# Theory ListApplication

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

section ‹Application of a term to a list of terms›

theory ListApplication imports Lambda begin

abbreviation
list_application :: "dB => dB list => dB"  (infixl "°°" 150) where
"t °° ts == foldl (op °) t ts"

lemma apps_eq_tail_conv [iff]: "(r °° ts = s °° ts) = (r = s)"
by (induct ts rule: rev_induct) auto

lemma Var_eq_apps_conv [iff]: "(Var m = s °° ss) = (Var m = s ∧ ss = [])"
by (induct ss arbitrary: s) auto

lemma Var_apps_eq_Var_apps_conv [iff]:
"(Var m °° rs = Var n °° ss) = (m = n ∧ rs = ss)"
apply (induct rs arbitrary: ss rule: rev_induct)
apply simp
apply blast
apply (induct_tac ss rule: rev_induct)
apply auto
done

lemma App_eq_foldl_conv:
"(r ° s = t °° ts) =
(if ts = [] then r ° s = t
else (∃ss. ts = ss @ [s] ∧ r = t °° ss))"
apply (rule_tac xs = ts in rev_exhaust)
apply auto
done

lemma Abs_eq_apps_conv [iff]:
"(Abs r = s °° ss) = (Abs r = s ∧ ss = [])"
by (induct ss rule: rev_induct) auto

lemma apps_eq_Abs_conv [iff]: "(s °° ss = Abs r) = (s = Abs r ∧ ss = [])"
by (induct ss rule: rev_induct) auto

lemma Abs_apps_eq_Abs_apps_conv [iff]:
"(Abs r °° rs = Abs s °° ss) = (r = s ∧ rs = ss)"
apply (induct rs arbitrary: ss rule: rev_induct)
apply simp
apply blast
apply (induct_tac ss rule: rev_induct)
apply auto
done

lemma Abs_App_neq_Var_apps [iff]:
"Abs s ° t ≠ Var n °° ss"
by (induct ss arbitrary: s t rule: rev_induct) auto

lemma Var_apps_neq_Abs_apps [iff]:
"Var n °° ts ≠ Abs r °° ss"
apply (induct ss arbitrary: ts rule: rev_induct)
apply simp
apply (induct_tac ts rule: rev_induct)
apply auto
done

"∃ts h. t = h °° ts ∧ ((∃n. h = Var n) ∨ (∃u. h = Abs u))"
apply (induct t)
apply (rule_tac x = "[]" in exI)
apply simp
apply clarify
apply (rename_tac ts1 ts2 h1 h2)
apply (rule_tac x = "ts1 @ [h2 °° ts2]" in exI)
apply simp
apply simp
done

lemma size_apps [simp]:
"size (r °° rs) = size r + foldl (op +) 0 (map size rs) + length rs"
by (induct rs rule: rev_induct) auto

lemma lem0: "[| (0::nat) < k; m <= n |] ==> m < n + k"
by simp

lemma lift_map [simp]:
"lift (t °° ts) i = lift t i °° map (λt. lift t i) ts"
by (induct ts arbitrary: t) simp_all

lemma subst_map [simp]:
"subst (t °° ts) u i = subst t u i °° map (λt. subst t u i) ts"
by (induct ts arbitrary: t) simp_all

lemma app_last: "(t °° ts) ° u = t °° (ts @ [u])"
by simp

text ‹\medskip A customized induction schema for ‹°°›.›

lemma lem:
assumes "!!n ts. ∀t ∈ set ts. P t ==> P (Var n °° ts)"
and "!!u ts. [| P u; ∀t ∈ set ts. P t |] ==> P (Abs u °° ts)"
shows "size t = n ⟹ P t"
apply (induct n arbitrary: t rule: nat_less_induct)
apply (cut_tac t = t in ex_head_tail)
apply clarify
apply (erule disjE)
apply clarify
apply (rule assms)
apply clarify
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule mp, rule refl)
apply simp
apply (simp only: foldl_conv_fold add.commute fold_plus_sum_list_rev)
apply clarify
apply (rule assms)
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule mp, rule refl)
apply simp
apply clarify
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule mp, rule refl)
apply simp
apply (rule le_imp_less_Suc)
apply (simp only: foldl_conv_fold add.commute fold_plus_sum_list_rev)
done

theorem Apps_dB_induct:
assumes "!!n ts. ∀t ∈ set ts. P t ==> P (Var n °° ts)"
and "!!u ts. [| P u; ∀t ∈ set ts. P t |] ==> P (Abs u °° ts)"
shows "P t"
apply (rule_tac t = t in lem)
prefer 3
apply (rule refl)
using assms apply iprover+
done

end

```