(* Title: ZF/Sum
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Disjoint sums in Zermelo-Fraenkel Set Theory
*)
open Sum;
(*** Rules for the Part primitive ***)
goalw Sum.thy [Part_def]
"a : Part(A,h) <-> a:A & (EX y. a=h(y))";
by (rtac separation 1);
qed "Part_iff";
goalw Sum.thy [Part_def]
"!!a b A h. [| a : A; a=h(b) |] ==> a : Part(A,h)";
by (REPEAT (ares_tac [exI,CollectI] 1));
qed "Part_eqI";
val PartI = refl RSN (2,Part_eqI);
val major::prems = goalw Sum.thy [Part_def]
"[| a : Part(A,h); !!z. [| a : A; a=h(z) |] ==> P \
\ |] ==> P";
by (rtac (major RS CollectE) 1);
by (etac exE 1);
by (REPEAT (ares_tac prems 1));
qed "PartE";
goalw Sum.thy [Part_def] "Part(A,h) <= A";
by (rtac Collect_subset 1);
qed "Part_subset";
(*** Rules for Disjoint Sums ***)
val sum_defs = [sum_def,Inl_def,Inr_def,case_def];
goalw Sum.thy (bool_def::sum_defs) "Sigma(bool,C) = C(0) + C(1)";
by (fast_tac eq_cs 1);
qed "Sigma_bool";
(** Introduction rules for the injections **)
goalw Sum.thy sum_defs "!!a A B. a : A ==> Inl(a) : A+B";
by (REPEAT (ares_tac [UnI1,SigmaI,singletonI] 1));
qed "InlI";
goalw Sum.thy sum_defs "!!b A B. b : B ==> Inr(b) : A+B";
by (REPEAT (ares_tac [UnI2,SigmaI,singletonI] 1));
qed "InrI";
(** Elimination rules **)
val major::prems = goalw Sum.thy sum_defs
"[| u: A+B; \
\ !!x. [| x:A; u=Inl(x) |] ==> P; \
\ !!y. [| y:B; u=Inr(y) |] ==> P \
\ |] ==> P";
by (rtac (major RS UnE) 1);
by (REPEAT (rtac refl 1
ORELSE eresolve_tac (prems@[SigmaE,singletonE,ssubst]) 1));
qed "sumE";
(** Injection and freeness equivalences, for rewriting **)
goalw Sum.thy sum_defs "Inl(a)=Inl(b) <-> a=b";
by (simp_tac ZF_ss 1);
qed "Inl_iff";
goalw Sum.thy sum_defs "Inr(a)=Inr(b) <-> a=b";
by (simp_tac ZF_ss 1);
qed "Inr_iff";
goalw Sum.thy sum_defs "Inl(a)=Inr(b) <-> False";
by (simp_tac (ZF_ss addsimps [one_not_0 RS not_sym]) 1);
qed "Inl_Inr_iff";
goalw Sum.thy sum_defs "Inr(b)=Inl(a) <-> False";
by (simp_tac (ZF_ss addsimps [one_not_0]) 1);
qed "Inr_Inl_iff";
(*Injection and freeness rules*)
bind_thm ("Inl_inject", (Inl_iff RS iffD1));
bind_thm ("Inr_inject", (Inr_iff RS iffD1));
bind_thm ("Inl_neq_Inr", (Inl_Inr_iff RS iffD1 RS FalseE));
bind_thm ("Inr_neq_Inl", (Inr_Inl_iff RS iffD1 RS FalseE));
val sum_cs = ZF_cs addSIs [PartI, InlI, InrI]
addSEs [PartE, sumE, Inl_neq_Inr, Inr_neq_Inl]
addSDs [Inl_inject, Inr_inject];
goal Sum.thy "!!A B. Inl(a): A+B ==> a: A";
by (fast_tac sum_cs 1);
qed "InlD";
goal Sum.thy "!!A B. Inr(b): A+B ==> b: B";
by (fast_tac sum_cs 1);
qed "InrD";
goal Sum.thy "u: A+B <-> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))";
by (fast_tac sum_cs 1);
qed "sum_iff";
goal Sum.thy "A+B <= C+D <-> A<=C & B<=D";
by (fast_tac sum_cs 1);
qed "sum_subset_iff";
goal Sum.thy "A+B = C+D <-> A=C & B=D";
by (simp_tac (ZF_ss addsimps [extension,sum_subset_iff]) 1);
by (fast_tac ZF_cs 1);
qed "sum_equal_iff";
(*** Eliminator -- case ***)
goalw Sum.thy sum_defs "case(c, d, Inl(a)) = c(a)";
by (simp_tac (ZF_ss addsimps [cond_0]) 1);
qed "case_Inl";
goalw Sum.thy sum_defs "case(c, d, Inr(b)) = d(b)";
by (simp_tac (ZF_ss addsimps [cond_1]) 1);
qed "case_Inr";
val major::prems = goal Sum.thy
"[| u: A+B; \
\ !!x. x: A ==> c(x): C(Inl(x)); \
\ !!y. y: B ==> d(y): C(Inr(y)) \
\ |] ==> case(c,d,u) : C(u)";
by (rtac (major RS sumE) 1);
by (ALLGOALS (etac ssubst));
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps
(prems@[case_Inl,case_Inr]))));
qed "case_type";
goal Sum.thy
"!!u. u: A+B ==> \
\ R(case(c,d,u)) <-> \
\ ((ALL x:A. u = Inl(x) --> R(c(x))) & \
\ (ALL y:B. u = Inr(y) --> R(d(y))))";
by (etac sumE 1);
by (asm_simp_tac (ZF_ss addsimps [case_Inl]) 1);
by (fast_tac sum_cs 1);
by (asm_simp_tac (ZF_ss addsimps [case_Inr]) 1);
by (fast_tac sum_cs 1);
qed "expand_case";
val major::prems = goal Sum.thy
"[| z: A+B; \
\ !!x. x:A ==> c(x)=c'(x); \
\ !!y. y:B ==> d(y)=d'(y) \
\ |] ==> case(c,d,z) = case(c',d',z)";
by (resolve_tac [major RS sumE] 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps ([case_Inl, case_Inr] @ prems))));
qed "case_cong";
val [major] = goal Sum.thy
"z: A+B ==> \
\ case(c, d, case(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) = \
\ case(%x. c(c'(x)), %y. d(d'(y)), z)";
by (resolve_tac [major RS sumE] 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [case_Inl, case_Inr])));
qed "case_case";
val sum_ss = ZF_ss addsimps [InlI, InrI, Inl_iff, Inr_iff,
Inl_Inr_iff, Inr_Inl_iff,
case_Inl, case_Inr];
(*** More rules for Part(A,h) ***)
goal Sum.thy "!!A B h. A<=B ==> Part(A,h)<=Part(B,h)";
by (fast_tac sum_cs 1);
qed "Part_mono";
goal Sum.thy "Part(Collect(A,P), h) = Collect(Part(A,h), P)";
by (fast_tac (sum_cs addSIs [equalityI]) 1);
qed "Part_Collect";
bind_thm ("Part_CollectE", Part_Collect RS equalityD1 RS subsetD RS CollectE);
goal Sum.thy "Part(A+B,Inl) = {Inl(x). x: A}";
by (fast_tac (sum_cs addIs [equalityI]) 1);
qed "Part_Inl";
goal Sum.thy "Part(A+B,Inr) = {Inr(y). y: B}";
by (fast_tac (sum_cs addIs [equalityI]) 1);
qed "Part_Inr";
goalw Sum.thy [Part_def] "!!a. a : Part(A,h) ==> a : A";
by (etac CollectD1 1);
qed "PartD1";
goal Sum.thy "Part(A,%x.x) = A";
by (fast_tac (sum_cs addIs [equalityI]) 1);
qed "Part_id";
goal Sum.thy "Part(A+B, %x.Inr(h(x))) = {Inr(y). y: Part(B,h)}";
by (fast_tac (sum_cs addIs [equalityI]) 1);
qed "Part_Inr2";
goal Sum.thy "!!A B C. C <= A+B ==> Part(C,Inl) Un Part(C,Inr) = C";
by (fast_tac (sum_cs addIs [equalityI]) 1);
qed "Part_sum_equality";