# Theory func

```(*  Title:      ZF/func.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

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)"

lemma relation_converse_converse [simp]:
"relation(r) ⟹ converse(converse(r)) = r"

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. ∃!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)"

(*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')"

(*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 -> *)

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)
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)
done

lemma function_apply_Pair: "⟦function(f);  a ∈ domain(f)⟧ ⟹ <a,f`a>: f"
apply (subgoal_tac "f`a = y", blast)
done

lemma apply_Pair: "⟦f ∈ Pi(A,B);  a ∈ A⟧ ⟹ <a,f`a>: f"
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, λx. {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))"
unfolding lam_def
apply (erule RepFunI)
done

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 [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))"

lemma relation_lam: "relation (λx∈A. b(x))"

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"

lemma domain_lam [simp]: "domain(Lambda(A,b)) = A"

(*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. ∀x∈A. Q(x, f`x)"
apply (rule_tac x = "λx∈A. THE y. Q (x,y)" in exI)
apply simp
apply (blast intro: theI)
done

lemma 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 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) ⟹ (λx∈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)⟧ ⟹ (∀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)+
done

subsection‹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 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 ⋃x∈C. {f`x} *)
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}"
apply (blast intro: image_function)
done

lemma image_eq_UN:
assumes f: "f ∈ Pi(A,B)" "C ⊆ A" shows "f``C = (⋃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⟩)"

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
done

lemma restrict_idem: "f ⊆ Sigma(A,B) ⟹ restrict(f,A) = f"

(*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)
done

lemma restrict_if [simp]: "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))"
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:
"⟦∀x∈S. function(x);
∀x∈S. ∀y∈S. x<=y | y<=x⟧
⟹ function(⋃(S))"
by (unfold function_def, blast)

lemma fun_Union:
"⟦∀f∈S. ∃C D. f ∈ C->D;
∀f∈S. ∀y∈S. f<=y | y<=f⟧ ⟹
⋃(S) ∈ domain(⋃(S)) -> range(⋃(S))"
unfolding Pi_def
apply (blast intro!: rel_Union function_Union)
done

lemma gen_relation_Union:
"(⋀f. f∈F ⟹ relation(f)) ⟹ relation(⋃(F))"

(** 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)
done

lemma fun_disjoint_apply1: "a ∉ domain(g) ⟹ (f ∪ g)`a = f`a"

lemma fun_disjoint_apply2: "c ∉ domain(f) ⟹ (f ∪ g)`c = g`c"

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)
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)"

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)
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 = (⋃f ∈ A->B. ⋃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 = (⋃f ∈ n->B. ⋃b∈B. {cons(⟨n,b⟩, f)})"
by (simp add: succ_def mem_not_refl cons_fun_eq)

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 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)"                       == "CONST update(f,x,y)"

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

lemma update_idem: "⟦f`x = y;  f ∈ Pi(A,B);  x ∈ A⟧ ⟹ f(x:=y) = f"
unfolding update_def
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)"
unfolding 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 ⟹ ⋃(A) ⊆ ⋃(B)"
by blast

lemma UN_mono:
"⟦A<=C;  ⋀x. x ∈ A ⟹ B(x)<=D(x)⟧ ⟹ (⋃x∈A. B(x)) ⊆ (⋃x∈C. D(x))"
by blast

(*Intersection is ANTI-monotonic.  There are TWO premises! *)
lemma Inter_anti_mono: "⟦A<=B;  A≠0⟧ ⟹ ⋂(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)
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

(* 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