--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Nominal/LamEx2.thy Mon Feb 01 15:57:37 2010 +0100
@@ -0,0 +1,632 @@
+theory LamEx
+imports Nominal "../QuotMain" "../QuotList"
+begin
+
+atom_decl name
+
+datatype rlam =
+ rVar "name"
+| rApp "rlam" "rlam"
+| rLam "name" "rlam"
+
+fun
+ rfv :: "rlam \<Rightarrow> name set"
+where
+ rfv_var: "rfv (rVar a) = {a}"
+| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
+| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
+
+overloading
+ perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked)
+begin
+
+fun
+ perm_rlam
+where
+ "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"
+| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"
+| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"
+
+end
+
+declare perm_rlam.simps[eqvt]
+
+instance rlam::pt_name
+ apply(default)
+ apply(induct_tac [!] x rule: rlam.induct)
+ apply(simp_all add: pt_name2 pt_name3)
+ done
+
+instance rlam::fs_name
+ apply(default)
+ apply(induct_tac [!] x rule: rlam.induct)
+ apply(simp add: supp_def)
+ apply(fold supp_def)
+ apply(simp add: supp_atm)
+ apply(simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ apply(simp add: supp_def)
+ apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric])
+ apply(fold supp_def)
+ apply(simp add: supp_atm)
+ done
+
+declare set_diff_eqvt[eqvt]
+
+lemma rfv_eqvt[eqvt]:
+ fixes pi::"name prm"
+ shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
+apply(induct t)
+apply(simp_all)
+apply(simp add: perm_set_eq)
+apply(simp add: union_eqvt)
+apply(simp add: set_diff_eqvt)
+apply(simp add: perm_set_eq)
+done
+
+inductive
+ alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
+where
+ a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
+| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
+| a3: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b)
+ \<Longrightarrow> rLam a t \<approx> rLam b s"
+
+
+
+
+(* should be automatic with new version of eqvt-machinery *)
+lemma alpha_eqvt:
+ fixes pi::"name prm"
+ shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)"
+apply(induct rule: alpha.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(simp)
+apply(rule a3)
+apply(erule conjE)
+apply(erule exE)
+apply(erule conjE)
+apply(rule_tac x="pi \<bullet> pia" in exI)
+apply(rule conjI)
+apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1])
+apply(perm_simp add: eqvts)
+apply(rule conjI)
+apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1])
+apply(perm_simp add: eqvts)
+apply(rule conjI)
+apply(subst perm_compose[symmetric])
+apply(simp)
+apply(subst perm_compose[symmetric])
+apply(simp)
+done
+
+lemma alpha_refl:
+ shows "t \<approx> t"
+apply(induct t rule: rlam.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(rule a3)
+apply(rule_tac x="[]" in exI)
+apply(simp_all add: fresh_star_def fresh_list_nil)
+done
+
+lemma alpha_sym:
+ shows "t \<approx> s \<Longrightarrow> s \<approx> t"
+apply(induct rule: alpha.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(rule a3)
+apply(erule exE)
+apply(rule_tac x="rev pi" in exI)
+apply(simp)
+apply(simp add: fresh_star_def fresh_list_rev)
+apply(rule conjI)
+apply(erule conjE)+
+apply(rotate_tac 3)
+apply(drule_tac pi="rev pi" in alpha_eqvt)
+apply(perm_simp)
+apply(rule pt_bij2[OF pt_name_inst at_name_inst])
+apply(simp)
+done
+
+lemma alpha_trans:
+ shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
+apply(induct arbitrary: t3 rule: alpha.induct)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(simp add: a1)
+apply(rotate_tac 4)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(simp add: a2)
+apply(rotate_tac 1)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(erule conjE)+
+apply(erule exE)+
+apply(erule conjE)+
+apply(rule a3)
+apply(rule_tac x="pia @ pi" in exI)
+apply(simp add: fresh_star_def fresh_list_append)
+apply(simp add: pt_name2)
+apply(drule_tac x="rev pia \<bullet> sa" in spec)
+apply(drule mp)
+apply(rotate_tac 8)
+apply(drule_tac pi="rev pia" in alpha_eqvt)
+apply(perm_simp)
+apply(rotate_tac 11)
+apply(drule_tac pi="pia" in alpha_eqvt)
+apply(perm_simp)
+done
+
+lemma alpha_equivp:
+ shows "equivp alpha"
+apply(rule equivpI)
+unfolding reflp_def symp_def transp_def
+apply(auto intro: alpha_refl alpha_sym alpha_trans)
+done
+
+lemma alpha_rfv:
+ shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
+apply(induct rule: alpha.induct)
+apply(simp)
+apply(simp)
+apply(simp)
+done
+
+quotient_type lam = rlam / alpha
+ by (rule alpha_equivp)
+
+
+quotient_definition
+ "Var :: name \<Rightarrow> lam"
+as
+ "rVar"
+
+quotient_definition
+ "App :: lam \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "rApp"
+
+quotient_definition
+ "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "rLam"
+
+quotient_definition
+ "fv :: lam \<Rightarrow> name set"
+as
+ "rfv"
+
+(* definition of overloaded permutation function *)
+(* for the lifted type lam *)
+overloading
+ perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked)
+begin
+
+quotient_definition
+ "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"
+
+end
+
+lemma perm_rsp[quot_respect]:
+ "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
+ apply(auto)
+ (* this is propably true if some type conditions are imposed ;o) *)
+ sorry
+
+lemma fresh_rsp:
+ "(op = ===> alpha ===> op =) fresh fresh"
+ apply(auto)
+ (* this is probably only true if some type conditions are imposed *)
+ sorry
+
+lemma rVar_rsp[quot_respect]:
+ "(op = ===> alpha) rVar rVar"
+ by (auto intro: a1)
+
+lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp"
+ by (auto intro: a2)
+
+lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"
+ apply(auto)
+ apply(rule a3)
+ apply(rule_tac x="[]" in exI)
+ unfolding fresh_star_def
+ apply(simp add: fresh_list_nil)
+ apply(simp add: alpha_rfv)
+ done
+
+lemma rfv_rsp[quot_respect]:
+ "(alpha ===> op =) rfv rfv"
+apply(simp add: alpha_rfv)
+done
+
+section {* lifted theorems *}
+
+lemma lam_induct:
+ "\<lbrakk>\<And>name. P (Var name);
+ \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
+ \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk>
+ \<Longrightarrow> P lam"
+ by (lifting rlam.induct)
+
+lemma perm_lam [simp]:
+ fixes pi::"'a prm"
+ shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
+ and "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
+ and "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
+apply(lifting perm_rlam.simps)
+done
+
+instance lam::pt_name
+apply(default)
+apply(induct_tac [!] x rule: lam_induct)
+apply(simp_all add: pt_name2 pt_name3)
+done
+
+lemma fv_lam [simp]:
+ shows "fv (Var a) = {a}"
+ and "fv (App t1 t2) = fv t1 \<union> fv t2"
+ and "fv (Lam a t) = fv t - {a}"
+apply(lifting rfv_var rfv_app rfv_lam)
+done
+
+
+lemma a1:
+ "a = b \<Longrightarrow> Var a = Var b"
+ by (lifting a1)
+
+lemma a2:
+ "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
+ by (lifting a2)
+
+lemma a3:
+ "\<lbrakk>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk>
+ \<Longrightarrow> Lam a t = Lam b s"
+ by (lifting a3)
+
+lemma alpha_cases:
+ "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
+ \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
+ \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s;
+ \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> P\<rbrakk>
+ \<Longrightarrow> P"
+ by (lifting alpha.cases)
+
+lemma alpha_induct:
+ "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);
+ \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
+ \<And>t a s b.
+ \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and>
+ (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
+ \<Longrightarrow> qxb qx qxa"
+ by (lifting alpha.induct)
+
+lemma lam_inject [simp]:
+ shows "(Var a = Var b) = (a = b)"
+ and "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
+apply(lifting rlam.inject(1) rlam.inject(2))
+apply(auto)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(simp add: alpha.a1)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(rule alpha.a2)
+apply(simp_all)
+done
+
+lemma rlam_distinct:
+ shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"
+ and "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"
+ and "\<not>(rVar nam \<approx> rLam nam' rlam')"
+ and "\<not>(rLam nam' rlam' \<approx> rVar nam)"
+ and "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"
+ and "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"
+apply auto
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+done
+
+lemma lam_distinct[simp]:
+ shows "Var nam \<noteq> App lam1' lam2'"
+ and "App lam1' lam2' \<noteq> Var nam"
+ and "Var nam \<noteq> Lam nam' lam'"
+ and "Lam nam' lam' \<noteq> Var nam"
+ and "App lam1 lam2 \<noteq> Lam nam' lam'"
+ and "Lam nam' lam' \<noteq> App lam1 lam2"
+apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6))
+done
+
+lemma var_supp1:
+ shows "(supp (Var a)) = ((supp a)::name set)"
+ by (simp add: supp_def)
+
+lemma var_supp:
+ shows "(supp (Var a)) = {a::name}"
+ using var_supp1 by (simp add: supp_atm)
+
+lemma app_supp:
+ shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"
+apply(simp only: perm_lam supp_def lam_inject)
+apply(simp add: Collect_imp_eq Collect_neg_eq)
+done
+
+lemma lam_supp:
+ shows "supp (Lam x t) = ((supp ([x].t))::name set)"
+apply(simp add: supp_def)
+apply(simp add: abs_perm)
+sorry
+
+
+instance lam::fs_name
+apply(default)
+apply(induct_tac x rule: lam_induct)
+apply(simp add: var_supp)
+apply(simp add: app_supp)
+apply(simp add: lam_supp abs_supp)
+done
+
+lemma fresh_lam:
+ "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)"
+apply(simp add: fresh_def)
+apply(simp add: lam_supp abs_supp)
+apply(auto)
+done
+
+lemma lam_induct_strong:
+ fixes a::"'a::fs_name"
+ assumes a1: "\<And>name b. P b (Var name)"
+ and a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"
+ and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
+ shows "P a lam"
+proof -
+ have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)"
+ proof (induct lam rule: lam_induct)
+ case (1 name pi)
+ show "P a (pi \<bullet> Var name)"
+ apply (simp)
+ apply (rule a1)
+ done
+ next
+ case (2 lam1 lam2 pi)
+ have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact
+ have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact
+ show "P a (pi \<bullet> App lam1 lam2)"
+ apply (simp)
+ apply (rule a2)
+ apply (rule b1)
+ apply (rule b2)
+ done
+ next
+ case (3 name lam pi a)
+ have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact
+ obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
+ apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"])
+ apply(simp_all add: fs_name1)
+ done
+ from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))"
+ apply -
+ apply(rule a3)
+ apply(blast)
+ apply(simp)
+ done
+ have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
+ apply(rule perm_fresh_fresh)
+ using fr
+ apply(simp add: fresh_lam)
+ apply(simp add: fresh_lam)
+ done
+ show "P a (pi \<bullet> Lam name lam)"
+ apply (simp)
+ apply(subst eq[symmetric])
+ using p
+ apply(simp only: perm_lam pt_name2 swap_simps)
+ done
+ qed
+ then have "P a (([]::name prm) \<bullet> lam)" by blast
+ then show "P a lam" by simp
+qed
+
+
+lemma var_fresh:
+ fixes a::"name"
+ shows "(a \<sharp> (Var b)) = (a \<sharp> b)"
+ apply(simp add: fresh_def)
+ apply(simp add: var_supp1)
+ done
+
+(* lemma hom_reg: *)
+
+lemma rlam_rec_eqvt:
+ fixes pi::"name prm"
+ and f1::"name \<Rightarrow> ('a::pt_name)"
+ shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
+apply(induct t)
+apply(simp_all)
+apply(simp add: perm_fun_def)
+apply(perm_simp)
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+back
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(simp)
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+back
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(simp)
+done
+
+
+lemma rlam_rec_respects:
+ assumes f1: "f_var \<in> Respects (op= ===> op=)"
+ and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
+ and f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)"
+ shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)"
+apply(simp add: mem_def)
+apply(simp add: Respects_def)
+apply(rule allI)
+apply(rule allI)
+apply(rule impI)
+apply(erule alpha.induct)
+apply(simp)
+apply(simp)
+using f2
+apply(simp add: mem_def)
+apply(simp add: Respects_def)
+using f3[simplified mem_def Respects_def]
+apply(simp)
+apply(case_tac "a=b")
+apply(clarify)
+apply(simp)
+(* probably true *)
+sorry
+
+function
+ term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
+ (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
+ ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a"
+where
+ "term1_hom var app abs' (rVar x) = (var x)"
+| "term1_hom var app abs' (rApp t u) =
+ app t u (term1_hom var app abs' t) (term1_hom var app abs' u)"
+| "term1_hom var app abs' (rLam x u) =
+ abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))"
+apply(pat_completeness)
+apply(auto)
+done
+
+lemma pi_size:
+ fixes pi::"name prm"
+ and t::"rlam"
+ shows "size (pi \<bullet> t) = size t"
+apply(induct t)
+apply(auto)
+done
+
+termination term1_hom
+ apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)")
+apply(auto simp add: pi_size)
+done
+
+lemma lam_exhaust:
+ "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk>
+ \<Longrightarrow> P"
+apply(lifting rlam.exhaust)
+done
+
+(* THIS IS NOT TRUE, but it lets prove the existence of the hom function *)
+lemma lam_inject':
+ "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))"
+sorry
+
+function
+ hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
+ (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
+ ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a"
+where
+ "hom f_var f_app f_lam (Var x) = f_var x"
+| "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)"
+| "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))"
+defer
+apply(simp_all add: lam_inject') (* inject, distinct *)
+apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+apply(rule refl)
+apply(rule ext)
+apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+apply simp_all
+apply(erule conjE)+
+apply(rule_tac x="b" in cong)
+apply simp_all
+apply auto
+apply(rule_tac y="b" in lam_exhaust)
+apply simp_all
+apply auto
+apply meson
+apply(simp_all add: lam_inject')
+apply metis
+done
+
+termination hom
+ apply -
+(*
+ML_prf {* Size.size_thms @{theory} "LamEx.lam" *}
+*)
+sorry
+
+thm hom.simps
+
+lemma term1_hom_rsp:
+ "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk>
+ \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)"
+apply(simp)
+apply(rule allI)+
+apply(rule impI)
+apply(erule alpha.induct)
+apply(auto)[1]
+apply(auto)[1]
+apply(simp)
+apply(erule conjE)+
+apply(erule exE)+
+apply(erule conjE)+
+apply(clarify)
+sorry
+
+lemma hom: "
+\<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =).
+\<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =).
+\<exists>hom\<in>Respects (alpha ===> op =).
+ ((\<forall>x. hom (rVar x) = f_var x) \<and>
+ (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+apply(rule allI)
+apply(rule ballI)+
+apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI)
+apply(simp_all)
+apply(simp only: in_respects)
+apply(rule term1_hom_rsp)
+apply(assumption)+
+done
+
+lemma hom':
+"\<exists>hom.
+ ((\<forall>x. hom (Var x) = f_var x) \<and>
+ (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+apply (lifting hom)
+done
+
+(* test test
+lemma raw_hom_correct:
+ assumes f1: "f_var \<in> Respects (op= ===> op=)"
+ and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
+ and f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)"
+ shows "\<exists>!hom\<in>Respects (alpha ===> op =).
+ ((\<forall>x. hom (rVar x) = f_var x) \<and>
+ (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+unfolding Bex1_def
+apply(rule ex1I)
+sorry
+*)
+
+
+end
+