# Theory ParRed

theory ParRed
imports Lambda Commutation
```(*  Title:      HOL/Proofs/Lambda/ParRed.thy
Author:     Tobias Nipkow

Properties of => and "cd", in particular the diamond property of => and
confluence of beta.
*)

section ‹Parallel reduction and a complete developments›

theory ParRed imports Lambda Commutation begin

subsection ‹Parallel reduction›

inductive par_beta :: "[dB, dB] => bool"  (infixl "=>" 50)
where
var [simp, intro!]: "Var n => Var n"
| abs [simp, intro!]: "s => t ==> Abs s => Abs t"
| app [simp, intro!]: "[| s => s'; t => t' |] ==> s ° t => s' ° t'"
| beta [simp, intro!]: "[| s => s'; t => t' |] ==> (Abs s) ° t => s'[t'/0]"

inductive_cases par_beta_cases [elim!]:
"Var n => t"
"Abs s => Abs t"
"(Abs s) ° t => u"
"s ° t => u"
"Abs s => t"

subsection ‹Inclusions›

text ‹‹beta ⊆ par_beta ⊆ beta⇧*› \medskip›

lemma par_beta_varL [simp]:
"(Var n => t) = (t = Var n)"
by blast

lemma par_beta_refl [simp]: "t => t"  (* par_beta_refl [intro!] causes search to blow up *)
by (induct t) simp_all

lemma beta_subset_par_beta: "beta <= par_beta"
apply (rule predicate2I)
apply (erule beta.induct)
apply (blast intro!: par_beta_refl)+
done

lemma par_beta_subset_beta: "par_beta ≤ beta⇧*⇧*"
apply (rule predicate2I)
apply (erule par_beta.induct)
apply blast
apply (blast del: rtranclp.rtrancl_refl intro: rtranclp.rtrancl_into_rtrancl)+
― ‹@{thm[source] rtrancl_refl} complicates the proof by increasing the branching factor›
done

subsection ‹Misc properties of ‹par_beta››

lemma par_beta_lift [simp]:
"t => t' ⟹ lift t n => lift t' n"
by (induct t arbitrary: t' n) fastforce+

lemma par_beta_subst:
"s => s' ⟹ t => t' ⟹ t[s/n] => t'[s'/n]"
apply (induct t arbitrary: s s' t' n)
apply (erule par_beta_cases)
apply simp
apply (fastforce intro!: par_beta_lift)
apply fastforce
done

subsection ‹Confluence (directly)›

lemma diamond_par_beta: "diamond par_beta"
apply (unfold diamond_def commute_def square_def)
apply (rule impI [THEN allI [THEN allI]])
apply (erule par_beta.induct)
apply (blast intro!: par_beta_subst)+
done

subsection ‹Complete developments›

fun
cd :: "dB => dB"
where
"cd (Var n) = Var n"
| "cd (Var n ° t) = Var n ° cd t"
| "cd ((s1 ° s2) ° t) = cd (s1 ° s2) ° cd t"
| "cd (Abs u ° t) = (cd u)[cd t/0]"
| "cd (Abs s) = Abs (cd s)"

lemma par_beta_cd: "s => t ⟹ t => cd s"
apply (induct s arbitrary: t rule: cd.induct)
apply auto
apply (fast intro!: par_beta_subst)
done

subsection ‹Confluence (via complete developments)›

lemma diamond_par_beta2: "diamond par_beta"
apply (unfold diamond_def commute_def square_def)
apply (blast intro: par_beta_cd)
done

theorem beta_confluent: "confluent beta"
apply (rule diamond_par_beta2 diamond_to_confluence
par_beta_subset_beta beta_subset_par_beta)+
done

end
```