berghofe@5181
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(* Title: HOL/Datatype.thy
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berghofe@5181
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ID: $Id$
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wenzelm@20819
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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wenzelm@11954
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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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berghofe@5181
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*)
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berghofe@5181
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wenzelm@21669
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header {* Analogues of the Cartesian Product and Disjoint Sum for Datatypes *}
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nipkow@15131
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theory Datatype
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haftmann@27981
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imports Nat Relation
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nipkow@15131
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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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Push :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
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Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
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ndepth :: "('a, 'b) node => nat"
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Atom :: "('a + nat) => ('a, 'b) dtree"
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Leaf :: "'a => ('a, 'b) dtree"
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Numb :: "nat => ('a, 'b) dtree"
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Scons :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
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In0 :: "('a, 'b) dtree => ('a, 'b) dtree"
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In1 :: "('a, 'b) dtree => ('a, 'b) dtree"
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Lim :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
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ntrunc :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
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uprod :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
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usum :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
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Split :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
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Case :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
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dprod :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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=> (('a, 'b) dtree * ('a, 'b) dtree)set"
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dsum :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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=> (('a, 'b) dtree * ('a, 'b) dtree)set"
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defs
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Push_Node_def: "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
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(*crude "lists" of nats -- needed for the constructions*)
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Push_def: "Push == (%b h. nat_case b h)"
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(** operations on S-expressions -- sets of nodes **)
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(*S-expression constructors*)
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Atom_def: "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
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Scons_def: "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
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(*Leaf nodes, with arbitrary or nat labels*)
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Leaf_def: "Leaf == Atom o Inl"
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Numb_def: "Numb == Atom o Inr"
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(*Injections of the "disjoint sum"*)
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In0_def: "In0(M) == Scons (Numb 0) M"
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In1_def: "In1(M) == Scons (Numb 1) M"
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(*Function spaces*)
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Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
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(*the set of nodes with depth less than k*)
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ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
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ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
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(*products and sums for the "universe"*)
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uprod_def: "uprod A B == UN x:A. UN y:B. { Scons x y }"
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usum_def: "usum A B == In0`A Un In1`B"
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(*the corresponding eliminators*)
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Split_def: "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
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Case_def: "Case c d M == THE u. (EX x . M = In0(x) & u = c(x))
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| (EX y . M = In1(y) & u = d(y))"
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(** equality for the "universe" **)
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dprod_def: "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
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dsum_def: "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
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(UN (y,y'):s. {(In1(y),In1(y'))})"
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lemma apfst_convE:
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"[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> R
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|] ==> R"
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by (force simp add: apfst_def)
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(** Push -- an injection, analogous to Cons on lists **)
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lemma Push_inject1: "Push i f = Push j g ==> i=j"
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apply (simp add: Push_def expand_fun_eq)
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apply (drule_tac x=0 in spec, simp)
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done
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lemma Push_inject2: "Push i f = Push j g ==> f=g"
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apply (auto simp add: Push_def expand_fun_eq)
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apply (drule_tac x="Suc x" in spec, simp)
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done
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lemma Push_inject:
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"[| Push i f =Push j g; [| i=j; f=g |] ==> P |] ==> P"
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by (blast dest: Push_inject1 Push_inject2)
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lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
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by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
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lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
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(*** Introduction rules for Node ***)
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lemma Node_K0_I: "(%k. Inr 0, a) : Node"
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by (simp add: Node_def)
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lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
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apply (simp add: Node_def Push_def)
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apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
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done
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subsection{*Freeness: Distinctness of Constructors*}
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(** Scons vs Atom **)
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lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
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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 [iff] = Scons_not_Atom [THEN not_sym, standard]
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(*** Injectiveness ***)
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(** Atomic nodes **)
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lemma inj_Atom: "inj(Atom)"
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apply (simp add: Atom_def)
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apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
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done
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lemmas Atom_inject = inj_Atom [THEN injD, standard]
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lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
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by (blast dest!: Atom_inject)
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lemma inj_Leaf: "inj(Leaf)"
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apply (simp add: Leaf_def o_def)
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apply (rule inj_onI)
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apply (erule Atom_inject [THEN Inl_inject])
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done
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lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD, standard]
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lemma inj_Numb: "inj(Numb)"
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apply (simp add: Numb_def o_def)
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apply (rule inj_onI)
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apply (erule Atom_inject [THEN Inr_inject])
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done
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lemmas Numb_inject [dest!] = inj_Numb [THEN injD, standard]
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(** Injectiveness of Push_Node **)
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lemma Push_Node_inject:
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"[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P
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|] ==> P"
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apply (simp add: Push_Node_def)
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apply (erule Abs_Node_inj [THEN apfst_convE])
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apply (rule Rep_Node [THEN Node_Push_I])+
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apply (erule sym [THEN apfst_convE])
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apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
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done
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(** Injectiveness of Scons **)
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lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
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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 [iff] = Scons_not_Leaf [THEN not_sym, standard]
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(** Scons vs Numb **)
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lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
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by (simp add: Numb_def o_def Scons_not_Atom)
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lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym, standard]
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(** Leaf vs Numb **)
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lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
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by (simp add: Leaf_def Numb_def)
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lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym, standard]
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(*** ndepth -- the depth of a node ***)
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lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
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by (simp add: ndepth_def Node_K0_I [THEN Abs_Node_inverse] Least_equality)
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lemma ndepth_Push_Node_aux:
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"nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
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apply (induct_tac "k", auto)
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apply (erule Least_le)
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done
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lemma ndepth_Push_Node:
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"ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
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apply (insert Rep_Node [of n, unfolded Node_def])
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wenzelm@20819
|
264 |
apply (auto simp add: ndepth_def Push_Node_def
|
wenzelm@20819
|
265 |
Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
|
wenzelm@20819
|
266 |
apply (rule Least_equality)
|
wenzelm@20819
|
267 |
apply (auto simp add: Push_def ndepth_Push_Node_aux)
|
wenzelm@20819
|
268 |
apply (erule LeastI)
|
wenzelm@20819
|
269 |
done
|
wenzelm@20819
|
270 |
|
wenzelm@20819
|
271 |
|
wenzelm@20819
|
272 |
(*** ntrunc applied to the various node sets ***)
|
wenzelm@20819
|
273 |
|
wenzelm@20819
|
274 |
lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
|
wenzelm@20819
|
275 |
by (simp add: ntrunc_def)
|
wenzelm@20819
|
276 |
|
wenzelm@20819
|
277 |
lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
|
wenzelm@20819
|
278 |
by (auto simp add: Atom_def ntrunc_def ndepth_K0)
|
wenzelm@20819
|
279 |
|
wenzelm@20819
|
280 |
lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
|
wenzelm@20819
|
281 |
by (simp add: Leaf_def o_def ntrunc_Atom)
|
wenzelm@20819
|
282 |
|
wenzelm@20819
|
283 |
lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
|
wenzelm@20819
|
284 |
by (simp add: Numb_def o_def ntrunc_Atom)
|
wenzelm@20819
|
285 |
|
wenzelm@20819
|
286 |
lemma ntrunc_Scons [simp]:
|
wenzelm@20819
|
287 |
"ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
|
wenzelm@20819
|
288 |
by (auto simp add: Scons_def ntrunc_def One_nat_def ndepth_Push_Node)
|
wenzelm@20819
|
289 |
|
wenzelm@20819
|
290 |
|
wenzelm@20819
|
291 |
|
wenzelm@20819
|
292 |
(** Injection nodes **)
|
wenzelm@20819
|
293 |
|
wenzelm@20819
|
294 |
lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
|
wenzelm@20819
|
295 |
apply (simp add: In0_def)
|
wenzelm@20819
|
296 |
apply (simp add: Scons_def)
|
wenzelm@20819
|
297 |
done
|
wenzelm@20819
|
298 |
|
wenzelm@20819
|
299 |
lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
|
wenzelm@20819
|
300 |
by (simp add: In0_def)
|
wenzelm@20819
|
301 |
|
wenzelm@20819
|
302 |
lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
|
wenzelm@20819
|
303 |
apply (simp add: In1_def)
|
wenzelm@20819
|
304 |
apply (simp add: Scons_def)
|
wenzelm@20819
|
305 |
done
|
wenzelm@20819
|
306 |
|
wenzelm@20819
|
307 |
lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
|
wenzelm@20819
|
308 |
by (simp add: In1_def)
|
wenzelm@20819
|
309 |
|
wenzelm@20819
|
310 |
|
wenzelm@20819
|
311 |
subsection{*Set Constructions*}
|
wenzelm@20819
|
312 |
|
wenzelm@20819
|
313 |
|
wenzelm@20819
|
314 |
(*** Cartesian Product ***)
|
wenzelm@20819
|
315 |
|
wenzelm@20819
|
316 |
lemma uprodI [intro!]: "[| M:A; N:B |] ==> Scons M N : uprod A B"
|
wenzelm@20819
|
317 |
by (simp add: uprod_def)
|
wenzelm@20819
|
318 |
|
wenzelm@20819
|
319 |
(*The general elimination rule*)
|
wenzelm@20819
|
320 |
lemma uprodE [elim!]:
|
wenzelm@20819
|
321 |
"[| c : uprod A B;
|
wenzelm@20819
|
322 |
!!x y. [| x:A; y:B; c = Scons x y |] ==> P
|
wenzelm@20819
|
323 |
|] ==> P"
|
wenzelm@20819
|
324 |
by (auto simp add: uprod_def)
|
wenzelm@20819
|
325 |
|
wenzelm@20819
|
326 |
|
wenzelm@20819
|
327 |
(*Elimination of a pair -- introduces no eigenvariables*)
|
wenzelm@20819
|
328 |
lemma uprodE2: "[| Scons M N : uprod A B; [| M:A; N:B |] ==> P |] ==> P"
|
wenzelm@20819
|
329 |
by (auto simp add: uprod_def)
|
wenzelm@20819
|
330 |
|
wenzelm@20819
|
331 |
|
wenzelm@20819
|
332 |
(*** Disjoint Sum ***)
|
wenzelm@20819
|
333 |
|
wenzelm@20819
|
334 |
lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
|
wenzelm@20819
|
335 |
by (simp add: usum_def)
|
wenzelm@20819
|
336 |
|
wenzelm@20819
|
337 |
lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
|
wenzelm@20819
|
338 |
by (simp add: usum_def)
|
wenzelm@20819
|
339 |
|
wenzelm@20819
|
340 |
lemma usumE [elim!]:
|
wenzelm@20819
|
341 |
"[| u : usum A B;
|
wenzelm@20819
|
342 |
!!x. [| x:A; u=In0(x) |] ==> P;
|
wenzelm@20819
|
343 |
!!y. [| y:B; u=In1(y) |] ==> P
|
wenzelm@20819
|
344 |
|] ==> P"
|
wenzelm@20819
|
345 |
by (auto simp add: usum_def)
|
wenzelm@20819
|
346 |
|
wenzelm@20819
|
347 |
|
wenzelm@20819
|
348 |
(** Injection **)
|
wenzelm@20819
|
349 |
|
wenzelm@20819
|
350 |
lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
|
wenzelm@20819
|
351 |
by (auto simp add: In0_def In1_def One_nat_def)
|
wenzelm@20819
|
352 |
|
haftmann@21407
|
353 |
lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym, standard]
|
wenzelm@20819
|
354 |
|
wenzelm@20819
|
355 |
lemma In0_inject: "In0(M) = In0(N) ==> M=N"
|
wenzelm@20819
|
356 |
by (simp add: In0_def)
|
wenzelm@20819
|
357 |
|
wenzelm@20819
|
358 |
lemma In1_inject: "In1(M) = In1(N) ==> M=N"
|
wenzelm@20819
|
359 |
by (simp add: In1_def)
|
wenzelm@20819
|
360 |
|
wenzelm@20819
|
361 |
lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
|
wenzelm@20819
|
362 |
by (blast dest!: In0_inject)
|
wenzelm@20819
|
363 |
|
wenzelm@20819
|
364 |
lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
|
wenzelm@20819
|
365 |
by (blast dest!: In1_inject)
|
wenzelm@20819
|
366 |
|
wenzelm@20819
|
367 |
lemma inj_In0: "inj In0"
|
wenzelm@20819
|
368 |
by (blast intro!: inj_onI)
|
wenzelm@20819
|
369 |
|
wenzelm@20819
|
370 |
lemma inj_In1: "inj In1"
|
wenzelm@20819
|
371 |
by (blast intro!: inj_onI)
|
wenzelm@20819
|
372 |
|
wenzelm@20819
|
373 |
|
wenzelm@20819
|
374 |
(*** Function spaces ***)
|
wenzelm@20819
|
375 |
|
wenzelm@20819
|
376 |
lemma Lim_inject: "Lim f = Lim g ==> f = g"
|
wenzelm@20819
|
377 |
apply (simp add: Lim_def)
|
wenzelm@20819
|
378 |
apply (rule ext)
|
wenzelm@20819
|
379 |
apply (blast elim!: Push_Node_inject)
|
wenzelm@20819
|
380 |
done
|
wenzelm@20819
|
381 |
|
wenzelm@20819
|
382 |
|
wenzelm@20819
|
383 |
(*** proving equality of sets and functions using ntrunc ***)
|
wenzelm@20819
|
384 |
|
wenzelm@20819
|
385 |
lemma ntrunc_subsetI: "ntrunc k M <= M"
|
wenzelm@20819
|
386 |
by (auto simp add: ntrunc_def)
|
wenzelm@20819
|
387 |
|
wenzelm@20819
|
388 |
lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
|
wenzelm@20819
|
389 |
by (auto simp add: ntrunc_def)
|
wenzelm@20819
|
390 |
|
wenzelm@20819
|
391 |
(*A generalized form of the take-lemma*)
|
wenzelm@20819
|
392 |
lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
|
wenzelm@20819
|
393 |
apply (rule equalityI)
|
wenzelm@20819
|
394 |
apply (rule_tac [!] ntrunc_subsetD)
|
wenzelm@20819
|
395 |
apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto)
|
wenzelm@20819
|
396 |
done
|
wenzelm@20819
|
397 |
|
wenzelm@20819
|
398 |
lemma ntrunc_o_equality:
|
wenzelm@20819
|
399 |
"[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
|
wenzelm@20819
|
400 |
apply (rule ntrunc_equality [THEN ext])
|
wenzelm@20819
|
401 |
apply (simp add: expand_fun_eq)
|
wenzelm@20819
|
402 |
done
|
wenzelm@20819
|
403 |
|
wenzelm@20819
|
404 |
|
wenzelm@20819
|
405 |
(*** Monotonicity ***)
|
wenzelm@20819
|
406 |
|
wenzelm@20819
|
407 |
lemma uprod_mono: "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'"
|
wenzelm@20819
|
408 |
by (simp add: uprod_def, blast)
|
wenzelm@20819
|
409 |
|
wenzelm@20819
|
410 |
lemma usum_mono: "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'"
|
wenzelm@20819
|
411 |
by (simp add: usum_def, blast)
|
wenzelm@20819
|
412 |
|
wenzelm@20819
|
413 |
lemma Scons_mono: "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'"
|
wenzelm@20819
|
414 |
by (simp add: Scons_def, blast)
|
wenzelm@20819
|
415 |
|
wenzelm@20819
|
416 |
lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
|
wenzelm@20819
|
417 |
by (simp add: In0_def subset_refl Scons_mono)
|
wenzelm@20819
|
418 |
|
wenzelm@20819
|
419 |
lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
|
wenzelm@20819
|
420 |
by (simp add: In1_def subset_refl Scons_mono)
|
wenzelm@20819
|
421 |
|
wenzelm@20819
|
422 |
|
wenzelm@20819
|
423 |
(*** Split and Case ***)
|
wenzelm@20819
|
424 |
|
wenzelm@20819
|
425 |
lemma Split [simp]: "Split c (Scons M N) = c M N"
|
wenzelm@20819
|
426 |
by (simp add: Split_def)
|
wenzelm@20819
|
427 |
|
wenzelm@20819
|
428 |
lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
|
wenzelm@20819
|
429 |
by (simp add: Case_def)
|
wenzelm@20819
|
430 |
|
wenzelm@20819
|
431 |
lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
|
wenzelm@20819
|
432 |
by (simp add: Case_def)
|
wenzelm@20819
|
433 |
|
wenzelm@20819
|
434 |
|
wenzelm@20819
|
435 |
|
wenzelm@20819
|
436 |
(**** UN x. B(x) rules ****)
|
wenzelm@20819
|
437 |
|
wenzelm@20819
|
438 |
lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
|
wenzelm@20819
|
439 |
by (simp add: ntrunc_def, blast)
|
wenzelm@20819
|
440 |
|
wenzelm@20819
|
441 |
lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
|
wenzelm@20819
|
442 |
by (simp add: Scons_def, blast)
|
wenzelm@20819
|
443 |
|
wenzelm@20819
|
444 |
lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
|
wenzelm@20819
|
445 |
by (simp add: Scons_def, blast)
|
wenzelm@20819
|
446 |
|
wenzelm@20819
|
447 |
lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
|
wenzelm@20819
|
448 |
by (simp add: In0_def Scons_UN1_y)
|
wenzelm@20819
|
449 |
|
wenzelm@20819
|
450 |
lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
|
wenzelm@20819
|
451 |
by (simp add: In1_def Scons_UN1_y)
|
wenzelm@20819
|
452 |
|
wenzelm@20819
|
453 |
|
wenzelm@20819
|
454 |
(*** Equality for Cartesian Product ***)
|
wenzelm@20819
|
455 |
|
wenzelm@20819
|
456 |
lemma dprodI [intro!]:
|
wenzelm@20819
|
457 |
"[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
|
wenzelm@20819
|
458 |
by (auto simp add: dprod_def)
|
wenzelm@20819
|
459 |
|
wenzelm@20819
|
460 |
(*The general elimination rule*)
|
wenzelm@20819
|
461 |
lemma dprodE [elim!]:
|
wenzelm@20819
|
462 |
"[| c : dprod r s;
|
wenzelm@20819
|
463 |
!!x y x' y'. [| (x,x') : r; (y,y') : s;
|
wenzelm@20819
|
464 |
c = (Scons x y, Scons x' y') |] ==> P
|
wenzelm@20819
|
465 |
|] ==> P"
|
wenzelm@20819
|
466 |
by (auto simp add: dprod_def)
|
wenzelm@20819
|
467 |
|
wenzelm@20819
|
468 |
|
wenzelm@20819
|
469 |
(*** Equality for Disjoint Sum ***)
|
wenzelm@20819
|
470 |
|
wenzelm@20819
|
471 |
lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
|
wenzelm@20819
|
472 |
by (auto simp add: dsum_def)
|
wenzelm@20819
|
473 |
|
wenzelm@20819
|
474 |
lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
|
wenzelm@20819
|
475 |
by (auto simp add: dsum_def)
|
wenzelm@20819
|
476 |
|
wenzelm@20819
|
477 |
lemma dsumE [elim!]:
|
wenzelm@20819
|
478 |
"[| w : dsum r s;
|
wenzelm@20819
|
479 |
!!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P;
|
wenzelm@20819
|
480 |
!!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P
|
wenzelm@20819
|
481 |
|] ==> P"
|
wenzelm@20819
|
482 |
by (auto simp add: dsum_def)
|
wenzelm@20819
|
483 |
|
wenzelm@20819
|
484 |
|
wenzelm@20819
|
485 |
(*** Monotonicity ***)
|
wenzelm@20819
|
486 |
|
wenzelm@20819
|
487 |
lemma dprod_mono: "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'"
|
wenzelm@20819
|
488 |
by blast
|
wenzelm@20819
|
489 |
|
wenzelm@20819
|
490 |
lemma dsum_mono: "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'"
|
wenzelm@20819
|
491 |
by blast
|
wenzelm@20819
|
492 |
|
wenzelm@20819
|
493 |
|
wenzelm@20819
|
494 |
(*** Bounding theorems ***)
|
wenzelm@20819
|
495 |
|
wenzelm@20819
|
496 |
lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
|
wenzelm@20819
|
497 |
by blast
|
wenzelm@20819
|
498 |
|
wenzelm@20819
|
499 |
lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
|
wenzelm@20819
|
500 |
|
wenzelm@20819
|
501 |
(*Dependent version*)
|
wenzelm@20819
|
502 |
lemma dprod_subset_Sigma2:
|
wenzelm@20819
|
503 |
"(dprod (Sigma A B) (Sigma C D)) <=
|
wenzelm@20819
|
504 |
Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
|
wenzelm@20819
|
505 |
by auto
|
wenzelm@20819
|
506 |
|
wenzelm@20819
|
507 |
lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
|
wenzelm@20819
|
508 |
by blast
|
wenzelm@20819
|
509 |
|
wenzelm@20819
|
510 |
lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
|
wenzelm@20819
|
511 |
|
wenzelm@20819
|
512 |
|
wenzelm@20819
|
513 |
(*** Domain ***)
|
wenzelm@20819
|
514 |
|
wenzelm@20819
|
515 |
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
|
wenzelm@20819
|
516 |
by auto
|
wenzelm@20819
|
517 |
|
wenzelm@20819
|
518 |
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
|
wenzelm@20819
|
519 |
by auto
|
wenzelm@20819
|
520 |
|
wenzelm@20819
|
521 |
|
haftmann@24162
|
522 |
text {* hides popular names *}
|
haftmann@24162
|
523 |
hide (open) type node item
|
wenzelm@20819
|
524 |
hide (open) const Push Node Atom Leaf Numb Lim Split Case
|
wenzelm@20819
|
525 |
|
wenzelm@20819
|
526 |
|
wenzelm@20819
|
527 |
section {* Datatypes *}
|
wenzelm@20819
|
528 |
|
haftmann@24699
|
529 |
subsection {* Representing sums *}
|
wenzelm@12918
|
530 |
|
haftmann@27104
|
531 |
rep_datatype (sum) Inl Inr
|
haftmann@27104
|
532 |
proof -
|
haftmann@27104
|
533 |
fix P
|
haftmann@27104
|
534 |
fix s :: "'a + 'b"
|
haftmann@27104
|
535 |
assume x: "\<And>x\<Colon>'a. P (Inl x)" and y: "\<And>y\<Colon>'b. P (Inr y)"
|
haftmann@27104
|
536 |
then show "P s" by (auto intro: sumE [of s])
|
haftmann@27104
|
537 |
qed simp_all
|
haftmann@24194
|
538 |
|
nipkow@22230
|
539 |
lemma sum_case_KK[simp]: "sum_case (%x. a) (%x. a) = (%x. a)"
|
nipkow@22230
|
540 |
by (rule ext) (simp split: sum.split)
|
nipkow@22230
|
541 |
|
wenzelm@12918
|
542 |
lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
|
wenzelm@12918
|
543 |
apply (rule_tac s = s in sumE)
|
wenzelm@12918
|
544 |
apply (erule ssubst)
|
wenzelm@20798
|
545 |
apply (rule sum.cases(1))
|
wenzelm@12918
|
546 |
apply (erule ssubst)
|
wenzelm@20798
|
547 |
apply (rule sum.cases(2))
|
wenzelm@12918
|
548 |
done
|
wenzelm@12918
|
549 |
|
wenzelm@12918
|
550 |
lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
|
wenzelm@12918
|
551 |
-- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
|
wenzelm@20798
|
552 |
by simp
|
wenzelm@12918
|
553 |
|
wenzelm@12918
|
554 |
lemma sum_case_inject:
|
wenzelm@12918
|
555 |
"sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
|
wenzelm@12918
|
556 |
proof -
|
wenzelm@12918
|
557 |
assume a: "sum_case f1 f2 = sum_case g1 g2"
|
wenzelm@12918
|
558 |
assume r: "f1 = g1 ==> f2 = g2 ==> P"
|
wenzelm@12918
|
559 |
show P
|
wenzelm@12918
|
560 |
apply (rule r)
|
wenzelm@12918
|
561 |
apply (rule ext)
|
paulson@14208
|
562 |
apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
|
wenzelm@12918
|
563 |
apply (rule ext)
|
paulson@14208
|
564 |
apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
|
wenzelm@12918
|
565 |
done
|
wenzelm@12918
|
566 |
qed
|
wenzelm@12918
|
567 |
|
berghofe@13635
|
568 |
constdefs
|
berghofe@13635
|
569 |
Suml :: "('a => 'c) => 'a + 'b => 'c"
|
haftmann@28524
|
570 |
"Suml == (%f. sum_case f undefined)"
|
berghofe@13635
|
571 |
|
berghofe@13635
|
572 |
Sumr :: "('b => 'c) => 'a + 'b => 'c"
|
haftmann@28524
|
573 |
"Sumr == sum_case undefined"
|
berghofe@13635
|
574 |
|
berghofe@13635
|
575 |
lemma Suml_inject: "Suml f = Suml g ==> f = g"
|
berghofe@13635
|
576 |
by (unfold Suml_def) (erule sum_case_inject)
|
berghofe@13635
|
577 |
|
berghofe@13635
|
578 |
lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
|
berghofe@13635
|
579 |
by (unfold Sumr_def) (erule sum_case_inject)
|
berghofe@13635
|
580 |
|
wenzelm@20798
|
581 |
hide (open) const Suml Sumr
|
berghofe@13635
|
582 |
|
wenzelm@12918
|
583 |
|
haftmann@24194
|
584 |
subsection {* The option datatype *}
|
haftmann@24194
|
585 |
|
haftmann@24194
|
586 |
datatype 'a option = None | Some 'a
|
haftmann@24194
|
587 |
|
haftmann@24194
|
588 |
lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
|
haftmann@24194
|
589 |
by (induct x) auto
|
haftmann@24194
|
590 |
|
haftmann@24194
|
591 |
lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
|
haftmann@24194
|
592 |
by (induct x) auto
|
haftmann@24194
|
593 |
|
haftmann@24194
|
594 |
text{*Although it may appear that both of these equalities are helpful
|
haftmann@24194
|
595 |
only when applied to assumptions, in practice it seems better to give
|
haftmann@24194
|
596 |
them the uniform iff attribute. *}
|
haftmann@24194
|
597 |
|
haftmann@24194
|
598 |
lemma option_caseE:
|
haftmann@24194
|
599 |
assumes c: "(case x of None => P | Some y => Q y)"
|
haftmann@24194
|
600 |
obtains
|
haftmann@24194
|
601 |
(None) "x = None" and P
|
haftmann@24194
|
602 |
| (Some) y where "x = Some y" and "Q y"
|
haftmann@24194
|
603 |
using c by (cases x) simp_all
|
haftmann@24194
|
604 |
|
haftmann@24728
|
605 |
lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
|
haftmann@24728
|
606 |
by (rule set_ext, case_tac x) auto
|
haftmann@24728
|
607 |
|
haftmann@24194
|
608 |
|
haftmann@24194
|
609 |
subsubsection {* Operations *}
|
haftmann@24194
|
610 |
|
haftmann@24194
|
611 |
consts
|
haftmann@24194
|
612 |
the :: "'a option => 'a"
|
haftmann@24194
|
613 |
primrec
|
haftmann@24194
|
614 |
"the (Some x) = x"
|
haftmann@24194
|
615 |
|
haftmann@24194
|
616 |
consts
|
haftmann@24194
|
617 |
o2s :: "'a option => 'a set"
|
haftmann@24194
|
618 |
primrec
|
haftmann@24194
|
619 |
"o2s None = {}"
|
haftmann@24194
|
620 |
"o2s (Some x) = {x}"
|
haftmann@24194
|
621 |
|
haftmann@24194
|
622 |
lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
|
haftmann@24194
|
623 |
by simp
|
haftmann@24194
|
624 |
|
wenzelm@26339
|
625 |
declaration {* fn _ =>
|
wenzelm@26339
|
626 |
Classical.map_cs (fn cs => cs addSD2 ("ospec", thm "ospec"))
|
wenzelm@26339
|
627 |
*}
|
haftmann@24194
|
628 |
|
haftmann@24194
|
629 |
lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
|
haftmann@24194
|
630 |
by (cases xo) auto
|
haftmann@24194
|
631 |
|
haftmann@24194
|
632 |
lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
|
haftmann@24194
|
633 |
by (cases xo) auto
|
haftmann@24194
|
634 |
|
haftmann@25511
|
635 |
definition
|
haftmann@25511
|
636 |
option_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option"
|
haftmann@25511
|
637 |
where
|
haftmann@28562
|
638 |
[code del]: "option_map = (%f y. case y of None => None | Some x => Some (f x))"
|
haftmann@24194
|
639 |
|
haftmann@24194
|
640 |
lemma option_map_None [simp, code]: "option_map f None = None"
|
haftmann@24194
|
641 |
by (simp add: option_map_def)
|
haftmann@24194
|
642 |
|
haftmann@24194
|
643 |
lemma option_map_Some [simp, code]: "option_map f (Some x) = Some (f x)"
|
haftmann@24194
|
644 |
by (simp add: option_map_def)
|
haftmann@24194
|
645 |
|
haftmann@24194
|
646 |
lemma option_map_is_None [iff]:
|
haftmann@24194
|
647 |
"(option_map f opt = None) = (opt = None)"
|
haftmann@24194
|
648 |
by (simp add: option_map_def split add: option.split)
|
haftmann@24194
|
649 |
|
haftmann@24194
|
650 |
lemma option_map_eq_Some [iff]:
|
haftmann@24194
|
651 |
"(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
|
haftmann@24194
|
652 |
by (simp add: option_map_def split add: option.split)
|
haftmann@24194
|
653 |
|
haftmann@24194
|
654 |
lemma option_map_comp:
|
haftmann@24194
|
655 |
"option_map f (option_map g opt) = option_map (f o g) opt"
|
haftmann@24194
|
656 |
by (simp add: option_map_def split add: option.split)
|
haftmann@24194
|
657 |
|
haftmann@24194
|
658 |
lemma option_map_o_sum_case [simp]:
|
haftmann@24194
|
659 |
"option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
|
haftmann@24194
|
660 |
by (rule ext) (simp split: sum.split)
|
haftmann@24194
|
661 |
|
haftmann@24194
|
662 |
|
haftmann@24194
|
663 |
subsubsection {* Code generator setup *}
|
haftmann@24194
|
664 |
|
haftmann@24194
|
665 |
definition
|
haftmann@24194
|
666 |
is_none :: "'a option \<Rightarrow> bool" where
|
haftmann@24194
|
667 |
is_none_none [code post, symmetric, code inline]: "is_none x \<longleftrightarrow> x = None"
|
haftmann@24194
|
668 |
|
haftmann@24194
|
669 |
lemma is_none_code [code]:
|
haftmann@24194
|
670 |
shows "is_none None \<longleftrightarrow> True"
|
haftmann@24194
|
671 |
and "is_none (Some x) \<longleftrightarrow> False"
|
haftmann@24194
|
672 |
unfolding is_none_none [symmetric] by simp_all
|
haftmann@24194
|
673 |
|
haftmann@24194
|
674 |
hide (open) const is_none
|
haftmann@24194
|
675 |
|
haftmann@24194
|
676 |
code_type option
|
haftmann@24194
|
677 |
(SML "_ option")
|
haftmann@24194
|
678 |
(OCaml "_ option")
|
haftmann@24194
|
679 |
(Haskell "Maybe _")
|
haftmann@24194
|
680 |
|
haftmann@24194
|
681 |
code_const None and Some
|
haftmann@24194
|
682 |
(SML "NONE" and "SOME")
|
haftmann@24194
|
683 |
(OCaml "None" and "Some _")
|
haftmann@24194
|
684 |
(Haskell "Nothing" and "Just")
|
haftmann@24194
|
685 |
|
haftmann@24194
|
686 |
code_instance option :: eq
|
haftmann@24194
|
687 |
(Haskell -)
|
haftmann@24194
|
688 |
|
haftmann@28346
|
689 |
code_const "eq_class.eq \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"
|
haftmann@24194
|
690 |
(Haskell infixl 4 "==")
|
haftmann@24194
|
691 |
|
haftmann@24194
|
692 |
code_reserved SML
|
haftmann@24194
|
693 |
option NONE SOME
|
haftmann@24194
|
694 |
|
haftmann@24194
|
695 |
code_reserved OCaml
|
haftmann@24194
|
696 |
option None Some
|
haftmann@24194
|
697 |
|
haftmann@24194
|
698 |
code_modulename SML
|
haftmann@24194
|
699 |
Datatype Nat
|
haftmann@24194
|
700 |
|
haftmann@24194
|
701 |
code_modulename OCaml
|
haftmann@24194
|
702 |
Datatype Nat
|
haftmann@24194
|
703 |
|
haftmann@24194
|
704 |
code_modulename Haskell
|
haftmann@24194
|
705 |
Datatype Nat
|
haftmann@24194
|
706 |
|
berghofe@5181
|
707 |
end
|