(* Title: ZF/Inductive.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
(Co)Inductive Definitions for Zermelo-Fraenkel Set Theory
Inductive definitions use least fixedpoints with standard products and sums
Coinductive definitions use greatest fixedpoints with Quine products and sums
Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)
val iT = Ind_Syntax.iT
and oT = FOLogic.oT;
structure Lfp =
struct
val oper = Const("Fixedpt.lfp", [iT,iT-->iT]--->iT)
val bnd_mono = Const("Fixedpt.bnd_mono", [iT,iT-->iT]--->oT)
val bnd_monoI = bnd_monoI
val subs = def_lfp_subset
val Tarski = def_lfp_Tarski
val induct = def_induct
end;
structure Standard_Prod =
struct
val sigma = Const("Sigma", [iT, iT-->iT]--->iT)
val pair = Const("Pair", [iT,iT]--->iT)
val split_name = "split"
val pair_iff = Pair_iff
val split_eq = split
val fsplitI = splitI
val fsplitD = splitD
val fsplitE = splitE
end;
structure Standard_CP = CartProd_Fun (Standard_Prod);
structure Standard_Sum =
struct
val sum = Const("op +", [iT,iT]--->iT)
val inl = Const("Inl", iT-->iT)
val inr = Const("Inr", iT-->iT)
val elim = Const("case", [iT-->iT, iT-->iT, iT]--->iT)
val case_inl = case_Inl
val case_inr = case_Inr
val inl_iff = Inl_iff
val inr_iff = Inr_iff
val distinct = Inl_Inr_iff
val distinct' = Inr_Inl_iff
val free_SEs = Ind_Syntax.mk_free_SEs
[distinct, distinct', inl_iff, inr_iff, Standard_Prod.pair_iff]
end;
structure Ind_Package =
Add_inductive_def_Fun
(structure Fp=Lfp and Pr=Standard_Prod and CP=Standard_CP
and Su=Standard_Sum);
structure Gfp =
struct
val oper = Const("Fixedpt.gfp", [iT,iT-->iT]--->iT)
val bnd_mono = Const("Fixedpt.bnd_mono", [iT,iT-->iT]--->oT)
val bnd_monoI = bnd_monoI
val subs = def_gfp_subset
val Tarski = def_gfp_Tarski
val induct = def_Collect_coinduct
end;
structure Quine_Prod =
struct
val sigma = Const("QPair.QSigma", [iT, iT-->iT]--->iT)
val pair = Const("QPair.QPair", [iT,iT]--->iT)
val split_name = "QPair.qsplit"
val pair_iff = QPair_iff
val split_eq = qsplit
val fsplitI = qsplitI
val fsplitD = qsplitD
val fsplitE = qsplitE
end;
structure Quine_CP = CartProd_Fun (Quine_Prod);
structure Quine_Sum =
struct
val sum = Const("QPair.op <+>", [iT,iT]--->iT)
val inl = Const("QPair.QInl", iT-->iT)
val inr = Const("QPair.QInr", iT-->iT)
val elim = Const("QPair.qcase", [iT-->iT, iT-->iT, iT]--->iT)
val case_inl = qcase_QInl
val case_inr = qcase_QInr
val inl_iff = QInl_iff
val inr_iff = QInr_iff
val distinct = QInl_QInr_iff
val distinct' = QInr_QInl_iff
val free_SEs = Ind_Syntax.mk_free_SEs
[distinct, distinct', inl_iff, inr_iff, Quine_Prod.pair_iff]
end;
structure CoInd_Package =
Add_inductive_def_Fun(structure Fp=Gfp and Pr=Quine_Prod and CP=Quine_CP
and Su=Quine_Sum);