src/HOL/Datatype.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 36176 3fe7e97ccca8
child 39198 f967a16dfcdd
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
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(*  Title:      HOL/Datatype.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
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*)
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header {* Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
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theory Datatype
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imports Product_Type Sum_Type Nat
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uses
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  ("Tools/Datatype/datatype.ML")
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  ("Tools/inductive_realizer.ML")
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  ("Tools/Datatype/datatype_realizer.ML")
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begin
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subsection {* The datatype universe *}
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typedef (Node)
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  ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
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    --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
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  by auto
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text{*Datatypes will be represented by sets of type @{text node}*}
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types 'a item        = "('a, unit) node set"
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      ('a, 'b) dtree = "('a, 'b) node set"
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consts
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  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
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  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
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  ndepth    :: "('a, 'b) node => nat"
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  Atom      :: "('a + nat) => ('a, 'b) dtree"
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  Leaf      :: "'a => ('a, 'b) dtree"
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  Numb      :: "nat => ('a, 'b) dtree"
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  Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
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  In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
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  In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
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  Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
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  ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
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  uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
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  usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
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  Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
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  Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
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  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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                => (('a, 'b) dtree * ('a, 'b) dtree)set"
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  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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                => (('a, 'b) dtree * ('a, 'b) dtree)set"
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defs
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  Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
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  (*crude "lists" of nats -- needed for the constructions*)
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  Push_def:   "Push == (%b h. nat_case b h)"
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  (** operations on S-expressions -- sets of nodes **)
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  (*S-expression constructors*)
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  Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
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  Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
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  (*Leaf nodes, with arbitrary or nat labels*)
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  Leaf_def:   "Leaf == Atom o Inl"
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  Numb_def:   "Numb == Atom o Inr"
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  (*Injections of the "disjoint sum"*)
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  In0_def:    "In0(M) == Scons (Numb 0) M"
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  In1_def:    "In1(M) == Scons (Numb 1) M"
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  (*Function spaces*)
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  Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
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  (*the set of nodes with depth less than k*)
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  ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
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  ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
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  (*products and sums for the "universe"*)
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  uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
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  usum_def:   "usum A B == In0`A Un In1`B"
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  (*the corresponding eliminators*)
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  Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
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  Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
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                                  | (EX y . M = In1(y) & u = d(y))"
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  (** equality for the "universe" **)
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  dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
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  dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
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                          (UN (y,y'):s. {(In1(y),In1(y'))})"
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lemma apfst_convE: 
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    "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
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     |] ==> R"
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by (force simp add: apfst_def)
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(** Push -- an injection, analogous to Cons on lists **)
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lemma Push_inject1: "Push i f = Push j g  ==> i=j"
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apply (simp add: Push_def expand_fun_eq) 
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apply (drule_tac x=0 in spec, simp) 
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done
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lemma Push_inject2: "Push i f = Push j g  ==> f=g"
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apply (auto simp add: Push_def expand_fun_eq) 
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apply (drule_tac x="Suc x" in spec, simp) 
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done
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lemma Push_inject:
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    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
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by (blast dest: Push_inject1 Push_inject2) 
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lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
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by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
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lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
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(*** Introduction rules for Node ***)
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lemma Node_K0_I: "(%k. Inr 0, a) : Node"
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by (simp add: Node_def)
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lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
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apply (simp add: Node_def Push_def) 
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apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
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done
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subsection{*Freeness: Distinctness of Constructors*}
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(** Scons vs Atom **)
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lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
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unfolding Atom_def Scons_def Push_Node_def One_nat_def
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by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
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         dest!: Abs_Node_inj 
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         elim!: apfst_convE sym [THEN Push_neq_K0])  
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lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym, standard]
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(*** Injectiveness ***)
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(** Atomic nodes **)
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lemma inj_Atom: "inj(Atom)"
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apply (simp add: Atom_def)
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apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
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done
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lemmas Atom_inject = inj_Atom [THEN injD, standard]
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lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
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by (blast dest!: Atom_inject)
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lemma inj_Leaf: "inj(Leaf)"
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apply (simp add: Leaf_def o_def)
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apply (rule inj_onI)
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apply (erule Atom_inject [THEN Inl_inject])
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done
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lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD, standard]
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lemma inj_Numb: "inj(Numb)"
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apply (simp add: Numb_def o_def)
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apply (rule inj_onI)
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apply (erule Atom_inject [THEN Inr_inject])
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done
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lemmas Numb_inject [dest!] = inj_Numb [THEN injD, standard]
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(** Injectiveness of Push_Node **)
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lemma Push_Node_inject:
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    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
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     |] ==> P"
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apply (simp add: Push_Node_def)
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apply (erule Abs_Node_inj [THEN apfst_convE])
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apply (rule Rep_Node [THEN Node_Push_I])+
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apply (erule sym [THEN apfst_convE]) 
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apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
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done
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(** Injectiveness of Scons **)
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lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
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unfolding Scons_def One_nat_def
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by (blast dest!: Push_Node_inject)
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lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
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unfolding Scons_def One_nat_def
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by (blast dest!: Push_Node_inject)
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lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
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apply (erule equalityE)
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apply (iprover intro: equalityI Scons_inject_lemma1)
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done
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lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
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apply (erule equalityE)
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apply (iprover intro: equalityI Scons_inject_lemma2)
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done
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lemma Scons_inject:
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    "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
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by (iprover dest: Scons_inject1 Scons_inject2)
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lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
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by (blast elim!: Scons_inject)
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(*** Distinctness involving Leaf and Numb ***)
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(** Scons vs Leaf **)
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lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
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unfolding Leaf_def o_def by (rule Scons_not_Atom)
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lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym, standard]
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(** Scons vs Numb **)
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lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
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unfolding Numb_def o_def by (rule Scons_not_Atom)
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lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym, standard]
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(** Leaf vs Numb **)
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lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
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by (simp add: Leaf_def Numb_def)
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lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym, standard]
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(*** ndepth -- the depth of a node ***)
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lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
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by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
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lemma ndepth_Push_Node_aux:
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     "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
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apply (induct_tac "k", auto)
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apply (erule Least_le)
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done
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lemma ndepth_Push_Node: 
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    "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
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apply (insert Rep_Node [of n, unfolded Node_def])
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apply (auto simp add: ndepth_def Push_Node_def
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                 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
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apply (rule Least_equality)
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apply (auto simp add: Push_def ndepth_Push_Node_aux)
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apply (erule LeastI)
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done
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(*** ntrunc applied to the various node sets ***)
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lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
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by (simp add: ntrunc_def)
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lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
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by (auto simp add: Atom_def ntrunc_def ndepth_K0)
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lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
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unfolding Leaf_def o_def by (rule ntrunc_Atom)
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lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
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unfolding Numb_def o_def by (rule ntrunc_Atom)
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lemma ntrunc_Scons [simp]: 
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    "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
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unfolding Scons_def ntrunc_def One_nat_def
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by (auto simp add: ndepth_Push_Node)
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(** Injection nodes **)
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lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
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apply (simp add: In0_def)
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apply (simp add: Scons_def)
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done
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lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
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by (simp add: In0_def)
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lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
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apply (simp add: In1_def)
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apply (simp add: Scons_def)
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done
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lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
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by (simp add: In1_def)
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subsection{*Set Constructions*}
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(*** Cartesian Product ***)
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lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
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by (simp add: uprod_def)
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(*The general elimination rule*)
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lemma uprodE [elim!]:
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    "[| c : uprod A B;   
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        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
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     |] ==> P"
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by (auto simp add: uprod_def) 
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(*Elimination of a pair -- introduces no eigenvariables*)
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lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
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by (auto simp add: uprod_def)
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(*** Disjoint Sum ***)
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lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
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by (simp add: usum_def)
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lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
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by (simp add: usum_def)
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lemma usumE [elim!]: 
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    "[| u : usum A B;   
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        !!x. [| x:A;  u=In0(x) |] ==> P;  
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        !!y. [| y:B;  u=In1(y) |] ==> P  
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     |] ==> P"
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by (auto simp add: usum_def)
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(** Injection **)
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lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
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unfolding In0_def In1_def One_nat_def by auto
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lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym, standard]
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lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
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by (simp add: In0_def)
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lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
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by (simp add: In1_def)
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lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
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by (blast dest!: In0_inject)
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lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
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by (blast dest!: In1_inject)
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lemma inj_In0: "inj In0"
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by (blast intro!: inj_onI)
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lemma inj_In1: "inj In1"
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by (blast intro!: inj_onI)
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(*** Function spaces ***)
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lemma Lim_inject: "Lim f = Lim g ==> f = g"
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apply (simp add: Lim_def)
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apply (rule ext)
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apply (blast elim!: Push_Node_inject)
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done
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(*** proving equality of sets and functions using ntrunc ***)
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lemma ntrunc_subsetI: "ntrunc k M <= M"
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by (auto simp add: ntrunc_def)
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lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
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by (auto simp add: ntrunc_def)
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(*A generalized form of the take-lemma*)
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lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
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apply (rule equalityI)
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apply (rule_tac [!] ntrunc_subsetD)
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apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
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done
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lemma ntrunc_o_equality: 
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    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
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apply (rule ntrunc_equality [THEN ext])
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apply (simp add: expand_fun_eq) 
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done
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(*** Monotonicity ***)
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lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
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by (simp add: uprod_def, blast)
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lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
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by (simp add: usum_def, blast)
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lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
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by (simp add: Scons_def, blast)
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lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
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by (simp add: In0_def Scons_mono)
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lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
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by (simp add: In1_def Scons_mono)
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(*** Split and Case ***)
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lemma Split [simp]: "Split c (Scons M N) = c M N"
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by (simp add: Split_def)
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lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
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by (simp add: Case_def)
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lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
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by (simp add: Case_def)
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(**** UN x. B(x) rules ****)
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lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
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by (simp add: ntrunc_def, blast)
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lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
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by (simp add: Scons_def, blast)
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lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
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by (simp add: Scons_def, blast)
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lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
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by (simp add: In0_def Scons_UN1_y)
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lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
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by (simp add: In1_def Scons_UN1_y)
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   455
(*** Equality for Cartesian Product ***)
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lemma dprodI [intro!]: 
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    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
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by (auto simp add: dprod_def)
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(*The general elimination rule*)
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lemma dprodE [elim!]: 
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    "[| c : dprod r s;   
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        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
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                        c = (Scons x y, Scons x' y') |] ==> P  
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     |] ==> P"
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by (auto simp add: dprod_def)
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   469
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(*** Equality for Disjoint Sum ***)
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lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
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by (auto simp add: dsum_def)
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   475
lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
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by (auto simp add: dsum_def)
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   477
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lemma dsumE [elim!]: 
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   479
    "[| w : dsum r s;   
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   480
        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
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        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
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   482
     |] ==> P"
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   483
by (auto simp add: dsum_def)
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   484
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   485
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   486
(*** Monotonicity ***)
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   488
lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
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by blast
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   490
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   491
lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
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by blast
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   493
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   494
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   495
(*** Bounding theorems ***)
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lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
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by blast
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   499
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   500
lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
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   501
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   502
(*Dependent version*)
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lemma dprod_subset_Sigma2:
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   504
     "(dprod (Sigma A B) (Sigma C D)) <= 
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   505
      Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
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   506
by auto
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   507
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   508
lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
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   509
by blast
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   510
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   511
lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
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   513
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   514
text {* hides popular names *}
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   515
hide_type (open) node item
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   516
hide_const (open) Push Node Atom Leaf Numb Lim Split Case
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   517
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   518
use "Tools/Datatype/datatype.ML"
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   519
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   520
use "Tools/inductive_realizer.ML"
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   521
setup InductiveRealizer.setup
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   522
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   523
use "Tools/Datatype/datatype_realizer.ML"
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   524
setup Datatype_Realizer.setup
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   525
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   526
end