--- /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