Installation of new simplifier for ZF. Deleted all congruence rules not
involving local assumptions. NB the congruence rules for Sigma and Pi
(dependent type constructions) cause difficulties and are not used by
default.
(* Title: ZF/ordinal.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
For ordinal.thy. Ordinals in Zermelo-Fraenkel Set Theory
*)
open Ord;
(*** Rules for Transset ***)
(** Two neat characterisations of Transset **)
goalw Ord.thy [Transset_def] "Transset(A) <-> A<=Pow(A)";
by (fast_tac ZF_cs 1);
val Transset_iff_Pow = result();
goalw Ord.thy [Transset_def] "Transset(A) <-> Union(succ(A)) = A";
by (fast_tac (eq_cs addSEs [equalityE]) 1);
val Transset_iff_Union_succ = result();
(** Consequences of downwards closure **)
goalw Ord.thy [Transset_def]
"!!C a b. [| Transset(C); {a,b}: C |] ==> a:C & b: C";
by (fast_tac ZF_cs 1);
val Transset_doubleton_D = result();
val [prem1,prem2] = goalw Ord.thy [Pair_def]
"[| Transset(C); <a,b>: C |] ==> a:C & b: C";
by (cut_facts_tac [prem2] 1);
by (fast_tac (ZF_cs addSDs [prem1 RS Transset_doubleton_D]) 1);
val Transset_Pair_D = result();
val prem1::prems = goal Ord.thy
"[| Transset(C); A*B <= C; b: B |] ==> A <= C";
by (cut_facts_tac prems 1);
by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
val Transset_includes_domain = result();
val prem1::prems = goal Ord.thy
"[| Transset(C); A*B <= C; a: A |] ==> B <= C";
by (cut_facts_tac prems 1);
by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
val Transset_includes_range = result();
val [prem1,prem2] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
"[| Transset(C); A+B <= C |] ==> A <= C & B <= C";
br (prem2 RS (Un_subset_iff RS iffD1) RS conjE) 1;
by (REPEAT (etac (prem1 RS Transset_includes_range) 1
ORELSE resolve_tac [conjI, singletonI] 1));
val Transset_includes_summands = result();
val [prem] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
"Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)";
br (Int_Un_distrib RS ssubst) 1;
by (fast_tac (ZF_cs addSDs [prem RS Transset_Pair_D]) 1);
val Transset_sum_Int_subset = result();
(** Closure properties **)
goalw Ord.thy [Transset_def] "Transset(0)";
by (fast_tac ZF_cs 1);
val Transset_0 = result();
goalw Ord.thy [Transset_def]
"!!i j. [| Transset(i); Transset(j) |] ==> Transset(i Un j)";
by (fast_tac ZF_cs 1);
val Transset_Un = result();
goalw Ord.thy [Transset_def]
"!!i j. [| Transset(i); Transset(j) |] ==> Transset(i Int j)";
by (fast_tac ZF_cs 1);
val Transset_Int = result();
goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))";
by (fast_tac ZF_cs 1);
val Transset_succ = result();
goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))";
by (fast_tac ZF_cs 1);
val Transset_Pow = result();
goalw Ord.thy [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))";
by (fast_tac ZF_cs 1);
val Transset_Union = result();
val [Transprem] = goalw Ord.thy [Transset_def]
"[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))";
by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
val Transset_Union_family = result();
val [prem,Transprem] = goalw Ord.thy [Transset_def]
"[| j:A; !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))";
by (cut_facts_tac [prem] 1);
by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
val Transset_Inter_family = result();
(*** Natural Deduction rules for Ord ***)
val prems = goalw Ord.thy [Ord_def]
"[| Transset(i); !!x. x:i ==> Transset(x) |] ==> Ord(i) ";
by (REPEAT (ares_tac (prems@[ballI,conjI]) 1));
val OrdI = result();
val [major] = goalw Ord.thy [Ord_def]
"Ord(i) ==> Transset(i)";
by (rtac (major RS conjunct1) 1);
val Ord_is_Transset = result();
val [major,minor] = goalw Ord.thy [Ord_def]
"[| Ord(i); j:i |] ==> Transset(j) ";
by (rtac (minor RS (major RS conjunct2 RS bspec)) 1);
val Ord_contains_Transset = result();
(*** Lemmas for ordinals ***)
goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| Ord(i); j:i |] ==> Ord(j) ";
by (fast_tac ZF_cs 1);
val Ord_in_Ord = result();
goal Ord.thy "!!i j. [| Ord(i); Transset(j); j<=i |] ==> Ord(j)";
by (REPEAT (ares_tac [OrdI] 1
ORELSE eresolve_tac [Ord_contains_Transset, subsetD] 1));
val Ord_subset_Ord = result();
goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| j:i; Ord(i) |] ==> j<=i";
by (fast_tac ZF_cs 1);
val OrdmemD = result();
goal Ord.thy "!!i j k. [| i:j; j:k; Ord(k) |] ==> i:k";
by (REPEAT (ares_tac [OrdmemD RS subsetD] 1));
val Ord_trans = result();
goal Ord.thy "!!i j. [| i:j; Ord(j) |] ==> succ(i) <= j";
by (REPEAT (ares_tac [OrdmemD RSN (2,succ_subsetI)] 1));
val Ord_succ_subsetI = result();
(*** The construction of ordinals: 0, succ, Union ***)
goal Ord.thy "Ord(0)";
by (REPEAT (ares_tac [OrdI,Transset_0] 1 ORELSE etac emptyE 1));
val Ord_0 = result();
goal Ord.thy "!!i. Ord(i) ==> Ord(succ(i))";
by (REPEAT (ares_tac [OrdI,Transset_succ] 1
ORELSE eresolve_tac [succE,ssubst,Ord_is_Transset,
Ord_contains_Transset] 1));
val Ord_succ = result();
val nonempty::prems = goal Ord.thy
"[| j:A; !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))";
by (rtac (nonempty RS Transset_Inter_family RS OrdI) 1);
by (rtac Ord_is_Transset 1);
by (REPEAT (ares_tac ([Ord_contains_Transset,nonempty]@prems) 1
ORELSE etac InterD 1));
val Ord_Inter = result();
val jmemA::prems = goal Ord.thy
"[| j:A; !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))";
by (rtac (jmemA RS RepFunI RS Ord_Inter) 1);
by (etac RepFunE 1);
by (etac ssubst 1);
by (eresolve_tac prems 1);
val Ord_INT = result();
(*** Natural Deduction rules for Memrel ***)
goalw Ord.thy [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A";
by (fast_tac ZF_cs 1);
val Memrel_iff = result();
val prems = goal Ord.thy "[| a: b; a: A; b: A |] ==> <a,b> : Memrel(A)";
by (REPEAT (resolve_tac (prems@[conjI, Memrel_iff RS iffD2]) 1));
val MemrelI = result();
val [major,minor] = goal Ord.thy
"[| <a,b> : Memrel(A); \
\ [| a: A; b: A; a:b |] ==> P \
\ |] ==> P";
by (rtac (major RS (Memrel_iff RS iffD1) RS conjE) 1);
by (etac conjE 1);
by (rtac minor 1);
by (REPEAT (assume_tac 1));
val MemrelE = result();
(*The membership relation (as a set) is well-founded.
Proof idea: show A<=B by applying the foundation axiom to A-B *)
goalw Ord.thy [wf_def] "wf(Memrel(A))";
by (EVERY1 [rtac (foundation RS disjE RS allI),
etac disjI1,
etac bexE,
rtac (impI RS allI RS bexI RS disjI2),
etac MemrelE,
etac bspec,
REPEAT o assume_tac]);
val wf_Memrel = result();
(*** Transfinite induction ***)
(*Epsilon induction over a transitive set*)
val major::prems = goalw Ord.thy [Transset_def]
"[| i: k; Transset(k); \
\ !!x.[| x: k; ALL y:x. P(y) |] ==> P(x) \
\ |] ==> P(i)";
by (rtac (major RS (wf_Memrel RS wf_induct2)) 1);
by (fast_tac (ZF_cs addEs [MemrelE]) 1);
by (resolve_tac prems 1);
by (assume_tac 1);
by (cut_facts_tac prems 1);
by (fast_tac (ZF_cs addIs [MemrelI]) 1);
val Transset_induct = result();
(*Induction over an ordinal*)
val Ord_induct = Ord_is_Transset RSN (2, Transset_induct);
(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
val [major,indhyp] = goal Ord.thy
"[| Ord(i); \
\ !!x.[| Ord(x); ALL y:x. P(y) |] ==> P(x) \
\ |] ==> P(i)";
by (rtac (major RS Ord_succ RS (succI1 RS Ord_induct)) 1);
by (rtac indhyp 1);
by (rtac (major RS Ord_succ RS Ord_in_Ord) 1);
by (REPEAT (assume_tac 1));
val trans_induct = result();
(*Perform induction on i, then prove the Ord(i) subgoal using prems. *)
fun trans_ind_tac a prems i =
EVERY [res_inst_tac [("i",a)] trans_induct i,
rename_last_tac a ["1"] (i+1),
ares_tac prems i];
(*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
(*Finds contradictions for the following proof*)
val Ord_trans_tac = EVERY' [etac notE, etac Ord_trans, REPEAT o atac];
(** Proving that "member" is a linear ordering on the ordinals **)
val prems = goal Ord.thy
"Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)";
by (trans_ind_tac "i" prems 1);
by (rtac (impI RS allI) 1);
by (trans_ind_tac "j" [] 1);
by (DEPTH_SOLVE (swap_res_tac [disjCI,equalityI,subsetI] 1
ORELSE ball_tac 1
ORELSE eresolve_tac [impE,disjE,allE] 1
ORELSE hyp_subst_tac 1
ORELSE Ord_trans_tac 1));
val Ord_linear_lemma = result();
val ordi::ordj::prems = goal Ord.thy
"[| Ord(i); Ord(j); i:j ==> P; i=j ==> P; j:i ==> P |] \
\ ==> P";
by (rtac (ordi RS Ord_linear_lemma RS spec RS impE) 1);
by (rtac ordj 1);
by (REPEAT (eresolve_tac (prems@[asm_rl,disjE]) 1));
val Ord_linear = result();
val prems = goal Ord.thy
"[| Ord(i); Ord(j); i<=j ==> P; j<=i ==> P |] \
\ ==> P";
by (res_inst_tac [("i","i"),("j","j")] Ord_linear 1);
by (DEPTH_SOLVE (ares_tac (prems@[subset_refl]) 1
ORELSE eresolve_tac [asm_rl,OrdmemD,ssubst] 1));
val Ord_subset = result();
goal Ord.thy "!!i j. [| j<=i; ~ i<=j; Ord(i); Ord(j) |] ==> j:i";
by (etac Ord_linear 1);
by (REPEAT (ares_tac [subset_refl] 1
ORELSE eresolve_tac [notE,OrdmemD,ssubst] 1));
val Ord_member = result();
val [prem] = goal Ord.thy "Ord(i) ==> 0: succ(i)";
by (rtac (empty_subsetI RS Ord_member) 1);
by (fast_tac ZF_cs 1);
by (rtac (prem RS Ord_succ) 1);
by (rtac Ord_0 1);
val Ord_0_mem_succ = result();
goal Ord.thy "!!i j. [| Ord(i); Ord(j) |] ==> j:i <-> j<=i & ~(i<=j)";
by (rtac iffI 1);
by (rtac conjI 1);
by (etac OrdmemD 1);
by (rtac (mem_anti_refl RS notI) 2);
by (etac subsetD 2);
by (REPEAT (eresolve_tac [asm_rl, conjE, Ord_member] 1));
val Ord_member_iff = result();
goal Ord.thy "!!i. Ord(i) ==> 0:i <-> ~ i=0";
be (Ord_0 RSN (2,Ord_member_iff) RS iff_trans) 1;
by (fast_tac eq_cs 1);
val Ord_0_member_iff = result();
(** For ordinals, i: succ(j) means 'less-than or equals' **)
goal Ord.thy "!!i j. [| j<=i; Ord(i); Ord(j) |] ==> j : succ(i)";
by (rtac Ord_member 1);
by (REPEAT (ares_tac [Ord_succ] 3));
by (rtac (mem_anti_refl RS notI) 2);
by (etac subsetD 2);
by (ALLGOALS (fast_tac ZF_cs));
val member_succI = result();
goalw Ord.thy [Transset_def,Ord_def]
"!!i j. [| i : succ(j); Ord(j) |] ==> i<=j";
by (fast_tac ZF_cs 1);
val member_succD = result();
goal Ord.thy "!!i j. [| Ord(i); Ord(j) |] ==> j:succ(i) <-> j<=i";
by (fast_tac (subset_cs addSEs [member_succI, member_succD]) 1);
val member_succ_iff = result();
goal Ord.thy
"!!i j. [| Ord(i); Ord(j) |] ==> i<=succ(j) <-> i<=j | i=succ(j)";
by (asm_simp_tac (ZF_ss addsimps [member_succ_iff RS iff_sym, Ord_succ]) 1);
by (fast_tac ZF_cs 1);
val subset_succ_iff = result();
goal Ord.thy "!!i j. [| i:succ(j); j:k; Ord(k) |] ==> i:k";
by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
val Ord_trans1 = result();
goal Ord.thy "!!i j. [| i:j; j:succ(k); Ord(k) |] ==> i:k";
by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
val Ord_trans2 = result();
goal Ord.thy "!!i jk. [| i:j; j<=k; Ord(k) |] ==> i:k";
by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
val Ord_transsub2 = result();
goal Ord.thy "!!i j. [| i:j; Ord(j) |] ==> succ(i) : succ(j)";
by (rtac member_succI 1);
by (REPEAT (ares_tac [subsetI,Ord_succ,Ord_in_Ord] 1
ORELSE eresolve_tac [succE,Ord_trans,ssubst] 1));
val succ_mem_succI = result();
goal Ord.thy "!!i j. [| succ(i) : succ(j); Ord(j) |] ==> i:j";
by (REPEAT (eresolve_tac [asm_rl, make_elim member_succD, succ_subsetE] 1));
val succ_mem_succE = result();
goal Ord.thy "!!i j. Ord(j) ==> succ(i) : succ(j) <-> i:j";
by (REPEAT (ares_tac [iffI,succ_mem_succI,succ_mem_succE] 1));
val succ_mem_succ_iff = result();
goal Ord.thy "!!i j. [| i<=j; Ord(i); Ord(j) |] ==> succ(i) <= succ(j)";
by (rtac (member_succI RS succ_mem_succI RS member_succD) 1);
by (REPEAT (ares_tac [Ord_succ] 1));
val Ord_succ_mono = result();
goal Ord.thy "!!i j k. [| i:k; j:k; Ord(k) |] ==> i Un j : k";
by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
by (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iffD1]) 1);
by (rtac (Un_commute RS ssubst) 1);
by (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iffD1]) 1);
val Ord_member_UnI = result();
goal Ord.thy "!!i j k. [| i:k; j:k; Ord(k) |] ==> i Int j : k";
by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
by (asm_simp_tac (ZF_ss addsimps [subset_Int_iff RS iffD1]) 1);
by (rtac (Int_commute RS ssubst) 1);
by (asm_simp_tac (ZF_ss addsimps [subset_Int_iff RS iffD1]) 1);
val Ord_member_IntI = result();
(*** Results about limits ***)
val prems = goal Ord.thy "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))";
by (rtac (Ord_is_Transset RS Transset_Union_family RS OrdI) 1);
by (REPEAT (etac UnionE 1 ORELSE ares_tac ([Ord_contains_Transset]@prems) 1));
val Ord_Union = result();
val prems = goal Ord.thy "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))";
by (rtac Ord_Union 1);
by (etac RepFunE 1);
by (etac ssubst 1);
by (eresolve_tac prems 1);
val Ord_UN = result();
(*The upper bound must be a successor ordinal --
consider that (UN i:nat.i)=nat although nat is an upper bound of each i*)
val [ordi,limit] = goal Ord.thy
"[| Ord(i); !!y. y:A ==> f(y): succ(i) |] ==> (UN y:A. f(y)) : succ(i)";
by (rtac (member_succD RS UN_least RS member_succI) 1);
by (REPEAT (ares_tac [ordi, Ord_UN, ordi RS Ord_succ RS Ord_in_Ord,
limit] 1));
val sup_least = result();
val [jmemi,ordi,limit] = goal Ord.thy
"[| j: i; Ord(i); !!y. y:A ==> f(y): j |] ==> (UN y:A. succ(f(y))) : i";
by (cut_facts_tac [jmemi RS (ordi RS Ord_in_Ord)] 1);
by (rtac (sup_least RS Ord_trans2) 1);
by (REPEAT (ares_tac [jmemi, ordi, succ_mem_succI, limit] 1));
val sup_least2 = result();
goal Ord.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
by (fast_tac (eq_cs addSEs [Ord_trans1]) 1);
val Ord_equality = result();
(*Holds for all transitive sets, not just ordinals*)
goal Ord.thy "!!i. Ord(i) ==> Union(i) <= i";
by (fast_tac (ZF_cs addSEs [Ord_trans]) 1);
val Ord_Union_subset = result();