# Theory Datatype

Up to index of Isabelle/HOL-Proofs

theory Datatype
imports Product_Type Sum_Type Nat
`(*  Title:      HOL/Datatype.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen*)header {* Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}theory Datatypeimports Product_Type Sum_Type Natkeywords "datatype" :: thy_declbeginsubsection {* The datatype universe *}definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}"typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"  morphisms Rep_Node Abs_Node  unfolding Node_def by autotext{*Datatypes will be represented by sets of type @{text node}*}type_synonym 'a item        = "('a, unit) node set"type_synonym ('a, 'b) dtree = "('a, 'b) node set"consts  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"  ndepth    :: "('a, 'b) node => nat"  Atom      :: "('a + nat) => ('a, 'b) dtree"  Leaf      :: "'a => ('a, 'b) dtree"  Numb      :: "nat => ('a, 'b) dtree"  Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"  In0       :: "('a, 'b) dtree => ('a, 'b) dtree"  In1       :: "('a, 'b) dtree => ('a, 'b) dtree"  Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"  ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"  uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"  usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"  Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"  Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]                => (('a, 'b) dtree * ('a, 'b) dtree)set"  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]                => (('a, 'b) dtree * ('a, 'b) dtree)set"defs  Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"  (*crude "lists" of nats -- needed for the constructions*)  Push_def:   "Push == (%b h. nat_case b h)"  (** operations on S-expressions -- sets of nodes **)  (*S-expression constructors*)  Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"  Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"  (*Leaf nodes, with arbitrary or nat labels*)  Leaf_def:   "Leaf == Atom o Inl"  Numb_def:   "Numb == Atom o Inr"  (*Injections of the "disjoint sum"*)  In0_def:    "In0(M) == Scons (Numb 0) M"  In1_def:    "In1(M) == Scons (Numb 1) M"  (*Function spaces*)  Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"  (*the set of nodes with depth less than k*)  ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"  ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"  (*products and sums for the "universe"*)  uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"  usum_def:   "usum A B == In0`A Un In1`B"  (*the corresponding eliminators*)  Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"  Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))                                  | (EX y . M = In1(y) & u = d(y))"  (** equality for the "universe" **)  dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"  dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un                          (UN (y,y'):s. {(In1(y),In1(y'))})"lemma apfst_convE:     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R       |] ==> R"by (force simp add: apfst_def)(** Push -- an injection, analogous to Cons on lists **)lemma Push_inject1: "Push i f = Push j g  ==> i=j"apply (simp add: Push_def fun_eq_iff) apply (drule_tac x=0 in spec, simp) donelemma Push_inject2: "Push i f = Push j g  ==> f=g"apply (auto simp add: Push_def fun_eq_iff) apply (drule_tac x="Suc x" in spec, simp) donelemma Push_inject:    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"by (blast dest: Push_inject1 Push_inject2) lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"by (auto simp add: Push_def fun_eq_iff split: nat.split_asm)lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1](*** Introduction rules for Node ***)lemma Node_K0_I: "(%k. Inr 0, a) : Node"by (simp add: Node_def)lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"apply (simp add: Node_def Push_def) apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])donesubsection{*Freeness: Distinctness of Constructors*}(** Scons vs Atom **)lemma Scons_not_Atom [iff]: "Scons M N ≠ Atom(a)"unfolding Atom_def Scons_def Push_Node_def One_nat_defby (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I]          dest!: Abs_Node_inj          elim!: apfst_convE sym [THEN Push_neq_K0])  lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym](*** Injectiveness ***)(** Atomic nodes **)lemma inj_Atom: "inj(Atom)"apply (simp add: Atom_def)apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)donelemmas Atom_inject = inj_Atom [THEN injD]lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"by (blast dest!: Atom_inject)lemma inj_Leaf: "inj(Leaf)"apply (simp add: Leaf_def o_def)apply (rule inj_onI)apply (erule Atom_inject [THEN Inl_inject])donelemmas Leaf_inject [dest!] = inj_Leaf [THEN injD]lemma inj_Numb: "inj(Numb)"apply (simp add: Numb_def o_def)apply (rule inj_onI)apply (erule Atom_inject [THEN Inr_inject])donelemmas Numb_inject [dest!] = inj_Numb [THEN injD](** Injectiveness of Push_Node **)lemma Push_Node_inject:    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P       |] ==> P"apply (simp add: Push_Node_def)apply (erule Abs_Node_inj [THEN apfst_convE])apply (rule Rep_Node [THEN Node_Push_I])+apply (erule sym [THEN apfst_convE]) apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)done(** Injectiveness of Scons **)lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"unfolding Scons_def One_nat_defby (blast dest!: Push_Node_inject)lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"unfolding Scons_def One_nat_defby (blast dest!: Push_Node_inject)lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"apply (erule equalityE)apply (iprover intro: equalityI Scons_inject_lemma1)donelemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"apply (erule equalityE)apply (iprover intro: equalityI Scons_inject_lemma2)donelemma Scons_inject:    "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"by (iprover dest: Scons_inject1 Scons_inject2)lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"by (blast elim!: Scons_inject)(*** Distinctness involving Leaf and Numb ***)(** Scons vs Leaf **)lemma Scons_not_Leaf [iff]: "Scons M N ≠ Leaf(a)"unfolding Leaf_def o_def by (rule Scons_not_Atom)lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym](** Scons vs Numb **)lemma Scons_not_Numb [iff]: "Scons M N ≠ Numb(k)"unfolding Numb_def o_def by (rule Scons_not_Atom)lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym](** Leaf vs Numb **)lemma Leaf_not_Numb [iff]: "Leaf(a) ≠ Numb(k)"by (simp add: Leaf_def Numb_def)lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym](*** ndepth -- the depth of a node ***)lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)lemma ndepth_Push_Node_aux:     "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"apply (induct_tac "k", auto)apply (erule Least_le)donelemma ndepth_Push_Node:     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"apply (insert Rep_Node [of n, unfolded Node_def])apply (auto simp add: ndepth_def Push_Node_def                 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])apply (rule Least_equality)apply (auto simp add: Push_def ndepth_Push_Node_aux)apply (erule LeastI)done(*** ntrunc applied to the various node sets ***)lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"by (simp add: ntrunc_def)lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"by (auto simp add: Atom_def ntrunc_def ndepth_K0)lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"unfolding Leaf_def o_def by (rule ntrunc_Atom)lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"unfolding Numb_def o_def by (rule ntrunc_Atom)lemma ntrunc_Scons [simp]:     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"unfolding Scons_def ntrunc_def One_nat_defby (auto simp add: ndepth_Push_Node)(** Injection nodes **)lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"apply (simp add: In0_def)apply (simp add: Scons_def)donelemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"by (simp add: In0_def)lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"apply (simp add: In1_def)apply (simp add: Scons_def)donelemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"by (simp add: In1_def)subsection{*Set Constructions*}(*** Cartesian Product ***)lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"by (simp add: uprod_def)(*The general elimination rule*)lemma uprodE [elim!]:    "[| c : uprod A B;           !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P       |] ==> P"by (auto simp add: uprod_def) (*Elimination of a pair -- introduces no eigenvariables*)lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"by (auto simp add: uprod_def)(*** Disjoint Sum ***)lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"by (simp add: usum_def)lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"by (simp add: usum_def)lemma usumE [elim!]:     "[| u : usum A B;           !!x. [| x:A;  u=In0(x) |] ==> P;          !!y. [| y:B;  u=In1(y) |] ==> P       |] ==> P"by (auto simp add: usum_def)(** Injection **)lemma In0_not_In1 [iff]: "In0(M) ≠ In1(N)"unfolding In0_def In1_def One_nat_def by autolemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]lemma In0_inject: "In0(M) = In0(N) ==>  M=N"by (simp add: In0_def)lemma In1_inject: "In1(M) = In1(N) ==>  M=N"by (simp add: In1_def)lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"by (blast dest!: In0_inject)lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"by (blast dest!: In1_inject)lemma inj_In0: "inj In0"by (blast intro!: inj_onI)lemma inj_In1: "inj In1"by (blast intro!: inj_onI)(*** Function spaces ***)lemma Lim_inject: "Lim f = Lim g ==> f = g"apply (simp add: Lim_def)apply (rule ext)apply (blast elim!: Push_Node_inject)done(*** proving equality of sets and functions using ntrunc ***)lemma ntrunc_subsetI: "ntrunc k M <= M"by (auto simp add: ntrunc_def)lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"by (auto simp add: ntrunc_def)(*A generalized form of the take-lemma*)lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"apply (rule equalityI)apply (rule_tac [!] ntrunc_subsetD)apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) donelemma ntrunc_o_equality:     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"apply (rule ntrunc_equality [THEN ext])apply (simp add: fun_eq_iff) done(*** Monotonicity ***)lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"by (simp add: uprod_def, blast)lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"by (simp add: usum_def, blast)lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"by (simp add: Scons_def, blast)lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"by (simp add: In0_def Scons_mono)lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"by (simp add: In1_def Scons_mono)(*** Split and Case ***)lemma Split [simp]: "Split c (Scons M N) = c M N"by (simp add: Split_def)lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"by (simp add: Case_def)lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"by (simp add: Case_def)(**** UN x. B(x) rules ****)lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"by (simp add: ntrunc_def, blast)lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"by (simp add: Scons_def, blast)lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"by (simp add: Scons_def, blast)lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"by (simp add: In0_def Scons_UN1_y)lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"by (simp add: In1_def Scons_UN1_y)(*** Equality for Cartesian Product ***)lemma dprodI [intro!]:     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"by (auto simp add: dprod_def)(*The general elimination rule*)lemma dprodE [elim!]:     "[| c : dprod r s;           !!x y x' y'. [| (x,x') : r;  (y,y') : s;                          c = (Scons x y, Scons x' y') |] ==> P       |] ==> P"by (auto simp add: dprod_def)(*** Equality for Disjoint Sum ***)lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"by (auto simp add: dsum_def)lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"by (auto simp add: dsum_def)lemma dsumE [elim!]:     "[| w : dsum r s;           !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;          !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P       |] ==> P"by (auto simp add: dsum_def)(*** Monotonicity ***)lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"by blastlemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"by blast(*** Bounding theorems ***)lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"by blastlemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma](*Dependent version*)lemma dprod_subset_Sigma2:     "(dprod (Sigma A B) (Sigma C D)) <=       Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"by autolemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"by blastlemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]text {* hides popular names *}hide_type (open) node itemhide_const (open) Push Node Atom Leaf Numb Lim Split CaseML_file "Tools/Datatype/datatype.ML"ML_file "Tools/inductive_realizer.ML"setup InductiveRealizer.setupML_file "Tools/Datatype/datatype_realizer.ML"setup Datatype_Realizer.setupend`