Theory func (Isabelle2005: October 2005)

# Theory func

Up to index of Isabelle/ZF

theory func
imports equalities Sum
begin

    (*  Title:      ZF/func.thy
        ID:         $Id: func.thy,v 1.22 2005/06/17 14:15:10 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1991  University of Cambridge

    *)

    header{*Functions, Function Spaces, Lambda-Abstraction*}

    theory func imports equalities Sum begin

    subsection{*The Pi Operator: Dependent Function Space*}

    lemma subset_Sigma_imp_relation: "r <= Sigma(A,B) ==> relation(r)"
    by (simp add: relation_def, blast)

    lemma relation_converse_converse [simp]:
         "relation(r) ==> converse(converse(r)) = r"
    by (simp add: relation_def, blast)

    lemma relation_restrict [simp]:  "relation(restrict(r,A))"
    by (simp add: restrict_def relation_def, blast)

    lemma Pi_iff:
        "f: Pi(A,B) <-> function(f) & f<=Sigma(A,B) & A<=domain(f)"
    by (unfold Pi_def, blast)

    (*For upward compatibility with the former definition*)
    lemma Pi_iff_old:
        "f: Pi(A,B) <-> f<=Sigma(A,B) & (ALL x:A. EX! y. <x,y>: f)"
    by (unfold Pi_def function_def, blast)

    lemma fun_is_function: "f: Pi(A,B) ==> function(f)"
    by (simp only: Pi_iff)

    lemma function_imp_Pi:
         "[|function(f); relation(f)|] ==> f ∈ domain(f) -> range(f)"
    by (simp add: Pi_iff relation_def, blast)

    lemma functionI:
         "[| !!x y y'. [| <x,y>:r; <x,y'>:r |] ==> y=y' |] ==> function(r)"
    by (simp add: function_def, blast)

    (*Functions are relations*)
    lemma fun_is_rel: "f: Pi(A,B) ==> f <= Sigma(A,B)"
    by (unfold Pi_def, blast)

    lemma Pi_cong:
        "[| A=A';  !!x. x:A' ==> B(x)=B'(x) |] ==> Pi(A,B) = Pi(A',B')"
    by (simp add: Pi_def cong add: Sigma_cong)

    (*Sigma_cong, Pi_cong NOT given to Addcongs: they cause
      flex-flex pairs and the "Check your prover" error.  Most
      Sigmas and Pis are abbreviated as * or -> *)

    (*Weakening one function type to another; see also Pi_type*)
    lemma fun_weaken_type: "[| f: A->B;  B<=D |] ==> f: A->D"
    by (unfold Pi_def, best)

    subsection{*Function Application*}

    lemma apply_equality2: "[| <a,b>: f;  <a,c>: f;  f: Pi(A,B) |] ==> b=c"
    by (unfold Pi_def function_def, blast)

    lemma function_apply_equality: "[| <a,b>: f;  function(f) |] ==> f`a = b"
    by (unfold apply_def function_def, blast)

    lemma apply_equality: "[| <a,b>: f;  f: Pi(A,B) |] ==> f`a = b"
    apply (unfold Pi_def)
    apply (blast intro: function_apply_equality)
    done

    (*Applying a function outside its domain yields 0*)
    lemma apply_0: "a ~: domain(f) ==> f`a = 0"
    by (unfold apply_def, blast)

    lemma Pi_memberD: "[| f: Pi(A,B);  c: f |] ==> EX x:A.  c = <x,f`x>"
    apply (frule fun_is_rel)
    apply (blast dest: apply_equality)
    done

    lemma function_apply_Pair: "[| function(f);  a : domain(f)|] ==> <a,f`a>: f"
    apply (simp add: function_def, clarify)
    apply (subgoal_tac "f`a = y", blast)
    apply (simp add: apply_def, blast)
    done

    lemma apply_Pair: "[| f: Pi(A,B);  a:A |] ==> <a,f`a>: f"
    apply (simp add: Pi_iff)
    apply (blast intro: function_apply_Pair)
    done

    (*Conclusion is flexible -- use res_inst_tac or else apply_funtype below!*)
    lemma apply_type [TC]: "[| f: Pi(A,B);  a:A |] ==> f`a : B(a)"
    by (blast intro: apply_Pair dest: fun_is_rel)

    (*This version is acceptable to the simplifier*)
    lemma apply_funtype: "[| f: A->B;  a:A |] ==> f`a : B"
    by (blast dest: apply_type)

    lemma apply_iff: "f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b"
    apply (frule fun_is_rel)
    apply (blast intro!: apply_Pair apply_equality)
    done

    (*Refining one Pi type to another*)
    lemma Pi_type: "[| f: Pi(A,C);  !!x. x:A ==> f`x : B(x) |] ==> f : Pi(A,B)"
    apply (simp only: Pi_iff)
    apply (blast dest: function_apply_equality)
    done

    (*Such functions arise in non-standard datatypes, ZF/ex/Ntree for instance*)
    lemma Pi_Collect_iff:
         "(f : Pi(A, %x. {y:B(x). P(x,y)}))
          <->  f : Pi(A,B) & (ALL x: A. P(x, f`x))"
    by (blast intro: Pi_type dest: apply_type)

    lemma Pi_weaken_type:
            "[| f : Pi(A,B);  !!x. x:A ==> B(x)<=C(x) |] ==> f : Pi(A,C)"
    by (blast intro: Pi_type dest: apply_type)

    (** Elimination of membership in a function **)

    lemma domain_type: "[| <a,b> : f;  f: Pi(A,B) |] ==> a : A"
    by (blast dest: fun_is_rel)

    lemma range_type: "[| <a,b> : f;  f: Pi(A,B) |] ==> b : B(a)"
    by (blast dest: fun_is_rel)

    lemma Pair_mem_PiD: "[| <a,b>: f;  f: Pi(A,B) |] ==> a:A & b:B(a) & f`a = b"
    by (blast intro: domain_type range_type apply_equality)

    subsection{*Lambda Abstraction*}

    lemma lamI: "a:A ==> <a,b(a)> : (lam x:A. b(x))"
    apply (unfold lam_def)
    apply (erule RepFunI)
    done

    lemma lamE:
        "[| p: (lam x:A. b(x));  !!x.[| x:A; p=<x,b(x)> |] ==> P
         |] ==>  P"
    by (simp add: lam_def, blast)

    lemma lamD: "[| <a,c>: (lam x:A. b(x)) |] ==> c = b(a)"
    by (simp add: lam_def)

    lemma lam_type [TC]:
        "[| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)"
    by (simp add: lam_def Pi_def function_def, blast)

    lemma lam_funtype: "(lam x:A. b(x)) : A -> {b(x). x:A}"
    by (blast intro: lam_type)

    lemma function_lam: "function (lam x:A. b(x))"
    by (simp add: function_def lam_def)

    lemma relation_lam: "relation (lam x:A. b(x))"
    by (simp add: relation_def lam_def)

    lemma beta_if [simp]: "(lam x:A. b(x)) ` a = (if a : A then b(a) else 0)"
    by (simp add: apply_def lam_def, blast)

    lemma beta: "a : A ==> (lam x:A. b(x)) ` a = b(a)"
    by (simp add: apply_def lam_def, blast)

    lemma lam_empty [simp]: "(lam x:0. b(x)) = 0"
    by (simp add: lam_def)

    lemma domain_lam [simp]: "domain(Lambda(A,b)) = A"
    by (simp add: lam_def, blast)

    (*congruence rule for lambda abstraction*)
    lemma lam_cong [cong]:
        "[| A=A';  !!x. x:A' ==> b(x)=b'(x) |] ==> Lambda(A,b) = Lambda(A',b')"
    by (simp only: lam_def cong add: RepFun_cong)

    lemma lam_theI:
        "(!!x. x:A ==> EX! y. Q(x,y)) ==> EX f. ALL x:A. Q(x, f`x)"
    apply (rule_tac x = "lam x: A. THE y. Q (x,y)" in exI)
    apply simp
    apply (blast intro: theI)
    done

    lemma lam_eqE: "[| (lam x:A. f(x)) = (lam x:A. g(x));  a:A |] ==> f(a)=g(a)"
    by (fast intro!: lamI elim: equalityE lamE)

    (*Empty function spaces*)
    lemma Pi_empty1 [simp]: "Pi(0,A) = {0}"
    by (unfold Pi_def function_def, blast)

    (*The singleton function*)
    lemma singleton_fun [simp]: "{<a,b>} : {a} -> {b}"
    by (unfold Pi_def function_def, blast)

    lemma Pi_empty2 [simp]: "(A->0) = (if A=0 then {0} else 0)"
    by (unfold Pi_def function_def, force)

    lemma  fun_space_empty_iff [iff]: "(A->X)=0 <-> X=0 & (A ≠ 0)"
    apply auto
    apply (fast intro!: equals0I intro: lam_type)
    done

    subsection{*Extensionality*}

    (*Semi-extensionality!*)

    lemma fun_subset:
        "[| f : Pi(A,B);  g: Pi(C,D);  A<=C;
            !!x. x:A ==> f`x = g`x       |] ==> f<=g"
    by (force dest: Pi_memberD intro: apply_Pair)

    lemma fun_extension:
        "[| f : Pi(A,B);  g: Pi(A,D);
            !!x. x:A ==> f`x = g`x       |] ==> f=g"
    by (blast del: subsetI intro: subset_refl sym fun_subset)

    lemma eta [simp]: "f : Pi(A,B) ==> (lam x:A. f`x) = f"
    apply (rule fun_extension)
    apply (auto simp add: lam_type apply_type beta)
    done

    lemma fun_extension_iff:
         "[| f:Pi(A,B); g:Pi(A,C) |] ==> (ALL a:A. f`a = g`a) <-> f=g"
    by (blast intro: fun_extension)

    (*thm by Mark Staples, proof by lcp*)
    lemma fun_subset_eq: "[| f:Pi(A,B); g:Pi(A,C) |] ==> f <= g <-> (f = g)"
    by (blast dest: apply_Pair
              intro: fun_extension apply_equality [symmetric])

    (*Every element of Pi(A,B) may be expressed as a lambda abstraction!*)
    lemma Pi_lamE:
      assumes major: "f: Pi(A,B)"
          and minor: "!!b. [| ALL x:A. b(x):B(x);  f = (lam x:A. b(x)) |] ==> P"
      shows "P"
    apply (rule minor)
    apply (rule_tac [2] eta [symmetric])
    apply (blast intro: major apply_type)+
    done

    subsection{*Images of Functions*}

    lemma image_lam: "C <= A ==> (lam x:A. b(x)) `` C = {b(x). x:C}"
    by (unfold lam_def, blast)

    lemma Repfun_function_if:
         "function(f)
          ==> {f`x. x:C} = (if C <= domain(f) then f``C else cons(0,f``C))";
    apply simp
    apply (intro conjI impI)
     apply (blast dest: function_apply_equality intro: function_apply_Pair)
    apply (rule equalityI)
     apply (blast intro!: function_apply_Pair apply_0)
    apply (blast dest: function_apply_equality intro: apply_0 [symmetric])
    done

    (*For this lemma and the next, the right-hand side could equivalently
      be written \<Union>x∈C. {f`x} *)
    lemma image_function:
         "[| function(f);  C <= domain(f) |] ==> f``C = {f`x. x:C}";
    by (simp add: Repfun_function_if)

    lemma image_fun: "[| f : Pi(A,B);  C <= A |] ==> f``C = {f`x. x:C}"
    apply (simp add: Pi_iff)
    apply (blast intro: image_function)
    done

    lemma image_eq_UN:
      assumes f: "f ∈ Pi(A,B)" "C ⊆ A" shows "f``C = (\<Union>x∈C. {f ` x})"
    by (auto simp add: image_fun [OF f])

    lemma Pi_image_cons:
         "[| f: Pi(A,B);  x: A |] ==> f `` cons(x,y) = cons(f`x, f``y)"
    by (blast dest: apply_equality apply_Pair)

    subsection{*Properties of @{term "restrict(f,A)"}*}

    lemma restrict_subset: "restrict(f,A) <= f"
    by (unfold restrict_def, blast)

    lemma function_restrictI:
        "function(f) ==> function(restrict(f,A))"
    by (unfold restrict_def function_def, blast)

    lemma restrict_type2: "[| f: Pi(C,B);  A<=C |] ==> restrict(f,A) : Pi(A,B)"
    by (simp add: Pi_iff function_def restrict_def, blast)

    lemma restrict: "restrict(f,A) ` a = (if a : A then f`a else 0)"
    by (simp add: apply_def restrict_def, blast)

    lemma restrict_empty [simp]: "restrict(f,0) = 0"
    by (unfold restrict_def, simp)

    lemma restrict_iff: "z ∈ restrict(r,A) <-> z ∈ r & (∃x∈A. ∃y. z = ⟨x, y⟩)"
    by (simp add: restrict_def)

    lemma restrict_restrict [simp]:
         "restrict(restrict(r,A),B) = restrict(r, A Int B)"
    by (unfold restrict_def, blast)

    lemma domain_restrict [simp]: "domain(restrict(f,C)) = domain(f) Int C"
    apply (unfold restrict_def)
    apply (auto simp add: domain_def)
    done

    lemma restrict_idem: "f <= Sigma(A,B) ==> restrict(f,A) = f"
    by (simp add: restrict_def, blast)

    (*converse probably holds too*)
    lemma domain_restrict_idem:
         "[| domain(r) <= A; relation(r) |] ==> restrict(r,A) = r"
    by (simp add: restrict_def relation_def, blast)

    lemma domain_restrict_lam [simp]: "domain(restrict(Lambda(A,f),C)) = A Int C"
    apply (unfold restrict_def lam_def)
    apply (rule equalityI)
    apply (auto simp add: domain_iff)
    done

    lemma restrict_if [simp]: "restrict(f,A) ` a = (if a : A then f`a else 0)"
    by (simp add: restrict apply_0)

    lemma restrict_lam_eq:
        "A<=C ==> restrict(lam x:C. b(x), A) = (lam x:A. b(x))"
    by (unfold restrict_def lam_def, auto)

    lemma fun_cons_restrict_eq:
         "f : cons(a, b) -> B ==> f = cons(<a, f ` a>, restrict(f, b))"
    apply (rule equalityI)
     prefer 2 apply (blast intro: apply_Pair restrict_subset [THEN subsetD])
    apply (auto dest!: Pi_memberD simp add: restrict_def lam_def)
    done

    subsection{*Unions of Functions*}

    (** The Union of a set of COMPATIBLE functions is a function **)

    lemma function_Union:
        "[| ALL x:S. function(x);
            ALL x:S. ALL y:S. x<=y | y<=x  |]
         ==> function(Union(S))"
    by (unfold function_def, blast)

    lemma fun_Union:
        "[| ALL f:S. EX C D. f:C->D;
                 ALL f:S. ALL y:S. f<=y | y<=f  |] ==>
              Union(S) : domain(Union(S)) -> range(Union(S))"
    apply (unfold Pi_def)
    apply (blast intro!: rel_Union function_Union)
    done

    lemma gen_relation_Union [rule_format]:
         "∀f∈F. relation(f) ==> relation(Union(F))"
    by (simp add: relation_def)

    (** The Union of 2 disjoint functions is a function **)

    lemmas Un_rls = Un_subset_iff SUM_Un_distrib1 prod_Un_distrib2
                    subset_trans [OF _ Un_upper1]
                    subset_trans [OF _ Un_upper2]

    lemma fun_disjoint_Un:
         "[| f: A->B;  g: C->D;  A Int C = 0  |]
          ==> (f Un g) : (A Un C) -> (B Un D)"
    (*Prove the product and domain subgoals using distributive laws*)
    apply (simp add: Pi_iff extension Un_rls)
    apply (unfold function_def, blast)
    done

    lemma fun_disjoint_apply1: "a ∉ domain(g) ==> (f Un g)`a = f`a"
    by (simp add: apply_def, blast)

    lemma fun_disjoint_apply2: "c ∉ domain(f) ==> (f Un g)`c = g`c"
    by (simp add: apply_def, blast)

    subsection{*Domain and Range of a Function or Relation*}

    lemma domain_of_fun: "f : Pi(A,B) ==> domain(f)=A"
    by (unfold Pi_def, blast)

    lemma apply_rangeI: "[| f : Pi(A,B);  a: A |] ==> f`a : range(f)"
    by (erule apply_Pair [THEN rangeI], assumption)

    lemma range_of_fun: "f : Pi(A,B) ==> f : A->range(f)"
    by (blast intro: Pi_type apply_rangeI)

    subsection{*Extensions of Functions*}

    lemma fun_extend:
         "[| f: A->B;  c~:A |] ==> cons(<c,b>,f) : cons(c,A) -> cons(b,B)"
    apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
    apply (simp add: cons_eq)
    done

    lemma fun_extend3:
         "[| f: A->B;  c~:A;  b: B |] ==> cons(<c,b>,f) : cons(c,A) -> B"
    by (blast intro: fun_extend [THEN fun_weaken_type])

    lemma extend_apply:
         "c ~: domain(f) ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
    by (auto simp add: apply_def)

    lemma fun_extend_apply [simp]:
         "[| f: A->B;  c~:A |] ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
    apply (rule extend_apply)
    apply (simp add: Pi_def, blast)
    done

    lemmas singleton_apply = apply_equality [OF singletonI singleton_fun, simp]

    (*For Finite.ML.  Inclusion of right into left is easy*)
    lemma cons_fun_eq:
         "c ~: A ==> cons(c,A) -> B = (\<Union>f ∈ A->B. \<Union>b∈B. {cons(<c,b>, f)})"
    apply (rule equalityI)
    apply (safe elim!: fun_extend3)
    (*Inclusion of left into right*)
    apply (subgoal_tac "restrict (x, A) : A -> B")
     prefer 2 apply (blast intro: restrict_type2)
    apply (rule UN_I, assumption)
    apply (rule apply_funtype [THEN UN_I])
      apply assumption
     apply (rule consI1)
    apply (simp (no_asm))
    apply (rule fun_extension)
      apply assumption
     apply (blast intro: fun_extend)
    apply (erule consE, simp_all)
    done

    lemma succ_fun_eq: "succ(n) -> B = (\<Union>f ∈ n->B. \<Union>b∈B. {cons(<n,b>, f)})"
    by (simp add: succ_def mem_not_refl cons_fun_eq)

    subsection{*Function Updates*}

    constdefs
      update  :: "[i,i,i] => i"
       "update(f,a,b) == lam x: cons(a, domain(f)). if(x=a, b, f`x)"

    nonterminals
      updbinds  updbind

    syntax

      (* Let expressions *)

      "_updbind"    :: "[i, i] => updbind"               ("(2_ :=/ _)")
      ""            :: "updbind => updbinds"             ("_")
      "_updbinds"   :: "[updbind, updbinds] => updbinds" ("_,/ _")
      "_Update"     :: "[i, updbinds] => i"              ("_/'((_)')" [900,0] 900)

    translations
      "_Update (f, _updbinds(b,bs))"  == "_Update (_Update(f,b), bs)"
      "f(x:=y)"                       == "update(f,x,y)"

    lemma update_apply [simp]: "f(x:=y) ` z = (if z=x then y else f`z)"
    apply (simp add: update_def)
    apply (case_tac "z ∈ domain(f)")
    apply (simp_all add: apply_0)
    done

    lemma update_idem: "[| f`x = y;  f: Pi(A,B);  x: A |] ==> f(x:=y) = f"
    apply (unfold update_def)
    apply (simp add: domain_of_fun cons_absorb)
    apply (rule fun_extension)
    apply (best intro: apply_type if_type lam_type, assumption, simp)
    done

    (* [| f: Pi(A, B); x:A |] ==> f(x := f`x) = f *)
    declare refl [THEN update_idem, simp]

    lemma domain_update [simp]: "domain(f(x:=y)) = cons(x, domain(f))"
    by (unfold update_def, simp)

    lemma update_type: "[| f:Pi(A,B);  x : A;  y: B(x) |] ==> f(x:=y) : Pi(A, B)"
    apply (unfold update_def)
    apply (simp add: domain_of_fun cons_absorb apply_funtype lam_type)
    done

    subsection{*Monotonicity Theorems*}

    subsubsection{*Replacement in its Various Forms*}

    (*Not easy to express monotonicity in P, since any "bigger" predicate
      would have to be single-valued*)
    lemma Replace_mono: "A<=B ==> Replace(A,P) <= Replace(B,P)"
    by (blast elim!: ReplaceE)

    lemma RepFun_mono: "A<=B ==> {f(x). x:A} <= {f(x). x:B}"
    by blast

    lemma Pow_mono: "A<=B ==> Pow(A) <= Pow(B)"
    by blast

    lemma Union_mono: "A<=B ==> Union(A) <= Union(B)"
    by blast

    lemma UN_mono:
        "[| A<=C;  !!x. x:A ==> B(x)<=D(x) |] ==> (\<Union>x∈A. B(x)) <= (\<Union>x∈C. D(x))"
    by blast

    (*Intersection is ANTI-monotonic.  There are TWO premises! *)
    lemma Inter_anti_mono: "[| A<=B;  A≠0 |] ==> Inter(B) <= Inter(A)"
    by blast

    lemma cons_mono: "C<=D ==> cons(a,C) <= cons(a,D)"
    by blast

    lemma Un_mono: "[| A<=C;  B<=D |] ==> A Un B <= C Un D"
    by blast

    lemma Int_mono: "[| A<=C;  B<=D |] ==> A Int B <= C Int D"
    by blast

    lemma Diff_mono: "[| A<=C;  D<=B |] ==> A-B <= C-D"
    by blast

    subsubsection{*Standard Products, Sums and Function Spaces *}

    lemma Sigma_mono [rule_format]:
         "[| A<=C;  !!x. x:A --> B(x) <= D(x) |] ==> Sigma(A,B) <= Sigma(C,D)"
    by blast

    lemma sum_mono: "[| A<=C;  B<=D |] ==> A+B <= C+D"
    by (unfold sum_def, blast)

    (*Note that B->A and C->A are typically disjoint!*)
    lemma Pi_mono: "B<=C ==> A->B <= A->C"
    by (blast intro: lam_type elim: Pi_lamE)

    lemma lam_mono: "A<=B ==> Lambda(A,c) <= Lambda(B,c)"
    apply (unfold lam_def)
    apply (erule RepFun_mono)
    done

    subsubsection{*Converse, Domain, Range, Field*}

    lemma converse_mono: "r<=s ==> converse(r) <= converse(s)"
    by blast

    lemma domain_mono: "r<=s ==> domain(r)<=domain(s)"
    by blast

    lemmas domain_rel_subset = subset_trans [OF domain_mono domain_subset]

    lemma range_mono: "r<=s ==> range(r)<=range(s)"
    by blast

    lemmas range_rel_subset = subset_trans [OF range_mono range_subset]

    lemma field_mono: "r<=s ==> field(r)<=field(s)"
    by blast

    lemma field_rel_subset: "r <= A*A ==> field(r) <= A"
    by (erule field_mono [THEN subset_trans], blast)

    subsubsection{*Images*}

    lemma image_pair_mono:
        "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r``A <= s``B"
    by blast

    lemma vimage_pair_mono:
        "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r-``A <= s-``B"
    by blast

    lemma image_mono: "[| r<=s;  A<=B |] ==> r``A <= s``B"
    by blast

    lemma vimage_mono: "[| r<=s;  A<=B |] ==> r-``A <= s-``B"
    by blast

    lemma Collect_mono:
        "[| A<=B;  !!x. x:A ==> P(x) --> Q(x) |] ==> Collect(A,P) <= Collect(B,Q)"
    by blast

    (*Used in intr_elim.ML and in individual datatype definitions*)
    lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono
                         Collect_mono Part_mono in_mono

    ML
    {*
    val Pi_iff = thm "Pi_iff";
    val Pi_iff_old = thm "Pi_iff_old";
    val fun_is_function = thm "fun_is_function";
    val fun_is_rel = thm "fun_is_rel";
    val Pi_cong = thm "Pi_cong";
    val fun_weaken_type = thm "fun_weaken_type";
    val apply_equality2 = thm "apply_equality2";
    val function_apply_equality = thm "function_apply_equality";
    val apply_equality = thm "apply_equality";
    val apply_0 = thm "apply_0";
    val Pi_memberD = thm "Pi_memberD";
    val function_apply_Pair = thm "function_apply_Pair";
    val apply_Pair = thm "apply_Pair";
    val apply_type = thm "apply_type";
    val apply_funtype = thm "apply_funtype";
    val apply_iff = thm "apply_iff";
    val Pi_type = thm "Pi_type";
    val Pi_Collect_iff = thm "Pi_Collect_iff";
    val Pi_weaken_type = thm "Pi_weaken_type";
    val domain_type = thm "domain_type";
    val range_type = thm "range_type";
    val Pair_mem_PiD = thm "Pair_mem_PiD";
    val lamI = thm "lamI";
    val lamE = thm "lamE";
    val lamD = thm "lamD";
    val lam_type = thm "lam_type";
    val lam_funtype = thm "lam_funtype";
    val beta = thm "beta";
    val lam_empty = thm "lam_empty";
    val domain_lam = thm "domain_lam";
    val lam_cong = thm "lam_cong";
    val lam_theI = thm "lam_theI";
    val lam_eqE = thm "lam_eqE";
    val Pi_empty1 = thm "Pi_empty1";
    val singleton_fun = thm "singleton_fun";
    val Pi_empty2 = thm "Pi_empty2";
    val fun_space_empty_iff = thm "fun_space_empty_iff";
    val fun_subset = thm "fun_subset";
    val fun_extension = thm "fun_extension";
    val eta = thm "eta";
    val fun_extension_iff = thm "fun_extension_iff";
    val fun_subset_eq = thm "fun_subset_eq";
    val Pi_lamE = thm "Pi_lamE";
    val image_lam = thm "image_lam";
    val image_fun = thm "image_fun";
    val Pi_image_cons = thm "Pi_image_cons";
    val restrict_subset = thm "restrict_subset";
    val function_restrictI = thm "function_restrictI";
    val restrict_type2 = thm "restrict_type2";
    val restrict = thm "restrict";
    val restrict_empty = thm "restrict_empty";
    val domain_restrict_lam = thm "domain_restrict_lam";
    val restrict_restrict = thm "restrict_restrict";
    val domain_restrict = thm "domain_restrict";
    val restrict_idem = thm "restrict_idem";
    val restrict_if = thm "restrict_if";
    val restrict_lam_eq = thm "restrict_lam_eq";
    val fun_cons_restrict_eq = thm "fun_cons_restrict_eq";
    val function_Union = thm "function_Union";
    val fun_Union = thm "fun_Union";
    val fun_disjoint_Un = thm "fun_disjoint_Un";
    val fun_disjoint_apply1 = thm "fun_disjoint_apply1";
    val fun_disjoint_apply2 = thm "fun_disjoint_apply2";
    val domain_of_fun = thm "domain_of_fun";
    val apply_rangeI = thm "apply_rangeI";
    val range_of_fun = thm "range_of_fun";
    val fun_extend = thm "fun_extend";
    val fun_extend3 = thm "fun_extend3";
    val fun_extend_apply = thm "fun_extend_apply";
    val singleton_apply = thm "singleton_apply";
    val cons_fun_eq = thm "cons_fun_eq";

    val update_def = thm "update_def";
    val update_apply = thm "update_apply";
    val update_idem = thm "update_idem";
    val domain_update = thm "domain_update";
    val update_type = thm "update_type";

    val Replace_mono = thm "Replace_mono";
    val RepFun_mono = thm "RepFun_mono";
    val Pow_mono = thm "Pow_mono";
    val Union_mono = thm "Union_mono";
    val UN_mono = thm "UN_mono";
    val Inter_anti_mono = thm "Inter_anti_mono";
    val cons_mono = thm "cons_mono";
    val Un_mono = thm "Un_mono";
    val Int_mono = thm "Int_mono";
    val Diff_mono = thm "Diff_mono";
    val Sigma_mono = thm "Sigma_mono";
    val sum_mono = thm "sum_mono";
    val Pi_mono = thm "Pi_mono";
    val lam_mono = thm "lam_mono";
    val converse_mono = thm "converse_mono";
    val domain_mono = thm "domain_mono";
    val domain_rel_subset = thm "domain_rel_subset";
    val range_mono = thm "range_mono";
    val range_rel_subset = thm "range_rel_subset";
    val field_mono = thm "field_mono";
    val field_rel_subset = thm "field_rel_subset";
    val image_pair_mono = thm "image_pair_mono";
    val vimage_pair_mono = thm "vimage_pair_mono";
    val image_mono = thm "image_mono";
    val vimage_mono = thm "vimage_mono";
    val Collect_mono = thm "Collect_mono";

    val basic_monos = thms "basic_monos";
    *}

    end

### The Pi Operator: Dependent Function Space

lemma subset_Sigma_imp_relation:

      r ⊆ Sigma(A, B) ==> relation(r)

lemma relation_converse_converse:

      relation(r) ==> converse(converse(r)) = r

lemma relation_restrict:

      relation(restrict(r, A))

lemma Pi_iff:

      f ∈ Pi(A, B) <-> function(f) ∧ f ⊆ Sigma(A, B) ∧ A ⊆ domain(f)

lemma Pi_iff_old:

      f ∈ Pi(A, B) <-> f ⊆ Sigma(A, B) ∧ (∀x∈A. ∃!y. ⟨x, y⟩ ∈ f)

lemma fun_is_function:

      f ∈ Pi(A, B) ==> function(f)

lemma function_imp_Pi:

      [| function(f); relation(f) |] ==> f ∈ domain(f) -> range(f)

lemma functionI:

      (!!x y y'. [| ⟨x, y⟩ ∈ r; ⟨x, y'⟩ ∈ r |] ==> y = y') ==> function(r)

lemma fun_is_rel:

      f ∈ Pi(A, B) ==> f ⊆ Sigma(A, B)

lemma Pi_cong:

      [| A = A'; !!x. x ∈ A' ==> B(x) = B'(x) |] ==> Pi(A, B) = Pi(A', B')

lemma fun_weaken_type:

      [| f ∈ A -> B; B ⊆ D |] ==> f ∈ A -> D

### Function Application

lemma apply_equality2:

      [| ⟨a, b⟩ ∈ f; ⟨a, c⟩ ∈ f; f ∈ Pi(A, B) |] ==> b = c

lemma function_apply_equality:

      [| ⟨a, b⟩ ∈ f; function(f) |] ==> f ` a = b

lemma apply_equality:

      [| ⟨a, b⟩ ∈ f; f ∈ Pi(A, B) |] ==> f ` a = b

lemma apply_0:

      a ∉ domain(f) ==> f ` a = 0

lemma Pi_memberD:

      [| f ∈ Pi(A, B); c ∈ f |] ==> ∃x∈A. c = ⟨x, f ` x⟩

lemma function_apply_Pair:

      [| function(f); a ∈ domain(f) |] ==> ⟨a, f ` a⟩ ∈ f

lemma apply_Pair:

      [| f ∈ Pi(A, B); a ∈ A |] ==> ⟨a, f ` a⟩ ∈ f

lemma apply_type:

      [| f ∈ Pi(A, B); a ∈ A |] ==> f ` a ∈ B(a)

lemma apply_funtype:

      [| f ∈ A -> B; a ∈ A |] ==> f ` a ∈ B

lemma apply_iff:

      f ∈ Pi(A, B) ==> ⟨a, b⟩ ∈ f <-> a ∈ A ∧ f ` a = b

lemma Pi_type:

      [| f ∈ Pi(A, C); !!x. x ∈ A ==> f ` x ∈ B(x) |] ==> f ∈ Pi(A, B)

lemma Pi_Collect_iff:

      f ∈ (Πx∈A. {y ∈ B(x) . P(x, y)}) <-> f ∈ Pi(A, B) ∧ (∀x∈A. P(x, f ` x))

lemma Pi_weaken_type:

      [| f ∈ Pi(A, B); !!x. x ∈ A ==> B(x) ⊆ C(x) |] ==> f ∈ Pi(A, C)

lemma domain_type:

      [| ⟨a, b⟩ ∈ f; f ∈ Pi(A, B) |] ==> a ∈ A

lemma range_type:

      [| ⟨a, b⟩ ∈ f; f ∈ Pi(A, B) |] ==> b ∈ B(a)

lemma Pair_mem_PiD:

      [| ⟨a, b⟩ ∈ f; f ∈ Pi(A, B) |] ==> a ∈ A ∧ b ∈ B(a) ∧ f ` a = b

### Lambda Abstraction

lemma lamI:

      a ∈ A ==> ⟨a, b(a)⟩ ∈ (λx∈A. b(x))

lemma lamE:

      [| p ∈ (λx∈A. b(x)); !!x. [| x ∈ A; p = ⟨x, b(x)⟩ |] ==> P |] ==> P

lemma lamD:

      ⟨a, c⟩ ∈ (λx∈A. b(x)) ==> c = b(a)

lemma lam_type:

      (!!x. x ∈ A ==> b(x) ∈ B(x)) ==> (λx∈A. b(x)) ∈ Pi(A, B)

lemma lam_funtype:

      (λx∈A. b(x)) ∈ A -> {b(x) . x ∈ A}

lemma function_lam:

      function(λx∈A. b(x))

lemma relation_lam:

      relation(λx∈A. b(x))

lemma beta_if:

      (λx∈A. b(x)) ` a = (if a ∈ A then b(a) else 0)

lemma beta:

      a ∈ A ==> (λx∈A. b(x)) ` a = b(a)

lemma lam_empty:

      (λx∈0. b(x)) = 0

lemma domain_lam:

      domain(Lambda(A, b)) = A

lemma lam_cong:

      [| A = A'; !!x. x ∈ A' ==> b(x) = b'(x) |] ==> Lambda(A, b) = Lambda(A', b')

lemma lam_theI:

      (!!x. x ∈ A ==> ∃!y. Q(x, y)) ==> ∃f. ∀x∈A. Q(x, f ` x)

lemma lam_eqE:

      [| (λx∈A. f(x)) = (λx∈A. g(x)); a ∈ A |] ==> f(a) = g(a)

lemma Pi_empty1:

      Pi(0, A) = {0}

lemma singleton_fun:

      {⟨a, b⟩} ∈ {a} -> {b}

lemma Pi_empty2:

      A -> 0 = (if A = 0 then {0} else 0)

lemma fun_space_empty_iff:

      A -> X = 0 <-> X = 0 ∧ A ≠ 0

### Extensionality

lemma fun_subset:

      [| f ∈ Pi(A, B); g ∈ Pi(C, D); A ⊆ C; !!x. x ∈ A ==> f ` x = g ` x |] ==> f ⊆ g

lemma fun_extension:

      [| f ∈ Pi(A, B); g ∈ Pi(A, D); !!x. x ∈ A ==> f ` x = g ` x |] ==> f = g

lemma eta:

      f ∈ Pi(A, B) ==> (λx∈A. f ` x) = f

lemma fun_extension_iff:

      [| f ∈ Pi(A, B); g ∈ Pi(A, C) |] ==> (∀a∈A. f ` a = g ` a) <-> f = g

lemma fun_subset_eq:

      [| f ∈ Pi(A, B); g ∈ Pi(A, C) |] ==> f ⊆ g <-> f = g

lemma Pi_lamE:

      [| f ∈ Pi(A, B); !!b. [| ∀x∈A. b(x) ∈ B(x); f = (λx∈A. b(x)) |] ==> P |] ==> P

### Images of Functions

lemma image_lam:

      C ⊆ A ==> (λx∈A. b(x)) `` C = {b(x) . x ∈ C}

lemma Repfun_function_if:

      function(f)
      ==> {f ` x . x ∈ C} = (if C ⊆ domain(f) then f `` C else cons(0, f `` C))

lemma image_function:

      [| function(f); C ⊆ domain(f) |] ==> f `` C = {f ` x . x ∈ C}

lemma image_fun:

      [| f ∈ Pi(A, B); C ⊆ A |] ==> f `` C = {f ` x . x ∈ C}

lemma image_eq_UN:

      [| f ∈ Pi(A, B); C ⊆ A |] ==> f `` C = (\<Union>x∈C. {f ` x})

lemma Pi_image_cons:

      [| f ∈ Pi(A, B); x ∈ A |] ==> f `` cons(x, y) = cons(f ` x, f `` y)

### Properties of @{term "restrict(f,A)"}

lemma restrict_subset:

      restrict(f, A) ⊆ f

lemma function_restrictI:

      function(f) ==> function(restrict(f, A))

lemma restrict_type2:

      [| f ∈ Pi(C, B); A ⊆ C |] ==> restrict(f, A) ∈ Pi(A, B)

lemma restrict:

      restrict(f, A) ` a = (if a ∈ A then f ` a else 0)

lemma restrict_empty:

      restrict(f, 0) = 0

lemma restrict_iff:

      z ∈ restrict(r, A) <-> z ∈ r ∧ (∃x∈A. ∃y. z = ⟨x, y⟩)

lemma restrict_restrict:

      restrict(restrict(r, A), B) = restrict(r, A ∩ B)

lemma domain_restrict:

      domain(restrict(f, C)) = domain(f) ∩ C

lemma restrict_idem:

      f ⊆ Sigma(A, B) ==> restrict(f, A) = f

lemma domain_restrict_idem:

      [| domain(r) ⊆ A; relation(r) |] ==> restrict(r, A) = r

lemma domain_restrict_lam:

      domain(restrict(Lambda(A, f), C)) = A ∩ C

lemma restrict_if:

      restrict(f, A) ` a = (if a ∈ A then f ` a else 0)

lemma restrict_lam_eq:

      A ⊆ C ==> restrict(λx∈C. b(x), A) = (λx∈A. b(x))

lemma fun_cons_restrict_eq:

      f ∈ cons(a, b) -> B ==> f = cons(⟨a, f ` a⟩, restrict(f, b))

### Unions of Functions

lemma function_Union:

      [| ∀x∈S. function(x); ∀x∈S. ∀y∈S. x ⊆ y ∨ y ⊆ x |] ==> function(\<Union>S)

lemma fun_Union:

      [| ∀f∈S. ∃C D. f ∈ C -> D; ∀f∈S. ∀y∈S. f ⊆ y ∨ y ⊆ f |]
      ==> \<Union>S ∈ domain(\<Union>S) -> range(\<Union>S)

lemma gen_relation_Union:

      (!!x. x ∈ F ==> relation(x)) ==> relation(\<Union>F)

lemmas Un_rls:

      A ∪ B ⊆ C <-> A ⊆ C ∧ B ⊆ C

      (Σi∈I ∪ J. C(i)) = (Σi∈I. C(i)) ∪ (Σj∈J. C(j))

      I × (A ∪ B) = I × A ∪ I × B

      A ⊆ B ==> A ⊆ B ∪ B1

      A ⊆ B ==> A ⊆ A1 ∪ B

lemmas Un_rls:

      A ∪ B ⊆ C <-> A ⊆ C ∧ B ⊆ C

      (Σi∈I ∪ J. C(i)) = (Σi∈I. C(i)) ∪ (Σj∈J. C(j))

      I × (A ∪ B) = I × A ∪ I × B

      A ⊆ B ==> A ⊆ B ∪ B1

      A ⊆ B ==> A ⊆ A1 ∪ B

lemma fun_disjoint_Un:

      [| f ∈ A -> B; g ∈ C -> D; A ∩ C = 0 |] ==> f ∪ g ∈ A ∪ C -> B ∪ D

lemma fun_disjoint_apply1:

      a ∉ domain(g) ==> (f ∪ g) ` a = f ` a

lemma fun_disjoint_apply2:

      c ∉ domain(f) ==> (f ∪ g) ` c = g ` c

### Domain and Range of a Function or Relation

lemma domain_of_fun:

      f ∈ Pi(A, B) ==> domain(f) = A

lemma apply_rangeI:

      [| f ∈ Pi(A, B); a ∈ A |] ==> f ` a ∈ range(f)

lemma range_of_fun:

      f ∈ Pi(A, B) ==> f ∈ A -> range(f)

### Extensions of Functions

lemma fun_extend:

      [| f ∈ A -> B; c ∉ A |] ==> cons(⟨c, b⟩, f) ∈ cons(c, A) -> cons(b, B)

lemma fun_extend3:

      [| f ∈ A -> B; c ∉ A; b ∈ B |] ==> cons(⟨c, b⟩, f) ∈ cons(c, A) -> B

lemma extend_apply:

      c ∉ domain(f) ==> cons(⟨c, b⟩, f) ` a = (if a = c then b else f ` a)

lemma fun_extend_apply:

      [| f ∈ A -> B; c ∉ A |] ==> cons(⟨c, b⟩, f) ` a = (if a = c then b else f ` a)

lemmas singleton_apply:

      {⟨a, b⟩} ` a = b

lemmas singleton_apply:

      {⟨a, b⟩} ` a = b

lemma cons_fun_eq:

      c ∉ A ==> cons(c, A) -> B = (\<Union>f∈A -> B. \<Union>b∈B. {cons(⟨c, b⟩, f)})

lemma succ_fun_eq:

      succ(n) -> B = (\<Union>f∈n -> B. \<Union>b∈B. {cons(⟨n, b⟩, f)})

### Function Updates

lemma update_apply:

      f(x := y) ` z = (if z = x then y else f ` z)

lemma update_idem:

      [| f ` x = y; f ∈ Pi(A, B); x ∈ A |] ==> f(x := y) = f

lemma domain_update:

      domain(f(x := y)) = cons(x, domain(f))

lemma update_type:

      [| f ∈ Pi(A, B); x ∈ A; y ∈ B(x) |] ==> f(x := y) ∈ Pi(A, B)

### Monotonicity Theorems

#### Replacement in its Various Forms

lemma Replace_mono:

      A ⊆ B ==> Replace(A, P) ⊆ Replace(B, P)

lemma RepFun_mono:

      A ⊆ B ==> {f(x) . x ∈ A} ⊆ {f(x) . x ∈ B}

lemma Pow_mono:

      A ⊆ B ==> Pow(A) ⊆ Pow(B)

lemma Union_mono:

      A ⊆ B ==> \<Union>A ⊆ \<Union>B

lemma UN_mono:

      [| A ⊆ C; !!x. x ∈ A ==> B(x) ⊆ D(x) |]
      ==> (\<Union>x∈A. B(x)) ⊆ (\<Union>x∈C. D(x))

lemma Inter_anti_mono:

      [| A ⊆ B; A ≠ 0 |] ==> \<Inter>B ⊆ \<Inter>A

lemma cons_mono:

      C ⊆ D ==> cons(a, C) ⊆ cons(a, D)

lemma Un_mono:

      [| A ⊆ C; B ⊆ D |] ==> A ∪ B ⊆ C ∪ D

lemma Int_mono:

      [| A ⊆ C; B ⊆ D |] ==> A ∩ B ⊆ C ∩ D

lemma Diff_mono:

      [| A ⊆ C; D ⊆ B |] ==> A - B ⊆ C - D

#### Standard Products, Sums and Function Spaces

lemma Sigma_mono:

      [| A ⊆ C; !!x. x ∈ A ==> B(x) ⊆ D(x) |] ==> Sigma(A, B) ⊆ Sigma(C, D)

lemma sum_mono:

      [| A ⊆ C; B ⊆ D |] ==> A + B ⊆ C + D

lemma Pi_mono:

      B ⊆ C ==> A -> B ⊆ A -> C

lemma lam_mono:

      A ⊆ B ==> Lambda(A, c) ⊆ Lambda(B, c)

#### Converse, Domain, Range, Field

lemma converse_mono:

      r ⊆ s ==> converse(r) ⊆ converse(s)

lemma domain_mono:

      r ⊆ s ==> domain(r) ⊆ domain(s)

lemmas domain_rel_subset:

      r2 ⊆ Sigma(C, B1) ==> domain(r2) ⊆ C

lemmas domain_rel_subset:

      r2 ⊆ Sigma(C, B1) ==> domain(r2) ⊆ C

lemma range_mono:

      r ⊆ s ==> range(r) ⊆ range(s)

lemmas range_rel_subset:

      r2 ⊆ A1 × C ==> range(r2) ⊆ C

lemmas range_rel_subset:

      r2 ⊆ A1 × C ==> range(r2) ⊆ C

lemma field_mono:

      r ⊆ s ==> field(r) ⊆ field(s)

lemma field_rel_subset:

      r ⊆ A × A ==> field(r) ⊆ A

#### Images

lemma image_pair_mono:

      [| !!x y. ⟨x, y⟩ ∈ r ==> ⟨x, y⟩ ∈ s; A ⊆ B |] ==> r `` A ⊆ s `` B

lemma vimage_pair_mono:

      [| !!x y. ⟨x, y⟩ ∈ r ==> ⟨x, y⟩ ∈ s; A ⊆ B |] ==> r -`` A ⊆ s -`` B

lemma image_mono:

      [| r ⊆ s; A ⊆ B |] ==> r `` A ⊆ s `` B

lemma vimage_mono:

      [| r ⊆ s; A ⊆ B |] ==> r -`` A ⊆ s -`` B

lemma Collect_mono:

      [| A ⊆ B; !!x. x ∈ A ==> P(x) --> Q(x) |] ==> Collect(A, P) ⊆ Collect(B, Q)

lemmas basic_monos:

      A ⊆ A

      P --> P

      [| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∨ P2.0 --> Q1.0 ∨ Q2.0

      [| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∧ P2.0 --> Q1.0 ∧ Q2.0

      (!!x. P(x) --> Q(x)) ==> (∃x. P(x)) --> (∃x. Q(x))

      [| A ⊆ B; !!x. x ∈ A ==> P(x) --> Q(x) |] ==> Collect(A, P) ⊆ Collect(B, Q)

      A ⊆ B ==> Part(A, h) ⊆ Part(B, h)

      A ⊆ B ==> x ∈ A --> x ∈ B

lemmas basic_monos:

      A ⊆ A

      P --> P

      [| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∨ P2.0 --> Q1.0 ∨ Q2.0

      [| P1.0 --> Q1.0; P2.0 --> Q2.0 |] ==> P1.0 ∧ P2.0 --> Q1.0 ∧ Q2.0

      (!!x. P(x) --> Q(x)) ==> (∃x. P(x)) --> (∃x. Q(x))

      [| A ⊆ B; !!x. x ∈ A ==> P(x) --> Q(x) |] ==> Collect(A, P) ⊆ Collect(B, Q)

      A ⊆ B ==> Part(A, h) ⊆ Part(B, h)

      A ⊆ B ==> x ∈ A --> x ∈ B