diff -r 82db9ad610b9 -r 8081087096ad src/HOL/Datatype.thy --- a/src/HOL/Datatype.thy Mon Sep 01 16:17:46 2014 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,524 +0,0 @@ -(* Title: HOL/Datatype.thy - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Author: Stefan Berghofer and Markus Wenzel, TU Muenchen -*) - -header {* Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *} - -theory Datatype -imports Product_Type Sum_Type Nat -keywords "datatype" :: thy_decl -begin - -subsection {* The datatype universe *} - -definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}" - -typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set" - morphisms Rep_Node Abs_Node - unfolding Node_def by auto - -text{*Datatypes will be represented by sets of type @{text node}*} - -type_synonym 'a item = "('a, unit) node set" -type_synonym ('a, 'b) dtree = "('a, 'b) node set" - -consts - Push :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))" - - Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node" - ndepth :: "('a, 'b) node => nat" - - Atom :: "('a + nat) => ('a, 'b) dtree" - Leaf :: "'a => ('a, 'b) dtree" - Numb :: "nat => ('a, 'b) dtree" - Scons :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree" - In0 :: "('a, 'b) dtree => ('a, 'b) dtree" - In1 :: "('a, 'b) dtree => ('a, 'b) dtree" - Lim :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree" - - ntrunc :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree" - - uprod :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set" - usum :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set" - - Split :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c" - Case :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c" - - dprod :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] - => (('a, 'b) dtree * ('a, 'b) dtree)set" - dsum :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] - => (('a, 'b) dtree * ('a, 'b) dtree)set" - - -defs - - Push_Node_def: "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))" - - (*crude "lists" of nats -- needed for the constructions*) - Push_def: "Push == (%b h. case_nat b h)" - - (** operations on S-expressions -- sets of nodes **) - - (*S-expression constructors*) - Atom_def: "Atom == (%x. {Abs_Node((%k. Inr 0, x))})" - Scons_def: "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)" - - (*Leaf nodes, with arbitrary or nat labels*) - Leaf_def: "Leaf == Atom o Inl" - Numb_def: "Numb == Atom o Inr" - - (*Injections of the "disjoint sum"*) - In0_def: "In0(M) == Scons (Numb 0) M" - In1_def: "In1(M) == Scons (Numb 1) M" - - (*Function spaces*) - Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}" - - (*the set of nodes with depth less than k*) - ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)" - ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n) R - |] ==> R" -by (force simp add: apfst_def) - -(** Push -- an injection, analogous to Cons on lists **) - -lemma Push_inject1: "Push i f = Push j g ==> i=j" -apply (simp add: Push_def fun_eq_iff) -apply (drule_tac x=0 in spec, simp) -done - -lemma Push_inject2: "Push i f = Push j g ==> f=g" -apply (auto simp add: Push_def fun_eq_iff) -apply (drule_tac x="Suc x" in spec, simp) -done - -lemma Push_inject: - "[| Push i f =Push j g; [| i=j; f=g |] ==> P |] ==> P" -by (blast dest: Push_inject1 Push_inject2) - -lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P" -by (auto simp add: Push_def fun_eq_iff split: nat.split_asm) - -lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1] - - -(*** Introduction rules for Node ***) - -lemma Node_K0_I: "(%k. Inr 0, a) : Node" -by (simp add: Node_def) - -lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node" -apply (simp add: Node_def Push_def) -apply (fast intro!: apfst_conv nat.case(2)[THEN trans]) -done - - -subsection{*Freeness: Distinctness of Constructors*} - -(** Scons vs Atom **) - -lemma Scons_not_Atom [iff]: "Scons M N \ Atom(a)" -unfolding Atom_def Scons_def Push_Node_def One_nat_def -by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] - dest!: Abs_Node_inj - elim!: apfst_convE sym [THEN Push_neq_K0]) - -lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym] - - -(*** Injectiveness ***) - -(** Atomic nodes **) - -lemma inj_Atom: "inj(Atom)" -apply (simp add: Atom_def) -apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj) -done -lemmas Atom_inject = inj_Atom [THEN injD] - -lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)" -by (blast dest!: Atom_inject) - -lemma inj_Leaf: "inj(Leaf)" -apply (simp add: Leaf_def o_def) -apply (rule inj_onI) -apply (erule Atom_inject [THEN Inl_inject]) -done - -lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD] - -lemma inj_Numb: "inj(Numb)" -apply (simp add: Numb_def o_def) -apply (rule inj_onI) -apply (erule Atom_inject [THEN Inr_inject]) -done - -lemmas Numb_inject [dest!] = inj_Numb [THEN injD] - - -(** Injectiveness of Push_Node **) - -lemma Push_Node_inject: - "[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P - |] ==> P" -apply (simp add: Push_Node_def) -apply (erule Abs_Node_inj [THEN apfst_convE]) -apply (rule Rep_Node [THEN Node_Push_I])+ -apply (erule sym [THEN apfst_convE]) -apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject) -done - - -(** Injectiveness of Scons **) - -lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'" -unfolding Scons_def One_nat_def -by (blast dest!: Push_Node_inject) - -lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'" -unfolding Scons_def One_nat_def -by (blast dest!: Push_Node_inject) - -lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'" -apply (erule equalityE) -apply (iprover intro: equalityI Scons_inject_lemma1) -done - -lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'" -apply (erule equalityE) -apply (iprover intro: equalityI Scons_inject_lemma2) -done - -lemma Scons_inject: - "[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P |] ==> P" -by (iprover dest: Scons_inject1 Scons_inject2) - -lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')" -by (blast elim!: Scons_inject) - -(*** Distinctness involving Leaf and Numb ***) - -(** Scons vs Leaf **) - -lemma Scons_not_Leaf [iff]: "Scons M N \ Leaf(a)" -unfolding Leaf_def o_def by (rule Scons_not_Atom) - -lemmas Leaf_not_Scons [iff] = Scons_not_Leaf [THEN not_sym] - -(** Scons vs Numb **) - -lemma Scons_not_Numb [iff]: "Scons M N \ Numb(k)" -unfolding Numb_def o_def by (rule Scons_not_Atom) - -lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym] - - -(** Leaf vs Numb **) - -lemma Leaf_not_Numb [iff]: "Leaf(a) \ Numb(k)" -by (simp add: Leaf_def Numb_def) - -lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym] - - -(*** ndepth -- the depth of a node ***) - -lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0" -by (simp add: ndepth_def Node_K0_I [THEN Abs_Node_inverse] Least_equality) - -lemma ndepth_Push_Node_aux: - "case_nat (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k" -apply (induct_tac "k", auto) -apply (erule Least_le) -done - -lemma ndepth_Push_Node: - "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))" -apply (insert Rep_Node [of n, unfolded Node_def]) -apply (auto simp add: ndepth_def Push_Node_def - Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse]) -apply (rule Least_equality) -apply (auto simp add: Push_def ndepth_Push_Node_aux) -apply (erule LeastI) -done - - -(*** ntrunc applied to the various node sets ***) - -lemma ntrunc_0 [simp]: "ntrunc 0 M = {}" -by (simp add: ntrunc_def) - -lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)" -by (auto simp add: Atom_def ntrunc_def ndepth_K0) - -lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)" -unfolding Leaf_def o_def by (rule ntrunc_Atom) - -lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)" -unfolding Numb_def o_def by (rule ntrunc_Atom) - -lemma ntrunc_Scons [simp]: - "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)" -unfolding Scons_def ntrunc_def One_nat_def -by (auto simp add: ndepth_Push_Node) - - - -(** Injection nodes **) - -lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}" -apply (simp add: In0_def) -apply (simp add: Scons_def) -done - -lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)" -by (simp add: In0_def) - -lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}" -apply (simp add: In1_def) -apply (simp add: Scons_def) -done - -lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)" -by (simp add: In1_def) - - -subsection{*Set Constructions*} - - -(*** Cartesian Product ***) - -lemma uprodI [intro!]: "[| M:A; N:B |] ==> Scons M N : uprod A B" -by (simp add: uprod_def) - -(*The general elimination rule*) -lemma uprodE [elim!]: - "[| c : uprod A B; - !!x y. [| x:A; y:B; c = Scons x y |] ==> P - |] ==> P" -by (auto simp add: uprod_def) - - -(*Elimination of a pair -- introduces no eigenvariables*) -lemma uprodE2: "[| Scons M N : uprod A B; [| M:A; N:B |] ==> P |] ==> P" -by (auto simp add: uprod_def) - - -(*** Disjoint Sum ***) - -lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B" -by (simp add: usum_def) - -lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B" -by (simp add: usum_def) - -lemma usumE [elim!]: - "[| u : usum A B; - !!x. [| x:A; u=In0(x) |] ==> P; - !!y. [| y:B; u=In1(y) |] ==> P - |] ==> P" -by (auto simp add: usum_def) - - -(** Injection **) - -lemma In0_not_In1 [iff]: "In0(M) \ In1(N)" -unfolding In0_def In1_def One_nat_def by auto - -lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym] - -lemma In0_inject: "In0(M) = In0(N) ==> M=N" -by (simp add: In0_def) - -lemma In1_inject: "In1(M) = In1(N) ==> M=N" -by (simp add: In1_def) - -lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)" -by (blast dest!: In0_inject) - -lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)" -by (blast dest!: In1_inject) - -lemma inj_In0: "inj In0" -by (blast intro!: inj_onI) - -lemma inj_In1: "inj In1" -by (blast intro!: inj_onI) - - -(*** Function spaces ***) - -lemma Lim_inject: "Lim f = Lim g ==> f = g" -apply (simp add: Lim_def) -apply (rule ext) -apply (blast elim!: Push_Node_inject) -done - - -(*** proving equality of sets and functions using ntrunc ***) - -lemma ntrunc_subsetI: "ntrunc k M <= M" -by (auto simp add: ntrunc_def) - -lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N" -by (auto simp add: ntrunc_def) - -(*A generalized form of the take-lemma*) -lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N" -apply (rule equalityI) -apply (rule_tac [!] ntrunc_subsetD) -apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) -done - -lemma ntrunc_o_equality: - "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2" -apply (rule ntrunc_equality [THEN ext]) -apply (simp add: fun_eq_iff) -done - - -(*** Monotonicity ***) - -lemma uprod_mono: "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'" -by (simp add: uprod_def, blast) - -lemma usum_mono: "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'" -by (simp add: usum_def, blast) - -lemma Scons_mono: "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'" -by (simp add: Scons_def, blast) - -lemma In0_mono: "M<=N ==> In0(M) <= In0(N)" -by (simp add: In0_def Scons_mono) - -lemma In1_mono: "M<=N ==> In1(M) <= In1(N)" -by (simp add: In1_def Scons_mono) - - -(*** Split and Case ***) - -lemma Split [simp]: "Split c (Scons M N) = c M N" -by (simp add: Split_def) - -lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)" -by (simp add: Case_def) - -lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)" -by (simp add: Case_def) - - - -(**** UN x. B(x) rules ****) - -lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))" -by (simp add: ntrunc_def, blast) - -lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)" -by (simp add: Scons_def, blast) - -lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))" -by (simp add: Scons_def, blast) - -lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))" -by (simp add: In0_def Scons_UN1_y) - -lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))" -by (simp add: In1_def Scons_UN1_y) - - -(*** Equality for Cartesian Product ***) - -lemma dprodI [intro!]: - "[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s" -by (auto simp add: dprod_def) - -(*The general elimination rule*) -lemma dprodE [elim!]: - "[| c : dprod r s; - !!x y x' y'. [| (x,x') : r; (y,y') : s; - c = (Scons x y, Scons x' y') |] ==> P - |] ==> P" -by (auto simp add: dprod_def) - - -(*** Equality for Disjoint Sum ***) - -lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s" -by (auto simp add: dsum_def) - -lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s" -by (auto simp add: dsum_def) - -lemma dsumE [elim!]: - "[| w : dsum r s; - !!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P; - !!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P - |] ==> P" -by (auto simp add: dsum_def) - - -(*** Monotonicity ***) - -lemma dprod_mono: "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'" -by blast - -lemma dsum_mono: "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'" -by blast - - -(*** Bounding theorems ***) - -lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)" -by blast - -lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma] - -(*Dependent version*) -lemma dprod_subset_Sigma2: - "(dprod (Sigma A B) (Sigma C D)) <= - Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))" -by auto - -lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)" -by blast - -lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma] - - -text {* hides popular names *} -hide_type (open) node item -hide_const (open) Push Node Atom Leaf Numb Lim Split Case - -ML_file "Tools/Datatype/datatype.ML" - -ML_file "Tools/inductive_realizer.ML" -setup InductiveRealizer.setup - -ML_file "Tools/Datatype/datatype_realizer.ML" -setup Datatype_Realizer.setup - -end