# Theory func

Up to index of Isabelle/ZF

theory func
imports Sum
`(*  Title:      ZF/func.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1991  University of Cambridge*)header{*Functions, Function Spaces, Lambda-Abstraction*}theory func imports equalities Sum beginsubsection{*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) & (∀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 |] ==> ∃x∈A.  c = <x,f`x>"apply (frule fun_is_rel)apply (blast dest: apply_equality)donelemma 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)donelemma 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 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)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) & (∀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)> ∈ (λx∈A. b(x))"apply (unfold lam_def)apply (erule RepFunI)donelemma lamE:    "[| p: (λx∈A. b(x));  !!x.[| x ∈ A; p=<x,b(x)> |] ==> P     |] ==>  P"by (simp add: lam_def, blast)lemma lamD: "[| <a,c>: (λx∈A. b(x)) |] ==> c = b(a)"by (simp add: lam_def)lemma lam_type [TC]:    "[| !!x. x ∈ A ==> b(x): B(x) |] ==> (λx∈A. b(x)) ∈ Pi(A,B)"by (simp add: lam_def Pi_def function_def, blast)lemma lam_funtype: "(λx∈A. b(x)) ∈ A -> {b(x). x ∈ A}"by (blast intro: lam_type)lemma function_lam: "function (λx∈A. b(x))"by (simp add: function_def lam_def)lemma relation_lam: "relation (λx∈A. b(x))"by (simp add: relation_def lam_def)lemma beta_if [simp]: "(λx∈A. b(x)) ` a = (if a ∈ A then b(a) else 0)"by (simp add: apply_def lam_def, blast)lemma beta: "a ∈ A ==> (λx∈A. b(x)) ` a = b(a)"by (simp add: apply_def lam_def, blast)lemma lam_empty [simp]: "(λx∈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)) ==> ∃f. ∀x∈A. Q(x, f`x)"apply (rule_tac x = "λx∈A. THE y. Q (x,y)" in exI)apply simpapply (blast intro: theI)donelemma lam_eqE: "[| (λx∈A. f(x)) = (λx∈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 autoapply (fast intro!: equals0I intro: lam_type)donesubsection{*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) ==> (λx∈A. f`x) = f"apply (rule fun_extension)apply (auto simp add: lam_type apply_type beta)donelemma fun_extension_iff:     "[| f ∈ Pi(A,B); g ∈ Pi(A,C) |] ==> (∀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. [| ∀x∈A. b(x):B(x);  f = (λx∈A. b(x)) |] ==> P"  shows "P"apply (rule minor)apply (rule_tac [2] eta [symmetric])apply (blast intro: major apply_type)+donesubsection{*Images of Functions*}lemma image_lam: "C ⊆ A ==> (λx∈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 simpapply (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)donelemma 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 ∩ B)"by (unfold restrict_def, blast)lemma domain_restrict [simp]: "domain(restrict(f,C)) = domain(f) ∩ C"apply (unfold restrict_def)apply (auto simp add: domain_def)donelemma 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"apply (unfold restrict_def lam_def)apply (rule equalityI)apply (auto simp add: domain_iff)donelemma 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(λx∈C. b(x), A) = (λx∈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)donesubsection{*Unions of Functions*}(** The Union of a set of COMPATIBLE functions is a function **)lemma function_Union:    "[| ∀x∈S. function(x);        ∀x∈S. ∀y∈S. x<=y | y<=x  |]     ==> function(\<Union>(S))"by (unfold function_def, blast)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))"apply (unfold Pi_def)apply (blast intro!: rel_Union function_Union)donelemma 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 ∩ 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)donelemma 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;  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)donelemma 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)donelemmas 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)donelemma 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*}definition  update  :: "[i,i,i] => i"  where   "update(f,a,b) == λx∈cons(a, domain(f)). if(x=a, b, f`x)"nonterminal updbinds and updbindsyntax  (* 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)"                       == "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)donelemma 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)donesubsection{*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 blastlemma Pow_mono: "A<=B ==> Pow(A) ⊆ Pow(B)"by blastlemma Union_mono: "A<=B ==> \<Union>(A) ⊆ \<Union>(B)"by blastlemma 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 blastlemma cons_mono: "C<=D ==> cons(a,C) ⊆ cons(a,D)"by blastlemma Un_mono: "[| A<=C;  B<=D |] ==> A ∪ B ⊆ C ∪ D"by blastlemma Int_mono: "[| A<=C;  B<=D |] ==> A ∩ B ⊆ C ∩ D"by blastlemma Diff_mono: "[| A<=C;  D<=B |] ==> A-B ⊆ C-D"by blastsubsubsection{*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 blastlemma 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)donesubsubsection{*Converse, Domain, Range, Field*}lemma converse_mono: "r<=s ==> converse(r) ⊆ converse(s)"by blastlemma domain_mono: "r<=s ==> domain(r)<=domain(s)"by blastlemmas domain_rel_subset = subset_trans [OF domain_mono domain_subset]lemma range_mono: "r<=s ==> range(r)<=range(s)"by blastlemmas range_rel_subset = subset_trans [OF range_mono range_subset]lemma field_mono: "r<=s ==> field(r)<=field(s)"by blastlemma 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 blastlemma vimage_pair_mono:    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r-``A ⊆ s-``B"by blastlemma image_mono: "[| r<=s;  A<=B |] ==> r``A ⊆ s``B"by blastlemma vimage_mono: "[| r<=s;  A<=B |] ==> r-``A ⊆ s-``B"by blastlemma 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 |]  ==> (∃x∈f``A. P(x)) <-> (∃x∈A. P(f`x))"  apply safe   apply rule    prefer 2 apply assumption   apply (simp add: apply_equality)  apply (blast intro: apply_Pair)  donelemma ball_image_simp:  "[| f ∈ Pi(X, Y); A ⊆ X |]  ==> (∀x∈f``A. P(x)) <-> (∀x∈A. P(f`x))"  apply safe   apply (blast intro: apply_Pair)  apply (drule bspec) apply assumption  apply (simp add: apply_equality)  doneend`