# Theory ListBeta

theory ListBeta
imports ListApplication ListOrder
```(*  Title:      HOL/Proofs/Lambda/ListBeta.thy
Author:     Tobias Nipkow
*)

section ‹Lifting beta-reduction to lists›

theory ListBeta imports ListApplication ListOrder begin

text ‹
Lifting beta-reduction to lists of terms, reducing exactly one element.
›

abbreviation
list_beta :: "dB list => dB list => bool"  (infixl "=>" 50) where
"rs => ss == step1 beta rs ss"

"Var n °° rs →⇩β v ⟹ ∃ss. rs => ss ∧ v = Var n °° ss"
apply (induct u == "Var n °° rs" v arbitrary: rs set: beta)
apply simp
apply (rule_tac xs = rs in rev_exhaust)
apply simp
apply (atomize, force intro: append_step1I)
apply (rule_tac xs = rs in rev_exhaust)
apply simp
apply (auto 0 3 intro: disjI2 [THEN append_step1I])
done

lemma apps_betasE [elim!]:
assumes major: "r °° rs →⇩β s"
and cases: "!!r'. [| r →⇩β r'; s = r' °° rs |] ==> R"
"!!rs'. [| rs => rs'; s = r °° rs' |] ==> R"
"!!t u us. [| r = Abs t; rs = u # us; s = t[u/0] °° us |] ==> R"
shows R
proof -
from major have
"(∃r'. r →⇩β r' ∧ s = r' °° rs) ∨
(∃rs'. rs => rs' ∧ s = r °° rs') ∨
(∃t u us. r = Abs t ∧ rs = u # us ∧ s = t[u/0] °° us)"
apply (induct u == "r °° rs" s arbitrary: r rs set: beta)
apply (case_tac r)
apply simp
apply (split if_split_asm)
apply simp
apply blast
apply simp
apply (split if_split_asm)
apply simp
apply simp
apply (drule App_eq_foldl_conv [THEN iffD1])
apply (split if_split_asm)
apply simp
apply blast
apply (force intro!: disjI1 [THEN append_step1I])
apply (drule App_eq_foldl_conv [THEN iffD1])
apply (split if_split_asm)
apply simp
apply blast
apply (clarify, auto 0 3 intro!: exI intro: append_step1I)
done
with cases show ?thesis by blast
qed

lemma apps_preserves_beta [simp]:
"r →⇩β s ==> r °° ss →⇩β s °° ss"
by (induct ss rule: rev_induct) auto

lemma apps_preserves_beta2 [simp]:
"r →⇩β⇧* s ==> r °° ss →⇩β⇧* s °° ss"
apply (induct set: rtranclp)
apply blast
apply (blast intro: apps_preserves_beta rtranclp.rtrancl_into_rtrancl)
done

lemma apps_preserves_betas [simp]:
"rs => ss ⟹ r °° rs →⇩β r °° ss"
apply (induct rs arbitrary: ss rule: rev_induct)
apply simp
apply simp
apply (rule_tac xs = ss in rev_exhaust)
apply simp
apply simp
apply (drule Snoc_step1_SnocD)
apply blast
done

end
```