(* Title: ZF/func
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Functions in Zermelo-Fraenkel Set Theory
*)
(*** The Pi operator -- dependent function space ***)
goalw ZF.thy [Pi_def]
"f: Pi(A,B) <-> function(f) & f<=Sigma(A,B) & A<=domain(f)";
by (Blast_tac 1);
qed "Pi_iff";
(*For upward compatibility with the former definition*)
goalw ZF.thy [Pi_def, function_def]
"f: Pi(A,B) <-> f<=Sigma(A,B) & (ALL x:A. EX! y. <x,y>: f)";
by (Blast_tac 1);
qed "Pi_iff_old";
goal ZF.thy "!!f. f: Pi(A,B) ==> function(f)";
by (asm_full_simp_tac (FOL_ss addsimps [Pi_iff]) 1);
qed "fun_is_function";
(**Two "destruct" rules for Pi **)
val [major] = goalw ZF.thy [Pi_def] "f: Pi(A,B) ==> f <= Sigma(A,B)";
by (rtac (major RS CollectD1 RS PowD) 1);
qed "fun_is_rel";
goal ZF.thy "!!f. [| f: Pi(A,B); a:A |] ==> EX! y. <a,y>: f";
by (blast_tac (claset() addSDs [Pi_iff_old RS iffD1]) 1);
qed "fun_unique_Pair";
val prems = goalw ZF.thy [Pi_def]
"[| A=A'; !!x. x:A' ==> B(x)=B'(x) |] ==> Pi(A,B) = Pi(A',B')";
by (simp_tac (FOL_ss addsimps prems addcongs [Sigma_cong]) 1);
qed "Pi_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*)
goalw ZF.thy [Pi_def] "!!f. [| f: A->B; B<=D |] ==> f: A->D";
by (Best_tac 1);
qed "fun_weaken_type";
(*Empty function spaces*)
goalw ZF.thy [Pi_def, function_def] "Pi(0,A) = {0}";
by (Blast_tac 1);
qed "Pi_empty1";
goalw ZF.thy [Pi_def] "!!A a. a:A ==> A->0 = 0";
by (Blast_tac 1);
qed "Pi_empty2";
(*The empty function*)
goalw ZF.thy [Pi_def, function_def] "0: Pi(0,B)";
by (Blast_tac 1);
qed "empty_fun";
(*The singleton function*)
goalw ZF.thy [Pi_def, function_def] "{<a,b>} : {a} -> {b}";
by (Blast_tac 1);
qed "singleton_fun";
Addsimps [empty_fun, singleton_fun];
(*** Function Application ***)
goalw ZF.thy [Pi_def, function_def]
"!!a b f. [| <a,b>: f; <a,c>: f; f: Pi(A,B) |] ==> b=c";
by (Blast_tac 1);
qed "apply_equality2";
goalw ZF.thy [apply_def, function_def]
"!!a b f. [| <a,b>: f; function(f) |] ==> f`a = b";
by (blast_tac (claset() addIs [the_equality]) 1);
qed "function_apply_equality";
goalw ZF.thy [Pi_def] "!!a b f. [| <a,b>: f; f: Pi(A,B) |] ==> f`a = b";
by (blast_tac (claset() addIs [function_apply_equality]) 1);
qed "apply_equality";
(*Applying a function outside its domain yields 0*)
goalw ZF.thy [apply_def]
"!!a b f. [| a ~: domain(f); f: Pi(A,B) |] ==> f`a = 0";
by (rtac the_0 1);
by (Blast_tac 1);
qed "apply_0";
goal ZF.thy "!!f. [| f: Pi(A,B); c: f |] ==> EX x:A. c = <x,f`x>";
by (forward_tac [fun_is_rel] 1);
by (blast_tac (claset() addDs [apply_equality]) 1);
qed "Pi_memberD";
goal ZF.thy "!!f. [| f: Pi(A,B); a:A |] ==> <a,f`a>: f";
by (rtac (fun_unique_Pair RS ex1E) 1);
by (resolve_tac [apply_equality RS ssubst] 3);
by (REPEAT (assume_tac 1));
qed "apply_Pair";
(*Conclusion is flexible -- use res_inst_tac or else apply_funtype below!*)
goal ZF.thy "!!f. [| f: Pi(A,B); a:A |] ==> f`a : B(a)";
by (rtac (fun_is_rel RS subsetD RS SigmaE2) 1);
by (REPEAT (ares_tac [apply_Pair] 1));
qed "apply_type";
(*This version is acceptable to the simplifier*)
goal ZF.thy "!!f. [| f: A->B; a:A |] ==> f`a : B";
by (REPEAT (ares_tac [apply_type] 1));
qed "apply_funtype";
val [major] = goal ZF.thy
"f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b";
by (cut_facts_tac [major RS fun_is_rel] 1);
by (blast_tac (claset() addSIs [major RS apply_Pair,
major RSN (2,apply_equality)]) 1);
qed "apply_iff";
(*Refining one Pi type to another*)
val pi_prem::prems = goal ZF.thy
"[| f: Pi(A,C); !!x. x:A ==> f`x : B(x) |] ==> f : Pi(A,B)";
by (cut_facts_tac [pi_prem] 1);
by (asm_full_simp_tac (FOL_ss addsimps [Pi_iff]) 1);
by (blast_tac (claset() addIs prems addSDs [pi_prem RS Pi_memberD]) 1);
qed "Pi_type";
(** Elimination of membership in a function **)
goal ZF.thy "!!a A. [| <a,b> : f; f: Pi(A,B) |] ==> a : A";
by (REPEAT (ares_tac [fun_is_rel RS subsetD RS SigmaD1] 1));
qed "domain_type";
goal ZF.thy "!!b B a. [| <a,b> : f; f: Pi(A,B) |] ==> b : B(a)";
by (etac (fun_is_rel RS subsetD RS SigmaD2) 1);
by (assume_tac 1);
qed "range_type";
val prems = goal ZF.thy
"[| <a,b>: f; f: Pi(A,B); \
\ [| a:A; b:B(a); f`a = b |] ==> P \
\ |] ==> P";
by (cut_facts_tac prems 1);
by (resolve_tac prems 1);
by (REPEAT (eresolve_tac [asm_rl,domain_type,range_type,apply_equality] 1));
qed "Pair_mem_PiE";
(*** Lambda Abstraction ***)
goalw ZF.thy [lam_def] "!!A b. a:A ==> <a,b(a)> : (lam x:A. b(x))";
by (etac RepFunI 1);
qed "lamI";
val major::prems = goalw ZF.thy [lam_def]
"[| p: (lam x:A. b(x)); !!x.[| x:A; p=<x,b(x)> |] ==> P \
\ |] ==> P";
by (rtac (major RS RepFunE) 1);
by (REPEAT (ares_tac prems 1));
qed "lamE";
goal ZF.thy "!!a b c. [| <a,c>: (lam x:A. b(x)) |] ==> c = b(a)";
by (REPEAT (eresolve_tac [asm_rl,lamE,Pair_inject,ssubst] 1));
qed "lamD";
val prems = goalw ZF.thy [lam_def, Pi_def, function_def]
"[| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)";
by (blast_tac (claset() addIs prems) 1);
qed "lam_type";
goal ZF.thy "(lam x:A. b(x)) : A -> {b(x). x:A}";
by (REPEAT (ares_tac [refl,lam_type,RepFunI] 1));
qed "lam_funtype";
goal ZF.thy "!!a A. a : A ==> (lam x:A. b(x)) ` a = b(a)";
by (REPEAT (ares_tac [apply_equality,lam_funtype,lamI] 1));
qed "beta";
goalw ZF.thy [lam_def] "(lam x:0. b(x)) = 0";
by (Simp_tac 1);
qed "lam_empty";
Addsimps [beta, lam_empty];
(*congruence rule for lambda abstraction*)
val prems = goalw ZF.thy [lam_def]
"[| A=A'; !!x. x:A' ==> b(x)=b'(x) |] ==> Lambda(A,b) = Lambda(A',b')";
by (simp_tac (FOL_ss addsimps prems addcongs [RepFun_cong]) 1);
qed "lam_cong";
Addcongs [lam_cong];
val [major] = goal ZF.thy
"(!!x. x:A ==> EX! y. Q(x,y)) ==> EX f. ALL x:A. Q(x, f`x)";
by (res_inst_tac [("x", "lam x: A. THE y. Q(x,y)")] exI 1);
by (rtac ballI 1);
by (stac beta 1);
by (assume_tac 1);
by (etac (major RS theI) 1);
qed "lam_theI";
(** Extensionality **)
(*Semi-extensionality!*)
val prems = goal ZF.thy
"[| f : Pi(A,B); g: Pi(C,D); A<=C; \
\ !!x. x:A ==> f`x = g`x |] ==> f<=g";
by (rtac subsetI 1);
by (eresolve_tac (prems RL [Pi_memberD RS bexE]) 1);
by (etac ssubst 1);
by (resolve_tac (prems RL [ssubst]) 1);
by (REPEAT (ares_tac (prems@[apply_Pair,subsetD]) 1));
qed "fun_subset";
val prems = goal ZF.thy
"[| f : Pi(A,B); g: Pi(A,D); \
\ !!x. x:A ==> f`x = g`x |] ==> f=g";
by (REPEAT (ares_tac (prems @ (prems RL [sym]) @
[subset_refl,equalityI,fun_subset]) 1));
qed "fun_extension";
goal ZF.thy "!!f A B. f : Pi(A,B) ==> (lam x:A. f`x) = f";
by (rtac fun_extension 1);
by (REPEAT (ares_tac [lam_type,apply_type,beta] 1));
qed "eta";
Addsimps [eta];
(*Every element of Pi(A,B) may be expressed as a lambda abstraction!*)
val prems = goal ZF.thy
"[| f: Pi(A,B); \
\ !!b. [| ALL x:A. b(x):B(x); f = (lam x:A. b(x)) |] ==> P \
\ |] ==> P";
by (resolve_tac prems 1);
by (rtac (eta RS sym) 2);
by (REPEAT (ares_tac (prems@[ballI,apply_type]) 1));
qed "Pi_lamE";
(** Images of functions **)
goalw ZF.thy [lam_def] "!!C A. C <= A ==> (lam x:A. b(x)) `` C = {b(x). x:C}";
by (Blast_tac 1);
qed "image_lam";
goal ZF.thy "!!C A. [| f : Pi(A,B); C <= A |] ==> f``C = {f`x. x:C}";
by (etac (eta RS subst) 1);
by (asm_full_simp_tac (FOL_ss addsimps [beta, image_lam, subset_iff]
addcongs [RepFun_cong]) 1);
qed "image_fun";
goal thy "!!f. [| f: Pi(A,B); x: A |] ==> f `` cons(x,y) = cons(f`x, f``y)";
by (blast_tac (claset() addDs [apply_equality, apply_Pair]) 1);
qed "Pi_image_cons";
(*** properties of "restrict" ***)
goalw ZF.thy [restrict_def,lam_def]
"!!f A. [| f: Pi(C,B); A<=C |] ==> restrict(f,A) <= f";
by (blast_tac (claset() addIs [apply_Pair]) 1);
qed "restrict_subset";
val prems = goalw ZF.thy [restrict_def]
"[| !!x. x:A ==> f`x: B(x) |] ==> restrict(f,A) : Pi(A,B)";
by (rtac lam_type 1);
by (eresolve_tac prems 1);
qed "restrict_type";
val [pi,subs] = goal ZF.thy
"[| f: Pi(C,B); A<=C |] ==> restrict(f,A) : Pi(A,B)";
by (rtac (pi RS apply_type RS restrict_type) 1);
by (etac (subs RS subsetD) 1);
qed "restrict_type2";
goalw ZF.thy [restrict_def] "!!a A. a : A ==> restrict(f,A) ` a = f`a";
by (etac beta 1);
qed "restrict";
goalw ZF.thy [restrict_def] "restrict(f,0) = 0";
by (Simp_tac 1);
qed "restrict_empty";
Addsimps [restrict, restrict_empty];
(*NOT SAFE as a congruence rule for the simplifier! Can cause it to fail!*)
val prems = goalw ZF.thy [restrict_def]
"[| A=B; !!x. x:B ==> f`x=g`x |] ==> restrict(f,A) = restrict(g,B)";
by (REPEAT (ares_tac (prems@[lam_cong]) 1));
qed "restrict_eqI";
goalw ZF.thy [restrict_def, lam_def] "domain(restrict(f,C)) = C";
by (Blast_tac 1);
qed "domain_restrict";
val [prem] = goalw ZF.thy [restrict_def]
"A<=C ==> restrict(lam x:C. b(x), A) = (lam x:A. b(x))";
by (rtac (refl RS lam_cong) 1);
by (etac (prem RS subsetD RS beta) 1); (*easier than calling simp_tac*)
qed "restrict_lam_eq";
(*** Unions of functions ***)
(** The Union of a set of COMPATIBLE functions is a function **)
goalw ZF.thy [function_def]
"!!S. [| ALL x:S. function(x); \
\ ALL x:S. ALL y:S. x<=y | y<=x |] ==> \
\ function(Union(S))";
by (fast_tac (ZF_cs addSDs [bspec RS bspec]) 1);
(*by (Blast_tac 1); takes too long...*)
qed "function_Union";
goalw ZF.thy [Pi_def]
"!!S. [| 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))";
by (blast_tac (subset_cs addSIs [rel_Union, function_Union]) 1);
qed "fun_Union";
(** The Union of 2 disjoint functions is a function **)
val Un_rls = [Un_subset_iff, domain_Un_eq, SUM_Un_distrib1, prod_Un_distrib2,
Un_upper1 RSN (2, subset_trans),
Un_upper2 RSN (2, subset_trans)];
goal ZF.thy "!!f. [| 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*)
by (asm_full_simp_tac (simpset() addsimps [Pi_iff,extension]@Un_rls) 1);
by (rewtac function_def);
by (Blast_tac 1);
qed "fun_disjoint_Un";
goal ZF.thy
"!!f g a. [| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> \
\ (f Un g)`a = f`a";
by (rtac (apply_Pair RS UnI1 RS apply_equality) 1);
by (REPEAT (ares_tac [fun_disjoint_Un] 1));
qed "fun_disjoint_apply1";
goal ZF.thy
"!!f g c. [| c:C; f: A->B; g: C->D; A Int C = 0 |] ==> \
\ (f Un g)`c = g`c";
by (rtac (apply_Pair RS UnI2 RS apply_equality) 1);
by (REPEAT (ares_tac [fun_disjoint_Un] 1));
qed "fun_disjoint_apply2";
(** Domain and range of a function/relation **)
goalw ZF.thy [Pi_def] "!!f. f : Pi(A,B) ==> domain(f)=A";
by (Blast_tac 1);
qed "domain_of_fun";
goal ZF.thy "!!f. [| f : Pi(A,B); a: A |] ==> f`a : range(f)";
by (etac (apply_Pair RS rangeI) 1);
by (assume_tac 1);
qed "apply_rangeI";
val [major] = goal ZF.thy "f : Pi(A,B) ==> f : A->range(f)";
by (rtac (major RS Pi_type) 1);
by (etac (major RS apply_rangeI) 1);
qed "range_of_fun";
(*** Extensions of functions ***)
goal ZF.thy
"!!f A B. [| f: A->B; c~:A |] ==> cons(<c,b>,f) : cons(c,A) -> cons(b,B)";
by (forward_tac [singleton_fun RS fun_disjoint_Un] 1);
by (asm_full_simp_tac (FOL_ss addsimps [cons_eq]) 2);
by (Blast_tac 1);
qed "fun_extend";
goal ZF.thy
"!!f A B. [| f: A->B; c~:A; b: B |] ==> cons(<c,b>,f) : cons(c,A) -> B";
by (blast_tac (claset() addIs [fun_extend RS fun_weaken_type]) 1);
qed "fun_extend3";
goal ZF.thy "!!f A B. [| f: A->B; a:A; c~:A |] ==> cons(<c,b>,f)`a = f`a";
by (rtac (apply_Pair RS consI2 RS apply_equality) 1);
by (rtac fun_extend 3);
by (REPEAT (assume_tac 1));
qed "fun_extend_apply1";
goal ZF.thy "!!f A B. [| f: A->B; c~:A |] ==> cons(<c,b>,f)`c = b";
by (rtac (consI1 RS apply_equality) 1);
by (rtac fun_extend 1);
by (REPEAT (assume_tac 1));
qed "fun_extend_apply2";
bind_thm ("singleton_apply", [singletonI, singleton_fun] MRS apply_equality);
Addsimps [singleton_apply];
(*For Finite.ML. Inclusion of right into left is easy*)
goal ZF.thy
"!!c. c ~: A ==> cons(c,A) -> B = (UN f: A->B. UN b:B. {cons(<c,b>, f)})";
by (rtac equalityI 1);
by (safe_tac (claset() addSEs [fun_extend3]));
(*Inclusion of left into right*)
by (subgoal_tac "restrict(x, A) : A -> B" 1);
by (blast_tac (claset() addIs [restrict_type2]) 2);
by (rtac UN_I 1 THEN assume_tac 1);
by (rtac (apply_funtype RS UN_I) 1 THEN REPEAT (ares_tac [consI1] 1));
by (Simp_tac 1);
by (rtac fun_extension 1 THEN REPEAT (ares_tac [fun_extend] 1));
by (etac consE 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [restrict, fun_extend_apply1,
fun_extend_apply2])));
qed "cons_fun_eq";