# Theory SList

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

This theory is strictly of historical interest. It illustrates how
recursive datatypes can be constructed in higher-order logic from
first principles. Isabelle's datatype package automates a
construction of this sort.

Enriched theory of lists; mutual indirect recursive data-types.

Definition of type 'a list (strict lists) by a least fixed point

We use          list(A) == lfp(%Z. {NUMB(0)} <+> A × Z)
and not         list    == lfp(%Z. {NUMB(0)} <+> range(Leaf) × Z)

so that list can serve as a "functor" for defining other recursive types.

This enables the conservative construction of mutual recursive datatypes
such as

datatype 'a m = Node 'a * 'a m list
*)

section ‹Extended List Theory (old)›

theory SList
imports Sexp
begin

(*Hilbert_Choice is needed for the function "inv"*)

(* *********************************************************************** *)
(*                                                                         *)
(* Building up data type                                                   *)
(*                                                                         *)
(* *********************************************************************** *)

(* Defining the Concrete Constructors *)
definition
NIL  :: "'a item" where
"NIL = In0(Numb(0))"

definition
CONS :: "['a item, 'a item] => 'a item" where
"CONS M N = In1(Scons M N)"

inductive_set
list :: "'a item set => 'a item set"
for A :: "'a item set"
where
NIL_I:  "NIL: list A"
| CONS_I: "[| a: A;  M: list A |] ==> CONS a M : list A"

definition "List = list (range Leaf)"

typedef 'a list = "List :: 'a item set"
morphisms Rep_List Abs_List
unfolding List_def by (blast intro: list.NIL_I)

abbreviation "Case == Old_Datatype.Case"
abbreviation "Split == Old_Datatype.Split"

definition
List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b" where
"List_case c d = Case(%x. c)(Split(d))"

definition
List_rec  :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b" where
"List_rec M c d = wfrec (pred_sexp^+)
(%g. List_case c (%x y. d x y (g y))) M"

(* *********************************************************************** *)
(*                                                                         *)
(* Abstracting data type                                                   *)
(*                                                                         *)
(* *********************************************************************** *)

(*Declaring the abstract list constructors*)

no_translations
"[x, xs]" == "x#[xs]"
"[x]" == "x#[]"
no_notation
Nil  ("[]") and
Cons (infixr "#" 65)

definition
Nil       :: "'a list"                               ("[]") where
"Nil  = Abs_List(NIL)"

definition
"Cons"       :: "['a, 'a list] => 'a list"           (infixr "#" 65) where
"x#xs = Abs_List(CONS (Leaf x)(Rep_List xs))"

definition
(* list Recursion -- the trancl is Essential; see list.ML *)
list_rec  :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b" where
"list_rec l c d =
List_rec(Rep_List l) c (%x y r. d(inv Leaf x)(Abs_List y) r)"

definition
list_case :: "['b, ['a, 'a list]=>'b, 'a list] => 'b" where
"list_case a f xs = list_rec xs a (%x xs r. f x xs)"

(* list Enumeration *)
translations
"[x, xs]" == "x#[xs]"
"[x]"     == "x#[]"

"case xs of [] => a | y#ys => b" == "CONST list_case(a, %y ys. b, xs)"

(* *********************************************************************** *)
(*                                                                         *)
(* Generalized Map Functionals                                             *)
(*                                                                         *)
(* *********************************************************************** *)

(* Generalized Map Functionals *)

definition
Rep_map   :: "('b => 'a item) => ('b list => 'a item)" where
"Rep_map f xs = list_rec xs  NIL(%x l r. CONS(f x) r)"

definition
Abs_map   :: "('a item => 'b) => 'a item => 'b list" where
"Abs_map g M  = List_rec M Nil (%N L r. g(N)#r)"

definition
map       :: "('a=>'b) => ('a list => 'b list)" where
"map f xs = list_rec xs [] (%x l r. f(x)#r)"

primrec take :: "['a list,nat] => 'a list" where
take_0:  "take xs 0 = []"
| take_Suc: "take xs (Suc n) = list_case [] (%x l. x # take l n) xs"

lemma ListI: "x : list (range Leaf) ==> x : List"

lemma ListD: "x : List ==> x : list (range Leaf)"

lemma list_unfold: "list(A) = usum {Numb(0)} (uprod A (list(A)))"
by (fast intro!: list.intros [unfolded NIL_def CONS_def]
elim: list.cases [unfolded NIL_def CONS_def])

(*This justifies using list in other recursive type definitions*)
lemma list_mono: "A<=B ==> list(A) <= list(B)"
apply (rule subsetI)
apply (erule list.induct)
apply (auto intro!: list.intros)
done

(*Type checking -- list creates well-founded sets*)
lemma list_sexp: "list(sexp) <= sexp"
apply (rule subsetI)
apply (erule list.induct)
apply (unfold NIL_def CONS_def)
apply (auto intro: sexp.intros sexp_In0I sexp_In1I)
done

(* A <= sexp ==> list(A) <= sexp *)
lemmas list_subset_sexp = subset_trans [OF list_mono list_sexp]

(*Induction for the type 'a list *)
lemma list_induct:
"[| P(Nil);
!!x xs. P(xs) ==> P(x # xs) |]  ==> P(l)"
apply (unfold Nil_def Cons_def)
apply (rule Rep_List_inverse [THEN subst])
(*types force good instantiation*)
apply (rule Rep_List [unfolded List_def, THEN list.induct], simp)
apply (erule Abs_List_inverse [unfolded List_def, THEN subst], blast)
done

(*** Isomorphisms ***)

lemma inj_on_Abs_list: "inj_on Abs_List (list(range Leaf))"
apply (rule inj_on_inverseI)
apply (erule Abs_List_inverse [unfolded List_def])
done

(** Distinctness of constructors **)

lemma CONS_not_NIL [iff]: "CONS M N ~= NIL"

lemmas NIL_not_CONS [iff] = CONS_not_NIL [THEN not_sym]
lemmas CONS_neq_NIL = CONS_not_NIL [THEN notE]
lemmas NIL_neq_CONS = sym [THEN CONS_neq_NIL]

lemma Cons_not_Nil [iff]: "x # xs ~= Nil"
apply (unfold Nil_def Cons_def)
apply (rule CONS_not_NIL [THEN inj_on_Abs_list [THEN inj_on_contraD]])
apply (simp_all add: list.intros rangeI Rep_List [unfolded List_def])
done

lemmas Nil_not_Cons = Cons_not_Nil [THEN not_sym]
declare Nil_not_Cons [iff]
lemmas Cons_neq_Nil = Cons_not_Nil [THEN notE]
lemmas Nil_neq_Cons = sym [THEN Cons_neq_Nil]

(** Injectiveness of CONS and Cons **)

lemma CONS_CONS_eq [iff]: "(CONS K M)=(CONS L N) = (K=L & M=N)"

(*For reasoning about abstract list constructors*)
declare Rep_List [THEN ListD, intro] ListI [intro]
declare list.intros [intro,simp]
declare Leaf_inject [dest!]

lemma Cons_Cons_eq [iff]: "(x#xs=y#ys) = (x=y & xs=ys)"
apply (subst Abs_List_inject)
done

lemmas Cons_inject2 = Cons_Cons_eq [THEN iffD1, THEN conjE]

lemma CONS_D: "CONS M N: list(A) ==> M: A & N: list(A)"
by (induct L == "CONS M N" rule: list.induct) auto

lemma sexp_CONS_D: "CONS M N: sexp ==> M: sexp & N: sexp"
apply (fast dest!: Scons_D)
done

(*Reasoning about constructors and their freeness*)

lemma not_CONS_self: "N: list(A) ==> !M. N ~= CONS M N"
apply (erule list.induct) apply simp_all done

lemma not_Cons_self2: "∀x. l ~= x#l"
by (induct l rule: list_induct) simp_all

lemma neq_Nil_conv2: "(xs ~= []) = (∃y ys. xs = y#ys)"
by (induct xs rule: list_induct) auto

(** Conversion rules for List_case: case analysis operator **)

lemma List_case_NIL [simp]: "List_case c h NIL = c"

lemma List_case_CONS [simp]: "List_case c h (CONS M N) = h M N"

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

(* The trancl(pred_sexp) is essential because pred_sexp_CONS_I1,2 would not
hold if pred_sexp^+ were changed to pred_sexp. *)

lemma List_rec_unfold_lemma:
"(%M. List_rec M c d) ==
wfrec (pred_sexp^+) (%g. List_case c (%x y. d x y (g y)))"

lemmas List_rec_unfold =
def_wfrec [OF List_rec_unfold_lemma wf_pred_sexp [THEN wf_trancl]]

(** pred_sexp lemmas **)

lemma pred_sexp_CONS_I1:
"[| M: sexp;  N: sexp |] ==> (M, CONS M N) : pred_sexp^+"

lemma pred_sexp_CONS_I2:
"[| M: sexp;  N: sexp |] ==> (N, CONS M N) : pred_sexp^+"

lemma pred_sexp_CONS_D:
"(CONS M1 M2, N) : pred_sexp^+ ==>
(M1,N) : pred_sexp^+ & (M2,N) : pred_sexp^+"
apply (frule pred_sexp_subset_Sigma [THEN trancl_subset_Sigma, THEN subsetD])
apply (blast dest!: sexp_CONS_D intro: pred_sexp_CONS_I1 pred_sexp_CONS_I2
trans_trancl [THEN transD])
done

(** Conversion rules for List_rec **)

lemma List_rec_NIL [simp]: "List_rec NIL c h = c"
apply (rule List_rec_unfold [THEN trans])
done

lemma List_rec_CONS [simp]:
"[| M: sexp;  N: sexp |]
==> List_rec (CONS M N) c h = h M N (List_rec N c h)"
apply (rule List_rec_unfold [THEN trans])
done

(*** list_rec -- by List_rec ***)

lemmas Rep_List_in_sexp =
subsetD [OF range_Leaf_subset_sexp [THEN list_subset_sexp]
Rep_List [THEN ListD]]

lemma list_rec_Nil [simp]: "list_rec Nil c h = c"
by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Nil_def)

lemma list_rec_Cons [simp]: "list_rec (a#l) c h = h a l (list_rec l c h)"
by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Cons_def
Rep_List_inverse Rep_List [THEN ListD] inj_Leaf Rep_List_in_sexp)

(*Type checking.  Useful?*)
lemma List_rec_type:
"[| M: list(A);
A<=sexp;
c: C(NIL);
!!x y r. [| x: A;  y: list(A);  r: C(y) |] ==> h x y r: C(CONS x y)
|] ==> List_rec M c h : C(M :: 'a item)"
apply (erule list.induct, simp)
apply (insert list_subset_sexp)
apply (subst List_rec_CONS, blast+)
done

(** Generalized map functionals **)

lemma Rep_map_Nil [simp]: "Rep_map f Nil = NIL"

lemma Rep_map_Cons [simp]:
"Rep_map f(x#xs) = CONS(f x)(Rep_map f xs)"

lemma Rep_map_type: "(!!x. f(x): A) ==> Rep_map f xs: list(A)"
apply (rule list_induct, auto)
done

lemma Abs_map_NIL [simp]: "Abs_map g NIL = Nil"

lemma Abs_map_CONS [simp]:
"[| M: sexp;  N: sexp |] ==> Abs_map g (CONS M N) = g(M) # Abs_map g N"

(*Eases the use of primitive recursion.*)
lemma def_list_rec_NilCons:
"[| !!xs. f(xs) = list_rec xs c h  |]
==> f [] = c & f(x#xs) = h x xs (f xs)"
by simp

lemma Abs_map_inverse:
"[| M: list(A);  A<=sexp;  !!z. z: A ==> f(g(z)) = z |]
==> Rep_map f (Abs_map g M) = M"
apply (erule list.induct, simp_all)
apply (insert list_subset_sexp)
apply (subst Abs_map_CONS, blast)
apply blast
apply simp
done

(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)

(** list_case **)

(* setting up rewrite sets *)

text‹Better to have a single theorem with a conjunctive conclusion.›
declare def_list_rec_NilCons [OF list_case_def, simp]

(** list_case **)

lemma expand_list_case:
"P(list_case a f xs) = ((xs=[] --> P a ) & (!y ys. xs=y#ys --> P(f y ys)))"
by (induct xs rule: list_induct) simp_all

(**** Function definitions ****)

declare def_list_rec_NilCons [OF map_def, simp]

(** The functional "map" and the generalized functionals **)

lemma Abs_Rep_map:
"(!!x. f(x): sexp) ==>
Abs_map g (Rep_map f xs) = map (%t. g(f(t))) xs"
apply (induct xs rule: list_induct)
apply (simp_all add: Rep_map_type list_sexp [THEN subsetD])
done

lemma map_ident [simp]: "map(%x. x)(xs) = xs"
by (induct xs rule: list_induct) simp_all

lemma map_compose: "map(f o g)(xs) = map f (map g xs)"
apply (induct xs rule: list_induct)
apply simp_all
done

(** take **)

lemma take_Suc1 [simp]: "take [] (Suc x) = []"
by simp

lemma take_Suc2 [simp]: "take(a#xs)(Suc x) = a#take xs x"
by simp

lemma take_Nil [simp]: "take [] n = []"
by (induct n) simp_all

lemma take_take_eq [simp]: "∀n. take (take xs n) n = take xs n"
apply (induct xs rule: list_induct)
apply simp_all
apply (rule allI)
apply (induct_tac n)
apply auto
done

end
```