# Theory misc

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theory misc
imports Main
begin

    (*  Title:      ZF/ex/misc.ML
        ID:         $Id: misc.thy,v 1.5 2005/08/17 13:10:00 paulson Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1993  University of Cambridge

    Composition of homomorphisms, Pastre's examples, ...
    *)

    header{*Miscellaneous ZF Examples*}

    theory misc imports Main begin

    subsection{*Various Small Problems*}

    text{*The singleton problems are much harder in HOL.*}
    lemma singleton_example_1:
         "∀x ∈ S. ∀y ∈ S. x ⊆ y ==> ∃z. S ⊆ {z}"
      by blast

    lemma singleton_example_2:
         "∀x ∈ S. \<Union>S ⊆ x ==> ∃z. S ⊆ {z}"
      -- {*Variant of the problem above. *}
      by blast

    lemma "∃!x. f (g(x)) = x ==> ∃!y. g (f(y)) = y"
      -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text auto} all fail. *}
      apply (erule ex1E, rule ex1I, erule subst_context)
      apply (rule subst, assumption, erule allE, rule subst_context, erule mp)
      apply (erule subst_context)
      done

    text{*A weird property of ordered pairs.*}
    lemma "b≠c ==> <a,b> Int <a,c> = <a,a>"
    by (simp add: Pair_def Int_cons_left Int_cons_right doubleton_eq_iff, blast)

    text{*These two are cited in Benzmueller and Kohlhase's system description of
     LEO, CADE-15, 1998 (page 139-143) as theorems LEO could not prove.*}
    lemma "(X = Y Un Z) <-> (Y ⊆ X & Z ⊆ X & (∀V. Y ⊆ V & Z ⊆ V --> X ⊆ V))"
    by (blast intro!: equalityI)

    text{*the dual of the previous one}
    lemma "(X = Y Int Z) <-> (X ⊆ Y & X ⊆ Z & (∀V. V ⊆ Y & V ⊆ Z --> V ⊆ X))"
    by (blast intro!: equalityI)

    text{*trivial example of term synthesis: apparently hard for some provers!}
    lemma "a ≠ b ==> a:?X & b ∉ ?X"
    by blast

    text{*Nice Blast_tac benchmark.  Proved in 0.3s; old tactics can't manage it!}
    lemma "∀x ∈ S. ∀y ∈ S. x ⊆ y ==> ∃z. S ⊆ {z}"
    by blast

    text{*variant of the benchmark above}
    lemma "∀x ∈ S. Union(S) ⊆ x ==> ∃z. S ⊆ {z}"
    by blast

    (*Example 12 (credited to Peter Andrews) from
     W. Bledsoe.  A Maximal Method for Set Variables in Automatic Theorem-proving.
     In: J. Hayes and D. Michie and L. Mikulich, eds.  Machine Intelligence 9.
     Ellis Horwood, 53-100 (1979). *)
    lemma "(∀F. {x} ∈ F --> {y} ∈ F) --> (∀A. x ∈ A --> y ∈ A)"
    by best

    text{*A characterization of functions suggested by Tobias Nipkow*}
    lemma "r ∈ domain(r)->B  <->  r ⊆ domain(r)*B & (∀X. r `` (r -`` X) ⊆ X)"
    by (unfold Pi_def function_def, best)

    subsection{*Composition of homomorphisms is a Homomorphism*}

    text{*Given as a challenge problem in
      R. Boyer et al.,
      Set Theory in First-Order Logic: Clauses for G\"odel's Axioms,
      JAR 2 (1986), 287-327 *}

    text{*collecting the relevant lemmas}
    declare comp_fun [simp] SigmaI [simp] apply_funtype [simp]

    (*Force helps prove conditions of rewrites such as comp_fun_apply, since
      rewriting does not instantiate Vars.*)
    lemma "(∀A f B g. hom(A,f,B,g) =
               {H ∈ A->B. f ∈ A*A->A & g ∈ B*B->B &
                         (∀x ∈ A. ∀y ∈ A. H`(f`<x,y>) = g`<H`x,H`y>)}) -->
           J ∈ hom(A,f,B,g) & K ∈ hom(B,g,C,h) -->
           (K O J) ∈ hom(A,f,C,h)"
    by force

    text{*Another version, with meta-level rewriting}
    lemma "(!! A f B g. hom(A,f,B,g) ==
               {H ∈ A->B. f ∈ A*A->A & g ∈ B*B->B &
                         (∀x ∈ A. ∀y ∈ A. H`(f`<x,y>) = g`<H`x,H`y>)})
           ==> J ∈ hom(A,f,B,g) & K ∈ hom(B,g,C,h) --> (K O J) ∈ hom(A,f,C,h)"
    by force

    subsection{*Pastre's Examples*}

    text{*D Pastre.  Automatic theorem proving in set theory.
            Artificial Intelligence, 10:1--27, 1978.
    Previously, these were done using ML code, but blast manages fine.*}

    lemmas compIs [intro] = comp_surj comp_inj comp_fun [intro]
    lemmas compDs [dest] =  comp_mem_injD1 comp_mem_surjD1
                            comp_mem_injD2 comp_mem_surjD2

    lemma pastre1:
        "[| (h O g O f) ∈ inj(A,A);
            (f O h O g) ∈ surj(B,B);
            (g O f O h) ∈ surj(C,C);
            f ∈ A->B;  g ∈ B->C;  h ∈ C->A |] ==> h ∈ bij(C,A)";
    by (unfold bij_def, blast)

    lemma pastre3:
        "[| (h O g O f) ∈ surj(A,A);
            (f O h O g) ∈ surj(B,B);
            (g O f O h) ∈ inj(C,C);
            f ∈ A->B;  g ∈ B->C;  h ∈ C->A |] ==> h ∈ bij(C,A)"
    by (unfold bij_def, blast)

    lemma pastre4:
        "[| (h O g O f) ∈ surj(A,A);
            (f O h O g) ∈ inj(B,B);
            (g O f O h) ∈ inj(C,C);
            f ∈ A->B;  g ∈ B->C;  h ∈ C->A |] ==> h ∈ bij(C,A)"
    by (unfold bij_def, blast)

    lemma pastre5:
        "[| (h O g O f) ∈ inj(A,A);
            (f O h O g) ∈ surj(B,B);
            (g O f O h) ∈ inj(C,C);
            f ∈ A->B;  g ∈ B->C;  h ∈ C->A |] ==> h ∈ bij(C,A)"
    by (unfold bij_def, blast)

    lemma pastre6:
        "[| (h O g O f) ∈ inj(A,A);
            (f O h O g) ∈ inj(B,B);
            (g O f O h) ∈ surj(C,C);
            f ∈ A->B;  g ∈ B->C;  h ∈ C->A |] ==> h ∈ bij(C,A)"
    by (unfold bij_def, blast)

    end

### Various Small Problems

lemma singleton_example_1:

      ∀x∈S. ∀y∈S. x ⊆ y ==> ∃z. S ⊆ {z}

lemma singleton_example_2:

      ∀x∈S. \<Union>S ⊆ x ==> ∃z. S ⊆ {z}

lemma

      ∃!x. f(g(x)) = x ==> ∃!y. g(f(y)) = y

lemma

      b ≠ c ==> ⟨a, b⟩ ∩ ⟨a, c⟩ = ⟨a, a⟩

lemma

      X = Y ∪ Z <-> Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V)

lemma

      r ∈ domain(r) -> B <-> r ⊆ domain(r) × B ∧ (∀X. r `` (r -`` X) ⊆ X)

### Composition of homomorphisms is a Homomorphism

lemmas compIs:

      [| g ∈ surj(A, B); f ∈ surj(B, C) |] ==> f O g ∈ surj(A, C)

      [| g ∈ inj(A, B); f ∈ inj(B, C) |] ==> f O g ∈ inj(A, C)

      [| g ∈ A -> B; f ∈ B -> C |] ==> f O g ∈ A -> C

lemmas compIs:

      [| g ∈ surj(A, B); f ∈ surj(B, C) |] ==> f O g ∈ surj(A, C)

      [| g ∈ inj(A, B); f ∈ inj(B, C) |] ==> f O g ∈ inj(A, C)

      [| g ∈ A -> B; f ∈ B -> C |] ==> f O g ∈ A -> C

lemmas compDs:

      [| f O g ∈ inj(A, C); g ∈ A -> B; f ∈ B -> C |] ==> g ∈ inj(A, B)

      [| f O g ∈ surj(A, C); g ∈ A -> B; f ∈ B -> C |] ==> f ∈ surj(B, C)

      [| f O g ∈ inj(A, C); g ∈ surj(A, B); f ∈ B -> C |] ==> f ∈ inj(B, C)

      [| f O g ∈ surj(A, C); g ∈ A -> B; f ∈ inj(B, C) |] ==> g ∈ surj(A, B)

lemmas compDs:

      [| f O g ∈ inj(A, C); g ∈ A -> B; f ∈ B -> C |] ==> g ∈ inj(A, B)

      [| f O g ∈ surj(A, C); g ∈ A -> B; f ∈ B -> C |] ==> f ∈ surj(B, C)

      [| f O g ∈ inj(A, C); g ∈ surj(A, B); f ∈ B -> C |] ==> f ∈ inj(B, C)

      [| f O g ∈ surj(A, C); g ∈ A -> B; f ∈ inj(B, C) |] ==> g ∈ surj(A, B)

lemma pastre1:

      [| h O g O f ∈ inj(A, A); f O h O g ∈ surj(B, B); g O f O h ∈ surj(C, C);
         f ∈ A -> B; g ∈ B -> C; h ∈ C -> A |]
      ==> h ∈ bij(C, A)

lemma pastre3:

      [| h O g O f ∈ surj(A, A); f O h O g ∈ surj(B, B); g O f O h ∈ inj(C, C);
         f ∈ A -> B; g ∈ B -> C; h ∈ C -> A |]
      ==> h ∈ bij(C, A)

lemma pastre4:

      [| h O g O f ∈ surj(A, A); f O h O g ∈ inj(B, B); g O f O h ∈ inj(C, C);
         f ∈ A -> B; g ∈ B -> C; h ∈ C -> A |]
      ==> h ∈ bij(C, A)

lemma pastre5:

      [| h O g O f ∈ inj(A, A); f O h O g ∈ surj(B, B); g O f O h ∈ inj(C, C);
         f ∈ A -> B; g ∈ B -> C; h ∈ C -> A |]
      ==> h ∈ bij(C, A)

lemma pastre6:

      [| h O g O f ∈ inj(A, A); f O h O g ∈ inj(B, B); g O f O h ∈ surj(C, C);
         f ∈ A -> B; g ∈ B -> C; h ∈ C -> A |]
      ==> h ∈ bij(C, A)