src/HOL/Datatype.thy
author wenzelm
Sun Oct 01 22:19:21 2006 +0200 (2006-10-01)
changeset 20819 cb6ae81dd0be
parent 20798 3275b03e2fff
child 20847 7e8c724339e0
permissions -rw-r--r--
merged with theory Datatype_Universe;
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(*  Title:      HOL/Datatype.thy
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    ID:         $Id$
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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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Could <*> be generalized to a general summation (Sigma)?
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*)
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header{*Analogues of the Cartesian Product and Disjoint Sum for Datatypes*}
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theory Datatype
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imports NatArith Sum_Type
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begin
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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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  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
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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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  apfst_def:  "apfst == (%f (x,y). (f(x),y))"
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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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(** apfst -- can be used in similar type definitions **)
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lemma apfst_conv [simp]: "apfst f (a,b) = (f(a),b)"
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by (simp add: apfst_def)
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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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apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
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apply (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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done
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lemmas Atom_not_Scons = Scons_not_Atom [THEN not_sym, standard]
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declare Atom_not_Scons [iff]
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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 = inj_Leaf [THEN injD, standard]
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declare Leaf_inject [dest!]
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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 = inj_Numb [THEN injD, standard]
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declare Numb_inject [dest!]
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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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apply (simp add: Scons_def One_nat_def)
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apply (blast dest!: Push_Node_inject)
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done
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lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
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apply (simp add: Scons_def One_nat_def)
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apply (blast dest!: Push_Node_inject)
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done
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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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by (simp add: Leaf_def o_def Scons_not_Atom)
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lemmas Leaf_not_Scons = Scons_not_Leaf [THEN not_sym, standard]
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declare Leaf_not_Scons [iff]
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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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by (simp add: Numb_def o_def Scons_not_Atom)
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lemmas Numb_not_Scons = Scons_not_Numb [THEN not_sym, standard]
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declare Numb_not_Scons [iff]
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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 = Leaf_not_Numb [THEN not_sym, standard]
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declare Numb_not_Leaf [iff]
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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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by (simp add: Leaf_def o_def ntrunc_Atom)
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lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
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by (simp add: Numb_def o_def 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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by (auto simp add: Scons_def ntrunc_def One_nat_def 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)
wenzelm@20819
   319
done
wenzelm@20819
   320
wenzelm@20819
   321
lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
wenzelm@20819
   322
by (simp add: In1_def)
wenzelm@20819
   323
wenzelm@20819
   324
wenzelm@20819
   325
subsection{*Set Constructions*}
wenzelm@20819
   326
wenzelm@20819
   327
wenzelm@20819
   328
(*** Cartesian Product ***)
wenzelm@20819
   329
wenzelm@20819
   330
lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
wenzelm@20819
   331
by (simp add: uprod_def)
wenzelm@20819
   332
wenzelm@20819
   333
(*The general elimination rule*)
wenzelm@20819
   334
lemma uprodE [elim!]:
wenzelm@20819
   335
    "[| c : uprod A B;   
wenzelm@20819
   336
        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
wenzelm@20819
   337
     |] ==> P"
wenzelm@20819
   338
by (auto simp add: uprod_def) 
wenzelm@20819
   339
wenzelm@20819
   340
wenzelm@20819
   341
(*Elimination of a pair -- introduces no eigenvariables*)
wenzelm@20819
   342
lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
wenzelm@20819
   343
by (auto simp add: uprod_def)
wenzelm@20819
   344
wenzelm@20819
   345
wenzelm@20819
   346
(*** Disjoint Sum ***)
wenzelm@20819
   347
wenzelm@20819
   348
lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
wenzelm@20819
   349
by (simp add: usum_def)
wenzelm@20819
   350
wenzelm@20819
   351
lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
wenzelm@20819
   352
by (simp add: usum_def)
wenzelm@20819
   353
wenzelm@20819
   354
lemma usumE [elim!]: 
wenzelm@20819
   355
    "[| u : usum A B;   
wenzelm@20819
   356
        !!x. [| x:A;  u=In0(x) |] ==> P;  
wenzelm@20819
   357
        !!y. [| y:B;  u=In1(y) |] ==> P  
wenzelm@20819
   358
     |] ==> P"
wenzelm@20819
   359
by (auto simp add: usum_def)
wenzelm@20819
   360
wenzelm@20819
   361
wenzelm@20819
   362
(** Injection **)
wenzelm@20819
   363
wenzelm@20819
   364
lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
wenzelm@20819
   365
by (auto simp add: In0_def In1_def One_nat_def)
wenzelm@20819
   366
wenzelm@20819
   367
lemmas In1_not_In0 = In0_not_In1 [THEN not_sym, standard]
wenzelm@20819
   368
declare In1_not_In0 [iff]
wenzelm@20819
   369
wenzelm@20819
   370
lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
wenzelm@20819
   371
by (simp add: In0_def)
wenzelm@20819
   372
wenzelm@20819
   373
lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
wenzelm@20819
   374
by (simp add: In1_def)
wenzelm@20819
   375
wenzelm@20819
   376
lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
wenzelm@20819
   377
by (blast dest!: In0_inject)
wenzelm@20819
   378
wenzelm@20819
   379
lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
wenzelm@20819
   380
by (blast dest!: In1_inject)
wenzelm@20819
   381
wenzelm@20819
   382
lemma inj_In0: "inj In0"
wenzelm@20819
   383
by (blast intro!: inj_onI)
wenzelm@20819
   384
wenzelm@20819
   385
lemma inj_In1: "inj In1"
wenzelm@20819
   386
by (blast intro!: inj_onI)
wenzelm@20819
   387
wenzelm@20819
   388
wenzelm@20819
   389
(*** Function spaces ***)
wenzelm@20819
   390
wenzelm@20819
   391
lemma Lim_inject: "Lim f = Lim g ==> f = g"
wenzelm@20819
   392
apply (simp add: Lim_def)
wenzelm@20819
   393
apply (rule ext)
wenzelm@20819
   394
apply (blast elim!: Push_Node_inject)
wenzelm@20819
   395
done
wenzelm@20819
   396
wenzelm@20819
   397
wenzelm@20819
   398
(*** proving equality of sets and functions using ntrunc ***)
wenzelm@20819
   399
wenzelm@20819
   400
lemma ntrunc_subsetI: "ntrunc k M <= M"
wenzelm@20819
   401
by (auto simp add: ntrunc_def)
wenzelm@20819
   402
wenzelm@20819
   403
lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
wenzelm@20819
   404
by (auto simp add: ntrunc_def)
wenzelm@20819
   405
wenzelm@20819
   406
(*A generalized form of the take-lemma*)
wenzelm@20819
   407
lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
wenzelm@20819
   408
apply (rule equalityI)
wenzelm@20819
   409
apply (rule_tac [!] ntrunc_subsetD)
wenzelm@20819
   410
apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
wenzelm@20819
   411
done
wenzelm@20819
   412
wenzelm@20819
   413
lemma ntrunc_o_equality: 
wenzelm@20819
   414
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
wenzelm@20819
   415
apply (rule ntrunc_equality [THEN ext])
wenzelm@20819
   416
apply (simp add: expand_fun_eq) 
wenzelm@20819
   417
done
wenzelm@20819
   418
wenzelm@20819
   419
wenzelm@20819
   420
(*** Monotonicity ***)
wenzelm@20819
   421
wenzelm@20819
   422
lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
wenzelm@20819
   423
by (simp add: uprod_def, blast)
wenzelm@20819
   424
wenzelm@20819
   425
lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
wenzelm@20819
   426
by (simp add: usum_def, blast)
wenzelm@20819
   427
wenzelm@20819
   428
lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
wenzelm@20819
   429
by (simp add: Scons_def, blast)
wenzelm@20819
   430
wenzelm@20819
   431
lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
wenzelm@20819
   432
by (simp add: In0_def subset_refl Scons_mono)
wenzelm@20819
   433
wenzelm@20819
   434
lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
wenzelm@20819
   435
by (simp add: In1_def subset_refl Scons_mono)
wenzelm@20819
   436
wenzelm@20819
   437
wenzelm@20819
   438
(*** Split and Case ***)
wenzelm@20819
   439
wenzelm@20819
   440
lemma Split [simp]: "Split c (Scons M N) = c M N"
wenzelm@20819
   441
by (simp add: Split_def)
wenzelm@20819
   442
wenzelm@20819
   443
lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
wenzelm@20819
   444
by (simp add: Case_def)
wenzelm@20819
   445
wenzelm@20819
   446
lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
wenzelm@20819
   447
by (simp add: Case_def)
wenzelm@20819
   448
wenzelm@20819
   449
wenzelm@20819
   450
wenzelm@20819
   451
(**** UN x. B(x) rules ****)
wenzelm@20819
   452
wenzelm@20819
   453
lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
wenzelm@20819
   454
by (simp add: ntrunc_def, blast)
wenzelm@20819
   455
wenzelm@20819
   456
lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
wenzelm@20819
   457
by (simp add: Scons_def, blast)
wenzelm@20819
   458
wenzelm@20819
   459
lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
wenzelm@20819
   460
by (simp add: Scons_def, blast)
wenzelm@20819
   461
wenzelm@20819
   462
lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
wenzelm@20819
   463
by (simp add: In0_def Scons_UN1_y)
wenzelm@20819
   464
wenzelm@20819
   465
lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
wenzelm@20819
   466
by (simp add: In1_def Scons_UN1_y)
wenzelm@20819
   467
wenzelm@20819
   468
wenzelm@20819
   469
(*** Equality for Cartesian Product ***)
wenzelm@20819
   470
wenzelm@20819
   471
lemma dprodI [intro!]: 
wenzelm@20819
   472
    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
wenzelm@20819
   473
by (auto simp add: dprod_def)
wenzelm@20819
   474
wenzelm@20819
   475
(*The general elimination rule*)
wenzelm@20819
   476
lemma dprodE [elim!]: 
wenzelm@20819
   477
    "[| c : dprod r s;   
wenzelm@20819
   478
        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
wenzelm@20819
   479
                        c = (Scons x y, Scons x' y') |] ==> P  
wenzelm@20819
   480
     |] ==> P"
wenzelm@20819
   481
by (auto simp add: dprod_def)
wenzelm@20819
   482
wenzelm@20819
   483
wenzelm@20819
   484
(*** Equality for Disjoint Sum ***)
wenzelm@20819
   485
wenzelm@20819
   486
lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
wenzelm@20819
   487
by (auto simp add: dsum_def)
wenzelm@20819
   488
wenzelm@20819
   489
lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
wenzelm@20819
   490
by (auto simp add: dsum_def)
wenzelm@20819
   491
wenzelm@20819
   492
lemma dsumE [elim!]: 
wenzelm@20819
   493
    "[| w : dsum r s;   
wenzelm@20819
   494
        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
wenzelm@20819
   495
        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
wenzelm@20819
   496
     |] ==> P"
wenzelm@20819
   497
by (auto simp add: dsum_def)
wenzelm@20819
   498
wenzelm@20819
   499
wenzelm@20819
   500
(*** Monotonicity ***)
wenzelm@20819
   501
wenzelm@20819
   502
lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
wenzelm@20819
   503
by blast
wenzelm@20819
   504
wenzelm@20819
   505
lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
wenzelm@20819
   506
by blast
wenzelm@20819
   507
wenzelm@20819
   508
wenzelm@20819
   509
(*** Bounding theorems ***)
wenzelm@20819
   510
wenzelm@20819
   511
lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
wenzelm@20819
   512
by blast
wenzelm@20819
   513
wenzelm@20819
   514
lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
wenzelm@20819
   515
wenzelm@20819
   516
(*Dependent version*)
wenzelm@20819
   517
lemma dprod_subset_Sigma2:
wenzelm@20819
   518
     "(dprod (Sigma A B) (Sigma C D)) <= 
wenzelm@20819
   519
      Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
wenzelm@20819
   520
by auto
wenzelm@20819
   521
wenzelm@20819
   522
lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
wenzelm@20819
   523
by blast
wenzelm@20819
   524
wenzelm@20819
   525
lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
wenzelm@20819
   526
wenzelm@20819
   527
wenzelm@20819
   528
(*** Domain ***)
wenzelm@20819
   529
wenzelm@20819
   530
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
wenzelm@20819
   531
by auto
wenzelm@20819
   532
wenzelm@20819
   533
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
wenzelm@20819
   534
by auto
wenzelm@20819
   535
wenzelm@20819
   536
wenzelm@20819
   537
subsection {* Finishing the datatype package setup *}
wenzelm@20819
   538
wenzelm@20819
   539
text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
wenzelm@20819
   540
hide (open) const Push Node Atom Leaf Numb Lim Split Case
wenzelm@20819
   541
hide (open) type node item
wenzelm@20819
   542
wenzelm@20819
   543
wenzelm@20819
   544
section {* Datatypes *}
wenzelm@20819
   545
haftmann@20588
   546
setup "DatatypeCodegen.setup2"
haftmann@20588
   547
wenzelm@11954
   548
subsection {* Representing primitive types *}
berghofe@5181
   549
berghofe@5759
   550
rep_datatype bool
wenzelm@11954
   551
  distinct True_not_False False_not_True
wenzelm@11954
   552
  induction bool_induct
wenzelm@11954
   553
wenzelm@11954
   554
declare case_split [cases type: bool]
wenzelm@11954
   555
  -- "prefer plain propositional version"
wenzelm@11954
   556
wenzelm@11954
   557
rep_datatype unit
wenzelm@11954
   558
  induction unit_induct
berghofe@5181
   559
berghofe@5181
   560
rep_datatype prod
wenzelm@11954
   561
  inject Pair_eq
wenzelm@11954
   562
  induction prod_induct
wenzelm@11954
   563
wenzelm@12918
   564
rep_datatype sum
wenzelm@12918
   565
  distinct Inl_not_Inr Inr_not_Inl
wenzelm@12918
   566
  inject Inl_eq Inr_eq
wenzelm@12918
   567
  induction sum_induct
wenzelm@12918
   568
wenzelm@12918
   569
lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
wenzelm@12918
   570
  apply (rule_tac s = s in sumE)
wenzelm@12918
   571
   apply (erule ssubst)
wenzelm@20798
   572
   apply (rule sum.cases(1))
wenzelm@12918
   573
  apply (erule ssubst)
wenzelm@20798
   574
  apply (rule sum.cases(2))
wenzelm@12918
   575
  done
wenzelm@12918
   576
wenzelm@12918
   577
lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
wenzelm@12918
   578
  -- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
wenzelm@20798
   579
  by simp
wenzelm@12918
   580
wenzelm@12918
   581
lemma sum_case_inject:
wenzelm@12918
   582
  "sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
wenzelm@12918
   583
proof -
wenzelm@12918
   584
  assume a: "sum_case f1 f2 = sum_case g1 g2"
wenzelm@12918
   585
  assume r: "f1 = g1 ==> f2 = g2 ==> P"
wenzelm@12918
   586
  show P
wenzelm@12918
   587
    apply (rule r)
wenzelm@12918
   588
     apply (rule ext)
paulson@14208
   589
     apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
wenzelm@12918
   590
    apply (rule ext)
paulson@14208
   591
    apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
wenzelm@12918
   592
    done
wenzelm@12918
   593
qed
wenzelm@12918
   594
berghofe@13635
   595
constdefs
berghofe@13635
   596
  Suml :: "('a => 'c) => 'a + 'b => 'c"
berghofe@13635
   597
  "Suml == (%f. sum_case f arbitrary)"
berghofe@13635
   598
berghofe@13635
   599
  Sumr :: "('b => 'c) => 'a + 'b => 'c"
berghofe@13635
   600
  "Sumr == sum_case arbitrary"
berghofe@13635
   601
berghofe@13635
   602
lemma Suml_inject: "Suml f = Suml g ==> f = g"
berghofe@13635
   603
  by (unfold Suml_def) (erule sum_case_inject)
berghofe@13635
   604
berghofe@13635
   605
lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
berghofe@13635
   606
  by (unfold Sumr_def) (erule sum_case_inject)
berghofe@13635
   607
wenzelm@20798
   608
hide (open) const Suml Sumr
berghofe@13635
   609
wenzelm@12918
   610
wenzelm@12918
   611
subsection {* Further cases/induct rules for tuples *}
wenzelm@11954
   612
wenzelm@20798
   613
lemma prod_cases3 [cases type]:
wenzelm@20798
   614
  obtains (fields) a b c where "y = (a, b, c)"
wenzelm@20798
   615
  by (cases y, case_tac b) blast
wenzelm@11954
   616
wenzelm@11954
   617
lemma prod_induct3 [case_names fields, induct type]:
wenzelm@11954
   618
    "(!!a b c. P (a, b, c)) ==> P x"
wenzelm@11954
   619
  by (cases x) blast
wenzelm@11954
   620
wenzelm@20798
   621
lemma prod_cases4 [cases type]:
wenzelm@20798
   622
  obtains (fields) a b c d where "y = (a, b, c, d)"
wenzelm@20798
   623
  by (cases y, case_tac c) blast
wenzelm@11954
   624
wenzelm@11954
   625
lemma prod_induct4 [case_names fields, induct type]:
wenzelm@11954
   626
    "(!!a b c d. P (a, b, c, d)) ==> P x"
wenzelm@11954
   627
  by (cases x) blast
berghofe@5181
   628
wenzelm@20798
   629
lemma prod_cases5 [cases type]:
wenzelm@20798
   630
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
wenzelm@20798
   631
  by (cases y, case_tac d) blast
wenzelm@11954
   632
wenzelm@11954
   633
lemma prod_induct5 [case_names fields, induct type]:
wenzelm@11954
   634
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
wenzelm@11954
   635
  by (cases x) blast
wenzelm@11954
   636
wenzelm@20798
   637
lemma prod_cases6 [cases type]:
wenzelm@20798
   638
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
wenzelm@20798
   639
  by (cases y, case_tac e) blast
wenzelm@11954
   640
wenzelm@11954
   641
lemma prod_induct6 [case_names fields, induct type]:
wenzelm@11954
   642
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
wenzelm@11954
   643
  by (cases x) blast
wenzelm@11954
   644
wenzelm@20798
   645
lemma prod_cases7 [cases type]:
wenzelm@20798
   646
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
wenzelm@20798
   647
  by (cases y, case_tac f) blast
wenzelm@11954
   648
wenzelm@11954
   649
lemma prod_induct7 [case_names fields, induct type]:
wenzelm@11954
   650
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
wenzelm@11954
   651
  by (cases x) blast
berghofe@5759
   652
wenzelm@12918
   653
wenzelm@12918
   654
subsection {* The option type *}
wenzelm@12918
   655
wenzelm@12918
   656
datatype 'a option = None | Some 'a
wenzelm@12918
   657
wenzelm@20798
   658
lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
nipkow@18576
   659
  by (induct x) auto
nipkow@18576
   660
wenzelm@20798
   661
lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
paulson@18447
   662
  by (induct x) auto
paulson@18447
   663
nipkow@18576
   664
text{*Although it may appear that both of these equalities are helpful
nipkow@18576
   665
only when applied to assumptions, in practice it seems better to give
nipkow@18576
   666
them the uniform iff attribute. *}
wenzelm@12918
   667
wenzelm@12918
   668
lemma option_caseE:
wenzelm@20798
   669
  assumes c: "(case x of None => P | Some y => Q y)"
wenzelm@20798
   670
  obtains
wenzelm@20798
   671
    (None) "x = None" and P
wenzelm@20798
   672
  | (Some) y where "x = Some y" and "Q y"
wenzelm@20798
   673
  using c by (cases x) simp_all
wenzelm@12918
   674
wenzelm@12918
   675
wenzelm@12918
   676
subsubsection {* Operations *}
wenzelm@12918
   677
wenzelm@12918
   678
consts
wenzelm@12918
   679
  the :: "'a option => 'a"
wenzelm@12918
   680
primrec
wenzelm@12918
   681
  "the (Some x) = x"
wenzelm@12918
   682
wenzelm@12918
   683
consts
wenzelm@12918
   684
  o2s :: "'a option => 'a set"
wenzelm@12918
   685
primrec
wenzelm@12918
   686
  "o2s None = {}"
wenzelm@12918
   687
  "o2s (Some x) = {x}"
wenzelm@12918
   688
wenzelm@12918
   689
lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
wenzelm@12918
   690
  by simp
wenzelm@12918
   691
wenzelm@17876
   692
ML_setup {* change_claset (fn cs => cs addSD2 ("ospec", thm "ospec")) *}
wenzelm@12918
   693
wenzelm@12918
   694
lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
wenzelm@12918
   695
  by (cases xo) auto
wenzelm@12918
   696
wenzelm@12918
   697
lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
wenzelm@12918
   698
  by (cases xo) auto
wenzelm@12918
   699
wenzelm@12918
   700
wenzelm@12918
   701
constdefs
wenzelm@12918
   702
  option_map :: "('a => 'b) => ('a option => 'b option)"
wenzelm@12918
   703
  "option_map == %f y. case y of None => None | Some x => Some (f x)"
wenzelm@12918
   704
wenzelm@12918
   705
lemma option_map_None [simp]: "option_map f None = None"
wenzelm@12918
   706
  by (simp add: option_map_def)
wenzelm@12918
   707
wenzelm@12918
   708
lemma option_map_Some [simp]: "option_map f (Some x) = Some (f x)"
wenzelm@12918
   709
  by (simp add: option_map_def)
wenzelm@12918
   710
wenzelm@20798
   711
lemma option_map_is_None [iff]:
wenzelm@20798
   712
    "(option_map f opt = None) = (opt = None)"
wenzelm@20798
   713
  by (simp add: option_map_def split add: option.split)
nipkow@14187
   714
wenzelm@12918
   715
lemma option_map_eq_Some [iff]:
wenzelm@12918
   716
    "(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
wenzelm@20798
   717
  by (simp add: option_map_def split add: option.split)
nipkow@14187
   718
nipkow@14187
   719
lemma option_map_comp:
wenzelm@20798
   720
    "option_map f (option_map g opt) = option_map (f o g) opt"
wenzelm@20798
   721
  by (simp add: option_map_def split add: option.split)
wenzelm@12918
   722
wenzelm@12918
   723
lemma option_map_o_sum_case [simp]:
wenzelm@12918
   724
    "option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
wenzelm@20798
   725
  by (rule ext) (simp split: sum.split)
wenzelm@12918
   726
haftmann@19787
   727
haftmann@19817
   728
subsubsection {* Codegenerator setup *}
haftmann@19817
   729
haftmann@19787
   730
consts
haftmann@19787
   731
  is_none :: "'a option \<Rightarrow> bool"
haftmann@19787
   732
primrec
haftmann@19787
   733
  "is_none None = True"
haftmann@19787
   734
  "is_none (Some x) = False"
haftmann@19787
   735
haftmann@20105
   736
lemma is_none_none [code inline]:
wenzelm@20798
   737
    "(x = None) = (is_none x)" 
wenzelm@20798
   738
  by (cases x) simp_all
haftmann@19787
   739
wenzelm@17458
   740
lemmas [code] = imp_conv_disj
wenzelm@17458
   741
krauss@20523
   742
lemma [code func]:
haftmann@19138
   743
  "(\<not> True) = False" by (rule HOL.simp_thms)
haftmann@19138
   744
krauss@20523
   745
lemma [code func]:
haftmann@19138
   746
  "(\<not> False) = True" by (rule HOL.simp_thms)
haftmann@19138
   747
krauss@20523
   748
lemma [code func]:
nipkow@19179
   749
  shows "(False \<and> x) = False"
wenzelm@20798
   750
    and "(True \<and> x) = x"
wenzelm@20798
   751
    and "(x \<and> False) = False"
wenzelm@20798
   752
    and "(x \<and> True) = x" by simp_all
haftmann@19138
   753
krauss@20523
   754
lemma [code func]:
nipkow@19179
   755
  shows "(False \<or> x) = x"
wenzelm@20798
   756
    and "(True \<or> x) = True"
wenzelm@20798
   757
    and "(x \<or> False) = x"
wenzelm@20798
   758
    and "(x \<or> True) = True" by simp_all
haftmann@19138
   759
haftmann@19138
   760
declare
krauss@20523
   761
  if_True [code func]
krauss@20523
   762
  if_False [code func]
nipkow@19179
   763
  fst_conv [code]
nipkow@19179
   764
  snd_conv [code]
haftmann@19138
   765
haftmann@20105
   766
lemma split_is_prod_case [code inline]:
wenzelm@20798
   767
    "split = prod_case"
wenzelm@20798
   768
  by (simp add: expand_fun_eq split_def prod.cases)
haftmann@20105
   769
haftmann@20453
   770
code_type bool
haftmann@20453
   771
  (SML target_atom "bool")
haftmann@20453
   772
  (Haskell target_atom "Bool")
haftmann@19138
   773
haftmann@20453
   774
code_const True and False and Not and "op &" and "op |" and If
haftmann@20453
   775
  (SML target_atom "true" and target_atom "false" and target_atom "not"
haftmann@20453
   776
    and infixl 1 "andalso" and infixl 0 "orelse"
haftmann@20453
   777
    and target_atom "(if __/ then __/ else __)")
haftmann@20453
   778
  (Haskell target_atom "True" and target_atom "False" and target_atom "not"
haftmann@20453
   779
    and infixl 3 "&&" and infixl 2 "||"
haftmann@20453
   780
    and target_atom "(if __/ then __/ else __)")
haftmann@20453
   781
haftmann@20453
   782
code_type *
haftmann@20453
   783
  (SML infix 2 "*")
haftmann@20453
   784
  (Haskell target_atom "(__,/ __)")
haftmann@19138
   785
haftmann@20453
   786
code_const Pair
haftmann@20453
   787
  (SML target_atom "(__,/ __)")
haftmann@20453
   788
  (Haskell target_atom "(__,/ __)")
haftmann@18702
   789
haftmann@20453
   790
code_type unit
haftmann@20453
   791
  (SML target_atom "unit")
haftmann@20453
   792
  (Haskell target_atom "()")
haftmann@19150
   793
haftmann@20453
   794
code_const Unity
haftmann@20453
   795
  (SML target_atom "()")
haftmann@20453
   796
  (Haskell target_atom "()")
haftmann@19150
   797
haftmann@20453
   798
code_type option
haftmann@20453
   799
  (SML "_ option")
haftmann@20453
   800
  (Haskell "Maybe _")
haftmann@19150
   801
haftmann@20453
   802
code_const None and Some
haftmann@20453
   803
  (SML target_atom "NONE" and target_atom "SOME")
haftmann@20453
   804
  (Haskell target_atom "Nothing" and target_atom "Just")
haftmann@19150
   805
haftmann@20588
   806
code_instance option :: eq
haftmann@20588
   807
  (Haskell -)
haftmann@20588
   808
haftmann@20588
   809
code_const "OperationalEquality.eq \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"
haftmann@20588
   810
  (Haskell infixl 4 "==")
haftmann@20588
   811
wenzelm@20819
   812
ML
wenzelm@20819
   813
{*
wenzelm@20819
   814
val apfst_conv = thm "apfst_conv";
wenzelm@20819
   815
val apfst_convE = thm "apfst_convE";
wenzelm@20819
   816
val Push_inject1 = thm "Push_inject1";
wenzelm@20819
   817
val Push_inject2 = thm "Push_inject2";
wenzelm@20819
   818
val Push_inject = thm "Push_inject";
wenzelm@20819
   819
val Push_neq_K0 = thm "Push_neq_K0";
wenzelm@20819
   820
val Abs_Node_inj = thm "Abs_Node_inj";
wenzelm@20819
   821
val Node_K0_I = thm "Node_K0_I";
wenzelm@20819
   822
val Node_Push_I = thm "Node_Push_I";
wenzelm@20819
   823
val Scons_not_Atom = thm "Scons_not_Atom";
wenzelm@20819
   824
val Atom_not_Scons = thm "Atom_not_Scons";
wenzelm@20819
   825
val inj_Atom = thm "inj_Atom";
wenzelm@20819
   826
val Atom_inject = thm "Atom_inject";
wenzelm@20819
   827
val Atom_Atom_eq = thm "Atom_Atom_eq";
wenzelm@20819
   828
val inj_Leaf = thm "inj_Leaf";
wenzelm@20819
   829
val Leaf_inject = thm "Leaf_inject";
wenzelm@20819
   830
val inj_Numb = thm "inj_Numb";
wenzelm@20819
   831
val Numb_inject = thm "Numb_inject";
wenzelm@20819
   832
val Push_Node_inject = thm "Push_Node_inject";
wenzelm@20819
   833
val Scons_inject1 = thm "Scons_inject1";
wenzelm@20819
   834
val Scons_inject2 = thm "Scons_inject2";
wenzelm@20819
   835
val Scons_inject = thm "Scons_inject";
wenzelm@20819
   836
val Scons_Scons_eq = thm "Scons_Scons_eq";
wenzelm@20819
   837
val Scons_not_Leaf = thm "Scons_not_Leaf";
wenzelm@20819
   838
val Leaf_not_Scons = thm "Leaf_not_Scons";
wenzelm@20819
   839
val Scons_not_Numb = thm "Scons_not_Numb";
wenzelm@20819
   840
val Numb_not_Scons = thm "Numb_not_Scons";
wenzelm@20819
   841
val Leaf_not_Numb = thm "Leaf_not_Numb";
wenzelm@20819
   842
val Numb_not_Leaf = thm "Numb_not_Leaf";
wenzelm@20819
   843
val ndepth_K0 = thm "ndepth_K0";
wenzelm@20819
   844
val ndepth_Push_Node_aux = thm "ndepth_Push_Node_aux";
wenzelm@20819
   845
val ndepth_Push_Node = thm "ndepth_Push_Node";
wenzelm@20819
   846
val ntrunc_0 = thm "ntrunc_0";
wenzelm@20819
   847
val ntrunc_Atom = thm "ntrunc_Atom";
wenzelm@20819
   848
val ntrunc_Leaf = thm "ntrunc_Leaf";
wenzelm@20819
   849
val ntrunc_Numb = thm "ntrunc_Numb";
wenzelm@20819
   850
val ntrunc_Scons = thm "ntrunc_Scons";
wenzelm@20819
   851
val ntrunc_one_In0 = thm "ntrunc_one_In0";
wenzelm@20819
   852
val ntrunc_In0 = thm "ntrunc_In0";
wenzelm@20819
   853
val ntrunc_one_In1 = thm "ntrunc_one_In1";
wenzelm@20819
   854
val ntrunc_In1 = thm "ntrunc_In1";
wenzelm@20819
   855
val uprodI = thm "uprodI";
wenzelm@20819
   856
val uprodE = thm "uprodE";
wenzelm@20819
   857
val uprodE2 = thm "uprodE2";
wenzelm@20819
   858
val usum_In0I = thm "usum_In0I";
wenzelm@20819
   859
val usum_In1I = thm "usum_In1I";
wenzelm@20819
   860
val usumE = thm "usumE";
wenzelm@20819
   861
val In0_not_In1 = thm "In0_not_In1";
wenzelm@20819
   862
val In1_not_In0 = thm "In1_not_In0";
wenzelm@20819
   863
val In0_inject = thm "In0_inject";
wenzelm@20819
   864
val In1_inject = thm "In1_inject";
wenzelm@20819
   865
val In0_eq = thm "In0_eq";
wenzelm@20819
   866
val In1_eq = thm "In1_eq";
wenzelm@20819
   867
val inj_In0 = thm "inj_In0";
wenzelm@20819
   868
val inj_In1 = thm "inj_In1";
wenzelm@20819
   869
val Lim_inject = thm "Lim_inject";
wenzelm@20819
   870
val ntrunc_subsetI = thm "ntrunc_subsetI";
wenzelm@20819
   871
val ntrunc_subsetD = thm "ntrunc_subsetD";
wenzelm@20819
   872
val ntrunc_equality = thm "ntrunc_equality";
wenzelm@20819
   873
val ntrunc_o_equality = thm "ntrunc_o_equality";
wenzelm@20819
   874
val uprod_mono = thm "uprod_mono";
wenzelm@20819
   875
val usum_mono = thm "usum_mono";
wenzelm@20819
   876
val Scons_mono = thm "Scons_mono";
wenzelm@20819
   877
val In0_mono = thm "In0_mono";
wenzelm@20819
   878
val In1_mono = thm "In1_mono";
wenzelm@20819
   879
val Split = thm "Split";
wenzelm@20819
   880
val Case_In0 = thm "Case_In0";
wenzelm@20819
   881
val Case_In1 = thm "Case_In1";
wenzelm@20819
   882
val ntrunc_UN1 = thm "ntrunc_UN1";
wenzelm@20819
   883
val Scons_UN1_x = thm "Scons_UN1_x";
wenzelm@20819
   884
val Scons_UN1_y = thm "Scons_UN1_y";
wenzelm@20819
   885
val In0_UN1 = thm "In0_UN1";
wenzelm@20819
   886
val In1_UN1 = thm "In1_UN1";
wenzelm@20819
   887
val dprodI = thm "dprodI";
wenzelm@20819
   888
val dprodE = thm "dprodE";
wenzelm@20819
   889
val dsum_In0I = thm "dsum_In0I";
wenzelm@20819
   890
val dsum_In1I = thm "dsum_In1I";
wenzelm@20819
   891
val dsumE = thm "dsumE";
wenzelm@20819
   892
val dprod_mono = thm "dprod_mono";
wenzelm@20819
   893
val dsum_mono = thm "dsum_mono";
wenzelm@20819
   894
val dprod_Sigma = thm "dprod_Sigma";
wenzelm@20819
   895
val dprod_subset_Sigma = thm "dprod_subset_Sigma";
wenzelm@20819
   896
val dprod_subset_Sigma2 = thm "dprod_subset_Sigma2";
wenzelm@20819
   897
val dsum_Sigma = thm "dsum_Sigma";
wenzelm@20819
   898
val dsum_subset_Sigma = thm "dsum_subset_Sigma";
wenzelm@20819
   899
val Domain_dprod = thm "Domain_dprod";
wenzelm@20819
   900
val Domain_dsum = thm "Domain_dsum";
wenzelm@20819
   901
*}
wenzelm@20819
   902
berghofe@5181
   903
end