--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Atoms.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,46 @@
+
+theory Atoms = Main + Swap + Terms:
+
+consts atms   :: "(string \<times> string) list \<Rightarrow> string set"
+primrec
+"atms [] = {}"
+"atms (x#xs) = ((atms xs) \<union>  {fst(x),snd(x)})"
+
+lemma [simp]: 
+  "atms (xs@ys) = atms xs \<union> atms ys"
+apply(induct_tac xs, auto)
+done
+
+lemma [simp]: 
+  "atms (rev pi) = atms pi"
+apply(induct_tac pi, simp_all)
+done
+
+lemma a_not_in_atms: 
+  "a\<notin>atms pi \<longrightarrow> a = swapas pi a" 
+apply(induct_tac pi, auto)
+done
+
+lemma swapas_pi_ineq_a: 
+  "swapas pi a \<noteq> a \<longrightarrow> a\<in>atms pi"
+apply(case_tac "a\<in>atms pi")
+apply(simp)+
+apply(drule a_not_in_atms[THEN mp])
+apply(simp)
+done
+
+lemma a_ineq_swapas_pi: 
+  "a \<noteq> swapas pi a \<longrightarrow> a\<in>atms pi"
+apply(case_tac "a\<in>atms pi")
+apply(simp_all add: a_not_in_atms)
+done
+
+lemma swapas_pi_in_atms[THEN mp]: 
+  "a\<in>atms pi \<longrightarrow> swapas pi a\<in>atms pi"
+apply(subgoal_tac "\<forall>pi1. (a\<in>atms pi1) \<longrightarrow> swapas pi a\<in>(atms pi1 \<union> atms pi)")
+apply(force)
+apply(induct_tac pi)
+apply(auto)
+done
+
+end
\ No newline at end of file

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Disagreement.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,147 @@
+
+theory Disagreement = Main + Swap + Atoms:
+
+consts 
+  ds :: "(string \<times> string) list \<Rightarrow> (string \<times> string) list \<Rightarrow> string set"
+defs   
+  ds_def: "ds xs ys  \<equiv>  { a . a \<in> (atms xs \<union> atms ys) \<and> (swapas xs a \<noteq> swapas ys a) }"
+
+lemma 
+  ds_elem: "\<lbrakk>swapas pi a\<noteq>a\<rbrakk>\<Longrightarrow>a\<in>ds [] pi"
+apply(simp add: ds_def)
+apply(auto simp add: swapas_pi_ineq_a)
+done
+
+lemma 
+  elem_ds: "\<lbrakk>a\<in>ds [] pi\<rbrakk>\<Longrightarrow>a\<noteq>swapas pi a"
+apply(simp add: ds_def)
+done
+
+lemma 
+  ds_sym: "ds pi1 pi2 = ds pi2 pi1"
+apply(simp only: ds_def)
+apply(auto)
+done
+
+lemma 
+  ds_trans: "c\<in>ds pi1 pi3\<longrightarrow>(c\<in>ds pi1 pi2 \<or> c\<in>ds pi2 pi3)"
+apply(auto)
+apply(simp only: ds_def)
+apply(auto)
+apply(drule a_not_in_atms[THEN mp])+
+apply(simp)
+apply(drule a_not_in_atms[THEN mp])
+apply(simp)
+apply(drule swapas_pi_ineq_a[THEN mp])
+apply(assumption)
+done
+
+lemma ds_cancel_pi_left: 
+  "(c\<in> ds (pi1@pi) (pi2@pi)) \<longrightarrow> (swapas pi c\<in> ds pi1 pi2)"
+apply(simp only: ds_def)
+apply(auto)
+apply(simp_all add: swapas_append)
+apply(rule a_ineq_swapas_pi[THEN mp], clarify, drule a_not_in_atms[THEN mp], simp)+
+done
+
+lemma ds_cancel_pi_right: 
+  "(swapas pi c\<in> ds pi1 pi2) \<longrightarrow> (c\<in> ds (pi1@pi) (pi2@pi))"
+apply(simp only: ds_def)
+apply(auto)
+apply(simp_all add: swapas_append)
+apply(rule a_ineq_swapas_pi[THEN mp],clarify,
+      drule a_not_in_atms[THEN mp],drule a_not_in_atms[THEN mp],simp)+
+done
+
+lemma ds_equality: 
+  "(ds [] pi)-{a,swapas pi a} = (ds [] ((a,swapas pi a)#pi))-{swapas pi a}"
+apply(simp only: ds_def)
+apply(auto)
+done
+
+lemma ds_7: 
+  "\<lbrakk>b\<noteq> swapas pi b;a\<in>ds [] ((b,swapas pi b)#pi)\<rbrakk>\<Longrightarrow>a\<in>ds [] pi"
+apply(simp only: ds_def)
+apply(case_tac "b=a")
+apply(auto)
+apply(rule swapas_pi_in_atms)
+apply(drule a_ineq_swapas_pi[THEN mp])
+apply(assumption)
+apply(drule sym)
+apply(drule swapas_rev_pi_a)
+apply(simp)
+apply(case_tac "swapas pi b = a")
+apply(auto)
+apply(drule sym)
+apply(drule swapas_rev_pi_a)
+apply(simp)
+done
+
+lemma ds_cancel_pi_front: 
+  "ds (pi@pi1) (pi@pi2) = ds pi1 pi2"
+apply(simp only: ds_def)
+apply(auto)
+apply(simp_all add: swapas_append)
+apply(rule swapas_pi_ineq_a[THEN mp], clarify, drule a_not_in_atms[THEN mp], simp)+
+apply(drule swapas_rev_pi_a, simp)+
+done
+
+lemma ds_rev_pi_pi: 
+  "ds ((rev pi1)@pi1) pi2 = ds [] pi2"
+apply(simp only: ds_def)
+apply(auto)
+apply(simp_all add: swapas_append)
+apply(drule a_ineq_swapas_pi[THEN mp], assumption)+
+done
+
+lemma ds_rev: 
+  "ds [] ((rev pi1)@pi2) = ds pi1 pi2"
+apply(subgoal_tac "ds pi1 pi2 = ds ((rev pi1)@pi1) ((rev pi1)@pi2)")
+apply(simp add: ds_rev_pi_pi)
+apply(simp only: ds_cancel_pi_front)
+done
+
+lemma ds_acabbc: 
+  "\<lbrakk>a\<noteq>b;b\<noteq>c;c\<noteq>a\<rbrakk>\<Longrightarrow>ds [] [(a, c), (a, b), (b, c)] = {a, b}"
+apply(simp only: ds_def)
+apply(auto)
+done
+
+lemma ds_baab: 
+  "\<lbrakk>a\<noteq>b\<rbrakk>\<Longrightarrow>ds [] [(b, a), (a, b)] = {}"
+apply(simp only: ds_def)
+apply(auto)
+done
+
+lemma ds_abab: 
+  "\<lbrakk>a\<noteq>b\<rbrakk>\<Longrightarrow>ds [] [(a, b), (a, b)] = {}"
+apply(simp only: ds_def)
+apply(auto)
+done
+
+(* disagreement set as list *)
+
+consts flatten :: "(string \<times> string)list \<Rightarrow> string list"
+primrec 
+"flatten []     = []"
+"flatten (x#xs) = (fst x)#(snd x)#(flatten xs)"
+
+consts ds_list :: "(string \<times> string)list \<Rightarrow> (string \<times> string)list \<Rightarrow> string list"
+defs ds_list_def: "ds_list pi1 pi2 \<equiv> remdups ([x:(flatten (pi1@pi2)). x\<in>ds pi1 pi2])"
+
+lemma set_flatten_eq_atms: 
+  "set (flatten pi) = atms pi"
+apply(induct_tac pi)
+apply(auto)
+done
+
+lemma ds_list_equ_ds: 
+  "set (ds_list pi1 pi2) = ds pi1 pi2"
+apply(auto)
+apply(simp add: ds_list_def)
+apply(simp add: ds_list_def)
+apply(simp add: set_flatten_eq_atms)
+apply(simp add: ds_def)
+done
+
+end
\ No newline at end of file

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Equ.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,156 @@
+
+
+theory Equ = Main + Terms + Fresh + PreEqu:
+
+lemma equ_refl: 
+  "nabla\<turnstile>t\<approx>t"
+apply(induct_tac t)
+apply(auto simp add: ds_def)
+done
+
+
+lemma 
+  equ_sym:    "nabla\<turnstile>t1\<approx>t2 \<Longrightarrow> nabla\<turnstile>t2\<approx>t1" and
+  equ_trans:  "\<lbrakk>nabla\<turnstile>t1\<approx>t2;nabla\<turnstile>t2\<approx>t3\<rbrakk> \<Longrightarrow> nabla\<turnstile>t1\<approx>t3" and
+  equ_add_pi: "nabla\<turnstile>t1\<approx>t2 \<Longrightarrow> nabla\<turnstile>swap pi t1\<approx>swap pi t2"
+apply(insert big)
+apply(best)+
+done
+
+
+lemma equ_dec_pi: 
+  "nabla\<turnstile>swap pi t1\<approx>swap pi t2\<Longrightarrow>nabla\<turnstile>t1\<approx>t2"
+apply(drule_tac pi="(rev pi)" in equ_add_pi)
+apply(subgoal_tac "nabla\<turnstile>swap (rev pi) (swap pi t2)\<approx>t2")
+apply(drule_tac  "t1.0"="swap (rev pi) (swap pi t1)" and
+                 "t2.0"="swap (rev pi) (swap pi t2)" and 
+                 "t3.0"="t2" in equ_trans)
+apply(assumption)
+apply(subgoal_tac "nabla\<turnstile>t1\<approx>swap (rev pi) (swap pi t1)")
+apply(drule_tac  "t1.0"="t1" and
+                 "t2.0"="swap (rev pi) (swap pi t1)" and 
+                 "t3.0"="t2" in equ_trans)
+apply(assumption)+
+apply(rule equ_sym)
+apply(rule rev_pi_pi_equ)
+apply(rule rev_pi_pi_equ)
+done
+
+lemma equ_involutive_left: 
+  "nabla\<turnstile>swap (rev pi) (swap pi t1)\<approx>t2 = nabla\<turnstile>t1\<approx>t2"
+apply(auto)
+apply(subgoal_tac "nabla \<turnstile> t1\<approx> swap (rev pi) (swap pi t1)")
+apply(drule_tac  "t2.0"="swap (rev pi) (swap pi t1)" and
+                 "t1.0"="t1" and "t3.0"="t2" in equ_trans)
+apply(assumption)+
+apply(rule equ_sym)
+apply(rule rev_pi_pi_equ)
+apply(subgoal_tac "nabla \<turnstile> swap (rev pi) (swap pi t1)\<approx>t1")
+apply(drule_tac  "t1.0"="swap (rev pi) (swap pi t1)" and
+                 "t2.0"="t1" and "t3.0"="t2" in equ_trans)
+apply(assumption)+
+apply(rule rev_pi_pi_equ)
+done
+
+lemma equ_pi_to_left: 
+  "nabla\<turnstile>swap (rev pi) t1\<approx>t2 = nabla\<turnstile>t1\<approx> swap pi t2"
+apply(auto)
+apply(rule_tac pi="rev pi" in equ_dec_pi)
+apply(rule equ_sym)
+apply(simp only: equ_involutive_left)
+apply(rule equ_sym)
+apply(assumption)
+apply(drule_tac pi="rev pi" in equ_add_pi)
+apply(drule equ_sym)
+apply(simp only: equ_involutive_left)
+apply(drule equ_sym)
+apply(assumption)
+done
+
+lemma equ_pi_to_right: 
+  "nabla\<turnstile>t1\<approx>swap (rev pi) t2 = nabla\<turnstile>swap pi t1\<approx>t2"
+apply(auto)
+apply(rule_tac pi="rev pi" in equ_dec_pi)
+apply(simp only: equ_involutive_left)
+apply(drule_tac pi="rev pi" in equ_add_pi)
+apply(simp only: equ_involutive_left)
+done
+
+lemma equ_involutive_right: 
+  "nabla\<turnstile>t1\<approx>swap (rev pi) (swap pi t2) = nabla\<turnstile>t1\<approx>t2"
+apply(simp only: swap_append[THEN sym])
+apply(simp only: equ_pi_to_left[THEN sym])
+apply(simp)
+apply(simp only: swap_append)
+apply(simp only: equ_involutive_left)
+done
+
+lemma equ_pi1_pi2_add: 
+  "(\<forall>a\<in> ds pi1 pi2. nabla\<turnstile>a\<sharp>t)\<longrightarrow>(nabla\<turnstile>swap pi1 t \<approx> swap pi2 t)"
+apply(rule impI)
+apply(simp only: equ_pi_to_right[THEN sym])
+apply(simp only: swap_append[THEN sym])
+apply(rule equ_pi_right[THEN spec, THEN mp])
+apply(auto)
+apply(simp only: ds_rev)
+done
+
+lemma pi_right_equ: "(nabla\<turnstile>t \<approx> swap pi t)\<longrightarrow>(\<forall>a\<in> ds [] pi. nabla\<turnstile>a\<sharp>t)"
+apply(insert  pi_right_equ_help)
+apply(best)
+done
+
+lemma equ_pi1_pi2_dec:  
+  "(nabla\<turnstile>swap pi1 t \<approx> swap pi2 t)\<longrightarrow>(\<forall>a\<in> ds pi1 pi2. nabla\<turnstile>a\<sharp>t)"
+apply(rule impI)
+apply(simp only: equ_pi_to_right[THEN sym])
+apply(simp only: swap_append[THEN sym])
+apply(drule pi_right_equ[THEN mp])
+apply(simp only: ds_rev)
+done
+
+lemma equ_weak: 
+  "nabla1\<turnstile>t1\<approx>t2\<longrightarrow>(nabla1\<union>nabla2)\<turnstile>t1\<approx>t2"
+apply(rule impI)
+apply(erule equ.induct)
+apply(auto simp add: fresh_weak)
+done
+
+(* no term can be equal to one of its proper subterm *)
+lemma psub_trm_not_equ: 
+  "\<forall> t2\<in>psub_trms t1. (\<not>(\<exists> pi. (nabla\<turnstile>t1\<approx>swap pi t2)))"
+apply(induct t1)
+apply(auto)
+apply(simp add: equ_pi_to_left[THEN sym])
+apply(ind_cases "nabla \<turnstile> Abst (swapas (rev pi) list) (swap (rev pi) trm) \<approx> t2")
+apply(simp_all)
+apply(drule abst_psub_trms)
+apply(drule_tac x="t2a" in bspec)
+apply(simp)
+apply(simp add: equ_pi_to_left[THEN sym] swap_append[THEN sym])
+apply(drule_tac x="rev (((swapas (rev pi) list, b) # rev pi))" in spec)
+apply(simp)
+apply(drule abst_psub_trms)
+apply(drule_tac x="t2a" in bspec)
+apply(simp)
+apply(simp)
+apply(simp add: equ_pi_to_left[THEN sym])
+apply(ind_cases "nabla \<turnstile> Paar (swap (rev pi) trm1) (swap (rev pi) trm2) \<approx> t2")
+apply(simp)
+apply(drule paar_psub_trms)
+apply(best)
+apply(simp add: equ_pi_to_left[THEN sym])
+apply(ind_cases "nabla \<turnstile> Paar (swap (rev pi) trm1) (swap (rev pi) trm2) \<approx> t2")
+apply(simp)
+apply(drule paar_psub_trms)
+apply(best)
+apply(simp add: equ_pi_to_left[THEN sym])
+apply(ind_cases "nabla \<turnstile> Func list (swap (rev pi) trm) \<approx> t2")
+apply(simp_all)
+apply(drule func_psub_trms)
+apply(best)
+done
+
+
+
+end 

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Fresh.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,90 @@
+
+
+theory Fresh = Main + Swap + Terms + Disagreement:
+
+types fresh_envs = "(string \<times> string)set"
+
+consts 
+  fresh :: "(fresh_envs \<times> string \<times> trm) set"
+syntax 
+  "_fresh_judge" :: "fresh_envs \<Rightarrow> string \<Rightarrow> trm \<Rightarrow> bool" (" _ \<turnstile> _ \<sharp> _" [80,80,80] 80)
+translations 
+  "nabla \<turnstile> a \<sharp> t"  \<rightleftharpoons> "(nabla,a,t) \<in> fresh"
+
+inductive fresh  
+intros
+  fresh_abst_ab[intro!]: "\<lbrakk>(nabla\<turnstile>a\<sharp>t); a\<noteq>b\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>Abst b t)"
+  fresh_abst_aa[intro!]: "(nabla\<turnstile>a\<sharp>Abst a t)"
+  fresh_unit[intro!]:    "(nabla\<turnstile>a\<sharp>Unit)"
+  fresh_atom[intro!]:    "a\<noteq>b \<Longrightarrow> (nabla\<turnstile>a\<sharp>Atom b)"
+  fresh_susp[intro!]:    "(swapas (rev pi) a,X)\<in>nabla \<Longrightarrow> (nabla\<turnstile>a\<sharp>Susp pi X)"
+  fresh_paar[intro!]:    "\<lbrakk>(nabla\<turnstile>a\<sharp>t1);(nabla\<turnstile>a\<sharp>t2)\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>Paar t1 t2)"
+  fresh_func[intro!]:    "(nabla\<turnstile>a\<sharp>t) \<Longrightarrow> (nabla\<turnstile>a\<sharp>Func F t)"
+
+lemma fresh_atom_elim[elim!]: "(nabla\<turnstile>a\<sharp>Atom b) \<Longrightarrow> a\<noteq>b"
+apply(ind_cases "(nabla \<turnstile>a\<sharp>Atom b)")
+apply(auto)
+done
+lemma fresh_susp_elim[elim!]: "(nabla\<turnstile>a\<sharp>Susp pi X) \<Longrightarrow> (swapas (rev pi) a,X)\<in>nabla"
+apply(ind_cases "(nabla\<turnstile>a\<sharp>Susp pi X)")
+apply(auto)
+done
+lemma fresh_paar_elim[elim!]: "(nabla\<turnstile>a\<sharp>Paar t1 t2) \<Longrightarrow> (nabla\<turnstile>a\<sharp>t1)\<and>(nabla \<turnstile>a\<sharp>t2)"
+apply(ind_cases "(nabla\<turnstile>a\<sharp>Paar t1 t2)")
+apply(auto)
+done
+lemma fresh_func_elim[elim!]: "(nabla\<turnstile>a\<sharp>Func F t) \<Longrightarrow> (nabla\<turnstile>a\<sharp>t)"
+apply(ind_cases "nabla\<turnstile>a\<sharp>Func F t")
+apply(auto)
+done
+lemma fresh_abst_ab_elim[elim!]: "\<lbrakk>nabla\<turnstile>a\<sharp>Abst b t;a\<noteq>b\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>t)"
+apply(ind_cases "nabla\<turnstile>a\<sharp>Abst b t", auto)
+done
+
+lemma fresh_swap_left: "(nabla\<turnstile>a\<sharp>swap pi t) \<longrightarrow> (nabla\<turnstile>swapas (rev pi) a\<sharp>t)"
+apply(induct t)
+apply(simp_all)
+-- Abst
+apply(rule impI)
+apply(case_tac "swapas (rev pi) a = list")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+--Susp
+apply(force dest!: fresh_susp_elim simp add: swapas_append[THEN sym]) 
+--Unit
+apply(force)
+--Atom
+apply(force dest!: fresh_atom_elim)
+--Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force dest!: fresh_func_elim)
+done
+
+lemma fresh_swap_right: "(nabla\<turnstile>swapas (rev pi) a\<sharp>t) \<longrightarrow> (nabla\<turnstile>a\<sharp>swap pi t)"
+apply(induct t)
+apply(simp_all)
+-- Abst
+apply(rule impI)
+apply(case_tac "a = swapas pi list")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+--Susp
+apply(force dest!: fresh_susp_elim simp add: swapas_append[THEN sym])
+--Unit
+apply(force)
+--Atom
+apply(force dest!: fresh_atom_elim)
+--Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force dest!: fresh_func_elim)
+done
+
+lemma fresh_weak: "nabla1\<turnstile>a\<sharp>t\<longrightarrow>(nabla1\<union>nabla2)\<turnstile>a\<sharp>t"
+apply(rule impI)
+apply(erule fresh.induct)
+apply(auto)
+done
+
+end 

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Mgu.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,403 @@
+
+
+theory Mgu = Main + Terms + Fresh + Equ + Substs:
+
+(* unification problems *)
+
+syntax
+ "_equ_prob"   :: "trm \<Rightarrow> trm \<Rightarrow> (trm\<times>trm)"        ("_ \<approx>? _" [81,81] 81)
+ "_fresh_prob" :: "string \<Rightarrow> trm \<Rightarrow> (string\<times>trm)"  ("_ \<sharp>? _" [81,81] 81)
+
+translations 
+  "t1 \<approx>? t2" \<rightharpoonup> "(t1,t2)"
+  " a \<sharp>? t"  \<rightharpoonup> "(a,t)"
+
+(* all solutions for a unification problem *)
+
+types 
+  problem_type = "((trm\<times>trm)list) \<times> ((string\<times>trm)list)"
+  unifier_type = "fresh_envs \<times> substs"
+
+consts 
+  U :: "problem_type \<Rightarrow> (unifier_type set)"
+defs all_solutions_def : 
+  "U P  \<equiv> {(nabla,s). 
+             (\<forall> (t1,t2)\<in>set (fst P). nabla \<turnstile> subst s t1 \<approx> subst s t2) \<and> 
+             (\<forall>   (a,t)\<in>set (snd P). nabla \<turnstile> a \<sharp> subst s t) }"
+
+(* set of variables in unification problems *)
+
+consts
+  vars_fprobs :: "((string\<times>trm) list) \<Rightarrow> (string set)"
+  vars_eprobs :: "((trm\<times>trm)list) \<Rightarrow> (string set)"
+  vars_probs  :: "problem_type \<Rightarrow> nat"
+primrec
+  "vars_fprobs [] = {}"
+  "vars_fprobs (x#xs) = (vars_trm (snd x))\<union>(vars_fprobs xs)"
+primrec
+  "vars_eprobs [] = {}"
+  "vars_eprobs (x#xs) = (vars_trm (snd x))\<union>(vars_trm (fst x))\<union>(vars_eprobs xs)"
+defs
+  vars_probs_def: "vars_probs P \<equiv> card((vars_fprobs (snd P))\<union>(vars_eprobs (fst P)))"
+
+
+(* most general unifier *)
+
+consts 
+  mgu :: "problem_type \<Rightarrow> unifier_type \<Rightarrow> bool"
+defs mgu_def:
+  "mgu P unif \<equiv> 
+   \<forall> (nabla,s1)\<in> U P. (\<exists> s2. (nabla\<Turnstile>(subst s2) (fst unif)) \<and> 
+                              (nabla\<Turnstile>subst (s2 \<bullet>(snd unif)) \<approx> subst s1))"
+
+(* idempotency of a unifier *)
+
+consts 
+  idem :: "unifier_type \<Rightarrow> bool"
+defs 
+  idem_def: "idem unif \<equiv> (fst unif)\<Turnstile> subst ((snd unif)\<bullet>(snd unif)) \<approx> subst (snd unif)"
+
+(* application of a substitution to a problem *)
+
+consts 
+  apply_subst :: "substs \<Rightarrow> problem_type \<Rightarrow> problem_type"
+defs apply_subst_def: 
+  "apply_subst s P \<equiv> (map (\<lambda>(t1,t2). (subst s t1 \<approx>? subst s t2)) (fst P),
+                      map (\<lambda>(a,t).   (a \<sharp>? (subst s t)) ) (snd P))" 
+
+(* equality reductions *)
+
+consts 
+  s_red :: "(problem_type \<times> substs \<times> problem_type) set"
+syntax 
+  "_s_red" :: "problem_type \<Rightarrow> substs \<Rightarrow> problem_type \<Rightarrow> bool"  ("_ \<turnstile> _ \<leadsto> _ " [80,80,80] 80)
+translations "P1 \<turnstile>sigma\<leadsto> P2" \<rightleftharpoons> "(P1,sigma,P2)\<in>s_red"
+inductive s_red
+intros
+  unit_sred[intro!,dest!]:    "((Unit\<approx>?Unit)#xs,ys) \<turnstile>[]\<leadsto> (xs,ys)"
+  paar_sred[intro!,dest!]:    "((Paar t1 t2\<approx>?Paar s1 s2)#xs,ys) \<turnstile>[]\<leadsto> ((t1\<approx>?s1)#(t2\<approx>?s2)#xs,ys)"
+  func_sred[intro!,dest!]:    "((Func F t1\<approx>?Func F t2)#xs,ys) \<turnstile>[]\<leadsto> ((t1\<approx>?t2)#xs,ys)"
+  abst_aa_sred[intro!,dest!]: "((Abst a t1\<approx>?Abst a t2)#xs,ys) \<turnstile>[]\<leadsto> ((t1\<approx>?t2)#xs,ys)"
+  abst_ab_sred[intro!]: "a\<noteq>b\<Longrightarrow> 
+                       ((Abst a t1\<approx>?Abst b t2)#xs,ys) \<turnstile>[]\<leadsto> ((t1\<approx>?swap [(a,b)] t2)#xs,(a\<sharp>?t2)#ys)"
+  atom_sred[intro!,dest!]:    "((Atom a\<approx>?Atom a)#xs,ys) \<turnstile>[]\<leadsto> (xs,ys)"  
+  susp_sred[intro!,dest!]:    "((Susp pi1 X\<approx>?Susp pi2 X)#xs,ys) 
+                                \<turnstile>[]\<leadsto> (xs,(map (\<lambda>a. a\<sharp>? Susp [] X) (ds_list pi1 pi2))@ys)"
+  var_1_sred[intro!]:   "\<not>(occurs X t)\<Longrightarrow>((Susp pi X\<approx>?t)#xs,ys)
+                               \<turnstile>[(X,swap (rev pi) t)]\<leadsto> apply_subst [(X,swap (rev pi) t)] (xs,ys)"
+  var_2_sred[intro!]:   "\<not>(occurs X t)\<Longrightarrow>((t\<approx>?Susp pi X)#xs,ys) 
+                               \<turnstile>[(X,swap (rev pi) t)]\<leadsto> apply_subst [(X,swap (rev pi) t)] (xs,ys)"
+
+(* freshness reductions *)
+
+consts 
+  c_red :: "(problem_type \<times> fresh_envs \<times> problem_type) set"
+syntax 
+  "_c_red" :: "problem_type \<Rightarrow> fresh_envs \<Rightarrow> problem_type \<Rightarrow> bool" ("_ \<turnstile> _ \<rightarrow> _ " [80,80,80] 80)
+translations "P1 \<turnstile>nabla\<rightarrow> P2" \<rightleftharpoons> "(P1,nabla,P2)\<in>c_red"
+inductive c_red
+intros
+  unit_cred[intro!]:    "([],(a \<sharp>? Unit)#xs) \<turnstile>{}\<rightarrow> ([],xs)"
+  paar_cred[intro!]:    "([],(a \<sharp>? Paar t1 t2)#xs) \<turnstile>{}\<rightarrow> ([],(a\<sharp>?t1)#(a\<sharp>?t2)#xs)"
+  func_cred[intro!]:    "([],(a \<sharp>? Func F t)#xs) \<turnstile>{}\<rightarrow> ([],(a\<sharp>?t)#xs)"
+  abst_aa_cred[intro!]: "([],(a \<sharp>? Abst a t)#xs) \<turnstile>{}\<rightarrow> ([],xs)"
+  abst_ab_cred[intro!]: "a\<noteq>b\<Longrightarrow>([],(a \<sharp>? Abst b t)#xs) \<turnstile>{}\<rightarrow> ([],(a\<sharp>?t)#xs)"
+  atom_cred[intro!]:    "a\<noteq>b\<Longrightarrow>([],(a \<sharp>? Atom b)#xs) \<turnstile>{}\<rightarrow> ([],xs)"  
+  susp_cred[intro!]:    "([],(a \<sharp>? Susp pi X)#xs) \<turnstile>{((swapas (rev pi) a),X)}\<rightarrow> ([],xs)" 
+
+(* unification reduction sequence *)
+
+consts 
+  red_plus :: "(problem_type \<times> unifier_type \<times> problem_type) set"
+syntax 
+  red_plus :: "problem_type \<Rightarrow> unifier_type \<Rightarrow> problem_type \<Rightarrow> bool" ("_ \<Turnstile> _ \<Rightarrow> _ " [80,80,80] 80)
+
+translations "P1 \<Turnstile>(nabla,s)\<Rightarrow> P2" \<rightleftharpoons> "(P1,(nabla,s),P2)\<in>red_plus"
+inductive red_plus
+intros
+  sred_single[intro!]: "\<lbrakk>P1\<turnstile>s1\<leadsto>P2\<rbrakk>\<Longrightarrow>P1\<Turnstile>({},s1)\<Rightarrow>P2"
+  cred_single[intro!]: "\<lbrakk>P1\<turnstile>nabla1\<rightarrow>P2\<rbrakk>\<Longrightarrow>P1\<Turnstile>(nabla1,[])\<Rightarrow>P2"
+  sred_step[intro!]:   "\<lbrakk>P1\<turnstile>s1\<leadsto>P2;P2\<Turnstile>(nabla2,s2)\<Rightarrow>P3\<rbrakk>\<Longrightarrow>P1\<Turnstile>(nabla2,(s2\<bullet>s1))\<Rightarrow>P3"
+  cred_step[intro!]:   "\<lbrakk>P1\<turnstile>nabla1\<rightarrow>P2;P2\<Turnstile>(nabla2,[])\<Rightarrow>P3\<rbrakk>\<Longrightarrow>P1\<Turnstile>(nabla2\<union>nabla1,[])\<Rightarrow>P3"
+
+lemma mgu_idem: 
+  "\<lbrakk>(nabla1,s1)\<in>U P; 
+   \<forall>(nabla2,s2)\<in> U P. nabla2\<Turnstile>(subst s2) nabla1 \<and>  nabla2\<Turnstile>subst(s2 \<bullet> s1)\<approx>subst s2 \<rbrakk>\<Longrightarrow>
+   mgu P (nabla1,s1)\<and>idem (nabla1,s1)"
+apply(rule conjI)
+apply(simp only: mgu_def)
+apply(rule ballI)
+apply(simp)
+apply(drule_tac x="x" in bspec)
+apply(assumption)
+apply(force)
+apply(drule_tac x="(nabla1,s1)" in bspec)
+apply(assumption)
+apply(simp add: idem_def)
+done
+
+lemma problem_subst_comm: "((nabla,s2)\<in>U (apply_subst s1 P)) = ((nabla,(s2\<bullet>s1))\<in>U P)"
+apply(simp add: all_solutions_def apply_subst_def)
+apply(auto)
+apply(drule_tac x="(a,b)" in bspec, assumption, simp add: subst_comp_expand)+
+done
+
+lemma P1_to_P2_sred: 
+  "\<lbrakk>(nabla1,s1)\<in>U P1; P1 \<turnstile>s2\<leadsto> P2 \<rbrakk>\<Longrightarrow>((nabla1,s1)\<in>U P2) \<and> (nabla1\<Turnstile>subst (s1\<bullet>s2)\<approx>subst s1)"
+apply(ind_cases "P1 \<turnstile>s2\<leadsto> P2")
+apply(simp_all)
+--Unit
+apply(force intro!: equ_refl simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+--Paar
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+apply(force intro!: equ_refl dest!: equ_paar_elim)
+--Func
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+apply(force intro!: equ_refl dest!: equ_func_elim)
+--Abst.aa
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+apply(force intro!: equ_refl dest!: equ_abst_aa_elim)
+--Abst.ab
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+apply(force intro!: equ_refl dest!: equ_abst_ab_elim simp add: subst_swap_comm)
+--Atom
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def subst_susp)
+apply(force intro!: equ_refl)
+--Susp
+apply(rule conjI)
+apply(auto)
+apply(simp add: all_solutions_def)
+apply(erule conjE)+
+apply(simp add: ds_list_equ_ds)
+apply(simp only: subst_susp)
+apply(drule equ_pi1_pi2_dec[THEN mp])
+apply(auto)
+apply(drule_tac x="aa" in bspec)
+apply(assumption)
+apply(simp add: subst_susp)
+apply(simp add: subst_equ_def subst_susp)
+apply(rule ballI)
+apply(rule equ_refl)
+--Var.one
+apply(drule_tac "t2.1"="swap (rev pi) t" in subst_not_occurs[THEN mp])
+apply(simp only: problem_subst_comm)
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def)
+apply(rule conjI)
+apply(rule ballI)
+apply(erule conjE)+
+apply(drule unif_1)
+apply(clarify)
+apply(drule_tac x="(a,b)" in bspec)
+apply(assumption)
+apply(simp)
+apply(simp add: unif_2a)
+apply(erule conjE)+
+apply(drule unif_1)
+apply(rule ballI)
+apply(clarify)
+apply(drule_tac x="(a,b)" in bspec)
+apply(assumption)
+apply(simp)
+apply(simp add: unif_2b)
+apply(rule unif_1)
+apply(simp add: all_solutions_def)
+--Var.two
+apply(drule_tac "t2.1"="swap (rev pi) t" in subst_not_occurs[THEN mp])
+apply(simp only: problem_subst_comm)
+apply(simp add: all_solutions_def ext_subst_def subst_equ_def)
+apply(auto)
+apply(drule_tac x="(a,b)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule equ_sym)
+apply(drule unif_1)
+apply(simp add: unif_2a)
+apply(drule_tac x="(a,b)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule equ_sym)
+apply(drule unif_1)
+apply(simp add: unif_2b)
+apply(rule unif_1)
+apply(rule equ_sym)
+apply(simp add: all_solutions_def)
+done
+
+lemma P1_from_P2_sred: "\<lbrakk>(nabla1,s1)\<in>U P2; P1\<turnstile>s2\<leadsto>P2\<rbrakk>\<Longrightarrow>(nabla1,s1\<bullet>s2)\<in>U P1"
+apply(ind_cases "P1 \<turnstile>s2\<leadsto> P2")
+--Susp.Paar.Func.Abst.aa
+apply(simp add: all_solutions_def, force)
+apply(simp add: all_solutions_def, force)
+apply(simp add: all_solutions_def, force)
+apply(simp add: all_solutions_def, force)
+--Abst.ab
+apply(simp only: all_solutions_def)
+apply(force simp add: subst_swap_comm)
+--Atom
+apply(simp only: all_solutions_def, force)
+--Susp
+apply(simp)
+apply(auto)
+apply(simp add: all_solutions_def)
+apply(simp add: ds_list_equ_ds)
+apply(subgoal_tac "nabla1\<turnstile>(swap pi1 (subst s1 (Susp [] X)))\<approx>(swap pi2 (subst s1 (Susp [] X)))")
+apply(simp add: subst_susp subst_swap_comm)
+apply(simp add: subst_susp subst_swap_comm)
+apply(rule equ_pi1_pi2_add[THEN mp])
+apply(drule conjunct2)
+apply(auto)
+apply(drule_tac x="(a,Susp [] X)" in bspec)
+apply(auto)
+apply(simp add: subst_susp)
+--Var.one
+apply(simp only: problem_subst_comm)
+apply(simp only: all_solutions_def)
+apply(simp)
+apply(simp only: subst_comp_expand)
+apply(subgoal_tac "subst [(X, swap (rev pi) t)] t = t")--A
+apply(simp add: subst_susp)
+apply(simp only: subst_swap_comm)
+apply(simp only: equ_pi_to_right[THEN sym])
+apply(simp only: equ_involutive_right)
+apply(rule equ_refl)
+--A
+apply(rule subst_not_occurs[THEN mp])
+apply(assumption)
+--Var.two
+apply(simp only: problem_subst_comm)
+apply(simp only: all_solutions_def)
+apply(simp)
+apply(simp only: subst_comp_expand)
+apply(subgoal_tac "subst [(X, swap (rev pi) t)] t = t")--B
+apply(simp add: subst_susp)
+apply(simp only: subst_swap_comm)
+apply(simp only: equ_pi_to_left[THEN sym])
+apply(simp only: equ_involutive_left)
+apply(rule equ_refl)
+--B
+apply(rule subst_not_occurs[THEN mp])
+apply(assumption)
+done
+
+lemma P1_to_P2_cred: 
+  "\<lbrakk>(nabla1,s1)\<in>U P1; P1 \<turnstile>nabla2\<rightarrow> P2 \<rbrakk>\<Longrightarrow>((nabla1,s1)\<in>U P2) \<and> (nabla1\<Turnstile>(subst s1) nabla2)"
+apply(ind_cases " P1\<turnstile>nabla2\<rightarrow>P2")
+apply(simp_all)
+apply(auto simp add: ext_subst_def all_solutions_def)
+apply(rule fresh_swap_left[THEN mp])
+apply(simp add: subst_swap_comm[THEN sym] subst_susp)
+done
+
+lemma P1_from_P2_cred: 
+  "\<lbrakk>(nabla1,s1)\<in>U P2; P1 \<turnstile>nabla2\<rightarrow> P2; nabla3\<Turnstile>(subst s1) nabla2\<rbrakk>\<Longrightarrow>(nabla1\<union>nabla3,s1)\<in>U P1"
+apply(ind_cases "P1 \<turnstile>nabla2\<rightarrow> P2")
+apply(simp_all)
+apply(auto simp add: ext_subst_def all_solutions_def fresh_weak)
+apply(simp add: subst_susp)
+apply(rule fresh_swap_right[THEN mp])
+apply(drule_tac "nabla2.1"="nabla1" in fresh_weak[THEN mp])
+apply(subgoal_tac "nabla3 \<union> nabla1=nabla1 \<union> nabla3")--A
+apply(simp)
+apply(rule Un_commute)
+done
+
+lemma P1_to_P2_red_plus: "\<lbrakk>P1 \<Turnstile>(nabla,s)\<Rightarrow>P2\<rbrakk>\<Longrightarrow> (nabla1,s1)\<in>U P1 \<longrightarrow>
+  ((nabla1,s1)\<in>U P2)\<and>(nabla1\<Turnstile>subst (s1\<bullet>s)\<approx>subst s1)\<and>(nabla1\<Turnstile>(subst s1) nabla)"
+apply(erule red_plus.induct)
+-- sred
+apply(rule impI)
+apply(drule_tac "P2.0"="P2" and "s2.0"="s1a" in P1_to_P2_sred)
+apply(force)
+apply(rule conjI, force)+
+apply(force simp add: ext_subst_def)
+-- cred
+apply(rule impI)
+apply(drule_tac "P2.0"="P2" and "nabla2.0"="nabla1a" in P1_to_P2_cred)
+apply(assumption)
+apply(force intro!: equ_refl simp add: subst_equ_def)
+-- sred
+apply(rule impI)
+apply(drule_tac "P2.0"="P2" and "s2.0"="s1a" in P1_to_P2_sred)
+apply(assumption)
+apply(erule conjE)+
+apply(rule conjI)
+apply(force)
+apply(rule conjI)
+apply(drule mp)
+apply(assumption)
+apply(erule conjE)+
+apply(rule_tac "s2.0"="((s1\<bullet>s2)\<bullet>s1a)" in subst_trans)
+apply(simp only: subst_assoc subst_equ_def)
+apply(rule ballI)
+apply(rule equ_refl)
+apply(rule_tac "s2.0"="(s1\<bullet>s1a)" in subst_trans)
+apply(rule subst_cancel_right)
+apply(assumption)
+apply(assumption)
+apply(force)
+-- cred
+apply(rule impI)
+apply(drule_tac "P2.0"="P2" and "nabla2.0"="nabla1a" in P1_to_P2_cred)
+apply(auto simp add: ext_subst_def)
+done
+
+lemma P1_from_P2_red_plus: "\<lbrakk>P1 \<Turnstile>(nabla,s)\<Rightarrow>P2\<rbrakk>\<Longrightarrow>(nabla1,s1)\<in>U P2\<longrightarrow>
+        nabla3\<Turnstile>(subst s1)(nabla)\<longrightarrow>(nabla1\<union>nabla3,(s1\<bullet>s))\<in>U P1"
+apply(erule red_plus.induct)
+-- sred
+apply(rule impI)+
+apply(drule_tac "P1.0"="P1" and "s2.0"="s1a" in P1_from_P2_sred)
+apply(assumption)
+apply(force simp only: all_solutions_def equ_weak fresh_weak)
+-- cred
+apply(rule impI)+
+apply(drule_tac "P1.0"="P1" and "nabla3.0"="nabla3" and "nabla2.0"="nabla1a" in P1_from_P2_cred)
+apply(assumption)+
+apply(simp add: all_solutions_def)
+-- sred
+apply(rule impI)+
+apply(simp)
+apply(drule_tac "P1.0"="P1" and "P2.0"="P2" and "s2.0"="s1a" in P1_from_P2_sred)
+apply(assumption)
+apply(simp add: all_solutions_def subst_assoc)
+-- cred
+apply(rule impI)+
+apply(subgoal_tac "nabla3 \<Turnstile> (subst s1) nabla2")--A
+apply(simp)
+apply(drule_tac "P1.0"="P1" and "P2.0"="P2" and
+                "nabla2.0"="nabla1a" and "nabla3.0"="nabla3" in P1_from_P2_cred)
+apply(assumption)
+apply(simp)
+apply(simp add: ext_subst_def)
+apply(subgoal_tac "nabla1 \<union> nabla3 \<union> nabla3=nabla1 \<union> nabla3")--A
+apply(simp)
+--A
+apply(force)
+--B
+apply(simp add: ext_subst_def)
+done
+
+lemma mgu: "\<lbrakk>P \<Turnstile>(nabla,s)\<Rightarrow>([],[])\<rbrakk>\<Longrightarrow> mgu P (nabla,s) \<and> idem (nabla,s)"
+apply(frule_tac "nabla3.2"="nabla" and "nabla2"="nabla" and 
+                "s1.2"="[]" and "nabla1.2"="{}"
+                in P1_from_P2_red_plus[THEN mp,THEN mp])
+apply(force simp add: all_solutions_def)
+apply(force simp add: ext_subst_def)
+apply(rule mgu_idem)
+apply(simp add: all_solutions_def)
+apply(rule ballI)
+apply(clarify)
+apply(drule_tac  "nabla1.0"="a" and "s1.0"="b"in P1_to_P2_red_plus)
+apply(simp)
+done  
+
+end
+
+
+
+
+
+
+
+
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/PreEqu.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,579 @@
+
+
+theory PreEqu = Main + Swap + Terms + Disagreement + Fresh:
+
+consts 
+  equ :: "(fresh_envs \<times> trm \<times> trm) set"
+
+syntax 
+  "_equ_judge"   :: "fresh_envs \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool" (" _ \<turnstile> _ \<approx> _" [80,80,80] 80)
+translations 
+  "nabla \<turnstile> t1 \<approx> t2" \<rightleftharpoons> "(nabla,t1,t2) \<in> equ"
+
+inductive equ 
+intros
+equ_abst_ab[intro!]: "\<lbrakk>a\<noteq>b;(nabla \<turnstile> a \<sharp> t2);(nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<rbrakk> 
+                      \<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst b t2)"
+equ_abst_aa[intro!]: "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst a t2)" 
+equ_unit[intro!]:    "(nabla \<turnstile> Unit \<approx> Unit)"
+equ_atom[intro!]:    "a=b\<Longrightarrow>nabla \<turnstile> Atom a \<approx> Atom b"
+equ_susp[intro!]:    "(\<forall> c \<in> ds pi1 pi2. (c,X) \<in> nabla) \<Longrightarrow> (nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X)"
+equ_paar[intro!]:    "\<lbrakk>(nabla \<turnstile> t1\<approx>t2);(nabla \<turnstile> s1\<approx>s2)\<rbrakk> \<Longrightarrow> (nabla \<turnstile> Paar t1 s1 \<approx> Paar t2 s2)"
+equ_func[intro!]:    "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Func F t1 \<approx> Func F t2)" 
+
+lemma equ_atom_elim[elim!]: "nabla\<turnstile>Atom a \<approx> Atom b \<Longrightarrow> a=b"
+apply(ind_cases "nabla \<turnstile> Atom a \<approx> Atom b", auto)
+done
+
+lemma equ_susp_elim[elim!]: "(nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X) 
+                             \<Longrightarrow> (\<forall> c \<in> ds pi1 pi2. (c,X)\<in> nabla)"
+apply(ind_cases "nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X", auto)
+done
+lemma equ_paar_elim[elim!]: "(nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2) \<Longrightarrow> 
+                             (nabla \<turnstile> s1 \<approx> s2)\<and>(nabla \<turnstile> t1 \<approx> t2)"
+apply(ind_cases "nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2", auto)
+done
+lemma equ_func_elim[elim!]: "(nabla \<turnstile> Func F t1 \<approx> Func F t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"
+apply(ind_cases "nabla \<turnstile> Func F t1 \<approx> Func F t2", auto)
+done
+lemma equ_abst_aa_elim[elim!]: "(nabla \<turnstile> Abst a t1 \<approx> Abst a t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"
+apply(ind_cases "nabla \<turnstile> Abst a t1 \<approx> Abst a t2", auto)
+done
+lemma equ_abst_ab_elim[elim!]: "\<lbrakk>(nabla \<turnstile> Abst a t1 \<approx> Abst b t2);a\<noteq>b\<rbrakk> \<Longrightarrow> 
+                                (nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<and>(nabla\<turnstile>a\<sharp>t2)"
+apply(ind_cases "(nabla \<turnstile> Abst a t1 \<approx> Abst b t2)", auto)
+done
+
+lemma equ_depth: "nabla \<turnstile> t1 \<approx> t2 \<Longrightarrow> depth t1 = depth t2"
+apply(erule equ.induct)
+apply(simp_all)
+done
+
+lemma rev_pi_pi_equ: "(nabla\<turnstile>swap (rev pi) (swap pi t)\<approx>t)"
+apply(induct_tac t)
+apply(auto)
+-- Susp
+apply(drule_tac ds_cancel_pi_left[of _ "rev pi @ pi" _ "[]", THEN mp, simplified])
+apply(simp only: ds_rev_pi_pi)
+apply(simp only: ds_def)
+apply(force)
+done
+
+lemma equ_pi_right: "\<forall>pi. (\<forall>a\<in>ds [] pi. nabla\<turnstile>a\<sharp>t) \<longrightarrow> (nabla\<turnstile>t\<approx>swap pi t)"
+apply(induct_tac t)
+apply(simp_all)
+-- Abst
+apply(rule allI)
+apply(case_tac "(swapas pi list)=list") 
+apply(simp)
+apply(rule impI)
+apply(rule equ_abst_aa)
+apply(drule_tac x="pi" in spec)
+apply(subgoal_tac "(\<forall>a\<in>ds [] pi.  nabla \<turnstile> a \<sharp> trm)")--A
+apply(force)
+--A
+apply(rule ballI)
+apply(drule_tac x=a in bspec)
+apply(assumption)
+apply(case_tac "list\<noteq>a")
+apply(force dest!: fresh_abst_ab_elim)
+apply(simp add: ds_def)
+apply(rule impI)
+apply(rule equ_abst_ab)
+apply(force)
+apply(drule_tac x="swapas (rev pi) list" in bspec)
+apply(simp add: ds_def)
+apply(rule conjI)
+apply(subgoal_tac "swapas (rev pi) list \<in> atms (rev pi)") --B
+apply(simp)
+--B
+apply(drule swapas_pi_ineq_a[THEN mp])
+apply(rule swapas_pi_in_atms)
+apply(simp)
+apply(clarify)
+apply(drule swapas_rev_pi_b)
+apply(simp)
+apply(force dest!: fresh_abst_ab_elim  swapas_rev_pi_b intro!: fresh_swap_right[THEN mp])
+apply(drule_tac x="(list, swapas pi list)#pi" in spec)
+apply(subgoal_tac "(\<forall>a\<in>ds [] ((list, swapas pi list) # pi).  nabla \<turnstile> a \<sharp> trm)")--C
+apply(force simp add: swap_append[THEN sym])
+--C
+apply(rule ballI)
+apply(drule_tac x="a" in bspec)
+apply(rule_tac b="list" in ds_7)
+apply(force)
+apply(assumption)
+apply(case_tac "list=a")
+apply(simp)
+apply(simp only: ds_def mem_Collect_eq)
+apply(erule conjE)
+apply(subgoal_tac "a\<noteq>swapas pi a")
+apply(simp)
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+-- Susp
+apply(rule allI)
+apply(rule impI)
+apply(rule equ_susp)
+apply(rule ballI)
+apply(subgoal_tac "swapas list1 c\<in>ds [] pi")--A
+apply(force dest!: fresh_susp_elim)
+--A
+apply(rule ds_cancel_pi_left[THEN mp])
+apply(simp)
+-- Unit
+apply(force)
+-- Atom
+apply(rule allI)
+apply(rule impI)
+apply(case_tac "(swapas pi list) = list")
+apply(force)
+apply(drule ds_elem)
+apply(force dest!: fresh_atom_elim)
+-- Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force)
+done
+
+lemma pi_comm: "nabla\<turnstile>(swap (pi@[(a,b)]) t)\<approx>(swap ([(swapas pi a, swapas pi b)]@pi) t)"
+apply(induct_tac t)
+apply(simp_all)
+-- Abst
+apply(force simp add: swapas_comm)
+-- Susp
+apply(rule equ_susp)
+apply(rule ballI)
+apply(simp only: ds_def)
+apply(simp only: mem_Collect_eq)
+apply(erule conjE)
+apply(subgoal_tac "swapas (pi@[(a,b)]) (swapas list1 c) =
+                   swapas ([(swapas pi a,swapas pi b)]@pi) (swapas list1 c)")--A
+apply(simp add: swapas_append[THEN sym])
+--A
+apply(simp only: swapas_comm)
+-- Units
+apply(rule equ_unit)
+-- Atom
+apply(force dest!: swapas_rev_pi_b swapas_rev_pi_a simp add: swapas_append)
+--Paar
+apply(force)
+--Func
+apply(force)
+done
+
+
+lemma l3_jud: "(nabla \<turnstile> t1\<approx>t2) \<Longrightarrow> (nabla \<turnstile> a\<sharp>t1) \<longrightarrow> (nabla \<turnstile> a\<sharp>t2)"
+apply(erule equ.induct)
+apply(simp_all)
+--Abst.ab
+apply(rule impI)
+apply(case_tac "aa=a")
+apply(force)
+apply(case_tac "b=a")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim fresh_swap_left[THEN mp])
+-- Abst.aa
+apply(case_tac "a=aa")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+-- Susp
+apply(rule impI)
+apply(drule fresh_susp_elim, rule fresh_susp)
+apply(case_tac "swapas (rev pi1) a = swapas (rev pi2) a") 
+apply(simp)
+apply(drule_tac x="swapas (rev pi2) a" in bspec)
+apply(rule ds_cancel_pi_left[THEN mp])
+apply(subgoal_tac "swapas (pi1@(rev pi2)) a \<noteq> a")--A
+apply(drule ds_elem)
+apply(force simp add: ds_def swapas_append)
+--A
+apply(clarify)
+apply(simp only: swapas_append)
+apply(drule swapas_rev_pi_a)
+apply(force)
+apply(assumption)
+-- Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force dest!: fresh_func_elim)
+done
+
+lemma big: "\<forall>t1 t2 t3. (n=depth t1) \<longrightarrow>
+             (((nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>t2\<approx>t1))\<and>  
+              (\<forall>pi. (nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>swap pi t1\<approx>swap pi t2))\<and> 
+              ((nabla\<turnstile>t1\<approx>t2)\<and>(nabla\<turnstile>t2\<approx>t3)\<longrightarrow>(nabla\<turnstile>t1\<approx>t3)))" 
+apply(induct_tac n rule: nat_less_induct)
+apply(rule allI)+apply(rule impI)
+apply(rule conjI)
+-- SYMMETRY
+apply(rule impI)
+apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
+apply(simp_all)
+-- Abst.ab
+apply(rule equ_abst_ab)
+apply(force) --abst.ab.first.premise
+apply(rule_tac "t1.1"="swap [(a,b)] t2a" in l3_jud[THEN mp])
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(rule fresh_swap_right[THEN mp])
+apply(simp) --abst.ab.second.premise
+apply(subgoal_tac "nabla \<turnstile> swap [(b, a)] t1a \<approx> t2a")
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(subgoal_tac "nabla \<turnstile> swap [(b,a)] t1a \<approx> swap ([(b,a)]@[(a,b)]) t2a") --A
+apply(subgoal_tac "nabla \<turnstile> swap ([(b,a)]@[(a,b)]) t2a \<approx> t2a") --B
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="swap [(b,a)] t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="swap [(b,a),(a,b)] t2a" in spec)
+apply(force)
+--B
+apply(subgoal_tac "nabla\<turnstile>t2a \<approx> swap ([(b, a)] @ [(a, b)]) t2a")--C
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2a" in spec)
+apply(drule mp)
+apply(drule equ_depth)
+apply(force)
+apply(best)
+--C
+apply(rule equ_pi_right[THEN spec,THEN mp])
+apply(subgoal_tac "ds [] ([(b, a)] @ [(a, b)])={}")
+apply(simp)
+apply(simp add: ds_baab)
+--A
+apply(force simp only: swap_append)
+-- Abst.aa
+apply(force)
+-- Unit
+apply(rule equ_unit)
+-- Atom
+apply(force)
+-- Susp
+apply(force simp only: ds_sym)
+-- Paar
+apply(rule equ_paar)
+apply(drule_tac x="depth t1a" in spec)
+apply(simp add: Suc_max_left)
+apply(drule_tac x="depth s1" in spec)
+apply(simp add: Suc_max_right)
+-- Func
+apply(best)
+-- ADD.PI
+apply(rule conjI)
+apply(rule impI)
+apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
+apply(simp_all)
+-- Abst.ab
+apply(rule allI)
+apply(rule equ_abst_ab)
+-- abst.ab.first.premise
+apply(clarify)
+apply(drule swapas_rev_pi_a)
+apply(simp)
+-- abst.ab.second.premise
+apply(rule fresh_swap_right[THEN mp])
+apply(simp)
+-- abst.ab.third.premise
+apply(subgoal_tac "nabla \<turnstile> swap pi t1a \<approx> swap (pi@[(a,b)]) t2a") --A
+apply(subgoal_tac "nabla \<turnstile> swap (pi@[(a,b)]) t2a \<approx> swap ([(swapas pi a,swapas pi b)]@pi) t2a") --B
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="swap pi t1a" in spec)
+apply(simp (no_asm_use)) 
+apply(drule_tac x="swap (pi@[(a,b)]) t2a" in spec)
+apply(drule conjunct2)+
+apply(drule_tac x="swap ((swapas pi a, swapas pi b) # pi) t2a" in spec)
+apply(simp add: swap_append[THEN sym])
+--B
+apply(rule pi_comm)
+apply(force simp only: swap_append)
+-- A
+apply(force simp only: swap_append)
+-- Unit
+apply(rule equ_unit)
+-- Atom
+apply(force)
+-- Susp
+apply(force simp only: ds_cancel_pi_front)
+-- Paar
+apply(rule allI)
+apply(rule equ_paar)
+apply(drule_tac x="depth t1a" in spec)
+apply(simp only: Suc_max_left)
+apply(drule_tac x="depth s1" in spec)
+apply(simp only: Suc_max_right)
+-- Func
+apply(best)
+-- TRANSITIVITY
+apply(rule impI)
+apply(erule conjE)
+apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
+apply(simp_all)
+-- Abst.ab
+apply(ind_cases "nabla \<turnstile> Abst b t2a \<approx> t3")
+apply(simp)
+apply(case_tac "ba=a")
+apply(simp)
+apply(rule equ_abst_aa)
+apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a \<approx> t2b") --A
+apply(best)
+--A
+apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx> swap ([(a,b)]@[(b,a)]) t2b") --B
+apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,a)]) t2b \<approx> t2b") --C
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="swap [(a,b)] t2a" in spec)
+apply(drule equ_depth)
+apply(simp (no_asm_use))
+apply(best)
+--C
+apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(a,b)]@[(b,a)]) t2b")--D
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2b" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(best)
+--D
+apply(rule equ_pi_right[THEN spec, THEN mp])
+apply(simp add: ds_baab)
+--B
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2a" in spec)
+apply(drule equ_depth)
+apply(simp) 
+apply(drule_tac x="swap [(b, a)] t2b" in spec)
+apply(drule conjunct2)
+apply(drule conjunct1)
+apply(simp)
+apply(drule_tac x="[(a,b)]" in spec)
+apply(simp add: swap_append[THEN sym])
+-- Abst.ab
+apply(rule equ_abst_ab)
+-- abst.ab.first.premise
+apply(force)
+-- abst.ab.second.premise
+apply(rule_tac "t1.1"="swap [(b,ba)] t2a" in l3_jud[THEN mp])
+apply(subgoal_tac "nabla \<turnstile> swap [(b,ba)] t2a \<approx> swap ([(b,ba)]@[(b, ba)]) t2b") --A
+apply(subgoal_tac "nabla\<turnstile>swap ([(b,ba)]@[(b,ba)]) t2b \<approx> t2b") --B
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="swap [(b, ba)] t2a" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(best)
+--B
+apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(b,ba)] @ [(b,ba)]) t2b")--C
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2b" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(best)
+-- C
+apply(rule equ_pi_right[THEN spec, THEN mp])
+apply(simp add: ds_abab)
+--A
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2a" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(drule_tac x="swap [(b,ba)] t2b" in spec)
+apply(drule conjunct2)
+apply(drule conjunct1)
+apply(simp)
+apply(drule_tac x="[(b,ba)]" in spec)
+apply(simp add: swap_append[THEN sym])
+-- abst.ab.third.premise
+apply(force intro!: fresh_swap_right[THEN mp])
+-- very.complex
+apply(subgoal_tac "nabla\<turnstile>t1a \<approx> swap ([(a,b)]@[(b,ba)]) t2b") --A
+apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,ba)]) t2b \<approx> swap [(a,ba)] t2b") --B
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(best)
+--B
+apply(subgoal_tac "nabla\<turnstile>swap [(a, ba)] t2b \<approx> swap [(a,b),(b,ba)] t2b")--C
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="swap [(a, ba)] t2b" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(force)
+apply(subgoal_tac "nabla\<turnstile>swap [(a,ba)] t2b\<approx> swap [(a,ba)] (swap [(a,ba),(a,b),(b,ba)] t2b)") --D
+apply(subgoal_tac "nabla\<turnstile>swap (rev [(a,ba)]) (swap [(a,ba)] (swap [(a,b),(b,ba)] t2b)) 
+                        \<approx>swap [(a,b),(b,ba)] t2b") --E
+apply(simp add: swap_append[THEN sym])
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="swap [(a,ba)] t2b" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(drule_tac x="swap [(a, ba), (a, ba), (a, b), (b, ba)] t2b" in spec)
+apply(drule conjunct2)+
+apply(best)
+-- D
+apply(rule rev_pi_pi_equ)
+-- E
+apply(subgoal_tac "nabla\<turnstile>t2b\<approx>swap [(a, ba), (a, b), (b, ba)] t2b") --F
+apply(drule_tac x="depth t1a" in spec)
+apply(simp)
+apply(drule_tac x="t2b" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(best)
+--F
+apply(rule equ_pi_right[THEN spec, THEN mp])
+apply(subgoal_tac "ds [] [(a,ba),(a,b),(b,ba)]={a,b}") -- G
+apply(simp)
+apply(drule_tac "t1.1"="t2a" and "t2.1"="swap [(b, ba)] t2b" and a1="a" in l3_jud[THEN mp])
+apply(assumption)
+apply(subgoal_tac "nabla \<turnstile> swapas (rev [(b,ba)]) a \<sharp> t2b") --H
+apply(simp)
+apply(case_tac "b=a")
+apply(force)
+apply(force)
+--H
+apply(rule fresh_swap_left[THEN mp])
+apply(assumption)
+--G
+apply(rule ds_acabbc)
+apply(assumption)+
+--A
+apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx>swap [(a,b)] (swap [(b,ba)] t2b)")--I
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="swap [(a,b)] t2a" in spec)
+apply(drule conjunct2)+
+apply(drule_tac x="swap [(a, b)] (swap [(b, ba)] t2b)" in spec)
+apply(force simp add: swap_append[THEN sym])
+--I 
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="t2a" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(drule_tac x="swap [(b,ba)] t2b" in spec)
+apply(best)
+-- Abst.ab
+apply(simp)
+apply(rule equ_abst_ab)
+apply(assumption)
+apply(drule_tac "t1.1"="t2a" and "t2.1"="t2b" and a1="a" in l3_jud[THEN mp])
+apply(assumption)+
+apply(subgoal_tac "nabla\<turnstile>swap [(a, b)] t2a\<approx>swap [(a, b)] t2b") --A
+apply(best)
+--A
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use))
+apply(drule_tac x="t2a" in spec)
+apply(drule mp)
+apply(force dest!: equ_depth)
+apply(drule_tac x="t2b" in spec)
+apply(best)
+-- Abst
+apply(ind_cases "nabla \<turnstile> Abst a t2a \<approx> t3")
+apply(best)
+apply(best)
+-- Susp
+apply(ind_cases "nabla \<turnstile> Susp pi2 X \<approx> t3")
+apply(simp)
+apply(rule equ_susp)
+apply(rule ballI)
+apply(drule_tac "pi2.1"="pi2" in ds_trans[THEN mp])
+apply(force)
+-- Paar
+apply(ind_cases "nabla \<turnstile> Paar t2a s2 \<approx> t3")
+apply(simp)
+apply(rule equ_paar)
+apply(drule_tac x="depth t1a" in spec)
+apply(simp (no_asm_use) add: Suc_max_left)
+apply(best)
+apply(drule_tac x="depth s1" in spec)
+apply(simp (no_asm_use) add: Suc_max_right)
+apply(best)
+-- Func
+apply(ind_cases "nabla \<turnstile> Func F t2a \<approx> t3")
+apply(best)
+done
+
+lemma pi_right_equ_help: 
+      "\<forall>t. (n=depth t) \<longrightarrow> (\<forall>pi. nabla\<turnstile>t\<approx>swap pi t\<longrightarrow>(\<forall>a\<in> ds [] pi. nabla\<turnstile>a\<sharp>t))"
+apply(induct_tac n rule: nat_less_induct)
+apply(rule allI)+
+apply(rule impI)
+apply(rule allI)+
+apply(rule impI)
+apply(ind_cases "nabla \<turnstile> t \<approx> swap pi t")
+apply(simp_all)
+--Abst.ab
+apply(rule ballI)
+apply(case_tac "aa=a")
+apply(force)
+apply(rule fresh_abst_ab)
+apply(case_tac "aa=swapas (rev pi) a")
+apply(simp)
+apply(drule fresh_swap_left[THEN mp])
+apply(assumption)
+apply(drule_tac x="depth t1" in spec)
+apply(simp)
+apply(drule_tac x="t1" in spec)
+apply(simp add: swap_append[THEN sym])
+apply(drule_tac x="[(a, swapas pi a)] @ pi" in spec)
+apply(simp)
+apply(case_tac "aa=swapas pi a")
+apply(simp)
+apply(drule_tac x="swapas pi a" in bspec)
+apply(simp (no_asm) only: ds_def)
+apply(simp only: mem_Collect_eq)
+apply(rule conjI)
+apply(simp)
+apply(simp)
+apply(rule impI)
+apply(clarify)
+apply(drule sym)
+apply(drule swapas_rev_pi_a)
+apply(force)
+apply(assumption)
+apply(drule_tac x="aa" in bspec)
+apply(subgoal_tac "\<forall>A. aa\<in>A-{swapas pi a} \<and> aa\<noteq>swapas pi a \<longrightarrow>aa\<in>A")--A
+apply(drule_tac x="ds [] ((a,swapas pi a) # pi)" in spec)
+apply(simp add: ds_equality[THEN sym])
+--A
+apply(force)
+apply(assumption)+
+--Abst.aa
+apply(rule ballI)
+apply(case_tac "aa=a")
+apply(force)
+apply(best)
+--Unit
+apply(force)
+--Atom
+apply(force simp add: ds_def)
+--Susp
+apply(rule ballI)
+apply(rule fresh_susp)
+apply(drule_tac x="swapas (rev pi1) a" in bspec)
+apply(rule ds_cancel_pi_right[of _ _ "[]" _, simplified, THEN mp])
+apply(simp)
+apply(assumption)
+--Paar
+apply(rule ballI)
+apply(rule fresh_paar)
+apply(drule_tac x="depth t1" in spec)
+apply(force simp add: Suc_max_left)
+apply(drule_tac x="depth s1" in spec)
+apply(force simp add: Suc_max_right)
+--Func
+apply(best)
+done
+
+end 

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Substs.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,363 @@
+
+
+theory Substs = Main + Terms + PreEqu + Equ:
+
+(* substitutions *)
+
+types substs = "(string \<times> trm)list"
+
+consts 
+  look_up :: "string \<Rightarrow> substs \<Rightarrow> trm"
+primrec 
+  "look_up X []     = Susp [] X"
+  "look_up X (x#xs) = (if (X = fst x) then (snd x) else look_up X xs)"
+
+consts 
+  subst :: "substs \<Rightarrow> trm \<Rightarrow> trm"
+primrec
+  subst_unit: "subst s (Unit)          = Unit"
+  subst_susp: "subst s (Susp pi X)     = swap pi (look_up X s)"
+  subst_atom: "subst s (Atom a)        = Atom a" 
+  subst_abst: "subst s (Abst a t)      = Abst a (subst s t)"
+  subst_paar: "subst s (Paar t1 t2)    = Paar (subst s t1) (subst s t2)"
+  subst_func: "subst s (Func F t)      = Func F (subst s t)"
+
+declare subst_susp [simp del]
+
+(* composition of substitutions (adapted from Martin Coen) *)
+
+consts
+  alist_rec :: "substs \<Rightarrow> substs \<Rightarrow> (string\<Rightarrow>trm\<Rightarrow>substs\<Rightarrow>substs\<Rightarrow>substs) \<Rightarrow> substs"
+
+primrec
+  "alist_rec [] c d = c"
+  "alist_rec (p#al) c d = d (fst p) (snd p) al (alist_rec al c d)"
+
+consts 
+  "\<bullet>"   ::  "substs \<Rightarrow> substs \<Rightarrow> substs" (infixl 81)
+defs
+  comp_def: "s1 \<bullet> s2 \<equiv> alist_rec s2 s1 (\<lambda> x y xs g. (x,subst s1 y)#g)"
+
+(* domain of substitutions *)
+
+consts  
+  domn :: "(trm \<Rightarrow> trm) \<Rightarrow> string set"
+defs 
+  domn_def: "domn s \<equiv> {X. (s (Susp [] X)) \<noteq> (Susp [] X)}" 
+
+(* substitutions extending freshness environments *)
+
+consts 
+  ext_subst :: "fresh_envs \<Rightarrow> (trm \<Rightarrow> trm) \<Rightarrow> fresh_envs \<Rightarrow> bool" (" _ \<Turnstile> _ _ " [80,80,80] 80)
+defs 
+  ext_subst_def: "nabla1 \<Turnstile> s (nabla2) \<equiv> (\<forall>(a,X)\<in>nabla2. nabla1\<turnstile>a\<sharp> s (Susp [] X))"
+
+(* alpah-equality for substitutions *)
+
+consts 
+  subst_equ :: "fresh_envs \<Rightarrow> (trm\<Rightarrow>trm) \<Rightarrow> (trm\<Rightarrow>trm) \<Rightarrow> bool" (" _ \<Turnstile> _ \<approx> _" [80,80,80] 80)
+
+defs 
+  subst_equ_def: 
+  "nabla\<Turnstile> s1 \<approx> s2 \<equiv>  \<forall>X\<in>(domn s1\<union>domn s2). (nabla \<turnstile> s1 (Susp [] X) \<approx> s2 (Susp [] X))"
+
+lemma not_in_domn: "X\<notin>(domn s)\<Longrightarrow> (s (Susp [] X)) = (Susp [] X)"
+apply(auto simp only: domn_def)
+done
+
+lemma subst_swap_comm: "subst s (swap pi t) = swap pi (subst s t)"
+apply(induct_tac t)
+apply(auto simp add: swap_append subst_susp)
+done
+
+lemma subst_not_occurs: "\<not>(occurs X t) \<longrightarrow> subst [(X,t2)] t = t"
+apply(induct t)
+apply(auto simp add: subst_susp)
+done
+
+lemma [simp]: "subst [] t = t"
+apply(induct_tac t, auto simp add: subst_susp)
+done
+
+lemma [simp]: "subst (s\<bullet>[]) = subst s"
+apply(auto simp add: comp_def)
+done
+
+lemma [simp]: "subst ([]\<bullet>s) = subst s"
+apply(rule ext)
+apply(induct_tac x)
+apply(auto)
+apply(induct_tac s rule: alist_rec.induct)
+apply(auto simp add: comp_def subst_susp)
+done
+
+lemma subst_comp_expand: "subst (s1\<bullet>s2) t = subst s1 (subst s2 t)"
+apply(induct_tac t)
+apply(auto)
+apply(induct_tac s2 rule: alist_rec.induct)
+apply(auto simp add: comp_def subst_susp subst_swap_comm)
+done
+
+lemma subst_assoc: "subst (s1\<bullet>(s2\<bullet>s3))=subst ((s1\<bullet>s2)\<bullet>s3)"
+apply(rule ext)
+apply(induct_tac x)
+apply(auto)
+apply(simp add: subst_comp_expand)
+done
+
+lemma fresh_subst: "nabla1\<turnstile>a\<sharp>t\<Longrightarrow> nabla2\<Turnstile>(subst s) nabla1 \<longrightarrow> nabla2\<turnstile>a\<sharp>subst s t"
+apply(erule fresh.induct)
+apply(auto)
+--Susp
+apply(simp add: ext_subst_def subst_susp)
+apply(drule_tac x="(swapas (rev pi) a, X)" in bspec)
+apply(assumption)
+apply(simp)
+apply(rule fresh_swap_right[THEN mp])
+apply(assumption)
+done
+
+lemma equ_subst: "nabla1\<turnstile>t1\<approx>t2\<Longrightarrow> nabla2\<Turnstile> (subst s) nabla1 \<longrightarrow> nabla2\<turnstile>(subst s t1)\<approx>(subst s t2)"
+apply(erule equ.induct)
+apply(auto)
+apply(rule_tac "nabla1.1"="nabla" in fresh_subst[THEN mp])
+apply(assumption)+
+apply(simp add: subst_swap_comm)
+-- Susp
+apply(simp only: subst_susp)
+apply(rule equ_pi1_pi2_add[THEN mp])
+apply(simp only: ext_subst_def subst_susp)
+apply(force)
+done
+
+lemma unif_1: 
+  "\<lbrakk>nabla\<turnstile>subst s (Susp pi X)\<approx>subst s t\<rbrakk>\<Longrightarrow> nabla\<Turnstile> subst (s\<bullet>[(X,swap (rev pi) t)])\<approx>subst s"
+apply(simp only: subst_equ_def)
+apply(case_tac "pi=[]")
+apply(simp add: subst_susp comp_def)
+apply(force intro: equ_sym equ_refl)
+apply(subgoal_tac "domn (subst (s\<bullet>[(X,swap (rev pi) t)]))=domn(subst s)\<union>{X}")--A
+apply(simp)
+apply(rule conjI)
+apply(simp add: subst_comp_expand)
+apply(simp add: subst_susp)
+apply(simp only: subst_swap_comm)
+apply(simp only: equ_pi_to_left)
+apply(rule equ_sym)
+apply(assumption)
+apply(rule ballI)
+apply(simp only: subst_comp_expand)
+apply(simp add: subst_susp)
+apply(force intro: equ_sym equ_refl simp add: subst_swap_comm equ_pi_to_left)
+--A
+apply(simp only: domn_def)
+apply(auto)
+apply(simp add: subst_comp_expand)
+apply(simp add: subst_susp subst_swap_comm)
+apply(simp only: subst_comp_expand)
+apply(simp add: subst_susp subst_swap_comm)
+apply(drule swap_empty[THEN mp])
+apply(simp)
+apply(simp only: subst_comp_expand)
+apply(simp only: subst_susp)
+apply(auto)
+apply(case_tac "x=X")
+apply(simp)
+apply(simp add: subst_swap_comm)
+apply(drule swap_empty[THEN mp])
+apply(simp)
+apply(simp add: subst_susp)
+done
+
+lemma subst_equ_a:
+"\<lbrakk>nabla\<Turnstile>(subst s1) \<approx> (subst s2)\<rbrakk>\<Longrightarrow> \<forall>t2. nabla\<turnstile>(subst s2 t1)\<approx>t2 \<longrightarrow> nabla\<turnstile>(subst s1 t1)\<approx>t2"
+apply(rule allI)
+apply(induct t1)
+--Abst.ab
+apply(rule impI)
+apply(simp)
+apply(ind_cases "nabla \<turnstile> Abst list (subst s1 trm) \<approx> t2")
+apply(best)
+--Abst.aa
+apply(force)
+--Susp
+apply(rule impI)
+apply(simp only: subst_equ_def)
+apply(case_tac "list2\<in>domn (subst s1) \<union> domn (subst s2)")
+apply(drule_tac x="list2" in bspec)
+apply(assumption)
+apply(simp)
+apply(subgoal_tac "nabla \<turnstile> subst s2 (Susp [] list2) \<approx> swap (rev list1) t2")--A
+apply(drule_tac "t1.0"="subst s1 (Susp [] list2)" and
+                "t2.0"="subst s2 (Susp [] list2)" and 
+                "t3.0"="swap (rev list1) t2" in equ_trans)
+apply(assumption)
+apply(simp only: equ_pi_to_right)
+apply(simp add: subst_swap_comm[THEN sym])
+--A
+apply(simp only: equ_pi_to_right)
+apply(simp add: subst_swap_comm[THEN sym])
+apply(simp)
+apply(erule conjE)
+apply(drule not_in_domn)+
+apply(subgoal_tac "subst s1 (Susp list1 list2)=swap list1 (subst s1 (Susp [] list2))")--B
+apply(subgoal_tac "subst s2 (Susp list1 list2)=swap list1 (subst s2 (Susp [] list2))")--C
+apply(simp)
+--BC
+apply(simp (no_asm) add: subst_swap_comm[THEN sym])
+apply(simp (no_asm) add: subst_swap_comm[THEN sym])
+--Unit
+apply(force)
+--Atom
+apply(force)
+--Paar
+apply(rule impI)
+apply(simp)
+apply(ind_cases "nabla \<turnstile> Paar (subst sigma1 trm1) (subst sigma1 trm2) \<approx> t2")
+apply(best)
+--Func
+apply(rule impI)
+apply(simp)
+apply(ind_cases "nabla \<turnstile> Func list (subst sigma1 trm) \<approx> t2")
+apply(best)
+done
+
+lemma unif_2a: "\<lbrakk>nabla\<Turnstile>subst s1\<approx>subst s2\<rbrakk>\<Longrightarrow> 
+                (nabla\<turnstile>subst s2 t1 \<approx> subst s2 t2)\<longrightarrow>(nabla\<turnstile>subst s1 t1 \<approx> subst s1 t2)"
+apply(rule impI)
+apply(frule_tac "t1.0"="t1" in subst_equ_a)
+apply(drule_tac x="subst s2 t2" in spec)
+apply(simp)
+apply(drule equ_sym)
+apply(drule equ_sym) 
+apply(frule_tac "t1.0"="t2" in subst_equ_a)
+apply(drule_tac x="subst s1 t1" in spec)
+apply(rule equ_sym)
+apply(simp)
+done
+
+
+lemma unif_2b: "\<lbrakk>nabla\<Turnstile>subst s1\<approx> subst s2\<rbrakk>\<Longrightarrow>nabla\<turnstile>a\<sharp> subst s2 t\<longrightarrow>nabla\<turnstile>a\<sharp>subst s1 t"
+apply(induct t)
+--Abst
+apply(simp)
+apply(rule impI)
+apply(case_tac "a=list")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+--Susp
+apply(rule impI)
+apply(subgoal_tac "subst s1 (Susp list1 list2)= swap list1 (subst s1 (Susp [] list2))")--A
+apply(subgoal_tac "subst s2 (Susp list1 list2)= swap list1 (subst s2 (Susp [] list2))")--B
+apply(simp add: subst_equ_def)
+apply(case_tac "list2\<in>domn (subst s1) \<union> domn (subst s2)")
+apply(drule_tac x="list2" in bspec)
+apply(assumption)
+apply(rule fresh_swap_right[THEN mp])
+apply(rule_tac "t1.1"=" subst s2 (Susp [] list2)" in l3_jud[THEN mp])
+apply(rule equ_sym)
+apply(simp)
+apply(rule fresh_swap_left[THEN mp])
+apply(simp)
+apply(simp)
+apply(erule conjE)
+apply(drule not_in_domn)+
+apply(simp add: subst_swap_comm)
+--AB
+apply(simp (no_asm) add: subst_swap_comm[THEN sym])
+apply(simp (no_asm) add: subst_swap_comm[THEN sym])
+--Unit
+apply(force)
+--Atom
+apply(force)
+--Paar
+apply(force dest!: fresh_paar_elim)
+--Func
+apply(force dest!:  fresh_func_elim)
+done
+
+lemma subst_equ_to_trm: "nabla\<Turnstile>subst s1 \<approx> subst s2\<Longrightarrow> nabla\<turnstile>subst s1 t\<approx>subst s2 t"
+apply(induct_tac t)
+apply(auto simp add: subst_equ_def)
+apply(case_tac "list2\<in>domn (subst s1) \<union> domn (subst s2)")
+apply(simp only: subst_susp)
+apply(simp only: equ_pi_to_left[THEN sym])
+apply(simp only: equ_involutive_left)
+apply(simp)
+apply(erule conjE)
+apply(drule not_in_domn)+
+apply(subgoal_tac "subst s1 (Susp list1 list2)=swap list1 (subst s1 (Susp [] list2))")--A
+apply(subgoal_tac "subst s2 (Susp list1 list2)=swap list1 (subst s2 (Susp [] list2))")--B
+apply(simp)
+apply(rule equ_refl)
+apply(simp (no_asm_use) add: subst_swap_comm[THEN sym])+
+done
+
+lemma subst_cancel_right: "\<lbrakk>nabla\<Turnstile>(subst s1)\<approx>(subst s2)\<rbrakk>\<Longrightarrow>nabla\<Turnstile>(subst (s1\<bullet>s))\<approx>(subst (s2\<bullet>s))" 
+apply(simp (no_asm) only: subst_equ_def)
+apply(rule ballI)
+apply(simp only: subst_comp_expand)
+apply(rule subst_equ_to_trm)
+apply(assumption)
+done
+
+lemma subst_trans: "\<lbrakk>nabla\<Turnstile>subst s1\<approx>subst s2; nabla\<Turnstile>subst s2\<approx>subst s3\<rbrakk>\<Longrightarrow>nabla\<Turnstile>subst s1\<approx>subst s3"
+apply(simp add: subst_equ_def)
+apply(auto)
+apply(case_tac "X \<in>domn (subst s2)")
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(rule_tac "t2.0"="subst s2 (Susp [] X)" in equ_trans)
+apply(assumption)+
+apply(case_tac "X \<in>domn (subst s3)")
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(rule_tac "t2.0"="subst s2 (Susp [] X)" in equ_trans)
+apply(assumption)+
+apply(drule not_in_domn)+
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(simp)
+apply(case_tac "X \<in>domn (subst s1)")
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(rule_tac "t2.0"="subst s2 (Susp [] X)" in equ_trans)
+apply(assumption)+
+apply(case_tac "X \<in>domn (subst s2)")
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(rule_tac "t2.0"="subst s2 (Susp [] X)" in equ_trans)
+apply(assumption)+
+apply(drule not_in_domn)+
+apply(simp)
+apply(rotate_tac 1)
+apply(drule_tac x="X" in bspec)
+apply(force)
+apply(simp)
+done
+
+(* if occurs holds, then one subterm is equal to (subst s (Susp pi X)) *)
+
+lemma occurs_sub_trm_equ: 
+  "occurs X t1 \<longrightarrow> (\<exists>t2\<in>sub_trms (subst s t1). (\<exists>pi.(nabla\<turnstile>subst s (Susp pi1 X)\<approx>(swap pi t2))))" 
+apply(induct_tac t1)
+apply(auto)
+apply(simp only: subst_susp)
+apply(rule_tac x="swap list1 (look_up X s)" in bexI)
+apply(rule_tac x="pi1@(rev list1)" in exI)
+apply(simp add: swap_append)
+apply(simp add: equ_pi_to_left[THEN sym])
+apply(simp only: equ_involutive_left)
+apply(rule equ_refl)
+apply(rule t_sub_trms_t)
+done
+
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Swap.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,55 @@
+
+theory Swap = Main:
+
+consts 
+  swapa   :: "(string \<times> string) \<Rightarrow> string \<Rightarrow> string"
+  swapas  :: "(string \<times> string)list \<Rightarrow> string \<Rightarrow> string"
+primrec
+  "swapa (b,c) a = (if b = a then c else (if c = a then b else a))"
+
+primrec
+  "swapas []     a = a"
+  "swapas (x#pi) a = swapa x (swapas pi a)"
+
+lemma [simp]: 
+  "swapas [(a,a)] b=b"
+apply(auto)
+done
+
+lemma swapas_append: 
+  "swapas (pi1@pi2) a = swapas pi1 (swapas pi2 a)"
+apply(induct_tac pi1, auto)
+done 
+
+lemma [simp]: 
+  "swapas (rev pi) (swapas pi a) = a"
+apply(induct_tac pi)
+apply(simp_all add: swapas_append, auto)
+done
+
+lemma swapas_rev_pi_a: 
+  "swapas pi a = b \<Longrightarrow> swapas (rev pi) b = a"
+apply(auto)
+done
+
+lemma [simp]: 
+  "swapas pi (swapas (rev pi) a) = a"
+apply(rule swapas_rev_pi_a[of "(rev pi)" _ _,simplified])
+apply(simp)
+done
+
+lemma swapas_rev_pi_b: 
+  "swapas (rev pi) a = b \<Longrightarrow> swapas pi b = a"
+apply(auto)
+done
+
+lemma swapas_comm: 
+  "(swapas (pi@[(a,b)]) c) = (swapas ([(swapas pi a,swapas pi b)]@pi) c)"
+apply(auto)
+apply(simp_all only: swapas_append)
+apply(auto)
+apply(drule swapas_rev_pi_a, simp)
+apply(drule swapas_rev_pi_a, simp)
+done
+
+end
\ No newline at end of file

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Termination.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,205 @@
+
+
+theory Termination = Main + Terms + Fresh + Equ + Substs + Mgu:
+
+(* set of variables *)
+
+consts 
+  vars_trm    :: "trm \<Rightarrow> string set"
+  vars_eprobs :: "eprobs \<Rightarrow> (string set)"
+primrec
+  "vars_trm (Unit)       = {}"
+  "vars_trm (Atom a)     = {}"
+  "vars_trm (Susp pi X)  = {X}"
+  "vars_trm (Paar t1 t2) = (vars_trm t1)\<union>(vars_trm t2)"
+  "vars_trm (Abst a t)   = vars_trm t"
+  "vars_trm (Func F t)   = vars_trm t"
+primrec
+  "vars_eprobs [] = {}"
+  "vars_eprobs (x#xs) = (vars_trm (snd x))\<union>(vars_trm (fst x))\<union>(vars_eprobs xs)"
+
+lemma[simp]: "vars_trm (swap pi t) = vars_trm t"
+apply(induct_tac t)
+apply(auto)
+done
+
+consts 
+  size_trm    :: "trm \<Rightarrow> nat"
+  size_fprobs :: "fprobs \<Rightarrow> nat"
+  size_eprobs :: "eprobs \<Rightarrow> nat"
+  size_probs  :: "probs \<Rightarrow> nat"
+primrec
+  "size_trm (Unit)      = 1"
+  "size_trm (Atom a)    = 1"
+  "size_trm (Susp pi X) = 1"
+  "size_trm (Abst a t)  = 1 + size_trm t"
+  "size_trm (Func F t)  = 1 + size_trm t"
+  "size_trm (Paar t t') = 1 + (size_trm t) + (size_trm t')"
+primrec
+  "size_fprobs [] = 0"
+  "size_fprobs (x#xs) = (size_trm (snd x))+(size_fprobs xs)"
+primrec
+  "size_eprobs [] = 0"
+  "size_eprobs (x#xs) = (size_trm (fst x))+(size_trm (snd x))+(size_eprobs xs)"
+
+lemma[simp]: "size_trm (swap pi t) = size_trm t"
+apply(induct_tac t)
+apply(auto)
+done
+
+syntax        
+  "_measure_relation" :: "(nat\<times>nat\<times>nat) \<Rightarrow> (nat\<times>nat\<times>nat) \<Rightarrow> bool" ("_ \<lless> _" [80,80] 80)
+translations
+  "n1 \<lless> n2"  \<rightleftharpoons> "(n1,n2) \<in> (less_than<*lex*>less_than<*lex*>less_than)"
+
+consts 
+  rank :: "probs \<Rightarrow> (nat\<times>nat\<times>nat)"
+primrec
+  "rank (eprobs,fprobs) = (card (vars_eprobs eprobs),size_eprobs eprobs,size_fprobs fprobs)" 
+
+lemma[simp]: "finite (vars_trm t)"
+apply(induct t)
+apply(auto)
+done
+
+lemma[simp]: "finite (vars_eprobs P)"
+apply(induct_tac P)
+apply(auto)
+done
+
+lemma union_comm: "A\<union>(B\<union>C)=(A\<union>B)\<union>C"
+apply(auto)
+done
+
+lemma card_union: "\<lbrakk>finite A; finite B\<rbrakk>\<Longrightarrow>(card B < card (A\<union>B)) \<or> (card B = card (A\<union>B))"
+apply(auto)
+apply(rule psubset_card_mono)
+apply(auto)
+done
+
+lemma card_insert: "\<lbrakk>finite B\<rbrakk>\<Longrightarrow>(card B < card (insert X B)) \<or> (card B = card (insert X B))"
+apply(auto)
+apply(rule psubset_card_mono)
+apply(auto)
+done
+
+lemma subseteq_card: "\<lbrakk>A\<subseteq>B; finite B\<rbrakk>\<Longrightarrow>(card A \<le> card B)"
+apply(drule_tac A="A" in card_mono)
+apply(auto simp add: le_eq_less_or_eq)
+done
+
+lemma not_occurs_trm: "\<not>occurs X t \<longrightarrow> X\<notin> vars_trm t"
+apply(induct_tac t)
+apply(auto)
+done
+
+lemma not_occurs_subst: "\<not>occurs X t1\<longrightarrow> X\<notin> vars_trm (subst [(X,swap pi2 t1)] t2)" 
+apply(induct_tac t2)
+apply(auto simp add: subst_susp not_occurs_trm)
+done
+
+lemma not_occurs_list: "\<not>occurs X t \<longrightarrow>
+  X\<notin> vars_eprobs (apply_subst_eprobs [(X, swap pi t)] xs)"
+apply(induct_tac xs)
+apply(simp)
+apply(case_tac a)
+apply(auto simp add: not_occurs_subst)
+done
+
+lemma vars_equ: "\<not>occurs X t1 \<and> occurs X t2\<longrightarrow> 
+  vars_trm (subst [(X, swap pi t1)] t2)=(vars_trm t1\<union>vars_trm t2)-{X}"
+apply(induct_tac t2)
+apply(force)
+apply(case_tac "X=list2")
+apply(simp add: subst_susp not_occurs_trm)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(rule conjI)
+apply(case_tac "occurs X trm2")
+apply(force)
+apply(force dest: not_occurs_trm[THEN mp] simp add: subst_not_occurs)
+apply(force dest: not_occurs_trm[THEN mp] simp add: subst_not_occurs)
+apply(force)
+done
+
+lemma vars_subseteq: "\<not>occurs X t \<longrightarrow>
+  vars_eprobs (apply_subst_eprobs [(X, swap pi t)] xs) \<subseteq> (vars_trm t \<union> vars_eprobs xs)"
+apply(induct_tac xs)
+apply(simp)
+apply(rule impI)
+apply(simp)
+apply(case_tac "occurs X (fst a)")
+apply(case_tac "occurs X (snd a)")
+apply(simp add: vars_equ[THEN mp])
+apply(force)
+apply(simp add: subst_not_occurs)
+apply(force simp add: vars_equ)
+apply(case_tac "occurs X (snd a)")
+apply(simp add: vars_equ[THEN mp])
+apply(force simp add: subst_not_occurs)
+apply(force simp add: subst_not_occurs)
+done
+
+lemma vars_decrease: 
+  "\<not>occurs X t\<longrightarrow> card (vars_eprobs (apply_subst_eprobs [(X, swap pi t)] xs))
+                 <card (insert X (vars_trm t \<union> vars_eprobs xs))"
+apply(rule impI)
+apply(case_tac "X\<in>(vars_trm t \<union> vars_eprobs xs)")
+(* first case *)
+apply(subgoal_tac "insert X (vars_trm t \<union> vars_eprobs xs) = (vars_trm t \<union> vars_eprobs xs)") (*A*)
+apply(simp)
+apply(frule_tac pi1="pi" and xs1="xs" in vars_subseteq[THEN mp])
+apply(frule_tac pi1="pi" and xs1="xs" in not_occurs_list[THEN mp])
+apply(subgoal_tac "vars_eprobs (apply_subst_eprobs [(X, swap pi t)] xs)
+                   \<subset>  vars_trm t \<union> vars_eprobs xs") (* B *)
+apply(simp add: psubset_card_mono)
+(* B *)
+apply(force)
+(* A *)
+apply(force)
+(* second case *)
+apply(subgoal_tac "finite (vars_trm t \<union> vars_eprobs xs)")
+apply(drule_tac x="X" in card_insert_disjoint)
+apply(simp)
+apply(simp)
+apply(frule_tac pi1="pi" and xs1="xs" in vars_subseteq[THEN mp])
+apply(auto dest: subseteq_card)
+done
+
+lemma rank_cred: "\<lbrakk>P1\<turnstile>(nabla)\<rightarrow>P2\<rbrakk>\<Longrightarrow>(rank P2) \<lless> (rank P1)"
+apply(ind_cases "P1 \<turnstile> nabla \<rightarrow> P2")
+apply(simp_all add: lex_prod_def)
+done
+
+lemma rank_sred: "\<lbrakk>P1\<turnstile>(s)\<leadsto>P2\<rbrakk>\<Longrightarrow>(rank P2) \<lless> (rank P1)"
+apply(ind_cases "P1 \<turnstile> s \<leadsto> P2")
+apply(simp_all add: lex_prod_def union_comm)
+apply(subgoal_tac 
+  "vars_trm s1\<union>vars_trm t1\<union>vars_trm s2\<union>vars_trm t2\<union>vars_eprobs xs = 
+   vars_trm s1\<union>vars_trm s2\<union>vars_trm t1\<union>vars_trm t2\<union>vars_eprobs xs") (*A*)
+apply(simp)
+(* A *)
+apply(force)
+(* Susp-Susp case *)
+apply(simp add: card_insert)
+(* variable elimination cases *)
+apply(simp_all add: apply_subst_def vars_decrease)
+done
+
+
+lemma rank_trans: "\<lbrakk>rank P1 \<lless> rank P2; rank P2 \<lless> rank P3\<rbrakk>\<Longrightarrow>rank P1 \<lless> rank P3"
+apply(simp add: lex_prod_def)
+apply(auto)
+done
+
+(* all reduction are well-founded under \<lless> *)
+
+lemma rank_red_plus: "\<lbrakk>P1\<Turnstile>(s,nabla)\<Rightarrow>P2\<rbrakk>\<Longrightarrow>(rank P2) \<lless> (rank P1)"
+apply(erule red_plus.induct)
+apply(auto dest: rank_sred rank_cred rank_trans)
+done
+
+end
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Terms.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,142 @@
+
+theory Terms = Main + Swap:
+
+(* terms *)
+
+datatype trm = Abst  string  trm   
+             | Susp  "(string \<times> string)list" string
+             | Unit                                   
+             | Atom  string                           
+             | Paar  trm trm                          
+             | Func string trm                       
+
+(* swapping operation on terms *)
+
+consts swap  :: "(string \<times> string)list \<Rightarrow> trm \<Rightarrow> trm"
+primrec
+"swap  pi (Unit)        = Unit"
+"swap  pi (Atom a)      = Atom (swapas pi a)"
+"swap  pi (Susp pi' X)  = Susp (pi@pi') X"
+"swap  pi (Abst a t)    = Abst (swapas pi a) (swap pi t)"
+"swap  pi (Paar t1 t2)  = Paar (swap pi t1) (swap pi t2)"
+"swap  pi (Func F t)    = Func F (swap pi t)"
+
+lemma [simp]: "swap [] t = t"
+apply(induct_tac t)
+apply(auto)
+done
+
+lemma swap_append[rule_format]: "\<forall>pi1 pi2. swap (pi1@pi2) t = swap pi1 (swap pi2 t)"
+apply(induct_tac t, auto)
+apply(induct_tac pi1,auto)
+apply(induct_tac pi1,auto)
+done
+
+lemma swap_empty: "swap pi t=Susp [] X\<longrightarrow> pi=[]"
+apply(induct_tac t)
+apply(auto)
+done
+
+(* depyh of terms *)
+
+consts depth :: "trm \<Rightarrow> nat"
+primrec
+"depth (Unit)      = 0"
+"depth (Atom a)    = 0"
+"depth (Susp pi X) = 0"
+"depth (Abst a t)  = 1 + depth t"
+"depth (Func F t)  = 1 + depth t"
+"depth (Paar t t') = 1 + (max (depth t) (depth t'))" 
+
+lemma 
+  Suc_max_left:  "Suc(max x y) = z \<longrightarrow> x < z" and
+  Suc_max_right: "Suc(max x y) = z \<longrightarrow> y < z"
+apply(auto)
+apply(simp_all only: max_Suc_Suc[THEN sym] less_max_iff_disj)
+apply(simp_all)
+done
+
+lemma [simp]: "depth (swap pi t) = depth t" 
+apply(induct_tac t,auto)
+done
+
+(* occurs predicate and variables in terms *)
+
+consts occurs :: "string \<Rightarrow> trm \<Rightarrow> bool"
+primrec 
+"occurs X (Unit)       = False"
+"occurs X (Atom a)     = False"
+"occurs X (Susp pi Y)  = (if X=Y then True else False)"
+"occurs X (Abst a t)   = occurs X t"
+"occurs X (Paar t1 t2) = (if (occurs X t1) then True else (occurs X t2))"
+"occurs X (Func F t)   = occurs X t"
+
+consts vars_trm :: "trm \<Rightarrow> string set"
+primrec
+"vars_trm (Unit)       = {}"
+"vars_trm (Atom a)     = {}"
+"vars_trm (Susp pi X)  = {X}"
+"vars_trm (Paar t1 t2) = (vars_trm t1)\<union>(vars_trm t2)"
+"vars_trm (Abst a t)   = vars_trm t"
+"vars_trm (Func F t)   = vars_trm t"
+
+
+(* subterms and proper subterms *)
+
+consts sub_trms :: "trm \<Rightarrow> trm set"
+primrec 
+"sub_trms (Unit)       = {Unit}"
+"sub_trms (Atom a)     = {Atom a}"
+"sub_trms (Susp pi Y)  = {Susp pi Y}"
+"sub_trms (Abst a t)   = {Abst a t}\<union>sub_trms t"
+"sub_trms (Paar t1 t2) = {Paar t1 t2}\<union>sub_trms t1 \<union>sub_trms t2"
+"sub_trms (Func F t)   = {Func F t}\<union>sub_trms t"
+
+consts psub_trms :: "trm \<Rightarrow> trm set"
+primrec
+"psub_trms (Unit)       = {}"
+"psub_trms (Atom a)     = {}"
+"psub_trms (Susp pi X)  = {}"
+"psub_trms (Paar t1 t2) = sub_trms t1 \<union> sub_trms t2"
+"psub_trms (Abst a t)   = sub_trms t"
+"psub_trms (Func F t)   = sub_trms t"
+
+lemma psub_sub_trms[rule_format]: 
+  "\<forall>t1 . t1 \<in> psub_trms t2 \<longrightarrow>  t1 \<in> sub_trms t2"
+apply(induct t2)
+apply(auto)
+done
+
+lemma t_sub_trms_t: 
+  "t\<in> sub_trms t"
+apply(induct t)
+apply(auto)
+done
+
+lemma abst_psub_trms[rule_format]: 
+  "\<forall> t1. Abst a t1 \<in> sub_trms t2 \<longrightarrow> t1 \<in> psub_trms t2"
+apply(induct_tac t2)
+apply(auto)
+apply(simp only: t_sub_trms_t)
+apply(best dest!: psub_sub_trms)+
+done
+
+lemma func_psub_trms[rule_format]: 
+  "\<forall> t1. Func F t1 \<in> sub_trms t2 \<longrightarrow> t1 \<in> psub_trms t2"
+apply(induct_tac t2)
+apply(auto)
+apply(best dest!: psub_sub_trms)+
+apply(simp add: t_sub_trms_t)
+apply(best dest!: psub_sub_trms)
+done
+
+lemma paar_psub_trms[rule_format]: 
+  "\<forall> t1. Paar t1 t2 \<in> sub_trms t3\<longrightarrow>(t1 \<in> psub_trms t3 \<and> t2 \<in> psub_trms t3)"
+apply(induct_tac t3)
+apply(auto)
+apply(best dest!: psub_sub_trms)+
+apply(simp add: t_sub_trms_t)+
+apply(best dest!: psub_sub_trms)+
+done
+
+end
\ No newline at end of file

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Unification.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,625 @@
+
+
+theory Unification = Main + Terms + Fresh + Equ + Substs + Mgu:
+
+(* problems to which no reduction applies *)
+
+consts stuck :: "problem_type set"
+defs
+  stuck_def: "stuck \<equiv> { P1. \<not>(\<exists>P2 nabla s. P1 \<Turnstile>(nabla,s)\<Rightarrow>P2)}"
+
+(* all problems which are stuck and have no unifier *)
+
+consts fail :: "problem_type set"
+inductive fail
+intros
+[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Abst a t#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Func F t#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Paar t1 t2#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Abst a t\<approx>?Susp pi X#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Func F t\<approx>?Susp pi X#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Paar t1 t2\<approx>?Susp pi X#xs,ys)\<in>fail"
+[intro!,dest!]: "([],a\<sharp>? Atom a#ys)\<in>fail"
+[intro!]: "a\<noteq>b\<Longrightarrow>(Atom a\<approx>? Atom b#xs,ys)\<in>fail"
+[intro!,dest!]: "(Abst a t\<approx>?Unit#xs,ys)\<in>fail"
+[intro!,dest!]: "(Unit\<approx>?Abst a t#xs,ys)\<in>fail"
+[intro!,dest!]: "(Abst a t\<approx>?Atom b#xs,ys)\<in>fail"
+[intro!,dest!]: "(Atom b\<approx>?Abst a t#xs,ys)\<in>fail"
+[intro!,dest!]: "(Abst a t\<approx>?Paar t1 t2#xs,ys)\<in>fail"
+[intro!,dest!]: "(Paar t1 t2\<approx>?Abst a t#xs,ys)\<in>fail"
+[intro!,dest!]: "(Abst a t\<approx>?Func F t1#xs,ys)\<in>fail"
+[intro!,dest!]: "(Func F t1\<approx>?Abst a t#xs,ys)\<in>fail"
+[intro!,dest!]: "(Unit\<approx>?Atom b#xs,ys)\<in>fail"
+[intro!,dest!]: "(Atom b\<approx>?Unit#xs,ys)\<in>fail"
+[intro!,dest!]: "(Unit\<approx>?Paar t1 t2#xs,ys)\<in>fail"
+[intro!,dest!]: "(Paar t1 t2\<approx>?Unit#xs,ys)\<in>fail"
+[intro!,dest!]: "(Unit\<approx>?Func F t1#xs,ys)\<in>fail"
+[intro!,dest!]: "(Func F t1\<approx>?Unit#xs,ys)\<in>fail"
+[intro!,dest!]: "(Atom a\<approx>?Paar t1 t2#xs,ys)\<in>fail"
+[intro!,dest!]: "(Paar t1 t2\<approx>?Atom a#xs,ys)\<in>fail"
+[intro!,dest!]: "(Atom a\<approx>?Func F t1#xs,ys)\<in>fail"
+[intro!,dest!]: "(Func F t1\<approx>?Atom a#xs,ys)\<in>fail"
+[intro!,dest!]: "(Func F t\<approx>?Paar t1 t2#xs,ys)\<in>fail"
+[intro!,dest!]: "(Paar t1 t2\<approx>?Func F t#xs,ys)\<in>fail"
+[intro!]: "\<lbrakk>F1\<noteq>F2\<rbrakk>\<Longrightarrow>(Func F1 t1\<approx>?Func F2 t2#xs,ys)\<in>fail"
+
+(* the results that are interesting are the stuck ones *)
+
+consts 
+  results :: "problem_type \<Rightarrow> problem_type set"
+defs
+  results_def: 
+  "results P1 \<equiv> if P1\<in>stuck then {P1} else {P2. \<exists>nabla s. P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<and> P2\<in>stuck}"
+
+(* a "failed" problem has no unifier *)
+
+lemma fail_then_empty: 
+  "(P1\<in>fail) \<Longrightarrow> (U P1={})"
+apply(erule fail.cases)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t1\<in>psub_trms (Abst a (subst b t)).\<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t1)")
+apply(simp)
+apply(drule equ_sym)
+apply(clarify)
+apply(drule_tac "t1.0"="Abst a (subst b t)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)")
+apply(simp)
+apply(drule equ_sym)
+apply(clarify)
+apply(drule_tac "t1.0"="Func F (subst b t)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(erule disjE)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
+                                \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
+apply(simp)
+apply(drule equ_sym)
+apply(clarify)
+apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2a" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
+                                \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
+apply(simp)
+apply(drule equ_sym)
+apply(clarify)
+apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2a" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t3\<in>psub_trms (Abst a (subst b t)).
+                                \<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t3)")
+apply(simp)
+apply(clarify)
+apply(drule_tac "t1.0"="Abst a (subst b t)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)")
+apply(simp)
+apply(clarify)
+apply(drule_tac "t1.0"="Func F (subst b t)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def)
+apply(rule allI)+
+apply(rule impI)
+apply(erule disjE)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
+                                \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
+apply(simp)
+apply(clarify)
+apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2a" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
+apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
+                                \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
+apply(simp)
+apply(clarify)
+apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and 
+                "t2.0"="subst b (Susp pi X)" and 
+                "t3.0"="swap pia t2a" in equ_trans)
+apply(simp)
+apply(best)
+apply(rule psub_trm_not_equ)
+apply(simp add: all_solutions_def, fast dest!: fresh.cases)
+apply(simp add: all_solutions_def, fast dest!: equ.cases)+
+done
+
+(* the only stuck problems are the "failed" problems and the empty problem *)
+
+lemma stuck_equiv: "stuck = {([],[])}\<union>fail"
+apply(subgoal_tac "([],[])\<in>stuck")
+apply(subgoal_tac "\<forall>P\<in>fail. P\<in>stuck")
+apply(subgoal_tac "\<forall>P\<in>stuck. P=([],[]) \<or> P\<in>fail")
+apply(force)
+apply(rule ballI)
+apply(thin_tac "([], []) \<in> stuck")
+apply(thin_tac "\<forall>P\<in>fail. P \<in> stuck")
+apply(simp add: stuck_def)
+apply(clarify)
+apply(case_tac a)
+apply(simp)
+apply(case_tac b)
+apply(simp)
+apply(simp)
+apply(case_tac aa)
+apply(simp)
+apply(case_tac ba)
+apply(simp_all)
+apply(case_tac "ab=lista")
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(case_tac aa)
+apply(simp)
+apply(case_tac ab)
+apply(simp_all)
+apply(case_tac ba)
+apply(simp_all)
+apply(case_tac "lista=listb")
+apply(force)
+apply(force)
+apply(case_tac "occurs list2 (Abst lista trm)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(case_tac ba)
+apply(simp_all)
+apply(case_tac "occurs list2 (Abst lista trm)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac "list2=list2a")
+apply(force)
+apply(case_tac "occurs list2 (Susp list1a list2a)")
+apply(simp)
+apply(drule_tac 
+    x="fst (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec)
+apply(drule_tac 
+    x="snd (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Susp list1a list2a))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac "occurs list2 Unit")
+apply(simp)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac "occurs list2 (Atom lista)")
+apply(simp)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac "occurs list2 (Paar trm1 trm2)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac "occurs list2 (Func lista trm)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(case_tac ba)
+apply(simp_all)
+apply(force)
+apply(case_tac "occurs list2 Unit")
+apply(simp)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(case_tac ba)
+apply(simp_all)
+apply(force)
+apply(case_tac "occurs list2 (Atom lista)")
+apply(simp)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(case_tac ba)
+apply(simp_all)
+apply(force)
+apply(case_tac "occurs list2 (Paar trm1 trm2)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(case_tac ba)
+apply(simp_all)
+apply(force)
+apply(case_tac "occurs list2 (Func lista trm)")
+apply(force)
+apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
+apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
+apply(drule_tac x="{}" in spec)
+apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec)
+apply(simp only: surjective_pairing[THEN sym])
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(force)
+apply(rule ballI)
+apply(thin_tac "([], []) \<in> stuck")
+apply(simp add: stuck_def)
+apply(clarify)
+apply(ind_cases "((a, b), (nabla, s), aa, ba) \<in> red_plus")
+apply(ind_cases "((a, b), s, aa, ba) \<in> s_red")
+apply(simp_all)
+apply(ind_cases "((Unit, Unit) # aa, b) \<in> fail")
+apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail")
+apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail")
+apply(simp)
+apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail")
+apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail")
+apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail")
+apply(simp)
+apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail")
+apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail")
+apply(simp add: apply_subst_def)+
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail")
+apply(simp add: apply_subst_def)
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(ind_cases "((a, b), nabla, aa, ba) \<in> c_red")
+apply(simp_all)
+apply(ind_cases "([], (ab, Unit) # ba) \<in> fail")
+apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail")
+apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail")
+apply(simp)
+apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail")
+apply(ind_cases "(a, b) \<turnstile> s1 \<leadsto> P2")
+apply(simp_all)
+apply(ind_cases "((Unit, Unit) # xs, b) \<in> fail")
+apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail")
+apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail")
+apply(simp)
+apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail")
+apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail")
+apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail")
+apply(simp)
+apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail")
+apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail")
+apply(simp add: apply_subst_def)+
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail")
+apply(simp add: apply_subst_def)
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(case_tac "occurs X t1")
+apply(simp)
+apply(simp)
+apply(ind_cases "(a, b) \<turnstile> nabla1 \<rightarrow> P2")
+apply(simp_all)
+apply(ind_cases "([], (ab, Unit) # ba) \<in> fail")
+apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail")
+apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail")
+apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail")
+apply(simp)
+apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail")
+apply(simp add: stuck_def)
+apply(rule allI)+
+apply(clarify)
+apply(ind_cases "(([], []), (nabla, s), a, b) \<in> red_plus")
+apply(ind_cases "(([], []), s, a, b) \<in> s_red")
+apply(ind_cases "(([], []), nabla, a, b) \<in> c_red")
+apply(ind_cases "([], []) \<turnstile> s1 \<leadsto> P2")
+apply(ind_cases "([], []) \<turnstile> nabla1 \<rightarrow> P2")
+done
+
+lemma u_empty_sred: 
+  "P1\<turnstile>s\<leadsto>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
+apply(rule impI)
+apply(ind_cases "P1 \<turnstile> s \<leadsto> P2")
+apply(rule impI, simp add: all_solutions_def)
+apply(rule impI, simp add: all_solutions_def)
+apply(fast dest!: equ_paar_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(fast dest!: equ_func_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(fast dest!: equ_abst_aa_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(force dest!: equ_abst_ab_elim simp add: subst_swap_comm[THEN sym])
+apply(rule impI, simp add: all_solutions_def)
+apply(rule impI, simp add: all_solutions_def)
+apply(simp add: ds_list_equ_ds)
+apply(rule allI)+
+apply(rule impI)
+apply(drule_tac x="a" in spec)
+apply(drule_tac x="b" in spec)
+apply(erule disjE)
+apply(force)
+apply(simp add: subst_susp)
+apply(drule equ_pi1_pi2_dec[THEN mp])
+apply(force simp add: subst_susp)
+apply(auto)
+apply(simp add: all_solutions_def)
+apply(simp_all add: apply_subst_def)
+apply(auto)
+apply(drule_tac x="a" in spec)
+apply(drule_tac x="b" in spec)
+apply(drule unif_1)
+apply(auto)
+apply(drule_tac x="(aa,ba)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a)
+apply(simp add: subst_comp_expand)
+apply(drule_tac x="(aa,ba)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule_tac a="aa" and "t"="ba" in unif_2b)
+apply(simp add: subst_comp_expand)
+apply(simp add: all_solutions_def)
+apply(auto)
+apply(drule_tac x="a" in spec)
+apply(drule_tac x="b" in spec)
+apply(drule equ_sym)
+apply(drule unif_1)
+apply(auto)
+apply(drule_tac x="(aa,ba)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a)
+apply(simp add: subst_comp_expand)
+apply(drule_tac x="(aa,ba)" in bspec)
+apply(assumption)
+apply(simp)
+apply(drule_tac a="aa" and "t"="ba" in unif_2b)
+apply(simp add: subst_comp_expand)
+done
+
+lemma u_empty_cred: 
+  "P1\<turnstile>nabla\<rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
+apply(rule impI)
+apply(ind_cases "P1 \<turnstile>nabla\<rightarrow>P2")
+apply(rule impI, simp add: all_solutions_def)
+apply(rule impI, simp add: all_solutions_def)
+apply(fast dest!: fresh_paar_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(fast dest!: fresh_func_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(rule impI, simp add: all_solutions_def)
+apply(force dest!: fresh_abst_ab_elim)
+apply(rule impI, simp add: all_solutions_def)
+apply(rule impI, simp add: all_solutions_def)
+done
+
+lemma u_empty_red_plus: 
+  "P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
+apply(rule impI)
+apply(erule red_plus.induct)
+apply(drule u_empty_sred[THEN mp], assumption)
+apply(drule u_empty_cred[THEN mp], assumption)
+apply(drule u_empty_sred[THEN mp], force)
+apply(drule u_empty_cred[THEN mp], force)
+done
+
+(* all problems that cannot be solved produce "failed" problems only *)
+
+lemma empty_then_fail: "U P1={} \<longrightarrow> (\<forall>P\<in>results P1. P\<in>fail)"
+apply(simp add: results_def)
+apply(rule conjI)
+apply(rule impI)
+apply(rule impI)
+apply(simp add: stuck_equiv)
+apply(erule disjE)
+apply(subgoal_tac "({},[])\<in>U ([],[])")
+apply(simp)
+apply(simp add: all_solutions_def)
+apply(assumption)
+apply(rule impI)+
+apply(rule allI)+
+apply(rule impI)
+apply(erule conjE)
+apply(simp add: stuck_equiv)
+apply(auto)
+apply(subgoal_tac "({},[])\<in>U ([],[])")
+apply(drule_tac "nabla3.0"="nabla" and "nabla1.0"="{}" and "s1.0"="[]" in P1_from_P2_red_plus)
+apply(simp add: ext_subst_def)
+apply(auto)
+apply(simp add: all_solutions_def)
+done
+
+(* if a problem can be solved then no "failed" problem is produced *)
+
+lemma not_empty_then_not_fail: "U P1\<noteq>{} \<longrightarrow> \<not>(\<exists>P\<in>results P1. P\<in>fail)"
+apply(rule impI)
+apply(simp)
+apply(rule ballI)
+apply(clarify)
+apply(simp add: results_def)
+apply(case_tac "P1\<in>stuck")
+apply(simp_all)
+apply(drule fail_then_empty)
+apply(simp)
+apply(drule fail_then_empty)
+apply(erule conjE)
+apply(clarify)
+apply(drule u_empty_red_plus[THEN mp])
+apply(simp)
+done
+
+end
+
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/app-nomu.ps	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,4441 @@
+%!PS-Adobe-2.0
+%%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software
+%%Title: nomu-csl.dvi
+%%Pages: 4
+%%PageOrder: Ascend
+%%BoundingBox: 0 0 596 842
+%%DocumentFonts: Times-Bold Times-Roman Times-Italic CMMI9 CMMI10 CMSY10
+%%+ CMTT10 CMR10 CMSY7 CMBSY10 CMR7 CMMI7 CMR6 CMR9 Times-BoldItalic
+%%+ CMSY9 CMTI9 CMTT9 CMTI10 TeX-cmex9 CMSY6 CMSY8 CMR8 CMBXTI10 CMBX10
+%%+ CMMIB10 MSBM10 CMMI6 MSBM7 CMR5 CMMI5 EUSM10 CMMI8 CMTT8 CMBSY7
+%%+ Courier
+%%EndComments
+%DVIPSWebPage: (www.radicaleye.com)
+%DVIPSCommandLine: dvips -o nomu-csl.ps nomu-csl.dvi
+%DVIPSParameters: dpi=600, compressed
+%DVIPSSource:  TeX output 2003.04.09:1710
+%%BeginProcSet: texc.pro
+%!
+/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
+N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
+mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
+0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
+landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
+mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
+matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
+exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
+statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
+N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
+/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
+/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
+array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
+df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
+definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
+}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
+B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
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+0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
+sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
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+/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
+/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
+A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
+get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
+ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
+fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
+{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
+chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
+1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
+forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
+/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
+}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
+bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
+mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
+SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
+userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
+1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
+index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
+/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
+/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
+(LaserWriter 16/600)]{A length product length le{A length product exch 0
+exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
+end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
+grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
+imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
+exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
+fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
+delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
+B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
+p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
+rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
+
+%%EndProcSet
+%%BeginProcSet: 8r.enc
+% File  8r.enc as of 2002-03-12 for PSNFSS 9
+%
+% This is the encoding vector for Type1 and TrueType fonts to be used
+% with TeX.  This file is part of the PSNFSS bundle, version 9
+% 
+% Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt
+%
+% Idea is to have all the characters normally included in Type 1 fonts
+% available for typesetting. This is effectively the characters in Adobe
+% Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro.
+% 
+% Character code assignments were made as follows:
+% 
+% (1) the Windows ANSI characters are almost all in their Windows ANSI
+% positions, because some Windows users cannot easily reencode the
+% fonts, and it makes no difference on other systems. The only Windows
+% ANSI characters not available are those that make no sense for
+% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
+% (173). quotesingle and grave are moved just because it's such an
+% irritation not having them in TeX positions.
+% 
+% (2) Remaining characters are assigned arbitrarily to the lower part
+% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
+% 
+% (3) Y&Y Lucida Bright includes some extra text characters; in the
+% hopes that other PostScript fonts, perhaps created for public
+% consumption, will include them, they are included starting at 0x12.
+% 
+% (4) Remaining positions left undefined are for use in (hopefully)
+% upward-compatible revisions, if someday more characters are generally
+% available.
+% 
+% (5) hyphen appears twice for compatibility with both ASCII and Windows.
+%
+% (6) /Euro is assigned to 128, as in Windows ANSI
+% 
+/TeXBase1Encoding [
+% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
+  /.notdef /dotaccent /fi /fl
+  /fraction /hungarumlaut /Lslash /lslash
+  /ogonek /ring /.notdef
+  /breve /minus /.notdef 
+% These are the only two remaining unencoded characters, so may as
+% well include them.
+  /Zcaron /zcaron 
+% 0x10
+ /caron /dotlessi 
+% (unusual TeX characters available in, e.g., Lucida Bright)
+ /dotlessj /ff /ffi /ffl 
+ /.notdef /.notdef /.notdef /.notdef
+ /.notdef /.notdef /.notdef /.notdef
+ % very contentious; it's so painful not having quoteleft and quoteright
+ % at 96 and 145 that we move the things normally found there down to here.
+ /grave /quotesingle 
+% 0x20 (ASCII begins)
+ /space /exclam /quotedbl /numbersign
+ /dollar /percent /ampersand /quoteright
+ /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
+% 0x30
+ /zero /one /two /three /four /five /six /seven
+ /eight /nine /colon /semicolon /less /equal /greater /question
+% 0x40
+ /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
+% 0x50
+ /P /Q /R /S /T /U /V /W
+ /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
+% 0x60
+ /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
+% 0x70
+ /p /q /r /s /t /u /v /w
+ /x /y /z /braceleft /bar /braceright /asciitilde
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+% Math Symbol fonts were designed by the American Mathematical Society.
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+11 dict begin
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+% Computer Modern fonts were designed by Donald E. Knuth.
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (001.000) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMBSY7) readonly def
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+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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+%!PS-AdobeFont-1.1: CMTT8 1.0
+%%CreationDate: 1991 Aug 20 16:46:05
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
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+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
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+dup 0 /.notdef put
+readonly def
+/FontBBox{-5 -232 545 699}readonly def
+/UniqueID 5000830 def
+currentdict end
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+%!PS-AdobeFont-1.1: CMMI8 1.100
+%%CreationDate: 1996 Jul 23 07:53:54
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+%%CreationDate: 1991 Aug 15 07:22:10
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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+11 dict begin
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+%!PS-AdobeFont-1.1: CMTI9 1.0
+%%CreationDate: 1991 Aug 18 21:08:07
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMTI9) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.04 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMTI9 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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+%%CreationDate: 1991 Aug 15 07:22:27
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
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+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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+cleartomark
+%%EndFont 
+%%BeginFont: CMR9
+%!PS-AdobeFont-1.1: CMR9 1.0
+%%CreationDate: 1991 Aug 20 16:39:59
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR9) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR9 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/FontBBox{-39 -250 1036 750}readonly def
+/UniqueID 5000792 def
+currentdict end
+currentfile eexec
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+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR10 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/FontBBox{-251 -250 1009 969}readonly def
+/UniqueID 5000793 def
+currentdict end
+currentfile eexec
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+cleartomark
+%%EndFont 
+%%BeginFont: CMTT10
+%!PS-AdobeFont-1.1: CMTT10 1.00B
+%%CreationDate: 1992 Apr 26 10:42:42
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMTT10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch true def
+end readonly def
+/FontName /CMTT10 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/FontBBox{-4 -235 731 800}readonly def
+/UniqueID 5000832 def
+currentdict end
+currentfile eexec
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+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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+(the)g(nominal)f(uni\002cation)g(algorithm,)f(one)664
+2175 y(w)o(ould)20 b(certainly)f(hope)g(not)h(to)g(ha)n(v)o(e)g(to)g
+(use)h(this)f(roundabout)d(method)i(in)h(practice.)581
+2275 y FO(\017)41 b FW(An)15 b(encoding/decoding)c(does)k(not)g(gi)n(v)
+o(e)f(an)o(y)h(insight)f(into)h(the)h(properties)d(of)i(the)h
+FO(\031)p FW(-equi)n(v)n(alence.)664 2374 y(In)25 b(the)g(paper)f(we)h
+(ha)n(v)o(e)f(sho)n(wn)h(that)g FO(\031)g FW(is)h(indeed)e(an)g(equi)n
+(v)n(alence)f(relation)h(and)h(that)g(it)h(is)g(re-)664
+2474 y(spected)20 b(by)g(possibly-capturing)c(substitutions.)k(Both)g
+(f)o(acts)h(are)f(non-ob)o(vious.)3058 2444 y Fy(3)523
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+(solving)g(nominal)g(uni\002cation)f(problems)h(by)g(the)h(uni-)523
+2740 y(\002cation)d(algorithm)f(described)h(in)h([7])f(might)g(be)g
+(possible,)g(b)n(ut)h(this)g(is)h FP(no)e FW(substitute)h(for)f(ha)n
+(ving)f(the)523 2839 y(direct)i(algorithm)e(we)j(presented.)p
+523 5050 473 4 v 558 5105 a Fu(3)606 5137 y FV(See)e
+FB(http)j FF(:)g FS(==)p FB(www)p FS(:)p FB(cl)p FS(:)p
+FB(cam)p FS(:)p FB(ac)p FS(:)p FB(uk)p FS(=)1607 5129
+y Fw(\030)1664 5137 y FB(cu200)p FS(=)p FB(Unification)h
+FV(for)c(the)g(Isabelle)g(proof)g(scripts.)p eop end
+%%Trailer
+
+userdict /end-hook known{end-hook}if
+%%EOF