(* Title: ZF/func.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Functions in Zermelo-Fraenkel Set Theory
*)
theory func = equalities:
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)
(*** The Pi operator -- dependent function space ***)
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 \<in> 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)
(*** 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)
(*** 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 \<longleftrightarrow> X=0 & (A \<noteq> 0)"
apply auto
apply (fast intro!: equals0I intro: lam_type)
done
(** 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
(** 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 UN 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 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)
(*** properties of "restrict" ***)
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 \<in> restrict(r,A) \<longleftrightarrow> z \<in> r & (\<exists>x\<in>A. \<exists>y. z = \<langle>x, y\<rangle>)"
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
(*** 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]:
"\<forall>f\<in>F. relation(f) \<Longrightarrow> 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 \<notin> domain(g) ==> (f Un g)`a = f`a"
by (simp add: apply_def, blast)
lemma fun_disjoint_apply2: "c \<notin> domain(f) ==> (f Un g)`c = g`c"
by (simp add: apply_def, blast)
(** Domain and range of a function/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)
(*** 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 = (UN f: A->B. UN 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
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";
*}
end