berghofe@5181: (*  Title:      HOL/Datatype.thy
wenzelm@20819:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@11954:     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
wenzelm@20819: 
wenzelm@20819: Could <*> be generalized to a general summation (Sigma)?
berghofe@5181: *)
berghofe@5181: 
wenzelm@21669: header {* Analogues of the Cartesian Product and Disjoint Sum for Datatypes *}
wenzelm@11954: 
nipkow@15131: theory Datatype
haftmann@29609: imports Nat Product_Type
nipkow@15131: begin
wenzelm@11954: 
wenzelm@20819: typedef (Node)
wenzelm@20819:   ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
wenzelm@20819:     --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
wenzelm@20819:   by auto
wenzelm@20819: 
wenzelm@20819: text{*Datatypes will be represented by sets of type @{text node}*}
wenzelm@20819: 
wenzelm@20819: types 'a item        = "('a, unit) node set"
wenzelm@20819:       ('a, 'b) dtree = "('a, 'b) node set"
wenzelm@20819: 
wenzelm@20819: consts
wenzelm@20819:   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
wenzelm@20819: 
wenzelm@20819:   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
wenzelm@20819:   ndepth    :: "('a, 'b) node => nat"
wenzelm@20819: 
wenzelm@20819:   Atom      :: "('a + nat) => ('a, 'b) dtree"
wenzelm@20819:   Leaf      :: "'a => ('a, 'b) dtree"
wenzelm@20819:   Numb      :: "nat => ('a, 'b) dtree"
wenzelm@20819:   Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
wenzelm@20819:   In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
wenzelm@20819:   In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
wenzelm@20819:   Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
wenzelm@20819: 
wenzelm@20819:   ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
wenzelm@20819: 
wenzelm@20819:   uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
wenzelm@20819:   usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
wenzelm@20819: 
wenzelm@20819:   Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
wenzelm@20819:   Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
wenzelm@20819: 
wenzelm@20819:   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
wenzelm@20819:                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
wenzelm@20819:   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
wenzelm@20819:                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: defs
wenzelm@20819: 
wenzelm@20819:   Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
wenzelm@20819: 
wenzelm@20819:   (*crude "lists" of nats -- needed for the constructions*)
wenzelm@20819:   Push_def:   "Push == (%b h. nat_case b h)"
wenzelm@20819: 
wenzelm@20819:   (** operations on S-expressions -- sets of nodes **)
wenzelm@20819: 
wenzelm@20819:   (*S-expression constructors*)
wenzelm@20819:   Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
wenzelm@20819:   Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
wenzelm@20819: 
wenzelm@20819:   (*Leaf nodes, with arbitrary or nat labels*)
wenzelm@20819:   Leaf_def:   "Leaf == Atom o Inl"
wenzelm@20819:   Numb_def:   "Numb == Atom o Inr"
wenzelm@20819: 
wenzelm@20819:   (*Injections of the "disjoint sum"*)
wenzelm@20819:   In0_def:    "In0(M) == Scons (Numb 0) M"
wenzelm@20819:   In1_def:    "In1(M) == Scons (Numb 1) M"
wenzelm@20819: 
wenzelm@20819:   (*Function spaces*)
wenzelm@20819:   Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
wenzelm@20819: 
wenzelm@20819:   (*the set of nodes with depth less than k*)
wenzelm@20819:   ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
wenzelm@20819:   ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
wenzelm@20819: 
wenzelm@20819:   (*products and sums for the "universe"*)
wenzelm@20819:   uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
wenzelm@20819:   usum_def:   "usum A B == In0`A Un In1`B"
wenzelm@20819: 
wenzelm@20819:   (*the corresponding eliminators*)
wenzelm@20819:   Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
wenzelm@20819: 
wenzelm@20819:   Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
wenzelm@20819:                                   | (EX y . M = In1(y) & u = d(y))"
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819:   (** equality for the "universe" **)
wenzelm@20819: 
wenzelm@20819:   dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
wenzelm@20819: 
wenzelm@20819:   dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
wenzelm@20819:                           (UN (y,y'):s. {(In1(y),In1(y'))})"
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: lemma apfst_convE: 
wenzelm@20819:     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
wenzelm@20819:      |] ==> R"
wenzelm@20819: by (force simp add: apfst_def)
wenzelm@20819: 
wenzelm@20819: (** Push -- an injection, analogous to Cons on lists **)
wenzelm@20819: 
wenzelm@20819: lemma Push_inject1: "Push i f = Push j g  ==> i=j"
wenzelm@20819: apply (simp add: Push_def expand_fun_eq) 
wenzelm@20819: apply (drule_tac x=0 in spec, simp) 
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Push_inject2: "Push i f = Push j g  ==> f=g"
wenzelm@20819: apply (auto simp add: Push_def expand_fun_eq) 
wenzelm@20819: apply (drule_tac x="Suc x" in spec, simp) 
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Push_inject:
wenzelm@20819:     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
wenzelm@20819: by (blast dest: Push_inject1 Push_inject2) 
wenzelm@20819: 
wenzelm@20819: lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
wenzelm@20819: by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
wenzelm@20819: 
wenzelm@20819: lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Introduction rules for Node ***)
wenzelm@20819: 
wenzelm@20819: lemma Node_K0_I: "(%k. Inr 0, a) : Node"
wenzelm@20819: by (simp add: Node_def)
wenzelm@20819: 
wenzelm@20819: lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
wenzelm@20819: apply (simp add: Node_def Push_def) 
wenzelm@20819: apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: subsection{*Freeness: Distinctness of Constructors*}
wenzelm@20819: 
wenzelm@20819: (** Scons vs Atom **)
wenzelm@20819: 
wenzelm@20819: lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
wenzelm@20819: apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
wenzelm@20819: apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
wenzelm@20819:          dest!: Abs_Node_inj 
wenzelm@20819:          elim!: apfst_convE sym [THEN Push_neq_K0])  
wenzelm@20819: done
wenzelm@20819: 
haftmann@21407: lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym, standard]
haftmann@21407: 
wenzelm@20819: 
wenzelm@20819: (*** Injectiveness ***)
wenzelm@20819: 
wenzelm@20819: (** Atomic nodes **)
wenzelm@20819: 
wenzelm@20819: lemma inj_Atom: "inj(Atom)"
wenzelm@20819: apply (simp add: Atom_def)
wenzelm@20819: apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
wenzelm@20819: done
wenzelm@20819: lemmas Atom_inject = inj_Atom [THEN injD, standard]
wenzelm@20819: 
wenzelm@20819: lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
wenzelm@20819: by (blast dest!: Atom_inject)
wenzelm@20819: 
wenzelm@20819: lemma inj_Leaf: "inj(Leaf)"
wenzelm@20819: apply (simp add: Leaf_def o_def)
wenzelm@20819: apply (rule inj_onI)
wenzelm@20819: apply (erule Atom_inject [THEN Inl_inject])
wenzelm@20819: done
wenzelm@20819: 
haftmann@21407: lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD, standard]
wenzelm@20819: 
wenzelm@20819: lemma inj_Numb: "inj(Numb)"
wenzelm@20819: apply (simp add: Numb_def o_def)
wenzelm@20819: apply (rule inj_onI)
wenzelm@20819: apply (erule Atom_inject [THEN Inr_inject])
wenzelm@20819: done
wenzelm@20819: 
haftmann@21407: lemmas Numb_inject [dest!] = inj_Numb [THEN injD, standard]
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (** Injectiveness of Push_Node **)
wenzelm@20819: 
wenzelm@20819: lemma Push_Node_inject:
wenzelm@20819:     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
wenzelm@20819:      |] ==> P"
wenzelm@20819: apply (simp add: Push_Node_def)
wenzelm@20819: apply (erule Abs_Node_inj [THEN apfst_convE])
wenzelm@20819: apply (rule Rep_Node [THEN Node_Push_I])+
wenzelm@20819: apply (erule sym [THEN apfst_convE]) 
wenzelm@20819: apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (** Injectiveness of Scons **)
wenzelm@20819: 
wenzelm@20819: lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
wenzelm@20819: apply (simp add: Scons_def One_nat_def)
wenzelm@20819: apply (blast dest!: Push_Node_inject)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
wenzelm@20819: apply (simp add: Scons_def One_nat_def)
wenzelm@20819: apply (blast dest!: Push_Node_inject)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
wenzelm@20819: apply (erule equalityE)
wenzelm@20819: apply (iprover intro: equalityI Scons_inject_lemma1)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
wenzelm@20819: apply (erule equalityE)
wenzelm@20819: apply (iprover intro: equalityI Scons_inject_lemma2)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma Scons_inject:
wenzelm@20819:     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
wenzelm@20819: by (iprover dest: Scons_inject1 Scons_inject2)
wenzelm@20819: 
wenzelm@20819: lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
wenzelm@20819: by (blast elim!: Scons_inject)
wenzelm@20819: 
wenzelm@20819: (*** Distinctness involving Leaf and Numb ***)
wenzelm@20819: 
wenzelm@20819: (** Scons vs Leaf **)
wenzelm@20819: 
wenzelm@20819: lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
wenzelm@20819: by (simp add: Leaf_def o_def Scons_not_Atom)
wenzelm@20819: 
haftmann@21407: lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym, standard]
wenzelm@20819: 
wenzelm@20819: (** Scons vs Numb **)
wenzelm@20819: 
wenzelm@20819: lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
wenzelm@20819: by (simp add: Numb_def o_def Scons_not_Atom)
wenzelm@20819: 
haftmann@21407: lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym, standard]
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (** Leaf vs Numb **)
wenzelm@20819: 
wenzelm@20819: lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
wenzelm@20819: by (simp add: Leaf_def Numb_def)
wenzelm@20819: 
haftmann@21407: lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym, standard]
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** ndepth -- the depth of a node ***)
wenzelm@20819: 
wenzelm@20819: lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
wenzelm@20819: by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
wenzelm@20819: 
wenzelm@20819: lemma ndepth_Push_Node_aux:
wenzelm@20819:      "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
wenzelm@20819: apply (induct_tac "k", auto)
wenzelm@20819: apply (erule Least_le)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma ndepth_Push_Node: 
wenzelm@20819:     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
wenzelm@20819: apply (insert Rep_Node [of n, unfolded Node_def])
wenzelm@20819: apply (auto simp add: ndepth_def Push_Node_def
wenzelm@20819:                  Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
wenzelm@20819: apply (rule Least_equality)
wenzelm@20819: apply (auto simp add: Push_def ndepth_Push_Node_aux)
wenzelm@20819: apply (erule LeastI)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** ntrunc applied to the various node sets ***)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
wenzelm@20819: by (simp add: ntrunc_def)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
wenzelm@20819: by (auto simp add: Atom_def ntrunc_def ndepth_K0)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
wenzelm@20819: by (simp add: Leaf_def o_def ntrunc_Atom)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
wenzelm@20819: by (simp add: Numb_def o_def ntrunc_Atom)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_Scons [simp]: 
wenzelm@20819:     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
wenzelm@20819: by (auto simp add: Scons_def ntrunc_def One_nat_def ndepth_Push_Node) 
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (** Injection nodes **)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
wenzelm@20819: apply (simp add: In0_def)
wenzelm@20819: apply (simp add: Scons_def)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
wenzelm@20819: by (simp add: In0_def)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
wenzelm@20819: apply (simp add: In1_def)
wenzelm@20819: apply (simp add: Scons_def)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
wenzelm@20819: by (simp add: In1_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: subsection{*Set Constructions*}
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Cartesian Product ***)
wenzelm@20819: 
wenzelm@20819: lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
wenzelm@20819: by (simp add: uprod_def)
wenzelm@20819: 
wenzelm@20819: (*The general elimination rule*)
wenzelm@20819: lemma uprodE [elim!]:
wenzelm@20819:     "[| c : uprod A B;   
wenzelm@20819:         !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
wenzelm@20819:      |] ==> P"
wenzelm@20819: by (auto simp add: uprod_def) 
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*Elimination of a pair -- introduces no eigenvariables*)
wenzelm@20819: lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
wenzelm@20819: by (auto simp add: uprod_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Disjoint Sum ***)
wenzelm@20819: 
wenzelm@20819: lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
wenzelm@20819: by (simp add: usum_def)
wenzelm@20819: 
wenzelm@20819: lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
wenzelm@20819: by (simp add: usum_def)
wenzelm@20819: 
wenzelm@20819: lemma usumE [elim!]: 
wenzelm@20819:     "[| u : usum A B;   
wenzelm@20819:         !!x. [| x:A;  u=In0(x) |] ==> P;  
wenzelm@20819:         !!y. [| y:B;  u=In1(y) |] ==> P  
wenzelm@20819:      |] ==> P"
wenzelm@20819: by (auto simp add: usum_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (** Injection **)
wenzelm@20819: 
wenzelm@20819: lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
wenzelm@20819: by (auto simp add: In0_def In1_def One_nat_def)
wenzelm@20819: 
haftmann@21407: lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym, standard]
wenzelm@20819: 
wenzelm@20819: lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
wenzelm@20819: by (simp add: In0_def)
wenzelm@20819: 
wenzelm@20819: lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
wenzelm@20819: by (simp add: In1_def)
wenzelm@20819: 
wenzelm@20819: lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
wenzelm@20819: by (blast dest!: In0_inject)
wenzelm@20819: 
wenzelm@20819: lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
wenzelm@20819: by (blast dest!: In1_inject)
wenzelm@20819: 
wenzelm@20819: lemma inj_In0: "inj In0"
wenzelm@20819: by (blast intro!: inj_onI)
wenzelm@20819: 
wenzelm@20819: lemma inj_In1: "inj In1"
wenzelm@20819: by (blast intro!: inj_onI)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Function spaces ***)
wenzelm@20819: 
wenzelm@20819: lemma Lim_inject: "Lim f = Lim g ==> f = g"
wenzelm@20819: apply (simp add: Lim_def)
wenzelm@20819: apply (rule ext)
wenzelm@20819: apply (blast elim!: Push_Node_inject)
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** proving equality of sets and functions using ntrunc ***)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_subsetI: "ntrunc k M <= M"
wenzelm@20819: by (auto simp add: ntrunc_def)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
wenzelm@20819: by (auto simp add: ntrunc_def)
wenzelm@20819: 
wenzelm@20819: (*A generalized form of the take-lemma*)
wenzelm@20819: lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
wenzelm@20819: apply (rule equalityI)
wenzelm@20819: apply (rule_tac [!] ntrunc_subsetD)
wenzelm@20819: apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_o_equality: 
wenzelm@20819:     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
wenzelm@20819: apply (rule ntrunc_equality [THEN ext])
wenzelm@20819: apply (simp add: expand_fun_eq) 
wenzelm@20819: done
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Monotonicity ***)
wenzelm@20819: 
wenzelm@20819: lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
wenzelm@20819: by (simp add: uprod_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
wenzelm@20819: by (simp add: usum_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
wenzelm@20819: by (simp add: Scons_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
wenzelm@20819: by (simp add: In0_def subset_refl Scons_mono)
wenzelm@20819: 
wenzelm@20819: lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
wenzelm@20819: by (simp add: In1_def subset_refl Scons_mono)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Split and Case ***)
wenzelm@20819: 
wenzelm@20819: lemma Split [simp]: "Split c (Scons M N) = c M N"
wenzelm@20819: by (simp add: Split_def)
wenzelm@20819: 
wenzelm@20819: lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
wenzelm@20819: by (simp add: Case_def)
wenzelm@20819: 
wenzelm@20819: lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
wenzelm@20819: by (simp add: Case_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (**** UN x. B(x) rules ****)
wenzelm@20819: 
wenzelm@20819: lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
wenzelm@20819: by (simp add: ntrunc_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
wenzelm@20819: by (simp add: Scons_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
wenzelm@20819: by (simp add: Scons_def, blast)
wenzelm@20819: 
wenzelm@20819: lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
wenzelm@20819: by (simp add: In0_def Scons_UN1_y)
wenzelm@20819: 
wenzelm@20819: lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
wenzelm@20819: by (simp add: In1_def Scons_UN1_y)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Equality for Cartesian Product ***)
wenzelm@20819: 
wenzelm@20819: lemma dprodI [intro!]: 
wenzelm@20819:     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
wenzelm@20819: by (auto simp add: dprod_def)
wenzelm@20819: 
wenzelm@20819: (*The general elimination rule*)
wenzelm@20819: lemma dprodE [elim!]: 
wenzelm@20819:     "[| c : dprod r s;   
wenzelm@20819:         !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
wenzelm@20819:                         c = (Scons x y, Scons x' y') |] ==> P  
wenzelm@20819:      |] ==> P"
wenzelm@20819: by (auto simp add: dprod_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Equality for Disjoint Sum ***)
wenzelm@20819: 
wenzelm@20819: lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
wenzelm@20819: by (auto simp add: dsum_def)
wenzelm@20819: 
wenzelm@20819: lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
wenzelm@20819: by (auto simp add: dsum_def)
wenzelm@20819: 
wenzelm@20819: lemma dsumE [elim!]: 
wenzelm@20819:     "[| w : dsum r s;   
wenzelm@20819:         !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
wenzelm@20819:         !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
wenzelm@20819:      |] ==> P"
wenzelm@20819: by (auto simp add: dsum_def)
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Monotonicity ***)
wenzelm@20819: 
wenzelm@20819: lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
wenzelm@20819: by blast
wenzelm@20819: 
wenzelm@20819: lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
wenzelm@20819: by blast
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: (*** Bounding theorems ***)
wenzelm@20819: 
wenzelm@20819: lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
wenzelm@20819: by blast
wenzelm@20819: 
wenzelm@20819: lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
wenzelm@20819: 
wenzelm@20819: (*Dependent version*)
wenzelm@20819: lemma dprod_subset_Sigma2:
wenzelm@20819:      "(dprod (Sigma A B) (Sigma C D)) <= 
wenzelm@20819:       Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
wenzelm@20819: by auto
wenzelm@20819: 
wenzelm@20819: lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
wenzelm@20819: by blast
wenzelm@20819: 
wenzelm@20819: lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
wenzelm@20819: 
wenzelm@20819: 
haftmann@24162: text {* hides popular names *}
haftmann@24162: hide (open) type node item
wenzelm@20819: hide (open) const Push Node Atom Leaf Numb Lim Split Case
wenzelm@20819: 
wenzelm@20819: 
wenzelm@20819: section {* Datatypes *}
wenzelm@20819: 
haftmann@24699: subsection {* Representing sums *}
wenzelm@12918: 
haftmann@27104: rep_datatype (sum) Inl Inr
haftmann@27104: proof -
haftmann@27104:   fix P
haftmann@27104:   fix s :: "'a + 'b"
haftmann@27104:   assume x: "\<And>x\<Colon>'a. P (Inl x)" and y: "\<And>y\<Colon>'b. P (Inr y)"
haftmann@27104:   then show "P s" by (auto intro: sumE [of s])
haftmann@27104: qed simp_all
haftmann@24194: 
nipkow@22230: lemma sum_case_KK[simp]: "sum_case (%x. a) (%x. a) = (%x. a)"
nipkow@22230:   by (rule ext) (simp split: sum.split)
nipkow@22230: 
wenzelm@12918: lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
wenzelm@12918:   apply (rule_tac s = s in sumE)
wenzelm@12918:    apply (erule ssubst)
wenzelm@20798:    apply (rule sum.cases(1))
wenzelm@12918:   apply (erule ssubst)
wenzelm@20798:   apply (rule sum.cases(2))
wenzelm@12918:   done
wenzelm@12918: 
wenzelm@12918: lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
wenzelm@12918:   -- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
wenzelm@20798:   by simp
wenzelm@12918: 
wenzelm@12918: lemma sum_case_inject:
wenzelm@12918:   "sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
wenzelm@12918: proof -
wenzelm@12918:   assume a: "sum_case f1 f2 = sum_case g1 g2"
wenzelm@12918:   assume r: "f1 = g1 ==> f2 = g2 ==> P"
wenzelm@12918:   show P
wenzelm@12918:     apply (rule r)
wenzelm@12918:      apply (rule ext)
paulson@14208:      apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
wenzelm@12918:     apply (rule ext)
paulson@14208:     apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
wenzelm@12918:     done
wenzelm@12918: qed
wenzelm@12918: 
berghofe@13635: constdefs
berghofe@13635:   Suml :: "('a => 'c) => 'a + 'b => 'c"
haftmann@28524:   "Suml == (%f. sum_case f undefined)"
berghofe@13635: 
berghofe@13635:   Sumr :: "('b => 'c) => 'a + 'b => 'c"
haftmann@28524:   "Sumr == sum_case undefined"
berghofe@13635: 
berghofe@13635: lemma Suml_inject: "Suml f = Suml g ==> f = g"
berghofe@13635:   by (unfold Suml_def) (erule sum_case_inject)
berghofe@13635: 
berghofe@13635: lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
berghofe@13635:   by (unfold Sumr_def) (erule sum_case_inject)
berghofe@13635: 
krauss@29183: primrec Projl :: "'a + 'b => 'a"
krauss@29183: where Projl_Inl: "Projl (Inl x) = x"
krauss@29183: 
krauss@29183: primrec Projr :: "'a + 'b => 'b"
krauss@29183: where Projr_Inr: "Projr (Inr x) = x"
krauss@29183: 
krauss@29183: hide (open) const Suml Sumr Projl Projr
berghofe@13635: 
berghofe@5181: end