(* Title: ZF/Induct/Datatypes.thy ID: $Id: Datatypes.thy,v 1.3 2005/06/17 14:15:11 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) header {* Sample datatype definitions *} theory Datatypes imports Main begin subsection {* A type with four constructors *} text {* It has four contructors, of arities 0--3, and two parameters @{text A} and @{text B}. *} consts data :: "[i, i] => i" datatype "data(A, B)" = Con0 | Con1 ("a ∈ A") | Con2 ("a ∈ A", "b ∈ B") | Con3 ("a ∈ A", "b ∈ B", "d ∈ data(A, B)") lemma data_unfold: "data(A, B) = ({0} + A) + (A × B + A × B × data(A, B))" by (fast intro!: data.intros [unfolded data.con_defs] elim: data.cases [unfolded data.con_defs]) text {* \medskip Lemmas to justify using @{term data} in other recursive type definitions. *} lemma data_mono: "[| A ⊆ C; B ⊆ D |] ==> data(A, B) ⊆ data(C, D)" apply (unfold data.defs) apply (rule lfp_mono) apply (rule data.bnd_mono)+ apply (rule univ_mono Un_mono basic_monos | assumption)+ done lemma data_univ: "data(univ(A), univ(A)) ⊆ univ(A)" apply (unfold data.defs data.con_defs) apply (rule lfp_lowerbound) apply (rule_tac [2] subset_trans [OF A_subset_univ Un_upper1, THEN univ_mono]) apply (fast intro!: zero_in_univ Inl_in_univ Inr_in_univ Pair_in_univ) done lemma data_subset_univ: "[| A ⊆ univ(C); B ⊆ univ(C) |] ==> data(A, B) ⊆ univ(C)" by (rule subset_trans [OF data_mono data_univ]) subsection {* Example of a big enumeration type *} text {* Can go up to at least 100 constructors, but it takes nearly 7 minutes \dots\ (back in 1994 that is). *} consts enum :: i datatype enum = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09 | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19 | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29 | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39 | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49 | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59 end
lemma data_unfold:
data(A, B) = ({0} + A) + A × B + A × B × data(A, B)
lemma data_mono:
[| A ⊆ C; B ⊆ D |] ==> data(A, B) ⊆ data(C, D)
lemma data_univ:
data(univ(A), univ(A)) ⊆ univ(A)
lemma data_subset_univ:
[| A ⊆ univ(C); B ⊆ univ(C) |] ==> data(A, B) ⊆ univ(C)