fresh_fun_simp) apply(fresh_fun_simp) apply(rule sym) apply(rule trans) apply(rule better_Cut_substc) apply(simp add: subst_fresh fresh_atm fresh_prod) apply(simp add: abs_fresh subst_fresh) apply(auto simp add: fresh_atm)[1] apply(simp add: trm.inject alpha forget) apply(rule trans) apply(rule substc.simps) apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(auto simp add: fresh_prod fresh_atm subst_fresh)[1] apply(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] done qed end