src/HOL/HOL.thy
author Andreas Lochbihler
Fri Jul 29 09:49:23 2016 +0200 (2016-07-29)
changeset 63561 fba08009ff3e
parent 62958 b41c1cb5e251
child 63575 b9bd9e61fd63
permissions -rw-r--r--
add lemmas contributed by Peter Gammie
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(*  Title:      HOL/HOL.thy
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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section \<open>The basis of Higher-Order Logic\<close>
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theory HOL
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imports Pure "~~/src/Tools/Code_Generator"
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keywords
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  "try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
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    "print_induct_rules" :: diag and
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  "quickcheck_params" :: thy_decl
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begin
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ML_file "~~/src/Tools/misc_legacy.ML"
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ML_file "~~/src/Tools/try.ML"
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ML_file "~~/src/Tools/quickcheck.ML"
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ML_file "~~/src/Tools/solve_direct.ML"
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ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
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ML_file "~~/src/Tools/IsaPlanner/isand.ML"
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ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
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ML_file "~~/src/Provers/hypsubst.ML"
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ML_file "~~/src/Provers/splitter.ML"
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ML_file "~~/src/Provers/classical.ML"
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ML_file "~~/src/Provers/blast.ML"
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ML_file "~~/src/Provers/clasimp.ML"
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ML_file "~~/src/Tools/eqsubst.ML"
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ML_file "~~/src/Provers/quantifier1.ML"
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ML_file "~~/src/Tools/atomize_elim.ML"
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ML_file "~~/src/Tools/cong_tac.ML"
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ML_file "~~/src/Tools/intuitionistic.ML" setup \<open>Intuitionistic.method_setup @{binding iprover}\<close>
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ML_file "~~/src/Tools/project_rule.ML"
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ML_file "~~/src/Tools/subtyping.ML"
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ML_file "~~/src/Tools/case_product.ML"
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ML \<open>Plugin_Name.declare_setup @{binding extraction}\<close>
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ML \<open>
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  Plugin_Name.declare_setup @{binding quickcheck_random};
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  Plugin_Name.declare_setup @{binding quickcheck_exhaustive};
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  Plugin_Name.declare_setup @{binding quickcheck_bounded_forall};
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  Plugin_Name.declare_setup @{binding quickcheck_full_exhaustive};
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  Plugin_Name.declare_setup @{binding quickcheck_narrowing};
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\<close>
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ML \<open>
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  Plugin_Name.define_setup @{binding quickcheck}
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   [@{plugin quickcheck_exhaustive},
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    @{plugin quickcheck_random},
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    @{plugin quickcheck_bounded_forall},
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    @{plugin quickcheck_full_exhaustive},
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    @{plugin quickcheck_narrowing}]
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\<close>
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subsection \<open>Primitive logic\<close>
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subsubsection \<open>Core syntax\<close>
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setup \<open>Axclass.class_axiomatization (@{binding type}, [])\<close>
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default_sort type
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setup \<open>Object_Logic.add_base_sort @{sort type}\<close>
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axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
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instance "fun" :: (type, type) type by (rule fun_arity)
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axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
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instance itself :: (type) type by (rule itself_arity)
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typedecl bool
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judgment Trueprop :: "bool \<Rightarrow> prop"  ("(_)" 5)
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axiomatization implies :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longrightarrow>" 25)
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  and eq :: "['a, 'a] \<Rightarrow> bool"  (infixl "=" 50)
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  and The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
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subsubsection \<open>Defined connectives and quantifiers\<close>
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definition True :: bool
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  where "True \<equiv> ((\<lambda>x::bool. x) = (\<lambda>x. x))"
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definition All :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<forall>" 10)
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  where "All P \<equiv> (P = (\<lambda>x. True))"
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definition Ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>" 10)
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  where "Ex P \<equiv> \<forall>Q. (\<forall>x. P x \<longrightarrow> Q) \<longrightarrow> Q"
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definition False :: bool
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  where "False \<equiv> (\<forall>P. P)"
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definition Not :: "bool \<Rightarrow> bool"  ("\<not> _" [40] 40)
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  where not_def: "\<not> P \<equiv> P \<longrightarrow> False"
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definition conj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<and>" 35)
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  where and_def: "P \<and> Q \<equiv> \<forall>R. (P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> R"
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definition disj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<or>" 30)
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  where or_def: "P \<or> Q \<equiv> \<forall>R. (P \<longrightarrow> R) \<longrightarrow> (Q \<longrightarrow> R) \<longrightarrow> R"
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definition Ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>!" 10)
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  where "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"
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subsubsection \<open>Additional concrete syntax\<close>
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abbreviation Not_Ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<nexists>" 10)
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  where "\<nexists>x. P x \<equiv> \<not> (\<exists>x. P x)"
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abbreviation Not_Ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<nexists>!" 10)
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  where "\<nexists>!x. P x \<equiv> \<not> (\<exists>!x. P x)"
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abbreviation not_equal :: "['a, 'a] \<Rightarrow> bool"  (infixl "\<noteq>" 50)
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  where "x \<noteq> y \<equiv> \<not> (x = y)"
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notation (output)
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  eq  (infix "=" 50) and
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  not_equal  (infix "\<noteq>" 50)
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notation (ASCII output)
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  not_equal  (infix "~=" 50)
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notation (ASCII)
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  Not  ("~ _" [40] 40) and
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  conj  (infixr "&" 35) and
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  disj  (infixr "|" 30) and
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  implies  (infixr "-->" 25) and
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  not_equal  (infixl "~=" 50)
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abbreviation (iff)
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  iff :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longleftrightarrow>" 25)
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  where "A \<longleftrightarrow> B \<equiv> A = B"
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syntax "_The" :: "[pttrn, bool] \<Rightarrow> 'a"  ("(3THE _./ _)" [0, 10] 10)
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translations "THE x. P" \<rightleftharpoons> "CONST The (\<lambda>x. P)"
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print_translation \<open>
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  [(@{const_syntax The}, fn _ => fn [Abs abs] =>
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      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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      in Syntax.const @{syntax_const "_The"} $ x $ t end)]
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\<close>  \<comment> \<open>To avoid eta-contraction of body\<close>
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nonterminal letbinds and letbind
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syntax
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  "_bind"       :: "[pttrn, 'a] \<Rightarrow> letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind \<Rightarrow> letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] \<Rightarrow> letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] \<Rightarrow> 'a"                ("(let (_)/ in (_))" [0, 10] 10)
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nonterminal case_syn and cases_syn
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syntax
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  "_case_syntax" :: "['a, cases_syn] \<Rightarrow> 'b"  ("(case _ of/ _)" 10)
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  "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ \<Rightarrow>/ _)" 10)
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  "" :: "case_syn \<Rightarrow> cases_syn"  ("_")
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  "_case2" :: "[case_syn, cases_syn] \<Rightarrow> cases_syn"  ("_/ | _")
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syntax (ASCII)
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  "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ =>/ _)" 10)
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notation (ASCII)
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  All  (binder "ALL " 10) and
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  Ex  (binder "EX " 10) and
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  Ex1  (binder "EX! " 10)
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notation (input)
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  All  (binder "! " 10) and
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  Ex  (binder "? " 10) and
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  Ex1  (binder "?! " 10)
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subsubsection \<open>Axioms and basic definitions\<close>
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axiomatization where
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  refl: "t = (t::'a)" and
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  subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
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  ext: "(\<And>x::'a. (f x ::'b) = g x) \<Longrightarrow> (\<lambda>x. f x) = (\<lambda>x. g x)"
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    \<comment> \<open>Extensionality is built into the meta-logic, and this rule expresses
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         a related property.  It is an eta-expanded version of the traditional
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         rule, and similar to the ABS rule of HOL\<close> and
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  the_eq_trivial: "(THE x. x = a) = (a::'a)"
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axiomatization where
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  impI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q" and
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  mp: "\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q" and
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  iff: "(P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P) \<longrightarrow> (P = Q)" and
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  True_or_False: "(P = True) \<or> (P = False)"
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definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
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  where "If P x y \<equiv> (THE z::'a. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y))"
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definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
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  where "Let s f \<equiv> f s"
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translations
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  "_Let (_binds b bs) e"  \<rightleftharpoons> "_Let b (_Let bs e)"
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  "let x = a in e"        \<rightleftharpoons> "CONST Let a (\<lambda>x. e)"
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axiomatization undefined :: 'a
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class default = fixes default :: 'a
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subsection \<open>Fundamental rules\<close>
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subsubsection \<open>Equality\<close>
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lemma sym: "s = t \<Longrightarrow> t = s"
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  by (erule subst) (rule refl)
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lemma ssubst: "t = s \<Longrightarrow> P s \<Longrightarrow> P t"
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  by (drule sym) (erule subst)
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lemma trans: "\<lbrakk>r = s; s = t\<rbrakk> \<Longrightarrow> r = t"
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  by (erule subst)
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lemma trans_sym [Pure.elim?]: "r = s \<Longrightarrow> t = s \<Longrightarrow> r = t"
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  by (rule trans [OF _ sym])
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lemma meta_eq_to_obj_eq:
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  assumes meq: "A \<equiv> B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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text \<open>Useful with \<open>erule\<close> for proving equalities from known equalities.\<close>
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     (* a = b
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        |   |
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        c = d   *)
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lemma box_equals: "\<lbrakk>a = b; a = c; b = d\<rbrakk> \<Longrightarrow> c = d"
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apply (rule trans)
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apply (rule trans)
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apply (rule sym)
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apply assumption+
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done
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text \<open>For calculational reasoning:\<close>
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lemma forw_subst: "a = b \<Longrightarrow> P b \<Longrightarrow> P a"
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  by (rule ssubst)
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lemma back_subst: "P a \<Longrightarrow> a = b \<Longrightarrow> P b"
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  by (rule subst)
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subsubsection \<open>Congruence rules for application\<close>
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text \<open>Similar to \<open>AP_THM\<close> in Gordon's HOL.\<close>
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lemma fun_cong: "(f :: 'a \<Rightarrow> 'b) = g \<Longrightarrow> f x = g x"
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apply (erule subst)
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apply (rule refl)
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done
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text \<open>Similar to \<open>AP_TERM\<close> in Gordon's HOL and FOL's \<open>subst_context\<close>.\<close>
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lemma arg_cong: "x = y \<Longrightarrow> f x = f y"
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apply (erule subst)
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apply (rule refl)
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done
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lemma arg_cong2: "\<lbrakk>a = b; c = d\<rbrakk> \<Longrightarrow> f a c = f b d"
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apply (erule ssubst)+
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apply (rule refl)
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done
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lemma cong: "\<lbrakk>f = g; (x::'a) = y\<rbrakk> \<Longrightarrow> f x = g y"
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apply (erule subst)+
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apply (rule refl)
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done
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ML \<open>fun cong_tac ctxt = Cong_Tac.cong_tac ctxt @{thm cong}\<close>
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subsubsection \<open>Equality of booleans -- iff\<close>
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lemma iffI: assumes "P \<Longrightarrow> Q" and "Q \<Longrightarrow> P" shows "P = Q"
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  by (iprover intro: iff [THEN mp, THEN mp] impI assms)
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lemma iffD2: "\<lbrakk>P = Q; Q\<rbrakk> \<Longrightarrow> P"
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  by (erule ssubst)
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lemma rev_iffD2: "\<lbrakk>Q; P = Q\<rbrakk> \<Longrightarrow> P"
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  by (erule iffD2)
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lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
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  by (drule sym) (rule iffD2)
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lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
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  by (drule sym) (rule rev_iffD2)
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lemma iffE:
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  assumes major: "P = Q"
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    and minor: "\<lbrakk>P \<longrightarrow> Q; Q \<longrightarrow> P\<rbrakk> \<Longrightarrow> R"
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  shows R
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  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
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subsubsection \<open>True\<close>
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lemma TrueI: "True"
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  unfolding True_def by (rule refl)
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lemma eqTrueI: "P \<Longrightarrow> P = True"
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  by (iprover intro: iffI TrueI)
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lemma eqTrueE: "P = True \<Longrightarrow> P"
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  by (erule iffD2) (rule TrueI)
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subsubsection \<open>Universal quantifier\<close>
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lemma allI: assumes "\<And>x::'a. P x" shows "\<forall>x. P x"
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  unfolding All_def by (iprover intro: ext eqTrueI assms)
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lemma spec: "\<forall>x::'a. P x \<Longrightarrow> P x"
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apply (unfold All_def)
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apply (rule eqTrueE)
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apply (erule fun_cong)
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done
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lemma allE:
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  assumes major: "\<forall>x. P x"
wenzelm@60759
   321
    and minor: "P x \<Longrightarrow> R"
wenzelm@21504
   322
  shows R
wenzelm@21504
   323
  by (iprover intro: minor major [THEN spec])
paulson@15411
   324
paulson@15411
   325
lemma all_dupE:
wenzelm@60759
   326
  assumes major: "\<forall>x. P x"
wenzelm@60759
   327
    and minor: "\<lbrakk>P x; \<forall>x. P x\<rbrakk> \<Longrightarrow> R"
wenzelm@21504
   328
  shows R
wenzelm@21504
   329
  by (iprover intro: minor major major [THEN spec])
paulson@15411
   330
paulson@15411
   331
wenzelm@60758
   332
subsubsection \<open>False\<close>
wenzelm@21504
   333
wenzelm@60758
   334
text \<open>
wenzelm@61799
   335
  Depends upon \<open>spec\<close>; it is impossible to do propositional
wenzelm@21504
   336
  logic before quantifiers!
wenzelm@60758
   337
\<close>
paulson@15411
   338
wenzelm@60759
   339
lemma FalseE: "False \<Longrightarrow> P"
wenzelm@21504
   340
  apply (unfold False_def)
wenzelm@21504
   341
  apply (erule spec)
wenzelm@21504
   342
  done
paulson@15411
   343
wenzelm@60759
   344
lemma False_neq_True: "False = True \<Longrightarrow> P"
wenzelm@21504
   345
  by (erule eqTrueE [THEN FalseE])
paulson@15411
   346
paulson@15411
   347
wenzelm@60758
   348
subsubsection \<open>Negation\<close>
paulson@15411
   349
paulson@15411
   350
lemma notI:
wenzelm@60759
   351
  assumes "P \<Longrightarrow> False"
wenzelm@60759
   352
  shows "\<not> P"
wenzelm@21504
   353
  apply (unfold not_def)
wenzelm@21504
   354
  apply (iprover intro: impI assms)
wenzelm@21504
   355
  done
paulson@15411
   356
wenzelm@60759
   357
lemma False_not_True: "False \<noteq> True"
wenzelm@21504
   358
  apply (rule notI)
wenzelm@21504
   359
  apply (erule False_neq_True)
wenzelm@21504
   360
  done
paulson@15411
   361
wenzelm@60759
   362
lemma True_not_False: "True \<noteq> False"
wenzelm@21504
   363
  apply (rule notI)
wenzelm@21504
   364
  apply (drule sym)
wenzelm@21504
   365
  apply (erule False_neq_True)
wenzelm@21504
   366
  done
paulson@15411
   367
wenzelm@60759
   368
lemma notE: "\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R"
wenzelm@21504
   369
  apply (unfold not_def)
wenzelm@21504
   370
  apply (erule mp [THEN FalseE])
wenzelm@21504
   371
  apply assumption
wenzelm@21504
   372
  done
paulson@15411
   373
wenzelm@21504
   374
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
wenzelm@21504
   375
  by (erule notE [THEN notI]) (erule meta_mp)
paulson@15411
   376
paulson@15411
   377
wenzelm@60758
   378
subsubsection \<open>Implication\<close>
paulson@15411
   379
paulson@15411
   380
lemma impE:
wenzelm@60759
   381
  assumes "P \<longrightarrow> Q" P "Q \<Longrightarrow> R"
wenzelm@60759
   382
  shows R
wenzelm@23553
   383
by (iprover intro: assms mp)
paulson@15411
   384
wenzelm@60759
   385
(* Reduces Q to P \<longrightarrow> Q, allowing substitution in P. *)
wenzelm@60759
   386
lemma rev_mp: "\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
nipkow@17589
   387
by (iprover intro: mp)
paulson@15411
   388
paulson@15411
   389
lemma contrapos_nn:
wenzelm@60759
   390
  assumes major: "\<not> Q"
wenzelm@60759
   391
      and minor: "P \<Longrightarrow> Q"
wenzelm@60759
   392
  shows "\<not> P"
nipkow@17589
   393
by (iprover intro: notI minor major [THEN notE])
paulson@15411
   394
paulson@15411
   395
(*not used at all, but we already have the other 3 combinations *)
paulson@15411
   396
lemma contrapos_pn:
paulson@15411
   397
  assumes major: "Q"
wenzelm@60759
   398
      and minor: "P \<Longrightarrow> \<not> Q"
wenzelm@60759
   399
  shows "\<not> P"
nipkow@17589
   400
by (iprover intro: notI minor major notE)
paulson@15411
   401
wenzelm@60759
   402
lemma not_sym: "t \<noteq> s \<Longrightarrow> s \<noteq> t"
haftmann@21250
   403
  by (erule contrapos_nn) (erule sym)
haftmann@21250
   404
wenzelm@60759
   405
lemma eq_neq_eq_imp_neq: "\<lbrakk>x = a; a \<noteq> b; b = y\<rbrakk> \<Longrightarrow> x \<noteq> y"
haftmann@21250
   406
  by (erule subst, erule ssubst, assumption)
paulson@15411
   407
paulson@15411
   408
wenzelm@60758
   409
subsubsection \<open>Existential quantifier\<close>
paulson@15411
   410
wenzelm@60759
   411
lemma exI: "P x \<Longrightarrow> \<exists>x::'a. P x"
paulson@15411
   412
apply (unfold Ex_def)
nipkow@17589
   413
apply (iprover intro: allI allE impI mp)
paulson@15411
   414
done
paulson@15411
   415
paulson@15411
   416
lemma exE:
wenzelm@60759
   417
  assumes major: "\<exists>x::'a. P x"
wenzelm@60759
   418
      and minor: "\<And>x. P x \<Longrightarrow> Q"
paulson@15411
   419
  shows "Q"
paulson@15411
   420
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
nipkow@17589
   421
apply (iprover intro: impI [THEN allI] minor)
paulson@15411
   422
done
paulson@15411
   423
paulson@15411
   424
wenzelm@60758
   425
subsubsection \<open>Conjunction\<close>
paulson@15411
   426
wenzelm@60759
   427
lemma conjI: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> P \<and> Q"
paulson@15411
   428
apply (unfold and_def)
nipkow@17589
   429
apply (iprover intro: impI [THEN allI] mp)
paulson@15411
   430
done
paulson@15411
   431
wenzelm@60759
   432
lemma conjunct1: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> P"
paulson@15411
   433
apply (unfold and_def)
nipkow@17589
   434
apply (iprover intro: impI dest: spec mp)
paulson@15411
   435
done
paulson@15411
   436
wenzelm@60759
   437
lemma conjunct2: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> Q"
paulson@15411
   438
apply (unfold and_def)
nipkow@17589
   439
apply (iprover intro: impI dest: spec mp)
paulson@15411
   440
done
paulson@15411
   441
paulson@15411
   442
lemma conjE:
wenzelm@60759
   443
  assumes major: "P \<and> Q"
wenzelm@60759
   444
      and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   445
  shows R
paulson@15411
   446
apply (rule minor)
paulson@15411
   447
apply (rule major [THEN conjunct1])
paulson@15411
   448
apply (rule major [THEN conjunct2])
paulson@15411
   449
done
paulson@15411
   450
paulson@15411
   451
lemma context_conjI:
wenzelm@60759
   452
  assumes P "P \<Longrightarrow> Q" shows "P \<and> Q"
wenzelm@23553
   453
by (iprover intro: conjI assms)
paulson@15411
   454
paulson@15411
   455
wenzelm@60758
   456
subsubsection \<open>Disjunction\<close>
paulson@15411
   457
wenzelm@60759
   458
lemma disjI1: "P \<Longrightarrow> P \<or> Q"
paulson@15411
   459
apply (unfold or_def)
nipkow@17589
   460
apply (iprover intro: allI impI mp)
paulson@15411
   461
done
paulson@15411
   462
wenzelm@60759
   463
lemma disjI2: "Q \<Longrightarrow> P \<or> Q"
paulson@15411
   464
apply (unfold or_def)
nipkow@17589
   465
apply (iprover intro: allI impI mp)
paulson@15411
   466
done
paulson@15411
   467
paulson@15411
   468
lemma disjE:
wenzelm@60759
   469
  assumes major: "P \<or> Q"
wenzelm@60759
   470
      and minorP: "P \<Longrightarrow> R"
wenzelm@60759
   471
      and minorQ: "Q \<Longrightarrow> R"
wenzelm@60759
   472
  shows R
nipkow@17589
   473
by (iprover intro: minorP minorQ impI
paulson@15411
   474
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411
   475
paulson@15411
   476
wenzelm@60758
   477
subsubsection \<open>Classical logic\<close>
paulson@15411
   478
paulson@15411
   479
lemma classical:
wenzelm@60759
   480
  assumes prem: "\<not> P \<Longrightarrow> P"
wenzelm@60759
   481
  shows P
paulson@15411
   482
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
paulson@15411
   483
apply assumption
paulson@15411
   484
apply (rule notI [THEN prem, THEN eqTrueI])
paulson@15411
   485
apply (erule subst)
paulson@15411
   486
apply assumption
paulson@15411
   487
done
paulson@15411
   488
wenzelm@45607
   489
lemmas ccontr = FalseE [THEN classical]
paulson@15411
   490
wenzelm@60759
   491
(*notE with premises exchanged; it discharges \<not> R so that it can be used to
paulson@15411
   492
  make elimination rules*)
paulson@15411
   493
lemma rev_notE:
wenzelm@60759
   494
  assumes premp: P
wenzelm@60759
   495
      and premnot: "\<not> R \<Longrightarrow> \<not> P"
wenzelm@60759
   496
  shows R
paulson@15411
   497
apply (rule ccontr)
paulson@15411
   498
apply (erule notE [OF premnot premp])
paulson@15411
   499
done
paulson@15411
   500
paulson@15411
   501
(*Double negation law*)
wenzelm@60759
   502
lemma notnotD: "\<not>\<not> P \<Longrightarrow> P"
paulson@15411
   503
apply (rule classical)
paulson@15411
   504
apply (erule notE)
paulson@15411
   505
apply assumption
paulson@15411
   506
done
paulson@15411
   507
paulson@15411
   508
lemma contrapos_pp:
wenzelm@60759
   509
  assumes p1: Q
wenzelm@60759
   510
      and p2: "\<not> P \<Longrightarrow> \<not> Q"
wenzelm@60759
   511
  shows P
nipkow@17589
   512
by (iprover intro: classical p1 p2 notE)
paulson@15411
   513
paulson@15411
   514
wenzelm@60758
   515
subsubsection \<open>Unique existence\<close>
paulson@15411
   516
paulson@15411
   517
lemma ex1I:
wenzelm@60759
   518
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a"
wenzelm@60759
   519
  shows "\<exists>!x. P x"
wenzelm@23553
   520
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
paulson@15411
   521
wenzelm@60758
   522
text\<open>Sometimes easier to use: the premises have no shared variables.  Safe!\<close>
paulson@15411
   523
lemma ex_ex1I:
wenzelm@60759
   524
  assumes ex_prem: "\<exists>x. P x"
wenzelm@60759
   525
      and eq: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y"
wenzelm@60759
   526
  shows "\<exists>!x. P x"
nipkow@17589
   527
by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411
   528
paulson@15411
   529
lemma ex1E:
wenzelm@60759
   530
  assumes major: "\<exists>!x. P x"
wenzelm@60759
   531
      and minor: "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   532
  shows R
paulson@15411
   533
apply (rule major [unfolded Ex1_def, THEN exE])
paulson@15411
   534
apply (erule conjE)
nipkow@17589
   535
apply (iprover intro: minor)
paulson@15411
   536
done
paulson@15411
   537
wenzelm@60759
   538
lemma ex1_implies_ex: "\<exists>!x. P x \<Longrightarrow> \<exists>x. P x"
paulson@15411
   539
apply (erule ex1E)
paulson@15411
   540
apply (rule exI)
paulson@15411
   541
apply assumption
paulson@15411
   542
done
paulson@15411
   543
paulson@15411
   544
wenzelm@60758
   545
subsubsection \<open>Classical intro rules for disjunction and existential quantifiers\<close>
paulson@15411
   546
paulson@15411
   547
lemma disjCI:
wenzelm@60759
   548
  assumes "\<not> Q \<Longrightarrow> P" shows "P \<or> Q"
paulson@15411
   549
apply (rule classical)
wenzelm@23553
   550
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
paulson@15411
   551
done
paulson@15411
   552
wenzelm@60759
   553
lemma excluded_middle: "\<not> P \<or> P"
nipkow@17589
   554
by (iprover intro: disjCI)
paulson@15411
   555
wenzelm@60758
   556
text \<open>
haftmann@20944
   557
  case distinction as a natural deduction rule.
wenzelm@60759
   558
  Note that @{term "\<not> P"} is the second case, not the first
wenzelm@60758
   559
\<close>
wenzelm@27126
   560
lemma case_split [case_names True False]:
wenzelm@60759
   561
  assumes prem1: "P \<Longrightarrow> Q"
wenzelm@60759
   562
      and prem2: "\<not> P \<Longrightarrow> Q"
wenzelm@60759
   563
  shows Q
paulson@15411
   564
apply (rule excluded_middle [THEN disjE])
paulson@15411
   565
apply (erule prem2)
paulson@15411
   566
apply (erule prem1)
paulson@15411
   567
done
wenzelm@27126
   568
wenzelm@60759
   569
(*Classical implies (\<longrightarrow>) elimination. *)
paulson@15411
   570
lemma impCE:
wenzelm@60759
   571
  assumes major: "P \<longrightarrow> Q"
wenzelm@60759
   572
      and minor: "\<not> P \<Longrightarrow> R" "Q \<Longrightarrow> R"
wenzelm@60759
   573
  shows R
paulson@15411
   574
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   575
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   576
done
paulson@15411
   577
wenzelm@60759
   578
(*This version of \<longrightarrow> elimination works on Q before P.  It works best for
paulson@15411
   579
  those cases in which P holds "almost everywhere".  Can't install as
paulson@15411
   580
  default: would break old proofs.*)
paulson@15411
   581
lemma impCE':
wenzelm@60759
   582
  assumes major: "P \<longrightarrow> Q"
wenzelm@60759
   583
      and minor: "Q \<Longrightarrow> R" "\<not> P \<Longrightarrow> R"
wenzelm@60759
   584
  shows R
paulson@15411
   585
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   586
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   587
done
paulson@15411
   588
paulson@15411
   589
(*Classical <-> elimination. *)
paulson@15411
   590
lemma iffCE:
wenzelm@60759
   591
  assumes major: "P = Q"
wenzelm@60759
   592
      and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R" "\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   593
  shows R
paulson@15411
   594
apply (rule major [THEN iffE])
nipkow@17589
   595
apply (iprover intro: minor elim: impCE notE)
paulson@15411
   596
done
paulson@15411
   597
paulson@15411
   598
lemma exCI:
wenzelm@60759
   599
  assumes "\<forall>x. \<not> P x \<Longrightarrow> P a"
wenzelm@60759
   600
  shows "\<exists>x. P x"
paulson@15411
   601
apply (rule ccontr)
wenzelm@23553
   602
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
paulson@15411
   603
done
paulson@15411
   604
paulson@15411
   605
wenzelm@60758
   606
subsubsection \<open>Intuitionistic Reasoning\<close>
wenzelm@12386
   607
wenzelm@12386
   608
lemma impE':
wenzelm@60759
   609
  assumes 1: "P \<longrightarrow> Q"
wenzelm@60759
   610
    and 2: "Q \<Longrightarrow> R"
wenzelm@60759
   611
    and 3: "P \<longrightarrow> Q \<Longrightarrow> P"
wenzelm@12937
   612
  shows R
wenzelm@12386
   613
proof -
wenzelm@12386
   614
  from 3 and 1 have P .
wenzelm@12386
   615
  with 1 have Q by (rule impE)
wenzelm@12386
   616
  with 2 show R .
wenzelm@12386
   617
qed
wenzelm@12386
   618
wenzelm@12386
   619
lemma allE':
wenzelm@60759
   620
  assumes 1: "\<forall>x. P x"
wenzelm@60759
   621
    and 2: "P x \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q"
wenzelm@12937
   622
  shows Q
wenzelm@12386
   623
proof -
wenzelm@12386
   624
  from 1 have "P x" by (rule spec)
wenzelm@12386
   625
  from this and 1 show Q by (rule 2)
wenzelm@12386
   626
qed
wenzelm@12386
   627
wenzelm@12937
   628
lemma notE':
wenzelm@60759
   629
  assumes 1: "\<not> P"
wenzelm@60759
   630
    and 2: "\<not> P \<Longrightarrow> P"
wenzelm@12937
   631
  shows R
wenzelm@12386
   632
proof -
wenzelm@12386
   633
  from 2 and 1 have P .
wenzelm@12386
   634
  with 1 show R by (rule notE)
wenzelm@12386
   635
qed
wenzelm@12386
   636
wenzelm@60759
   637
lemma TrueE: "True \<Longrightarrow> P \<Longrightarrow> P" .
wenzelm@60759
   638
lemma notFalseE: "\<not> False \<Longrightarrow> P \<Longrightarrow> P" .
dixon@22444
   639
dixon@22467
   640
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
wenzelm@15801
   641
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801
   642
  and [Pure.elim 2] = allE notE' impE'
wenzelm@15801
   643
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386
   644
wenzelm@12386
   645
lemmas [trans] = trans
wenzelm@12386
   646
  and [sym] = sym not_sym
wenzelm@15801
   647
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   648
wenzelm@11438
   649
wenzelm@60758
   650
subsubsection \<open>Atomizing meta-level connectives\<close>
wenzelm@11750
   651
haftmann@28513
   652
axiomatization where
haftmann@28513
   653
  eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
haftmann@28513
   654
wenzelm@60759
   655
lemma atomize_all [atomize]: "(\<And>x. P x) \<equiv> Trueprop (\<forall>x. P x)"
wenzelm@12003
   656
proof
wenzelm@60759
   657
  assume "\<And>x. P x"
wenzelm@60759
   658
  then show "\<forall>x. P x" ..
wenzelm@9488
   659
next
wenzelm@60759
   660
  assume "\<forall>x. P x"
wenzelm@60759
   661
  then show "\<And>x. P x" by (rule allE)
wenzelm@9488
   662
qed
wenzelm@9488
   663
wenzelm@60759
   664
lemma atomize_imp [atomize]: "(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)"
wenzelm@12003
   665
proof
wenzelm@60759
   666
  assume r: "A \<Longrightarrow> B"
wenzelm@60759
   667
  show "A \<longrightarrow> B" by (rule impI) (rule r)
wenzelm@9488
   668
next
wenzelm@60759
   669
  assume "A \<longrightarrow> B" and A
wenzelm@23553
   670
  then show B by (rule mp)
wenzelm@9488
   671
qed
wenzelm@9488
   672
wenzelm@60759
   673
lemma atomize_not: "(A \<Longrightarrow> False) \<equiv> Trueprop (\<not> A)"
paulson@14749
   674
proof
wenzelm@60759
   675
  assume r: "A \<Longrightarrow> False"
wenzelm@60759
   676
  show "\<not> A" by (rule notI) (rule r)
paulson@14749
   677
next
wenzelm@60759
   678
  assume "\<not> A" and A
wenzelm@23553
   679
  then show False by (rule notE)
paulson@14749
   680
qed
paulson@14749
   681
wenzelm@60759
   682
lemma atomize_eq [atomize, code]: "(x \<equiv> y) \<equiv> Trueprop (x = y)"
wenzelm@12003
   683
proof
wenzelm@60759
   684
  assume "x \<equiv> y"
wenzelm@60759
   685
  show "x = y" by (unfold \<open>x \<equiv> y\<close>) (rule refl)
wenzelm@10432
   686
next
wenzelm@10432
   687
  assume "x = y"
wenzelm@60759
   688
  then show "x \<equiv> y" by (rule eq_reflection)
wenzelm@10432
   689
qed
wenzelm@10432
   690
wenzelm@60759
   691
lemma atomize_conj [atomize]: "(A &&& B) \<equiv> Trueprop (A \<and> B)"
wenzelm@12003
   692
proof
wenzelm@28856
   693
  assume conj: "A &&& B"
wenzelm@60759
   694
  show "A \<and> B"
wenzelm@19121
   695
  proof (rule conjI)
wenzelm@19121
   696
    from conj show A by (rule conjunctionD1)
wenzelm@19121
   697
    from conj show B by (rule conjunctionD2)
wenzelm@19121
   698
  qed
wenzelm@11953
   699
next
wenzelm@60759
   700
  assume conj: "A \<and> B"
wenzelm@28856
   701
  show "A &&& B"
wenzelm@19121
   702
  proof -
wenzelm@19121
   703
    from conj show A ..
wenzelm@19121
   704
    from conj show B ..
wenzelm@11953
   705
  qed
wenzelm@11953
   706
qed
wenzelm@11953
   707
wenzelm@12386
   708
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832
   709
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386
   710
wenzelm@11750
   711
wenzelm@60758
   712
subsubsection \<open>Atomizing elimination rules\<close>
krauss@26580
   713
wenzelm@60759
   714
lemma atomize_exL[atomize_elim]: "(\<And>x. P x \<Longrightarrow> Q) \<equiv> ((\<exists>x. P x) \<Longrightarrow> Q)"
krauss@26580
   715
  by rule iprover+
krauss@26580
   716
wenzelm@60759
   717
lemma atomize_conjL[atomize_elim]: "(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)"
krauss@26580
   718
  by rule iprover+
krauss@26580
   719
wenzelm@60759
   720
lemma atomize_disjL[atomize_elim]: "((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)"
krauss@26580
   721
  by rule iprover+
krauss@26580
   722
wenzelm@60759
   723
lemma atomize_elimL[atomize_elim]: "(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop A" ..
krauss@26580
   724
krauss@26580
   725
wenzelm@60758
   726
subsection \<open>Package setup\<close>
haftmann@20944
   727
wenzelm@51314
   728
ML_file "Tools/hologic.ML"
wenzelm@51314
   729
wenzelm@51314
   730
wenzelm@60758
   731
subsubsection \<open>Sledgehammer setup\<close>
blanchet@35828
   732
wenzelm@60758
   733
text \<open>
blanchet@35828
   734
Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
blanchet@35828
   735
that are prolific (match too many equality or membership literals) and relate to
blanchet@35828
   736
seldom-used facts. Some duplicate other rules.
wenzelm@60758
   737
\<close>
blanchet@35828
   738
wenzelm@57963
   739
named_theorems no_atp "theorems that should be filtered out by Sledgehammer"
blanchet@35828
   740
blanchet@35828
   741
wenzelm@60758
   742
subsubsection \<open>Classical Reasoner setup\<close>
wenzelm@9529
   743
wenzelm@60759
   744
lemma imp_elim: "P \<longrightarrow> Q \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@26411
   745
  by (rule classical) iprover
wenzelm@26411
   746
wenzelm@60759
   747
lemma swap: "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"
wenzelm@26411
   748
  by (rule classical) iprover
wenzelm@26411
   749
wenzelm@62958
   750
lemma thin_refl: "\<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944
   751
wenzelm@60758
   752
ML \<open>
wenzelm@42799
   753
structure Hypsubst = Hypsubst
wenzelm@42799
   754
(
wenzelm@21218
   755
  val dest_eq = HOLogic.dest_eq
haftmann@21151
   756
  val dest_Trueprop = HOLogic.dest_Trueprop
haftmann@21151
   757
  val dest_imp = HOLogic.dest_imp
wenzelm@26411
   758
  val eq_reflection = @{thm eq_reflection}
wenzelm@26411
   759
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
wenzelm@26411
   760
  val imp_intr = @{thm impI}
wenzelm@26411
   761
  val rev_mp = @{thm rev_mp}
wenzelm@26411
   762
  val subst = @{thm subst}
wenzelm@26411
   763
  val sym = @{thm sym}
wenzelm@22129
   764
  val thin_refl = @{thm thin_refl};
wenzelm@42799
   765
);
wenzelm@21671
   766
open Hypsubst;
haftmann@21151
   767
wenzelm@42799
   768
structure Classical = Classical
wenzelm@42799
   769
(
wenzelm@26411
   770
  val imp_elim = @{thm imp_elim}
wenzelm@26411
   771
  val not_elim = @{thm notE}
wenzelm@26411
   772
  val swap = @{thm swap}
wenzelm@26411
   773
  val classical = @{thm classical}
haftmann@21151
   774
  val sizef = Drule.size_of_thm
haftmann@21151
   775
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
wenzelm@42799
   776
);
haftmann@21151
   777
wenzelm@58826
   778
structure Basic_Classical: BASIC_CLASSICAL = Classical;
wenzelm@33308
   779
open Basic_Classical;
wenzelm@60758
   780
\<close>
wenzelm@22129
   781
wenzelm@60758
   782
setup \<open>
wenzelm@35389
   783
  (*prevent substitution on bool*)
wenzelm@58826
   784
  let
wenzelm@58826
   785
    fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
wenzelm@58826
   786
      | non_bool_eq _ = false;
wenzelm@58826
   787
    fun hyp_subst_tac' ctxt =
wenzelm@58826
   788
      SUBGOAL (fn (goal, i) =>
wenzelm@58826
   789
        if Term.exists_Const non_bool_eq goal
wenzelm@58826
   790
        then Hypsubst.hyp_subst_tac ctxt i
wenzelm@58826
   791
        else no_tac);
wenzelm@58826
   792
  in
wenzelm@58826
   793
    Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
wenzelm@58826
   794
  end
wenzelm@60758
   795
\<close>
haftmann@21009
   796
haftmann@21009
   797
declare iffI [intro!]
haftmann@21009
   798
  and notI [intro!]
haftmann@21009
   799
  and impI [intro!]
haftmann@21009
   800
  and disjCI [intro!]
haftmann@21009
   801
  and conjI [intro!]
haftmann@21009
   802
  and TrueI [intro!]
haftmann@21009
   803
  and refl [intro!]
haftmann@21009
   804
haftmann@21009
   805
declare iffCE [elim!]
haftmann@21009
   806
  and FalseE [elim!]
haftmann@21009
   807
  and impCE [elim!]
haftmann@21009
   808
  and disjE [elim!]
haftmann@21009
   809
  and conjE [elim!]
haftmann@21009
   810
haftmann@21009
   811
declare ex_ex1I [intro!]
haftmann@21009
   812
  and allI [intro!]
haftmann@21009
   813
  and exI [intro]
haftmann@21009
   814
haftmann@21009
   815
declare exE [elim!]
haftmann@21009
   816
  allE [elim]
haftmann@21009
   817
wenzelm@60758
   818
ML \<open>val HOL_cs = claset_of @{context}\<close>
mengj@19162
   819
wenzelm@60759
   820
lemma contrapos_np: "\<not> Q \<Longrightarrow> (\<not> P \<Longrightarrow> Q) \<Longrightarrow> P"
wenzelm@20223
   821
  apply (erule swap)
wenzelm@20223
   822
  apply (erule (1) meta_mp)
wenzelm@20223
   823
  done
wenzelm@10383
   824
wenzelm@18689
   825
declare ex_ex1I [rule del, intro! 2]
wenzelm@18689
   826
  and ex1I [intro]
wenzelm@18689
   827
paulson@41865
   828
declare ext [intro]
paulson@41865
   829
wenzelm@12386
   830
lemmas [intro?] = ext
wenzelm@12386
   831
  and [elim?] = ex1_implies_ex
wenzelm@11977
   832
haftmann@20944
   833
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
haftmann@20973
   834
lemma alt_ex1E [elim!]:
haftmann@20944
   835
  assumes major: "\<exists>!x. P x"
haftmann@20944
   836
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
haftmann@20944
   837
  shows R
haftmann@20944
   838
apply (rule ex1E [OF major])
haftmann@20944
   839
apply (rule prem)
wenzelm@59499
   840
apply assumption
wenzelm@59499
   841
apply (rule allI)+
wenzelm@60758
   842
apply (tactic \<open>eresolve_tac @{context} [Classical.dup_elim @{context} @{thm allE}] 1\<close>)
wenzelm@22129
   843
apply iprover
wenzelm@22129
   844
done
haftmann@20944
   845
wenzelm@60758
   846
ML \<open>
wenzelm@42477
   847
  structure Blast = Blast
wenzelm@42477
   848
  (
wenzelm@42477
   849
    structure Classical = Classical
wenzelm@42802
   850
    val Trueprop_const = dest_Const @{const Trueprop}
wenzelm@42477
   851
    val equality_name = @{const_name HOL.eq}
wenzelm@42477
   852
    val not_name = @{const_name Not}
wenzelm@42477
   853
    val notE = @{thm notE}
wenzelm@42477
   854
    val ccontr = @{thm ccontr}
wenzelm@42477
   855
    val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
wenzelm@42477
   856
  );
wenzelm@42477
   857
  val blast_tac = Blast.blast_tac;
wenzelm@60758
   858
\<close>
haftmann@20944
   859
haftmann@20944
   860
wenzelm@60758
   861
subsubsection \<open>THE: definite description operator\<close>
lp15@59504
   862
lp15@59504
   863
lemma the_equality [intro]:
lp15@59504
   864
  assumes "P a"
wenzelm@60759
   865
      and "\<And>x. P x \<Longrightarrow> x = a"
lp15@59504
   866
  shows "(THE x. P x) = a"
lp15@59504
   867
  by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
lp15@59504
   868
lp15@59504
   869
lemma theI:
wenzelm@60759
   870
  assumes "P a" and "\<And>x. P x \<Longrightarrow> x = a"
lp15@59504
   871
  shows "P (THE x. P x)"
lp15@59504
   872
by (iprover intro: assms the_equality [THEN ssubst])
lp15@59504
   873
wenzelm@60759
   874
lemma theI': "\<exists>!x. P x \<Longrightarrow> P (THE x. P x)"
lp15@59504
   875
  by (blast intro: theI)
lp15@59504
   876
lp15@59504
   877
(*Easier to apply than theI: only one occurrence of P*)
lp15@59504
   878
lemma theI2:
wenzelm@60759
   879
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a" "\<And>x. P x \<Longrightarrow> Q x"
lp15@59504
   880
  shows "Q (THE x. P x)"
lp15@59504
   881
by (iprover intro: assms theI)
lp15@59504
   882
wenzelm@60759
   883
lemma the1I2: assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
lp15@59504
   884
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
lp15@59504
   885
           elim:allE impE)
lp15@59504
   886
wenzelm@60759
   887
lemma the1_equality [elim?]: "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> (THE x. P x) = a"
lp15@59504
   888
  by blast
lp15@59504
   889
wenzelm@60759
   890
lemma the_sym_eq_trivial: "(THE y. x = y) = x"
lp15@59504
   891
  by blast
lp15@59504
   892
lp15@59504
   893
wenzelm@60758
   894
subsubsection \<open>Simplifier\<close>
wenzelm@12281
   895
wenzelm@60759
   896
lemma eta_contract_eq: "(\<lambda>s. f s) = f" ..
wenzelm@12281
   897
wenzelm@12281
   898
lemma simp_thms:
wenzelm@60759
   899
  shows not_not: "(\<not> \<not> P) = P"
wenzelm@60759
   900
  and Not_eq_iff: "((\<not> P) = (\<not> Q)) = (P = Q)"
wenzelm@12937
   901
  and
wenzelm@60759
   902
    "(P \<noteq> Q) = (P = (\<not> Q))"
wenzelm@60759
   903
    "(P \<or> \<not>P) = True"    "(\<not> P \<or> P) = True"
wenzelm@12281
   904
    "(x = x) = True"
haftmann@32068
   905
  and not_True_eq_False [code]: "(\<not> True) = False"
haftmann@32068
   906
  and not_False_eq_True [code]: "(\<not> False) = True"
haftmann@20944
   907
  and
wenzelm@60759
   908
    "(\<not> P) \<noteq> P"  "P \<noteq> (\<not> P)"
wenzelm@60759
   909
    "(True = P) = P"
haftmann@20944
   910
  and eq_True: "(P = True) = P"
wenzelm@60759
   911
  and "(False = P) = (\<not> P)"
haftmann@20944
   912
  and eq_False: "(P = False) = (\<not> P)"
haftmann@20944
   913
  and
wenzelm@60759
   914
    "(True \<longrightarrow> P) = P"  "(False \<longrightarrow> P) = True"
wenzelm@60759
   915
    "(P \<longrightarrow> True) = True"  "(P \<longrightarrow> P) = True"
wenzelm@60759
   916
    "(P \<longrightarrow> False) = (\<not> P)"  "(P \<longrightarrow> \<not> P) = (\<not> P)"
wenzelm@60759
   917
    "(P \<and> True) = P"  "(True \<and> P) = P"
wenzelm@60759
   918
    "(P \<and> False) = False"  "(False \<and> P) = False"
wenzelm@60759
   919
    "(P \<and> P) = P"  "(P \<and> (P \<and> Q)) = (P \<and> Q)"
wenzelm@60759
   920
    "(P \<and> \<not> P) = False"    "(\<not> P \<and> P) = False"
wenzelm@60759
   921
    "(P \<or> True) = True"  "(True \<or> P) = True"
wenzelm@60759
   922
    "(P \<or> False) = P"  "(False \<or> P) = P"
wenzelm@60759
   923
    "(P \<or> P) = P"  "(P \<or> (P \<or> Q)) = (P \<or> Q)" and
wenzelm@60759
   924
    "(\<forall>x. P) = P"  "(\<exists>x. P) = P"  "\<exists>x. x = t"  "\<exists>x. t = x"
nipkow@31166
   925
  and
wenzelm@60759
   926
    "\<And>P. (\<exists>x. x = t \<and> P x) = P t"
wenzelm@60759
   927
    "\<And>P. (\<exists>x. t = x \<and> P x) = P t"
wenzelm@60759
   928
    "\<And>P. (\<forall>x. x = t \<longrightarrow> P x) = P t"
wenzelm@60759
   929
    "\<And>P. (\<forall>x. t = x \<longrightarrow> P x) = P t"
nipkow@17589
   930
  by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421
   931
wenzelm@60759
   932
lemma disj_absorb: "(A \<or> A) = A"
paulson@14201
   933
  by blast
paulson@14201
   934
wenzelm@60759
   935
lemma disj_left_absorb: "(A \<or> (A \<or> B)) = (A \<or> B)"
paulson@14201
   936
  by blast
paulson@14201
   937
wenzelm@60759
   938
lemma conj_absorb: "(A \<and> A) = A"
paulson@14201
   939
  by blast
paulson@14201
   940
wenzelm@60759
   941
lemma conj_left_absorb: "(A \<and> (A \<and> B)) = (A \<and> B)"
paulson@14201
   942
  by blast
paulson@14201
   943
wenzelm@12281
   944
lemma eq_ac:
haftmann@57512
   945
  shows eq_commute: "a = b \<longleftrightarrow> b = a"
haftmann@57512
   946
    and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
haftmann@57512
   947
    and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
haftmann@57512
   948
lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
wenzelm@12281
   949
wenzelm@12281
   950
lemma conj_comms:
wenzelm@60759
   951
  shows conj_commute: "(P \<and> Q) = (Q \<and> P)"
wenzelm@60759
   952
    and conj_left_commute: "(P \<and> (Q \<and> R)) = (Q \<and> (P \<and> R))" by iprover+
wenzelm@60759
   953
lemma conj_assoc: "((P \<and> Q) \<and> R) = (P \<and> (Q \<and> R))" by iprover
wenzelm@12281
   954
paulson@19174
   955
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174
   956
wenzelm@12281
   957
lemma disj_comms:
wenzelm@60759
   958
  shows disj_commute: "(P \<or> Q) = (Q \<or> P)"
wenzelm@60759
   959
    and disj_left_commute: "(P \<or> (Q \<or> R)) = (Q \<or> (P \<or> R))" by iprover+
wenzelm@60759
   960
lemma disj_assoc: "((P \<or> Q) \<or> R) = (P \<or> (Q \<or> R))" by iprover
wenzelm@12281
   961
paulson@19174
   962
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174
   963
wenzelm@60759
   964
lemma conj_disj_distribL: "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)" by iprover
wenzelm@60759
   965
lemma conj_disj_distribR: "((P \<or> Q) \<and> R) = (P \<and> R \<or> Q \<and> R)" by iprover
wenzelm@12281
   966
wenzelm@60759
   967
lemma disj_conj_distribL: "(P \<or> (Q \<and> R)) = ((P \<or> Q) \<and> (P \<or> R))" by iprover
wenzelm@60759
   968
lemma disj_conj_distribR: "((P \<and> Q) \<or> R) = ((P \<or> R) \<and> (Q \<or> R))" by iprover
wenzelm@12281
   969
wenzelm@60759
   970
lemma imp_conjR: "(P \<longrightarrow> (Q \<and> R)) = ((P \<longrightarrow> Q) \<and> (P \<longrightarrow> R))" by iprover
wenzelm@60759
   971
lemma imp_conjL: "((P \<and> Q) \<longrightarrow> R) = (P \<longrightarrow> (Q \<longrightarrow> R))" by iprover
wenzelm@60759
   972
lemma imp_disjL: "((P \<or> Q) \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" by iprover
wenzelm@12281
   973
wenzelm@61799
   974
text \<open>These two are specialized, but \<open>imp_disj_not1\<close> is useful in \<open>Auth/Yahalom\<close>.\<close>
wenzelm@60759
   975
lemma imp_disj_not1: "(P \<longrightarrow> Q \<or> R) = (\<not> Q \<longrightarrow> P \<longrightarrow> R)" by blast
wenzelm@60759
   976
lemma imp_disj_not2: "(P \<longrightarrow> Q \<or> R) = (\<not> R \<longrightarrow> P \<longrightarrow> Q)" by blast
wenzelm@12281
   977
wenzelm@60759
   978
lemma imp_disj1: "((P \<longrightarrow> Q) \<or> R) = (P \<longrightarrow> Q \<or> R)" by blast
wenzelm@60759
   979
lemma imp_disj2: "(Q \<or> (P \<longrightarrow> R)) = (P \<longrightarrow> Q \<or> R)" by blast
wenzelm@12281
   980
wenzelm@60759
   981
lemma imp_cong: "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<longrightarrow> Q) = (P' \<longrightarrow> Q'))"
haftmann@21151
   982
  by iprover
haftmann@21151
   983
wenzelm@60759
   984
lemma de_Morgan_disj: "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)" by iprover
wenzelm@60759
   985
lemma de_Morgan_conj: "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)" by blast
wenzelm@60759
   986
lemma not_imp: "(\<not> (P \<longrightarrow> Q)) = (P \<and> \<not> Q)" by blast
wenzelm@60759
   987
lemma not_iff: "(P \<noteq> Q) = (P = (\<not> Q))" by blast
wenzelm@60759
   988
lemma disj_not1: "(\<not> P \<or> Q) = (P \<longrightarrow> Q)" by blast
wenzelm@61799
   989
lemma disj_not2: "(P \<or> \<not> Q) = (Q \<longrightarrow> P)"  \<comment> \<open>changes orientation :-(\<close>
wenzelm@12281
   990
  by blast
wenzelm@60759
   991
lemma imp_conv_disj: "(P \<longrightarrow> Q) = ((\<not> P) \<or> Q)" by blast
Andreas@63561
   992
lemma disj_imp: "P \<or> Q \<longleftrightarrow> \<not> P \<longrightarrow> Q" by blast
wenzelm@12281
   993
wenzelm@60759
   994
lemma iff_conv_conj_imp: "(P = Q) = ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))" by iprover
wenzelm@12281
   995
wenzelm@12281
   996
wenzelm@60759
   997
lemma cases_simp: "((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q)) = Q"
nipkow@62390
   998
  \<comment> \<open>Avoids duplication of subgoals after \<open>if_split\<close>, when the true and false\<close>
wenzelm@61799
   999
  \<comment> \<open>cases boil down to the same thing.\<close>
wenzelm@12281
  1000
  by blast
wenzelm@12281
  1001
wenzelm@60759
  1002
lemma not_all: "(\<not> (\<forall>x. P x)) = (\<exists>x. \<not> P x)" by blast
wenzelm@60759
  1003
lemma imp_all: "((\<forall>x. P x) \<longrightarrow> Q) = (\<exists>x. P x \<longrightarrow> Q)" by blast
wenzelm@60759
  1004
lemma not_ex: "(\<not> (\<exists>x. P x)) = (\<forall>x. \<not> P x)" by iprover
wenzelm@60759
  1005
lemma imp_ex: "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)" by iprover
wenzelm@60759
  1006
lemma all_not_ex: "(\<forall>x. P x) = (\<not> (\<exists>x. \<not> P x ))" by blast
wenzelm@12281
  1007
blanchet@35828
  1008
declare All_def [no_atp]
paulson@24286
  1009
wenzelm@60759
  1010
lemma ex_disj_distrib: "(\<exists>x. P x \<or> Q x) = ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by iprover
wenzelm@60759
  1011
lemma all_conj_distrib: "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))" by iprover
wenzelm@12281
  1012
wenzelm@60758
  1013
text \<open>
wenzelm@61799
  1014
  \medskip The \<open>\<and>\<close> congruence rule: not included by default!
wenzelm@60758
  1015
  May slow rewrite proofs down by as much as 50\%\<close>
wenzelm@12281
  1016
wenzelm@12281
  1017
lemma conj_cong:
wenzelm@60759
  1018
    "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<and> Q) = (P' \<and> Q'))"
nipkow@17589
  1019
  by iprover
wenzelm@12281
  1020
wenzelm@12281
  1021
lemma rev_conj_cong:
wenzelm@60759
  1022
    "(Q = Q') \<Longrightarrow> (Q' \<Longrightarrow> (P = P')) \<Longrightarrow> ((P \<and> Q) = (P' \<and> Q'))"
nipkow@17589
  1023
  by iprover
wenzelm@12281
  1024
wenzelm@61799
  1025
text \<open>The \<open>|\<close> congruence rule: not included by default!\<close>
wenzelm@12281
  1026
wenzelm@12281
  1027
lemma disj_cong:
wenzelm@60759
  1028
    "(P = P') \<Longrightarrow> (\<not> P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<or> Q) = (P' \<or> Q'))"
wenzelm@12281
  1029
  by blast
wenzelm@12281
  1030
wenzelm@12281
  1031
wenzelm@60758
  1032
text \<open>\medskip if-then-else rules\<close>
wenzelm@12281
  1033
haftmann@32068
  1034
lemma if_True [code]: "(if True then x else y) = x"
haftmann@38525
  1035
  by (unfold If_def) blast
wenzelm@12281
  1036
haftmann@32068
  1037
lemma if_False [code]: "(if False then x else y) = y"
haftmann@38525
  1038
  by (unfold If_def) blast
wenzelm@12281
  1039
wenzelm@60759
  1040
lemma if_P: "P \<Longrightarrow> (if P then x else y) = x"
haftmann@38525
  1041
  by (unfold If_def) blast
wenzelm@12281
  1042
wenzelm@60759
  1043
lemma if_not_P: "\<not> P \<Longrightarrow> (if P then x else y) = y"
haftmann@38525
  1044
  by (unfold If_def) blast
wenzelm@12281
  1045
nipkow@62390
  1046
lemma if_split: "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
wenzelm@12281
  1047
  apply (rule case_split [of Q])
paulson@15481
  1048
   apply (simplesubst if_P)
paulson@15481
  1049
    prefer 3 apply (simplesubst if_not_P, blast+)
wenzelm@12281
  1050
  done
wenzelm@12281
  1051
nipkow@62390
  1052
lemma if_split_asm: "P (if Q then x else y) = (\<not> ((Q \<and> \<not> P x) \<or> (\<not> Q \<and> \<not> P y)))"
nipkow@62390
  1053
by (simplesubst if_split, blast)
wenzelm@12281
  1054
nipkow@62390
  1055
lemmas if_splits [no_atp] = if_split if_split_asm
wenzelm@12281
  1056
wenzelm@12281
  1057
lemma if_cancel: "(if c then x else x) = x"
nipkow@62390
  1058
by (simplesubst if_split, blast)
wenzelm@12281
  1059
wenzelm@12281
  1060
lemma if_eq_cancel: "(if x = y then y else x) = x"
nipkow@62390
  1061
by (simplesubst if_split, blast)
wenzelm@12281
  1062
wenzelm@60759
  1063
lemma if_bool_eq_conj: "(if P then Q else R) = ((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R))"
wenzelm@61799
  1064
  \<comment> \<open>This form is useful for expanding \<open>if\<close>s on the RIGHT of the \<open>\<Longrightarrow>\<close> symbol.\<close>
nipkow@62390
  1065
  by (rule if_split)
wenzelm@12281
  1066
wenzelm@60759
  1067
lemma if_bool_eq_disj: "(if P then Q else R) = ((P \<and> Q) \<or> (\<not> P \<and> R))"
wenzelm@61799
  1068
  \<comment> \<open>And this form is useful for expanding \<open>if\<close>s on the LEFT.\<close>
nipkow@62390
  1069
  by (simplesubst if_split) blast
wenzelm@12281
  1070
wenzelm@60759
  1071
lemma Eq_TrueI: "P \<Longrightarrow> P \<equiv> True" by (unfold atomize_eq) iprover
wenzelm@60759
  1072
lemma Eq_FalseI: "\<not> P \<Longrightarrow> P \<equiv> False" by (unfold atomize_eq) iprover
wenzelm@12281
  1073
wenzelm@60758
  1074
text \<open>\medskip let rules for simproc\<close>
schirmer@15423
  1075
wenzelm@60759
  1076
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
schirmer@15423
  1077
  by (unfold Let_def)
schirmer@15423
  1078
wenzelm@60759
  1079
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
schirmer@15423
  1080
  by (unfold Let_def)
schirmer@15423
  1081
wenzelm@60758
  1082
text \<open>
ballarin@16999
  1083
  The following copy of the implication operator is useful for
ballarin@16999
  1084
  fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999
  1085
  its premise.
wenzelm@60758
  1086
\<close>
berghofe@16633
  1087
wenzelm@60759
  1088
definition simp_implies :: "[prop, prop] \<Rightarrow> prop"  (infixr "=simp=>" 1) where
wenzelm@60759
  1089
  "simp_implies \<equiv> op \<Longrightarrow>"
berghofe@16633
  1090
wenzelm@18457
  1091
lemma simp_impliesI:
berghofe@16633
  1092
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633
  1093
  shows "PROP P =simp=> PROP Q"
berghofe@16633
  1094
  apply (unfold simp_implies_def)
berghofe@16633
  1095
  apply (rule PQ)
berghofe@16633
  1096
  apply assumption
berghofe@16633
  1097
  done
berghofe@16633
  1098
berghofe@16633
  1099
lemma simp_impliesE:
wenzelm@25388
  1100
  assumes PQ: "PROP P =simp=> PROP Q"
berghofe@16633
  1101
  and P: "PROP P"
berghofe@16633
  1102
  and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633
  1103
  shows "PROP R"
berghofe@16633
  1104
  apply (rule QR)
berghofe@16633
  1105
  apply (rule PQ [unfolded simp_implies_def])
berghofe@16633
  1106
  apply (rule P)
berghofe@16633
  1107
  done
berghofe@16633
  1108
berghofe@16633
  1109
lemma simp_implies_cong:
wenzelm@60759
  1110
  assumes PP' :"PROP P \<equiv> PROP P'"
wenzelm@60759
  1111
  and P'QQ': "PROP P' \<Longrightarrow> (PROP Q \<equiv> PROP Q')"
wenzelm@60759
  1112
  shows "(PROP P =simp=> PROP Q) \<equiv> (PROP P' =simp=> PROP Q')"
berghofe@16633
  1113
proof (unfold simp_implies_def, rule equal_intr_rule)
berghofe@16633
  1114
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
berghofe@16633
  1115
  and P': "PROP P'"
berghofe@16633
  1116
  from PP' [symmetric] and P' have "PROP P"
berghofe@16633
  1117
    by (rule equal_elim_rule1)
wenzelm@23553
  1118
  then have "PROP Q" by (rule PQ)
berghofe@16633
  1119
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633
  1120
next
berghofe@16633
  1121
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
berghofe@16633
  1122
  and P: "PROP P"
berghofe@16633
  1123
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
wenzelm@23553
  1124
  then have "PROP Q'" by (rule P'Q')
berghofe@16633
  1125
  with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633
  1126
    by (rule equal_elim_rule1)
berghofe@16633
  1127
qed
berghofe@16633
  1128
haftmann@20944
  1129
lemma uncurry:
haftmann@20944
  1130
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944
  1131
  shows "P \<and> Q \<longrightarrow> R"
wenzelm@23553
  1132
  using assms by blast
haftmann@20944
  1133
haftmann@20944
  1134
lemma iff_allI:
haftmann@20944
  1135
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1136
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
wenzelm@23553
  1137
  using assms by blast
haftmann@20944
  1138
haftmann@20944
  1139
lemma iff_exI:
haftmann@20944
  1140
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1141
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
wenzelm@23553
  1142
  using assms by blast
haftmann@20944
  1143
haftmann@20944
  1144
lemma all_comm:
haftmann@20944
  1145
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944
  1146
  by blast
haftmann@20944
  1147
haftmann@20944
  1148
lemma ex_comm:
haftmann@20944
  1149
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
haftmann@20944
  1150
  by blast
haftmann@20944
  1151
wenzelm@48891
  1152
ML_file "Tools/simpdata.ML"
wenzelm@60758
  1153
ML \<open>open Simpdata\<close>
wenzelm@42455
  1154
wenzelm@60758
  1155
setup \<open>
wenzelm@58826
  1156
  map_theory_simpset (put_simpset HOL_basic_ss) #>
wenzelm@58826
  1157
  Simplifier.method_setup Splitter.split_modifiers
wenzelm@60758
  1158
\<close>
wenzelm@42455
  1159
wenzelm@60759
  1160
simproc_setup defined_Ex ("\<exists>x. P x") = \<open>fn _ => Quantifier1.rearrange_ex\<close>
wenzelm@60759
  1161
simproc_setup defined_All ("\<forall>x. P x") = \<open>fn _ => Quantifier1.rearrange_all\<close>
wenzelm@21671
  1162
wenzelm@61799
  1163
text \<open>Simproc for proving \<open>(y = x) \<equiv> False\<close> from premise \<open>\<not> (x = y)\<close>:\<close>
wenzelm@24035
  1164
wenzelm@60758
  1165
simproc_setup neq ("x = y") = \<open>fn _ =>
wenzelm@24035
  1166
let
wenzelm@24035
  1167
  val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
wenzelm@24035
  1168
  fun is_neq eq lhs rhs thm =
wenzelm@24035
  1169
    (case Thm.prop_of thm of
wenzelm@24035
  1170
      _ $ (Not $ (eq' $ l' $ r')) =>
wenzelm@24035
  1171
        Not = HOLogic.Not andalso eq' = eq andalso
wenzelm@24035
  1172
        r' aconv lhs andalso l' aconv rhs
wenzelm@24035
  1173
    | _ => false);
wenzelm@24035
  1174
  fun proc ss ct =
wenzelm@24035
  1175
    (case Thm.term_of ct of
wenzelm@24035
  1176
      eq $ lhs $ rhs =>
wenzelm@43597
  1177
        (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
wenzelm@24035
  1178
          SOME thm => SOME (thm RS neq_to_EQ_False)
wenzelm@24035
  1179
        | NONE => NONE)
wenzelm@24035
  1180
     | _ => NONE);
wenzelm@24035
  1181
in proc end;
wenzelm@60758
  1182
\<close>
wenzelm@24035
  1183
wenzelm@60758
  1184
simproc_setup let_simp ("Let x f") = \<open>
wenzelm@24035
  1185
let
haftmann@28741
  1186
  fun count_loose (Bound i) k = if i >= k then 1 else 0
haftmann@28741
  1187
    | count_loose (s $ t) k = count_loose s k + count_loose t k
haftmann@28741
  1188
    | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
haftmann@28741
  1189
    | count_loose _ _ = 0;
haftmann@28741
  1190
  fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
wenzelm@59628
  1191
    (case t of
wenzelm@59628
  1192
      Abs (_, _, t') => count_loose t' 0 <= 1
wenzelm@59628
  1193
    | _ => true);
wenzelm@59628
  1194
in
wenzelm@59628
  1195
  fn _ => fn ctxt => fn ct =>
wenzelm@59628
  1196
    if is_trivial_let (Thm.term_of ct)
wenzelm@59628
  1197
    then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
wenzelm@59628
  1198
    else
wenzelm@59628
  1199
      let (*Norbert Schirmer's case*)
wenzelm@59628
  1200
        val t = Thm.term_of ct;
wenzelm@59628
  1201
        val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
wenzelm@59628
  1202
      in
wenzelm@59628
  1203
        Option.map (hd o Variable.export ctxt' ctxt o single)
wenzelm@59628
  1204
          (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
wenzelm@59628
  1205
            if is_Free x orelse is_Bound x orelse is_Const x
wenzelm@59628
  1206
            then SOME @{thm Let_def}
wenzelm@59628
  1207
            else
wenzelm@59628
  1208
              let
wenzelm@59628
  1209
                val n = case f of (Abs (x, _, _)) => x | _ => "x";
wenzelm@59628
  1210
                val cx = Thm.cterm_of ctxt x;
wenzelm@59628
  1211
                val xT = Thm.typ_of_cterm cx;
wenzelm@59628
  1212
                val cf = Thm.cterm_of ctxt f;
wenzelm@59628
  1213
                val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
wenzelm@59628
  1214
                val (_ $ _ $ g) = Thm.prop_of fx_g;
wenzelm@59628
  1215
                val g' = abstract_over (x, g);
wenzelm@59628
  1216
                val abs_g'= Abs (n, xT, g');
wenzelm@59628
  1217
              in
wenzelm@59628
  1218
                if g aconv g' then
wenzelm@59628
  1219
                  let
wenzelm@59628
  1220
                    val rl =
wenzelm@60781
  1221
                      infer_instantiate ctxt [(("f", 0), cf), (("x", 0), cx)] @{thm Let_unfold};
wenzelm@59628
  1222
                  in SOME (rl OF [fx_g]) end
wenzelm@59628
  1223
                else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
wenzelm@59628
  1224
                then NONE (*avoid identity conversion*)
wenzelm@59628
  1225
                else
wenzelm@59628
  1226
                  let
wenzelm@59628
  1227
                    val g'x = abs_g' $ x;
wenzelm@59628
  1228
                    val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
wenzelm@59628
  1229
                    val rl =
wenzelm@60781
  1230
                      @{thm Let_folded} |> infer_instantiate ctxt
wenzelm@60781
  1231
                        [(("f", 0), Thm.cterm_of ctxt f),
wenzelm@60781
  1232
                         (("x", 0), cx),
wenzelm@60781
  1233
                         (("g", 0), Thm.cterm_of ctxt abs_g')];
wenzelm@59628
  1234
                  in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
wenzelm@59628
  1235
              end
wenzelm@59628
  1236
          | _ => NONE)
wenzelm@59628
  1237
      end
wenzelm@60758
  1238
end\<close>
wenzelm@24035
  1239
haftmann@21151
  1240
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@21151
  1241
proof
wenzelm@23389
  1242
  assume "True \<Longrightarrow> PROP P"
wenzelm@23389
  1243
  from this [OF TrueI] show "PROP P" .
haftmann@21151
  1244
next
haftmann@21151
  1245
  assume "PROP P"
wenzelm@23389
  1246
  then show "PROP P" .
haftmann@21151
  1247
qed
haftmann@21151
  1248
nipkow@59864
  1249
lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
wenzelm@61169
  1250
  by standard (intro TrueI)
nipkow@59864
  1251
nipkow@59864
  1252
lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
wenzelm@61169
  1253
  by standard simp_all
nipkow@59864
  1254
nipkow@60183
  1255
(* This is not made a simp rule because it does not improve any proofs
nipkow@60183
  1256
   but slows some AFP entries down by 5% (cpu time). May 2015 *)
nipkow@60169
  1257
lemma implies_False_swap: "NO_MATCH (Trueprop False) P \<Longrightarrow>
nipkow@60169
  1258
  (False \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> False \<Longrightarrow> PROP Q)"
nipkow@60169
  1259
by(rule swap_prems_eq)
nipkow@60169
  1260
haftmann@21151
  1261
lemma ex_simps:
wenzelm@60759
  1262
  "\<And>P Q. (\<exists>x. P x \<and> Q)   = ((\<exists>x. P x) \<and> Q)"
wenzelm@60759
  1263
  "\<And>P Q. (\<exists>x. P \<and> Q x)   = (P \<and> (\<exists>x. Q x))"
wenzelm@60759
  1264
  "\<And>P Q. (\<exists>x. P x \<or> Q)   = ((\<exists>x. P x) \<or> Q)"
wenzelm@60759
  1265
  "\<And>P Q. (\<exists>x. P \<or> Q x)   = (P \<or> (\<exists>x. Q x))"
wenzelm@60759
  1266
  "\<And>P Q. (\<exists>x. P x \<longrightarrow> Q) = ((\<forall>x. P x) \<longrightarrow> Q)"
wenzelm@60759
  1267
  "\<And>P Q. (\<exists>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<exists>x. Q x))"
wenzelm@61799
  1268
  \<comment> \<open>Miniscoping: pushing in existential quantifiers.\<close>
haftmann@21151
  1269
  by (iprover | blast)+
haftmann@21151
  1270
haftmann@21151
  1271
lemma all_simps:
wenzelm@60759
  1272
  "\<And>P Q. (\<forall>x. P x \<and> Q)   = ((\<forall>x. P x) \<and> Q)"
wenzelm@60759
  1273
  "\<And>P Q. (\<forall>x. P \<and> Q x)   = (P \<and> (\<forall>x. Q x))"
wenzelm@60759
  1274
  "\<And>P Q. (\<forall>x. P x \<or> Q)   = ((\<forall>x. P x) \<or> Q)"
wenzelm@60759
  1275
  "\<And>P Q. (\<forall>x. P \<or> Q x)   = (P \<or> (\<forall>x. Q x))"
wenzelm@60759
  1276
  "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q) = ((\<exists>x. P x) \<longrightarrow> Q)"
wenzelm@60759
  1277
  "\<And>P Q. (\<forall>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<forall>x. Q x))"
wenzelm@61799
  1278
  \<comment> \<open>Miniscoping: pushing in universal quantifiers.\<close>
haftmann@21151
  1279
  by (iprover | blast)+
paulson@15481
  1280
wenzelm@21671
  1281
lemmas [simp] =
wenzelm@21671
  1282
  triv_forall_equality (*prunes params*)
nipkow@60143
  1283
  True_implies_equals implies_True_equals (*prune True in asms*)
nipkow@60183
  1284
  False_implies_equals (*prune False in asms*)
wenzelm@21671
  1285
  if_True
wenzelm@21671
  1286
  if_False
wenzelm@21671
  1287
  if_cancel
wenzelm@21671
  1288
  if_eq_cancel
wenzelm@21671
  1289
  imp_disjL
haftmann@20973
  1290
  (*In general it seems wrong to add distributive laws by default: they
haftmann@20973
  1291
    might cause exponential blow-up.  But imp_disjL has been in for a while
haftmann@20973
  1292
    and cannot be removed without affecting existing proofs.  Moreover,
wenzelm@60759
  1293
    rewriting by "(P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" might be justified on the
haftmann@20973
  1294
    grounds that it allows simplification of R in the two cases.*)
wenzelm@21671
  1295
  conj_assoc
wenzelm@21671
  1296
  disj_assoc
wenzelm@21671
  1297
  de_Morgan_conj
wenzelm@21671
  1298
  de_Morgan_disj
wenzelm@21671
  1299
  imp_disj1
wenzelm@21671
  1300
  imp_disj2
wenzelm@21671
  1301
  not_imp
wenzelm@21671
  1302
  disj_not1
wenzelm@21671
  1303
  not_all
wenzelm@21671
  1304
  not_ex
wenzelm@21671
  1305
  cases_simp
wenzelm@21671
  1306
  the_eq_trivial
wenzelm@21671
  1307
  the_sym_eq_trivial
wenzelm@21671
  1308
  ex_simps
wenzelm@21671
  1309
  all_simps
wenzelm@21671
  1310
  simp_thms
wenzelm@21671
  1311
wenzelm@21671
  1312
lemmas [cong] = imp_cong simp_implies_cong
nipkow@62390
  1313
lemmas [split] = if_split
haftmann@20973
  1314
wenzelm@60758
  1315
ML \<open>val HOL_ss = simpset_of @{context}\<close>
haftmann@20973
  1316
wenzelm@60761
  1317
text \<open>Simplifies @{term x} assuming @{prop c} and @{term y} assuming @{prop "\<not> c"}\<close>
haftmann@20944
  1318
lemma if_cong:
haftmann@20944
  1319
  assumes "b = c"
haftmann@20944
  1320
      and "c \<Longrightarrow> x = u"
haftmann@20944
  1321
      and "\<not> c \<Longrightarrow> y = v"
haftmann@20944
  1322
  shows "(if b then x else y) = (if c then u else v)"
haftmann@38525
  1323
  using assms by simp
haftmann@20944
  1324
wenzelm@60758
  1325
text \<open>Prevents simplification of x and y:
wenzelm@60758
  1326
  faster and allows the execution of functional programs.\<close>
haftmann@20944
  1327
lemma if_weak_cong [cong]:
haftmann@20944
  1328
  assumes "b = c"
haftmann@20944
  1329
  shows "(if b then x else y) = (if c then x else y)"
wenzelm@23553
  1330
  using assms by (rule arg_cong)
haftmann@20944
  1331
wenzelm@60758
  1332
text \<open>Prevents simplification of t: much faster\<close>
haftmann@20944
  1333
lemma let_weak_cong:
haftmann@20944
  1334
  assumes "a = b"
haftmann@20944
  1335
  shows "(let x = a in t x) = (let x = b in t x)"
wenzelm@23553
  1336
  using assms by (rule arg_cong)
haftmann@20944
  1337
wenzelm@60758
  1338
text \<open>To tidy up the result of a simproc.  Only the RHS will be simplified.\<close>
haftmann@20944
  1339
lemma eq_cong2:
haftmann@20944
  1340
  assumes "u = u'"
haftmann@20944
  1341
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
wenzelm@23553
  1342
  using assms by simp
haftmann@20944
  1343
haftmann@20944
  1344
lemma if_distrib:
haftmann@20944
  1345
  "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944
  1346
  by simp
haftmann@20944
  1347
wenzelm@60758
  1348
text\<open>As a simplification rule, it replaces all function equalities by
wenzelm@60758
  1349
  first-order equalities.\<close>
haftmann@44277
  1350
lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
haftmann@44277
  1351
  by auto
haftmann@44277
  1352
wenzelm@17459
  1353
wenzelm@60758
  1354
subsubsection \<open>Generic cases and induction\<close>
wenzelm@17459
  1355
wenzelm@60758
  1356
text \<open>Rule projections:\<close>
wenzelm@60758
  1357
ML \<open>
wenzelm@32172
  1358
structure Project_Rule = Project_Rule
wenzelm@25388
  1359
(
wenzelm@27126
  1360
  val conjunct1 = @{thm conjunct1}
wenzelm@27126
  1361
  val conjunct2 = @{thm conjunct2}
wenzelm@27126
  1362
  val mp = @{thm mp}
wenzelm@59929
  1363
);
wenzelm@60758
  1364
\<close>
wenzelm@17459
  1365
wenzelm@59940
  1366
context
wenzelm@59940
  1367
begin
wenzelm@59940
  1368
wenzelm@59990
  1369
qualified definition "induct_forall P \<equiv> \<forall>x. P x"
wenzelm@59990
  1370
qualified definition "induct_implies A B \<equiv> A \<longrightarrow> B"
wenzelm@59990
  1371
qualified definition "induct_equal x y \<equiv> x = y"
wenzelm@59990
  1372
qualified definition "induct_conj A B \<equiv> A \<and> B"
wenzelm@59990
  1373
qualified definition "induct_true \<equiv> True"
wenzelm@59990
  1374
qualified definition "induct_false \<equiv> False"
haftmann@35416
  1375
wenzelm@59929
  1376
lemma induct_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457
  1377
  by (unfold atomize_all induct_forall_def)
wenzelm@11824
  1378
wenzelm@59929
  1379
lemma induct_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (induct_implies A B)"
wenzelm@18457
  1380
  by (unfold atomize_imp induct_implies_def)
wenzelm@11824
  1381
wenzelm@59929
  1382
lemma induct_equal_eq: "(x \<equiv> y) \<equiv> Trueprop (induct_equal x y)"
wenzelm@18457
  1383
  by (unfold atomize_eq induct_equal_def)
wenzelm@18457
  1384
wenzelm@59929
  1385
lemma induct_conj_eq: "(A &&& B) \<equiv> Trueprop (induct_conj A B)"
wenzelm@18457
  1386
  by (unfold atomize_conj induct_conj_def)
wenzelm@18457
  1387
berghofe@34908
  1388
lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
berghofe@34908
  1389
lemmas induct_atomize = induct_atomize' induct_equal_eq
wenzelm@45607
  1390
lemmas induct_rulify' [symmetric] = induct_atomize'
wenzelm@45607
  1391
lemmas induct_rulify [symmetric] = induct_atomize
wenzelm@18457
  1392
lemmas induct_rulify_fallback =
wenzelm@18457
  1393
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
berghofe@34908
  1394
  induct_true_def induct_false_def
wenzelm@18457
  1395
wenzelm@11989
  1396
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
  1397
    induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589
  1398
  by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824
  1399
wenzelm@11989
  1400
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
  1401
    induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589
  1402
  by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989
  1403
wenzelm@59929
  1404
lemma induct_conj_curry: "(induct_conj A B \<Longrightarrow> PROP C) \<equiv> (A \<Longrightarrow> B \<Longrightarrow> PROP C)"
berghofe@13598
  1405
proof
wenzelm@59929
  1406
  assume r: "induct_conj A B \<Longrightarrow> PROP C"
wenzelm@59929
  1407
  assume ab: A B
wenzelm@59929
  1408
  show "PROP C" by (rule r) (simp add: induct_conj_def ab)
berghofe@13598
  1409
next
wenzelm@59929
  1410
  assume r: "A \<Longrightarrow> B \<Longrightarrow> PROP C"
wenzelm@59929
  1411
  assume ab: "induct_conj A B"
wenzelm@59929
  1412
  show "PROP C" by (rule r) (simp_all add: ab [unfolded induct_conj_def])
berghofe@13598
  1413
qed
wenzelm@11824
  1414
wenzelm@11989
  1415
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
  1416
berghofe@34908
  1417
lemma induct_trueI: "induct_true"
berghofe@34908
  1418
  by (simp add: induct_true_def)
wenzelm@11824
  1419
wenzelm@60758
  1420
text \<open>Method setup.\<close>
wenzelm@11824
  1421
wenzelm@58826
  1422
ML_file "~~/src/Tools/induct.ML"
wenzelm@60758
  1423
ML \<open>
wenzelm@32171
  1424
structure Induct = Induct
wenzelm@27126
  1425
(
wenzelm@27126
  1426
  val cases_default = @{thm case_split}
wenzelm@27126
  1427
  val atomize = @{thms induct_atomize}
berghofe@34908
  1428
  val rulify = @{thms induct_rulify'}
wenzelm@27126
  1429
  val rulify_fallback = @{thms induct_rulify_fallback}
berghofe@34988
  1430
  val equal_def = @{thm induct_equal_def}
berghofe@34908
  1431
  fun dest_def (Const (@{const_name induct_equal}, _) $ t $ u) = SOME (t, u)
berghofe@34908
  1432
    | dest_def _ = NONE
wenzelm@58957
  1433
  fun trivial_tac ctxt = match_tac ctxt @{thms induct_trueI}
wenzelm@27126
  1434
)
wenzelm@60758
  1435
\<close>
wenzelm@11824
  1436
wenzelm@48891
  1437
ML_file "~~/src/Tools/induction.ML"
nipkow@45014
  1438
wenzelm@60758
  1439
declaration \<open>
wenzelm@59940
  1440
  fn _ => Induct.map_simpset (fn ss => ss
berghofe@34908
  1441
    addsimprocs
wenzelm@61144
  1442
      [Simplifier.make_simproc @{context} "swap_induct_false"
wenzelm@61144
  1443
        {lhss = [@{term "induct_false \<Longrightarrow> PROP P \<Longrightarrow> PROP Q"}],
wenzelm@61144
  1444
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1445
          (case Thm.term_of ct of
wenzelm@61144
  1446
            _ $ (P as _ $ @{const induct_false}) $ (_ $ Q $ _) =>
wenzelm@61144
  1447
              if P <> Q then SOME Drule.swap_prems_eq else NONE
wenzelm@62913
  1448
          | _ => NONE)},
wenzelm@61144
  1449
       Simplifier.make_simproc @{context} "induct_equal_conj_curry"
wenzelm@61144
  1450
        {lhss = [@{term "induct_conj P Q \<Longrightarrow> PROP R"}],
wenzelm@61144
  1451
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1452
          (case Thm.term_of ct of
wenzelm@61144
  1453
            _ $ (_ $ P) $ _ =>
wenzelm@61144
  1454
              let
wenzelm@61144
  1455
                fun is_conj (@{const induct_conj} $ P $ Q) =
wenzelm@61144
  1456
                      is_conj P andalso is_conj Q
wenzelm@61144
  1457
                  | is_conj (Const (@{const_name induct_equal}, _) $ _ $ _) = true
wenzelm@61144
  1458
                  | is_conj @{const induct_true} = true
wenzelm@61144
  1459
                  | is_conj @{const induct_false} = true
wenzelm@61144
  1460
                  | is_conj _ = false
wenzelm@61144
  1461
              in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
wenzelm@62913
  1462
            | _ => NONE)}]
wenzelm@54742
  1463
    |> Simplifier.set_mksimps (fn ctxt =>
wenzelm@54742
  1464
        Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
wenzelm@59940
  1465
        map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback}))))
wenzelm@60758
  1466
\<close>
berghofe@34908
  1467
wenzelm@60758
  1468
text \<open>Pre-simplification of induction and cases rules\<close>
berghofe@34908
  1469
wenzelm@59929
  1470
lemma [induct_simp]: "(\<And>x. induct_equal x t \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908
  1471
  unfolding induct_equal_def
berghofe@34908
  1472
proof
wenzelm@59929
  1473
  assume r: "\<And>x. x = t \<Longrightarrow> PROP P x"
wenzelm@59929
  1474
  show "PROP P t" by (rule r [OF refl])
berghofe@34908
  1475
next
wenzelm@59929
  1476
  fix x
wenzelm@59929
  1477
  assume "PROP P t" "x = t"
berghofe@34908
  1478
  then show "PROP P x" by simp
berghofe@34908
  1479
qed
berghofe@34908
  1480
wenzelm@59929
  1481
lemma [induct_simp]: "(\<And>x. induct_equal t x \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908
  1482
  unfolding induct_equal_def
berghofe@34908
  1483
proof
wenzelm@59929
  1484
  assume r: "\<And>x. t = x \<Longrightarrow> PROP P x"
wenzelm@59929
  1485
  show "PROP P t" by (rule r [OF refl])
berghofe@34908
  1486
next
wenzelm@59929
  1487
  fix x
wenzelm@59929
  1488
  assume "PROP P t" "t = x"
berghofe@34908
  1489
  then show "PROP P x" by simp
berghofe@34908
  1490
qed
berghofe@34908
  1491
wenzelm@59929
  1492
lemma [induct_simp]: "(induct_false \<Longrightarrow> P) \<equiv> Trueprop induct_true"
berghofe@34908
  1493
  unfolding induct_false_def induct_true_def
berghofe@34908
  1494
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1495
wenzelm@59929
  1496
lemma [induct_simp]: "(induct_true \<Longrightarrow> PROP P) \<equiv> PROP P"
berghofe@34908
  1497
  unfolding induct_true_def
berghofe@34908
  1498
proof
wenzelm@59929
  1499
  assume "True \<Longrightarrow> PROP P"
wenzelm@59929
  1500
  then show "PROP P" using TrueI .
berghofe@34908
  1501
next
berghofe@34908
  1502
  assume "PROP P"
berghofe@34908
  1503
  then show "PROP P" .
berghofe@34908
  1504
qed
berghofe@34908
  1505
wenzelm@59929
  1506
lemma [induct_simp]: "(PROP P \<Longrightarrow> induct_true) \<equiv> Trueprop induct_true"
berghofe@34908
  1507
  unfolding induct_true_def
berghofe@34908
  1508
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1509
wenzelm@62958
  1510
lemma [induct_simp]: "(\<And>x::'a::{}. induct_true) \<equiv> Trueprop induct_true"
berghofe@34908
  1511
  unfolding induct_true_def
berghofe@34908
  1512
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1513
wenzelm@59929
  1514
lemma [induct_simp]: "induct_implies induct_true P \<equiv> P"
berghofe@34908
  1515
  by (simp add: induct_implies_def induct_true_def)
berghofe@34908
  1516
wenzelm@59929
  1517
lemma [induct_simp]: "x = x \<longleftrightarrow> True"
berghofe@34908
  1518
  by (rule simp_thms)
berghofe@34908
  1519
wenzelm@59940
  1520
end
wenzelm@18457
  1521
wenzelm@48891
  1522
ML_file "~~/src/Tools/induct_tacs.ML"
wenzelm@27126
  1523
haftmann@20944
  1524
wenzelm@60758
  1525
subsubsection \<open>Coherent logic\<close>
berghofe@28325
  1526
wenzelm@55632
  1527
ML_file "~~/src/Tools/coherent.ML"
wenzelm@60758
  1528
ML \<open>
wenzelm@32734
  1529
structure Coherent = Coherent
berghofe@28325
  1530
(
wenzelm@55632
  1531
  val atomize_elimL = @{thm atomize_elimL};
wenzelm@55632
  1532
  val atomize_exL = @{thm atomize_exL};
wenzelm@55632
  1533
  val atomize_conjL = @{thm atomize_conjL};
wenzelm@55632
  1534
  val atomize_disjL = @{thm atomize_disjL};
wenzelm@55632
  1535
  val operator_names = [@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}];
berghofe@28325
  1536
);
wenzelm@60758
  1537
\<close>
berghofe@28325
  1538
berghofe@28325
  1539
wenzelm@60758
  1540
subsubsection \<open>Reorienting equalities\<close>
huffman@31024
  1541
wenzelm@60758
  1542
ML \<open>
huffman@31024
  1543
signature REORIENT_PROC =
huffman@31024
  1544
sig
huffman@31024
  1545
  val add : (term -> bool) -> theory -> theory
wenzelm@51717
  1546
  val proc : morphism -> Proof.context -> cterm -> thm option
huffman@31024
  1547
end;
huffman@31024
  1548
wenzelm@33523
  1549
structure Reorient_Proc : REORIENT_PROC =
huffman@31024
  1550
struct
wenzelm@33523
  1551
  structure Data = Theory_Data
huffman@31024
  1552
  (
wenzelm@33523
  1553
    type T = ((term -> bool) * stamp) list;
wenzelm@33523
  1554
    val empty = [];
huffman@31024
  1555
    val extend = I;
wenzelm@33523
  1556
    fun merge data : T = Library.merge (eq_snd op =) data;
wenzelm@33523
  1557
  );
wenzelm@33523
  1558
  fun add m = Data.map (cons (m, stamp ()));
wenzelm@33523
  1559
  fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
huffman@31024
  1560
huffman@31024
  1561
  val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
wenzelm@51717
  1562
  fun proc phi ctxt ct =
huffman@31024
  1563
    let
wenzelm@42361
  1564
      val thy = Proof_Context.theory_of ctxt;
huffman@31024
  1565
    in
huffman@31024
  1566
      case Thm.term_of ct of
wenzelm@33523
  1567
        (_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
huffman@31024
  1568
      | _ => NONE
huffman@31024
  1569
    end;
huffman@31024
  1570
end;
wenzelm@60758
  1571
\<close>
huffman@31024
  1572
huffman@31024
  1573
wenzelm@60758
  1574
subsection \<open>Other simple lemmas and lemma duplicates\<close>
haftmann@20944
  1575
wenzelm@60759
  1576
lemma ex1_eq [iff]: "\<exists>!x. x = t" "\<exists>!x. t = x"
haftmann@20944
  1577
  by blast+
haftmann@20944
  1578
wenzelm@60759
  1579
lemma choice_eq: "(\<forall>x. \<exists>!y. P x y) = (\<exists>!f. \<forall>x. P x (f x))"
haftmann@20944
  1580
  apply (rule iffI)
wenzelm@60759
  1581
  apply (rule_tac a = "\<lambda>x. THE y. P x y" in ex1I)
haftmann@20944
  1582
  apply (fast dest!: theI')
huffman@44921
  1583
  apply (fast intro: the1_equality [symmetric])
haftmann@20944
  1584
  apply (erule ex1E)
haftmann@20944
  1585
  apply (rule allI)
haftmann@20944
  1586
  apply (rule ex1I)
haftmann@20944
  1587
  apply (erule spec)
wenzelm@60759
  1588
  apply (erule_tac x = "\<lambda>z. if z = x then y else f z" in allE)
haftmann@20944
  1589
  apply (erule impE)
haftmann@20944
  1590
  apply (rule allI)
wenzelm@27126
  1591
  apply (case_tac "xa = x")
haftmann@20944
  1592
  apply (drule_tac [3] x = x in fun_cong, simp_all)
haftmann@20944
  1593
  done
haftmann@20944
  1594
haftmann@22218
  1595
lemmas eq_sym_conv = eq_commute
haftmann@22218
  1596
chaieb@23037
  1597
lemma nnf_simps:
wenzelm@58826
  1598
  "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
wenzelm@58826
  1599
  "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
chaieb@23037
  1600
  "(\<not> \<not>(P)) = P"
chaieb@23037
  1601
by blast+
chaieb@23037
  1602
wenzelm@60758
  1603
subsection \<open>Basic ML bindings\<close>
wenzelm@21671
  1604
wenzelm@60758
  1605
ML \<open>
wenzelm@22129
  1606
val FalseE = @{thm FalseE}
wenzelm@22129
  1607
val Let_def = @{thm Let_def}
wenzelm@22129
  1608
val TrueI = @{thm TrueI}
wenzelm@22129
  1609
val allE = @{thm allE}
wenzelm@22129
  1610
val allI = @{thm allI}
wenzelm@22129
  1611
val all_dupE = @{thm all_dupE}
wenzelm@22129
  1612
val arg_cong = @{thm arg_cong}
wenzelm@22129
  1613
val box_equals = @{thm box_equals}
wenzelm@22129
  1614
val ccontr = @{thm ccontr}
wenzelm@22129
  1615
val classical = @{thm classical}
wenzelm@22129
  1616
val conjE = @{thm conjE}
wenzelm@22129
  1617
val conjI = @{thm conjI}
wenzelm@22129
  1618
val conjunct1 = @{thm conjunct1}
wenzelm@22129
  1619
val conjunct2 = @{thm conjunct2}
wenzelm@22129
  1620
val disjCI = @{thm disjCI}
wenzelm@22129
  1621
val disjE = @{thm disjE}
wenzelm@22129
  1622
val disjI1 = @{thm disjI1}
wenzelm@22129
  1623
val disjI2 = @{thm disjI2}
wenzelm@22129
  1624
val eq_reflection = @{thm eq_reflection}
wenzelm@22129
  1625
val ex1E = @{thm ex1E}
wenzelm@22129
  1626
val ex1I = @{thm ex1I}
wenzelm@22129
  1627
val ex1_implies_ex = @{thm ex1_implies_ex}
wenzelm@22129
  1628
val exE = @{thm exE}
wenzelm@22129
  1629
val exI = @{thm exI}
wenzelm@22129
  1630
val excluded_middle = @{thm excluded_middle}
wenzelm@22129
  1631
val ext = @{thm ext}
wenzelm@22129
  1632
val fun_cong = @{thm fun_cong}
wenzelm@22129
  1633
val iffD1 = @{thm iffD1}
wenzelm@22129
  1634
val iffD2 = @{thm iffD2}
wenzelm@22129
  1635
val iffI = @{thm iffI}
wenzelm@22129
  1636
val impE = @{thm impE}
wenzelm@22129
  1637
val impI = @{thm impI}
wenzelm@22129
  1638
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22129
  1639
val mp = @{thm mp}
wenzelm@22129
  1640
val notE = @{thm notE}
wenzelm@22129
  1641
val notI = @{thm notI}
wenzelm@22129
  1642
val not_all = @{thm not_all}
wenzelm@22129
  1643
val not_ex = @{thm not_ex}
wenzelm@22129
  1644
val not_iff = @{thm not_iff}
wenzelm@22129
  1645
val not_not = @{thm not_not}
wenzelm@22129
  1646
val not_sym = @{thm not_sym}
wenzelm@22129
  1647
val refl = @{thm refl}
wenzelm@22129
  1648
val rev_mp = @{thm rev_mp}
wenzelm@22129
  1649
val spec = @{thm spec}
wenzelm@22129
  1650
val ssubst = @{thm ssubst}
wenzelm@22129
  1651
val subst = @{thm subst}
wenzelm@22129
  1652
val sym = @{thm sym}
wenzelm@22129
  1653
val trans = @{thm trans}
wenzelm@60758
  1654
\<close>
wenzelm@21671
  1655
wenzelm@55239
  1656
ML_file "Tools/cnf.ML"
wenzelm@55239
  1657
wenzelm@21671
  1658
wenzelm@61799
  1659
section \<open>\<open>NO_MATCH\<close> simproc\<close>
hoelzl@58775
  1660
wenzelm@60758
  1661
text \<open>
hoelzl@58775
  1662
 The simplification procedure can be used to avoid simplification of terms of a certain form
wenzelm@60758
  1663
\<close>
hoelzl@58775
  1664
hoelzl@59779
  1665
definition NO_MATCH :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where "NO_MATCH pat val \<equiv> True"
hoelzl@58830
  1666
hoelzl@59779
  1667
lemma NO_MATCH_cong[cong]: "NO_MATCH pat val = NO_MATCH pat val" by (rule refl)
hoelzl@58775
  1668
hoelzl@58830
  1669
declare [[coercion_args NO_MATCH - -]]
hoelzl@58830
  1670
wenzelm@60758
  1671
simproc_setup NO_MATCH ("NO_MATCH pat val") = \<open>fn _ => fn ctxt => fn ct =>
hoelzl@58775
  1672
  let
hoelzl@58775
  1673
    val thy = Proof_Context.theory_of ctxt
hoelzl@58775
  1674
    val dest_binop = Term.dest_comb #> apfst (Term.dest_comb #> snd)
hoelzl@58775
  1675
    val m = Pattern.matches thy (dest_binop (Thm.term_of ct))
hoelzl@58775
  1676
  in if m then NONE else SOME @{thm NO_MATCH_def} end
wenzelm@60758
  1677
\<close>
hoelzl@58775
  1678
wenzelm@60758
  1679
text \<open>
hoelzl@59779
  1680
  This setup ensures that a rewrite rule of the form @{term "NO_MATCH pat val \<Longrightarrow> t"}
hoelzl@58775
  1681
  is only applied, if the pattern @{term pat} does not match the value @{term val}.
wenzelm@60758
  1682
\<close>
hoelzl@58775
  1683
hoelzl@58775
  1684
wenzelm@61222
  1685
text\<open>Tagging a premise of a simp rule with ASSUMPTION forces the simplifier
wenzelm@61222
  1686
not to simplify the argument and to solve it by an assumption.\<close>
nipkow@61202
  1687
nipkow@61202
  1688
definition ASSUMPTION :: "bool \<Rightarrow> bool" where
nipkow@61202
  1689
"ASSUMPTION A \<equiv> A"
nipkow@61202
  1690
nipkow@61202
  1691
lemma ASSUMPTION_cong[cong]: "ASSUMPTION A = ASSUMPTION A"
nipkow@61202
  1692
by (rule refl)
nipkow@61202
  1693
nipkow@61202
  1694
lemma ASSUMPTION_I: "A \<Longrightarrow> ASSUMPTION A"
nipkow@61202
  1695
by(simp add: ASSUMPTION_def)
nipkow@61202
  1696
nipkow@61202
  1697
lemma ASSUMPTION_D: "ASSUMPTION A \<Longrightarrow> A"
nipkow@61202
  1698
by(simp add: ASSUMPTION_def)
nipkow@61202
  1699
wenzelm@61222
  1700
setup \<open>
nipkow@61202
  1701
let
nipkow@61202
  1702
  val asm_sol = mk_solver "ASSUMPTION" (fn ctxt =>
nipkow@61202
  1703
    resolve_tac ctxt [@{thm ASSUMPTION_I}] THEN'
nipkow@61202
  1704
    resolve_tac ctxt (Simplifier.prems_of ctxt))
nipkow@61202
  1705
in
nipkow@61202
  1706
  map_theory_simpset (fn ctxt => Simplifier.addSolver (ctxt,asm_sol))
nipkow@61202
  1707
end
wenzelm@61222
  1708
\<close>
nipkow@61202
  1709
nipkow@61202
  1710
wenzelm@60758
  1711
subsection \<open>Code generator setup\<close>
haftmann@30929
  1712
wenzelm@60758
  1713
subsubsection \<open>Generic code generator preprocessor setup\<close>
haftmann@31151
  1714
haftmann@53146
  1715
lemma conj_left_cong:
haftmann@53146
  1716
  "P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
haftmann@53146
  1717
  by (fact arg_cong)
haftmann@53146
  1718
haftmann@53146
  1719
lemma disj_left_cong:
haftmann@53146
  1720
  "P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
haftmann@53146
  1721
  by (fact arg_cong)
haftmann@53146
  1722
wenzelm@60758
  1723
setup \<open>
wenzelm@58826
  1724
  Code_Preproc.map_pre (put_simpset HOL_basic_ss) #>
wenzelm@58826
  1725
  Code_Preproc.map_post (put_simpset HOL_basic_ss) #>
wenzelm@58826
  1726
  Code_Simp.map_ss (put_simpset HOL_basic_ss #>
wenzelm@58826
  1727
  Simplifier.add_cong @{thm conj_left_cong} #>
wenzelm@58826
  1728
  Simplifier.add_cong @{thm disj_left_cong})
wenzelm@60758
  1729
\<close>
haftmann@31151
  1730
haftmann@53146
  1731
wenzelm@60758
  1732
subsubsection \<open>Equality\<close>
haftmann@24844
  1733
haftmann@38857
  1734
class equal =
haftmann@38857
  1735
  fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@38857
  1736
  assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
haftmann@26513
  1737
begin
haftmann@26513
  1738
bulwahn@45231
  1739
lemma equal: "equal = (op =)"
haftmann@38857
  1740
  by (rule ext equal_eq)+
haftmann@28346
  1741
haftmann@38857
  1742
lemma equal_refl: "equal x x \<longleftrightarrow> True"
haftmann@38857
  1743
  unfolding equal by rule+
haftmann@28346
  1744
haftmann@38857
  1745
lemma eq_equal: "(op =) \<equiv> equal"
haftmann@38857
  1746
  by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
haftmann@30929
  1747
haftmann@26513
  1748
end
haftmann@26513
  1749
haftmann@38857
  1750
declare eq_equal [symmetric, code_post]
haftmann@38857
  1751
declare eq_equal [code]
haftmann@30966
  1752
wenzelm@60758
  1753
setup \<open>
wenzelm@51717
  1754
  Code_Preproc.map_pre (fn ctxt =>
wenzelm@61144
  1755
    ctxt addsimprocs
wenzelm@61144
  1756
      [Simplifier.make_simproc @{context} "equal"
wenzelm@61144
  1757
        {lhss = [@{term HOL.eq}],
wenzelm@61144
  1758
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1759
          (case Thm.term_of ct of
wenzelm@61144
  1760
            Const (_, Type (@{type_name fun}, [Type _, _])) => SOME @{thm eq_equal}
wenzelm@62913
  1761
          | _ => NONE)}])
wenzelm@60758
  1762
\<close>
haftmann@31151
  1763
haftmann@30966
  1764
wenzelm@60758
  1765
subsubsection \<open>Generic code generator foundation\<close>
haftmann@30929
  1766
wenzelm@60758
  1767
text \<open>Datatype @{typ bool}\<close>
haftmann@30929
  1768
haftmann@30929
  1769
code_datatype True False
haftmann@30929
  1770
haftmann@30929
  1771
lemma [code]:
haftmann@33185
  1772
  shows "False \<and> P \<longleftrightarrow> False"
haftmann@33185
  1773
    and "True \<and> P \<longleftrightarrow> P"
haftmann@33185
  1774
    and "P \<and> False \<longleftrightarrow> False"
haftmann@33185
  1775
    and "P \<and> True \<longleftrightarrow> P" by simp_all
haftmann@30929
  1776
haftmann@30929
  1777
lemma [code]:
haftmann@33185
  1778
  shows "False \<or> P \<longleftrightarrow> P"
haftmann@33185
  1779
    and "True \<or> P \<longleftrightarrow> True"
haftmann@33185
  1780
    and "P \<or> False \<longleftrightarrow> P"
haftmann@33185
  1781
    and "P \<or> True \<longleftrightarrow> True" by simp_all
haftmann@30929
  1782
haftmann@33185
  1783
lemma [code]:
haftmann@33185
  1784
  shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
haftmann@33185
  1785
    and "(True \<longrightarrow> P) \<longleftrightarrow> P"
haftmann@33185
  1786
    and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
haftmann@33185
  1787
    and "(P \<longrightarrow> True) \<longleftrightarrow> True" by simp_all
haftmann@30929
  1788
wenzelm@60758
  1789
text \<open>More about @{typ prop}\<close>
haftmann@39421
  1790
haftmann@39421
  1791
lemma [code nbe]:
wenzelm@58826
  1792
  shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
haftmann@39421
  1793
    and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
haftmann@39421
  1794
    and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)" by (auto intro!: equal_intr_rule)
haftmann@39421
  1795
haftmann@39421
  1796
lemma Trueprop_code [code]:
haftmann@39421
  1797
  "Trueprop True \<equiv> Code_Generator.holds"
haftmann@39421
  1798
  by (auto intro!: equal_intr_rule holds)
haftmann@39421
  1799
haftmann@39421
  1800
declare Trueprop_code [symmetric, code_post]
haftmann@39421
  1801
wenzelm@60758
  1802
text \<open>Equality\<close>
haftmann@39421
  1803
haftmann@39421
  1804
declare simp_thms(6) [code nbe]
haftmann@39421
  1805
haftmann@38857
  1806
instantiation itself :: (type) equal
haftmann@31132
  1807
begin
haftmann@31132
  1808
haftmann@38857
  1809
definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where
haftmann@38857
  1810
  "equal_itself x y \<longleftrightarrow> x = y"
haftmann@31132
  1811
haftmann@31132
  1812
instance proof
haftmann@38857
  1813
qed (fact equal_itself_def)
haftmann@31132
  1814
haftmann@31132
  1815
end
haftmann@31132
  1816
haftmann@38857
  1817
lemma equal_itself_code [code]:
haftmann@38857
  1818
  "equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
haftmann@38857
  1819
  by (simp add: equal)
haftmann@31132
  1820
wenzelm@61076
  1821
setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::type \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
haftmann@31956
  1822
haftmann@38857
  1823
lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)" (is "?ofclass \<equiv> ?equal")
haftmann@31956
  1824
proof
haftmann@31956
  1825
  assume "PROP ?ofclass"
haftmann@38857
  1826
  show "PROP ?equal"
wenzelm@60758
  1827
    by (tactic \<open>ALLGOALS (resolve_tac @{context} [Thm.unconstrainT @{thm eq_equal}])\<close>)
wenzelm@60758
  1828
      (fact \<open>PROP ?ofclass\<close>)
haftmann@31956
  1829
next
haftmann@38857
  1830
  assume "PROP ?equal"
haftmann@31956
  1831
  show "PROP ?ofclass" proof
wenzelm@60758
  1832
  qed (simp add: \<open>PROP ?equal\<close>)
haftmann@31956
  1833
qed
haftmann@31956
  1834
wenzelm@61076
  1835
setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::equal \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
wenzelm@58826
  1836
wenzelm@60758
  1837
setup \<open>Nbe.add_const_alias @{thm equal_alias_cert}\<close>
haftmann@30929
  1838
wenzelm@60758
  1839
text \<open>Cases\<close>
haftmann@30929
  1840
haftmann@30929
  1841
lemma Let_case_cert:
haftmann@30929
  1842
  assumes "CASE \<equiv> (\<lambda>x. Let x f)"
haftmann@30929
  1843
  shows "CASE x \<equiv> f x"
haftmann@30929
  1844
  using assms by simp_all
haftmann@30929
  1845
wenzelm@60758
  1846
setup \<open>
wenzelm@58826
  1847
  Code.add_case @{thm Let_case_cert} #>
wenzelm@58826
  1848
  Code.add_undefined @{const_name undefined}
wenzelm@60758
  1849
\<close>
haftmann@30929
  1850
haftmann@54890
  1851
declare [[code abort: undefined]]
haftmann@30929
  1852
haftmann@38972
  1853
wenzelm@60758
  1854
subsubsection \<open>Generic code generator target languages\<close>
haftmann@30929
  1855
wenzelm@60758
  1856
text \<open>type @{typ bool}\<close>
haftmann@30929
  1857
haftmann@52435
  1858
code_printing
haftmann@52435
  1859
  type_constructor bool \<rightharpoonup>
haftmann@52435
  1860
    (SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
haftmann@52435
  1861
| constant True \<rightharpoonup>
haftmann@52435
  1862
    (SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
haftmann@52435
  1863
| constant False \<rightharpoonup>
wenzelm@58826
  1864
    (SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"
haftmann@34294
  1865
haftmann@30929
  1866
code_reserved SML
haftmann@52435
  1867
  bool true false
haftmann@30929
  1868
haftmann@30929
  1869
code_reserved OCaml
haftmann@52435
  1870
  bool
haftmann@30929
  1871
haftmann@34294
  1872
code_reserved Scala
haftmann@34294
  1873
  Boolean
haftmann@34294
  1874
haftmann@52435
  1875
code_printing
haftmann@52435
  1876
  constant Not \<rightharpoonup>
haftmann@52435
  1877
    (SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
haftmann@52435
  1878
| constant HOL.conj \<rightharpoonup>
haftmann@52435
  1879
    (SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
haftmann@52435
  1880
| constant HOL.disj \<rightharpoonup>
haftmann@52435
  1881
    (SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
haftmann@52435
  1882
| constant HOL.implies \<rightharpoonup>
haftmann@52435
  1883
    (SML) "!(if (_)/ then (_)/ else true)"
haftmann@52435
  1884
    and (OCaml) "!(if (_)/ then (_)/ else true)"
haftmann@52435
  1885
    and (Haskell) "!(if (_)/ then (_)/ else True)"
haftmann@52435
  1886
    and (Scala) "!(if ((_))/ (_)/ else true)"
haftmann@52435
  1887
| constant If \<rightharpoonup>
haftmann@52435
  1888
    (SML) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1889
    and (OCaml) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1890
    and (Haskell) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1891
    and (Scala) "!(if ((_))/ (_)/ else (_))"
haftmann@52435
  1892
haftmann@52435
  1893
code_reserved SML
haftmann@52435
  1894
  not
haftmann@52435
  1895
haftmann@52435
  1896
code_reserved OCaml
haftmann@52435
  1897
  not
haftmann@52435
  1898
haftmann@52435
  1899
code_identifier
haftmann@52435
  1900
  code_module Pure \<rightharpoonup>
haftmann@52435
  1901
    (SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL
haftmann@39026
  1902
wenzelm@60758
  1903
text \<open>using built-in Haskell equality\<close>
haftmann@30929
  1904
haftmann@52435
  1905
code_printing
haftmann@52435
  1906
  type_class equal \<rightharpoonup> (Haskell) "Eq"
haftmann@52435
  1907
| constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@52435
  1908
| constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@30929
  1909
wenzelm@60758
  1910
text \<open>undefined\<close>
haftmann@30929
  1911
haftmann@52435
  1912
code_printing
haftmann@52435
  1913
  constant undefined \<rightharpoonup>
haftmann@52435
  1914
    (SML) "!(raise/ Fail/ \"undefined\")"
haftmann@52435
  1915
    and (OCaml) "failwith/ \"undefined\""
haftmann@52435
  1916
    and (Haskell) "error/ \"undefined\""
haftmann@52435
  1917
    and (Scala) "!sys.error(\"undefined\")"
haftmann@52435
  1918
haftmann@30929
  1919
wenzelm@60758
  1920
subsubsection \<open>Evaluation and normalization by evaluation\<close>
haftmann@30929
  1921
wenzelm@60758
  1922
method_setup eval = \<open>
wenzelm@58826
  1923
  let
wenzelm@58826
  1924
    fun eval_tac ctxt =
wenzelm@58826
  1925
      let val conv = Code_Runtime.dynamic_holds_conv ctxt
wenzelm@58839
  1926
      in
wenzelm@58839
  1927
        CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 conv)) ctxt) THEN'
wenzelm@59498
  1928
        resolve_tac ctxt [TrueI]
wenzelm@58839
  1929
      end
wenzelm@58826
  1930
  in
wenzelm@58826
  1931
    Scan.succeed (SIMPLE_METHOD' o eval_tac)
wenzelm@58826
  1932
  end
wenzelm@60758
  1933
\<close> "solve goal by evaluation"
haftmann@30929
  1934
wenzelm@60758
  1935
method_setup normalization = \<open>
wenzelm@46190
  1936
  Scan.succeed (fn ctxt =>
wenzelm@46190
  1937
    SIMPLE_METHOD'
wenzelm@46190
  1938
      (CHANGED_PROP o
haftmann@55757
  1939
        (CONVERSION (Nbe.dynamic_conv ctxt)
wenzelm@59498
  1940
          THEN_ALL_NEW (TRY o resolve_tac ctxt [TrueI]))))
wenzelm@60758
  1941
\<close> "solve goal by normalization"
haftmann@30929
  1942
wenzelm@31902
  1943
wenzelm@60758
  1944
subsection \<open>Counterexample Search Units\<close>
haftmann@33084
  1945
wenzelm@60758
  1946
subsubsection \<open>Quickcheck\<close>
haftmann@30929
  1947
haftmann@33084
  1948
quickcheck_params [size = 5, iterations = 50]
haftmann@33084
  1949
haftmann@30929
  1950
wenzelm@60758
  1951
subsubsection \<open>Nitpick setup\<close>
blanchet@30309
  1952
wenzelm@59028
  1953
named_theorems nitpick_unfold "alternative definitions of constants as needed by Nitpick"
wenzelm@59028
  1954
  and nitpick_simp "equational specification of constants as needed by Nitpick"
wenzelm@59028
  1955
  and nitpick_psimp "partial equational specification of constants as needed by Nitpick"
wenzelm@59028
  1956
  and nitpick_choice_spec "choice specification of constants as needed by Nitpick"
wenzelm@30980
  1957
blanchet@41792
  1958
declare if_bool_eq_conj [nitpick_unfold, no_atp]
blanchet@41792
  1959
        if_bool_eq_disj [no_atp]
blanchet@41792
  1960
blanchet@29863
  1961
wenzelm@60758
  1962
subsection \<open>Preprocessing for the predicate compiler\<close>
haftmann@33084
  1963
wenzelm@59028
  1964
named_theorems code_pred_def "alternative definitions of constants for the Predicate Compiler"
wenzelm@59028
  1965
  and code_pred_inline "inlining definitions for the Predicate Compiler"
wenzelm@59028
  1966
  and code_pred_simp "simplification rules for the optimisations in the Predicate Compiler"
haftmann@33084
  1967
haftmann@33084
  1968
wenzelm@60758
  1969
subsection \<open>Legacy tactics and ML bindings\<close>
wenzelm@21671
  1970
wenzelm@60758
  1971
ML \<open>
wenzelm@58826
  1972
  (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
wenzelm@58826
  1973
  local
wenzelm@58826
  1974
    fun wrong_prem (Const (@{const_name All}, _) $ Abs (_, _, t)) = wrong_prem t
wenzelm@58826
  1975
      | wrong_prem (Bound _) = true
wenzelm@58826
  1976
      | wrong_prem _ = false;
wenzelm@58826
  1977
    val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
haftmann@61914
  1978
    fun smp i = funpow i (fn m => filter_right ([spec] RL m)) [mp];
wenzelm@58826
  1979
  in
wenzelm@59498
  1980
    fun smp_tac ctxt j = EVERY' [dresolve_tac ctxt (smp j), assume_tac ctxt];
wenzelm@58826
  1981
  end;
haftmann@22839
  1982
wenzelm@58826
  1983
  local
wenzelm@58826
  1984
    val nnf_ss =
wenzelm@58826
  1985
      simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms simp_thms nnf_simps});
wenzelm@58826
  1986
  in
wenzelm@58826
  1987
    fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
wenzelm@58826
  1988
  end
wenzelm@60758
  1989
\<close>
wenzelm@21671
  1990
haftmann@38866
  1991
hide_const (open) eq equal
haftmann@38866
  1992
kleing@14357
  1993
end