src/FOL/FOL.thy
 author wenzelm Fri, 02 Oct 2009 22:15:08 +0200 changeset 32861 105f40051387 parent 32261 05e687ddbcee child 32960 69916a850301 permissions -rw-r--r--
```
(*  Title:      FOL/FOL.thy
Author:     Lawrence C Paulson and Markus Wenzel
*)

header {* Classical first-order logic *}

theory FOL
imports IFOL
uses
"~~/src/Provers/classical.ML"
"~~/src/Provers/blast.ML"
"~~/src/Provers/clasimp.ML"
"~~/src/Tools/induct.ML"
("simpdata.ML")
begin

subsection {* The classical axiom *}

axioms
classical: "(~P ==> P) ==> P"

subsection {* Lemmas and proof tools *}

lemma ccontr: "(\<not> P \<Longrightarrow> False) \<Longrightarrow> P"
by (erule FalseE [THEN classical])

(*** Classical introduction rules for | and EX ***)

lemma disjCI: "(~Q ==> P) ==> P|Q"
apply (rule classical)
apply (assumption | erule meta_mp | rule disjI1 notI)+
apply (erule notE disjI2)+
done

(*introduction rule involving only EX*)
lemma ex_classical:
assumes r: "~(EX x. P(x)) ==> P(a)"
shows "EX x. P(x)"
apply (rule classical)
apply (rule exI, erule r)
done

(*version of above, simplifying ~EX to ALL~ *)
lemma exCI:
assumes r: "ALL x. ~P(x) ==> P(a)"
shows "EX x. P(x)"
apply (rule ex_classical)
apply (rule notI [THEN allI, THEN r])
apply (erule notE)
apply (erule exI)
done

lemma excluded_middle: "~P | P"
apply (rule disjCI)
apply assumption
done

lemma case_split [case_names True False]:
assumes r1: "P ==> Q"
and r2: "~P ==> Q"
shows Q
apply (rule excluded_middle [THEN disjE])
apply (erule r2)
apply (erule r1)
done

ML {*
fun case_tac ctxt a = res_inst_tac ctxt [(("P", 0), a)] @{thm case_split}
*}

method_setup case_tac = {*
Args.goal_spec -- Scan.lift Args.name_source >>
(fn (quant, s) => fn ctxt => SIMPLE_METHOD'' quant (case_tac ctxt s))
*} "case_tac emulation (dynamic instantiation!)"

(*** Special elimination rules *)

(*Classical implies (-->) elimination. *)
lemma impCE:
assumes major: "P-->Q"
and r1: "~P ==> R"
and r2: "Q ==> R"
shows R
apply (rule excluded_middle [THEN disjE])
apply (erule r1)
apply (rule r2)
apply (erule major [THEN mp])
done

(*This version of --> elimination works on Q before P.  It works best for
those cases in which P holds "almost everywhere".  Can't install as
default: would break old proofs.*)
lemma impCE':
assumes major: "P-->Q"
and r1: "Q ==> R"
and r2: "~P ==> R"
shows R
apply (rule excluded_middle [THEN disjE])
apply (erule r2)
apply (rule r1)
apply (erule major [THEN mp])
done

(*Double negation law*)
lemma notnotD: "~~P ==> P"
apply (rule classical)
apply (erule notE)
apply assumption
done

lemma contrapos2:  "[| Q; ~ P ==> ~ Q |] ==> P"
apply (rule classical)
apply (drule (1) meta_mp)
apply (erule (1) notE)
done

(*** Tactics for implication and contradiction ***)

(*Classical <-> elimination.  Proof substitutes P=Q in
~P ==> ~Q    and    P ==> Q  *)
lemma iffCE:
assumes major: "P<->Q"
and r1: "[| P; Q |] ==> R"
and r2: "[| ~P; ~Q |] ==> R"
shows R
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: "EX! x. P(x)"
and r: "!!x. [| P(x);  ALL y y'. P(y) & P(y') --> y=y' |] ==> R"
shows R
using major
proof (rule ex1E)
fix x
assume * : "\<forall>y. P(y) \<longrightarrow> y = x"
assume "P(x)"
then show R
proof (rule r)
{ fix y y'
assume "P(y)" and "P(y')"
with * have "x = y" and "x = y'" by - (tactic "IntPr.fast_tac 1")+
then have "y = y'" by (rule subst)
} note r' = this
show "\<forall>y y'. P(y) \<and> P(y') \<longrightarrow> y = y'" by (intro strip, elim conjE) (rule r')
qed
qed

lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
by (rule classical) iprover

lemma swap: "~ P ==> (~ R ==> P) ==> R"
by (rule classical) iprover

section {* Classical Reasoner *}

setup Cla.setup
ML {* Context.>> (Cla.map_cs (K FOL_cs)) *}

ML {*
structure Blast = Blast
(
val thy = @{theory}
type claset	= Cla.claset
val equality_name = @{const_name "op ="}
val not_name = @{const_name Not}
val notE	= @{thm notE}
val ccontr	= @{thm ccontr}
val contr_tac = Cla.contr_tac
val dup_intr	= Cla.dup_intr
val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
val rep_cs = Cla.rep_cs
val cla_modifiers = Cla.cla_modifiers
val cla_meth' = Cla.cla_meth'
);
val blast_tac = Blast.blast_tac;
*}

setup Blast.setup

lemma ex1_functional: "[| EX! z. P(a,z);  P(a,b);  P(a,c) |] ==> b = c"
by blast

(* Elimination of True from asumptions: *)
lemma True_implies_equals: "(True ==> PROP P) == PROP P"
proof
assume "True \<Longrightarrow> PROP P"
from this and TrueI show "PROP P" .
next
assume "PROP P"
then show "PROP P" .
qed

lemma uncurry: "P --> Q --> R ==> P & Q --> R"
by blast

lemma iff_allI: "(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))"
by blast

lemma iff_exI: "(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))"
by blast

lemma all_comm: "(ALL x y. P(x,y)) <-> (ALL y x. P(x,y))" by blast

lemma ex_comm: "(EX x y. P(x,y)) <-> (EX y x. P(x,y))" by blast

(*** Classical simplification rules ***)

(*Avoids duplication of subgoals after expand_if, when the true and false
cases boil down to the same thing.*)
lemma cases_simp: "(P --> Q) & (~P --> Q) <-> Q" by blast

(*** Miniscoping: pushing quantifiers in
We do NOT distribute of ALL over &, or dually that of EX over |
Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
show that this step can increase proof length!
***)

(*existential miniscoping*)
lemma int_ex_simps:
"!!P Q. (EX x. P(x) & Q) <-> (EX x. P(x)) & Q"
"!!P Q. (EX x. P & Q(x)) <-> P & (EX x. Q(x))"
"!!P Q. (EX x. P(x) | Q) <-> (EX x. P(x)) | Q"
"!!P Q. (EX x. P | Q(x)) <-> P | (EX x. Q(x))"
by iprover+

(*classical rules*)
lemma cla_ex_simps:
"!!P Q. (EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q"
"!!P Q. (EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"
by blast+

lemmas ex_simps = int_ex_simps cla_ex_simps

(*universal miniscoping*)
lemma int_all_simps:
"!!P Q. (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q"
"!!P Q. (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))"
"!!P Q. (ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q"
"!!P Q. (ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"
by iprover+

(*classical rules*)
lemma cla_all_simps:
"!!P Q. (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q"
"!!P Q. (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"
by blast+

lemmas all_simps = int_all_simps cla_all_simps

(*** Named rewrite rules proved for IFOL ***)

lemma imp_disj1: "(P-->Q) | R <-> (P-->Q | R)" by blast
lemma imp_disj2: "Q | (P-->R) <-> (P-->Q | R)" by blast

lemma de_Morgan_conj: "(~(P & Q)) <-> (~P | ~Q)" by blast

lemma not_imp: "~(P --> Q) <-> (P & ~Q)" by blast
lemma not_iff: "~(P <-> Q) <-> (P <-> ~Q)" by blast

lemma not_all: "(~ (ALL x. P(x))) <-> (EX x.~P(x))" by blast
lemma imp_all: "((ALL x. P(x)) --> Q) <-> (EX x. P(x) --> Q)" by blast

lemmas meta_simps =
triv_forall_equality (* prunes params *)
True_implies_equals  (* prune asms `True' *)

lemmas IFOL_simps =
refl [THEN P_iff_T] conj_simps disj_simps not_simps
imp_simps iff_simps quant_simps

lemma notFalseI: "~False" by iprover

lemma cla_simps_misc:
"~(P&Q) <-> ~P | ~Q"
"P | ~P"
"~P | P"
"~ ~ P <-> P"
"(~P --> P) <-> P"
"(~P <-> ~Q) <-> (P<->Q)" 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

use "simpdata.ML"
setup simpsetup
setup "Simplifier.method_setup Splitter.split_modifiers"
setup Splitter.setup
setup clasimp_setup
setup EqSubst.setup

subsection {* Other simple lemmas *}

lemma [simp]: "((P-->R) <-> (Q-->R)) <-> ((P<->Q) | R)"
by blast

lemma [simp]: "((P-->Q) <-> (P-->R)) <-> (P --> (Q<->R))"
by blast

lemma not_disj_iff_imp: "~P | Q <-> (P-->Q)"
by blast

(** Monotonicity of implications **)

lemma conj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)"
by fast (*or (IntPr.fast_tac 1)*)

lemma disj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)"
by fast (*or (IntPr.fast_tac 1)*)

lemma imp_mono: "[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)"
by fast (*or (IntPr.fast_tac 1)*)

lemma imp_refl: "P-->P"
by (rule impI, assumption)

(*The quantifier monotonicity rules are also intuitionistically valid*)
lemma ex_mono: "(!!x. P(x) --> Q(x)) ==> (EX x. P(x)) --> (EX x. Q(x))"
by blast

lemma all_mono: "(!!x. P(x) --> Q(x)) ==> (ALL x. P(x)) --> (ALL x. Q(x))"
by blast

subsection {* Proof by cases and induction *}

text {* Proper handling of non-atomic rule statements. *}

constdefs
induct_forall where "induct_forall(P) == \<forall>x. P(x)"
induct_implies where "induct_implies(A, B) == A \<longrightarrow> B"
induct_equal where "induct_equal(x, y) == x = y"
induct_conj where "induct_conj(A, B) == A \<and> B"

lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(\<lambda>x. P(x)))"
unfolding atomize_all induct_forall_def .

lemma induct_implies_eq: "(A ==> B) == Trueprop(induct_implies(A, B))"
unfolding atomize_imp induct_implies_def .

lemma induct_equal_eq: "(x == y) == Trueprop(induct_equal(x, y))"
unfolding atomize_eq induct_equal_def .

lemma induct_conj_eq: "(A &&& B) == Trueprop(induct_conj(A, B))"
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, standard] = induct_atomize
lemmas induct_rulify_fallback =
induct_forall_def induct_implies_def induct_equal_def induct_conj_def

hide const induct_forall induct_implies induct_equal induct_conj

text {* Method setup. *}

ML {*
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}
);
*}

setup Induct.setup
declare case_split [cases type: o]

end
```