Theory func

(*  Title:      ZF/func.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

section‹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)  (xA. ∃!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"
  unfolding Pi_def
apply (blast intro: function_apply_equality)

(*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  xA.  c = <x,f`x>"
apply (frule fun_is_rel)
apply (blast dest: apply_equality)

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)

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

(*Conclusion is flexible -- use rule_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)

(*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)

(*Such functions arise in non-standard datatypes, ZF/ex/Ntree for instance*)
lemma Pi_Collect_iff:
     "(f  Pi(A, λx. {y  B(x). P(x,y)}))
        f  Pi(A,B)  (xA. 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)>  (λxA. b(x))"
  unfolding lam_def
apply (erule RepFunI)

lemma lamE:
    "p: (λxA. b(x));  x.x  A; p=<x,b(x)>  P
by (simp add: lam_def, blast)

lemma lamD: "a,c: (λxA. b(x))  c = b(a)"
by (simp add: lam_def)

lemma lam_type [TC]:
    "x. x  A  b(x): B(x)  (λxA. b(x))  Pi(A,B)"
by (simp add: lam_def Pi_def function_def, blast)

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

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

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

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

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

lemma lam_empty [simp]: "(λx0. 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  ∃!y. Q(x,y))  f. xA. Q(x, f`x)"
apply (rule_tac x = "λxA. THE y. Q (x,y)" in exI)
apply simp
apply (blast intro: theI)

lemma lam_eqE: "(λxA. f(x)) = (λxA. 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)



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)  (λxA. f`x) = f"
apply (rule fun_extension)
apply (auto simp add: lam_type apply_type beta)

lemma fun_extension_iff:
     "f  Pi(A,B); g  Pi(A,C)  (aA. 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. xA. b(x):B(x);  f = (λxA. b(x))  P"
  shows "P"
apply (rule minor)
apply (rule_tac [2] eta [symmetric])
apply (blast intro: major apply_type)+

subsection‹Images of Functions›

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

lemma Repfun_function_if:
       {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])

(*For this lemma and the next, the right-hand side could equivalently
  be written ⋃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)

lemma image_eq_UN:
  assumes f: "f  Pi(A,B)" "C  A" shows "f``C = (xC. {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 termrestrict(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  (xA. y. z = x, y)"
by (simp add: restrict_def)

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

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

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  C"
  unfolding restrict_def lam_def
apply (rule equalityI)
apply (auto simp add: domain_iff)

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(λxC. b(x), A) = (λxA. 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)

subsection‹Unions of Functions›

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

lemma function_Union:
    "xS. function(x);
        xS. yS. x<=y | y<=x
by (unfold function_def, blast)

lemma fun_Union:
    "fS. C D. f  C->D;
             fS. yS. f<=y | y<=f 
          (S)  domain((S)) -> range((S))"
  unfolding Pi_def
apply (blast intro!: rel_Union function_Union)

lemma gen_relation_Union:
     "(f. fF  relation(f))  relation((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  C = 0
       (f  g)  (A  C) -> (B  D)"
(*Prove the product and domain subgoals using distributive laws*)
apply (simp add: Pi_iff extension Un_rls)
apply (unfold function_def, blast)

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

lemma fun_disjoint_apply2: "c  domain(f)  (f  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;  cA  cons(c,b,f)  cons(c,A) -> cons(b,B)"
apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
apply (simp add: cons_eq)

lemma fun_extend3:
     "f  A->B;  cA;  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;  cA  cons(c,b,f)`a = (if a=c then b else f`a)"
apply (rule extend_apply)
apply (simp add: Pi_def, blast)

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 = (f  A->B. bB. {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)

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

subsection‹Function Updates›

  update  :: "[i,i,i]  i"  where
   "update(f,a,b)  λxcons(a, domain(f)). if(x=a, b, f`x)"

nonterminal updbinds and updbind


  (* Let expressions *)

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

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

lemma update_idem: "f`x = y;  f  Pi(A,B);  x  A  f(x:=y) = f"
  unfolding 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)

(* ⟦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)"
  unfolding update_def
apply (simp add: domain_of_fun cons_absorb apply_funtype lam_type)

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  (A)  (B)"
by blast

lemma UN_mono:
    "A<=C;  x. x  A  B(x)<=D(x)  (xA. B(x))  (xC. D(x))"
by blast

(*Intersection is ANTI-monotonic.  There are TWO premises! *)
lemma Inter_anti_mono: "A<=B;  A0  (B)  (A)"
by blast

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

lemma Un_mono: "A<=C;  B<=D  A  B  C  D"
by blast

lemma Int_mono: "A<=C;  B<=D  A  B  C  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)"
  unfolding lam_def
apply (erule RepFun_mono)

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)


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

(* Useful with simp; contributed by Clemens Ballarin. *)

lemma bex_image_simp:
  "f  Pi(X, Y); A  X   (xf``A. P(x))  (xA. P(f`x))"
  apply safe
   apply rule
    prefer 2 apply assumption
   apply (simp add: apply_equality)
  apply (blast intro: apply_Pair)

lemma ball_image_simp:
  "f  Pi(X, Y); A  X   (xf``A. P(x))  (xA. P(f`x))"
  apply safe
   apply (blast intro: apply_Pair)
  apply (drule bspec) apply assumption
  apply (simp add: apply_equality)