(* 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" (\<open>(\<open>notation=judgment\<close>_)\<close> 5)
axiomatization implies :: "[bool, bool] \<Rightarrow> bool" (infixr \<open>\<longrightarrow>\<close> 25)
and eq :: "['a, 'a] \<Rightarrow> bool"
and The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
notation (input)
eq (infixl \<open>=\<close> 50)
notation (output)
eq (infix \<open>=\<close> 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 \<open>\<forall>\<close> 10)
where "All P \<equiv> (P = (\<lambda>x. True))"
definition Ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder \<open>\<exists>\<close> 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" (\<open>\<not> _\<close> [40] 40)
where not_def: "\<not> P \<equiv> P \<longrightarrow> False"
definition conj :: "[bool, bool] \<Rightarrow> bool" (infixr \<open>\<and>\<close> 35)
where and_def: "P \<and> Q \<equiv> \<forall>R. (P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> R"
definition disj :: "[bool, bool] \<Rightarrow> bool" (infixr \<open>\<or>\<close> 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" (\<open>(\<open>indent=4 notation=\<open>binder ?<\<close>\<close>?< _./ _)\<close> [0, 10] 10)
syntax "_Uniq" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool" (\<open>(\<open>indent=2 notation=\<open>binder \<exists>\<^sub>\<le>\<^sub>1\<close>\<close>\<exists>\<^sub>\<le>\<^sub>1 _./ _)\<close> [0, 10] 10)
syntax_consts "_Uniq" \<rightleftharpoons> Uniq
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" (\<open>(\<open>indent=3 notation=\<open>binder EX!\<close>\<close>EX! _./ _)\<close> [0, 10] 10)
syntax (input)
"_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool" (\<open>(\<open>indent=3 notation=\<open>binder ?!\<close>\<close>?! _./ _)\<close> [0, 10] 10)
syntax "_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool" (\<open>(\<open>indent=3 notation=\<open>binder \<exists>!\<close>\<close>\<exists>!_./ _)\<close> [0, 10] 10)
syntax_consts "_Ex1" \<rightleftharpoons> Ex1
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" (\<open>(\<open>indent=3 notation=\<open>binder \<nexists>\<close>\<close>\<nexists>_./ _)\<close> [0, 10] 10)
"_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool" (\<open>(\<open>indent=3 notation=\<open>binder \<nexists>!\<close>\<close>\<nexists>!_./ _)\<close> [0, 10] 10)
syntax_consts
"_Not_Ex" \<rightleftharpoons> Ex and
"_Not_Ex1" \<rightleftharpoons> Ex1
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 \<open>\<noteq>\<close> 50)
where "x \<noteq> y \<equiv> \<not> (x = y)"
notation (ASCII)
Not (\<open>~ _\<close> [40] 40) and
conj (infixr \<open>&\<close> 35) and
disj (infixr \<open>|\<close> 30) and
implies (infixr \<open>-->\<close> 25) and
not_equal (infix \<open>~=\<close> 50)
abbreviation (iff)
iff :: "[bool, bool] \<Rightarrow> bool" (infixr \<open>\<longleftrightarrow>\<close> 25)
where "A \<longleftrightarrow> B \<equiv> A = B"
syntax "_The" :: "[pttrn, bool] \<Rightarrow> 'a" (\<open>(\<open>indent=3 notation=\<open>binder THE\<close>\<close>THE _./ _)\<close> [0, 10] 10)
syntax_consts "_The" \<rightleftharpoons> The
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 case_syn and cases_syn
syntax
"_case_syntax" :: "['a, cases_syn] \<Rightarrow> 'b" (\<open>(\<open>notation=\<open>mixfix case expression\<close>\<close>case _ of/ _)\<close> 10)
"_case1" :: "['a, 'b] \<Rightarrow> case_syn" (\<open>(\<open>indent=2 notation=\<open>mixfix case pattern\<close>\<close>_ \<Rightarrow>/ _)\<close> 10)
"" :: "case_syn \<Rightarrow> cases_syn" (\<open>_\<close>)
"_case2" :: "[case_syn, cases_syn] \<Rightarrow> cases_syn" (\<open>_/ | _\<close>)
syntax (ASCII)
"_case1" :: "['a, 'b] \<Rightarrow> case_syn" (\<open>(\<open>indent=2 notation=\<open>mixfix case pattern\<close>\<close>_ =>/ _)\<close> 10)
notation (ASCII)
All (binder \<open>ALL \<close> 10) and
Ex (binder \<open>EX \<close> 10)
notation (input)
All (binder \<open>! \<close> 10) and
Ex (binder \<open>? \<close> 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" (\<open>(\<open>notation=\<open>mixfix if expression\<close>\<close>if (_)/ then (_)/ else (_))\<close> [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"
nonterminal letbinds and letbind
syntax
"_bind" :: "[pttrn, 'a] \<Rightarrow> letbind" (\<open>(\<open>indent=2 notation=\<open>mixfix let binding\<close>\<close>_ =/ _)\<close> 10)
"" :: "letbind \<Rightarrow> letbinds" (\<open>_\<close>)
"_binds" :: "[letbind, letbinds] \<Rightarrow> letbinds" (\<open>_;/ _\<close>)
"_Let" :: "[letbinds, 'a] \<Rightarrow> 'a" (\<open>(\<open>notation=\<open>mixfix let expression\<close>\<close>let (_)/ in (_))\<close> [0, 10] 10)
syntax_consts
"_bind" "_binds" "_Let" \<rightleftharpoons> Let
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>The analogous introduction rule for conjunction, above, is even constructive\<close>
lemma context_disjE:
assumes major: "P \<or> Q" and minor: "P \<Longrightarrow> R" "\<not>P \<Longrightarrow> Q \<Longrightarrow> R"
shows R
by (iprover intro: disjE [OF major] disjE [OF excluded_middle] assms)
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_>\<open>HOL.eq T\<close> = T <> \<^Type>\<open>bool\<close>
| non_bool_eq _ = false;
fun hyp_subst_tac' ctxt =
SUBGOAL (fn (goal, i) =>
if Term.exists_subterm 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
lemma all_imp_conj_distrib: "(\<forall>x. P x \<longrightarrow> Q x \<and> R x) \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<and> (\<forall>x. P x \<longrightarrow> R 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 \<open>=simp=>\<close> 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_>\<open>Let _ _ for x t\<close> =
(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_>\<open>Let _ _ for x f\<close> => (* 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)
simproc_setup eliminate_false_implies ("False \<Longrightarrow> PROP P") = \<open>
let
fun conv n =
if n > 1 then
Conv.rewr_conv @{thm Pure.swap_prems_eq}
then_conv Conv.arg_conv (conv (n - 1))
then_conv Conv.rewr_conv @{thm HOL.implies_True_equals}
else
Conv.rewr_conv @{thm HOL.False_implies_equals}
in
fn _ => fn _ => fn ct =>
let
val t = Thm.term_of ct
val n = length (Logic.strip_imp_prems t)
in
(case Logic.strip_imp_concl t of
\<^Const_>\<open>Trueprop for _\<close> => SOME (conv n ct)
| _ => NONE)
end
end
\<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_>\<open>induct_equal _ for t u\<close> = 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
(Simplifier.add_proc \<^simproc>\<open>swap_induct_false\<close> #>
Simplifier.add_proc \<^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 : Simplifier.proc
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_>\<open>HOL.eq T\<close> => if is_Type T then SOME @{thm eq_equal} else NONE
| _ => NONE)\<close>
setup \<open>Code_Preproc.map_pre (Simplifier.add_proc \<^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) "!((_) match {/ case true => (_)/ case false => true/ })"
| constant If \<rightharpoonup>
(SML) "!(if (_)/ then (_)/ else (_))"
and (OCaml) "!(if (_)/ then (_)/ else (_))"
and (Haskell) "!(if (_)/ then (_)/ else (_))"
and (Scala) "!((_) match {/ case true => (_)/ case false => (_)/ })"
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_>\<open>All _ for \<open>Abs (_, _, t)\<close>\<close> = 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