src/HOL/ex/Unification.thy
author nipkow
Fri Mar 06 17:38:47 2009 +0100 (2009-03-06)
changeset 30313 b2441b0c8d38
parent 24444 448978b61556
child 30909 bd4f255837e5
permissions -rw-r--r--
added lemmas
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(*  ID:         $Id$
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    Author:     Alexander Krauss, Technische Universitaet Muenchen
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*)
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header {* Case study: Unification Algorithm *}
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theory Unification
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imports Main
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begin
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text {* 
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  This is a formalization of a first-order unification
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  algorithm. It uses the new "function" package to define recursive
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  functions, which allows a better treatment of nested recursion. 
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  This is basically a modernized version of a previous formalization
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  by Konrad Slind (see: HOL/Subst/Unify.thy), which itself builds on
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  previous work by Paulson and Manna \& Waldinger (for details, see
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  there).
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  Unlike that formalization, where the proofs of termination and
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  some partial correctness properties are intertwined, we can prove
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  partial correctness and termination separately.
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*}
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subsection {* Basic definitions *}
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datatype 'a trm = 
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  Var 'a 
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  | Const 'a
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  | App "'a trm" "'a trm" (infix "\<cdot>" 60)
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types
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  'a subst = "('a \<times> 'a trm) list"
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text {* Applying a substitution to a variable: *}
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fun assoc :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> 'b"
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where
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  "assoc x d [] = d"
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| "assoc x d ((p,q)#t) = (if x = p then q else assoc x d t)"
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text {* Applying a substitution to a term: *}
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fun apply_subst :: "'a trm \<Rightarrow> 'a subst \<Rightarrow> 'a trm" (infixl "\<triangleleft>" 60)
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where
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  "(Var v) \<triangleleft> s = assoc v (Var v) s"
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| "(Const c) \<triangleleft> s = (Const c)"
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| "(M \<cdot> N) \<triangleleft> s = (M \<triangleleft> s) \<cdot> (N \<triangleleft> s)"
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text {* Composition of substitutions: *}
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fun
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  "compose" :: "'a subst \<Rightarrow> 'a subst \<Rightarrow> 'a subst" (infixl "\<bullet>" 80)
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where
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  "[] \<bullet> bl = bl"
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| "((a,b) # al) \<bullet> bl = (a, b \<triangleleft> bl) # (al \<bullet> bl)"
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text {* Equivalence of substitutions: *}
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definition eqv (infix "=\<^sub>s" 50)
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where
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  "s1 =\<^sub>s s2 \<equiv> \<forall>t. t \<triangleleft> s1 = t \<triangleleft> s2" 
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subsection {* Basic lemmas *}
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lemma apply_empty[simp]: "t \<triangleleft> [] = t"
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by (induct t) auto
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lemma compose_empty[simp]: "\<sigma> \<bullet> [] = \<sigma>"
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by (induct \<sigma>) auto
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lemma apply_compose[simp]: "t \<triangleleft> (s1 \<bullet> s2) = t \<triangleleft> s1 \<triangleleft> s2"
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proof (induct t)
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  case App thus ?case by simp
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next 
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  case Const thus ?case by simp
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next
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  case (Var v) thus ?case
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  proof (induct s1)
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    case Nil show ?case by simp
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  next
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    case (Cons p s1s) thus ?case by (cases p, simp)
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  qed
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qed
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lemma eqv_refl[intro]: "s =\<^sub>s s"
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  by (auto simp:eqv_def)
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lemma eqv_trans[trans]: "\<lbrakk>s1 =\<^sub>s s2; s2 =\<^sub>s s3\<rbrakk> \<Longrightarrow> s1 =\<^sub>s s3"
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  by (auto simp:eqv_def)
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lemma eqv_sym[sym]: "\<lbrakk>s1 =\<^sub>s s2\<rbrakk> \<Longrightarrow> s2 =\<^sub>s s1"
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  by (auto simp:eqv_def)
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lemma eqv_intro[intro]: "(\<And>t. t \<triangleleft> \<sigma> = t \<triangleleft> \<theta>) \<Longrightarrow> \<sigma> =\<^sub>s \<theta>"
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  by (auto simp:eqv_def)
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lemma eqv_dest[dest]: "s1 =\<^sub>s s2 \<Longrightarrow> t \<triangleleft> s1 = t \<triangleleft> s2"
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  by (auto simp:eqv_def)
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lemma compose_eqv: "\<lbrakk>\<sigma> =\<^sub>s \<sigma>'; \<theta> =\<^sub>s \<theta>'\<rbrakk> \<Longrightarrow> (\<sigma> \<bullet> \<theta>) =\<^sub>s (\<sigma>' \<bullet> \<theta>')"
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  by (auto simp:eqv_def)
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lemma compose_assoc: "(a \<bullet> b) \<bullet> c =\<^sub>s a \<bullet> (b \<bullet> c)"
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  by auto
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subsection {* Specification: Most general unifiers *}
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definition
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  "Unifier \<sigma> t u \<equiv> (t\<triangleleft>\<sigma> = u\<triangleleft>\<sigma>)"
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definition
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  "MGU \<sigma> t u \<equiv> Unifier \<sigma> t u \<and> (\<forall>\<theta>. Unifier \<theta> t u 
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  \<longrightarrow> (\<exists>\<gamma>. \<theta> =\<^sub>s \<sigma> \<bullet> \<gamma>))"
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lemma MGUI[intro]:
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  "\<lbrakk>t \<triangleleft> \<sigma> = u \<triangleleft> \<sigma>; \<And>\<theta>. t \<triangleleft> \<theta> = u \<triangleleft> \<theta> \<Longrightarrow> \<exists>\<gamma>. \<theta> =\<^sub>s \<sigma> \<bullet> \<gamma>\<rbrakk>
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  \<Longrightarrow> MGU \<sigma> t u"
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  by (simp only:Unifier_def MGU_def, auto)
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lemma MGU_sym[sym]:
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  "MGU \<sigma> s t \<Longrightarrow> MGU \<sigma> t s"
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  by (auto simp:MGU_def Unifier_def)
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subsection {* The unification algorithm *}
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text {* Occurs check: Proper subterm relation *}
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fun occ :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"
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where
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  "occ u (Var v) = False"
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| "occ u (Const c) = False"
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| "occ u (M \<cdot> N) = (u = M \<or> u = N \<or> occ u M \<or> occ u N)"
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text {* The unification algorithm: *}
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function unify :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> 'a subst option"
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where
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  "unify (Const c) (M \<cdot> N)   = None"
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| "unify (M \<cdot> N)   (Const c) = None"
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| "unify (Const c) (Var v)   = Some [(v, Const c)]"
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| "unify (M \<cdot> N)   (Var v)   = (if (occ (Var v) (M \<cdot> N)) 
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                                        then None
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                                        else Some [(v, M \<cdot> N)])"
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| "unify (Var v)   M         = (if (occ (Var v) M) 
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                                        then None
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                                        else Some [(v, M)])"
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| "unify (Const c) (Const d) = (if c=d then Some [] else None)"
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| "unify (M \<cdot> N) (M' \<cdot> N') = (case unify M M' of
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                                    None \<Rightarrow> None |
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                                    Some \<theta> \<Rightarrow> (case unify (N \<triangleleft> \<theta>) (N' \<triangleleft> \<theta>)
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                                      of None \<Rightarrow> None |
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                                         Some \<sigma> \<Rightarrow> Some (\<theta> \<bullet> \<sigma>)))"
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  by pat_completeness auto
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subsection {* Partial correctness *}
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text {* Some lemmas about occ and MGU: *}
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lemma subst_no_occ: "\<not>occ (Var v) t \<Longrightarrow> Var v \<noteq> t
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  \<Longrightarrow> t \<triangleleft> [(v,s)] = t"
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by (induct t) auto
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lemma MGU_Var[intro]: 
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  assumes no_occ: "\<not>occ (Var v) t"
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  shows "MGU [(v,t)] (Var v) t"
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proof (intro MGUI exI)
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  show "Var v \<triangleleft> [(v,t)] = t \<triangleleft> [(v,t)]" using no_occ
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    by (cases "Var v = t", auto simp:subst_no_occ)
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next
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  fix \<theta> assume th: "Var v \<triangleleft> \<theta> = t \<triangleleft> \<theta>" 
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  show "\<theta> =\<^sub>s [(v,t)] \<bullet> \<theta>" 
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  proof
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    fix s show "s \<triangleleft> \<theta> = s \<triangleleft> [(v,t)] \<bullet> \<theta>" using th 
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      by (induct s) auto
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  qed
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qed
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declare MGU_Var[symmetric, intro]
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lemma MGU_Const[simp]: "MGU [] (Const c) (Const d) = (c = d)"
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  unfolding MGU_def Unifier_def
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  by auto
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text {* If unification terminates, then it computes most general unifiers: *}
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lemma unify_partial_correctness:
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  assumes "unify_dom (M, N)"
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  assumes "unify M N = Some \<sigma>"
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  shows "MGU \<sigma> M N"
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using assms
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proof (induct M N arbitrary: \<sigma>)
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  case (7 M N M' N' \<sigma>) -- "The interesting case"
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  then obtain \<theta>1 \<theta>2 
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    where "unify M M' = Some \<theta>1"
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    and "unify (N \<triangleleft> \<theta>1) (N' \<triangleleft> \<theta>1) = Some \<theta>2"
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    and \<sigma>: "\<sigma> = \<theta>1 \<bullet> \<theta>2"
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    and MGU_inner: "MGU \<theta>1 M M'" 
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    and MGU_outer: "MGU \<theta>2 (N \<triangleleft> \<theta>1) (N' \<triangleleft> \<theta>1)"
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    by (auto split:option.split_asm)
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  show ?case
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  proof
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    from MGU_inner and MGU_outer
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    have "M \<triangleleft> \<theta>1 = M' \<triangleleft> \<theta>1" 
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      and "N \<triangleleft> \<theta>1 \<triangleleft> \<theta>2 = N' \<triangleleft> \<theta>1 \<triangleleft> \<theta>2"
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      unfolding MGU_def Unifier_def
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      by auto
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    thus "M \<cdot> N \<triangleleft> \<sigma> = M' \<cdot> N' \<triangleleft> \<sigma>" unfolding \<sigma>
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      by simp
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  next
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    fix \<sigma>' assume "M \<cdot> N \<triangleleft> \<sigma>' = M' \<cdot> N' \<triangleleft> \<sigma>'"
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    hence "M \<triangleleft> \<sigma>' = M' \<triangleleft> \<sigma>'"
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      and Ns: "N \<triangleleft> \<sigma>' = N' \<triangleleft> \<sigma>'" by auto
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    with MGU_inner obtain \<delta>
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      where eqv: "\<sigma>' =\<^sub>s \<theta>1 \<bullet> \<delta>"
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      unfolding MGU_def Unifier_def
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      by auto
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    from Ns have "N \<triangleleft> \<theta>1 \<triangleleft> \<delta> = N' \<triangleleft> \<theta>1 \<triangleleft> \<delta>"
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      by (simp add:eqv_dest[OF eqv])
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    with MGU_outer obtain \<rho>
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      where eqv2: "\<delta> =\<^sub>s \<theta>2 \<bullet> \<rho>"
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      unfolding MGU_def Unifier_def
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      by auto
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    have "\<sigma>' =\<^sub>s \<sigma> \<bullet> \<rho>" unfolding \<sigma>
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      by (rule eqv_intro, auto simp:eqv_dest[OF eqv] 
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	eqv_dest[OF eqv2])
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    thus "\<exists>\<gamma>. \<sigma>' =\<^sub>s \<sigma> \<bullet> \<gamma>" ..
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  qed
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qed (auto split:split_if_asm) -- "Solve the remaining cases automatically"
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subsection {* Properties used in termination proof *}
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text {* The variables of a term: *}
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fun vars_of:: "'a trm \<Rightarrow> 'a set"
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where
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  "vars_of (Var v) = { v }"
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| "vars_of (Const c) = {}"
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| "vars_of (M \<cdot> N) = vars_of M \<union> vars_of N"
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lemma vars_of_finite[intro]: "finite (vars_of t)"
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  by (induct t) simp_all
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text {* Elimination of variables by a substitution: *}
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definition
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  "elim \<sigma> v \<equiv> \<forall>t. v \<notin> vars_of (t \<triangleleft> \<sigma>)"
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lemma elim_intro[intro]: "(\<And>t. v \<notin> vars_of (t \<triangleleft> \<sigma>)) \<Longrightarrow> elim \<sigma> v"
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  by (auto simp:elim_def)
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lemma elim_dest[dest]: "elim \<sigma> v \<Longrightarrow> v \<notin> vars_of (t \<triangleleft> \<sigma>)"
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  by (auto simp:elim_def)
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lemma elim_eqv: "\<sigma> =\<^sub>s \<theta> \<Longrightarrow> elim \<sigma> x = elim \<theta> x"
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  by (auto simp:elim_def eqv_def)
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text {* Replacing a variable by itself yields an identity subtitution: *}
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lemma var_self[intro]: "[(v, Var v)] =\<^sub>s []"
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proof
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  fix t show "t \<triangleleft> [(v, Var v)] = t \<triangleleft> []"
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    by (induct t) simp_all
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qed
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lemma var_same: "(t = Var v) = ([(v, t)] =\<^sub>s [])"
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proof
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  assume t_v: "t = Var v"
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  thus "[(v, t)] =\<^sub>s []"
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    by auto
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next
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  assume id: "[(v, t)] =\<^sub>s []"
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  show "t = Var v"
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  proof -
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    have "t = Var v \<triangleleft> [(v, t)]" by simp
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    also from id have "\<dots> = Var v \<triangleleft> []" ..
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    finally show ?thesis by simp
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  qed
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qed
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text {* A lemma about occ and elim *}
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lemma remove_var:
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  assumes [simp]: "v \<notin> vars_of s"
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  shows "v \<notin> vars_of (t \<triangleleft> [(v, s)])"
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  by (induct t) simp_all
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lemma occ_elim: "\<not>occ (Var v) t 
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  \<Longrightarrow> elim [(v,t)] v \<or> [(v,t)] =\<^sub>s []"
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proof (induct t)
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  case (Var x)
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  show ?case
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  proof cases
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    assume "v = x"
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    thus ?thesis
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      by (simp add:var_same[symmetric])
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  next
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    assume neq: "v \<noteq> x"
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    have "elim [(v, Var x)] v"
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      by (auto intro!:remove_var simp:neq)
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    thus ?thesis ..
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  qed
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next
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  case (Const c)
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  have "elim [(v, Const c)] v"
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    by (auto intro!:remove_var)
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  thus ?case ..
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next
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  case (App M N)
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   324
  
krauss@22999
   325
  hence ih1: "elim [(v, M)] v \<or> [(v, M)] =\<^sub>s []"
krauss@22999
   326
    and ih2: "elim [(v, N)] v \<or> [(v, N)] =\<^sub>s []"
krauss@22999
   327
    and nonocc: "Var v \<noteq> M" "Var v \<noteq> N"
krauss@22999
   328
    by auto
krauss@22999
   329
krauss@22999
   330
  from nonocc have "\<not> [(v,M)] =\<^sub>s []"
krauss@22999
   331
    by (simp add:var_same[symmetric])
krauss@22999
   332
  with ih1 have "elim [(v, M)] v" by blast
krauss@22999
   333
  hence "v \<notin> vars_of (Var v \<triangleleft> [(v,M)])" ..
krauss@22999
   334
  hence not_in_M: "v \<notin> vars_of M" by simp
krauss@22999
   335
krauss@22999
   336
  from nonocc have "\<not> [(v,N)] =\<^sub>s []"
krauss@22999
   337
    by (simp add:var_same[symmetric])
krauss@22999
   338
  with ih2 have "elim [(v, N)] v" by blast
krauss@22999
   339
  hence "v \<notin> vars_of (Var v \<triangleleft> [(v,N)])" ..
krauss@22999
   340
  hence not_in_N: "v \<notin> vars_of N" by simp
krauss@22999
   341
krauss@22999
   342
  have "elim [(v, M \<cdot> N)] v"
krauss@22999
   343
  proof 
krauss@22999
   344
    fix t 
krauss@22999
   345
    show "v \<notin> vars_of (t \<triangleleft> [(v, M \<cdot> N)])"
krauss@22999
   346
    proof (induct t)
krauss@22999
   347
      case (Var x) thus ?case by (simp add: not_in_M not_in_N)
krauss@22999
   348
    qed auto
krauss@22999
   349
  qed
krauss@22999
   350
  thus ?case ..
krauss@22999
   351
qed
krauss@22999
   352
krauss@22999
   353
text {* The result of a unification never introduces new variables: *}
krauss@22999
   354
krauss@22999
   355
lemma unify_vars: 
krauss@22999
   356
  assumes "unify_dom (M, N)"
krauss@22999
   357
  assumes "unify M N = Some \<sigma>"
krauss@22999
   358
  shows "vars_of (t \<triangleleft> \<sigma>) \<subseteq> vars_of M \<union> vars_of N \<union> vars_of t"
krauss@22999
   359
  (is "?P M N \<sigma> t")
wenzelm@24444
   360
using assms
krauss@22999
   361
proof (induct M N arbitrary:\<sigma> t)
krauss@22999
   362
  case (3 c v) 
krauss@22999
   363
  hence "\<sigma> = [(v, Const c)]" by simp
wenzelm@24444
   364
  thus ?case by (induct t) auto
krauss@22999
   365
next
krauss@22999
   366
  case (4 M N v) 
krauss@22999
   367
  hence "\<not>occ (Var v) (M\<cdot>N)" by (cases "occ (Var v) (M\<cdot>N)", auto)
wenzelm@24444
   368
  with 4 have "\<sigma> = [(v, M\<cdot>N)]" by simp
wenzelm@24444
   369
  thus ?case by (induct t) auto
krauss@22999
   370
next
krauss@22999
   371
  case (5 v M)
krauss@22999
   372
  hence "\<not>occ (Var v) M" by (cases "occ (Var v) M", auto)
wenzelm@24444
   373
  with 5 have "\<sigma> = [(v, M)]" by simp
wenzelm@24444
   374
  thus ?case by (induct t) auto
krauss@22999
   375
next
krauss@22999
   376
  case (7 M N M' N' \<sigma>)
krauss@22999
   377
  then obtain \<theta>1 \<theta>2 
krauss@22999
   378
    where "unify M M' = Some \<theta>1"
krauss@22999
   379
    and "unify (N \<triangleleft> \<theta>1) (N' \<triangleleft> \<theta>1) = Some \<theta>2"
krauss@22999
   380
    and \<sigma>: "\<sigma> = \<theta>1 \<bullet> \<theta>2"
krauss@22999
   381
    and ih1: "\<And>t. ?P M M' \<theta>1 t"
krauss@22999
   382
    and ih2: "\<And>t. ?P (N\<triangleleft>\<theta>1) (N'\<triangleleft>\<theta>1) \<theta>2 t"
krauss@22999
   383
    by (auto split:option.split_asm)
krauss@22999
   384
krauss@22999
   385
  show ?case
krauss@22999
   386
  proof
krauss@22999
   387
    fix v assume a: "v \<in> vars_of (t \<triangleleft> \<sigma>)"
krauss@22999
   388
    
krauss@22999
   389
    show "v \<in> vars_of (M \<cdot> N) \<union> vars_of (M' \<cdot> N') \<union> vars_of t"
krauss@22999
   390
    proof (cases "v \<notin> vars_of M \<and> v \<notin> vars_of M'
krauss@22999
   391
	    \<and> v \<notin> vars_of N \<and> v \<notin> vars_of N'")
krauss@22999
   392
      case True
krauss@22999
   393
      with ih1 have l:"\<And>t. v \<in> vars_of (t \<triangleleft> \<theta>1) \<Longrightarrow> v \<in> vars_of t"
krauss@22999
   394
	    by auto
krauss@22999
   395
      
krauss@22999
   396
      from a and ih2[where t="t \<triangleleft> \<theta>1"]
krauss@22999
   397
      have "v \<in> vars_of (N \<triangleleft> \<theta>1) \<union> vars_of (N' \<triangleleft> \<theta>1) 
krauss@22999
   398
        \<or> v \<in> vars_of (t \<triangleleft> \<theta>1)" unfolding \<sigma>
krauss@22999
   399
	    by auto
krauss@22999
   400
      hence "v \<in> vars_of t"
krauss@22999
   401
      proof
krauss@22999
   402
	    assume "v \<in> vars_of (N \<triangleleft> \<theta>1) \<union> vars_of (N' \<triangleleft> \<theta>1)"
krauss@22999
   403
	    with True show ?thesis by (auto dest:l)
krauss@22999
   404
      next
krauss@22999
   405
	    assume "v \<in> vars_of (t \<triangleleft> \<theta>1)" 
krauss@22999
   406
	    thus ?thesis by (rule l)
krauss@22999
   407
      qed
krauss@22999
   408
      
krauss@22999
   409
      thus ?thesis by auto
krauss@22999
   410
    qed auto
krauss@22999
   411
  qed
krauss@22999
   412
qed (auto split: split_if_asm)
krauss@22999
   413
krauss@22999
   414
krauss@22999
   415
text {* The result of a unification is either the identity
krauss@22999
   416
substitution or it eliminates a variable from one of the terms: *}
krauss@22999
   417
krauss@22999
   418
lemma unify_eliminates: 
krauss@22999
   419
  assumes "unify_dom (M, N)"
krauss@22999
   420
  assumes "unify M N = Some \<sigma>"
krauss@22999
   421
  shows "(\<exists>v\<in>vars_of M \<union> vars_of N. elim \<sigma> v) \<or> \<sigma> =\<^sub>s []"
krauss@22999
   422
  (is "?P M N \<sigma>")
wenzelm@24444
   423
using assms
krauss@22999
   424
proof (induct M N arbitrary:\<sigma>)
krauss@22999
   425
  case 1 thus ?case by simp
krauss@22999
   426
next
krauss@22999
   427
  case 2 thus ?case by simp
krauss@22999
   428
next
krauss@22999
   429
  case (3 c v)
krauss@22999
   430
  have no_occ: "\<not> occ (Var v) (Const c)" by simp
wenzelm@24444
   431
  with 3 have "\<sigma> = [(v, Const c)]" by simp
krauss@22999
   432
  with occ_elim[OF no_occ]
krauss@22999
   433
  show ?case by auto
krauss@22999
   434
next
krauss@22999
   435
  case (4 M N v)
krauss@22999
   436
  hence no_occ: "\<not>occ (Var v) (M\<cdot>N)" by (cases "occ (Var v) (M\<cdot>N)", auto)
wenzelm@24444
   437
  with 4 have "\<sigma> = [(v, M\<cdot>N)]" by simp
krauss@22999
   438
  with occ_elim[OF no_occ]
krauss@22999
   439
  show ?case by auto 
krauss@22999
   440
next
krauss@22999
   441
  case (5 v M) 
krauss@22999
   442
  hence no_occ: "\<not>occ (Var v) M" by (cases "occ (Var v) M", auto)
wenzelm@24444
   443
  with 5 have "\<sigma> = [(v, M)]" by simp
krauss@22999
   444
  with occ_elim[OF no_occ]
krauss@22999
   445
  show ?case by auto 
krauss@22999
   446
next 
krauss@22999
   447
  case (6 c d) thus ?case
krauss@22999
   448
    by (cases "c = d") auto
krauss@22999
   449
next
krauss@22999
   450
  case (7 M N M' N' \<sigma>)
krauss@22999
   451
  then obtain \<theta>1 \<theta>2 
krauss@22999
   452
    where "unify M M' = Some \<theta>1"
krauss@22999
   453
    and "unify (N \<triangleleft> \<theta>1) (N' \<triangleleft> \<theta>1) = Some \<theta>2"
krauss@22999
   454
    and \<sigma>: "\<sigma> = \<theta>1 \<bullet> \<theta>2"
krauss@22999
   455
    and ih1: "?P M M' \<theta>1"
krauss@22999
   456
    and ih2: "?P (N\<triangleleft>\<theta>1) (N'\<triangleleft>\<theta>1) \<theta>2"
krauss@22999
   457
    by (auto split:option.split_asm)
krauss@22999
   458
krauss@22999
   459
  from `unify_dom (M \<cdot> N, M' \<cdot> N')`
krauss@22999
   460
  have "unify_dom (M, M')"
berghofe@23777
   461
    by (rule accp_downward) (rule unify_rel.intros)
krauss@22999
   462
  hence no_new_vars: 
krauss@22999
   463
    "\<And>t. vars_of (t \<triangleleft> \<theta>1) \<subseteq> vars_of M \<union> vars_of M' \<union> vars_of t"
wenzelm@23373
   464
    by (rule unify_vars) (rule `unify M M' = Some \<theta>1`)
krauss@22999
   465
krauss@22999
   466
  from ih2 show ?case 
krauss@22999
   467
  proof 
krauss@22999
   468
    assume "\<exists>v\<in>vars_of (N \<triangleleft> \<theta>1) \<union> vars_of (N' \<triangleleft> \<theta>1). elim \<theta>2 v"
krauss@22999
   469
    then obtain v 
krauss@22999
   470
      where "v\<in>vars_of (N \<triangleleft> \<theta>1) \<union> vars_of (N' \<triangleleft> \<theta>1)"
krauss@22999
   471
      and el: "elim \<theta>2 v" by auto
krauss@22999
   472
    with no_new_vars show ?thesis unfolding \<sigma> 
krauss@22999
   473
      by (auto simp:elim_def)
krauss@22999
   474
  next
krauss@22999
   475
    assume empty[simp]: "\<theta>2 =\<^sub>s []"
krauss@22999
   476
krauss@22999
   477
    have "\<sigma> =\<^sub>s (\<theta>1 \<bullet> [])" unfolding \<sigma>
krauss@22999
   478
      by (rule compose_eqv) auto
krauss@22999
   479
    also have "\<dots> =\<^sub>s \<theta>1" by auto
krauss@22999
   480
    finally have "\<sigma> =\<^sub>s \<theta>1" .
krauss@22999
   481
krauss@22999
   482
    from ih1 show ?thesis
krauss@22999
   483
    proof
krauss@22999
   484
      assume "\<exists>v\<in>vars_of M \<union> vars_of M'. elim \<theta>1 v"
krauss@22999
   485
      with elim_eqv[OF `\<sigma> =\<^sub>s \<theta>1`]
krauss@22999
   486
      show ?thesis by auto
krauss@22999
   487
    next
krauss@22999
   488
      note `\<sigma> =\<^sub>s \<theta>1`
krauss@22999
   489
      also assume "\<theta>1 =\<^sub>s []"
krauss@22999
   490
      finally show ?thesis ..
krauss@22999
   491
    qed
krauss@22999
   492
  qed
krauss@22999
   493
qed
krauss@22999
   494
krauss@22999
   495
krauss@22999
   496
subsection {* Termination proof *}
krauss@22999
   497
krauss@22999
   498
termination unify
krauss@22999
   499
proof 
krauss@22999
   500
  let ?R = "measures [\<lambda>(M,N). card (vars_of M \<union> vars_of N),
krauss@22999
   501
                           \<lambda>(M, N). size M]"
krauss@22999
   502
  show "wf ?R" by simp
krauss@22999
   503
krauss@22999
   504
  fix M N M' N' 
krauss@22999
   505
  show "((M, M'), (M \<cdot> N, M' \<cdot> N')) \<in> ?R" -- "Inner call"
krauss@22999
   506
    by (rule measures_lesseq) (auto intro: card_mono)
krauss@22999
   507
krauss@22999
   508
  fix \<theta>                                   -- "Outer call"
krauss@22999
   509
  assume inner: "unify_dom (M, M')"
krauss@22999
   510
    "unify M M' = Some \<theta>"
krauss@22999
   511
krauss@22999
   512
  from unify_eliminates[OF inner]
krauss@22999
   513
  show "((N \<triangleleft> \<theta>, N' \<triangleleft> \<theta>), (M \<cdot> N, M' \<cdot> N')) \<in>?R"
krauss@22999
   514
  proof
krauss@22999
   515
    -- {* Either a variable is eliminated \ldots *}
krauss@22999
   516
    assume "(\<exists>v\<in>vars_of M \<union> vars_of M'. elim \<theta> v)"
krauss@22999
   517
    then obtain v 
krauss@22999
   518
	  where "elim \<theta> v" 
krauss@22999
   519
	  and "v\<in>vars_of M \<union> vars_of M'" by auto
krauss@22999
   520
    with unify_vars[OF inner]
krauss@22999
   521
    have "vars_of (N\<triangleleft>\<theta>) \<union> vars_of (N'\<triangleleft>\<theta>)
krauss@22999
   522
	  \<subset> vars_of (M\<cdot>N) \<union> vars_of (M'\<cdot>N')"
krauss@22999
   523
	  by auto
krauss@22999
   524
    
krauss@22999
   525
    thus ?thesis
krauss@22999
   526
      by (auto intro!: measures_less intro: psubset_card_mono)
krauss@22999
   527
  next
krauss@22999
   528
    -- {* Or the substitution is empty *}
krauss@22999
   529
    assume "\<theta> =\<^sub>s []"
krauss@22999
   530
    hence "N \<triangleleft> \<theta> = N" 
krauss@22999
   531
	  and "N' \<triangleleft> \<theta> = N'" by auto
krauss@22999
   532
    thus ?thesis 
krauss@22999
   533
       by (auto intro!: measures_less intro: psubset_card_mono)
krauss@22999
   534
  qed
krauss@22999
   535
qed
krauss@22999
   536
wenzelm@23219
   537
end