src/HOL/HOL.thy
author nipkow
Sat Jun 17 15:41:19 2017 +0200 (2017-06-17)
changeset 66109 e034a563ed7d
parent 63912 9f8325206465
child 66251 cd935b7cb3fb
permissions -rw-r--r--
added simp rules
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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"
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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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syntax (ASCII)
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  "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3EX! _./ _)" [0, 10] 10)
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syntax (input)
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  "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3?! _./ _)" [0, 10] 10)
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syntax "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<exists>!_./ _)" [0, 10] 10)
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translations "\<exists>!x. P" \<rightleftharpoons> "CONST Ex1 (\<lambda>x. P)"
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print_translation \<open>
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 [Syntax_Trans.preserve_binder_abs_tr' @{const_syntax Ex1} @{syntax_const "_Ex1"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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syntax
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  "_Not_Ex" :: "idts \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>_./ _)" [0, 10] 10)
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  "_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>!_./ _)" [0, 10] 10)
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translations
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  "\<nexists>x. P" \<rightleftharpoons> "\<not> (\<exists>x. P)"
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  "\<nexists>!x. P" \<rightleftharpoons> "\<not> (\<exists>!x. P)"
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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)
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notation (input)
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  All  (binder "! " 10) and
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  Ex  (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 "A \<equiv> B"
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  shows "A = B"
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  unfolding assms by (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"
wenzelm@21504
   317
  by (erule iffD2) (rule TrueI)
paulson@15411
   318
paulson@15411
   319
wenzelm@60758
   320
subsubsection \<open>Universal quantifier\<close>
paulson@15411
   321
wenzelm@63575
   322
lemma allI:
wenzelm@63575
   323
  assumes "\<And>x::'a. P x"
wenzelm@63575
   324
  shows "\<forall>x. P x"
wenzelm@21504
   325
  unfolding All_def by (iprover intro: ext eqTrueI assms)
paulson@15411
   326
wenzelm@60759
   327
lemma spec: "\<forall>x::'a. P x \<Longrightarrow> P x"
wenzelm@63575
   328
  apply (unfold All_def)
wenzelm@63575
   329
  apply (rule eqTrueE)
wenzelm@63575
   330
  apply (erule fun_cong)
wenzelm@63575
   331
  done
paulson@15411
   332
paulson@15411
   333
lemma allE:
wenzelm@60759
   334
  assumes major: "\<forall>x. P x"
wenzelm@60759
   335
    and minor: "P x \<Longrightarrow> R"
wenzelm@21504
   336
  shows R
wenzelm@21504
   337
  by (iprover intro: minor major [THEN spec])
paulson@15411
   338
paulson@15411
   339
lemma all_dupE:
wenzelm@60759
   340
  assumes major: "\<forall>x. P x"
wenzelm@60759
   341
    and minor: "\<lbrakk>P x; \<forall>x. P x\<rbrakk> \<Longrightarrow> R"
wenzelm@21504
   342
  shows R
wenzelm@21504
   343
  by (iprover intro: minor major major [THEN spec])
paulson@15411
   344
paulson@15411
   345
wenzelm@60758
   346
subsubsection \<open>False\<close>
wenzelm@21504
   347
wenzelm@60758
   348
text \<open>
wenzelm@61799
   349
  Depends upon \<open>spec\<close>; it is impossible to do propositional
wenzelm@21504
   350
  logic before quantifiers!
wenzelm@60758
   351
\<close>
paulson@15411
   352
wenzelm@60759
   353
lemma FalseE: "False \<Longrightarrow> P"
wenzelm@21504
   354
  apply (unfold False_def)
wenzelm@21504
   355
  apply (erule spec)
wenzelm@21504
   356
  done
paulson@15411
   357
wenzelm@60759
   358
lemma False_neq_True: "False = True \<Longrightarrow> P"
wenzelm@21504
   359
  by (erule eqTrueE [THEN FalseE])
paulson@15411
   360
paulson@15411
   361
wenzelm@60758
   362
subsubsection \<open>Negation\<close>
paulson@15411
   363
paulson@15411
   364
lemma notI:
wenzelm@60759
   365
  assumes "P \<Longrightarrow> False"
wenzelm@60759
   366
  shows "\<not> P"
wenzelm@21504
   367
  apply (unfold not_def)
wenzelm@21504
   368
  apply (iprover intro: impI assms)
wenzelm@21504
   369
  done
paulson@15411
   370
wenzelm@60759
   371
lemma False_not_True: "False \<noteq> True"
wenzelm@21504
   372
  apply (rule notI)
wenzelm@21504
   373
  apply (erule False_neq_True)
wenzelm@21504
   374
  done
paulson@15411
   375
wenzelm@60759
   376
lemma True_not_False: "True \<noteq> False"
wenzelm@21504
   377
  apply (rule notI)
wenzelm@21504
   378
  apply (drule sym)
wenzelm@21504
   379
  apply (erule False_neq_True)
wenzelm@21504
   380
  done
paulson@15411
   381
wenzelm@60759
   382
lemma notE: "\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R"
wenzelm@21504
   383
  apply (unfold not_def)
wenzelm@21504
   384
  apply (erule mp [THEN FalseE])
wenzelm@21504
   385
  apply assumption
wenzelm@21504
   386
  done
paulson@15411
   387
wenzelm@21504
   388
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
wenzelm@21504
   389
  by (erule notE [THEN notI]) (erule meta_mp)
paulson@15411
   390
paulson@15411
   391
wenzelm@60758
   392
subsubsection \<open>Implication\<close>
paulson@15411
   393
paulson@15411
   394
lemma impE:
wenzelm@60759
   395
  assumes "P \<longrightarrow> Q" P "Q \<Longrightarrow> R"
wenzelm@60759
   396
  shows R
wenzelm@63575
   397
  by (iprover intro: assms mp)
paulson@15411
   398
wenzelm@63575
   399
text \<open>Reduces \<open>Q\<close> to \<open>P \<longrightarrow> Q\<close>, allowing substitution in \<open>P\<close>.\<close>
wenzelm@60759
   400
lemma rev_mp: "\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
wenzelm@63575
   401
  by (iprover intro: mp)
paulson@15411
   402
paulson@15411
   403
lemma contrapos_nn:
wenzelm@60759
   404
  assumes major: "\<not> Q"
wenzelm@63575
   405
    and minor: "P \<Longrightarrow> Q"
wenzelm@60759
   406
  shows "\<not> P"
wenzelm@63575
   407
  by (iprover intro: notI minor major [THEN notE])
paulson@15411
   408
wenzelm@63575
   409
text \<open>Not used at all, but we already have the other 3 combinations.\<close>
paulson@15411
   410
lemma contrapos_pn:
paulson@15411
   411
  assumes major: "Q"
wenzelm@63575
   412
    and minor: "P \<Longrightarrow> \<not> Q"
wenzelm@60759
   413
  shows "\<not> P"
wenzelm@63575
   414
  by (iprover intro: notI minor major notE)
paulson@15411
   415
wenzelm@60759
   416
lemma not_sym: "t \<noteq> s \<Longrightarrow> s \<noteq> t"
haftmann@21250
   417
  by (erule contrapos_nn) (erule sym)
haftmann@21250
   418
wenzelm@60759
   419
lemma eq_neq_eq_imp_neq: "\<lbrakk>x = a; a \<noteq> b; b = y\<rbrakk> \<Longrightarrow> x \<noteq> y"
haftmann@21250
   420
  by (erule subst, erule ssubst, assumption)
paulson@15411
   421
paulson@15411
   422
wenzelm@60758
   423
subsubsection \<open>Existential quantifier\<close>
paulson@15411
   424
wenzelm@60759
   425
lemma exI: "P x \<Longrightarrow> \<exists>x::'a. P x"
wenzelm@63575
   426
  unfolding Ex_def by (iprover intro: allI allE impI mp)
paulson@15411
   427
paulson@15411
   428
lemma exE:
wenzelm@60759
   429
  assumes major: "\<exists>x::'a. P x"
wenzelm@63575
   430
    and minor: "\<And>x. P x \<Longrightarrow> Q"
paulson@15411
   431
  shows "Q"
wenzelm@63575
   432
  by (rule major [unfolded Ex_def, THEN spec, THEN mp]) (iprover intro: impI [THEN allI] minor)
paulson@15411
   433
paulson@15411
   434
wenzelm@60758
   435
subsubsection \<open>Conjunction\<close>
paulson@15411
   436
wenzelm@60759
   437
lemma conjI: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> P \<and> Q"
wenzelm@63575
   438
  unfolding and_def by (iprover intro: impI [THEN allI] mp)
paulson@15411
   439
wenzelm@60759
   440
lemma conjunct1: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> P"
wenzelm@63575
   441
  unfolding and_def by (iprover intro: impI dest: spec mp)
paulson@15411
   442
wenzelm@60759
   443
lemma conjunct2: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> Q"
wenzelm@63575
   444
  unfolding and_def by (iprover intro: impI dest: spec mp)
paulson@15411
   445
paulson@15411
   446
lemma conjE:
wenzelm@60759
   447
  assumes major: "P \<and> Q"
wenzelm@63575
   448
    and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   449
  shows R
wenzelm@63575
   450
  apply (rule minor)
wenzelm@63575
   451
   apply (rule major [THEN conjunct1])
wenzelm@63575
   452
  apply (rule major [THEN conjunct2])
wenzelm@63575
   453
  done
paulson@15411
   454
paulson@15411
   455
lemma context_conjI:
wenzelm@63575
   456
  assumes P "P \<Longrightarrow> Q"
wenzelm@63575
   457
  shows "P \<and> Q"
wenzelm@63575
   458
  by (iprover intro: conjI assms)
paulson@15411
   459
paulson@15411
   460
wenzelm@60758
   461
subsubsection \<open>Disjunction\<close>
paulson@15411
   462
wenzelm@60759
   463
lemma disjI1: "P \<Longrightarrow> P \<or> Q"
wenzelm@63575
   464
  unfolding or_def by (iprover intro: allI impI mp)
paulson@15411
   465
wenzelm@60759
   466
lemma disjI2: "Q \<Longrightarrow> P \<or> Q"
wenzelm@63575
   467
  unfolding or_def by (iprover intro: allI impI mp)
paulson@15411
   468
paulson@15411
   469
lemma disjE:
wenzelm@60759
   470
  assumes major: "P \<or> Q"
wenzelm@63575
   471
    and minorP: "P \<Longrightarrow> R"
wenzelm@63575
   472
    and minorQ: "Q \<Longrightarrow> R"
wenzelm@60759
   473
  shows R
wenzelm@63575
   474
  by (iprover intro: minorP minorQ impI
wenzelm@63575
   475
      major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411
   476
paulson@15411
   477
wenzelm@60758
   478
subsubsection \<open>Classical logic\<close>
paulson@15411
   479
paulson@15411
   480
lemma classical:
wenzelm@60759
   481
  assumes prem: "\<not> P \<Longrightarrow> P"
wenzelm@60759
   482
  shows P
wenzelm@63575
   483
  apply (rule True_or_False [THEN disjE, THEN eqTrueE])
wenzelm@63575
   484
   apply assumption
wenzelm@63575
   485
  apply (rule notI [THEN prem, THEN eqTrueI])
wenzelm@63575
   486
  apply (erule subst)
wenzelm@63575
   487
  apply assumption
wenzelm@63575
   488
  done
paulson@15411
   489
wenzelm@45607
   490
lemmas ccontr = FalseE [THEN classical]
paulson@15411
   491
wenzelm@63575
   492
text \<open>\<open>notE\<close> with premises exchanged; it discharges \<open>\<not> R\<close> so that it can be used to
wenzelm@63575
   493
  make elimination rules.\<close>
paulson@15411
   494
lemma rev_notE:
wenzelm@60759
   495
  assumes premp: P
wenzelm@63575
   496
    and premnot: "\<not> R \<Longrightarrow> \<not> P"
wenzelm@60759
   497
  shows R
wenzelm@63575
   498
  apply (rule ccontr)
wenzelm@63575
   499
  apply (erule notE [OF premnot premp])
wenzelm@63575
   500
  done
paulson@15411
   501
wenzelm@63575
   502
text \<open>Double negation law.\<close>
wenzelm@60759
   503
lemma notnotD: "\<not>\<not> P \<Longrightarrow> P"
wenzelm@63575
   504
  apply (rule classical)
wenzelm@63575
   505
  apply (erule notE)
wenzelm@63575
   506
  apply assumption
wenzelm@63575
   507
  done
paulson@15411
   508
paulson@15411
   509
lemma contrapos_pp:
wenzelm@60759
   510
  assumes p1: Q
wenzelm@63575
   511
    and p2: "\<not> P \<Longrightarrow> \<not> Q"
wenzelm@60759
   512
  shows P
wenzelm@63575
   513
  by (iprover intro: classical p1 p2 notE)
paulson@15411
   514
paulson@15411
   515
wenzelm@60758
   516
subsubsection \<open>Unique existence\<close>
paulson@15411
   517
paulson@15411
   518
lemma ex1I:
wenzelm@60759
   519
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a"
wenzelm@60759
   520
  shows "\<exists>!x. P x"
wenzelm@63575
   521
  unfolding Ex1_def by (iprover intro: assms exI conjI allI impI)
paulson@15411
   522
wenzelm@63575
   523
text \<open>Sometimes easier to use: the premises have no shared variables. Safe!\<close>
paulson@15411
   524
lemma ex_ex1I:
wenzelm@60759
   525
  assumes ex_prem: "\<exists>x. P x"
wenzelm@63575
   526
    and eq: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y"
wenzelm@60759
   527
  shows "\<exists>!x. P x"
wenzelm@63575
   528
  by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411
   529
paulson@15411
   530
lemma ex1E:
wenzelm@60759
   531
  assumes major: "\<exists>!x. P x"
wenzelm@63575
   532
    and minor: "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   533
  shows R
wenzelm@63575
   534
  apply (rule major [unfolded Ex1_def, THEN exE])
wenzelm@63575
   535
  apply (erule conjE)
wenzelm@63575
   536
  apply (iprover intro: minor)
wenzelm@63575
   537
  done
paulson@15411
   538
wenzelm@60759
   539
lemma ex1_implies_ex: "\<exists>!x. P x \<Longrightarrow> \<exists>x. P x"
wenzelm@63575
   540
  apply (erule ex1E)
wenzelm@63575
   541
  apply (rule exI)
wenzelm@63575
   542
  apply assumption
wenzelm@63575
   543
  done
paulson@15411
   544
paulson@15411
   545
wenzelm@60758
   546
subsubsection \<open>Classical intro rules for disjunction and existential quantifiers\<close>
paulson@15411
   547
paulson@15411
   548
lemma disjCI:
wenzelm@63575
   549
  assumes "\<not> Q \<Longrightarrow> P"
wenzelm@63575
   550
  shows "P \<or> Q"
wenzelm@63575
   551
  by (rule classical) (iprover intro: assms disjI1 disjI2 notI elim: notE)
paulson@15411
   552
wenzelm@60759
   553
lemma excluded_middle: "\<not> P \<or> P"
wenzelm@63575
   554
  by (iprover intro: disjCI)
paulson@15411
   555
wenzelm@60758
   556
text \<open>
haftmann@20944
   557
  case distinction as a natural deduction rule.
wenzelm@63575
   558
  Note that \<open>\<not> P\<close> 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@63575
   562
    and prem2: "\<not> P \<Longrightarrow> Q"
wenzelm@60759
   563
  shows Q
wenzelm@63575
   564
  apply (rule excluded_middle [THEN disjE])
wenzelm@63575
   565
   apply (erule prem2)
wenzelm@63575
   566
  apply (erule prem1)
wenzelm@63575
   567
  done
wenzelm@27126
   568
wenzelm@63575
   569
text \<open>Classical implies (\<open>\<longrightarrow>\<close>) elimination.\<close>
paulson@15411
   570
lemma impCE:
wenzelm@60759
   571
  assumes major: "P \<longrightarrow> Q"
wenzelm@63575
   572
    and minor: "\<not> P \<Longrightarrow> R" "Q \<Longrightarrow> R"
wenzelm@60759
   573
  shows R
wenzelm@63575
   574
  apply (rule excluded_middle [of P, THEN disjE])
wenzelm@63575
   575
   apply (iprover intro: minor major [THEN mp])+
wenzelm@63575
   576
  done
paulson@15411
   577
wenzelm@63575
   578
text \<open>
wenzelm@63575
   579
  This version of \<open>\<longrightarrow>\<close> elimination works on \<open>Q\<close> before \<open>P\<close>.  It works best for
wenzelm@63575
   580
  those cases in which \<open>P\<close> holds "almost everywhere".  Can't install as
wenzelm@63575
   581
  default: would break old proofs.
wenzelm@63575
   582
\<close>
paulson@15411
   583
lemma impCE':
wenzelm@60759
   584
  assumes major: "P \<longrightarrow> Q"
wenzelm@63575
   585
    and minor: "Q \<Longrightarrow> R" "\<not> P \<Longrightarrow> R"
wenzelm@60759
   586
  shows R
wenzelm@63575
   587
  apply (rule excluded_middle [of P, THEN disjE])
wenzelm@63575
   588
   apply (iprover intro: minor major [THEN mp])+
wenzelm@63575
   589
  done
paulson@15411
   590
wenzelm@63575
   591
text \<open>Classical \<open>\<longleftrightarrow>\<close> elimination.\<close>
paulson@15411
   592
lemma iffCE:
wenzelm@60759
   593
  assumes major: "P = Q"
wenzelm@63575
   594
    and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R" "\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> R"
wenzelm@60759
   595
  shows R
wenzelm@63575
   596
  by (rule major [THEN iffE]) (iprover intro: minor elim: impCE notE)
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"
wenzelm@63575
   601
  by (rule ccontr) (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
paulson@15411
   602
paulson@15411
   603
wenzelm@60758
   604
subsubsection \<open>Intuitionistic Reasoning\<close>
wenzelm@12386
   605
wenzelm@12386
   606
lemma impE':
wenzelm@60759
   607
  assumes 1: "P \<longrightarrow> Q"
wenzelm@60759
   608
    and 2: "Q \<Longrightarrow> R"
wenzelm@60759
   609
    and 3: "P \<longrightarrow> Q \<Longrightarrow> P"
wenzelm@12937
   610
  shows R
wenzelm@12386
   611
proof -
wenzelm@12386
   612
  from 3 and 1 have P .
wenzelm@12386
   613
  with 1 have Q by (rule impE)
wenzelm@12386
   614
  with 2 show R .
wenzelm@12386
   615
qed
wenzelm@12386
   616
wenzelm@12386
   617
lemma allE':
wenzelm@60759
   618
  assumes 1: "\<forall>x. P x"
wenzelm@60759
   619
    and 2: "P x \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q"
wenzelm@12937
   620
  shows Q
wenzelm@12386
   621
proof -
wenzelm@12386
   622
  from 1 have "P x" by (rule spec)
wenzelm@12386
   623
  from this and 1 show Q by (rule 2)
wenzelm@12386
   624
qed
wenzelm@12386
   625
wenzelm@12937
   626
lemma notE':
wenzelm@60759
   627
  assumes 1: "\<not> P"
wenzelm@60759
   628
    and 2: "\<not> P \<Longrightarrow> P"
wenzelm@12937
   629
  shows R
wenzelm@12386
   630
proof -
wenzelm@12386
   631
  from 2 and 1 have P .
wenzelm@12386
   632
  with 1 show R by (rule notE)
wenzelm@12386
   633
qed
wenzelm@12386
   634
wenzelm@60759
   635
lemma TrueE: "True \<Longrightarrow> P \<Longrightarrow> P" .
wenzelm@60759
   636
lemma notFalseE: "\<not> False \<Longrightarrow> P \<Longrightarrow> P" .
dixon@22444
   637
dixon@22467
   638
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
wenzelm@15801
   639
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801
   640
  and [Pure.elim 2] = allE notE' impE'
wenzelm@15801
   641
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386
   642
wenzelm@12386
   643
lemmas [trans] = trans
wenzelm@12386
   644
  and [sym] = sym not_sym
wenzelm@15801
   645
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   646
wenzelm@11438
   647
wenzelm@60758
   648
subsubsection \<open>Atomizing meta-level connectives\<close>
wenzelm@11750
   649
haftmann@28513
   650
axiomatization where
wenzelm@63575
   651
  eq_reflection: "x = y \<Longrightarrow> x \<equiv> y"  \<comment> \<open>admissible axiom\<close>
haftmann@28513
   652
wenzelm@60759
   653
lemma atomize_all [atomize]: "(\<And>x. P x) \<equiv> Trueprop (\<forall>x. P x)"
wenzelm@12003
   654
proof
wenzelm@60759
   655
  assume "\<And>x. P x"
wenzelm@60759
   656
  then show "\<forall>x. P x" ..
wenzelm@9488
   657
next
wenzelm@60759
   658
  assume "\<forall>x. P x"
wenzelm@60759
   659
  then show "\<And>x. P x" by (rule allE)
wenzelm@9488
   660
qed
wenzelm@9488
   661
wenzelm@60759
   662
lemma atomize_imp [atomize]: "(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)"
wenzelm@12003
   663
proof
wenzelm@60759
   664
  assume r: "A \<Longrightarrow> B"
wenzelm@60759
   665
  show "A \<longrightarrow> B" by (rule impI) (rule r)
wenzelm@9488
   666
next
wenzelm@60759
   667
  assume "A \<longrightarrow> B" and A
wenzelm@23553
   668
  then show B by (rule mp)
wenzelm@9488
   669
qed
wenzelm@9488
   670
wenzelm@60759
   671
lemma atomize_not: "(A \<Longrightarrow> False) \<equiv> Trueprop (\<not> A)"
paulson@14749
   672
proof
wenzelm@60759
   673
  assume r: "A \<Longrightarrow> False"
wenzelm@60759
   674
  show "\<not> A" by (rule notI) (rule r)
paulson@14749
   675
next
wenzelm@60759
   676
  assume "\<not> A" and A
wenzelm@23553
   677
  then show False by (rule notE)
paulson@14749
   678
qed
paulson@14749
   679
wenzelm@60759
   680
lemma atomize_eq [atomize, code]: "(x \<equiv> y) \<equiv> Trueprop (x = y)"
wenzelm@12003
   681
proof
wenzelm@60759
   682
  assume "x \<equiv> y"
wenzelm@60759
   683
  show "x = y" by (unfold \<open>x \<equiv> y\<close>) (rule refl)
wenzelm@10432
   684
next
wenzelm@10432
   685
  assume "x = y"
wenzelm@60759
   686
  then show "x \<equiv> y" by (rule eq_reflection)
wenzelm@10432
   687
qed
wenzelm@10432
   688
wenzelm@60759
   689
lemma atomize_conj [atomize]: "(A &&& B) \<equiv> Trueprop (A \<and> B)"
wenzelm@12003
   690
proof
wenzelm@28856
   691
  assume conj: "A &&& B"
wenzelm@60759
   692
  show "A \<and> B"
wenzelm@19121
   693
  proof (rule conjI)
wenzelm@19121
   694
    from conj show A by (rule conjunctionD1)
wenzelm@19121
   695
    from conj show B by (rule conjunctionD2)
wenzelm@19121
   696
  qed
wenzelm@11953
   697
next
wenzelm@60759
   698
  assume conj: "A \<and> B"
wenzelm@28856
   699
  show "A &&& B"
wenzelm@19121
   700
  proof -
wenzelm@19121
   701
    from conj show A ..
wenzelm@19121
   702
    from conj show B ..
wenzelm@11953
   703
  qed
wenzelm@11953
   704
qed
wenzelm@11953
   705
wenzelm@12386
   706
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832
   707
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386
   708
wenzelm@11750
   709
wenzelm@60758
   710
subsubsection \<open>Atomizing elimination rules\<close>
krauss@26580
   711
wenzelm@60759
   712
lemma atomize_exL[atomize_elim]: "(\<And>x. P x \<Longrightarrow> Q) \<equiv> ((\<exists>x. P x) \<Longrightarrow> Q)"
krauss@26580
   713
  by rule iprover+
krauss@26580
   714
wenzelm@60759
   715
lemma atomize_conjL[atomize_elim]: "(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)"
krauss@26580
   716
  by rule iprover+
krauss@26580
   717
wenzelm@60759
   718
lemma atomize_disjL[atomize_elim]: "((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)"
krauss@26580
   719
  by rule iprover+
krauss@26580
   720
wenzelm@60759
   721
lemma atomize_elimL[atomize_elim]: "(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop A" ..
krauss@26580
   722
krauss@26580
   723
wenzelm@60758
   724
subsection \<open>Package setup\<close>
haftmann@20944
   725
wenzelm@51314
   726
ML_file "Tools/hologic.ML"
wenzelm@51314
   727
wenzelm@51314
   728
wenzelm@60758
   729
subsubsection \<open>Sledgehammer setup\<close>
blanchet@35828
   730
wenzelm@60758
   731
text \<open>
wenzelm@63575
   732
  Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
wenzelm@63575
   733
  that are prolific (match too many equality or membership literals) and relate to
wenzelm@63575
   734
  seldom-used facts. Some duplicate other rules.
wenzelm@60758
   735
\<close>
blanchet@35828
   736
wenzelm@57963
   737
named_theorems no_atp "theorems that should be filtered out by Sledgehammer"
blanchet@35828
   738
blanchet@35828
   739
wenzelm@60758
   740
subsubsection \<open>Classical Reasoner setup\<close>
wenzelm@9529
   741
wenzelm@60759
   742
lemma imp_elim: "P \<longrightarrow> Q \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@26411
   743
  by (rule classical) iprover
wenzelm@26411
   744
wenzelm@60759
   745
lemma swap: "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"
wenzelm@26411
   746
  by (rule classical) iprover
wenzelm@26411
   747
wenzelm@62958
   748
lemma thin_refl: "\<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944
   749
wenzelm@60758
   750
ML \<open>
wenzelm@42799
   751
structure Hypsubst = Hypsubst
wenzelm@42799
   752
(
wenzelm@21218
   753
  val dest_eq = HOLogic.dest_eq
haftmann@21151
   754
  val dest_Trueprop = HOLogic.dest_Trueprop
haftmann@21151
   755
  val dest_imp = HOLogic.dest_imp
wenzelm@26411
   756
  val eq_reflection = @{thm eq_reflection}
wenzelm@26411
   757
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
wenzelm@26411
   758
  val imp_intr = @{thm impI}
wenzelm@26411
   759
  val rev_mp = @{thm rev_mp}
wenzelm@26411
   760
  val subst = @{thm subst}
wenzelm@26411
   761
  val sym = @{thm sym}
wenzelm@22129
   762
  val thin_refl = @{thm thin_refl};
wenzelm@42799
   763
);
wenzelm@21671
   764
open Hypsubst;
haftmann@21151
   765
wenzelm@42799
   766
structure Classical = Classical
wenzelm@42799
   767
(
wenzelm@26411
   768
  val imp_elim = @{thm imp_elim}
wenzelm@26411
   769
  val not_elim = @{thm notE}
wenzelm@26411
   770
  val swap = @{thm swap}
wenzelm@26411
   771
  val classical = @{thm classical}
haftmann@21151
   772
  val sizef = Drule.size_of_thm
haftmann@21151
   773
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
wenzelm@42799
   774
);
haftmann@21151
   775
wenzelm@58826
   776
structure Basic_Classical: BASIC_CLASSICAL = Classical;
wenzelm@33308
   777
open Basic_Classical;
wenzelm@60758
   778
\<close>
wenzelm@22129
   779
wenzelm@60758
   780
setup \<open>
wenzelm@35389
   781
  (*prevent substitution on bool*)
wenzelm@58826
   782
  let
wenzelm@58826
   783
    fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
wenzelm@58826
   784
      | non_bool_eq _ = false;
wenzelm@58826
   785
    fun hyp_subst_tac' ctxt =
wenzelm@58826
   786
      SUBGOAL (fn (goal, i) =>
wenzelm@58826
   787
        if Term.exists_Const non_bool_eq goal
wenzelm@58826
   788
        then Hypsubst.hyp_subst_tac ctxt i
wenzelm@58826
   789
        else no_tac);
wenzelm@58826
   790
  in
wenzelm@58826
   791
    Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
wenzelm@58826
   792
  end
wenzelm@60758
   793
\<close>
haftmann@21009
   794
haftmann@21009
   795
declare iffI [intro!]
haftmann@21009
   796
  and notI [intro!]
haftmann@21009
   797
  and impI [intro!]
haftmann@21009
   798
  and disjCI [intro!]
haftmann@21009
   799
  and conjI [intro!]
haftmann@21009
   800
  and TrueI [intro!]
haftmann@21009
   801
  and refl [intro!]
haftmann@21009
   802
haftmann@21009
   803
declare iffCE [elim!]
haftmann@21009
   804
  and FalseE [elim!]
haftmann@21009
   805
  and impCE [elim!]
haftmann@21009
   806
  and disjE [elim!]
haftmann@21009
   807
  and conjE [elim!]
haftmann@21009
   808
haftmann@21009
   809
declare ex_ex1I [intro!]
haftmann@21009
   810
  and allI [intro!]
haftmann@21009
   811
  and exI [intro]
haftmann@21009
   812
haftmann@21009
   813
declare exE [elim!]
haftmann@21009
   814
  allE [elim]
haftmann@21009
   815
wenzelm@60758
   816
ML \<open>val HOL_cs = claset_of @{context}\<close>
mengj@19162
   817
wenzelm@60759
   818
lemma contrapos_np: "\<not> Q \<Longrightarrow> (\<not> P \<Longrightarrow> Q) \<Longrightarrow> P"
wenzelm@20223
   819
  apply (erule swap)
wenzelm@20223
   820
  apply (erule (1) meta_mp)
wenzelm@20223
   821
  done
wenzelm@10383
   822
wenzelm@18689
   823
declare ex_ex1I [rule del, intro! 2]
wenzelm@18689
   824
  and ex1I [intro]
wenzelm@18689
   825
paulson@41865
   826
declare ext [intro]
paulson@41865
   827
wenzelm@12386
   828
lemmas [intro?] = ext
wenzelm@12386
   829
  and [elim?] = ex1_implies_ex
wenzelm@11977
   830
wenzelm@63575
   831
text \<open>Better than \<open>ex1E\<close> for classical reasoner: needs no quantifier duplication!\<close>
haftmann@20973
   832
lemma alt_ex1E [elim!]:
haftmann@20944
   833
  assumes major: "\<exists>!x. P x"
wenzelm@63575
   834
    and prem: "\<And>x. \<lbrakk>P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y'\<rbrakk> \<Longrightarrow> R"
haftmann@20944
   835
  shows R
wenzelm@63575
   836
  apply (rule ex1E [OF major])
wenzelm@63575
   837
  apply (rule prem)
wenzelm@63575
   838
   apply assumption
wenzelm@63575
   839
  apply (rule allI)+
wenzelm@63575
   840
  apply (tactic \<open>eresolve_tac @{context} [Classical.dup_elim @{context} @{thm allE}] 1\<close>)
wenzelm@63575
   841
  apply iprover
wenzelm@63575
   842
  done
haftmann@20944
   843
wenzelm@60758
   844
ML \<open>
wenzelm@42477
   845
  structure Blast = Blast
wenzelm@42477
   846
  (
wenzelm@42477
   847
    structure Classical = Classical
wenzelm@42802
   848
    val Trueprop_const = dest_Const @{const Trueprop}
wenzelm@42477
   849
    val equality_name = @{const_name HOL.eq}
wenzelm@42477
   850
    val not_name = @{const_name Not}
wenzelm@42477
   851
    val notE = @{thm notE}
wenzelm@42477
   852
    val ccontr = @{thm ccontr}
wenzelm@42477
   853
    val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
wenzelm@42477
   854
  );
wenzelm@42477
   855
  val blast_tac = Blast.blast_tac;
wenzelm@60758
   856
\<close>
haftmann@20944
   857
haftmann@20944
   858
wenzelm@60758
   859
subsubsection \<open>THE: definite description operator\<close>
lp15@59504
   860
lp15@59504
   861
lemma the_equality [intro]:
lp15@59504
   862
  assumes "P a"
wenzelm@63575
   863
    and "\<And>x. P x \<Longrightarrow> x = a"
lp15@59504
   864
  shows "(THE x. P x) = a"
lp15@59504
   865
  by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
lp15@59504
   866
lp15@59504
   867
lemma theI:
wenzelm@63575
   868
  assumes "P a"
wenzelm@63575
   869
    and "\<And>x. P x \<Longrightarrow> x = a"
lp15@59504
   870
  shows "P (THE x. P x)"
wenzelm@63575
   871
  by (iprover intro: assms the_equality [THEN ssubst])
lp15@59504
   872
wenzelm@60759
   873
lemma theI': "\<exists>!x. P x \<Longrightarrow> P (THE x. P x)"
lp15@59504
   874
  by (blast intro: theI)
lp15@59504
   875
wenzelm@63575
   876
text \<open>Easier to apply than \<open>theI\<close>: only one occurrence of \<open>P\<close>.\<close>
lp15@59504
   877
lemma theI2:
wenzelm@60759
   878
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a" "\<And>x. P x \<Longrightarrow> Q x"
lp15@59504
   879
  shows "Q (THE x. P x)"
wenzelm@63575
   880
  by (iprover intro: assms theI)
lp15@59504
   881
wenzelm@63575
   882
lemma the1I2:
wenzelm@63575
   883
  assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x"
wenzelm@63575
   884
  shows "Q (THE x. P x)"
wenzelm@63575
   885
  by (iprover intro: assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)] 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@66109
   930
    "(\<forall>x. x \<noteq> t) = False"  "(\<forall>x. t \<noteq> x) = False"
nipkow@17589
   931
  by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421
   932
wenzelm@63575
   933
lemma disj_absorb: "A \<or> A \<longleftrightarrow> A"
paulson@14201
   934
  by blast
paulson@14201
   935
wenzelm@63575
   936
lemma disj_left_absorb: "A \<or> (A \<or> B) \<longleftrightarrow> A \<or> B"
paulson@14201
   937
  by blast
paulson@14201
   938
wenzelm@63575
   939
lemma conj_absorb: "A \<and> A \<longleftrightarrow> A"
paulson@14201
   940
  by blast
paulson@14201
   941
wenzelm@63575
   942
lemma conj_left_absorb: "A \<and> (A \<and> B) \<longleftrightarrow> A \<and> B"
paulson@14201
   943
  by blast
paulson@14201
   944
wenzelm@12281
   945
lemma eq_ac:
haftmann@57512
   946
  shows eq_commute: "a = b \<longleftrightarrow> b = a"
haftmann@57512
   947
    and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
wenzelm@63575
   948
    and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))"
wenzelm@63575
   949
  by (iprover, blast+)
wenzelm@63575
   950
haftmann@57512
   951
lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
wenzelm@12281
   952
wenzelm@12281
   953
lemma conj_comms:
wenzelm@63575
   954
  shows conj_commute: "P \<and> Q \<longleftrightarrow> Q \<and> P"
wenzelm@63575
   955
    and conj_left_commute: "P \<and> (Q \<and> R) \<longleftrightarrow> Q \<and> (P \<and> R)" by iprover+
wenzelm@63575
   956
lemma conj_assoc: "(P \<and> Q) \<and> R \<longleftrightarrow> P \<and> (Q \<and> R)" by iprover
wenzelm@12281
   957
paulson@19174
   958
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174
   959
wenzelm@12281
   960
lemma disj_comms:
wenzelm@63575
   961
  shows disj_commute: "P \<or> Q \<longleftrightarrow> Q \<or> P"
wenzelm@63575
   962
    and disj_left_commute: "P \<or> (Q \<or> R) \<longleftrightarrow> Q \<or> (P \<or> R)" by iprover+
wenzelm@63575
   963
lemma disj_assoc: "(P \<or> Q) \<or> R \<longleftrightarrow> P \<or> (Q \<or> R)" by iprover
wenzelm@12281
   964
paulson@19174
   965
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174
   966
wenzelm@63575
   967
lemma conj_disj_distribL: "P \<and> (Q \<or> R) \<longleftrightarrow> P \<and> Q \<or> P \<and> R" by iprover
wenzelm@63575
   968
lemma conj_disj_distribR: "(P \<or> Q) \<and> R \<longleftrightarrow> P \<and> R \<or> Q \<and> R" by iprover
wenzelm@12281
   969
wenzelm@63575
   970
lemma disj_conj_distribL: "P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)" by iprover
wenzelm@63575
   971
lemma disj_conj_distribR: "(P \<and> Q) \<or> R \<longleftrightarrow> (P \<or> R) \<and> (Q \<or> R)" by iprover
wenzelm@12281
   972
wenzelm@60759
   973
lemma imp_conjR: "(P \<longrightarrow> (Q \<and> R)) = ((P \<longrightarrow> Q) \<and> (P \<longrightarrow> R))" by iprover
wenzelm@60759
   974
lemma imp_conjL: "((P \<and> Q) \<longrightarrow> R) = (P \<longrightarrow> (Q \<longrightarrow> R))" by iprover
wenzelm@60759
   975
lemma imp_disjL: "((P \<or> Q) \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" by iprover
wenzelm@12281
   976
wenzelm@61799
   977
text \<open>These two are specialized, but \<open>imp_disj_not1\<close> is useful in \<open>Auth/Yahalom\<close>.\<close>
wenzelm@63575
   978
lemma imp_disj_not1: "(P \<longrightarrow> Q \<or> R) \<longleftrightarrow> (\<not> Q \<longrightarrow> P \<longrightarrow> R)" by blast
wenzelm@63575
   979
lemma imp_disj_not2: "(P \<longrightarrow> Q \<or> R) \<longleftrightarrow> (\<not> R \<longrightarrow> P \<longrightarrow> Q)" by blast
wenzelm@12281
   980
wenzelm@63575
   981
lemma imp_disj1: "((P \<longrightarrow> Q) \<or> R) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)" by blast
wenzelm@63575
   982
lemma imp_disj2: "(Q \<or> (P \<longrightarrow> R)) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)" by blast
wenzelm@12281
   983
wenzelm@63575
   984
lemma imp_cong: "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<longrightarrow> Q) \<longleftrightarrow> (P' \<longrightarrow> Q'))"
haftmann@21151
   985
  by iprover
haftmann@21151
   986
wenzelm@63575
   987
lemma de_Morgan_disj: "\<not> (P \<or> Q) \<longleftrightarrow> \<not> P \<and> \<not> Q" by iprover
wenzelm@63575
   988
lemma de_Morgan_conj: "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q" by blast
wenzelm@63575
   989
lemma not_imp: "\<not> (P \<longrightarrow> Q) \<longleftrightarrow> P \<and> \<not> Q" by blast
wenzelm@63575
   990
lemma not_iff: "P \<noteq> Q \<longleftrightarrow> (P \<longleftrightarrow> \<not> Q)" by blast
wenzelm@63575
   991
lemma disj_not1: "\<not> P \<or> Q \<longleftrightarrow> (P \<longrightarrow> Q)" by blast
wenzelm@63575
   992
lemma disj_not2: "P \<or> \<not> Q \<longleftrightarrow> (Q \<longrightarrow> P)" by blast  \<comment> \<open>changes orientation :-(\<close>
wenzelm@63575
   993
lemma imp_conv_disj: "(P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P) \<or> Q" by blast
Andreas@63561
   994
lemma disj_imp: "P \<or> Q \<longleftrightarrow> \<not> P \<longrightarrow> Q" by blast
wenzelm@12281
   995
wenzelm@63575
   996
lemma iff_conv_conj_imp: "(P \<longleftrightarrow> Q) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)" by iprover
wenzelm@12281
   997
wenzelm@12281
   998
wenzelm@63575
   999
lemma cases_simp: "(P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q) \<longleftrightarrow> Q"
nipkow@62390
  1000
  \<comment> \<open>Avoids duplication of subgoals after \<open>if_split\<close>, when the true and false\<close>
wenzelm@61799
  1001
  \<comment> \<open>cases boil down to the same thing.\<close>
wenzelm@12281
  1002
  by blast
wenzelm@12281
  1003
wenzelm@63575
  1004
lemma not_all: "\<not> (\<forall>x. P x) \<longleftrightarrow> (\<exists>x. \<not> P x)" by blast
wenzelm@63575
  1005
lemma imp_all: "((\<forall>x. P x) \<longrightarrow> Q) \<longleftrightarrow> (\<exists>x. P x \<longrightarrow> Q)" by blast
wenzelm@63575
  1006
lemma not_ex: "\<not> (\<exists>x. P x) \<longleftrightarrow> (\<forall>x. \<not> P x)" by iprover
wenzelm@63575
  1007
lemma imp_ex: "((\<exists>x. P x) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q)" by iprover
wenzelm@63575
  1008
lemma all_not_ex: "(\<forall>x. P x) \<longleftrightarrow> \<not> (\<exists>x. \<not> P x)" by blast
wenzelm@12281
  1009
blanchet@35828
  1010
declare All_def [no_atp]
paulson@24286
  1011
wenzelm@63575
  1012
lemma ex_disj_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>x. P x) \<or> (\<exists>x. Q x)" by iprover
wenzelm@63575
  1013
lemma all_conj_distrib: "(\<forall>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>x. P x) \<and> (\<forall>x. Q x)" by iprover
wenzelm@12281
  1014
wenzelm@60758
  1015
text \<open>
wenzelm@63575
  1016
  \<^medskip> The \<open>\<and>\<close> congruence rule: not included by default!
wenzelm@60758
  1017
  May slow rewrite proofs down by as much as 50\%\<close>
wenzelm@12281
  1018
wenzelm@63575
  1019
lemma conj_cong: "P = P' \<Longrightarrow> (P' \<Longrightarrow> Q = Q') \<Longrightarrow> (P \<and> Q) = (P' \<and> Q')"
nipkow@17589
  1020
  by iprover
wenzelm@12281
  1021
wenzelm@63575
  1022
lemma rev_conj_cong: "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@63575
  1027
lemma disj_cong: "P = P' \<Longrightarrow> (\<not> P' \<Longrightarrow> Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
wenzelm@12281
  1028
  by blast
wenzelm@12281
  1029
wenzelm@12281
  1030
wenzelm@63575
  1031
text \<open>\<^medskip> if-then-else rules\<close>
wenzelm@12281
  1032
haftmann@32068
  1033
lemma if_True [code]: "(if True then x else y) = x"
wenzelm@63575
  1034
  unfolding If_def by blast
wenzelm@12281
  1035
haftmann@32068
  1036
lemma if_False [code]: "(if False then x else y) = y"
wenzelm@63575
  1037
  unfolding If_def by blast
wenzelm@12281
  1038
wenzelm@60759
  1039
lemma if_P: "P \<Longrightarrow> (if P then x else y) = x"
wenzelm@63575
  1040
  unfolding If_def by blast
wenzelm@12281
  1041
wenzelm@60759
  1042
lemma if_not_P: "\<not> P \<Longrightarrow> (if P then x else y) = y"
wenzelm@63575
  1043
  unfolding If_def by blast
wenzelm@12281
  1044
nipkow@62390
  1045
lemma if_split: "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
wenzelm@12281
  1046
  apply (rule case_split [of Q])
paulson@15481
  1047
   apply (simplesubst if_P)
wenzelm@63575
  1048
    prefer 3
wenzelm@63575
  1049
    apply (simplesubst if_not_P)
wenzelm@63575
  1050
     apply blast+
wenzelm@12281
  1051
  done
wenzelm@12281
  1052
nipkow@62390
  1053
lemma if_split_asm: "P (if Q then x else y) = (\<not> ((Q \<and> \<not> P x) \<or> (\<not> Q \<and> \<not> P y)))"
wenzelm@63575
  1054
  by (simplesubst if_split) blast
wenzelm@12281
  1055
nipkow@62390
  1056
lemmas if_splits [no_atp] = if_split if_split_asm
wenzelm@12281
  1057
wenzelm@12281
  1058
lemma if_cancel: "(if c then x else x) = x"
wenzelm@63575
  1059
  by (simplesubst if_split) blast
wenzelm@12281
  1060
wenzelm@12281
  1061
lemma if_eq_cancel: "(if x = y then y else x) = x"
wenzelm@63575
  1062
  by (simplesubst if_split) blast
wenzelm@12281
  1063
wenzelm@60759
  1064
lemma if_bool_eq_conj: "(if P then Q else R) = ((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R))"
wenzelm@61799
  1065
  \<comment> \<open>This form is useful for expanding \<open>if\<close>s on the RIGHT of the \<open>\<Longrightarrow>\<close> symbol.\<close>
nipkow@62390
  1066
  by (rule if_split)
wenzelm@12281
  1067
wenzelm@60759
  1068
lemma if_bool_eq_disj: "(if P then Q else R) = ((P \<and> Q) \<or> (\<not> P \<and> R))"
wenzelm@61799
  1069
  \<comment> \<open>And this form is useful for expanding \<open>if\<close>s on the LEFT.\<close>
nipkow@62390
  1070
  by (simplesubst if_split) blast
wenzelm@12281
  1071
wenzelm@63575
  1072
lemma Eq_TrueI: "P \<Longrightarrow> P \<equiv> True" unfolding atomize_eq by iprover
wenzelm@63575
  1073
lemma Eq_FalseI: "\<not> P \<Longrightarrow> P \<equiv> False" unfolding atomize_eq by iprover
wenzelm@12281
  1074
wenzelm@63575
  1075
text \<open>\<^medskip> let rules for simproc\<close>
schirmer@15423
  1076
wenzelm@60759
  1077
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
schirmer@15423
  1078
  by (unfold Let_def)
schirmer@15423
  1079
wenzelm@60759
  1080
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
schirmer@15423
  1081
  by (unfold Let_def)
schirmer@15423
  1082
wenzelm@60758
  1083
text \<open>
ballarin@16999
  1084
  The following copy of the implication operator is useful for
ballarin@16999
  1085
  fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999
  1086
  its premise.
wenzelm@60758
  1087
\<close>
berghofe@16633
  1088
wenzelm@63575
  1089
definition simp_implies :: "prop \<Rightarrow> prop \<Rightarrow> prop"  (infixr "=simp=>" 1)
wenzelm@63575
  1090
  where "simp_implies \<equiv> op \<Longrightarrow>"
berghofe@16633
  1091
wenzelm@18457
  1092
lemma simp_impliesI:
berghofe@16633
  1093
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633
  1094
  shows "PROP P =simp=> PROP Q"
berghofe@16633
  1095
  apply (unfold simp_implies_def)
berghofe@16633
  1096
  apply (rule PQ)
berghofe@16633
  1097
  apply assumption
berghofe@16633
  1098
  done
berghofe@16633
  1099
berghofe@16633
  1100
lemma simp_impliesE:
wenzelm@25388
  1101
  assumes PQ: "PROP P =simp=> PROP Q"
wenzelm@63575
  1102
    and P: "PROP P"
wenzelm@63575
  1103
    and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633
  1104
  shows "PROP R"
berghofe@16633
  1105
  apply (rule QR)
berghofe@16633
  1106
  apply (rule PQ [unfolded simp_implies_def])
berghofe@16633
  1107
  apply (rule P)
berghofe@16633
  1108
  done
berghofe@16633
  1109
berghofe@16633
  1110
lemma simp_implies_cong:
wenzelm@60759
  1111
  assumes PP' :"PROP P \<equiv> PROP P'"
wenzelm@63575
  1112
    and P'QQ': "PROP P' \<Longrightarrow> (PROP Q \<equiv> PROP Q')"
wenzelm@60759
  1113
  shows "(PROP P =simp=> PROP Q) \<equiv> (PROP P' =simp=> PROP Q')"
wenzelm@63575
  1114
  unfolding simp_implies_def
wenzelm@63575
  1115
proof (rule equal_intr_rule)
berghofe@16633
  1116
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
wenzelm@63575
  1117
    and P': "PROP P'"
berghofe@16633
  1118
  from PP' [symmetric] and P' have "PROP P"
berghofe@16633
  1119
    by (rule equal_elim_rule1)
wenzelm@23553
  1120
  then have "PROP Q" by (rule PQ)
berghofe@16633
  1121
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633
  1122
next
berghofe@16633
  1123
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
wenzelm@63575
  1124
    and P: "PROP P"
berghofe@16633
  1125
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
wenzelm@23553
  1126
  then have "PROP Q'" by (rule P'Q')
berghofe@16633
  1127
  with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633
  1128
    by (rule equal_elim_rule1)
berghofe@16633
  1129
qed
berghofe@16633
  1130
haftmann@20944
  1131
lemma uncurry:
haftmann@20944
  1132
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944
  1133
  shows "P \<and> Q \<longrightarrow> R"
wenzelm@23553
  1134
  using assms by blast
haftmann@20944
  1135
haftmann@20944
  1136
lemma iff_allI:
haftmann@20944
  1137
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1138
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
wenzelm@23553
  1139
  using assms by blast
haftmann@20944
  1140
haftmann@20944
  1141
lemma iff_exI:
haftmann@20944
  1142
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1143
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
wenzelm@23553
  1144
  using assms by blast
haftmann@20944
  1145
wenzelm@63575
  1146
lemma all_comm: "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944
  1147
  by blast
haftmann@20944
  1148
wenzelm@63575
  1149
lemma ex_comm: "(\<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@63575
  1166
  let
wenzelm@63575
  1167
    val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
wenzelm@63575
  1168
    fun is_neq eq lhs rhs thm =
wenzelm@63575
  1169
      (case Thm.prop_of thm of
wenzelm@63575
  1170
        _ $ (Not $ (eq' $ l' $ r')) =>
wenzelm@63575
  1171
          Not = HOLogic.Not andalso eq' = eq andalso
wenzelm@63575
  1172
          r' aconv lhs andalso l' aconv rhs
wenzelm@63575
  1173
      | _ => false);
wenzelm@63575
  1174
    fun proc ss ct =
wenzelm@63575
  1175
      (case Thm.term_of ct of
wenzelm@63575
  1176
        eq $ lhs $ rhs =>
wenzelm@63575
  1177
          (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
wenzelm@63575
  1178
            SOME thm => SOME (thm RS neq_to_EQ_False)
wenzelm@63575
  1179
          | NONE => NONE)
wenzelm@63575
  1180
       | _ => NONE);
wenzelm@63575
  1181
  in proc end;
wenzelm@60758
  1182
\<close>
wenzelm@24035
  1183
wenzelm@60758
  1184
simproc_setup let_simp ("Let x f") = \<open>
wenzelm@63575
  1185
  let
wenzelm@63575
  1186
    fun count_loose (Bound i) k = if i >= k then 1 else 0
wenzelm@63575
  1187
      | count_loose (s $ t) k = count_loose s k + count_loose t k
wenzelm@63575
  1188
      | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
wenzelm@63575
  1189
      | count_loose _ _ = 0;
wenzelm@63575
  1190
    fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
wenzelm@63575
  1191
      (case t of
wenzelm@63575
  1192
        Abs (_, _, t') => count_loose t' 0 <= 1
wenzelm@63575
  1193
      | _ => true);
wenzelm@63575
  1194
  in
wenzelm@63575
  1195
    fn _ => fn ctxt => fn ct =>
wenzelm@63575
  1196
      if is_trivial_let (Thm.term_of ct)
wenzelm@63575
  1197
      then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
wenzelm@63575
  1198
      else
wenzelm@63575
  1199
        let (*Norbert Schirmer's case*)
wenzelm@63575
  1200
          val t = Thm.term_of ct;
wenzelm@63575
  1201
          val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
wenzelm@63575
  1202
        in
wenzelm@63575
  1203
          Option.map (hd o Variable.export ctxt' ctxt o single)
wenzelm@63575
  1204
            (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
wenzelm@63575
  1205
              if is_Free x orelse is_Bound x orelse is_Const x
wenzelm@63575
  1206
              then SOME @{thm Let_def}
wenzelm@63575
  1207
              else
wenzelm@63575
  1208
                let
wenzelm@63575
  1209
                  val n = case f of (Abs (x, _, _)) => x | _ => "x";
wenzelm@63575
  1210
                  val cx = Thm.cterm_of ctxt x;
wenzelm@63575
  1211
                  val xT = Thm.typ_of_cterm cx;
wenzelm@63575
  1212
                  val cf = Thm.cterm_of ctxt f;
wenzelm@63575
  1213
                  val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
wenzelm@63575
  1214
                  val (_ $ _ $ g) = Thm.prop_of fx_g;
wenzelm@63575
  1215
                  val g' = abstract_over (x, g);
wenzelm@63575
  1216
                  val abs_g'= Abs (n, xT, g');
wenzelm@63575
  1217
                in
wenzelm@63575
  1218
                  if g aconv g' then
wenzelm@63575
  1219
                    let
wenzelm@63575
  1220
                      val rl =
wenzelm@63575
  1221
                        infer_instantiate ctxt [(("f", 0), cf), (("x", 0), cx)] @{thm Let_unfold};
wenzelm@63575
  1222
                    in SOME (rl OF [fx_g]) end
wenzelm@63575
  1223
                  else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
wenzelm@63575
  1224
                  then NONE (*avoid identity conversion*)
wenzelm@63575
  1225
                  else
wenzelm@63575
  1226
                    let
wenzelm@63575
  1227
                      val g'x = abs_g' $ x;
wenzelm@63575
  1228
                      val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
wenzelm@63575
  1229
                      val rl =
wenzelm@63575
  1230
                        @{thm Let_folded} |> infer_instantiate ctxt
wenzelm@63575
  1231
                          [(("f", 0), Thm.cterm_of ctxt f),
wenzelm@63575
  1232
                           (("x", 0), cx),
wenzelm@63575
  1233
                           (("g", 0), Thm.cterm_of ctxt abs_g')];
wenzelm@63575
  1234
                    in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
wenzelm@63575
  1235
                end
wenzelm@63575
  1236
            | _ => NONE)
wenzelm@63575
  1237
        end
wenzelm@63575
  1238
  end
wenzelm@63575
  1239
\<close>
wenzelm@24035
  1240
haftmann@21151
  1241
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@21151
  1242
proof
wenzelm@23389
  1243
  assume "True \<Longrightarrow> PROP P"
wenzelm@23389
  1244
  from this [OF TrueI] show "PROP P" .
haftmann@21151
  1245
next
haftmann@21151
  1246
  assume "PROP P"
wenzelm@23389
  1247
  then show "PROP P" .
haftmann@21151
  1248
qed
haftmann@21151
  1249
nipkow@59864
  1250
lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
wenzelm@61169
  1251
  by standard (intro TrueI)
nipkow@59864
  1252
nipkow@59864
  1253
lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
wenzelm@61169
  1254
  by standard simp_all
nipkow@59864
  1255
nipkow@60183
  1256
(* This is not made a simp rule because it does not improve any proofs
nipkow@60183
  1257
   but slows some AFP entries down by 5% (cpu time). May 2015 *)
wenzelm@63575
  1258
lemma implies_False_swap:
wenzelm@63575
  1259
  "NO_MATCH (Trueprop False) P \<Longrightarrow>
wenzelm@63575
  1260
    (False \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> False \<Longrightarrow> PROP Q)"
wenzelm@63575
  1261
  by (rule swap_prems_eq)
nipkow@60169
  1262
haftmann@21151
  1263
lemma ex_simps:
wenzelm@60759
  1264
  "\<And>P Q. (\<exists>x. P x \<and> Q)   = ((\<exists>x. P x) \<and> Q)"
wenzelm@60759
  1265
  "\<And>P Q. (\<exists>x. P \<and> Q x)   = (P \<and> (\<exists>x. Q x))"
wenzelm@60759
  1266
  "\<And>P Q. (\<exists>x. P x \<or> Q)   = ((\<exists>x. P x) \<or> Q)"
wenzelm@60759
  1267
  "\<And>P Q. (\<exists>x. P \<or> Q x)   = (P \<or> (\<exists>x. Q x))"
wenzelm@60759
  1268
  "\<And>P Q. (\<exists>x. P x \<longrightarrow> Q) = ((\<forall>x. P x) \<longrightarrow> Q)"
wenzelm@60759
  1269
  "\<And>P Q. (\<exists>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<exists>x. Q x))"
wenzelm@61799
  1270
  \<comment> \<open>Miniscoping: pushing in existential quantifiers.\<close>
haftmann@21151
  1271
  by (iprover | blast)+
haftmann@21151
  1272
haftmann@21151
  1273
lemma all_simps:
wenzelm@60759
  1274
  "\<And>P Q. (\<forall>x. P x \<and> Q)   = ((\<forall>x. P x) \<and> Q)"
wenzelm@60759
  1275
  "\<And>P Q. (\<forall>x. P \<and> Q x)   = (P \<and> (\<forall>x. Q x))"
wenzelm@60759
  1276
  "\<And>P Q. (\<forall>x. P x \<or> Q)   = ((\<forall>x. P x) \<or> Q)"
wenzelm@60759
  1277
  "\<And>P Q. (\<forall>x. P \<or> Q x)   = (P \<or> (\<forall>x. Q x))"
wenzelm@60759
  1278
  "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q) = ((\<exists>x. P x) \<longrightarrow> Q)"
wenzelm@60759
  1279
  "\<And>P Q. (\<forall>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<forall>x. Q x))"
wenzelm@61799
  1280
  \<comment> \<open>Miniscoping: pushing in universal quantifiers.\<close>
haftmann@21151
  1281
  by (iprover | blast)+
paulson@15481
  1282
wenzelm@21671
  1283
lemmas [simp] =
wenzelm@63575
  1284
  triv_forall_equality  \<comment> \<open>prunes params\<close>
wenzelm@63575
  1285
  True_implies_equals implies_True_equals  \<comment> \<open>prune \<open>True\<close> in asms\<close>
wenzelm@63575
  1286
  False_implies_equals  \<comment> \<open>prune \<open>False\<close> in asms\<close>
wenzelm@21671
  1287
  if_True
wenzelm@21671
  1288
  if_False
wenzelm@21671
  1289
  if_cancel
wenzelm@21671
  1290
  if_eq_cancel
wenzelm@63575
  1291
  imp_disjL \<comment>
wenzelm@63575
  1292
   \<open>In general it seems wrong to add distributive laws by default: they
wenzelm@63575
  1293
    might cause exponential blow-up.  But \<open>imp_disjL\<close> has been in for a while
haftmann@20973
  1294
    and cannot be removed without affecting existing proofs.  Moreover,
wenzelm@63575
  1295
    rewriting by \<open>(P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))\<close> might be justified on the
wenzelm@63575
  1296
    grounds that it allows simplification of \<open>R\<close> in the two cases.\<close>
wenzelm@21671
  1297
  conj_assoc
wenzelm@21671
  1298
  disj_assoc
wenzelm@21671
  1299
  de_Morgan_conj
wenzelm@21671
  1300
  de_Morgan_disj
wenzelm@21671
  1301
  imp_disj1
wenzelm@21671
  1302
  imp_disj2
wenzelm@21671
  1303
  not_imp
wenzelm@21671
  1304
  disj_not1
wenzelm@21671
  1305
  not_all
wenzelm@21671
  1306
  not_ex
wenzelm@21671
  1307
  cases_simp
wenzelm@21671
  1308
  the_eq_trivial
wenzelm@21671
  1309
  the_sym_eq_trivial
wenzelm@21671
  1310
  ex_simps
wenzelm@21671
  1311
  all_simps
wenzelm@21671
  1312
  simp_thms
wenzelm@21671
  1313
wenzelm@21671
  1314
lemmas [cong] = imp_cong simp_implies_cong
nipkow@62390
  1315
lemmas [split] = if_split
haftmann@20973
  1316
wenzelm@60758
  1317
ML \<open>val HOL_ss = simpset_of @{context}\<close>
haftmann@20973
  1318
wenzelm@63575
  1319
text \<open>Simplifies \<open>x\<close> assuming \<open>c\<close> and \<open>y\<close> assuming \<open>\<not> c\<close>.\<close>
haftmann@20944
  1320
lemma if_cong:
haftmann@20944
  1321
  assumes "b = c"
wenzelm@63575
  1322
    and "c \<Longrightarrow> x = u"
wenzelm@63575
  1323
    and "\<not> c \<Longrightarrow> y = v"
haftmann@20944
  1324
  shows "(if b then x else y) = (if c then u else v)"
haftmann@38525
  1325
  using assms by simp
haftmann@20944
  1326
wenzelm@63575
  1327
text \<open>Prevents simplification of \<open>x\<close> and \<open>y\<close>:
wenzelm@60758
  1328
  faster and allows the execution of functional programs.\<close>
haftmann@20944
  1329
lemma if_weak_cong [cong]:
haftmann@20944
  1330
  assumes "b = c"
haftmann@20944
  1331
  shows "(if b then x else y) = (if c then x else y)"
wenzelm@23553
  1332
  using assms by (rule arg_cong)
haftmann@20944
  1333
wenzelm@60758
  1334
text \<open>Prevents simplification of t: much faster\<close>
haftmann@20944
  1335
lemma let_weak_cong:
haftmann@20944
  1336
  assumes "a = b"
haftmann@20944
  1337
  shows "(let x = a in t x) = (let x = b in t x)"
wenzelm@23553
  1338
  using assms by (rule arg_cong)
haftmann@20944
  1339
wenzelm@60758
  1340
text \<open>To tidy up the result of a simproc.  Only the RHS will be simplified.\<close>
haftmann@20944
  1341
lemma eq_cong2:
haftmann@20944
  1342
  assumes "u = u'"
haftmann@20944
  1343
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
wenzelm@23553
  1344
  using assms by simp
haftmann@20944
  1345
wenzelm@63575
  1346
lemma if_distrib: "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944
  1347
  by simp
haftmann@20944
  1348
wenzelm@63575
  1349
text \<open>As a simplification rule, it replaces all function equalities by
wenzelm@60758
  1350
  first-order equalities.\<close>
haftmann@44277
  1351
lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
haftmann@44277
  1352
  by auto
haftmann@44277
  1353
wenzelm@17459
  1354
wenzelm@60758
  1355
subsubsection \<open>Generic cases and induction\<close>
wenzelm@17459
  1356
wenzelm@60758
  1357
text \<open>Rule projections:\<close>
wenzelm@60758
  1358
ML \<open>
wenzelm@32172
  1359
structure Project_Rule = Project_Rule
wenzelm@25388
  1360
(
wenzelm@27126
  1361
  val conjunct1 = @{thm conjunct1}
wenzelm@27126
  1362
  val conjunct2 = @{thm conjunct2}
wenzelm@27126
  1363
  val mp = @{thm mp}
wenzelm@59929
  1364
);
wenzelm@60758
  1365
\<close>
wenzelm@17459
  1366
wenzelm@59940
  1367
context
wenzelm@59940
  1368
begin
wenzelm@59940
  1369
wenzelm@59990
  1370
qualified definition "induct_forall P \<equiv> \<forall>x. P x"
wenzelm@59990
  1371
qualified definition "induct_implies A B \<equiv> A \<longrightarrow> B"
wenzelm@59990
  1372
qualified definition "induct_equal x y \<equiv> x = y"
wenzelm@59990
  1373
qualified definition "induct_conj A B \<equiv> A \<and> B"
wenzelm@59990
  1374
qualified definition "induct_true \<equiv> True"
wenzelm@59990
  1375
qualified definition "induct_false \<equiv> False"
haftmann@35416
  1376
wenzelm@59929
  1377
lemma induct_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457
  1378
  by (unfold atomize_all induct_forall_def)
wenzelm@11824
  1379
wenzelm@59929
  1380
lemma induct_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (induct_implies A B)"
wenzelm@18457
  1381
  by (unfold atomize_imp induct_implies_def)
wenzelm@11824
  1382
wenzelm@59929
  1383
lemma induct_equal_eq: "(x \<equiv> y) \<equiv> Trueprop (induct_equal x y)"
wenzelm@18457
  1384
  by (unfold atomize_eq induct_equal_def)
wenzelm@18457
  1385
wenzelm@59929
  1386
lemma induct_conj_eq: "(A &&& B) \<equiv> Trueprop (induct_conj A B)"
wenzelm@18457
  1387
  by (unfold atomize_conj induct_conj_def)
wenzelm@18457
  1388
berghofe@34908
  1389
lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
berghofe@34908
  1390
lemmas induct_atomize = induct_atomize' induct_equal_eq
wenzelm@45607
  1391
lemmas induct_rulify' [symmetric] = induct_atomize'
wenzelm@45607
  1392
lemmas induct_rulify [symmetric] = induct_atomize
wenzelm@18457
  1393
lemmas induct_rulify_fallback =
wenzelm@18457
  1394
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
berghofe@34908
  1395
  induct_true_def induct_false_def
wenzelm@18457
  1396
wenzelm@11989
  1397
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
  1398
    induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589
  1399
  by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824
  1400
wenzelm@11989
  1401
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
  1402
    induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589
  1403
  by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989
  1404
wenzelm@59929
  1405
lemma induct_conj_curry: "(induct_conj A B \<Longrightarrow> PROP C) \<equiv> (A \<Longrightarrow> B \<Longrightarrow> PROP C)"
berghofe@13598
  1406
proof
wenzelm@59929
  1407
  assume r: "induct_conj A B \<Longrightarrow> PROP C"
wenzelm@59929
  1408
  assume ab: A B
wenzelm@59929
  1409
  show "PROP C" by (rule r) (simp add: induct_conj_def ab)
berghofe@13598
  1410
next
wenzelm@59929
  1411
  assume r: "A \<Longrightarrow> B \<Longrightarrow> PROP C"
wenzelm@59929
  1412
  assume ab: "induct_conj A B"
wenzelm@59929
  1413
  show "PROP C" by (rule r) (simp_all add: ab [unfolded induct_conj_def])
berghofe@13598
  1414
qed
wenzelm@11824
  1415
wenzelm@11989
  1416
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
  1417
berghofe@34908
  1418
lemma induct_trueI: "induct_true"
berghofe@34908
  1419
  by (simp add: induct_true_def)
wenzelm@11824
  1420
wenzelm@60758
  1421
text \<open>Method setup.\<close>
wenzelm@11824
  1422
wenzelm@58826
  1423
ML_file "~~/src/Tools/induct.ML"
wenzelm@60758
  1424
ML \<open>
wenzelm@32171
  1425
structure Induct = Induct
wenzelm@27126
  1426
(
wenzelm@27126
  1427
  val cases_default = @{thm case_split}
wenzelm@27126
  1428
  val atomize = @{thms induct_atomize}
berghofe@34908
  1429
  val rulify = @{thms induct_rulify'}
wenzelm@27126
  1430
  val rulify_fallback = @{thms induct_rulify_fallback}
berghofe@34988
  1431
  val equal_def = @{thm induct_equal_def}
berghofe@34908
  1432
  fun dest_def (Const (@{const_name induct_equal}, _) $ t $ u) = SOME (t, u)
berghofe@34908
  1433
    | dest_def _ = NONE
wenzelm@58957
  1434
  fun trivial_tac ctxt = match_tac ctxt @{thms induct_trueI}
wenzelm@27126
  1435
)
wenzelm@60758
  1436
\<close>
wenzelm@11824
  1437
wenzelm@48891
  1438
ML_file "~~/src/Tools/induction.ML"
nipkow@45014
  1439
wenzelm@60758
  1440
declaration \<open>
wenzelm@59940
  1441
  fn _ => Induct.map_simpset (fn ss => ss
berghofe@34908
  1442
    addsimprocs
wenzelm@61144
  1443
      [Simplifier.make_simproc @{context} "swap_induct_false"
wenzelm@61144
  1444
        {lhss = [@{term "induct_false \<Longrightarrow> PROP P \<Longrightarrow> PROP Q"}],
wenzelm@61144
  1445
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1446
          (case Thm.term_of ct of
wenzelm@61144
  1447
            _ $ (P as _ $ @{const induct_false}) $ (_ $ Q $ _) =>
wenzelm@61144
  1448
              if P <> Q then SOME Drule.swap_prems_eq else NONE
wenzelm@62913
  1449
          | _ => NONE)},
wenzelm@61144
  1450
       Simplifier.make_simproc @{context} "induct_equal_conj_curry"
wenzelm@61144
  1451
        {lhss = [@{term "induct_conj P Q \<Longrightarrow> PROP R"}],
wenzelm@61144
  1452
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1453
          (case Thm.term_of ct of
wenzelm@61144
  1454
            _ $ (_ $ P) $ _ =>
wenzelm@61144
  1455
              let
wenzelm@61144
  1456
                fun is_conj (@{const induct_conj} $ P $ Q) =
wenzelm@61144
  1457
                      is_conj P andalso is_conj Q
wenzelm@61144
  1458
                  | is_conj (Const (@{const_name induct_equal}, _) $ _ $ _) = true
wenzelm@61144
  1459
                  | is_conj @{const induct_true} = true
wenzelm@61144
  1460
                  | is_conj @{const induct_false} = true
wenzelm@61144
  1461
                  | is_conj _ = false
wenzelm@61144
  1462
              in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
wenzelm@62913
  1463
            | _ => NONE)}]
wenzelm@54742
  1464
    |> Simplifier.set_mksimps (fn ctxt =>
wenzelm@54742
  1465
        Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
wenzelm@59940
  1466
        map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback}))))
wenzelm@60758
  1467
\<close>
berghofe@34908
  1468
wenzelm@60758
  1469
text \<open>Pre-simplification of induction and cases rules\<close>
berghofe@34908
  1470
wenzelm@59929
  1471
lemma [induct_simp]: "(\<And>x. induct_equal x t \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908
  1472
  unfolding induct_equal_def
berghofe@34908
  1473
proof
wenzelm@59929
  1474
  assume r: "\<And>x. x = t \<Longrightarrow> PROP P x"
wenzelm@59929
  1475
  show "PROP P t" by (rule r [OF refl])
berghofe@34908
  1476
next
wenzelm@59929
  1477
  fix x
wenzelm@59929
  1478
  assume "PROP P t" "x = t"
berghofe@34908
  1479
  then show "PROP P x" by simp
berghofe@34908
  1480
qed
berghofe@34908
  1481
wenzelm@59929
  1482
lemma [induct_simp]: "(\<And>x. induct_equal t x \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908
  1483
  unfolding induct_equal_def
berghofe@34908
  1484
proof
wenzelm@59929
  1485
  assume r: "\<And>x. t = x \<Longrightarrow> PROP P x"
wenzelm@59929
  1486
  show "PROP P t" by (rule r [OF refl])
berghofe@34908
  1487
next
wenzelm@59929
  1488
  fix x
wenzelm@59929
  1489
  assume "PROP P t" "t = x"
berghofe@34908
  1490
  then show "PROP P x" by simp
berghofe@34908
  1491
qed
berghofe@34908
  1492
wenzelm@59929
  1493
lemma [induct_simp]: "(induct_false \<Longrightarrow> P) \<equiv> Trueprop induct_true"
berghofe@34908
  1494
  unfolding induct_false_def induct_true_def
berghofe@34908
  1495
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1496
wenzelm@59929
  1497
lemma [induct_simp]: "(induct_true \<Longrightarrow> PROP P) \<equiv> PROP P"
berghofe@34908
  1498
  unfolding induct_true_def
berghofe@34908
  1499
proof
wenzelm@59929
  1500
  assume "True \<Longrightarrow> PROP P"
wenzelm@59929
  1501
  then show "PROP P" using TrueI .
berghofe@34908
  1502
next
berghofe@34908
  1503
  assume "PROP P"
berghofe@34908
  1504
  then show "PROP P" .
berghofe@34908
  1505
qed
berghofe@34908
  1506
wenzelm@59929
  1507
lemma [induct_simp]: "(PROP P \<Longrightarrow> induct_true) \<equiv> Trueprop induct_true"
berghofe@34908
  1508
  unfolding induct_true_def
berghofe@34908
  1509
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1510
wenzelm@62958
  1511
lemma [induct_simp]: "(\<And>x::'a::{}. induct_true) \<equiv> Trueprop induct_true"
berghofe@34908
  1512
  unfolding induct_true_def
berghofe@34908
  1513
  by (iprover intro: equal_intr_rule)
berghofe@34908
  1514
wenzelm@59929
  1515
lemma [induct_simp]: "induct_implies induct_true P \<equiv> P"
berghofe@34908
  1516
  by (simp add: induct_implies_def induct_true_def)
berghofe@34908
  1517
wenzelm@59929
  1518
lemma [induct_simp]: "x = x \<longleftrightarrow> True"
berghofe@34908
  1519
  by (rule simp_thms)
berghofe@34908
  1520
wenzelm@59940
  1521
end
wenzelm@18457
  1522
wenzelm@48891
  1523
ML_file "~~/src/Tools/induct_tacs.ML"
wenzelm@27126
  1524
haftmann@20944
  1525
wenzelm@60758
  1526
subsubsection \<open>Coherent logic\<close>
berghofe@28325
  1527
wenzelm@55632
  1528
ML_file "~~/src/Tools/coherent.ML"
wenzelm@60758
  1529
ML \<open>
wenzelm@32734
  1530
structure Coherent = Coherent
berghofe@28325
  1531
(
wenzelm@55632
  1532
  val atomize_elimL = @{thm atomize_elimL};
wenzelm@55632
  1533
  val atomize_exL = @{thm atomize_exL};
wenzelm@55632
  1534
  val atomize_conjL = @{thm atomize_conjL};
wenzelm@55632
  1535
  val atomize_disjL = @{thm atomize_disjL};
wenzelm@55632
  1536
  val operator_names = [@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}];
berghofe@28325
  1537
);
wenzelm@60758
  1538
\<close>
berghofe@28325
  1539
berghofe@28325
  1540
wenzelm@60758
  1541
subsubsection \<open>Reorienting equalities\<close>
huffman@31024
  1542
wenzelm@60758
  1543
ML \<open>
huffman@31024
  1544
signature REORIENT_PROC =
huffman@31024
  1545
sig
huffman@31024
  1546
  val add : (term -> bool) -> theory -> theory
wenzelm@51717
  1547
  val proc : morphism -> Proof.context -> cterm -> thm option
huffman@31024
  1548
end;
huffman@31024
  1549
wenzelm@33523
  1550
structure Reorient_Proc : REORIENT_PROC =
huffman@31024
  1551
struct
wenzelm@33523
  1552
  structure Data = Theory_Data
huffman@31024
  1553
  (
wenzelm@33523
  1554
    type T = ((term -> bool) * stamp) list;
wenzelm@33523
  1555
    val empty = [];
huffman@31024
  1556
    val extend = I;
wenzelm@33523
  1557
    fun merge data : T = Library.merge (eq_snd op =) data;
wenzelm@33523
  1558
  );
wenzelm@33523
  1559
  fun add m = Data.map (cons (m, stamp ()));
wenzelm@33523
  1560
  fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
huffman@31024
  1561
huffman@31024
  1562
  val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
wenzelm@51717
  1563
  fun proc phi ctxt ct =
huffman@31024
  1564
    let
wenzelm@42361
  1565
      val thy = Proof_Context.theory_of ctxt;
huffman@31024
  1566
    in
huffman@31024
  1567
      case Thm.term_of ct of
wenzelm@33523
  1568
        (_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
huffman@31024
  1569
      | _ => NONE
huffman@31024
  1570
    end;
huffman@31024
  1571
end;
wenzelm@60758
  1572
\<close>
huffman@31024
  1573
huffman@31024
  1574
wenzelm@60758
  1575
subsection \<open>Other simple lemmas and lemma duplicates\<close>
haftmann@20944
  1576
wenzelm@60759
  1577
lemma ex1_eq [iff]: "\<exists>!x. x = t" "\<exists>!x. t = x"
haftmann@20944
  1578
  by blast+
haftmann@20944
  1579
wenzelm@60759
  1580
lemma choice_eq: "(\<forall>x. \<exists>!y. P x y) = (\<exists>!f. \<forall>x. P x (f x))"
haftmann@20944
  1581
  apply (rule iffI)
wenzelm@63575
  1582
   apply (rule_tac a = "\<lambda>x. THE y. P x y" in ex1I)
wenzelm@63575
  1583
    apply (fast dest!: theI')
wenzelm@63575
  1584
   apply (fast intro: the1_equality [symmetric])
haftmann@20944
  1585
  apply (erule ex1E)
haftmann@20944
  1586
  apply (rule allI)
haftmann@20944
  1587
  apply (rule ex1I)
wenzelm@63575
  1588
   apply (erule spec)
wenzelm@60759
  1589
  apply (erule_tac x = "\<lambda>z. if z = x then y else f z" in allE)
haftmann@20944
  1590
  apply (erule impE)
wenzelm@63575
  1591
   apply (rule allI)
wenzelm@63575
  1592
   apply (case_tac "xa = x")
wenzelm@63575
  1593
    apply (drule_tac [3] x = x in fun_cong)
wenzelm@63575
  1594
    apply simp_all
haftmann@20944
  1595
  done
haftmann@20944
  1596
haftmann@22218
  1597
lemmas eq_sym_conv = eq_commute
haftmann@22218
  1598
chaieb@23037
  1599
lemma nnf_simps:
wenzelm@63575
  1600
  "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
wenzelm@63575
  1601
  "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
wenzelm@63575
  1602
  "(P \<longrightarrow> Q) = (\<not> P \<or> Q)"
wenzelm@63575
  1603
  "(P = Q) = ((P \<and> Q) \<or> (\<not> P \<and> \<not> Q))"
wenzelm@63575
  1604
  "(\<not> (P = Q)) = ((P \<and> \<not> Q) \<or> (\<not> P \<and> Q))"
wenzelm@63575
  1605
  "(\<not> \<not> P) = P"
wenzelm@63575
  1606
  by blast+
wenzelm@63575
  1607
chaieb@23037
  1608
wenzelm@60758
  1609
subsection \<open>Basic ML bindings\<close>
wenzelm@21671
  1610
wenzelm@60758
  1611
ML \<open>
wenzelm@22129
  1612
val FalseE = @{thm FalseE}
wenzelm@22129
  1613
val Let_def = @{thm Let_def}
wenzelm@22129
  1614
val TrueI = @{thm TrueI}
wenzelm@22129
  1615
val allE = @{thm allE}
wenzelm@22129
  1616
val allI = @{thm allI}
wenzelm@22129
  1617
val all_dupE = @{thm all_dupE}
wenzelm@22129
  1618
val arg_cong = @{thm arg_cong}
wenzelm@22129
  1619
val box_equals = @{thm box_equals}
wenzelm@22129
  1620
val ccontr = @{thm ccontr}
wenzelm@22129
  1621
val classical = @{thm classical}
wenzelm@22129
  1622
val conjE = @{thm conjE}
wenzelm@22129
  1623
val conjI = @{thm conjI}
wenzelm@22129
  1624
val conjunct1 = @{thm conjunct1}
wenzelm@22129
  1625
val conjunct2 = @{thm conjunct2}
wenzelm@22129
  1626
val disjCI = @{thm disjCI}
wenzelm@22129
  1627
val disjE = @{thm disjE}
wenzelm@22129
  1628
val disjI1 = @{thm disjI1}
wenzelm@22129
  1629
val disjI2 = @{thm disjI2}
wenzelm@22129
  1630
val eq_reflection = @{thm eq_reflection}
wenzelm@22129
  1631
val ex1E = @{thm ex1E}
wenzelm@22129
  1632
val ex1I = @{thm ex1I}
wenzelm@22129
  1633
val ex1_implies_ex = @{thm ex1_implies_ex}
wenzelm@22129
  1634
val exE = @{thm exE}
wenzelm@22129
  1635
val exI = @{thm exI}
wenzelm@22129
  1636
val excluded_middle = @{thm excluded_middle}
wenzelm@22129
  1637
val ext = @{thm ext}
wenzelm@22129
  1638
val fun_cong = @{thm fun_cong}
wenzelm@22129
  1639
val iffD1 = @{thm iffD1}
wenzelm@22129
  1640
val iffD2 = @{thm iffD2}
wenzelm@22129
  1641
val iffI = @{thm iffI}
wenzelm@22129
  1642
val impE = @{thm impE}
wenzelm@22129
  1643
val impI = @{thm impI}
wenzelm@22129
  1644
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22129
  1645
val mp = @{thm mp}
wenzelm@22129
  1646
val notE = @{thm notE}
wenzelm@22129
  1647
val notI = @{thm notI}
wenzelm@22129
  1648
val not_all = @{thm not_all}
wenzelm@22129
  1649
val not_ex = @{thm not_ex}
wenzelm@22129
  1650
val not_iff = @{thm not_iff}
wenzelm@22129
  1651
val not_not = @{thm not_not}
wenzelm@22129
  1652
val not_sym = @{thm not_sym}
wenzelm@22129
  1653
val refl = @{thm refl}
wenzelm@22129
  1654
val rev_mp = @{thm rev_mp}
wenzelm@22129
  1655
val spec = @{thm spec}
wenzelm@22129
  1656
val ssubst = @{thm ssubst}
wenzelm@22129
  1657
val subst = @{thm subst}
wenzelm@22129
  1658
val sym = @{thm sym}
wenzelm@22129
  1659
val trans = @{thm trans}
wenzelm@60758
  1660
\<close>
wenzelm@21671
  1661
wenzelm@55239
  1662
ML_file "Tools/cnf.ML"
wenzelm@55239
  1663
wenzelm@21671
  1664
wenzelm@61799
  1665
section \<open>\<open>NO_MATCH\<close> simproc\<close>
hoelzl@58775
  1666
wenzelm@60758
  1667
text \<open>
wenzelm@63575
  1668
  The simplification procedure can be used to avoid simplification of terms
wenzelm@63575
  1669
  of a certain form.
wenzelm@60758
  1670
\<close>
hoelzl@58775
  1671
wenzelm@63575
  1672
definition NO_MATCH :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
wenzelm@63575
  1673
  where "NO_MATCH pat val \<equiv> True"
hoelzl@58830
  1674
wenzelm@63575
  1675
lemma NO_MATCH_cong[cong]: "NO_MATCH pat val = NO_MATCH pat val"
wenzelm@63575
  1676
  by (rule refl)
hoelzl@58775
  1677
hoelzl@58830
  1678
declare [[coercion_args NO_MATCH - -]]
hoelzl@58830
  1679
wenzelm@60758
  1680
simproc_setup NO_MATCH ("NO_MATCH pat val") = \<open>fn _ => fn ctxt => fn ct =>
hoelzl@58775
  1681
  let
hoelzl@58775
  1682
    val thy = Proof_Context.theory_of ctxt
hoelzl@58775
  1683
    val dest_binop = Term.dest_comb #> apfst (Term.dest_comb #> snd)
hoelzl@58775
  1684
    val m = Pattern.matches thy (dest_binop (Thm.term_of ct))
hoelzl@58775
  1685
  in if m then NONE else SOME @{thm NO_MATCH_def} end
wenzelm@60758
  1686
\<close>
hoelzl@58775
  1687
wenzelm@60758
  1688
text \<open>
hoelzl@59779
  1689
  This setup ensures that a rewrite rule of the form @{term "NO_MATCH pat val \<Longrightarrow> t"}
wenzelm@63575
  1690
  is only applied, if the pattern \<open>pat\<close> does not match the value \<open>val\<close>.
wenzelm@60758
  1691
\<close>
hoelzl@58775
  1692
hoelzl@58775
  1693
wenzelm@63575
  1694
text\<open>
wenzelm@63575
  1695
  Tagging a premise of a simp rule with ASSUMPTION forces the simplifier
wenzelm@63575
  1696
  not to simplify the argument and to solve it by an assumption.
wenzelm@63575
  1697
\<close>
nipkow@61202
  1698
wenzelm@63575
  1699
definition ASSUMPTION :: "bool \<Rightarrow> bool"
wenzelm@63575
  1700
  where "ASSUMPTION A \<equiv> A"
nipkow@61202
  1701
nipkow@61202
  1702
lemma ASSUMPTION_cong[cong]: "ASSUMPTION A = ASSUMPTION A"
wenzelm@63575
  1703
  by (rule refl)
nipkow@61202
  1704
nipkow@61202
  1705
lemma ASSUMPTION_I: "A \<Longrightarrow> ASSUMPTION A"
wenzelm@63575
  1706
  by (simp add: ASSUMPTION_def)
nipkow@61202
  1707
nipkow@61202
  1708
lemma ASSUMPTION_D: "ASSUMPTION A \<Longrightarrow> A"
wenzelm@63575
  1709
  by (simp add: ASSUMPTION_def)
nipkow@61202
  1710
wenzelm@61222
  1711
setup \<open>
nipkow@61202
  1712
let
nipkow@61202
  1713
  val asm_sol = mk_solver "ASSUMPTION" (fn ctxt =>
nipkow@61202
  1714
    resolve_tac ctxt [@{thm ASSUMPTION_I}] THEN'
nipkow@61202
  1715
    resolve_tac ctxt (Simplifier.prems_of ctxt))
nipkow@61202
  1716
in
nipkow@61202
  1717
  map_theory_simpset (fn ctxt => Simplifier.addSolver (ctxt,asm_sol))
nipkow@61202
  1718
end
wenzelm@61222
  1719
\<close>
nipkow@61202
  1720
nipkow@61202
  1721
wenzelm@60758
  1722
subsection \<open>Code generator setup\<close>
haftmann@30929
  1723
wenzelm@60758
  1724
subsubsection \<open>Generic code generator preprocessor setup\<close>
haftmann@31151
  1725
wenzelm@63575
  1726
lemma conj_left_cong: "P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
haftmann@53146
  1727
  by (fact arg_cong)
haftmann@53146
  1728
wenzelm@63575
  1729
lemma disj_left_cong: "P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
haftmann@53146
  1730
  by (fact arg_cong)
haftmann@53146
  1731
wenzelm@60758
  1732
setup \<open>
wenzelm@58826
  1733
  Code_Preproc.map_pre (put_simpset HOL_basic_ss) #>
wenzelm@58826
  1734
  Code_Preproc.map_post (put_simpset HOL_basic_ss) #>
wenzelm@58826
  1735
  Code_Simp.map_ss (put_simpset HOL_basic_ss #>
wenzelm@58826
  1736
  Simplifier.add_cong @{thm conj_left_cong} #>
wenzelm@58826
  1737
  Simplifier.add_cong @{thm disj_left_cong})
wenzelm@60758
  1738
\<close>
haftmann@31151
  1739
haftmann@53146
  1740
wenzelm@60758
  1741
subsubsection \<open>Equality\<close>
haftmann@24844
  1742
haftmann@38857
  1743
class equal =
haftmann@38857
  1744
  fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@38857
  1745
  assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
haftmann@26513
  1746
begin
haftmann@26513
  1747
bulwahn@45231
  1748
lemma equal: "equal = (op =)"
haftmann@38857
  1749
  by (rule ext equal_eq)+
haftmann@28346
  1750
haftmann@38857
  1751
lemma equal_refl: "equal x x \<longleftrightarrow> True"
haftmann@38857
  1752
  unfolding equal by rule+
haftmann@28346
  1753
haftmann@38857
  1754
lemma eq_equal: "(op =) \<equiv> equal"
haftmann@38857
  1755
  by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
haftmann@30929
  1756
haftmann@26513
  1757
end
haftmann@26513
  1758
haftmann@38857
  1759
declare eq_equal [symmetric, code_post]
haftmann@38857
  1760
declare eq_equal [code]
haftmann@30966
  1761
wenzelm@60758
  1762
setup \<open>
wenzelm@51717
  1763
  Code_Preproc.map_pre (fn ctxt =>
wenzelm@61144
  1764
    ctxt addsimprocs
wenzelm@61144
  1765
      [Simplifier.make_simproc @{context} "equal"
wenzelm@61144
  1766
        {lhss = [@{term HOL.eq}],
wenzelm@61144
  1767
         proc = fn _ => fn _ => fn ct =>
wenzelm@61144
  1768
          (case Thm.term_of ct of
wenzelm@61144
  1769
            Const (_, Type (@{type_name fun}, [Type _, _])) => SOME @{thm eq_equal}
wenzelm@62913
  1770
          | _ => NONE)}])
wenzelm@60758
  1771
\<close>
haftmann@31151
  1772
haftmann@30966
  1773
wenzelm@60758
  1774
subsubsection \<open>Generic code generator foundation\<close>
haftmann@30929
  1775
wenzelm@60758
  1776
text \<open>Datatype @{typ bool}\<close>
haftmann@30929
  1777
haftmann@30929
  1778
code_datatype True False
haftmann@30929
  1779
haftmann@30929
  1780
lemma [code]:
haftmann@33185
  1781
  shows "False \<and> P \<longleftrightarrow> False"
haftmann@33185
  1782
    and "True \<and> P \<longleftrightarrow> P"
haftmann@33185
  1783
    and "P \<and> False \<longleftrightarrow> False"
wenzelm@63575
  1784
    and "P \<and> True \<longleftrightarrow> P"
wenzelm@63575
  1785
  by simp_all
haftmann@30929
  1786
haftmann@30929
  1787
lemma [code]:
haftmann@33185
  1788
  shows "False \<or> P \<longleftrightarrow> P"
haftmann@33185
  1789
    and "True \<or> P \<longleftrightarrow> True"
haftmann@33185
  1790
    and "P \<or> False \<longleftrightarrow> P"
wenzelm@63575
  1791
    and "P \<or> True \<longleftrightarrow> True"
wenzelm@63575
  1792
  by simp_all
haftmann@30929
  1793
haftmann@33185
  1794
lemma [code]:
haftmann@33185
  1795
  shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
haftmann@33185
  1796
    and "(True \<longrightarrow> P) \<longleftrightarrow> P"
haftmann@33185
  1797
    and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
wenzelm@63575
  1798
    and "(P \<longrightarrow> True) \<longleftrightarrow> True"
wenzelm@63575
  1799
  by simp_all
haftmann@30929
  1800
wenzelm@60758
  1801
text \<open>More about @{typ prop}\<close>
haftmann@39421
  1802
haftmann@39421
  1803
lemma [code nbe]:
wenzelm@58826
  1804
  shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
haftmann@39421
  1805
    and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
wenzelm@63575
  1806
    and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)"
wenzelm@63575
  1807
  by (auto intro!: equal_intr_rule)
haftmann@39421
  1808
wenzelm@63575
  1809
lemma Trueprop_code [code]: "Trueprop True \<equiv> Code_Generator.holds"
haftmann@39421
  1810
  by (auto intro!: equal_intr_rule holds)
haftmann@39421
  1811
haftmann@39421
  1812
declare Trueprop_code [symmetric, code_post]
haftmann@39421
  1813
wenzelm@60758
  1814
text \<open>Equality\<close>
haftmann@39421
  1815
haftmann@39421
  1816
declare simp_thms(6) [code nbe]
haftmann@39421
  1817
haftmann@38857
  1818
instantiation itself :: (type) equal
haftmann@31132
  1819
begin
haftmann@31132
  1820
wenzelm@63575
  1821
definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool"
wenzelm@63575
  1822
  where "equal_itself x y \<longleftrightarrow> x = y"
haftmann@31132
  1823
wenzelm@63575
  1824
instance
wenzelm@63575
  1825
  by standard (fact equal_itself_def)
haftmann@31132
  1826
haftmann@31132
  1827
end
haftmann@31132
  1828
wenzelm@63575
  1829
lemma equal_itself_code [code]: "equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
haftmann@38857
  1830
  by (simp add: equal)
haftmann@31132
  1831
wenzelm@61076
  1832
setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::type \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
haftmann@31956
  1833
wenzelm@63575
  1834
lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)"
wenzelm@63575
  1835
  (is "?ofclass \<equiv> ?equal")
haftmann@31956
  1836
proof
haftmann@31956
  1837
  assume "PROP ?ofclass"
haftmann@38857
  1838
  show "PROP ?equal"
wenzelm@60758
  1839
    by (tactic \<open>ALLGOALS (resolve_tac @{context} [Thm.unconstrainT @{thm eq_equal}])\<close>)
wenzelm@60758
  1840
      (fact \<open>PROP ?ofclass\<close>)
haftmann@31956
  1841
next
haftmann@38857
  1842
  assume "PROP ?equal"
haftmann@31956
  1843
  show "PROP ?ofclass" proof
wenzelm@60758
  1844
  qed (simp add: \<open>PROP ?equal\<close>)
haftmann@31956
  1845
qed
haftmann@31956
  1846
wenzelm@61076
  1847
setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::equal \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
wenzelm@58826
  1848
wenzelm@60758
  1849
setup \<open>Nbe.add_const_alias @{thm equal_alias_cert}\<close>
haftmann@30929
  1850
wenzelm@60758
  1851
text \<open>Cases\<close>
haftmann@30929
  1852
haftmann@30929
  1853
lemma Let_case_cert:
haftmann@30929
  1854
  assumes "CASE \<equiv> (\<lambda>x. Let x f)"
haftmann@30929
  1855
  shows "CASE x \<equiv> f x"
haftmann@30929
  1856
  using assms by simp_all
haftmann@30929
  1857
wenzelm@60758
  1858
setup \<open>
wenzelm@58826
  1859
  Code.add_case @{thm Let_case_cert} #>
wenzelm@58826
  1860
  Code.add_undefined @{const_name undefined}
wenzelm@60758
  1861
\<close>
haftmann@30929
  1862
haftmann@54890
  1863
declare [[code abort: undefined]]
haftmann@30929
  1864
haftmann@38972
  1865
wenzelm@60758
  1866
subsubsection \<open>Generic code generator target languages\<close>
haftmann@30929
  1867
wenzelm@60758
  1868
text \<open>type @{typ bool}\<close>
haftmann@30929
  1869
haftmann@52435
  1870
code_printing
haftmann@52435
  1871
  type_constructor bool \<rightharpoonup>
haftmann@52435
  1872
    (SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
haftmann@52435
  1873
| constant True \<rightharpoonup>
haftmann@52435
  1874
    (SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
haftmann@52435
  1875
| constant False \<rightharpoonup>
wenzelm@58826
  1876
    (SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"
haftmann@34294
  1877
haftmann@30929
  1878
code_reserved SML
haftmann@52435
  1879
  bool true false
haftmann@30929
  1880
haftmann@30929
  1881
code_reserved OCaml
haftmann@52435
  1882
  bool
haftmann@30929
  1883
haftmann@34294
  1884
code_reserved Scala
haftmann@34294
  1885
  Boolean
haftmann@34294
  1886
haftmann@52435
  1887
code_printing
haftmann@52435
  1888
  constant Not \<rightharpoonup>
haftmann@52435
  1889
    (SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
haftmann@52435
  1890
| constant HOL.conj \<rightharpoonup>
haftmann@52435
  1891
    (SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
haftmann@52435
  1892
| constant HOL.disj \<rightharpoonup>
haftmann@52435
  1893
    (SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
haftmann@52435
  1894
| constant HOL.implies \<rightharpoonup>
haftmann@52435
  1895
    (SML) "!(if (_)/ then (_)/ else true)"
haftmann@52435
  1896
    and (OCaml) "!(if (_)/ then (_)/ else true)"
haftmann@52435
  1897
    and (Haskell) "!(if (_)/ then (_)/ else True)"
haftmann@52435
  1898
    and (Scala) "!(if ((_))/ (_)/ else true)"
haftmann@52435
  1899
| constant If \<rightharpoonup>
haftmann@52435
  1900
    (SML) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1901
    and (OCaml) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1902
    and (Haskell) "!(if (_)/ then (_)/ else (_))"
haftmann@52435
  1903
    and (Scala) "!(if ((_))/ (_)/ else (_))"
haftmann@52435
  1904
haftmann@52435
  1905
code_reserved SML
haftmann@52435
  1906
  not
haftmann@52435
  1907
haftmann@52435
  1908
code_reserved OCaml
haftmann@52435
  1909
  not
haftmann@52435
  1910
haftmann@52435
  1911
code_identifier
haftmann@52435
  1912
  code_module Pure \<rightharpoonup>
haftmann@52435
  1913
    (SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL
haftmann@39026
  1914
wenzelm@63575
  1915
text \<open>Using built-in Haskell equality.\<close>
haftmann@52435
  1916
code_printing
haftmann@52435
  1917
  type_class equal \<rightharpoonup> (Haskell) "Eq"
haftmann@52435
  1918
| constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@52435
  1919
| constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@30929
  1920
wenzelm@63575
  1921
text \<open>\<open>undefined\<close>\<close>
haftmann@52435
  1922
code_printing
haftmann@52435
  1923
  constant undefined \<rightharpoonup>
haftmann@52435
  1924
    (SML) "!(raise/ Fail/ \"undefined\")"
haftmann@52435
  1925
    and (OCaml) "failwith/ \"undefined\""
haftmann@52435
  1926
    and (Haskell) "error/ \"undefined\""
haftmann@52435
  1927
    and (Scala) "!sys.error(\"undefined\")"
haftmann@52435
  1928
haftmann@30929
  1929
wenzelm@60758
  1930
subsubsection \<open>Evaluation and normalization by evaluation\<close>
haftmann@30929
  1931
wenzelm@60758
  1932
method_setup eval = \<open>
wenzelm@58826
  1933
  let
wenzelm@58826
  1934
    fun eval_tac ctxt =
wenzelm@58826
  1935
      let val conv = Code_Runtime.dynamic_holds_conv ctxt
wenzelm@58839
  1936
      in
wenzelm@58839
  1937
        CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 conv)) ctxt) THEN'
wenzelm@59498
  1938
        resolve_tac ctxt [TrueI]
wenzelm@58839
  1939
      end
wenzelm@58826
  1940
  in
wenzelm@58826
  1941
    Scan.succeed (SIMPLE_METHOD' o eval_tac)
wenzelm@58826
  1942
  end
wenzelm@60758
  1943
\<close> "solve goal by evaluation"
haftmann@30929
  1944
wenzelm@60758
  1945
method_setup normalization = \<open>
wenzelm@46190
  1946
  Scan.succeed (fn ctxt =>
wenzelm@46190
  1947
    SIMPLE_METHOD'
wenzelm@46190
  1948
      (CHANGED_PROP o
haftmann@55757
  1949
        (CONVERSION (Nbe.dynamic_conv ctxt)
wenzelm@59498
  1950
          THEN_ALL_NEW (TRY o resolve_tac ctxt [TrueI]))))
wenzelm@60758
  1951
\<close> "solve goal by normalization"
haftmann@30929
  1952
wenzelm@31902
  1953
wenzelm@60758
  1954
subsection \<open>Counterexample Search Units\<close>
haftmann@33084
  1955
wenzelm@60758
  1956
subsubsection \<open>Quickcheck\<close>
haftmann@30929
  1957
haftmann@33084
  1958
quickcheck_params [size = 5, iterations = 50]
haftmann@33084
  1959
haftmann@30929
  1960
wenzelm@60758
  1961
subsubsection \<open>Nitpick setup\<close>
blanchet@30309
  1962
wenzelm@59028
  1963
named_theorems nitpick_unfold "alternative definitions of constants as needed by Nitpick"
wenzelm@59028
  1964
  and nitpick_simp "equational specification of constants as needed by Nitpick"
wenzelm@59028
  1965
  and nitpick_psimp "partial equational specification of constants as needed by Nitpick"
wenzelm@59028
  1966
  and nitpick_choice_spec "choice specification of constants as needed by Nitpick"
wenzelm@30980
  1967
blanchet@41792
  1968
declare if_bool_eq_conj [nitpick_unfold, no_atp]
wenzelm@63575
  1969
  and if_bool_eq_disj [no_atp]
blanchet@41792
  1970
blanchet@29863
  1971
wenzelm@60758
  1972
subsection \<open>Preprocessing for the predicate compiler\<close>
haftmann@33084
  1973
wenzelm@59028
  1974
named_theorems code_pred_def "alternative definitions of constants for the Predicate Compiler"
wenzelm@59028
  1975
  and code_pred_inline "inlining definitions for the Predicate Compiler"
wenzelm@59028
  1976
  and code_pred_simp "simplification rules for the optimisations in the Predicate Compiler"
haftmann@33084
  1977
haftmann@33084
  1978
wenzelm@60758
  1979
subsection \<open>Legacy tactics and ML bindings\<close>
wenzelm@21671
  1980
wenzelm@60758
  1981
ML \<open>
wenzelm@58826
  1982
  (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
wenzelm@58826
  1983
  local
wenzelm@58826
  1984
    fun wrong_prem (Const (@{const_name All}, _) $ Abs (_, _, t)) = wrong_prem t
wenzelm@58826
  1985
      | wrong_prem (Bound _) = true
wenzelm@58826
  1986
      | wrong_prem _ = false;
wenzelm@58826
  1987
    val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
haftmann@61914
  1988
    fun smp i = funpow i (fn m => filter_right ([spec] RL m)) [mp];
wenzelm@58826
  1989
  in
wenzelm@59498
  1990
    fun smp_tac ctxt j = EVERY' [dresolve_tac ctxt (smp j), assume_tac ctxt];
wenzelm@58826
  1991
  end;
haftmann@22839
  1992
wenzelm@58826
  1993
  local
wenzelm@58826
  1994
    val nnf_ss =
wenzelm@58826
  1995
      simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms simp_thms nnf_simps});
wenzelm@58826
  1996
  in
wenzelm@58826
  1997
    fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
wenzelm@58826
  1998
  end
wenzelm@60758
  1999
\<close>
wenzelm@21671
  2000
haftmann@38866
  2001
hide_const (open) eq equal
haftmann@38866
  2002
kleing@14357
  2003
end