src/HOL/HOL.thy

clarified syntax and order of parameters;

(*  Title:      HOL/HOL.thy
    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)

section \<open>The basis of Higher-Order Logic\<close>

theory HOL
imports Pure Tools.Code_Generator
keywords
  "try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
    "print_induct_rules" :: diag and
  "quickcheck_params" :: thy_decl
abbrevs "?<" = "\<exists>\<^sub>\<le>\<^sub>1"
begin

ML_file \<open>~~/src/Tools/misc_legacy.ML\<close>
ML_file \<open>~~/src/Tools/try.ML\<close>
ML_file \<open>~~/src/Tools/quickcheck.ML\<close>
ML_file \<open>~~/src/Tools/solve_direct.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/zipper.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/isand.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/rw_inst.ML\<close>
ML_file \<open>~~/src/Provers/hypsubst.ML\<close>
ML_file \<open>~~/src/Provers/splitter.ML\<close>
ML_file \<open>~~/src/Provers/classical.ML\<close>
ML_file \<open>~~/src/Provers/blast.ML\<close>
ML_file \<open>~~/src/Provers/clasimp.ML\<close>
ML_file \<open>~~/src/Tools/eqsubst.ML\<close>
ML_file \<open>~~/src/Provers/quantifier1.ML\<close>
ML_file \<open>~~/src/Tools/atomize_elim.ML\<close>
ML_file \<open>~~/src/Tools/cong_tac.ML\<close>
ML_file \<open>~~/src/Tools/intuitionistic.ML\<close> setup \<open>Intuitionistic.method_setup \<^binding>\<open>iprover\<close>\<close>
ML_file \<open>~~/src/Tools/project_rule.ML\<close>
ML_file \<open>~~/src/Tools/subtyping.ML\<close>
ML_file \<open>~~/src/Tools/case_product.ML\<close>


ML \<open>Plugin_Name.declare_setup \<^binding>\<open>extraction\<close>\<close>

ML \<open>
  Plugin_Name.declare_setup \<^binding>\<open>quickcheck_random\<close>;
  Plugin_Name.declare_setup \<^binding>\<open>quickcheck_exhaustive\<close>;
  Plugin_Name.declare_setup \<^binding>\<open>quickcheck_bounded_forall\<close>;
  Plugin_Name.declare_setup \<^binding>\<open>quickcheck_full_exhaustive\<close>;
  Plugin_Name.declare_setup \<^binding>\<open>quickcheck_narrowing\<close>;
\<close>
ML \<open>
  Plugin_Name.define_setup \<^binding>\<open>quickcheck\<close>
   [\<^plugin>\<open>quickcheck_exhaustive\<close>,
    \<^plugin>\<open>quickcheck_random\<close>,
    \<^plugin>\<open>quickcheck_bounded_forall\<close>,
    \<^plugin>\<open>quickcheck_full_exhaustive\<close>,
    \<^plugin>\<open>quickcheck_narrowing\<close>]
\<close>


subsection \<open>Primitive logic\<close>

text \<open>
The definition of the logic is based on Mike Gordon's technical report \<^cite>\<open>"Gordon-TR68"\<close> that
describes the first implementation of HOL. However, there are a number of differences.
In particular, we start with the definite description operator and introduce Hilbert's \<open>\<epsilon>\<close> operator
only much later. Moreover, axiom \<open>(P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P) \<longrightarrow> (P = Q)\<close> is derived from the other
axioms. The fact that this axiom is derivable was first noticed by Bruno Barras (for Mike Gordon's
line of HOL systems) and later independently by Alexander Maletzky (for Isabelle/HOL).
\<close>

subsubsection \<open>Core syntax\<close>

setup \<open>Axclass.class_axiomatization (\<^binding>\<open>type\<close>, [])\<close>
default_sort type
setup \<open>Object_Logic.add_base_sort \<^sort>\<open>type\<close>\<close>

setup \<open>Proofterm.set_preproc (Proof_Rewrite_Rules.standard_preproc [])\<close>

axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
instance "fun" :: (type, type) type by (rule fun_arity)

axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
instance itself :: (type) type by (rule itself_arity)

typedecl bool

judgment Trueprop :: "bool \<Rightarrow> prop"  ("(_)" 5)

axiomatization implies :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longrightarrow>" 25)
  and eq :: "['a, 'a] \<Rightarrow> bool"
  and The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"

notation (input)
  eq  (infixl "=" 50)
notation (output)
  eq  (infix "=" 50)

text \<open>The input syntax for \<open>eq\<close> is more permissive than the output syntax
because of the large amount of material that relies on infixl.\<close>

subsubsection \<open>Defined connectives and quantifiers\<close>

definition True :: bool
  where "True \<equiv> ((\<lambda>x::bool. x) = (\<lambda>x. x))"

definition All :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<forall>" 10)
  where "All P \<equiv> (P = (\<lambda>x. True))"

definition Ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>" 10)
  where "Ex P \<equiv> \<forall>Q. (\<forall>x. P x \<longrightarrow> Q) \<longrightarrow> Q"

definition False :: bool
  where "False \<equiv> (\<forall>P. P)"

definition Not :: "bool \<Rightarrow> bool"  ("\<not> _" [40] 40)
  where not_def: "\<not> P \<equiv> P \<longrightarrow> False"

definition conj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<and>" 35)
  where and_def: "P \<and> Q \<equiv> \<forall>R. (P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> R"

definition disj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<or>" 30)
  where or_def: "P \<or> Q \<equiv> \<forall>R. (P \<longrightarrow> R) \<longrightarrow> (Q \<longrightarrow> R) \<longrightarrow> R"

definition Uniq :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
  where "Uniq P \<equiv> (\<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x)"

definition Ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
  where "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"


subsubsection \<open>Additional concrete syntax\<close>

syntax (ASCII) "_Uniq" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(4?< _./ _)" [0, 10] 10)
syntax "_Uniq" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(2\<exists>\<^sub>\<le>\<^sub>1 _./ _)" [0, 10] 10)
translations "\<exists>\<^sub>\<le>\<^sub>1x. P" \<rightleftharpoons> "CONST Uniq (\<lambda>x. P)"

print_translation \<open>
 [Syntax_Trans.preserve_binder_abs_tr' \<^const_syntax>\<open>Uniq\<close> \<^syntax_const>\<open>_Uniq\<close>]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>


syntax (ASCII)
  "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3EX! _./ _)" [0, 10] 10)
syntax (input)
  "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3?! _./ _)" [0, 10] 10)
syntax "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<exists>!_./ _)" [0, 10] 10)
translations "\<exists>!x. P" \<rightleftharpoons> "CONST Ex1 (\<lambda>x. P)"

print_translation \<open>
 [Syntax_Trans.preserve_binder_abs_tr' \<^const_syntax>\<open>Ex1\<close> \<^syntax_const>\<open>_Ex1\<close>]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>


syntax
  "_Not_Ex" :: "idts \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>_./ _)" [0, 10] 10)
  "_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>!_./ _)" [0, 10] 10)
translations
  "\<nexists>x. P" \<rightleftharpoons> "\<not> (\<exists>x. P)"
  "\<nexists>!x. P" \<rightleftharpoons> "\<not> (\<exists>!x. P)"


abbreviation not_equal :: "['a, 'a] \<Rightarrow> bool"  (infix "\<noteq>" 50)
  where "x \<noteq> y \<equiv> \<not> (x = y)"

notation (ASCII)
  Not  ("~ _" [40] 40) and
  conj  (infixr "&" 35) and
  disj  (infixr "|" 30) and
  implies  (infixr "-->" 25) and
  not_equal  (infix "~=" 50)

abbreviation (iff)
  iff :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longleftrightarrow>" 25)
  where "A \<longleftrightarrow> B \<equiv> A = B"

syntax "_The" :: "[pttrn, bool] \<Rightarrow> 'a"  ("(3THE _./ _)" [0, 10] 10)
translations "THE x. P" \<rightleftharpoons> "CONST The (\<lambda>x. P)"
print_translation \<open>
  [(\<^const_syntax>\<open>The\<close>, fn _ => fn [Abs abs] =>
      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
      in Syntax.const \<^syntax_const>\<open>_The\<close> $ x $ t end)]
\<close>  \<comment> \<open>To avoid eta-contraction of body\<close>

nonterminal letbinds and letbind
syntax
  "_bind"       :: "[pttrn, 'a] \<Rightarrow> letbind"              ("(2_ =/ _)" 10)
  ""            :: "letbind \<Rightarrow> letbinds"                 ("_")
  "_binds"      :: "[letbind, letbinds] \<Rightarrow> letbinds"     ("_;/ _")
  "_Let"        :: "[letbinds, 'a] \<Rightarrow> 'a"                ("(let (_)/ in (_))" [0, 10] 10)

nonterminal case_syn and cases_syn
syntax
  "_case_syntax" :: "['a, cases_syn] \<Rightarrow> 'b"  ("(case _ of/ _)" 10)
  "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ \<Rightarrow>/ _)" 10)
  "" :: "case_syn \<Rightarrow> cases_syn"  ("_")
  "_case2" :: "[case_syn, cases_syn] \<Rightarrow> cases_syn"  ("_/ | _")
syntax (ASCII)
  "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ =>/ _)" 10)

notation (ASCII)
  All  (binder "ALL " 10) and
  Ex  (binder "EX " 10)

notation (input)
  All  (binder "! " 10) and
  Ex  (binder "? " 10)


subsubsection \<open>Axioms and basic definitions\<close>

axiomatization where
  refl: "t = (t::'a)" and
  subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
  ext: "(\<And>x::'a. (f x ::'b) = g x) \<Longrightarrow> (\<lambda>x. f x) = (\<lambda>x. g x)"
    \<comment> \<open>Extensionality is built into the meta-logic, and this rule expresses
         a related property.  It is an eta-expanded version of the traditional
         rule, and similar to the ABS rule of HOL\<close> and

  the_eq_trivial: "(THE x. x = a) = (a::'a)"

axiomatization where
  impI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q" and
  mp: "\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q" and

  True_or_False: "(P = True) \<or> (P = False)"

definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
  where "If P x y \<equiv> (THE z::'a. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y))"

definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
  where "Let s f \<equiv> f s"

translations
  "_Let (_binds b bs) e"  \<rightleftharpoons> "_Let b (_Let bs e)"
  "let x = a in e"        \<rightleftharpoons> "CONST Let a (\<lambda>x. e)"

axiomatization undefined :: 'a

class default = fixes default :: 'a


subsection \<open>Fundamental rules\<close>

subsubsection \<open>Equality\<close>

lemma sym: "s = t \<Longrightarrow> t = s"
  by (erule subst) (rule refl)

lemma ssubst: "t = s \<Longrightarrow> P s \<Longrightarrow> P t"
  by (drule sym) (erule subst)

lemma trans: "\<lbrakk>r = s; s = t\<rbrakk> \<Longrightarrow> r = t"
  by (erule subst)

lemma trans_sym [Pure.elim?]: "r = s \<Longrightarrow> t = s \<Longrightarrow> r = t"
  by (rule trans [OF _ sym])

lemma meta_eq_to_obj_eq:
  assumes "A \<equiv> B"
  shows "A = B"
  unfolding assms by (rule refl)

text \<open>Useful with \<open>erule\<close> for proving equalities from known equalities.\<close>
     (* a = b
        |   |
        c = d   *)
lemma box_equals: "\<lbrakk>a = b; a = c; b = d\<rbrakk> \<Longrightarrow> c = d"
  by (iprover intro: sym trans)

text \<open>For calculational reasoning:\<close>

lemma forw_subst: "a = b \<Longrightarrow> P b \<Longrightarrow> P a"
  by (rule ssubst)

lemma back_subst: "P a \<Longrightarrow> a = b \<Longrightarrow> P b"
  by (rule subst)


subsubsection \<open>Congruence rules for application\<close>

text \<open>Similar to \<open>AP_THM\<close> in Gordon's HOL.\<close>
lemma fun_cong: "(f :: 'a \<Rightarrow> 'b) = g \<Longrightarrow> f x = g x"
  by (iprover intro: refl elim: subst)

text \<open>Similar to \<open>AP_TERM\<close> in Gordon's HOL and FOL's \<open>subst_context\<close>.\<close>
lemma arg_cong: "x = y \<Longrightarrow> f x = f y"
  by (iprover intro: refl elim: subst)

lemma arg_cong2: "\<lbrakk>a = b; c = d\<rbrakk> \<Longrightarrow> f a c = f b d"
  by (iprover intro: refl elim: subst)

lemma cong: "\<lbrakk>f = g; (x::'a) = y\<rbrakk> \<Longrightarrow> f x = g y"
  by (iprover intro: refl elim: subst)

ML \<open>fun cong_tac ctxt = Cong_Tac.cong_tac ctxt @{thm cong}\<close>


subsubsection \<open>Equality of booleans -- iff\<close>

lemma iffD2: "\<lbrakk>P = Q; Q\<rbrakk> \<Longrightarrow> P"
  by (erule ssubst)

lemma rev_iffD2: "\<lbrakk>Q; P = Q\<rbrakk> \<Longrightarrow> P"
  by (erule iffD2)

lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
  by (drule sym) (rule iffD2)

lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
  by (drule sym) (rule rev_iffD2)

lemma iffE:
  assumes major: "P = Q"
    and minor: "\<lbrakk>P \<longrightarrow> Q; Q \<longrightarrow> P\<rbrakk> \<Longrightarrow> R"
  shows R
  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])


subsubsection \<open>True (1)\<close>

lemma TrueI: True
  unfolding True_def by (rule refl)

lemma eqTrueE: "P = True \<Longrightarrow> P"
  by (erule iffD2) (rule TrueI)


subsubsection \<open>Universal quantifier (1)\<close>

lemma spec: "\<forall>x::'a. P x \<Longrightarrow> P x"
  unfolding All_def by (iprover intro: eqTrueE fun_cong)

lemma allE:
  assumes major: "\<forall>x. P x" and minor: "P x \<Longrightarrow> R"
  shows R
  by (iprover intro: minor major [THEN spec])

lemma all_dupE:
  assumes major: "\<forall>x. P x" and minor: "\<lbrakk>P x; \<forall>x. P x\<rbrakk> \<Longrightarrow> R"
  shows R
  by (iprover intro: minor major major [THEN spec])


subsubsection \<open>False\<close>

text \<open>
  Depends upon \<open>spec\<close>; it is impossible to do propositional
  logic before quantifiers!
\<close>

lemma FalseE: "False \<Longrightarrow> P"
  unfolding False_def by (erule spec)

lemma False_neq_True: "False = True \<Longrightarrow> P"
  by (erule eqTrueE [THEN FalseE])


subsubsection \<open>Negation\<close>

lemma notI:
  assumes "P \<Longrightarrow> False"
  shows "\<not> P"
  unfolding not_def by (iprover intro: impI assms)

lemma False_not_True: "False \<noteq> True"
  by (iprover intro: notI elim: False_neq_True)

lemma True_not_False: "True \<noteq> False"
  by (iprover intro: notI dest: sym elim: False_neq_True)

lemma notE: "\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R"
  unfolding not_def
  by (iprover intro: mp [THEN FalseE])


subsubsection \<open>Implication\<close>

lemma impE:
  assumes "P \<longrightarrow> Q" P "Q \<Longrightarrow> R"
  shows R
  by (iprover intro: assms mp)

text \<open>Reduces \<open>Q\<close> to \<open>P \<longrightarrow> Q\<close>, allowing substitution in \<open>P\<close>.\<close>
lemma rev_mp: "\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
  by (rule mp)

lemma contrapos_nn:
  assumes major: "\<not> Q"
    and minor: "P \<Longrightarrow> Q"
  shows "\<not> P"
  by (iprover intro: notI minor major [THEN notE])

text \<open>Not used at all, but we already have the other 3 combinations.\<close>
lemma contrapos_pn:
  assumes major: "Q"
    and minor: "P \<Longrightarrow> \<not> Q"
  shows "\<not> P"
  by (iprover intro: notI minor major notE)

lemma not_sym: "t \<noteq> s \<Longrightarrow> s \<noteq> t"
  by (erule contrapos_nn) (erule sym)

lemma eq_neq_eq_imp_neq: "\<lbrakk>x = a; a \<noteq> b; b = y\<rbrakk> \<Longrightarrow> x \<noteq> y"
  by (erule subst, erule ssubst, assumption)


subsubsection \<open>Disjunction (1)\<close>

lemma disjE:
  assumes major: "P \<or> Q"
    and minorP: "P \<Longrightarrow> R"
    and minorQ: "Q \<Longrightarrow> R"
  shows R
  by (iprover intro: minorP minorQ impI
      major [unfolded or_def, THEN spec, THEN mp, THEN mp])


subsubsection \<open>Derivation of \<open>iffI\<close>\<close>

text \<open>In an intuitionistic version of HOL \<open>iffI\<close> needs to be an axiom.\<close>

lemma iffI:
  assumes "P \<Longrightarrow> Q" and "Q \<Longrightarrow> P"
  shows "P = Q"
proof (rule disjE[OF True_or_False[of P]])
  assume 1: "P = True"
  note Q = assms(1)[OF eqTrueE[OF this]]
  from 1 show ?thesis
  proof (rule ssubst)
    from True_or_False[of Q] show "True = Q"
    proof (rule disjE)
      assume "Q = True"
      thus ?thesis by(rule sym)
    next
      assume "Q = False"
      with Q have False by (rule rev_iffD1)
      thus ?thesis by (rule FalseE)
    qed
  qed
next
  assume 2: "P = False"
  thus ?thesis
  proof (rule ssubst)
    from True_or_False[of Q] show "False = Q"
    proof (rule disjE)
      assume "Q = True"
      from 2 assms(2)[OF eqTrueE[OF this]] have False by (rule iffD1)
      thus ?thesis by (rule FalseE)
    next
      assume "Q = False"
      thus ?thesis by(rule sym)
    qed
  qed
qed


subsubsection \<open>True (2)\<close>

lemma eqTrueI: "P \<Longrightarrow> P = True"
  by (iprover intro: iffI TrueI)


subsubsection \<open>Universal quantifier (2)\<close>

lemma allI:
  assumes "\<And>x::'a. P x"
  shows "\<forall>x. P x"
  unfolding All_def by (iprover intro: ext eqTrueI assms)


subsubsection \<open>Existential quantifier\<close>

lemma exI: "P x \<Longrightarrow> \<exists>x::'a. P x"
  unfolding Ex_def by (iprover intro: allI allE impI mp)

lemma exE:
  assumes major: "\<exists>x::'a. P x"
    and minor: "\<And>x. P x \<Longrightarrow> Q"
  shows "Q"
  by (rule major [unfolded Ex_def, THEN spec, THEN mp]) (iprover intro: impI [THEN allI] minor)


subsubsection \<open>Conjunction\<close>

lemma conjI: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> P \<and> Q"
  unfolding and_def by (iprover intro: impI [THEN allI] mp)

lemma conjunct1: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> P"
  unfolding and_def by (iprover intro: impI dest: spec mp)

lemma conjunct2: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> Q"
  unfolding and_def by (iprover intro: impI dest: spec mp)

lemma conjE:
  assumes major: "P \<and> Q"
    and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R"
  shows R
proof (rule minor)
  show P by (rule major [THEN conjunct1])
  show Q by (rule major [THEN conjunct2])
qed

lemma context_conjI:
  assumes P "P \<Longrightarrow> Q"
  shows "P \<and> Q"
  by (iprover intro: conjI assms)


subsubsection \<open>Disjunction (2)\<close>

lemma disjI1: "P \<Longrightarrow> P \<or> Q"
  unfolding or_def by (iprover intro: allI impI mp)

lemma disjI2: "Q \<Longrightarrow> P \<or> Q"
  unfolding or_def by (iprover intro: allI impI mp)


subsubsection \<open>Classical logic\<close>

lemma classical:
  assumes "\<not> P \<Longrightarrow> P"
  shows P
proof (rule True_or_False [THEN disjE])
  show P if "P = True"
    using that by (iprover intro: eqTrueE)
  show P if "P = False"
  proof (intro notI assms)
    assume P
    with that show False
      by (iprover elim: subst)
  qed
qed

lemmas ccontr = FalseE [THEN classical]

text \<open>\<open>notE\<close> with premises exchanged; it discharges \<open>\<not> R\<close> so that it can be used to
  make elimination rules.\<close>
lemma rev_notE:
  assumes premp: P
    and premnot: "\<not> R \<Longrightarrow> \<not> P"
  shows R
  by (iprover intro: ccontr notE [OF premnot premp])


text \<open>Double negation law.\<close>
lemma notnotD: "\<not>\<not> P \<Longrightarrow> P"
  by (iprover intro: ccontr notE )

lemma contrapos_pp:
  assumes p1: Q
    and p2: "\<not> P \<Longrightarrow> \<not> Q"
  shows P
  by (iprover intro: classical p1 p2 notE)


subsubsection \<open>Unique existence\<close>

lemma Uniq_I [intro?]:
  assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x"
  shows "Uniq P"
  unfolding Uniq_def by (iprover intro: assms allI impI)

lemma Uniq_D [dest?]: "\<lbrakk>Uniq P; P a; P b\<rbrakk> \<Longrightarrow> a=b"
  unfolding Uniq_def by (iprover dest: spec mp)

lemma ex1I:
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a"
  shows "\<exists>!x. P x"
  unfolding Ex1_def by (iprover intro: assms exI conjI allI impI)

text \<open>Sometimes easier to use: the premises have no shared variables. Safe!\<close>
lemma ex_ex1I:
  assumes ex_prem: "\<exists>x. P x"
    and eq: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y"
  shows "\<exists>!x. P x"
  by (iprover intro: ex_prem [THEN exE] ex1I eq)

lemma ex1E:
  assumes major: "\<exists>!x. P x" and minor: "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R"
  shows R
proof (rule major [unfolded Ex1_def, THEN exE])
  show "\<And>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x) \<Longrightarrow> R"
    by (iprover intro: minor elim: conjE)
qed

lemma ex1_implies_ex: "\<exists>!x. P x \<Longrightarrow> \<exists>x. P x"
  by (iprover intro: exI elim: ex1E)

subsubsection \<open>Classical intro rules for disjunction and existential quantifiers\<close>

lemma disjCI:
  assumes "\<not> Q \<Longrightarrow> P"
  shows "P \<or> Q"
  by (rule classical) (iprover intro: assms disjI1 disjI2 notI elim: notE)

lemma excluded_middle: "\<not> P \<or> P"
  by (iprover intro: disjCI)

text \<open>
  case distinction as a natural deduction rule.
  Note that \<open>\<not> P\<close> is the second case, not the first.
\<close>
lemma case_split [case_names True False]:
  assumes "P \<Longrightarrow> Q" "\<not> P \<Longrightarrow> Q"
  shows Q
  using excluded_middle [of P]
    by (iprover intro: assms elim: disjE)

text \<open>Classical implies (\<open>\<longrightarrow>\<close>) elimination.\<close>
lemma impCE:
  assumes major: "P \<longrightarrow> Q"
    and minor: "\<not> P \<Longrightarrow> R" "Q \<Longrightarrow> R"
  shows R
  using excluded_middle [of P]
  by (iprover intro: minor major [THEN mp] elim: disjE)+

text \<open>
  This version of \<open>\<longrightarrow>\<close> elimination works on \<open>Q\<close> before \<open>P\<close>.  It works best for
  those cases in which \<open>P\<close> holds "almost everywhere".  Can't install as
  default: would break old proofs.
\<close>
lemma impCE':
  assumes major: "P \<longrightarrow> Q"
    and minor: "Q \<Longrightarrow> R" "\<not> P \<Longrightarrow> R"
  shows R
  using assms by (elim impCE)


text \<open>Classical \<open>\<longleftrightarrow>\<close> elimination.\<close>
lemma iffCE:
  assumes major: "P = Q"
    and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R" "\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> R"
  shows R
  by (rule major [THEN iffE]) (iprover intro: minor elim: impCE notE)

lemma exCI:
  assumes "\<forall>x. \<not> P x \<Longrightarrow> P a"
  shows "\<exists>x. P x"
  by (rule ccontr) (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])


subsubsection \<open>Intuitionistic Reasoning\<close>

lemma impE':
  assumes 1: "P \<longrightarrow> Q"
    and 2: "Q \<Longrightarrow> R"
    and 3: "P \<longrightarrow> Q \<Longrightarrow> P"
  shows R
proof -
  from 3 and 1 have P .
  with 1 have Q by (rule impE)
  with 2 show R .
qed

lemma allE':
  assumes 1: "\<forall>x. P x"
    and 2: "P x \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q"
  shows Q
proof -
  from 1 have "P x" by (rule spec)
  from this and 1 show Q by (rule 2)
qed

lemma notE':
  assumes 1: "\<not> P"
    and 2: "\<not> P \<Longrightarrow> P"
  shows R
proof -
  from 2 and 1 have P .
  with 1 show R by (rule notE)
qed

lemma TrueE: "True \<Longrightarrow> P \<Longrightarrow> P" .
lemma notFalseE: "\<not> False \<Longrightarrow> P \<Longrightarrow> P" .

lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
  and [Pure.elim 2] = allE notE' impE'
  and [Pure.intro] = exI disjI2 disjI1

lemmas [trans] = trans
  and [sym] = sym not_sym
  and [Pure.elim?] = iffD1 iffD2 impE


subsubsection \<open>Atomizing meta-level connectives\<close>

axiomatization where
  eq_reflection: "x = y \<Longrightarrow> x \<equiv> y"  \<comment> \<open>admissible axiom\<close>

lemma atomize_all [atomize]: "(\<And>x. P x) \<equiv> Trueprop (\<forall>x. P x)"
proof
  assume "\<And>x. P x"
  then show "\<forall>x. P x" ..
next
  assume "\<forall>x. P x"
  then show "\<And>x. P x" by (rule allE)
qed

lemma atomize_imp [atomize]: "(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)"
proof
  assume r: "A \<Longrightarrow> B"
  show "A \<longrightarrow> B" by (rule impI) (rule r)
next
  assume "A \<longrightarrow> B" and A
  then show B by (rule mp)
qed

lemma atomize_not: "(A \<Longrightarrow> False) \<equiv> Trueprop (\<not> A)"
proof
  assume r: "A \<Longrightarrow> False"
  show "\<not> A" by (rule notI) (rule r)
next
  assume "\<not> A" and A
  then show False by (rule notE)
qed

lemma atomize_eq [atomize, code]: "(x \<equiv> y) \<equiv> Trueprop (x = y)"
proof
  assume "x \<equiv> y"
  show "x = y" by (unfold \<open>x \<equiv> y\<close>) (rule refl)
next
  assume "x = y"
  then show "x \<equiv> y" by (rule eq_reflection)
qed

lemma atomize_conj [atomize]: "(A &&& B) \<equiv> Trueprop (A \<and> B)"
proof
  assume conj: "A &&& B"
  show "A \<and> B"
  proof (rule conjI)
    from conj show A by (rule conjunctionD1)
    from conj show B by (rule conjunctionD2)
  qed
next
  assume conj: "A \<and> B"
  show "A &&& B"
  proof -
    from conj show A ..
    from conj show B ..
  qed
qed

lemmas [symmetric, rulify] = atomize_all atomize_imp
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq


subsubsection \<open>Atomizing elimination rules\<close>

lemma atomize_exL[atomize_elim]: "(\<And>x. P x \<Longrightarrow> Q) \<equiv> ((\<exists>x. P x) \<Longrightarrow> Q)"
  by (rule equal_intr_rule) iprover+

lemma atomize_conjL[atomize_elim]: "(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)"
  by (rule equal_intr_rule) iprover+

lemma atomize_disjL[atomize_elim]: "((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)"
  by (rule equal_intr_rule) iprover+

lemma atomize_elimL[atomize_elim]: "(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop A" ..


subsection \<open>Package setup\<close>

ML_file \<open>Tools/hologic.ML\<close>
ML_file \<open>Tools/rewrite_hol_proof.ML\<close>

setup \<open>Proofterm.set_preproc (Proof_Rewrite_Rules.standard_preproc Rewrite_HOL_Proof.rews)\<close>


subsubsection \<open>Sledgehammer setup\<close>

text \<open>
  Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
  that are prolific (match too many equality or membership literals) and relate to
  seldom-used facts. Some duplicate other rules.
\<close>

named_theorems no_atp "theorems that should be filtered out by Sledgehammer"


subsubsection \<open>Classical Reasoner setup\<close>

lemma imp_elim: "P \<longrightarrow> Q \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
  by (rule classical) iprover

lemma swap: "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"
  by (rule classical) iprover

lemma thin_refl: "\<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W" .

ML \<open>
structure Hypsubst = Hypsubst
(
  val dest_eq = HOLogic.dest_eq
  val dest_Trueprop = HOLogic.dest_Trueprop
  val dest_imp = HOLogic.dest_imp
  val eq_reflection = @{thm eq_reflection}
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
  val imp_intr = @{thm impI}
  val rev_mp = @{thm rev_mp}
  val subst = @{thm subst}
  val sym = @{thm sym}
  val thin_refl = @{thm thin_refl};
);
open Hypsubst;

structure Classical = Classical
(
  val imp_elim = @{thm imp_elim}
  val not_elim = @{thm notE}
  val swap = @{thm swap}
  val classical = @{thm classical}
  val sizef = Drule.size_of_thm
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
);

structure Basic_Classical: BASIC_CLASSICAL = Classical;
open Basic_Classical;
\<close>

setup \<open>
  (*prevent substitution on bool*)
  let
    fun non_bool_eq (\<^const_name>\<open>HOL.eq\<close>, Type (_, [T, _])) = T <> \<^typ>\<open>bool\<close>
      | non_bool_eq _ = false;
    fun hyp_subst_tac' ctxt =
      SUBGOAL (fn (goal, i) =>
        if Term.exists_Const non_bool_eq goal
        then Hypsubst.hyp_subst_tac ctxt i
        else no_tac);
  in
    Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
  end
\<close>

declare iffI [intro!]
  and notI [intro!]
  and impI [intro!]
  and disjCI [intro!]
  and conjI [intro!]
  and TrueI [intro!]
  and refl [intro!]

declare iffCE [elim!]
  and FalseE [elim!]
  and impCE [elim!]
  and disjE [elim!]
  and conjE [elim!]

declare ex_ex1I [intro!]
  and allI [intro!]
  and exI [intro]

declare exE [elim!]
  allE [elim]

ML \<open>val HOL_cs = claset_of \<^context>\<close>

lemma contrapos_np: "\<not> Q \<Longrightarrow> (\<not> P \<Longrightarrow> Q) \<Longrightarrow> P"
  by (erule swap)

declare ex_ex1I [rule del, intro! 2]
  and ex1I [intro]

declare ext [intro]

lemmas [intro?] = ext
  and [elim?] = ex1_implies_ex

text \<open>Better than \<open>ex1E\<close> for classical reasoner: needs no quantifier duplication!\<close>
lemma alt_ex1E [elim!]:
  assumes major: "\<exists>!x. P x"
    and minor: "\<And>x. \<lbrakk>P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y'\<rbrakk> \<Longrightarrow> R"
  shows R
proof (rule ex1E [OF major minor])
  show "\<forall>y y'. P y \<and> P y' \<longrightarrow> y = y'" if "P x" and \<section>: "\<forall>y. P y \<longrightarrow> y = x" for x
    using \<open>P x\<close> \<section> \<section> by fast
qed assumption

text \<open>And again using Uniq\<close>
lemma alt_ex1E':
  assumes  "\<exists>!x. P x" "\<And>x. \<lbrakk>P x; \<exists>\<^sub>\<le>\<^sub>1x. P x\<rbrakk> \<Longrightarrow> R"
  shows R
  using assms unfolding Uniq_def by fast

lemma ex1_iff_ex_Uniq: "(\<exists>!x. P x) \<longleftrightarrow> (\<exists>x. P x) \<and> (\<exists>\<^sub>\<le>\<^sub>1x. P x)"
  unfolding Uniq_def by fast


ML \<open>
  structure Blast = Blast
  (
    structure Classical = Classical
    val Trueprop_const = dest_Const \<^Const>\<open>Trueprop\<close>
    val equality_name = \<^const_name>\<open>HOL.eq\<close>
    val not_name = \<^const_name>\<open>Not\<close>
    val notE = @{thm notE}
    val ccontr = @{thm ccontr}
    val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
  );
  val blast_tac = Blast.blast_tac;
\<close>


subsubsection \<open>THE: definite description operator\<close>

lemma the_equality [intro]:
  assumes "P a"
    and "\<And>x. P x \<Longrightarrow> x = a"
  shows "(THE x. P x) = a"
  by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])

lemma theI:
  assumes "P a"
    and "\<And>x. P x \<Longrightarrow> x = a"
  shows "P (THE x. P x)"
  by (iprover intro: assms the_equality [THEN ssubst])

lemma theI': "\<exists>!x. P x \<Longrightarrow> P (THE x. P x)"
  by (blast intro: theI)

text \<open>Easier to apply than \<open>theI\<close>: only one occurrence of \<open>P\<close>.\<close>
lemma theI2:
  assumes "P a" "\<And>x. P x \<Longrightarrow> x = a" "\<And>x. P x \<Longrightarrow> Q x"
  shows "Q (THE x. P x)"
  by (iprover intro: assms theI)

lemma the1I2:
  assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x"
  shows "Q (THE x. P x)"
  by (iprover intro: assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)] elim: allE impE)

lemma the1_equality [elim?]: "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> (THE x. P x) = a"
  by blast

lemma the1_equality': "\<lbrakk>\<exists>\<^sub>\<le>\<^sub>1x. P x; P a\<rbrakk> \<Longrightarrow> (THE x. P x) = a"
  unfolding Uniq_def by blast

lemma the_sym_eq_trivial: "(THE y. x = y) = x"
  by blast


subsubsection \<open>Simplifier\<close>

lemma eta_contract_eq: "(\<lambda>s. f s) = f" ..

lemma subst_all:
  \<open>(\<And>x. x = a \<Longrightarrow> PROP P x) \<equiv> PROP P a\<close>
  \<open>(\<And>x. a = x \<Longrightarrow> PROP P x) \<equiv> PROP P a\<close>
proof -
  show \<open>(\<And>x. x = a \<Longrightarrow> PROP P x) \<equiv> PROP P a\<close>
  proof (rule equal_intr_rule)
    assume *: \<open>\<And>x. x = a \<Longrightarrow> PROP P x\<close>
    show \<open>PROP P a\<close>
      by (rule *) (rule refl)
  next
    fix x
    assume \<open>PROP P a\<close> and \<open>x = a\<close>
    from \<open>x = a\<close> have \<open>x \<equiv> a\<close>
      by (rule eq_reflection)
    with \<open>PROP P a\<close> show \<open>PROP P x\<close>
      by simp
  qed
  show \<open>(\<And>x. a = x \<Longrightarrow> PROP P x) \<equiv> PROP P a\<close>
  proof (rule equal_intr_rule)
    assume *: \<open>\<And>x. a = x \<Longrightarrow> PROP P x\<close>
    show \<open>PROP P a\<close>
      by (rule *) (rule refl)
  next
    fix x
    assume \<open>PROP P a\<close> and \<open>a = x\<close>
    from \<open>a = x\<close> have \<open>a \<equiv> x\<close>
      by (rule eq_reflection)
    with \<open>PROP P a\<close> show \<open>PROP P x\<close>
      by simp
  qed
qed

lemma simp_thms:
  shows not_not: "(\<not> \<not> P) = P"
  and Not_eq_iff: "((\<not> P) = (\<not> Q)) = (P = Q)"
  and
    "(P \<noteq> Q) = (P = (\<not> Q))"
    "(P \<or> \<not>P) = True"    "(\<not> P \<or> P) = True"
    "(x = x) = True"
  and not_True_eq_False [code]: "(\<not> True) = False"
  and not_False_eq_True [code]: "(\<not> False) = True"
  and
    "(\<not> P) \<noteq> P"  "P \<noteq> (\<not> P)"
    "(True = P) = P"
  and eq_True: "(P = True) = P"
  and "(False = P) = (\<not> P)"
  and eq_False: "(P = False) = (\<not> P)"
  and
    "(True \<longrightarrow> P) = P"  "(False \<longrightarrow> P) = True"
    "(P \<longrightarrow> True) = True"  "(P \<longrightarrow> P) = True"
    "(P \<longrightarrow> False) = (\<not> P)"  "(P \<longrightarrow> \<not> P) = (\<not> P)"
    "(P \<and> True) = P"  "(True \<and> P) = P"
    "(P \<and> False) = False"  "(False \<and> P) = False"
    "(P \<and> P) = P"  "(P \<and> (P \<and> Q)) = (P \<and> Q)"
    "(P \<and> \<not> P) = False"    "(\<not> P \<and> P) = False"
    "(P \<or> True) = True"  "(True \<or> P) = True"
    "(P \<or> False) = P"  "(False \<or> P) = P"
    "(P \<or> P) = P"  "(P \<or> (P \<or> Q)) = (P \<or> Q)" and
    "(\<forall>x. P) = P"  "(\<exists>x. P) = P"  "\<exists>x. x = t"  "\<exists>x. t = x"
  and
    "\<And>P. (\<exists>x. x = t \<and> P x) = P t"
    "\<And>P. (\<exists>x. t = x \<and> P x) = P t"
    "\<And>P. (\<forall>x. x = t \<longrightarrow> P x) = P t"
    "\<And>P. (\<forall>x. t = x \<longrightarrow> P x) = P t"
    "(\<forall>x. x \<noteq> t) = False"  "(\<forall>x. t \<noteq> x) = False"
  by (blast, blast, blast, blast, blast, iprover+)

lemma disj_absorb: "A \<or> A \<longleftrightarrow> A"
  by blast

lemma disj_left_absorb: "A \<or> (A \<or> B) \<longleftrightarrow> A \<or> B"
  by blast

lemma conj_absorb: "A \<and> A \<longleftrightarrow> A"
  by blast

lemma conj_left_absorb: "A \<and> (A \<and> B) \<longleftrightarrow> A \<and> B"
  by blast

lemma eq_ac:
  shows eq_commute: "a = b \<longleftrightarrow> b = a"
    and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
    and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))"
  by (iprover, blast+)

lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover

lemma conj_comms:
  shows conj_commute: "P \<and> Q \<longleftrightarrow> Q \<and> P"
    and conj_left_commute: "P \<and> (Q \<and> R) \<longleftrightarrow> Q \<and> (P \<and> R)" by iprover+
lemma conj_assoc: "(P \<and> Q) \<and> R \<longleftrightarrow> P \<and> (Q \<and> R)" by iprover

lemmas conj_ac = conj_commute conj_left_commute conj_assoc

lemma disj_comms:
  shows disj_commute: "P \<or> Q \<longleftrightarrow> Q \<or> P"
    and disj_left_commute: "P \<or> (Q \<or> R) \<longleftrightarrow> Q \<or> (P \<or> R)" by iprover+
lemma disj_assoc: "(P \<or> Q) \<or> R \<longleftrightarrow> P \<or> (Q \<or> R)" by iprover

lemmas disj_ac = disj_commute disj_left_commute disj_assoc

lemma conj_disj_distribL: "P \<and> (Q \<or> R) \<longleftrightarrow> P \<and> Q \<or> P \<and> R" by iprover
lemma conj_disj_distribR: "(P \<or> Q) \<and> R \<longleftrightarrow> P \<and> R \<or> Q \<and> R" by iprover

lemma disj_conj_distribL: "P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)" by iprover
lemma disj_conj_distribR: "(P \<and> Q) \<or> R \<longleftrightarrow> (P \<or> R) \<and> (Q \<or> R)" by iprover

lemma imp_conjR: "(P \<longrightarrow> (Q \<and> R)) = ((P \<longrightarrow> Q) \<and> (P \<longrightarrow> R))" by iprover
lemma imp_conjL: "((P \<and> Q) \<longrightarrow> R) = (P \<longrightarrow> (Q \<longrightarrow> R))" by iprover
lemma imp_disjL: "((P \<or> Q) \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" by iprover

text \<open>These two are specialized, but \<open>imp_disj_not1\<close> is useful in \<open>Auth/Yahalom\<close>.\<close>
lemma imp_disj_not1: "(P \<longrightarrow> Q \<or> R) \<longleftrightarrow> (\<not> Q \<longrightarrow> P \<longrightarrow> R)" by blast
lemma imp_disj_not2: "(P \<longrightarrow> Q \<or> R) \<longleftrightarrow> (\<not> R \<longrightarrow> P \<longrightarrow> Q)" by blast

lemma imp_disj1: "((P \<longrightarrow> Q) \<or> R) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)" by blast
lemma imp_disj2: "(Q \<or> (P \<longrightarrow> R)) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)" by blast

lemma imp_cong: "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<longrightarrow> Q) \<longleftrightarrow> (P' \<longrightarrow> Q'))"
  by iprover

lemma de_Morgan_disj: "\<not> (P \<or> Q) \<longleftrightarrow> \<not> P \<and> \<not> Q" by iprover
lemma de_Morgan_conj: "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q" by blast
lemma not_imp: "\<not> (P \<longrightarrow> Q) \<longleftrightarrow> P \<and> \<not> Q" by blast
lemma not_iff: "P \<noteq> Q \<longleftrightarrow> (P \<longleftrightarrow> \<not> Q)" by blast
lemma disj_not1: "\<not> P \<or> Q \<longleftrightarrow> (P \<longrightarrow> Q)" by blast
lemma disj_not2: "P \<or> \<not> Q \<longleftrightarrow> (Q \<longrightarrow> P)" by blast  \<comment> \<open>changes orientation :-(\<close>
lemma imp_conv_disj: "(P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P) \<or> Q" by blast
lemma disj_imp: "P \<or> Q \<longleftrightarrow> \<not> P \<longrightarrow> Q" by blast

lemma iff_conv_conj_imp: "(P \<longleftrightarrow> Q) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)" by iprover


lemma cases_simp: "(P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q) \<longleftrightarrow> Q"
  \<comment> \<open>Avoids duplication of subgoals after \<open>if_split\<close>, when the true and false\<close>
  \<comment> \<open>cases boil down to the same thing.\<close>
  by blast

lemma not_all: "\<not> (\<forall>x. P x) \<longleftrightarrow> (\<exists>x. \<not> P x)" by blast
lemma imp_all: "((\<forall>x. P x) \<longrightarrow> Q) \<longleftrightarrow> (\<exists>x. P x \<longrightarrow> Q)" by blast
lemma not_ex: "\<not> (\<exists>x. P x) \<longleftrightarrow> (\<forall>x. \<not> P x)" by iprover
lemma imp_ex: "((\<exists>x. P x) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q)" by iprover
lemma all_not_ex: "(\<forall>x. P x) \<longleftrightarrow> \<not> (\<exists>x. \<not> P x)" by blast

declare All_def [no_atp]

lemma ex_disj_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>x. P x) \<or> (\<exists>x. Q x)" by iprover
lemma all_conj_distrib: "(\<forall>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>x. P x) \<and> (\<forall>x. Q x)" by iprover

text \<open>
  \<^medskip> The \<open>\<and>\<close> congruence rule: not included by default!
  May slow rewrite proofs down by as much as 50\%\<close>

lemma conj_cong: "P = P' \<Longrightarrow> (P' \<Longrightarrow> Q = Q') \<Longrightarrow> (P \<and> Q) = (P' \<and> Q')"
  by iprover

lemma rev_conj_cong: "Q = Q' \<Longrightarrow> (Q' \<Longrightarrow> P = P') \<Longrightarrow> (P \<and> Q) = (P' \<and> Q')"
  by iprover

text \<open>The \<open>|\<close> congruence rule: not included by default!\<close>

lemma disj_cong: "P = P' \<Longrightarrow> (\<not> P' \<Longrightarrow> Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
  by blast


text \<open>\<^medskip> if-then-else rules\<close>

lemma if_True [code]: "(if True then x else y) = x"
  unfolding If_def by blast

lemma if_False [code]: "(if False then x else y) = y"
  unfolding If_def by blast

lemma if_P: "P \<Longrightarrow> (if P then x else y) = x"
  unfolding If_def by blast

lemma if_not_P: "\<not> P \<Longrightarrow> (if P then x else y) = y"
  unfolding If_def by blast

lemma if_split: "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
proof (rule case_split [of Q])
  show ?thesis if Q
    using that by (simplesubst if_P) blast+
  show ?thesis if "\<not> Q"
    using that by (simplesubst if_not_P) blast+
qed

lemma if_split_asm: "P (if Q then x else y) = (\<not> ((Q \<and> \<not> P x) \<or> (\<not> Q \<and> \<not> P y)))"
  by (simplesubst if_split) blast

lemmas if_splits [no_atp] = if_split if_split_asm

lemma if_cancel: "(if c then x else x) = x"
  by (simplesubst if_split) blast

lemma if_eq_cancel: "(if x = y then y else x) = x"
  by (simplesubst if_split) blast

lemma if_bool_eq_conj: "(if P then Q else R) = ((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R))"
  \<comment> \<open>This form is useful for expanding \<open>if\<close>s on the RIGHT of the \<open>\<Longrightarrow>\<close> symbol.\<close>
  by (rule if_split)

lemma if_bool_eq_disj: "(if P then Q else R) = ((P \<and> Q) \<or> (\<not> P \<and> R))"
  \<comment> \<open>And this form is useful for expanding \<open>if\<close>s on the LEFT.\<close>
  by (simplesubst if_split) blast

lemma Eq_TrueI: "P \<Longrightarrow> P \<equiv> True" unfolding atomize_eq by iprover
lemma Eq_FalseI: "\<not> P \<Longrightarrow> P \<equiv> False" unfolding atomize_eq by iprover

text \<open>\<^medskip> let rules for simproc\<close>

lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
  by (unfold Let_def)

lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
  by (unfold Let_def)

text \<open>
  The following copy of the implication operator is useful for
  fine-tuning congruence rules.  It instructs the simplifier to simplify
  its premise.
\<close>

definition simp_implies :: "prop \<Rightarrow> prop \<Rightarrow> prop"  (infixr "=simp=>" 1)
  where "simp_implies \<equiv> (\<Longrightarrow>)"

lemma simp_impliesI:
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
  shows "PROP P =simp=> PROP Q"
  unfolding simp_implies_def
  by (iprover intro: PQ)

lemma simp_impliesE:
  assumes PQ: "PROP P =simp=> PROP Q"
    and P: "PROP P"
    and QR: "PROP Q \<Longrightarrow> PROP R"
  shows "PROP R"
  by (iprover intro: QR P PQ [unfolded simp_implies_def])

lemma simp_implies_cong:
  assumes PP' :"PROP P \<equiv> PROP P'"
    and P'QQ': "PROP P' \<Longrightarrow> (PROP Q \<equiv> PROP Q')"
  shows "(PROP P =simp=> PROP Q) \<equiv> (PROP P' =simp=> PROP Q')"
  unfolding simp_implies_def
proof (rule equal_intr_rule)
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
    and P': "PROP P'"
  from PP' [symmetric] and P' have "PROP P"
    by (rule equal_elim_rule1)
  then have "PROP Q" by (rule PQ)
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
next
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
    and P: "PROP P"
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
  then have "PROP Q'" by (rule P'Q')
  with P'QQ' [OF P', symmetric] show "PROP Q"
    by (rule equal_elim_rule1)
qed

lemma uncurry:
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
  shows "P \<and> Q \<longrightarrow> R"
  using assms by blast

lemma iff_allI:
  assumes "\<And>x. P x = Q x"
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
  using assms by blast

lemma iff_exI:
  assumes "\<And>x. P x = Q x"
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
  using assms by blast

lemma all_comm: "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
  by blast

lemma ex_comm: "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
  by blast

ML_file \<open>Tools/simpdata.ML\<close>
ML \<open>open Simpdata\<close>

setup \<open>
  map_theory_simpset (put_simpset HOL_basic_ss) #>
  Simplifier.method_setup Splitter.split_modifiers
\<close>

simproc_setup defined_Ex ("\<exists>x. P x") = \<open>K Quantifier1.rearrange_Ex\<close>
simproc_setup defined_All ("\<forall>x. P x") = \<open>K Quantifier1.rearrange_All\<close>
simproc_setup defined_all("\<And>x. PROP P x") = \<open>K Quantifier1.rearrange_all\<close>

text \<open>Simproc for proving \<open>(y = x) \<equiv> False\<close> from premise \<open>\<not> (x = y)\<close>:\<close>

simproc_setup neq ("x = y") = \<open>
  let
    val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
    fun is_neq eq lhs rhs thm =
      (case Thm.prop_of thm of
        _ $ (Not $ (eq' $ l' $ r')) =>
          Not = HOLogic.Not andalso eq' = eq andalso
          r' aconv lhs andalso l' aconv rhs
      | _ => false);
    fun proc ss ct =
      (case Thm.term_of ct of
        eq $ lhs $ rhs =>
          (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
            SOME thm => SOME (thm RS neq_to_EQ_False)
          | NONE => NONE)
       | _ => NONE);
  in K proc end
\<close>

simproc_setup let_simp ("Let x f") = \<open>
  let
    fun count_loose (Bound i) k = if i >= k then 1 else 0
      | count_loose (s $ t) k = count_loose s k + count_loose t k
      | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
      | count_loose _ _ = 0;
    fun is_trivial_let (Const (\<^const_name>\<open>Let\<close>, _) $ x $ t) =
      (case t of
        Abs (_, _, t') => count_loose t' 0 <= 1
      | _ => true);
  in
    K (fn ctxt => fn ct =>
      if is_trivial_let (Thm.term_of ct)
      then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
      else
        let (*Norbert Schirmer's case*)
          val t = Thm.term_of ct;
          val (t', ctxt') = yield_singleton (Variable.import_terms false) t ctxt;
        in
          Option.map (hd o Variable.export ctxt' ctxt o single)
            (case t' of Const (\<^const_name>\<open>Let\<close>,_) $ x $ f => (* x and f are already in normal form *)
              if is_Free x orelse is_Bound x orelse is_Const x
              then SOME @{thm Let_def}
              else
                let
                  val n = case f of (Abs (x, _, _)) => x | _ => "x";
                  val cx = Thm.cterm_of ctxt x;
                  val xT = Thm.typ_of_cterm cx;
                  val cf = Thm.cterm_of ctxt f;
                  val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
                  val (_ $ _ $ g) = Thm.prop_of fx_g;
                  val g' = abstract_over (x, g);
                  val abs_g'= Abs (n, xT, g');
                in
                  if g aconv g' then
                    let
                      val rl =
                        infer_instantiate ctxt [(("f", 0), cf), (("x", 0), cx)] @{thm Let_unfold};
                    in SOME (rl OF [fx_g]) end
                  else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
                  then NONE (*avoid identity conversion*)
                  else
                    let
                      val g'x = abs_g' $ x;
                      val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
                      val rl =
                        @{thm Let_folded} |> infer_instantiate ctxt
                          [(("f", 0), Thm.cterm_of ctxt f),
                           (("x", 0), cx),
                           (("g", 0), Thm.cterm_of ctxt abs_g')];
                    in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
                end
            | _ => NONE)
        end)
  end
\<close>

lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
proof
  assume "True \<Longrightarrow> PROP P"
  from this [OF TrueI] show "PROP P" .
next
  assume "PROP P"
  then show "PROP P" .
qed

lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
  by standard (intro TrueI)

lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
  by standard simp_all

(* It seems that making this a simp rule is slower than using the simproc below *)
lemma implies_False_swap:
  "(False \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> False \<Longrightarrow> PROP Q)"
  by (rule swap_prems_eq)

ML \<open>
fun eliminate_false_implies ct =
  let
    val (prems, concl) = Logic.strip_horn (Thm.term_of ct)
    fun go n =
      if n > 1 then
        Conv.rewr_conv @{thm Pure.swap_prems_eq}
        then_conv Conv.arg_conv (go (n - 1))
        then_conv Conv.rewr_conv @{thm HOL.implies_True_equals}
      else
        Conv.rewr_conv @{thm HOL.False_implies_equals}
  in
    case concl of
      Const (@{const_name HOL.Trueprop}, _) $ _ => SOME (go (length prems) ct)
    | _ => NONE
  end
\<close>

simproc_setup eliminate_false_implies ("False \<Longrightarrow> PROP P") = \<open>K (K eliminate_false_implies)\<close>


lemma ex_simps:
  "\<And>P Q. (\<exists>x. P x \<and> Q)   = ((\<exists>x. P x) \<and> Q)"
  "\<And>P Q. (\<exists>x. P \<and> Q x)   = (P \<and> (\<exists>x. Q x))"
  "\<And>P Q. (\<exists>x. P x \<or> Q)   = ((\<exists>x. P x) \<or> Q)"
  "\<And>P Q. (\<exists>x. P \<or> Q x)   = (P \<or> (\<exists>x. Q x))"
  "\<And>P Q. (\<exists>x. P x \<longrightarrow> Q) = ((\<forall>x. P x) \<longrightarrow> Q)"
  "\<And>P Q. (\<exists>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<exists>x. Q x))"
  \<comment> \<open>Miniscoping: pushing in existential quantifiers.\<close>
  by (iprover | blast)+

lemma all_simps:
  "\<And>P Q. (\<forall>x. P x \<and> Q)   = ((\<forall>x. P x) \<and> Q)"
  "\<And>P Q. (\<forall>x. P \<and> Q x)   = (P \<and> (\<forall>x. Q x))"
  "\<And>P Q. (\<forall>x. P x \<or> Q)   = ((\<forall>x. P x) \<or> Q)"
  "\<And>P Q. (\<forall>x. P \<or> Q x)   = (P \<or> (\<forall>x. Q x))"
  "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q) = ((\<exists>x. P x) \<longrightarrow> Q)"
  "\<And>P Q. (\<forall>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<forall>x. Q x))"
  \<comment> \<open>Miniscoping: pushing in universal quantifiers.\<close>
  by (iprover | blast)+

lemmas [simp] =
  triv_forall_equality  \<comment> \<open>prunes params\<close>
  True_implies_equals implies_True_equals  \<comment> \<open>prune \<open>True\<close> in asms\<close>
  False_implies_equals  \<comment> \<open>prune \<open>False\<close> in asms\<close>
  if_True
  if_False
  if_cancel
  if_eq_cancel
  imp_disjL \<comment> \<open>In general it seems wrong to add distributive laws by default: they
    might cause exponential blow-up.  But \<open>imp_disjL\<close> has been in for a while
    and cannot be removed without affecting existing proofs.  Moreover,
    rewriting by \<open>(P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))\<close> might be justified on the
    grounds that it allows simplification of \<open>R\<close> in the two cases.\<close>
  conj_assoc
  disj_assoc
  de_Morgan_conj
  de_Morgan_disj
  imp_disj1
  imp_disj2
  not_imp
  disj_not1
  not_all
  not_ex
  cases_simp
  the_eq_trivial
  the_sym_eq_trivial
  ex_simps
  all_simps
  simp_thms
  subst_all

lemmas [cong] = imp_cong simp_implies_cong
lemmas [split] = if_split

ML \<open>val HOL_ss = simpset_of \<^context>\<close>

text \<open>Simplifies \<open>x\<close> assuming \<open>c\<close> and \<open>y\<close> assuming \<open>\<not> c\<close>.\<close>
lemma if_cong:
  assumes "b = c"
    and "c \<Longrightarrow> x = u"
    and "\<not> c \<Longrightarrow> y = v"
  shows "(if b then x else y) = (if c then u else v)"
  using assms by simp

text \<open>Prevents simplification of \<open>x\<close> and \<open>y\<close>:
  faster and allows the execution of functional programs.\<close>
lemma if_weak_cong [cong]:
  assumes "b = c"
  shows "(if b then x else y) = (if c then x else y)"
  using assms by (rule arg_cong)

text \<open>Prevents simplification of t: much faster\<close>
lemma let_weak_cong:
  assumes "a = b"
  shows "(let x = a in t x) = (let x = b in t x)"
  using assms by (rule arg_cong)

text \<open>To tidy up the result of a simproc.  Only the RHS will be simplified.\<close>
lemma eq_cong2:
  assumes "u = u'"
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
  using assms by simp

lemma if_distrib: "f (if c then x else y) = (if c then f x else f y)"
  by simp

lemma if_distribR: "(if b then f else g) x = (if b then f x else g x)"
  by simp

lemma all_if_distrib: "(\<forall>x. if x = a then P x else Q x) \<longleftrightarrow> P a \<and> (\<forall>x. x\<noteq>a \<longrightarrow> Q x)"
  by auto

lemma ex_if_distrib: "(\<exists>x. if x = a then P x else Q x) \<longleftrightarrow> P a \<or> (\<exists>x. x\<noteq>a \<and> Q x)"
  by auto

lemma if_if_eq_conj: "(if P then if Q then x else y else y) = (if P \<and> Q then x else y)"
  by simp

text \<open>As a simplification rule, it replaces all function equalities by
  first-order equalities.\<close>
lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
  by auto


subsubsection \<open>Generic cases and induction\<close>

text \<open>Rule projections:\<close>
ML \<open>
structure Project_Rule = Project_Rule
(
  val conjunct1 = @{thm conjunct1}
  val conjunct2 = @{thm conjunct2}
  val mp = @{thm mp}
);
\<close>

context
begin

qualified definition "induct_forall P \<equiv> \<forall>x. P x"
qualified definition "induct_implies A B \<equiv> A \<longrightarrow> B"
qualified definition "induct_equal x y \<equiv> x = y"
qualified definition "induct_conj A B \<equiv> A \<and> B"
qualified definition "induct_true \<equiv> True"
qualified definition "induct_false \<equiv> False"

lemma induct_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (induct_forall (\<lambda>x. P x))"
  by (unfold atomize_all induct_forall_def)

lemma induct_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (induct_implies A B)"
  by (unfold atomize_imp induct_implies_def)

lemma induct_equal_eq: "(x \<equiv> y) \<equiv> Trueprop (induct_equal x y)"
  by (unfold atomize_eq induct_equal_def)

lemma induct_conj_eq: "(A &&& B) \<equiv> Trueprop (induct_conj A B)"
  by (unfold atomize_conj induct_conj_def)

lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
lemmas induct_atomize = induct_atomize' induct_equal_eq
lemmas induct_rulify' [symmetric] = induct_atomize'
lemmas induct_rulify [symmetric] = induct_atomize
lemmas induct_rulify_fallback =
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
  induct_true_def induct_false_def

lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
    induct_conj (induct_forall A) (induct_forall B)"
  by (unfold induct_forall_def induct_conj_def) iprover

lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
    induct_conj (induct_implies C A) (induct_implies C B)"
  by (unfold induct_implies_def induct_conj_def) iprover

lemma induct_conj_curry: "(induct_conj A B \<Longrightarrow> PROP C) \<equiv> (A \<Longrightarrow> B \<Longrightarrow> PROP C)"
proof
  assume r: "induct_conj A B \<Longrightarrow> PROP C"
  assume ab: A B
  show "PROP C" by (rule r) (simp add: induct_conj_def ab)
next
  assume r: "A \<Longrightarrow> B \<Longrightarrow> PROP C"
  assume ab: "induct_conj A B"
  show "PROP C" by (rule r) (simp_all add: ab [unfolded induct_conj_def])
qed

lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry

lemma induct_trueI: "induct_true"
  by (simp add: induct_true_def)

text \<open>Method setup.\<close>

ML_file \<open>~~/src/Tools/induct.ML\<close>
ML \<open>
structure Induct = Induct
(
  val cases_default = @{thm case_split}
  val atomize = @{thms induct_atomize}
  val rulify = @{thms induct_rulify'}
  val rulify_fallback = @{thms induct_rulify_fallback}
  val equal_def = @{thm induct_equal_def}
  fun dest_def (Const (\<^const_name>\<open>induct_equal\<close>, _) $ t $ u) = SOME (t, u)
    | dest_def _ = NONE
  fun trivial_tac ctxt = match_tac ctxt @{thms induct_trueI}
)
\<close>

ML_file \<open>~~/src/Tools/induction.ML\<close>

simproc_setup passive swap_induct_false ("induct_false \<Longrightarrow> PROP P \<Longrightarrow> PROP Q") =
  \<open>fn _ => fn _ => fn ct =>
    (case Thm.term_of ct of
      _ $ (P as _ $ \<^Const_>\<open>induct_false\<close>) $ (_ $ Q $ _) =>
        if P <> Q then SOME Drule.swap_prems_eq else NONE
    | _ => NONE)\<close>

simproc_setup passive induct_equal_conj_curry ("induct_conj P Q \<Longrightarrow> PROP R") =
  \<open>fn _ => fn _ => fn ct =>
    (case Thm.term_of ct of
      _ $ (_ $ P) $ _ =>
        let
          fun is_conj \<^Const_>\<open>induct_conj for P Q\<close> = is_conj P andalso is_conj Q
            | is_conj \<^Const_>\<open>induct_equal _ for _ _\<close> = true
            | is_conj \<^Const_>\<open>induct_true\<close> = true
            | is_conj \<^Const_>\<open>induct_false\<close> = true
            | is_conj _ = false
        in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
      | _ => NONE)\<close>

declaration \<open>
  K (Induct.map_simpset (fn ss => ss
      addsimprocs [\<^simproc>\<open>swap_induct_false\<close>, \<^simproc>\<open>induct_equal_conj_curry\<close>]
    |> Simplifier.set_mksimps (fn ctxt =>
      Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
      map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback})))))
\<close>

text \<open>Pre-simplification of induction and cases rules\<close>

lemma [induct_simp]: "(\<And>x. induct_equal x t \<Longrightarrow> PROP P x) \<equiv> PROP P t"
  unfolding induct_equal_def
proof
  assume r: "\<And>x. x = t \<Longrightarrow> PROP P x"
  show "PROP P t" by (rule r [OF refl])
next
  fix x
  assume "PROP P t" "x = t"
  then show "PROP P x" by simp
qed

lemma [induct_simp]: "(\<And>x. induct_equal t x \<Longrightarrow> PROP P x) \<equiv> PROP P t"
  unfolding induct_equal_def
proof
  assume r: "\<And>x. t = x \<Longrightarrow> PROP P x"
  show "PROP P t" by (rule r [OF refl])
next
  fix x
  assume "PROP P t" "t = x"
  then show "PROP P x" by simp
qed

lemma [induct_simp]: "(induct_false \<Longrightarrow> P) \<equiv> Trueprop induct_true"
  unfolding induct_false_def induct_true_def
  by (iprover intro: equal_intr_rule)

lemma [induct_simp]: "(induct_true \<Longrightarrow> PROP P) \<equiv> PROP P"
  unfolding induct_true_def
proof
  assume "True \<Longrightarrow> PROP P"
  then show "PROP P" using TrueI .
next
  assume "PROP P"
  then show "PROP P" .
qed

lemma [induct_simp]: "(PROP P \<Longrightarrow> induct_true) \<equiv> Trueprop induct_true"
  unfolding induct_true_def
  by (iprover intro: equal_intr_rule)

lemma [induct_simp]: "(\<And>x::'a::{}. induct_true) \<equiv> Trueprop induct_true"
  unfolding induct_true_def
  by (iprover intro: equal_intr_rule)

lemma [induct_simp]: "induct_implies induct_true P \<equiv> P"
  by (simp add: induct_implies_def induct_true_def)

lemma [induct_simp]: "x = x \<longleftrightarrow> True"
  by (rule simp_thms)

end

ML_file \<open>~~/src/Tools/induct_tacs.ML\<close>


subsubsection \<open>Coherent logic\<close>

ML_file \<open>~~/src/Tools/coherent.ML\<close>
ML \<open>
structure Coherent = Coherent
(
  val atomize_elimL = @{thm atomize_elimL};
  val atomize_exL = @{thm atomize_exL};
  val atomize_conjL = @{thm atomize_conjL};
  val atomize_disjL = @{thm atomize_disjL};
  val operator_names = [\<^const_name>\<open>HOL.disj\<close>, \<^const_name>\<open>HOL.conj\<close>, \<^const_name>\<open>Ex\<close>];
);
\<close>


subsubsection \<open>Reorienting equalities\<close>

ML \<open>
signature REORIENT_PROC =
sig
  val add : (term -> bool) -> theory -> theory
  val proc : Proof.context -> cterm -> thm option
end;

structure Reorient_Proc : REORIENT_PROC =
struct
  structure Data = Theory_Data
  (
    type T = ((term -> bool) * stamp) list;
    val empty = [];
    fun merge data : T = Library.merge (eq_snd (op =)) data;
  );
  fun add m = Data.map (cons (m, stamp ()));
  fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);

  val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
  fun proc ctxt ct =
    let
      val thy = Proof_Context.theory_of ctxt;
    in
      case Thm.term_of ct of
        (_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
      | _ => NONE
    end;
end;
\<close>


subsection \<open>Other simple lemmas and lemma duplicates\<close>

lemma eq_iff_swap: "(x = y \<longleftrightarrow> P) \<Longrightarrow> (y = x \<longleftrightarrow> P)"
by blast

lemma all_cong1: "(\<And>x. P x = P' x) \<Longrightarrow> (\<forall>x. P x) = (\<forall>x. P' x)"
  by auto

lemma ex_cong1: "(\<And>x. P x = P' x) \<Longrightarrow> (\<exists>x. P x) = (\<exists>x. P' x)"
  by auto

lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> (\<forall>x. Q x \<longrightarrow> P x) = (\<forall>x. Q x \<longrightarrow> P' x)"
  by auto

lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> (\<exists>x. Q x \<and> P x) = (\<exists>x. Q x \<and> P' x)"
  by auto

lemma ex1_eq [iff]: "\<exists>!x. x = t" "\<exists>!x. t = x"
  by blast+

lemma choice_eq: "(\<forall>x. \<exists>!y. P x y) = (\<exists>!f. \<forall>x. P x (f x))" (is "?lhs = ?rhs")
proof (intro iffI allI)
  assume L: ?lhs
  then have \<section>: "\<forall>x. P x (THE y. P x y)"
    by (best intro: theI')
  show ?rhs
    by (rule ex1I) (use L \<section> in \<open>fast+\<close>)
next
  fix x
  assume R: ?rhs
  then obtain f where f: "\<forall>x. P x (f x)" and f1: "\<And>y. (\<forall>x. P x (y x)) \<Longrightarrow> y = f"
    by (blast elim: ex1E)
  show "\<exists>!y. P x y"
  proof (rule ex1I)
    show "P x (f x)"
      using f by blast
    show "y = f x" if "P x y" for y
    proof -
      have "P z (if z = x then y else f z)" for z
        using f that by (auto split: if_split)
      with f1 [of "\<lambda>z. if z = x then y else f z"] f
      show ?thesis
        by (auto simp add: split: if_split_asm dest: fun_cong)
    qed
  qed
qed

lemmas eq_sym_conv = eq_commute

lemma nnf_simps:
  "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
  "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
  "(P \<longrightarrow> Q) = (\<not> P \<or> Q)"
  "(P = Q) = ((P \<and> Q) \<or> (\<not> P \<and> \<not> Q))"
  "(\<not> (P = Q)) = ((P \<and> \<not> Q) \<or> (\<not> P \<and> Q))"
  "(\<not> \<not> P) = P"
  by blast+


subsection \<open>Basic ML bindings\<close>

ML \<open>
val FalseE = @{thm FalseE}
val Let_def = @{thm Let_def}
val TrueI = @{thm TrueI}
val allE = @{thm allE}
val allI = @{thm allI}
val all_dupE = @{thm all_dupE}
val arg_cong = @{thm arg_cong}
val box_equals = @{thm box_equals}
val ccontr = @{thm ccontr}
val classical = @{thm classical}
val conjE = @{thm conjE}
val conjI = @{thm conjI}
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val disjCI = @{thm disjCI}
val disjE = @{thm disjE}
val disjI1 = @{thm disjI1}
val disjI2 = @{thm disjI2}
val eq_reflection = @{thm eq_reflection}
val ex1E = @{thm ex1E}
val ex1I = @{thm ex1I}
val ex1_implies_ex = @{thm ex1_implies_ex}
val exE = @{thm exE}
val exI = @{thm exI}
val excluded_middle = @{thm excluded_middle}
val ext = @{thm ext}
val fun_cong = @{thm fun_cong}
val iffD1 = @{thm iffD1}
val iffD2 = @{thm iffD2}
val iffI = @{thm iffI}
val impE = @{thm impE}
val impI = @{thm impI}
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
val mp = @{thm mp}
val notE = @{thm notE}
val notI = @{thm notI}
val not_all = @{thm not_all}
val not_ex = @{thm not_ex}
val not_iff = @{thm not_iff}
val not_not = @{thm not_not}
val not_sym = @{thm not_sym}
val refl = @{thm refl}
val rev_mp = @{thm rev_mp}
val spec = @{thm spec}
val ssubst = @{thm ssubst}
val subst = @{thm subst}
val sym = @{thm sym}
val trans = @{thm trans}
\<close>

locale cnf
begin

lemma clause2raw_notE: "\<lbrakk>P; \<not>P\<rbrakk> \<Longrightarrow> False" by auto
lemma clause2raw_not_disj: "\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> \<not> (P \<or> Q)" by auto
lemma clause2raw_not_not: "P \<Longrightarrow> \<not>\<not> P" by auto

lemma iff_refl: "(P::bool) = P" by auto
lemma iff_trans: "[| (P::bool) = Q; Q = R |] ==> P = R" by auto
lemma conj_cong: "[| P = P'; Q = Q' |] ==> (P \<and> Q) = (P' \<and> Q')" by auto
lemma disj_cong: "[| P = P'; Q = Q' |] ==> (P \<or> Q) = (P' \<or> Q')" by auto

lemma make_nnf_imp: "[| (\<not>P) = P'; Q = Q' |] ==> (P \<longrightarrow> Q) = (P' \<or> Q')" by auto
lemma make_nnf_iff: "[| P = P'; (\<not>P) = NP; Q = Q'; (\<not>Q) = NQ |] ==> (P = Q) = ((P' \<or> NQ) \<and> (NP \<or> Q'))" by auto
lemma make_nnf_not_false: "(\<not>False) = True" by auto
lemma make_nnf_not_true: "(\<not>True) = False" by auto
lemma make_nnf_not_conj: "[| (\<not>P) = P'; (\<not>Q) = Q' |] ==> (\<not>(P \<and> Q)) = (P' \<or> Q')" by auto
lemma make_nnf_not_disj: "[| (\<not>P) = P'; (\<not>Q) = Q' |] ==> (\<not>(P \<or> Q)) = (P' \<and> Q')" by auto
lemma make_nnf_not_imp: "[| P = P'; (\<not>Q) = Q' |] ==> (\<not>(P \<longrightarrow> Q)) = (P' \<and> Q')" by auto
lemma make_nnf_not_iff: "[| P = P'; (\<not>P) = NP; Q = Q'; (\<not>Q) = NQ |] ==> (\<not>(P = Q)) = ((P' \<or> Q') \<and> (NP \<or> NQ))" by auto
lemma make_nnf_not_not: "P = P' ==> (\<not>\<not>P) = P'" by auto

lemma simp_TF_conj_True_l: "[| P = True; Q = Q' |] ==> (P \<and> Q) = Q'" by auto
lemma simp_TF_conj_True_r: "[| P = P'; Q = True |] ==> (P \<and> Q) = P'" by auto
lemma simp_TF_conj_False_l: "P = False ==> (P \<and> Q) = False" by auto
lemma simp_TF_conj_False_r: "Q = False ==> (P \<and> Q) = False" by auto
lemma simp_TF_disj_True_l: "P = True ==> (P \<or> Q) = True" by auto
lemma simp_TF_disj_True_r: "Q = True ==> (P \<or> Q) = True" by auto
lemma simp_TF_disj_False_l: "[| P = False; Q = Q' |] ==> (P \<or> Q) = Q'" by auto
lemma simp_TF_disj_False_r: "[| P = P'; Q = False |] ==> (P \<or> Q) = P'" by auto

lemma make_cnf_disj_conj_l: "[| (P \<or> R) = PR; (Q \<or> R) = QR |] ==> ((P \<and> Q) \<or> R) = (PR \<and> QR)" by auto
lemma make_cnf_disj_conj_r: "[| (P \<or> Q) = PQ; (P \<or> R) = PR |] ==> (P \<or> (Q \<and> R)) = (PQ \<and> PR)" by auto

lemma make_cnfx_disj_ex_l: "((\<exists>(x::bool). P x) \<or> Q) = (\<exists>x. P x \<or> Q)" by auto
lemma make_cnfx_disj_ex_r: "(P \<or> (\<exists>(x::bool). Q x)) = (\<exists>x. P \<or> Q x)" by auto
lemma make_cnfx_newlit: "(P \<or> Q) = (\<exists>x. (P \<or> x) \<and> (Q \<or> \<not>x))" by auto
lemma make_cnfx_ex_cong: "(\<forall>(x::bool). P x = Q x) \<Longrightarrow> (\<exists>x. P x) = (\<exists>x. Q x)" by auto

lemma weakening_thm: "[| P; Q |] ==> Q" by auto

lemma cnftac_eq_imp: "[| P = Q; P |] ==> Q" by auto

end

ML_file \<open>Tools/cnf.ML\<close>


section \<open>\<open>NO_MATCH\<close> simproc\<close>

text \<open>
  The simplification procedure can be used to avoid simplification of terms
  of a certain form.
\<close>

definition NO_MATCH :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
  where "NO_MATCH pat val \<equiv> True"

lemma NO_MATCH_cong[cong]: "NO_MATCH pat val = NO_MATCH pat val"
  by (rule refl)

declare [[coercion_args NO_MATCH - -]]

simproc_setup NO_MATCH ("NO_MATCH pat val") = \<open>K (fn ctxt => fn ct =>
  let
    val thy = Proof_Context.theory_of ctxt
    val dest_binop = Term.dest_comb #> apfst (Term.dest_comb #> snd)
    val m = Pattern.matches thy (dest_binop (Thm.term_of ct))
  in if m then NONE else SOME @{thm NO_MATCH_def} end)
\<close>

text \<open>
  This setup ensures that a rewrite rule of the form \<^term>\<open>NO_MATCH pat val \<Longrightarrow> t\<close>
  is only applied, if the pattern \<open>pat\<close> does not match the value \<open>val\<close>.
\<close>


text\<open>
  Tagging a premise of a simp rule with ASSUMPTION forces the simplifier
  not to simplify the argument and to solve it by an assumption.
\<close>

definition ASSUMPTION :: "bool \<Rightarrow> bool"
  where "ASSUMPTION A \<equiv> A"

lemma ASSUMPTION_cong[cong]: "ASSUMPTION A = ASSUMPTION A"
  by (rule refl)

lemma ASSUMPTION_I: "A \<Longrightarrow> ASSUMPTION A"
  by (simp add: ASSUMPTION_def)

lemma ASSUMPTION_D: "ASSUMPTION A \<Longrightarrow> A"
  by (simp add: ASSUMPTION_def)

setup \<open>
let
  val asm_sol = mk_solver "ASSUMPTION" (fn ctxt =>
    resolve_tac ctxt [@{thm ASSUMPTION_I}] THEN'
    resolve_tac ctxt (Simplifier.prems_of ctxt))
in
  map_theory_simpset (fn ctxt => Simplifier.addSolver (ctxt,asm_sol))
end
\<close>


subsection \<open>Code generator setup\<close>

subsubsection \<open>Generic code generator preprocessor setup\<close>

lemma conj_left_cong: "P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
  by (fact arg_cong)

lemma disj_left_cong: "P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
  by (fact arg_cong)

setup \<open>
  Code_Preproc.map_pre (put_simpset HOL_basic_ss) #>
  Code_Preproc.map_post (put_simpset HOL_basic_ss) #>
  Code_Simp.map_ss (put_simpset HOL_basic_ss #>
  Simplifier.add_cong @{thm conj_left_cong} #>
  Simplifier.add_cong @{thm disj_left_cong})
\<close>


subsubsection \<open>Equality\<close>

class equal =
  fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
  assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
begin

lemma equal: "equal = (=)"
  by (rule ext equal_eq)+

lemma equal_refl: "equal x x \<longleftrightarrow> True"
  unfolding equal by (rule iffI TrueI refl)+

lemma eq_equal: "(=) \<equiv> equal"
  by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)

end

declare eq_equal [symmetric, code_post]
declare eq_equal [code]

simproc_setup passive equal (HOL.eq) =
  \<open>fn _ => fn _ => fn ct =>
    (case Thm.term_of ct of
      Const (_, Type (\<^type_name>\<open>fun\<close>, [Type _, _])) => SOME @{thm eq_equal}
    | _ => NONE)\<close>

setup \<open>Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs [\<^simproc>\<open>equal\<close>])\<close>


subsubsection \<open>Generic code generator foundation\<close>

text \<open>Datatype \<^typ>\<open>bool\<close>\<close>

code_datatype True False

lemma [code]:
  shows "False \<and> P \<longleftrightarrow> False"
    and "True \<and> P \<longleftrightarrow> P"
    and "P \<and> False \<longleftrightarrow> False"
    and "P \<and> True \<longleftrightarrow> P"
  by simp_all

lemma [code]:
  shows "False \<or> P \<longleftrightarrow> P"
    and "True \<or> P \<longleftrightarrow> True"
    and "P \<or> False \<longleftrightarrow> P"
    and "P \<or> True \<longleftrightarrow> True"
  by simp_all

lemma [code]:
  shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
    and "(True \<longrightarrow> P) \<longleftrightarrow> P"
    and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
    and "(P \<longrightarrow> True) \<longleftrightarrow> True"
  by simp_all

text \<open>More about \<^typ>\<open>prop\<close>\<close>

lemma [code nbe]:
  shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
    and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
    and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)"
  by (auto intro!: equal_intr_rule)

lemma Trueprop_code [code]: "Trueprop True \<equiv> Code_Generator.holds"
  by (auto intro!: equal_intr_rule holds)

declare Trueprop_code [symmetric, code_post]

text \<open>Equality\<close>

declare simp_thms(6) [code nbe]

instantiation itself :: (type) equal
begin

definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool"
  where "equal_itself x y \<longleftrightarrow> x = y"

instance
  by standard (fact equal_itself_def)

end

lemma equal_itself_code [code]: "equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
  by (simp add: equal)

setup \<open>Sign.add_const_constraint (\<^const_name>\<open>equal\<close>, SOME \<^typ>\<open>'a::type \<Rightarrow> 'a \<Rightarrow> bool\<close>)\<close>

lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> (((=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)"
  (is "?ofclass \<equiv> ?equal")
proof
  assume "PROP ?ofclass"
  show "PROP ?equal"
    by (tactic \<open>ALLGOALS (resolve_tac \<^context> [Thm.unconstrainT @{thm eq_equal}])\<close>)
      (fact \<open>PROP ?ofclass\<close>)
next
  assume "PROP ?equal"
  show "PROP ?ofclass" proof
  qed (simp add: \<open>PROP ?equal\<close>)
qed

setup \<open>Sign.add_const_constraint (\<^const_name>\<open>equal\<close>, SOME \<^typ>\<open>'a::equal \<Rightarrow> 'a \<Rightarrow> bool\<close>)\<close>

setup \<open>Nbe.add_const_alias @{thm equal_alias_cert}\<close>

text \<open>Cases\<close>

lemma Let_case_cert:
  assumes "CASE \<equiv> (\<lambda>x. Let x f)"
  shows "CASE x \<equiv> f x"
  using assms by simp_all

setup \<open>
  Code.declare_case_global @{thm Let_case_cert} #>
  Code.declare_undefined_global \<^const_name>\<open>undefined\<close>
\<close>

declare [[code abort: undefined]]


subsubsection \<open>Generic code generator target languages\<close>

text \<open>type \<^typ>\<open>bool\<close>\<close>

code_printing
  type_constructor bool \<rightharpoonup>
    (SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
| constant True \<rightharpoonup>
    (SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
| constant False \<rightharpoonup>
    (SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"

code_reserved SML
  bool true false

code_reserved OCaml
  bool

code_reserved Scala
  Boolean

code_printing
  constant Not \<rightharpoonup>
    (SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
| constant HOL.conj \<rightharpoonup>
    (SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
| constant HOL.disj \<rightharpoonup>
    (SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
| constant HOL.implies \<rightharpoonup>
    (SML) "!(if (_)/ then (_)/ else true)"
    and (OCaml) "!(if (_)/ then (_)/ else true)"
    and (Haskell) "!(if (_)/ then (_)/ else True)"
    and (Scala) "!(if ((_))/ (_)/ else true)"
| constant If \<rightharpoonup>
    (SML) "!(if (_)/ then (_)/ else (_))"
    and (OCaml) "!(if (_)/ then (_)/ else (_))"
    and (Haskell) "!(if (_)/ then (_)/ else (_))"
    and (Scala) "!(if ((_))/ (_)/ else (_))"

code_reserved SML
  not

code_reserved OCaml
  not

code_identifier
  code_module Pure \<rightharpoonup>
    (SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL

text \<open>Using built-in Haskell equality.\<close>
code_printing
  type_class equal \<rightharpoonup> (Haskell) "Eq"
| constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
| constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="

text \<open>\<open>undefined\<close>\<close>
code_printing
  constant undefined \<rightharpoonup>
    (SML) "!(raise/ Fail/ \"undefined\")"
    and (OCaml) "failwith/ \"undefined\""
    and (Haskell) "error/ \"undefined\""
    and (Scala) "!sys.error(\"undefined\")"


subsubsection \<open>Evaluation and normalization by evaluation\<close>

method_setup eval = \<open>
  let
    fun eval_tac ctxt =
      let val conv = Code_Runtime.dynamic_holds_conv
      in
        CONVERSION (Conv.params_conv ~1 (Conv.concl_conv ~1 o conv) ctxt) THEN'
        resolve_tac ctxt [TrueI]
      end
  in
    Scan.succeed (SIMPLE_METHOD' o eval_tac)
  end
\<close> "solve goal by evaluation"

method_setup normalization = \<open>
  Scan.succeed (fn ctxt =>
    SIMPLE_METHOD'
      (CHANGED_PROP o
        (CONVERSION (Nbe.dynamic_conv ctxt)
          THEN_ALL_NEW (TRY o resolve_tac ctxt [TrueI]))))
\<close> "solve goal by normalization"


subsection \<open>Counterexample Search Units\<close>

subsubsection \<open>Quickcheck\<close>

quickcheck_params [size = 5, iterations = 50]


subsubsection \<open>Nitpick setup\<close>

named_theorems nitpick_unfold "alternative definitions of constants as needed by Nitpick"
  and nitpick_simp "equational specification of constants as needed by Nitpick"
  and nitpick_psimp "partial equational specification of constants as needed by Nitpick"
  and nitpick_choice_spec "choice specification of constants as needed by Nitpick"

declare if_bool_eq_conj [nitpick_unfold, no_atp]
  and if_bool_eq_disj [no_atp]


subsection \<open>Preprocessing for the predicate compiler\<close>

named_theorems code_pred_def "alternative definitions of constants for the Predicate Compiler"
  and code_pred_inline "inlining definitions for the Predicate Compiler"
  and code_pred_simp "simplification rules for the optimisations in the Predicate Compiler"


subsection \<open>Legacy tactics and ML bindings\<close>

ML \<open>
  (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
  local
    fun wrong_prem (Const (\<^const_name>\<open>All\<close>, _) $ Abs (_, _, t)) = wrong_prem t
      | wrong_prem (Bound _) = true
      | wrong_prem _ = false;
    val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
    fun smp i = funpow i (fn m => filter_right ([spec] RL m)) [mp];
  in
    fun smp_tac ctxt j = EVERY' [dresolve_tac ctxt (smp j), assume_tac ctxt];
  end;

  local
    val nnf_ss =
      simpset_of (put_simpset HOL_basic_ss \<^context> addsimps @{thms simp_thms nnf_simps});
  in
    fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
  end
\<close>

hide_const (open) eq equal

end