Theory Sexp

theory Sexp
imports Old_Datatype
```(*  Title:      HOL/Induct/Sexp.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1992  University of Cambridge

S-expressions, general binary trees for defining recursive data
structures by hand.
*)

theory Sexp
imports "HOL-Library.Old_Datatype"
begin

type_synonym 'a item = "'a Old_Datatype.item"
abbreviation "Leaf == Old_Datatype.Leaf"
abbreviation "Numb == Old_Datatype.Numb"

inductive_set
sexp      :: "'a item set"
where
LeafI:  "Leaf(a) ∈ sexp"
| NumbI:  "Numb(i) ∈ sexp"
| SconsI: "[| M ∈ sexp;  N ∈ sexp |] ==> Scons M N ∈ sexp"

definition
sexp_case :: "['a=>'b, nat=>'b, ['a item, 'a item]=>'b,
'a item] => 'b" where
"sexp_case c d e M = (THE z. (EX x.   M=Leaf(x) & z=c(x))
| (EX k.   M=Numb(k) & z=d(k))
| (EX N1 N2. M = Scons N1 N2  & z=e N1 N2))"

definition
pred_sexp :: "('a item * 'a item)set" where
"pred_sexp = (⋃M ∈ sexp. ⋃N ∈ sexp. {(M, Scons M N), (N, Scons M N)})"

definition
sexp_rec  :: "['a item, 'a=>'b, nat=>'b,
['a item, 'a item, 'b, 'b]=>'b] => 'b" where
"sexp_rec M c d e = wfrec pred_sexp
(%g. sexp_case c d (%N1 N2. e N1 N2 (g N1) (g N2))) M"

(** sexp_case **)

lemma sexp_case_Leaf [simp]: "sexp_case c d e (Leaf a) = c(a)"
by (simp add: sexp_case_def, blast)

lemma sexp_case_Numb [simp]: "sexp_case c d e (Numb k) = d(k)"
by (simp add: sexp_case_def, blast)

lemma sexp_case_Scons [simp]: "sexp_case c d e (Scons M N) = e M N"
by (simp add: sexp_case_def)

(** Introduction rules for sexp constructors **)

lemma sexp_In0I: "M ∈ sexp ==> In0(M) ∈ sexp"
apply (simp add: In0_def)
apply (erule sexp.NumbI [THEN sexp.SconsI])
done

lemma sexp_In1I: "M ∈ sexp ==> In1(M) ∈ sexp"
apply (simp add: In1_def)
apply (erule sexp.NumbI [THEN sexp.SconsI])
done

declare sexp.intros [intro,simp]

lemma range_Leaf_subset_sexp: "range(Leaf) <= sexp"
by blast

lemma Scons_D: "Scons M N ∈ sexp ==> M ∈ sexp & N ∈ sexp"
by (induct S == "Scons M N" set: sexp) auto

(** Introduction rules for 'pred_sexp' **)

lemma pred_sexp_subset_Sigma: "pred_sexp <= sexp × sexp"
by (simp add: pred_sexp_def) blast

(* (a,b) ∈ pred_sexp^+ ==> a ∈ sexp *)
lemmas trancl_pred_sexpD1 =
pred_sexp_subset_Sigma
[THEN trancl_subset_Sigma, THEN subsetD, THEN SigmaD1]
and trancl_pred_sexpD2 =
pred_sexp_subset_Sigma
[THEN trancl_subset_Sigma, THEN subsetD, THEN SigmaD2]

lemma pred_sexpI1:
"[| M ∈ sexp;  N ∈ sexp |] ==> (M, Scons M N) ∈ pred_sexp"
by (simp add: pred_sexp_def, blast)

lemma pred_sexpI2:
"[| M ∈ sexp;  N ∈ sexp |] ==> (N, Scons M N) ∈ pred_sexp"
by (simp add: pred_sexp_def, blast)

(*Combinations involving transitivity and the rules above*)
lemmas pred_sexp_t1 [simp] = pred_sexpI1 [THEN r_into_trancl]
and    pred_sexp_t2 [simp] = pred_sexpI2 [THEN r_into_trancl]

lemmas pred_sexp_trans1 [simp] = trans_trancl [THEN transD, OF _ pred_sexp_t1]
and    pred_sexp_trans2 [simp] = trans_trancl [THEN transD, OF _ pred_sexp_t2]

(*Proves goals of the form (M,N):pred_sexp^+ provided M,N:sexp*)
declare cut_apply [simp]

lemma pred_sexpE:
"[| p ∈ pred_sexp;
!!M N. [| p = (M, Scons M N);  M ∈ sexp;  N ∈ sexp |] ==> R;
!!M N. [| p = (N, Scons M N);  M ∈ sexp;  N ∈ sexp |] ==> R
|] ==> R"
by (simp add: pred_sexp_def, blast)

lemma wf_pred_sexp: "wf(pred_sexp)"
apply (rule pred_sexp_subset_Sigma [THEN wfI])
apply (erule sexp.induct)
apply (blast elim!: pred_sexpE)+
done

(*** sexp_rec -- by wf recursion on pred_sexp ***)

lemma sexp_rec_unfold_lemma:
"(%M. sexp_rec M c d e) ==
wfrec pred_sexp (%g. sexp_case c d (%N1 N2. e N1 N2 (g N1) (g N2)))"
by (simp add: sexp_rec_def)

lemmas sexp_rec_unfold = def_wfrec [OF sexp_rec_unfold_lemma wf_pred_sexp]

(* sexp_rec a c d e =
sexp_case c d
(%N1 N2.
e N1 N2 (cut (%M. sexp_rec M c d e) pred_sexp a N1)
(cut (%M. sexp_rec M c d e) pred_sexp a N2)) a
*)

(** conversion rules **)

lemma sexp_rec_Leaf: "sexp_rec (Leaf a) c d h = c(a)"
apply (subst sexp_rec_unfold)
apply (rule sexp_case_Leaf)
done

lemma sexp_rec_Numb: "sexp_rec (Numb k) c d h = d(k)"
apply (subst sexp_rec_unfold)
apply (rule sexp_case_Numb)
done

lemma sexp_rec_Scons: "[| M ∈ sexp;  N ∈ sexp |] ==>
sexp_rec (Scons M N) c d h = h M N (sexp_rec M c d h) (sexp_rec N c d h)"
apply (rule sexp_rec_unfold [THEN trans])
apply (simp add: pred_sexpI1 pred_sexpI2)
done

end
```