(* Title: HOL/Datatype.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Could <*> be generalized to a general summation (Sigma)?
*)
header {* Analogues of the Cartesian Product and Disjoint Sum for Datatypes *}
theory Datatype
imports Nat Sum_Type
begin
typedef (Node)
('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
--{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
by auto
text{*Datatypes will be represented by sets of type @{text node}*}
types 'a item = "('a, unit) node set"
('a, 'b) dtree = "('a, 'b) node set"
consts
apfst :: "['a=>'c, 'a*'b] => 'c*'b"
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*)
apfst_def: "apfst == (%f (x,y). (f(x),y))"
Push_def: "Push == (%b h. nat_case 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)<k}"
(*products and sums for the "universe"*)
uprod_def: "uprod A B == UN x:A. UN y:B. { Scons x y }"
usum_def: "usum A B == In0`A Un In1`B"
(*the corresponding eliminators*)
Split_def: "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
Case_def: "Case c d M == THE u. (EX x . M = In0(x) & u = c(x))
| (EX y . M = In1(y) & u = d(y))"
(** equality for the "universe" **)
dprod_def: "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
dsum_def: "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
(UN (y,y'):s. {(In1(y),In1(y'))})"
(** apfst -- can be used in similar type definitions **)
lemma apfst_conv [simp, code]: "apfst f (a, b) = (f a, b)"
by (simp add: apfst_def)
lemma apfst_convE:
"[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> 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 expand_fun_eq)
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 expand_fun_eq)
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 expand_fun_eq split: nat.split_asm)
lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
(*** 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_Suc [THEN trans])
done
subsection{*Freeness: Distinctness of Constructors*}
(** Scons vs Atom **)
lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I]
dest!: Abs_Node_inj
elim!: apfst_convE sym [THEN Push_neq_K0])
done
lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym, standard]
(*** 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, standard]
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, standard]
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, standard]
(** 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'"
apply (simp add: Scons_def One_nat_def)
apply (blast dest!: Push_Node_inject)
done
lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
apply (simp add: Scons_def One_nat_def)
apply (blast dest!: Push_Node_inject)
done
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 \<noteq> Leaf(a)"
by (simp add: Leaf_def o_def Scons_not_Atom)
lemmas Leaf_not_Scons [iff] = Scons_not_Leaf [THEN not_sym, standard]
(** Scons vs Numb **)
lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
by (simp add: Numb_def o_def Scons_not_Atom)
lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym, standard]
(** Leaf vs Numb **)
lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
by (simp add: Leaf_def Numb_def)
lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym, standard]
(*** 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:
"nat_case (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)"
by (simp add: Leaf_def o_def ntrunc_Atom)
lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
by (simp add: Numb_def o_def ntrunc_Atom)
lemma ntrunc_Scons [simp]:
"ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
by (auto simp add: Scons_def ntrunc_def One_nat_def 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) \<noteq> In1(N)"
by (auto simp add: In0_def In1_def One_nat_def)
lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym, standard]
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: expand_fun_eq)
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 subset_refl Scons_mono)
lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
by (simp add: In1_def subset_refl 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, standard]
(*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, standard]
(*** Domain ***)
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
by auto
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
by auto
subsection {* Finishing the datatype package setup *}
text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
setup "DatatypeCodegen.setup_hooks"
hide (open) const Push Node Atom Leaf Numb Lim Split Case
hide (open) type node item
section {* Datatypes *}
subsection {* Representing primitive types *}
rep_datatype bool
distinct True_not_False False_not_True
induction bool_induct
declare case_split [cases type: bool]
-- "prefer plain propositional version"
rep_datatype unit
induction unit_induct
rep_datatype prod
inject Pair_eq
induction prod_induct
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
by auto
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
by (auto simp: split_tupled_all)
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
by (induct p) auto
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
by (induct p) auto
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
by (simp add: expand_fun_eq)
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
declare prod_caseE' [elim!] prod_caseE [elim!]
rep_datatype sum
distinct Inl_not_Inr Inr_not_Inl
inject Inl_eq Inr_eq
induction sum_induct
lemma sum_case_KK[simp]: "sum_case (%x. a) (%x. a) = (%x. a)"
by (rule ext) (simp split: sum.split)
lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
apply (rule_tac s = s in sumE)
apply (erule ssubst)
apply (rule sum.cases(1))
apply (erule ssubst)
apply (rule sum.cases(2))
done
lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
-- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
by simp
lemma sum_case_inject:
"sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
proof -
assume a: "sum_case f1 f2 = sum_case g1 g2"
assume r: "f1 = g1 ==> f2 = g2 ==> P"
show P
apply (rule r)
apply (rule ext)
apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
apply (rule ext)
apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
done
qed
constdefs
Suml :: "('a => 'c) => 'a + 'b => 'c"
"Suml == (%f. sum_case f arbitrary)"
Sumr :: "('b => 'c) => 'a + 'b => 'c"
"Sumr == sum_case arbitrary"
lemma Suml_inject: "Suml f = Suml g ==> f = g"
by (unfold Suml_def) (erule sum_case_inject)
lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
by (unfold Sumr_def) (erule sum_case_inject)
hide (open) const Suml Sumr
subsection {* Further cases/induct rules for tuples *}
lemma prod_cases3 [cases type]:
obtains (fields) a b c where "y = (a, b, c)"
by (cases y, case_tac b) blast
lemma prod_induct3 [case_names fields, induct type]:
"(!!a b c. P (a, b, c)) ==> P x"
by (cases x) blast
lemma prod_cases4 [cases type]:
obtains (fields) a b c d where "y = (a, b, c, d)"
by (cases y, case_tac c) blast
lemma prod_induct4 [case_names fields, induct type]:
"(!!a b c d. P (a, b, c, d)) ==> P x"
by (cases x) blast
lemma prod_cases5 [cases type]:
obtains (fields) a b c d e where "y = (a, b, c, d, e)"
by (cases y, case_tac d) blast
lemma prod_induct5 [case_names fields, induct type]:
"(!!a b c d e. P (a, b, c, d, e)) ==> P x"
by (cases x) blast
lemma prod_cases6 [cases type]:
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
by (cases y, case_tac e) blast
lemma prod_induct6 [case_names fields, induct type]:
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
by (cases x) blast
lemma prod_cases7 [cases type]:
obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
by (cases y, case_tac f) blast
lemma prod_induct7 [case_names fields, induct type]:
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
by (cases x) blast
subsection {* The option type *}
datatype 'a option = None | Some 'a
lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
by (induct x) auto
lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
by (induct x) auto
text{*Although it may appear that both of these equalities are helpful
only when applied to assumptions, in practice it seems better to give
them the uniform iff attribute. *}
lemma option_caseE:
assumes c: "(case x of None => P | Some y => Q y)"
obtains
(None) "x = None" and P
| (Some) y where "x = Some y" and "Q y"
using c by (cases x) simp_all
subsubsection {* Operations *}
consts
the :: "'a option => 'a"
primrec
"the (Some x) = x"
consts
o2s :: "'a option => 'a set"
primrec
"o2s None = {}"
"o2s (Some x) = {x}"
lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
by simp
ML_setup {* change_claset (fn cs => cs addSD2 ("ospec", thm "ospec")) *}
lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
by (cases xo) auto
lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
by (cases xo) auto
constdefs
option_map :: "('a => 'b) => ('a option => 'b option)"
[code func del]: "option_map == %f y. case y of None => None | Some x => Some (f x)"
lemma option_map_None [simp, code]: "option_map f None = None"
by (simp add: option_map_def)
lemma option_map_Some [simp, code]: "option_map f (Some x) = Some (f x)"
by (simp add: option_map_def)
lemma option_map_is_None [iff]:
"(option_map f opt = None) = (opt = None)"
by (simp add: option_map_def split add: option.split)
lemma option_map_eq_Some [iff]:
"(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
by (simp add: option_map_def split add: option.split)
lemma option_map_comp:
"option_map f (option_map g opt) = option_map (f o g) opt"
by (simp add: option_map_def split add: option.split)
lemma option_map_o_sum_case [simp]:
"option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
by (rule ext) (simp split: sum.split)
subsubsection {* Code generator setup *}
definition
is_none :: "'a option \<Rightarrow> bool" where
is_none_none [normal post, symmetric, code inline]: "is_none x \<longleftrightarrow> x = None"
lemma is_none_code [code]:
shows "is_none None \<longleftrightarrow> True"
and "is_none (Some x) \<longleftrightarrow> False"
unfolding is_none_none [symmetric] by simp_all
lemma split_is_prod_case [code inline]:
"split = prod_case"
by (simp add: expand_fun_eq split_def prod.cases)
hide (open) const is_none
code_type option
(SML "_ option")
(OCaml "_ option")
(Haskell "Maybe _")
code_const None and Some
(SML "NONE" and "SOME")
(OCaml "None" and "Some _")
(Haskell "Nothing" and "Just")
code_instance option :: eq
(Haskell -)
code_const "op = \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"
(Haskell infixl 4 "==")
code_reserved SML
option NONE SOME
code_reserved OCaml
option None Some
end