# Theory FOL

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theory FOL
imports IFOL
uses (FOL_lemmas1.ML) (cladata.ML) (blastdata.ML) (simpdata.ML) (eqrule_FOL_data.ML) ($ISABELLE_HOME/src/Provers/eqsubst.ML) [FOL.ML]
begin

    (*  Title:      FOL/FOL.thy
        ID:         $Id: FOL.thy,v 1.32 2005/06/17 14:15:08 haftmann Exp $
        Author:     Lawrence C Paulson and Markus Wenzel
    *)

    header {* Classical first-order logic *}

    theory FOL
    imports IFOL
    uses ("FOL_lemmas1.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML")
          ("eqrule_FOL_data.ML")
          ("~~/src/Provers/eqsubst.ML")
    begin

    subsection {* The classical axiom *}

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

    subsection {* Lemmas and proof tools *}

    use "FOL_lemmas1.ML"
    theorems case_split = case_split_thm [case_names True False, cases type: o]

    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

    ML {*
    val ex1_functional = thm "ex1_functional";
    *}

    use "simpdata.ML"
    setup simpsetup
    setup "Simplifier.method_setup Splitter.split_modifiers"
    setup Splitter.setup
    setup Clasimp.setup

    subsection {* Lucas Dixon's eqstep tactic *}

    use "~~/src/Provers/eqsubst.ML";
    use "eqrule_FOL_data.ML";

    setup EQSubstTac.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 :: "('a => o) => o"
      "induct_forall(P) == ∀x. P(x)"
      induct_implies :: "o => o => o"
      "induct_implies(A, B) == A --> B"
      induct_equal :: "'a => 'a => o"
      "induct_equal(x, y) == x = y"

    lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(λx. P(x)))"
      by (simp only: atomize_all induct_forall_def)

    lemma induct_implies_eq: "(A ==> B) == Trueprop(induct_implies(A, B))"
      by (simp only: atomize_imp induct_implies_def)

    lemma induct_equal_eq: "(x == y) == Trueprop(induct_equal(x, y))"
      by (simp only: atomize_eq induct_equal_def)

    lemma induct_impliesI: "(A ==> B) ==> induct_implies(A, B)"
      by (simp add: induct_implies_def)

    lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
    lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
    lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def

    lemma all_conj_eq: "(ALL x. P(x)) & (ALL y. Q(y)) == (ALL x y. P(x) & Q(y))"
      by simp

    hide const induct_forall induct_implies induct_equal

    text {* Method setup. *}

    ML {*
      structure InductMethod = InductMethodFun
      (struct
        val dest_concls = FOLogic.dest_concls;
        val cases_default = thm "case_split";
        val local_impI = thm "induct_impliesI";
        val conjI = thm "conjI";
        val atomize = thms "induct_atomize";
        val rulify1 = thms "induct_rulify1";
        val rulify2 = thms "induct_rulify2";
        val localize = [Thm.symmetric (thm "induct_implies_def"),
          Thm.symmetric (thm "atomize_all"), thm "all_conj_eq"];
      end);
    *}

    setup InductMethod.setup

    end

### The classical axiom

### Lemmas and proof tools

theorem ccontr:

      (¬ P ==> False) ==> P

theorem disjCI:

      (¬ Q ==> P) ==> P ∨ Q

theorem ex_classical:

      (¬ (∃x. P(x)) ==> P(a)) ==> ∃x. P(x)

theorem exCI:

      (∀x. ¬ P(x) ==> P(a)) ==> ∃x. P(x)

theorem excluded_middle:

      ¬ P ∨ P

theorem case_split_thm:

      [| P ==> Q; ¬ P ==> Q |] ==> Q

theorem impCE:

      [| P --> Q; ¬ P ==> R; Q ==> R |] ==> R

theorem impCE':

      [| P --> Q; Q ==> R; ¬ P ==> R |] ==> R

theorem notnotD:

      ¬ ¬ P ==> P

theorem contrapos2:

      [| Q; ¬ P ==> ¬ Q |] ==> P

theorem iffCE:

      [| P <-> Q; [| P; Q |] ==> R; [| ¬ P; ¬ Q |] ==> R |] ==> R

theorems case_split:

      [| P ==> Q; ¬ P ==> Q |] ==> Q

theorems case_split:

      [| P ==> Q; ¬ P ==> Q |] ==> Q

## Classical Reasoner

theorem imp_elim:

      [| P --> Q; ¬ Z_ ==> P; Q ==> Z_ |] ==> Z_

theorem swap:

      [| ¬ Pa; ¬ P ==> Pa |] ==> P

theorem alt_ex1E:

      [| ∃!x. P(x); !!x. [| P(x); ∀y y'. P(y) ∧ P(y') --> y = y' |] ==> R |] ==> R

lemma ex1_functional:

      [| ∃!z. P(a, z); P(a, b); P(a, c) |] ==> b = c

theorem True_implies_equals:

      (True ==> PROP P) == PROP P

theorems conj_simps:

      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

theorems disj_simps:

      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

theorems not_simps:

      ¬ (P ∨ Q) <-> ¬ P ∧ ¬ Q

      ¬ False <-> True

      ¬ True <-> False

theorems imp_simps:

      (P --> False) <-> ¬ P

      (P --> True) <-> True

      (False --> P) <-> True

      (True --> P) <-> P

      (P --> P) <-> True

      (P --> ¬ P) <-> ¬ P

theorems iff_simps:

      (True <-> P) <-> P

      (P <-> True) <-> P

      (P <-> P) <-> True

      (False <-> P) <-> ¬ P

      (P <-> False) <-> ¬ P

theorems quant_simps:

      (∀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)

theorems distrib_simps:

      P ∧ (Q ∨ R) <-> P ∧ Q ∨ P ∧ R

      (Q ∨ R) ∧ P <-> Q ∧ P ∨ R ∧ P

      (P ∨ Q --> R) <-> (P --> R) ∧ (Q --> R)

theorem P_iff_F:

      ¬ P ==> P <-> False

theorem iff_reflection_F:

      ¬ P ==> P == False

theorem P_iff_T:

      P ==> P <-> True

theorem iff_reflection_T:

      P ==> P == True

theorem cases_simp:

      (P --> Q) ∧ (¬ P --> Q) <-> Q

theorems 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))

theorems cla_ex_simps:

      (∃x. P(x) --> Q) <-> (∀x. P(x)) --> Q

      (∃x. P --> Q(x)) <-> P --> (∃x. Q(x))

theorems 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))

theorems 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))

theorems cla_all_simps:

      (∀x. P(x) ∨ Q) <-> (∀x. P(x)) ∨ Q

      (∀x. P ∨ Q(x)) <-> P ∨ (∀x. Q(x))

theorems 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))

theorem conj_commute:

      P ∧ Q <-> Q ∧ P

theorem conj_left_commute:

      P ∧ Q ∧ R <-> Q ∧ P ∧ R

theorems conj_comms:

      P ∧ Q <-> Q ∧ P

      P ∧ Q ∧ R <-> Q ∧ P ∧ R

theorem disj_commute:

      P ∨ Q <-> Q ∨ P

theorem disj_left_commute:

      P ∨ Q ∨ R <-> Q ∨ P ∨ R

theorems disj_comms:

      P ∨ Q <-> Q ∨ P

      P ∨ Q ∨ R <-> Q ∨ P ∨ R

theorem conj_disj_distribL:

      P ∧ (Q ∨ R) <-> P ∧ Q ∨ P ∧ R

theorem conj_disj_distribR:

      (P ∨ Q) ∧ R <-> P ∧ R ∨ Q ∧ R

theorem disj_conj_distribL:

      P ∨ Q ∧ R <-> (P ∨ Q) ∧ (P ∨ R)

theorem disj_conj_distribR:

      P ∧ Q ∨ R <-> (P ∨ R) ∧ (Q ∨ R)

theorem imp_conj_distrib:

      (P --> Q ∧ R) <-> (P --> Q) ∧ (P --> R)

theorem imp_conj:

      (P ∧ Q --> R) <-> P --> Q --> R

theorem imp_disj:

      (P ∨ Q --> R) <-> (P --> R) ∧ (Q --> R)

theorem imp_disj1:

      (P --> Q) ∨ R <-> P --> Q ∨ R

theorem imp_disj2:

      Q ∨ (P --> R) <-> P --> Q ∨ R

theorem de_Morgan_disj:

      ¬ (P ∨ Q) <-> ¬ P ∧ ¬ Q

theorem de_Morgan_conj:

      ¬ (P ∧ Q) <-> ¬ P ∨ ¬ Q

theorem not_imp:

      ¬ (P --> Q) <-> P ∧ ¬ Q

theorem not_iff:

      ¬ (P <-> Q) <-> P <-> ¬ Q

theorem not_all:

      ¬ (∀x. P(x)) <-> (∃x. ¬ P(x))

theorem imp_all:

      ((∀x. P(x)) --> Q) <-> (∃x. P(x) --> Q)

theorem not_ex:

      ¬ (∃x. P(x)) <-> (∀x. ¬ P(x))

theorem imp_ex:

      ((∃x. P(x)) --> Q) <-> (∀x. P(x) --> Q)

theorem ex_disj_distrib:

      (∃x. P(x) ∨ Q(x)) <-> (∃x. P(x)) ∨ (∃x. Q(x))

theorem all_conj_distrib:

      (∀x. P(x) ∧ Q(x)) <-> (∀x. P(x)) ∧ (∀x. Q(x))

theorem meta_eq_to_iff:

      x == y ==> x <-> y

theorems meta_simps:

      (!!x. PROP V) == PROP V

      (True ==> PROP P) == PROP P

theorems IFOL_simps:

      a = a <-> 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)

theorem notFalseI:

      ¬ False

theorems triv_rls:

      True

      a = a

      x == x

      P <-> P

      ¬ False

theorems 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

### Lucas Dixon's eqstep tactic

### Other simple lemmas

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))

### Proof by cases and induction

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_impliesI:

      (A ==> B) ==> induct_implies(A, B)

lemmas induct_atomize:

      (A && B) == A ∧ B

      (!!x. P(x)) == induct_forall(P)

      (A ==> B) == induct_implies(A, B)

      (x == y) == induct_equal(x, y)

lemmas induct_atomize:

      (A && B) == A ∧ B

      (!!x. P(x)) == induct_forall(P)

      (A ==> B) == induct_implies(A, B)

      (x == y) == induct_equal(x, y)

lemmas induct_rulify1:

      induct_forall(P) == (!!x. P(x))

      induct_implies(A, B) == (A ==> B)

      induct_equal(x, y) == x == y

lemmas induct_rulify1:

      induct_forall(P) == (!!x. P(x))

      induct_implies(A, B) == (A ==> B)

      induct_equal(x, y) == x == y

lemmas induct_rulify2:

      induct_forall(P) == ∀x. P(x)

      induct_implies(A, B) == A --> B

      induct_equal(x, y) == x = y

lemmas induct_rulify2:

      induct_forall(P) == ∀x. P(x)

      induct_implies(A, B) == A --> B

      induct_equal(x, y) == x = y

lemma all_conj_eq:

      (∀x. P(x)) ∧ (∀y. Q(y)) == ∀x y. P(x) ∧ Q(y)