(* Title: FOL/FOL.thy
Author: Lawrence C Paulson and Markus Wenzel
*)
section \<open>Classical first-order logic\<close>
theory FOL
imports IFOL
keywords "print_claset" "print_induct_rules" :: diag
begin
ML_file \<open>~~/src/Provers/classical.ML\<close>
ML_file \<open>~~/src/Provers/blast.ML\<close>
ML_file \<open>~~/src/Provers/clasimp.ML\<close>
subsection \<open>The classical axiom\<close>
axiomatization where
classical: \<open>(\<not> P \<Longrightarrow> P) \<Longrightarrow> P\<close>
subsection \<open>Lemmas and proof tools\<close>
lemma ccontr: \<open>(\<not> P \<Longrightarrow> False) \<Longrightarrow> P\<close>
by (erule FalseE [THEN classical])
subsubsection \<open>Classical introduction rules for \<open>\<or>\<close> and \<open>\<exists>\<close>\<close>
lemma disjCI: \<open>(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q\<close>
apply (rule classical)
apply (assumption | erule meta_mp | rule disjI1 notI)+
apply (erule notE disjI2)+
done
text \<open>Introduction rule involving only \<open>\<exists>\<close>\<close>
lemma ex_classical:
assumes r: \<open>\<not> (\<exists>x. P(x)) \<Longrightarrow> P(a)\<close>
shows \<open>\<exists>x. P(x)\<close>
apply (rule classical)
apply (rule exI, erule r)
done
text \<open>Version of above, simplifying \<open>\<not>\<exists>\<close> to \<open>\<forall>\<not>\<close>.\<close>
lemma exCI:
assumes r: \<open>\<forall>x. \<not> P(x) \<Longrightarrow> P(a)\<close>
shows \<open>\<exists>x. P(x)\<close>
apply (rule ex_classical)
apply (rule notI [THEN allI, THEN r])
apply (erule notE)
apply (erule exI)
done
lemma excluded_middle: \<open>\<not> P \<or> P\<close>
apply (rule disjCI)
apply assumption
done
lemma case_split [case_names True False]:
assumes r1: \<open>P \<Longrightarrow> Q\<close>
and r2: \<open>\<not> P \<Longrightarrow> Q\<close>
shows \<open>Q\<close>
apply (rule excluded_middle [THEN disjE])
apply (erule r2)
apply (erule r1)
done
ML \<open>
fun case_tac ctxt a fixes =
Rule_Insts.res_inst_tac ctxt [((("P", 0), Position.none), a)] fixes @{thm case_split};
\<close>
method_setup case_tac = \<open>
Args.goal_spec -- Scan.lift (Args.embedded_inner_syntax -- Parse.for_fixes) >>
(fn (quant, (s, fixes)) => fn ctxt => SIMPLE_METHOD'' quant (case_tac ctxt s fixes))
\<close> "case_tac emulation (dynamic instantiation!)"
subsection \<open>Special elimination rules\<close>
text \<open>Classical implies (\<open>\<longrightarrow>\<close>) elimination.\<close>
lemma impCE:
assumes major: \<open>P \<longrightarrow> Q\<close>
and r1: \<open>\<not> P \<Longrightarrow> R\<close>
and r2: \<open>Q \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule excluded_middle [THEN disjE])
apply (erule r1)
apply (rule r2)
apply (erule major [THEN mp])
done
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 P holds ``almost everywhere''.
Can't install as default: would break old proofs.
\<close>
lemma impCE':
assumes major: \<open>P \<longrightarrow> Q\<close>
and r1: \<open>Q \<Longrightarrow> R\<close>
and r2: \<open>\<not> P \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule excluded_middle [THEN disjE])
apply (erule r2)
apply (rule r1)
apply (erule major [THEN mp])
done
text \<open>Double negation law.\<close>
lemma notnotD: \<open>\<not> \<not> P \<Longrightarrow> P\<close>
apply (rule classical)
apply (erule notE)
apply assumption
done
lemma contrapos2: \<open>\<lbrakk>Q; \<not> P \<Longrightarrow> \<not> Q\<rbrakk> \<Longrightarrow> P\<close>
apply (rule classical)
apply (drule (1) meta_mp)
apply (erule (1) notE)
done
subsubsection \<open>Tactics for implication and contradiction\<close>
text \<open>
Classical \<open>\<longleftrightarrow>\<close> elimination. Proof substitutes \<open>P = Q\<close> in
\<open>\<not> P \<Longrightarrow> \<not> Q\<close> and \<open>P \<Longrightarrow> Q\<close>.
\<close>
lemma iffCE:
assumes major: \<open>P \<longleftrightarrow> Q\<close>
and r1: \<open>\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R\<close>
and r2: \<open>\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule major [unfolded iff_def, THEN conjE])
apply (elim impCE)
apply (erule (1) r2)
apply (erule (1) notE)+
apply (erule (1) r1)
done
(*Better for fast_tac: needs no quantifier duplication!*)
lemma alt_ex1E:
assumes major: \<open>\<exists>! x. P(x)\<close>
and r: \<open>\<And>x. \<lbrakk>P(x); \<forall>y y'. P(y) \<and> P(y') \<longrightarrow> y = y'\<rbrakk> \<Longrightarrow> R\<close>
shows \<open>R\<close>
using major
proof (rule ex1E)
fix x
assume * : \<open>\<forall>y. P(y) \<longrightarrow> y = x\<close>
assume \<open>P(x)\<close>
then show \<open>R\<close>
proof (rule r)
{
fix y y'
assume \<open>P(y)\<close> and \<open>P(y')\<close>
with * have \<open>x = y\<close> and \<open>x = y'\<close>
by - (tactic "IntPr.fast_tac \<^context> 1")+
then have \<open>y = y'\<close> by (rule subst)
} note r' = this
show \<open>\<forall>y y'. P(y) \<and> P(y') \<longrightarrow> y = y'\<close>
by (intro strip, elim conjE) (rule r')
qed
qed
lemma imp_elim: \<open>P \<longrightarrow> Q \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R\<close>
by (rule classical) iprover
lemma swap: \<open>\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R\<close>
by (rule classical) iprover
section \<open>Classical Reasoner\<close>
ML \<open>
structure Cla = Classical
(
val imp_elim = @{thm imp_elim}
val not_elim = @{thm notE}
val swap = @{thm swap}
val classical = @{thm classical}
val sizef = size_of_thm
val hyp_subst_tacs = [hyp_subst_tac]
);
structure Basic_Classical: BASIC_CLASSICAL = Cla;
open Basic_Classical;
\<close>
(*Propositional rules*)
lemmas [intro!] = refl TrueI conjI disjCI impI notI iffI
and [elim!] = conjE disjE impCE FalseE iffCE
ML \<open>val prop_cs = claset_of \<^context>\<close>
(*Quantifier rules*)
lemmas [intro!] = allI ex_ex1I
and [intro] = exI
and [elim!] = exE alt_ex1E
and [elim] = allE
ML \<open>val FOL_cs = claset_of \<^context>\<close>
ML \<open>
structure Blast = Blast
(
structure Classical = Cla
val Trueprop_const = dest_Const \<^const>\<open>Trueprop\<close>
val equality_name = \<^const_name>\<open>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>
lemma ex1_functional: \<open>\<lbrakk>\<exists>! z. P(a,z); P(a,b); P(a,c)\<rbrakk> \<Longrightarrow> b = c\<close>
by blast
text \<open>Elimination of \<open>True\<close> from assumptions:\<close>
lemma True_implies_equals: \<open>(True \<Longrightarrow> PROP P) \<equiv> PROP P\<close>
proof
assume \<open>True \<Longrightarrow> PROP P\<close>
from this and TrueI show \<open>PROP P\<close> .
next
assume \<open>PROP P\<close>
then show \<open>PROP P\<close> .
qed
lemma uncurry: \<open>P \<longrightarrow> Q \<longrightarrow> R \<Longrightarrow> P \<and> Q \<longrightarrow> R\<close>
by blast
lemma iff_allI: \<open>(\<And>x. P(x) \<longleftrightarrow> Q(x)) \<Longrightarrow> (\<forall>x. P(x)) \<longleftrightarrow> (\<forall>x. Q(x))\<close>
by blast
lemma iff_exI: \<open>(\<And>x. P(x) \<longleftrightarrow> Q(x)) \<Longrightarrow> (\<exists>x. P(x)) \<longleftrightarrow> (\<exists>x. Q(x))\<close>
by blast
lemma all_comm: \<open>(\<forall>x y. P(x,y)) \<longleftrightarrow> (\<forall>y x. P(x,y))\<close>
by blast
lemma ex_comm: \<open>(\<exists>x y. P(x,y)) \<longleftrightarrow> (\<exists>y x. P(x,y))\<close>
by blast
subsection \<open>Classical simplification rules\<close>
text \<open>
Avoids duplication of subgoals after \<open>expand_if\<close>, when the true and
false cases boil down to the same thing.
\<close>
lemma cases_simp: \<open>(P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q) \<longleftrightarrow> Q\<close>
by blast
subsubsection \<open>Miniscoping: pushing quantifiers in\<close>
text \<open>
We do NOT distribute of \<open>\<forall>\<close> over \<open>\<and>\<close>, or dually that of
\<open>\<exists>\<close> over \<open>\<or>\<close>.
Baaz and Leitsch, On Skolemization and Proof Complexity (1994) show that
this step can increase proof length!
\<close>
text \<open>Existential miniscoping.\<close>
lemma int_ex_simps:
\<open>\<And>P Q. (\<exists>x. P(x) \<and> Q) \<longleftrightarrow> (\<exists>x. P(x)) \<and> Q\<close>
\<open>\<And>P Q. (\<exists>x. P \<and> Q(x)) \<longleftrightarrow> P \<and> (\<exists>x. Q(x))\<close>
\<open>\<And>P Q. (\<exists>x. P(x) \<or> Q) \<longleftrightarrow> (\<exists>x. P(x)) \<or> Q\<close>
\<open>\<And>P Q. (\<exists>x. P \<or> Q(x)) \<longleftrightarrow> P \<or> (\<exists>x. Q(x))\<close>
by iprover+
text \<open>Classical rules.\<close>
lemma cla_ex_simps:
\<open>\<And>P Q. (\<exists>x. P(x) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<longrightarrow> Q\<close>
\<open>\<And>P Q. (\<exists>x. P \<longrightarrow> Q(x)) \<longleftrightarrow> P \<longrightarrow> (\<exists>x. Q(x))\<close>
by blast+
lemmas ex_simps = int_ex_simps cla_ex_simps
text \<open>Universal miniscoping.\<close>
lemma int_all_simps:
\<open>\<And>P Q. (\<forall>x. P(x) \<and> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<and> Q\<close>
\<open>\<And>P Q. (\<forall>x. P \<and> Q(x)) \<longleftrightarrow> P \<and> (\<forall>x. Q(x))\<close>
\<open>\<And>P Q. (\<forall>x. P(x) \<longrightarrow> Q) \<longleftrightarrow> (\<exists> x. P(x)) \<longrightarrow> Q\<close>
\<open>\<And>P Q. (\<forall>x. P \<longrightarrow> Q(x)) \<longleftrightarrow> P \<longrightarrow> (\<forall>x. Q(x))\<close>
by iprover+
text \<open>Classical rules.\<close>
lemma cla_all_simps:
\<open>\<And>P Q. (\<forall>x. P(x) \<or> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<or> Q\<close>
\<open>\<And>P Q. (\<forall>x. P \<or> Q(x)) \<longleftrightarrow> P \<or> (\<forall>x. Q(x))\<close>
by blast+
lemmas all_simps = int_all_simps cla_all_simps
subsubsection \<open>Named rewrite rules proved for IFOL\<close>
lemma imp_disj1: \<open>(P \<longrightarrow> Q) \<or> R \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)\<close> by blast
lemma imp_disj2: \<open>Q \<or> (P \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)\<close> by blast
lemma de_Morgan_conj: \<open>(\<not> (P \<and> Q)) \<longleftrightarrow> (\<not> P \<or> \<not> Q)\<close> by blast
lemma not_imp: \<open>\<not> (P \<longrightarrow> Q) \<longleftrightarrow> (P \<and> \<not> Q)\<close> by blast
lemma not_iff: \<open>\<not> (P \<longleftrightarrow> Q) \<longleftrightarrow> (P \<longleftrightarrow> \<not> Q)\<close> by blast
lemma not_all: \<open>(\<not> (\<forall>x. P(x))) \<longleftrightarrow> (\<exists>x. \<not> P(x))\<close> by blast
lemma imp_all: \<open>((\<forall>x. P(x)) \<longrightarrow> Q) \<longleftrightarrow> (\<exists>x. P(x) \<longrightarrow> Q)\<close> by blast
lemmas meta_simps =
triv_forall_equality \<comment> \<open>prunes params\<close>
True_implies_equals \<comment> \<open>prune asms \<open>True\<close>\<close>
lemmas IFOL_simps =
refl [THEN P_iff_T] conj_simps disj_simps not_simps
imp_simps iff_simps quant_simps
lemma notFalseI: \<open>\<not> False\<close> by iprover
lemma cla_simps_misc:
\<open>\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q\<close>
\<open>P \<or> \<not> P\<close>
\<open>\<not> P \<or> P\<close>
\<open>\<not> \<not> P \<longleftrightarrow> P\<close>
\<open>(\<not> P \<longrightarrow> P) \<longleftrightarrow> P\<close>
\<open>(\<not> P \<longleftrightarrow> \<not> Q) \<longleftrightarrow> (P \<longleftrightarrow> Q)\<close> by blast+
lemmas cla_simps =
de_Morgan_conj de_Morgan_disj imp_disj1 imp_disj2
not_imp not_all not_ex cases_simp cla_simps_misc
ML_file \<open>simpdata.ML\<close>
simproc_setup defined_Ex (\<open>\<exists>x. P(x)\<close>) = \<open>K Quantifier1.rearrange_Ex\<close>
simproc_setup defined_All (\<open>\<forall>x. P(x)\<close>) = \<open>K Quantifier1.rearrange_All\<close>
simproc_setup defined_all("\x. PROP P(x)") = \K Quantifier1.rearrange_all\
ML \<open>
(*intuitionistic simprules only*)
val IFOL_ss =
put_simpset FOL_basic_ss \<^context>
addsimps @{thms meta_simps IFOL_simps int_ex_simps int_all_simps subst_all}
addsimprocs [\<^simproc>\<open>defined_All\<close>, \<^simproc>\<open>defined_Ex\<close>]
|> Simplifier.add_cong @{thm imp_cong}
|> simpset_of;
(*classical simprules too*)
val FOL_ss =
put_simpset IFOL_ss \<^context>
addsimps @{thms cla_simps cla_ex_simps cla_all_simps}
|> simpset_of;
\<close>
setup \<open>
map_theory_simpset (put_simpset FOL_ss) #>
Simplifier.method_setup Splitter.split_modifiers
\<close>
ML_file \<open>~~/src/Tools/eqsubst.ML\<close>
subsection \<open>Other simple lemmas\<close>
lemma [simp]: \<open>((P \<longrightarrow> R) \<longleftrightarrow> (Q \<longrightarrow> R)) \<longleftrightarrow> ((P \<longleftrightarrow> Q) \<or> R)\<close>
by blast
lemma [simp]: \<open>((P \<longrightarrow> Q) \<longleftrightarrow> (P \<longrightarrow> R)) \<longleftrightarrow> (P \<longrightarrow> (Q \<longleftrightarrow> R))\<close>
by blast
lemma not_disj_iff_imp: \<open>\<not> P \<or> Q \<longleftrightarrow> (P \<longrightarrow> Q)\<close>
by blast
subsubsection \<open>Monotonicity of implications\<close>
lemma conj_mono: \<open>\<lbrakk>P1 \<longrightarrow> Q1; P2 \<longrightarrow> Q2\<rbrakk> \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)\<close>
by fast (*or (IntPr.fast_tac 1)*)
lemma disj_mono: \<open>\<lbrakk>P1 \<longrightarrow> Q1; P2 \<longrightarrow> Q2\<rbrakk> \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)\<close>
by fast (*or (IntPr.fast_tac 1)*)
lemma imp_mono: \<open>\<lbrakk>Q1 \<longrightarrow> P1; P2 \<longrightarrow> Q2\<rbrakk> \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)\<close>
by fast (*or (IntPr.fast_tac 1)*)
lemma imp_refl: \<open>P \<longrightarrow> P\<close>
by (rule impI)
text \<open>The quantifier monotonicity rules are also intuitionistically valid.\<close>
lemma ex_mono: \<open>(\<And>x. P(x) \<longrightarrow> Q(x)) \<Longrightarrow> (\<exists>x. P(x)) \<longrightarrow> (\<exists>x. Q(x))\<close>
by blast
lemma all_mono: \<open>(\<And>x. P(x) \<longrightarrow> Q(x)) \<Longrightarrow> (\<forall>x. P(x)) \<longrightarrow> (\<forall>x. Q(x))\<close>
by blast
subsection \<open>Proof by cases and induction\<close>
text \<open>Proper handling of non-atomic rule statements.\<close>
context
begin
qualified definition \<open>induct_forall(P) \<equiv> \<forall>x. P(x)\<close>
qualified definition \<open>induct_implies(A, B) \<equiv> A \<longrightarrow> B\<close>
qualified definition \<open>induct_equal(x, y) \<equiv> x = y\<close>
qualified definition \<open>induct_conj(A, B) \<equiv> A \<and> B\<close>
lemma induct_forall_eq: \<open>(\<And>x. P(x)) \<equiv> Trueprop(induct_forall(\<lambda>x. P(x)))\<close>
unfolding atomize_all induct_forall_def .
lemma induct_implies_eq: \<open>(A \<Longrightarrow> B) \<equiv> Trueprop(induct_implies(A, B))\<close>
unfolding atomize_imp induct_implies_def .
lemma induct_equal_eq: \<open>(x \<equiv> y) \<equiv> Trueprop(induct_equal(x, y))\<close>
unfolding atomize_eq induct_equal_def .
lemma induct_conj_eq: \<open>(A &&& B) \<equiv> Trueprop(induct_conj(A, B))\<close>
unfolding atomize_conj induct_conj_def .
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
lemmas induct_rulify [symmetric] = induct_atomize
lemmas induct_rulify_fallback =
induct_forall_def induct_implies_def induct_equal_def induct_conj_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 _ = NONE
fun trivial_tac _ _ = no_tac
);
\<close>
declare case_split [cases type: o]
end
ML_file \<open>~~/src/Tools/case_product.ML\<close>
hide_const (open) eq
end
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