(* Title: HOL/ex/Unification.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Author: Konrad Slind, TUM & Cambridge University Computer Laboratory
Author: Alexander Krauss, TUM
*)
section \<open>Substitution and Unification\<close>
theory Unification
imports Main
begin
text \<open>
Implements Manna \& Waldinger's formalization, with Paulson's
simplifications, and some new simplifications by Slind and Krauss.
Z Manna \& R Waldinger, Deductive Synthesis of the Unification
Algorithm. SCP 1 (1981), 5-48
L C Paulson, Verifying the Unification Algorithm in LCF. SCP 5
(1985), 143-170
K Slind, Reasoning about Terminating Functional Programs,
Ph.D. thesis, TUM, 1999, Sect. 5.8
A Krauss, Partial and Nested Recursive Function Definitions in
Higher-Order Logic, JAR 44(4):303-336, 2010. Sect. 6.3
\<close>
subsection \<open>Terms\<close>
text \<open>Binary trees with leaves that are constants or variables.\<close>
datatype 'a trm =
Var 'a
| Const 'a
| Comb "'a trm" "'a trm" (infix "\<cdot>" 60)
primrec vars_of :: "'a trm \<Rightarrow> 'a set"
where
"vars_of (Var v) = {v}"
| "vars_of (Const c) = {}"
| "vars_of (M \<cdot> N) = vars_of M \<union> vars_of N"
fun occs :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool" (infixl "\<prec>" 54)
where
"u \<prec> Var v \<longleftrightarrow> False"
| "u \<prec> Const c \<longleftrightarrow> False"
| "u \<prec> M \<cdot> N \<longleftrightarrow> u = M \<or> u = N \<or> u \<prec> M \<or> u \<prec> N"
lemma finite_vars_of[intro]: "finite (vars_of t)"
by (induct t) simp_all
lemma vars_iff_occseq: "x \<in> vars_of t \<longleftrightarrow> Var x \<prec> t \<or> Var x = t"
by (induct t) auto
lemma occs_vars_subset: "M \<prec> N \<Longrightarrow> vars_of M \<subseteq> vars_of N"
by (induct N) auto
subsection \<open>Substitutions\<close>
type_synonym 'a subst = "('a \<times> 'a trm) list"
fun assoc :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> 'b"
where
"assoc x d [] = d"
| "assoc x d ((p,q)#t) = (if x = p then q else assoc x d t)"
primrec subst :: "'a trm \<Rightarrow> 'a subst \<Rightarrow> 'a trm" (infixl "\<lhd>" 55)
where
"(Var v) \<lhd> s = assoc v (Var v) s"
| "(Const c) \<lhd> s = (Const c)"
| "(M \<cdot> N) \<lhd> s = (M \<lhd> s) \<cdot> (N \<lhd> s)"
definition subst_eq (infixr "\<doteq>" 52)
where
"s1 \<doteq> s2 \<longleftrightarrow> (\<forall>t. t \<lhd> s1 = t \<lhd> s2)"
fun comp :: "'a subst \<Rightarrow> 'a subst \<Rightarrow> 'a subst" (infixl "\<lozenge>" 56)
where
"[] \<lozenge> bl = bl"
| "((a,b) # al) \<lozenge> bl = (a, b \<lhd> bl) # (al \<lozenge> bl)"
lemma subst_Nil[simp]: "t \<lhd> [] = t"
by (induct t) auto
lemma subst_mono: "t \<prec> u \<Longrightarrow> t \<lhd> s \<prec> u \<lhd> s"
by (induct u) auto
lemma agreement: "(t \<lhd> r = t \<lhd> s) \<longleftrightarrow> (\<forall>v \<in> vars_of t. Var v \<lhd> r = Var v \<lhd> s)"
by (induct t) auto
lemma repl_invariance: "v \<notin> vars_of t \<Longrightarrow> t \<lhd> (v,u) # s = t \<lhd> s"
by (simp add: agreement)
lemma remove_var: "v \<notin> vars_of s \<Longrightarrow> v \<notin> vars_of (t \<lhd> [(v, s)])"
by (induct t) simp_all
lemma subst_refl[iff]: "s \<doteq> s"
by (auto simp:subst_eq_def)
lemma subst_sym[sym]: "\<lbrakk>s1 \<doteq> s2\<rbrakk> \<Longrightarrow> s2 \<doteq> s1"
by (auto simp:subst_eq_def)
lemma subst_trans[trans]: "\<lbrakk>s1 \<doteq> s2; s2 \<doteq> s3\<rbrakk> \<Longrightarrow> s1 \<doteq> s3"
by (auto simp:subst_eq_def)
lemma subst_no_occs: "\<not> Var v \<prec> t \<Longrightarrow> Var v \<noteq> t
\<Longrightarrow> t \<lhd> [(v,s)] = t"
by (induct t) auto
lemma comp_Nil[simp]: "\<sigma> \<lozenge> [] = \<sigma>"
by (induct \<sigma>) auto
lemma subst_comp[simp]: "t \<lhd> (r \<lozenge> s) = t \<lhd> r \<lhd> s"
proof (induct t)
case (Var v) thus ?case
by (induct r) auto
qed auto
lemma subst_eq_intro[intro]: "(\<And>t. t \<lhd> \<sigma> = t \<lhd> \<theta>) \<Longrightarrow> \<sigma> \<doteq> \<theta>"
by (auto simp:subst_eq_def)
lemma subst_eq_dest[dest]: "s1 \<doteq> s2 \<Longrightarrow> t \<lhd> s1 = t \<lhd> s2"
by (auto simp:subst_eq_def)
lemma comp_assoc: "(a \<lozenge> b) \<lozenge> c \<doteq> a \<lozenge> (b \<lozenge> c)"
by auto
lemma subst_cong: "\<lbrakk>\<sigma> \<doteq> \<sigma>'; \<theta> \<doteq> \<theta>'\<rbrakk> \<Longrightarrow> (\<sigma> \<lozenge> \<theta>) \<doteq> (\<sigma>' \<lozenge> \<theta>')"
by (auto simp: subst_eq_def)
lemma var_self: "[(v, Var v)] \<doteq> []"
proof
fix t show "t \<lhd> [(v, Var v)] = t \<lhd> []"
by (induct t) simp_all
qed
lemma var_same[simp]: "[(v, t)] \<doteq> [] \<longleftrightarrow> t = Var v"
by (metis assoc.simps(2) subst.simps(1) subst_eq_def var_self)
subsection \<open>Unifiers and Most General Unifiers\<close>
definition Unifier :: "'a subst \<Rightarrow> 'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"
where "Unifier \<sigma> t u \<longleftrightarrow> (t \<lhd> \<sigma> = u \<lhd> \<sigma>)"
definition MGU :: "'a subst \<Rightarrow> 'a trm \<Rightarrow> 'a trm \<Rightarrow> bool" where
"MGU \<sigma> t u \<longleftrightarrow>
Unifier \<sigma> t u \<and> (\<forall>\<theta>. Unifier \<theta> t u \<longrightarrow> (\<exists>\<gamma>. \<theta> \<doteq> \<sigma> \<lozenge> \<gamma>))"
lemma MGUI[intro]:
"\<lbrakk>t \<lhd> \<sigma> = u \<lhd> \<sigma>; \<And>\<theta>. t \<lhd> \<theta> = u \<lhd> \<theta> \<Longrightarrow> \<exists>\<gamma>. \<theta> \<doteq> \<sigma> \<lozenge> \<gamma>\<rbrakk>
\<Longrightarrow> MGU \<sigma> t u"
by (simp only:Unifier_def MGU_def, auto)
lemma MGU_sym[sym]:
"MGU \<sigma> s t \<Longrightarrow> MGU \<sigma> t s"
by (auto simp:MGU_def Unifier_def)
lemma MGU_is_Unifier: "MGU \<sigma> t u \<Longrightarrow> Unifier \<sigma> t u"
unfolding MGU_def by (rule conjunct1)
lemma MGU_Var:
assumes "\<not> Var v \<prec> t"
shows "MGU [(v,t)] (Var v) t"
proof (intro MGUI exI)
show "Var v \<lhd> [(v,t)] = t \<lhd> [(v,t)]" using assms
by (metis assoc.simps(2) repl_invariance subst.simps(1) subst_Nil vars_iff_occseq)
next
fix \<theta> assume th: "Var v \<lhd> \<theta> = t \<lhd> \<theta>"
show "\<theta> \<doteq> [(v,t)] \<lozenge> \<theta>"
proof
fix s show "s \<lhd> \<theta> = s \<lhd> [(v,t)] \<lozenge> \<theta>" using th
by (induct s) auto
qed
qed
lemma MGU_Const: "MGU [] (Const c) (Const d) \<longleftrightarrow> c = d"
by (auto simp: MGU_def Unifier_def)
subsection \<open>The unification algorithm\<close>
function unify :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> 'a subst option"
where
"unify (Const c) (M \<cdot> N) = None"
| "unify (M \<cdot> N) (Const c) = None"
| "unify (Const c) (Var v) = Some [(v, Const c)]"
| "unify (M \<cdot> N) (Var v) = (if Var v \<prec> M \<cdot> N
then None
else Some [(v, M \<cdot> N)])"
| "unify (Var v) M = (if Var v \<prec> M
then None
else Some [(v, M)])"
| "unify (Const c) (Const d) = (if c=d then Some [] else None)"
| "unify (M \<cdot> N) (M' \<cdot> N') = (case unify M M' of
None \<Rightarrow> None |
Some \<theta> \<Rightarrow> (case unify (N \<lhd> \<theta>) (N' \<lhd> \<theta>)
of None \<Rightarrow> None |
Some \<sigma> \<Rightarrow> Some (\<theta> \<lozenge> \<sigma>)))"
by pat_completeness auto
subsection \<open>Properties used in termination proof\<close>
text \<open>Elimination of variables by a substitution:\<close>
definition
"elim \<sigma> v \<equiv> \<forall>t. v \<notin> vars_of (t \<lhd> \<sigma>)"
lemma elim_intro[intro]: "(\<And>t. v \<notin> vars_of (t \<lhd> \<sigma>)) \<Longrightarrow> elim \<sigma> v"
by (auto simp:elim_def)
lemma elim_dest[dest]: "elim \<sigma> v \<Longrightarrow> v \<notin> vars_of (t \<lhd> \<sigma>)"
by (auto simp:elim_def)
lemma elim_eq: "\<sigma> \<doteq> \<theta> \<Longrightarrow> elim \<sigma> x = elim \<theta> x"
by (auto simp:elim_def subst_eq_def)
lemma occs_elim: "\<not> Var v \<prec> t
\<Longrightarrow> elim [(v,t)] v \<or> [(v,t)] \<doteq> []"
by (metis elim_intro remove_var var_same vars_iff_occseq)
text \<open>The result of a unification never introduces new variables:\<close>
declare unify.psimps[simp]
lemma unify_vars:
assumes "unify_dom (M, N)"
assumes "unify M N = Some \<sigma>"
shows "vars_of (t \<lhd> \<sigma>) \<subseteq> vars_of M \<union> vars_of N \<union> vars_of t"
(is "?P M N \<sigma> t")
using assms
proof (induct M N arbitrary:\<sigma> t)
case (3 c v)
hence "\<sigma> = [(v, Const c)]" by simp
thus ?case by (induct t) auto
next
case (4 M N v)
hence "\<not> Var v \<prec> M \<cdot> N" by auto
with 4 have "\<sigma> = [(v, M\<cdot>N)]" by simp
thus ?case by (induct t) auto
next
case (5 v M)
hence "\<not> Var v \<prec> M" by auto
with 5 have "\<sigma> = [(v, M)]" by simp
thus ?case by (induct t) auto
next
case (7 M N M' N' \<sigma>)
then obtain \<theta>1 \<theta>2
where "unify M M' = Some \<theta>1"
and "unify (N \<lhd> \<theta>1) (N' \<lhd> \<theta>1) = Some \<theta>2"
and \<sigma>: "\<sigma> = \<theta>1 \<lozenge> \<theta>2"
and ih1: "\<And>t. ?P M M' \<theta>1 t"
and ih2: "\<And>t. ?P (N\<lhd>\<theta>1) (N'\<lhd>\<theta>1) \<theta>2 t"
by (auto split:option.split_asm)
show ?case
proof
fix v assume a: "v \<in> vars_of (t \<lhd> \<sigma>)"
show "v \<in> vars_of (M \<cdot> N) \<union> vars_of (M' \<cdot> N') \<union> vars_of t"
proof (cases "v \<notin> vars_of M \<and> v \<notin> vars_of M'
\<and> v \<notin> vars_of N \<and> v \<notin> vars_of N'")
case True
with ih1 have l:"\<And>t. v \<in> vars_of (t \<lhd> \<theta>1) \<Longrightarrow> v \<in> vars_of t"
by auto
from a and ih2[where t="t \<lhd> \<theta>1"]
have "v \<in> vars_of (N \<lhd> \<theta>1) \<union> vars_of (N' \<lhd> \<theta>1)
\<or> v \<in> vars_of (t \<lhd> \<theta>1)" unfolding \<sigma>
by auto
hence "v \<in> vars_of t"
proof
assume "v \<in> vars_of (N \<lhd> \<theta>1) \<union> vars_of (N' \<lhd> \<theta>1)"
with True show ?thesis by (auto dest:l)
next
assume "v \<in> vars_of (t \<lhd> \<theta>1)"
thus ?thesis by (rule l)
qed
thus ?thesis by auto
qed auto
qed
qed (auto split: if_split_asm)
text \<open>The result of a unification is either the identity
substitution or it eliminates a variable from one of the terms:\<close>
lemma unify_eliminates:
assumes "unify_dom (M, N)"
assumes "unify M N = Some \<sigma>"
shows "(\<exists>v\<in>vars_of M \<union> vars_of N. elim \<sigma> v) \<or> \<sigma> \<doteq> []"
(is "?P M N \<sigma>")
using assms
proof (induct M N arbitrary:\<sigma>)
case 1 thus ?case by simp
next
case 2 thus ?case by simp
next
case (3 c v)
have no_occs: "\<not> Var v \<prec> Const c" by simp
with 3 have "\<sigma> = [(v, Const c)]" by simp
with occs_elim[OF no_occs]
show ?case by auto
next
case (4 M N v)
hence no_occs: "\<not> Var v \<prec> M \<cdot> N" by auto
with 4 have "\<sigma> = [(v, M\<cdot>N)]" by simp
with occs_elim[OF no_occs]
show ?case by auto
next
case (5 v M)
hence no_occs: "\<not> Var v \<prec> M" by auto
with 5 have "\<sigma> = [(v, M)]" by simp
with occs_elim[OF no_occs]
show ?case by auto
next
case (6 c d) thus ?case
by (cases "c = d") auto
next
case (7 M N M' N' \<sigma>)
then obtain \<theta>1 \<theta>2
where "unify M M' = Some \<theta>1"
and "unify (N \<lhd> \<theta>1) (N' \<lhd> \<theta>1) = Some \<theta>2"
and \<sigma>: "\<sigma> = \<theta>1 \<lozenge> \<theta>2"
and ih1: "?P M M' \<theta>1"
and ih2: "?P (N\<lhd>\<theta>1) (N'\<lhd>\<theta>1) \<theta>2"
by (auto split:option.split_asm)
from \<open>unify_dom (M \<cdot> N, M' \<cdot> N')\<close>
have "unify_dom (M, M')"
by (rule accp_downward) (rule unify_rel.intros)
hence no_new_vars:
"\<And>t. vars_of (t \<lhd> \<theta>1) \<subseteq> vars_of M \<union> vars_of M' \<union> vars_of t"
by (rule unify_vars) (rule \<open>unify M M' = Some \<theta>1\<close>)
from ih2 show ?case
proof
assume "\<exists>v\<in>vars_of (N \<lhd> \<theta>1) \<union> vars_of (N' \<lhd> \<theta>1). elim \<theta>2 v"
then obtain v
where "v\<in>vars_of (N \<lhd> \<theta>1) \<union> vars_of (N' \<lhd> \<theta>1)"
and el: "elim \<theta>2 v" by auto
with no_new_vars show ?thesis unfolding \<sigma>
by (auto simp:elim_def)
next
assume empty[simp]: "\<theta>2 \<doteq> []"
have "\<sigma> \<doteq> (\<theta>1 \<lozenge> [])" unfolding \<sigma>
by (rule subst_cong) auto
also have "\<dots> \<doteq> \<theta>1" by auto
finally have "\<sigma> \<doteq> \<theta>1" .
from ih1 show ?thesis
proof
assume "\<exists>v\<in>vars_of M \<union> vars_of M'. elim \<theta>1 v"
with elim_eq[OF \<open>\<sigma> \<doteq> \<theta>1\<close>]
show ?thesis by auto
next
note \<open>\<sigma> \<doteq> \<theta>1\<close>
also assume "\<theta>1 \<doteq> []"
finally show ?thesis ..
qed
qed
qed
declare unify.psimps[simp del]
subsection \<open>Termination proof\<close>
termination unify
proof
let ?R = "measures [\<lambda>(M,N). card (vars_of M \<union> vars_of N),
\<lambda>(M, N). size M]"
show "wf ?R" by simp
fix M N M' N' :: "'a trm"
show "((M, M'), (M \<cdot> N, M' \<cdot> N')) \<in> ?R" \<comment> "Inner call"
by (rule measures_lesseq) (auto intro: card_mono)
fix \<theta> \<comment> "Outer call"
assume inner: "unify_dom (M, M')"
"unify M M' = Some \<theta>"
from unify_eliminates[OF inner]
show "((N \<lhd> \<theta>, N' \<lhd> \<theta>), (M \<cdot> N, M' \<cdot> N')) \<in>?R"
proof
\<comment> \<open>Either a variable is eliminated \ldots\<close>
assume "(\<exists>v\<in>vars_of M \<union> vars_of M'. elim \<theta> v)"
then obtain v
where "elim \<theta> v"
and "v\<in>vars_of M \<union> vars_of M'" by auto
with unify_vars[OF inner]
have "vars_of (N\<lhd>\<theta>) \<union> vars_of (N'\<lhd>\<theta>)
\<subset> vars_of (M\<cdot>N) \<union> vars_of (M'\<cdot>N')"
by auto
thus ?thesis
by (auto intro!: measures_less intro: psubset_card_mono)
next
\<comment> \<open>Or the substitution is empty\<close>
assume "\<theta> \<doteq> []"
hence "N \<lhd> \<theta> = N"
and "N' \<lhd> \<theta> = N'" by auto
thus ?thesis
by (auto intro!: measures_less intro: psubset_card_mono)
qed
qed
subsection \<open>Unification returns a Most General Unifier\<close>
lemma unify_computes_MGU:
"unify M N = Some \<sigma> \<Longrightarrow> MGU \<sigma> M N"
proof (induct M N arbitrary: \<sigma> rule: unify.induct)
case (7 M N M' N' \<sigma>) \<comment> "The interesting case"
then obtain \<theta>1 \<theta>2
where "unify M M' = Some \<theta>1"
and "unify (N \<lhd> \<theta>1) (N' \<lhd> \<theta>1) = Some \<theta>2"
and \<sigma>: "\<sigma> = \<theta>1 \<lozenge> \<theta>2"
and MGU_inner: "MGU \<theta>1 M M'"
and MGU_outer: "MGU \<theta>2 (N \<lhd> \<theta>1) (N' \<lhd> \<theta>1)"
by (auto split:option.split_asm)
show ?case
proof
from MGU_inner and MGU_outer
have "M \<lhd> \<theta>1 = M' \<lhd> \<theta>1"
and "N \<lhd> \<theta>1 \<lhd> \<theta>2 = N' \<lhd> \<theta>1 \<lhd> \<theta>2"
unfolding MGU_def Unifier_def
by auto
thus "M \<cdot> N \<lhd> \<sigma> = M' \<cdot> N' \<lhd> \<sigma>" unfolding \<sigma>
by simp
next
fix \<sigma>' assume "M \<cdot> N \<lhd> \<sigma>' = M' \<cdot> N' \<lhd> \<sigma>'"
hence "M \<lhd> \<sigma>' = M' \<lhd> \<sigma>'"
and Ns: "N \<lhd> \<sigma>' = N' \<lhd> \<sigma>'" by auto
with MGU_inner obtain \<delta>
where eqv: "\<sigma>' \<doteq> \<theta>1 \<lozenge> \<delta>"
unfolding MGU_def Unifier_def
by auto
from Ns have "N \<lhd> \<theta>1 \<lhd> \<delta> = N' \<lhd> \<theta>1 \<lhd> \<delta>"
by (simp add:subst_eq_dest[OF eqv])
with MGU_outer obtain \<rho>
where eqv2: "\<delta> \<doteq> \<theta>2 \<lozenge> \<rho>"
unfolding MGU_def Unifier_def
by auto
have "\<sigma>' \<doteq> \<sigma> \<lozenge> \<rho>" unfolding \<sigma>
by (rule subst_eq_intro, auto simp:subst_eq_dest[OF eqv] subst_eq_dest[OF eqv2])
thus "\<exists>\<gamma>. \<sigma>' \<doteq> \<sigma> \<lozenge> \<gamma>" ..
qed
qed (auto simp: MGU_Const intro: MGU_Var MGU_Var[symmetric] split: if_split_asm)
subsection \<open>Unification returns Idempotent Substitution\<close>
definition Idem :: "'a subst \<Rightarrow> bool"
where "Idem s \<longleftrightarrow> (s \<lozenge> s) \<doteq> s"
lemma Idem_Nil [iff]: "Idem []"
by (simp add: Idem_def)
lemma Var_Idem:
assumes "~ (Var v \<prec> t)" shows "Idem [(v,t)]"
unfolding Idem_def
proof
from assms have [simp]: "t \<lhd> [(v, t)] = t"
by (metis assoc.simps(2) subst.simps(1) subst_no_occs)
fix s show "s \<lhd> [(v, t)] \<lozenge> [(v, t)] = s \<lhd> [(v, t)]"
by (induct s) auto
qed
lemma Unifier_Idem_subst:
"Idem(r) \<Longrightarrow> Unifier s (t \<lhd> r) (u \<lhd> r) \<Longrightarrow>
Unifier (r \<lozenge> s) (t \<lhd> r) (u \<lhd> r)"
by (simp add: Idem_def Unifier_def subst_eq_def)
lemma Idem_comp:
"Idem r \<Longrightarrow> Unifier s (t \<lhd> r) (u \<lhd> r) \<Longrightarrow>
(!!q. Unifier q (t \<lhd> r) (u \<lhd> r) \<Longrightarrow> s \<lozenge> q \<doteq> q) \<Longrightarrow>
Idem (r \<lozenge> s)"
apply (frule Unifier_Idem_subst, blast)
apply (force simp add: Idem_def subst_eq_def)
done
theorem unify_gives_Idem:
"unify M N = Some \<sigma> \<Longrightarrow> Idem \<sigma>"
proof (induct M N arbitrary: \<sigma> rule: unify.induct)
case (7 M M' N N' \<sigma>)
then obtain \<theta>1 \<theta>2
where "unify M N = Some \<theta>1"
and \<theta>2: "unify (M' \<lhd> \<theta>1) (N' \<lhd> \<theta>1) = Some \<theta>2"
and \<sigma>: "\<sigma> = \<theta>1 \<lozenge> \<theta>2"
and "Idem \<theta>1"
and "Idem \<theta>2"
by (auto split: option.split_asm)
from \<theta>2 have "Unifier \<theta>2 (M' \<lhd> \<theta>1) (N' \<lhd> \<theta>1)"
by (rule unify_computes_MGU[THEN MGU_is_Unifier])
with \<open>Idem \<theta>1\<close>
show "Idem \<sigma>" unfolding \<sigma>
proof (rule Idem_comp)
fix \<sigma> assume "Unifier \<sigma> (M' \<lhd> \<theta>1) (N' \<lhd> \<theta>1)"
with \<theta>2 obtain \<gamma> where \<sigma>: "\<sigma> \<doteq> \<theta>2 \<lozenge> \<gamma>"
using unify_computes_MGU MGU_def by blast
have "\<theta>2 \<lozenge> \<sigma> \<doteq> \<theta>2 \<lozenge> (\<theta>2 \<lozenge> \<gamma>)" by (rule subst_cong) (auto simp: \<sigma>)
also have "... \<doteq> (\<theta>2 \<lozenge> \<theta>2) \<lozenge> \<gamma>" by (rule comp_assoc[symmetric])
also have "... \<doteq> \<theta>2 \<lozenge> \<gamma>" by (rule subst_cong) (auto simp: \<open>Idem \<theta>2\<close>[unfolded Idem_def])
also have "... \<doteq> \<sigma>" by (rule \<sigma>[symmetric])
finally show "\<theta>2 \<lozenge> \<sigma> \<doteq> \<sigma>" .
qed
qed (auto intro!: Var_Idem split: option.splits if_splits)
end