src/HOL/Inductive.thy
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(*  Title:      HOL/Inductive.thy
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    ID:         $Id$
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    Author:     Markus Wenzel, TU Muenchen
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*)
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header {* Support for inductive sets and types *}
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theory Inductive 
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imports FixedPoint Sum_Type
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uses
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  ("Tools/inductive_package.ML")
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  "Tools/dseq.ML"
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  ("Tools/inductive_codegen.ML")
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  ("Tools/datatype_aux.ML")
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  ("Tools/datatype_prop.ML")
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  ("Tools/datatype_rep_proofs.ML")
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  ("Tools/datatype_abs_proofs.ML")
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  ("Tools/datatype_case.ML")
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  ("Tools/datatype_package.ML")
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  ("Tools/primrec_package.ML")
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begin
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subsection {* Inductive predicates and sets *}
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text {* Inversion of injective functions. *}
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constdefs
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  myinv :: "('a => 'b) => ('b => 'a)"
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  "myinv (f :: 'a => 'b) == \<lambda>y. THE x. f x = y"
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lemma myinv_f_f: "inj f ==> myinv f (f x) = x"
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proof -
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  assume "inj f"
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  hence "(THE x'. f x' = f x) = (THE x'. x' = x)"
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    by (simp only: inj_eq)
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  also have "... = x" by (rule the_eq_trivial)
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  finally show ?thesis by (unfold myinv_def)
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qed
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lemma f_myinv_f: "inj f ==> y \<in> range f ==> f (myinv f y) = y"
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proof (unfold myinv_def)
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  assume inj: "inj f"
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  assume "y \<in> range f"
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  then obtain x where "y = f x" ..
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  hence x: "f x = y" ..
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  thus "f (THE x. f x = y) = y"
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  proof (rule theI)
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    fix x' assume "f x' = y"
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    with x have "f x' = f x" by simp
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    with inj show "x' = x" by (rule injD)
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  qed
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qed
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hide const myinv
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text {* Package setup. *}
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theorems basic_monos =
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  subset_refl imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
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  Collect_mono in_mono vimage_mono
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  imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
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  not_all not_ex
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  Ball_def Bex_def
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  induct_rulify_fallback
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use "Tools/inductive_package.ML"
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setup InductivePackage.setup
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theorems [mono] =
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  imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
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  imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
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  not_all not_ex
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  Ball_def Bex_def
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  induct_rulify_fallback
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lemma False_meta_all:
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  "Trueprop False \<equiv> (\<And>P\<Colon>bool. P)"
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proof
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  fix P
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  assume False
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  then show P ..
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next
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  assume "\<And>P\<Colon>bool. P"
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  then show False .
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qed
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lemma not_eq_False:
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  assumes not_eq: "x \<noteq> y"
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  and eq: "x \<equiv> y"
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  shows False
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  using not_eq eq by auto
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lemmas not_eq_quodlibet =
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  not_eq_False [simplified False_meta_all]
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subsection {* Inductive datatypes and primitive recursion *}
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text {* Package setup. *}
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use "Tools/datatype_aux.ML"
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use "Tools/datatype_prop.ML"
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use "Tools/datatype_rep_proofs.ML"
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use "Tools/datatype_abs_proofs.ML"
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use "Tools/datatype_case.ML"
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use "Tools/datatype_package.ML"
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setup DatatypePackage.setup
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use "Tools/primrec_package.ML"
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use "Tools/inductive_codegen.ML"
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setup InductiveCodegen.setup
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text{* Lambda-abstractions with pattern matching: *}
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syntax
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  "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(%_)" 10)
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syntax (xsymbols)
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  "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(\<lambda>_)" 10)
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parse_translation (advanced) {*
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let
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  fun fun_tr ctxt [cs] =
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    let
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      val x = Free (Name.variant (add_term_free_names (cs, [])) "x", dummyT);
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      val ft = DatatypeCase.case_tr true DatatypePackage.datatype_of_constr
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                 ctxt [x, cs]
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    in lambda x ft end
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in [("_lam_pats_syntax", fun_tr)] end
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*}
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end