(* Title: HOL/Inductive.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
*)
theory Inductive = Gfp + Sum_Type + NatDef
files
("Tools/induct_method.ML")
("Tools/inductive_package.ML")
("Tools/datatype_aux.ML")
("Tools/datatype_prop.ML")
("Tools/datatype_rep_proofs.ML")
("Tools/datatype_abs_proofs.ML")
("Tools/datatype_package.ML")
("Tools/primrec_package.ML"):
constdefs
forall :: "('a => bool) => bool"
"forall P == \<forall>x. P x"
implies :: "bool => bool => bool"
"implies A B == A --> B"
lemma forall_eq: "(!!x. P x) == Trueprop (forall (\<lambda>x. P x))"
proof
assume "!!x. P x"
thus "forall (\<lambda>x. P x)" by (unfold forall_def) blast
next
fix x
assume "forall (\<lambda>x. P x)"
thus "P x" by (unfold forall_def) blast
qed
lemma implies_eq: "(A ==> B) == Trueprop (implies A B)"
proof
assume "A ==> B"
thus "implies A B" by (unfold implies_def) blast
next
assume "implies A B" and A
thus B by (unfold implies_def) blast
qed
lemmas inductive_atomize = forall_eq implies_eq
lemmas inductive_rulify = inductive_atomize [symmetric, standard]
hide const forall implies
use "Tools/induct_method.ML"
setup InductMethod.setup
use "Tools/inductive_package.ML"
setup InductivePackage.setup
use "Tools/datatype_aux.ML"
use "Tools/datatype_prop.ML"
use "Tools/datatype_rep_proofs.ML"
use "Tools/datatype_abs_proofs.ML"
use "Tools/datatype_package.ML"
setup DatatypePackage.setup
use "Tools/primrec_package.ML"
setup PrimrecPackage.setup
theorems basic_monos [mono] =
subset_refl imp_refl disj_mono conj_mono ex_mono all_mono
Collect_mono in_mono vimage_mono
imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
not_all not_ex
Ball_def Bex_def
(*belongs to theory Transitive_Closure*)
declare rtrancl_induct [induct set: rtrancl]
declare rtranclE [cases set: rtrancl]
declare trancl_induct [induct set: trancl]
declare tranclE [cases set: trancl]
end