src/HOL/Library/Old_Datatype.thy
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Thu, 18 Sep 2014 16:47:40 +0200
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(*  Title:      HOL/Library/Old_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 {* Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
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theory Old_Datatype
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imports "../Main"
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keywords "old_datatype" :: thy_decl
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begin
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ML_file "~~/src/HOL/Tools/Old_Datatype/old_size.ML"
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ML_file "~~/src/HOL/Tools/datatype_realizer.ML"
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subsection {* The datatype universe *}
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definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}"
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typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"
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  morphisms Rep_Node Abs_Node
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  unfolding Node_def by auto
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text{*Datatypes will be represented by sets of type @{text node}*}
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type_synonym 'a item        = "('a, unit) node set"
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type_synonym ('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. case_nat 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 fun_eq_iff) 
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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 fun_eq_iff) 
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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 fun_eq_iff split: nat.split_asm)
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lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1]
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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(2)[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]
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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]
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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]
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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]
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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]
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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]
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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]
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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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     "case_nat (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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   306
apply (simp add: In1_def)
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   307
apply (simp add: Scons_def)
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   308
done
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   309
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   310
lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
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   311
by (simp add: In1_def)
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   312
cb6ae81dd0be merged with theory Datatype_Universe;
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   313
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   314
subsection{*Set Constructions*}
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   315
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   316
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   317
(*** Cartesian Product ***)
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   318
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   319
lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
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   320
by (simp add: uprod_def)
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diff changeset
   321
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   322
(*The general elimination rule*)
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   323
lemma uprodE [elim!]:
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   324
    "[| c : uprod A B;   
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   325
        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
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diff changeset
   326
     |] ==> P"
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   327
by (auto simp add: uprod_def) 
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diff changeset
   328
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   329
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   330
(*Elimination of a pair -- introduces no eigenvariables*)
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   331
lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
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   332
by (auto simp add: uprod_def)
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diff changeset
   333
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   334
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   335
(*** Disjoint Sum ***)
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diff changeset
   336
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   337
lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
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   338
by (simp add: usum_def)
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diff changeset
   339
cb6ae81dd0be merged with theory Datatype_Universe;
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   340
lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
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   341
by (simp add: usum_def)
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diff changeset
   342
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   343
lemma usumE [elim!]: 
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   344
    "[| u : usum A B;   
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   345
        !!x. [| x:A;  u=In0(x) |] ==> P;  
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   346
        !!y. [| y:B;  u=In1(y) |] ==> P  
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diff changeset
   347
     |] ==> P"
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   348
by (auto simp add: usum_def)
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diff changeset
   349
cb6ae81dd0be merged with theory Datatype_Universe;
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   350
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   351
(** Injection **)
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   352
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   353
lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 33968
diff changeset
   354
unfolding In0_def In1_def One_nat_def by auto
20819
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   355
45607
16b4f5774621 eliminated obsolete "standard";
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diff changeset
   356
lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]
20819
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diff changeset
   357
cb6ae81dd0be merged with theory Datatype_Universe;
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   358
lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
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parents: 20798
diff changeset
   359
by (simp add: In0_def)
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wenzelm
parents: 20798
diff changeset
   360
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   361
lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
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parents: 20798
diff changeset
   362
by (simp add: In1_def)
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wenzelm
parents: 20798
diff changeset
   363
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   364
lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
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wenzelm
parents: 20798
diff changeset
   365
by (blast dest!: In0_inject)
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parents: 20798
diff changeset
   366
cb6ae81dd0be merged with theory Datatype_Universe;
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   367
lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
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wenzelm
parents: 20798
diff changeset
   368
by (blast dest!: In1_inject)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   369
cb6ae81dd0be merged with theory Datatype_Universe;
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diff changeset
   370
lemma inj_In0: "inj In0"
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diff changeset
   371
by (blast intro!: inj_onI)
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wenzelm
parents: 20798
diff changeset
   372
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diff changeset
   373
lemma inj_In1: "inj In1"
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diff changeset
   374
by (blast intro!: inj_onI)
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wenzelm
parents: 20798
diff changeset
   375
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   376
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   377
(*** Function spaces ***)
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wenzelm
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diff changeset
   378
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
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diff changeset
   379
lemma Lim_inject: "Lim f = Lim g ==> f = g"
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wenzelm
parents: 20798
diff changeset
   380
apply (simp add: Lim_def)
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wenzelm
parents: 20798
diff changeset
   381
apply (rule ext)
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wenzelm
parents: 20798
diff changeset
   382
apply (blast elim!: Push_Node_inject)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   383
done
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   384
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   385
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   386
(*** proving equality of sets and functions using ntrunc ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   387
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   388
lemma ntrunc_subsetI: "ntrunc k M <= M"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   389
by (auto simp add: ntrunc_def)
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wenzelm
parents: 20798
diff changeset
   390
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   391
lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   392
by (auto simp add: ntrunc_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   393
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   394
(*A generalized form of the take-lemma*)
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wenzelm
parents: 20798
diff changeset
   395
lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   396
apply (rule equalityI)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   397
apply (rule_tac [!] ntrunc_subsetD)
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wenzelm
parents: 20798
diff changeset
   398
apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   399
done
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   400
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   401
lemma ntrunc_o_equality: 
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wenzelm
parents: 20798
diff changeset
   402
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   403
apply (rule ntrunc_equality [THEN ext])
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   404
apply (simp add: fun_eq_iff) 
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   405
done
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   406
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   407
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
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diff changeset
   408
(*** Monotonicity ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   409
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   410
lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   411
by (simp add: uprod_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   412
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   413
lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   414
by (simp add: usum_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   415
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   416
lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   417
by (simp add: Scons_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   418
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   419
lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 33968
diff changeset
   420
by (simp add: In0_def Scons_mono)
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   421
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   422
lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 33968
diff changeset
   423
by (simp add: In1_def Scons_mono)
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   424
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   425
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   426
(*** Split and Case ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   427
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   428
lemma Split [simp]: "Split c (Scons M N) = c M N"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   429
by (simp add: Split_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   430
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   431
lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   432
by (simp add: Case_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   433
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   434
lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   435
by (simp add: Case_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   436
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   437
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   438
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   439
(**** UN x. B(x) rules ****)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   440
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   441
lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   442
by (simp add: ntrunc_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   443
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   444
lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   445
by (simp add: Scons_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   446
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   447
lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   448
by (simp add: Scons_def, blast)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   449
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   450
lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   451
by (simp add: In0_def Scons_UN1_y)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   452
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   453
lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   454
by (simp add: In1_def Scons_UN1_y)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   455
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   456
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   457
(*** Equality for Cartesian Product ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   458
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   459
lemma dprodI [intro!]: 
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   460
    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   461
by (auto simp add: dprod_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   462
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   463
(*The general elimination rule*)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   464
lemma dprodE [elim!]: 
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   465
    "[| c : dprod r s;   
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   466
        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   467
                        c = (Scons x y, Scons x' y') |] ==> P  
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   468
     |] ==> P"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   469
by (auto simp add: dprod_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   470
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   471
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   472
(*** Equality for Disjoint Sum ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   473
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   474
lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   475
by (auto simp add: dsum_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   476
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   477
lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   478
by (auto simp add: dsum_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   479
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   480
lemma dsumE [elim!]: 
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   481
    "[| w : dsum r s;   
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   482
        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   483
        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   484
     |] ==> P"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   485
by (auto simp add: dsum_def)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   486
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   487
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   488
(*** Monotonicity ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   489
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   490
lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   491
by blast
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   492
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   493
lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   494
by blast
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   495
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   496
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   497
(*** Bounding theorems ***)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   498
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   499
lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   500
by blast
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   501
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 42163
diff changeset
   502
lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma]
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   503
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   504
(*Dependent version*)
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   505
lemma dprod_subset_Sigma2:
58112
8081087096ad renamed modules defining old datatypes, as a step towards having 'datatype_new' take 'datatype's place
blanchet
parents: 55642
diff changeset
   506
    "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   507
by auto
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   508
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   509
lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   510
by blast
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   511
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 42163
diff changeset
   512
lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   513
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   514
58157
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   515
(*** Domain theorems ***)
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   516
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   517
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   518
  by auto
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   519
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   520
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   521
  by auto
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   522
c376c43c346c moved old datatype material around
blanchet
parents: 58112
diff changeset
   523
24162
8dfd5dd65d82 split off theory Option for benefit of code generator
haftmann
parents: 22886
diff changeset
   524
text {* hides popular names *}
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 35216
diff changeset
   525
hide_type (open) node item
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 35216
diff changeset
   526
hide_const (open) Push Node Atom Leaf Numb Lim Split Case
20819
cb6ae81dd0be merged with theory Datatype_Universe;
wenzelm
parents: 20798
diff changeset
   527
58372
bfd497f2f4c2 moved 'old_datatype' out of 'Main' (but put it in 'HOL-Proofs' because of the inductive realizer)
blanchet
parents: 58305
diff changeset
   528
ML_file "~~/src/HOL/Tools/Old_Datatype/old_datatype.ML"
bfd497f2f4c2 moved 'old_datatype' out of 'Main' (but put it in 'HOL-Proofs' because of the inductive realizer)
blanchet
parents: 58305
diff changeset
   529
ML_file "~~/src/HOL/Tools/inductive_realizer.ML"
13635
c41e88151b54 Added functions Suml and Sumr which are useful for constructing
berghofe
parents: 12918
diff changeset
   530
5181
4ba3787d9709 New theory Datatype. Needed as an ancestor when defining datatypes.
berghofe
parents:
diff changeset
   531
end