Up to index of Isabelle/FOL
theory FOL(* Title: FOL/FOL.thy ID: $Id: FOL.thy,v 1.48 2008/03/29 21:55:49 wenzelm Exp $ 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" ("cladata.ML") ("blastdata.ML") ("simpdata.ML") begin subsection {* The classical axiom *} axioms classical: "(~P ==> P) ==> P" subsection {* Lemmas and proof tools *} lemma ccontr: "(¬ P ==> False) ==> 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 (*For disjunctive case analysis*) ML {* fun excluded_middle_tac sP = res_inst_tac [("Q",sP)] (@{thm excluded_middle} RS @{thm disjE}) *} lemma case_split_thm: assumes r1: "P ==> Q" and r2: "~P ==> Q" shows Q apply (rule excluded_middle [THEN disjE]) apply (erule r2) apply (erule r1) done lemmas case_split = case_split_thm [case_names True False] (*HOL's more natural case analysis tactic*) ML {* fun case_tac a = res_inst_tac [("P",a)] @{thm case_split_thm} *} (*** 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 * : "∀y. P(y) --> 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 "∀y y'. P(y) ∧ P(y') --> 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 use "cladata.ML" setup Cla.setup setup cla_setup setup case_setup use "blastdata.ML" 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 ==> 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) == ∀x. P(x)" induct_implies where "induct_implies(A, B) == A --> B" induct_equal where "induct_equal(x, y) == x = y" induct_conj where "induct_conj(A, B) == A ∧ B" lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(λx. 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: includes meta_conjunction_syntax shows "(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 = InductFun ( 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
lemma ccontr:
(¬ P ==> False) ==> P
lemma disjCI:
(¬ Q ==> P) ==> P ∨ Q
lemma ex_classical:
(¬ (∃x. P(x)) ==> P(a)) ==> ∃x. P(x)
lemma exCI:
(∀x. ¬ P(x) ==> P(a)) ==> ∃x. P(x)
lemma excluded_middle:
¬ P ∨ P
lemma case_split_thm:
[| P ==> Q; ¬ P ==> Q |] ==> Q
lemma case_split:
[| P ==> Q; ¬ P ==> Q |] ==> Q
lemma impCE:
[| P --> Q; ¬ P ==> R; Q ==> R |] ==> R
lemma impCE':
[| P --> Q; Q ==> R; ¬ P ==> R |] ==> R
lemma notnotD:
¬ ¬ P ==> P
lemma contrapos2:
[| Q; ¬ P ==> ¬ Q |] ==> P
lemma iffCE:
[| P <-> Q; [| P; Q |] ==> R; [| ¬ P; ¬ Q |] ==> R |] ==> R
lemma alt_ex1E:
[| ∃!x. P(x); !!x. [| P(x); ∀y y'. P(y) ∧ P(y') --> y = y' |] ==> R |] ==> R
lemma imp_elim:
[| P --> Q; ¬ R ==> P; Q ==> R |] ==> R
lemma swap:
[| ¬ P; ¬ R ==> P |] ==> R
lemma ex1_functional:
[| ∃!z. P(a, z); P(a, b); P(a, c) |] ==> b = c
lemma True_implies_equals:
(True ==> PROP P) == PROP P
lemma uncurry:
P --> Q --> R ==> P ∧ Q --> R
lemma iff_allI:
(!!x. P(x) <-> Q(x)) ==> (∀x. P(x)) <-> (∀x. Q(x))
lemma iff_exI:
(!!x. P(x) <-> Q(x)) ==> (∃x. P(x)) <-> (∃x. Q(x))
lemma all_comm:
(∀x y. P(x, y)) <-> (∀y x. P(x, y))
lemma ex_comm:
(∃x y. P(x, y)) <-> (∃y x. P(x, y))
lemma cases_simp:
(P --> Q) ∧ (¬ P --> Q) <-> Q
lemma int_ex_simps:
(∃x. P(x) ∧ Q) <-> (∃x. P(x)) ∧ Q
(∃x. P ∧ Q(x)) <-> P ∧ (∃x. Q(x))
(∃x. P(x) ∨ Q) <-> (∃x. P(x)) ∨ Q
(∃x. P ∨ Q(x)) <-> P ∨ (∃x. Q(x))
lemma cla_ex_simps:
(∃x. P(x) --> Q) <-> (∀x. P(x)) --> Q
(∃x. P --> Q(x)) <-> P --> (∃x. Q(x))
lemma ex_simps:
(∃x. P(x) ∧ Q) <-> (∃x. P(x)) ∧ Q
(∃x. P ∧ Q(x)) <-> P ∧ (∃x. Q(x))
(∃x. P(x) ∨ Q) <-> (∃x. P(x)) ∨ Q
(∃x. P ∨ Q(x)) <-> P ∨ (∃x. Q(x))
(∃x. P(x) --> Q) <-> (∀x. P(x)) --> Q
(∃x. P --> Q(x)) <-> P --> (∃x. Q(x))
lemma int_all_simps:
(∀x. P(x) ∧ Q) <-> (∀x. P(x)) ∧ Q
(∀x. P ∧ Q(x)) <-> P ∧ (∀x. Q(x))
(∀x. P(x) --> Q) <-> (∃x. P(x)) --> Q
(∀x. P --> Q(x)) <-> P --> (∀x. Q(x))
lemma cla_all_simps:
(∀x. P(x) ∨ Q) <-> (∀x. P(x)) ∨ Q
(∀x. P ∨ Q(x)) <-> P ∨ (∀x. Q(x))
lemma all_simps:
(∀x. P(x) ∧ Q) <-> (∀x. P(x)) ∧ Q
(∀x. P ∧ Q(x)) <-> P ∧ (∀x. Q(x))
(∀x. P(x) --> Q) <-> (∃x. P(x)) --> Q
(∀x. P --> Q(x)) <-> P --> (∀x. Q(x))
(∀x. P(x) ∨ Q) <-> (∀x. P(x)) ∨ Q
(∀x. P ∨ Q(x)) <-> P ∨ (∀x. Q(x))
lemma imp_disj1:
(P --> Q) ∨ R <-> P --> Q ∨ R
lemma imp_disj2:
Q ∨ (P --> R) <-> P --> Q ∨ R
lemma de_Morgan_conj:
¬ (P ∧ Q) <-> ¬ P ∨ ¬ Q
lemma not_imp:
¬ (P --> Q) <-> P ∧ ¬ Q
lemma not_iff:
¬ (P <-> Q) <-> P <-> ¬ Q
lemma not_all:
¬ (∀x. P(x)) <-> (∃x. ¬ P(x))
lemma imp_all:
((∀x. P(x)) --> Q) <-> (∃x. P(x) --> Q)
lemma meta_simps:
(!!x. PROP V) == PROP V
(True ==> PROP P) == PROP P
lemma IFOL_simps:
a1 = a1 <-> True
P ∧ True <-> P
True ∧ P <-> P
P ∧ False <-> False
False ∧ P <-> False
P ∧ P <-> P
P ∧ P ∧ Q <-> P ∧ Q
P ∧ ¬ P <-> False
¬ P ∧ P <-> False
(P ∧ Q) ∧ R <-> P ∧ Q ∧ R
P ∨ True <-> True
True ∨ P <-> True
P ∨ False <-> P
False ∨ P <-> P
P ∨ P <-> P
P ∨ P ∨ Q <-> P ∨ Q
(P ∨ Q) ∨ R <-> P ∨ Q ∨ R
¬ (P ∨ Q) <-> ¬ P ∧ ¬ Q
¬ False <-> True
¬ True <-> False
(P --> False) <-> ¬ P
(P --> True) <-> True
(False --> P) <-> True
(True --> P) <-> P
(P --> P) <-> True
(P --> ¬ P) <-> ¬ P
(True <-> P) <-> P
(P <-> True) <-> P
(P <-> P) <-> True
(False <-> P) <-> ¬ P
(P <-> False) <-> ¬ P
(∀x. P) <-> P
(∀x. x = t --> P(x)) <-> P(t)
(∀x. t = x --> P(x)) <-> P(t)
(∃x. P) <-> P
∃x. x = t
∃x. t = x
(∃x. x = t ∧ P(x)) <-> P(t)
(∃x. t = x ∧ P(x)) <-> P(t)
lemma notFalseI:
¬ False
lemma cla_simps_misc:
¬ (P ∧ Q) <-> ¬ P ∨ ¬ Q
P ∨ ¬ P
¬ P ∨ P
¬ ¬ P <-> P
(¬ P --> P) <-> P
(¬ P <-> ¬ Q) <-> P <-> Q
lemma cla_simps:
¬ (P ∧ Q) <-> ¬ P ∨ ¬ Q
¬ (P ∨ Q) <-> ¬ P ∧ ¬ Q
(P --> Q) ∨ R <-> P --> Q ∨ R
Q ∨ (P --> R) <-> P --> Q ∨ R
¬ (P --> Q) <-> P ∧ ¬ Q
¬ (∀x. P(x)) <-> (∃x. ¬ P(x))
¬ (∃x. P(x)) <-> (∀x. ¬ P(x))
(P --> Q) ∧ (¬ P --> Q) <-> Q
¬ (P ∧ Q) <-> ¬ P ∨ ¬ Q
P ∨ ¬ P
¬ P ∨ P
¬ ¬ P <-> P
(¬ P --> P) <-> P
(¬ P <-> ¬ Q) <-> P <-> Q
lemma
((P --> R) <-> Q --> R) <-> (P <-> Q) ∨ R
lemma
((P --> Q) <-> P --> R) <-> P --> Q <-> R
lemma not_disj_iff_imp:
¬ P ∨ Q <-> P --> Q
lemma conj_mono:
[| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∧ P2.0 --> Q1.0 ∧ Q2.0
lemma disj_mono:
[| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∨ P2.0 --> Q1.0 ∨ Q2.0
lemma imp_mono:
[| Q1.0 --> P1.0; P2.0 --> Q2.0 |] ==> (P1.0 --> P2.0) --> Q1.0 --> Q2.0
lemma imp_refl:
P --> P
lemma ex_mono:
(!!x. P(x) --> Q(x)) ==> (∃x. P(x)) --> (∃x. Q(x))
lemma all_mono:
(!!x. P(x) --> Q(x)) ==> (∀x. P(x)) --> (∀x. Q(x))
lemma induct_forall_eq:
(!!x. P(x)) == induct_forall(P)
lemma induct_implies_eq:
(A ==> B) == induct_implies(A, B)
lemma induct_equal_eq:
(x == y) == induct_equal(x, y)
lemma induct_conj_eq:
(A && B) == induct_conj(A, B)
lemma induct_atomize:
(!!x. P(x)) == induct_forall(P)
(A ==> B) == induct_implies(A, B)
(x == y) == induct_equal(x, y)
(A && B) == induct_conj(A, B)
lemma induct_rulify:
induct_forall(P) == (!!x. P(x))
induct_implies(A, B) == (A ==> B)
induct_equal(x, y) == x == y
induct_conj(A, B) == A && B
lemma induct_rulify_fallback:
induct_forall(P) == ∀x. P(x)
induct_implies(A, B) == A --> B
induct_equal(x, y) == x = y
induct_conj(A, B) == A ∧ B