src/ZF/Ord.ML
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(*  Title: 	ZF/ordinal.thy
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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For ordinal.thy.  Ordinals in Zermelo-Fraenkel Set Theory 
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*)
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open Ord;
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(*** Rules for Transset ***)
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(** Two neat characterisations of Transset **)
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goalw Ord.thy [Transset_def] "Transset(A) <-> A<=Pow(A)";
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by (fast_tac ZF_cs 1);
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val Transset_iff_Pow = result();
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goalw Ord.thy [Transset_def] "Transset(A) <-> Union(succ(A)) = A";
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by (fast_tac (eq_cs addSEs [equalityE]) 1);
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val Transset_iff_Union_succ = result();
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(** Consequences of downwards closure **)
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goalw Ord.thy [Transset_def]
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    "!!C a b. [| Transset(C); {a,b}: C |] ==> a:C & b: C";
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by (fast_tac ZF_cs 1);
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val Transset_doubleton_D = result();
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val [prem1,prem2] = goalw Ord.thy [Pair_def]
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    "[| Transset(C); <a,b>: C |] ==> a:C & b: C";
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by (cut_facts_tac [prem2] 1);	
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by (fast_tac (ZF_cs addSDs [prem1 RS Transset_doubleton_D]) 1);
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val Transset_Pair_D = result();
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val prem1::prems = goal Ord.thy
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    "[| Transset(C); A*B <= C; b: B |] ==> A <= C";
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by (cut_facts_tac prems 1);
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by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
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val Transset_includes_domain = result();
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val prem1::prems = goal Ord.thy
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    "[| Transset(C); A*B <= C; a: A |] ==> B <= C";
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by (cut_facts_tac prems 1);
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by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
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val Transset_includes_range = result();
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val [prem1,prem2] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
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    "[| Transset(C); A+B <= C |] ==> A <= C & B <= C";
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br (prem2 RS (Un_subset_iff RS iffD1) RS conjE) 1;
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by (REPEAT (etac (prem1 RS Transset_includes_range) 1
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     ORELSE resolve_tac [conjI, singletonI] 1));
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val Transset_includes_summands = result();
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val [prem] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
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    "Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)";
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br (Int_Un_distrib RS ssubst) 1;
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by (fast_tac (ZF_cs addSDs [prem RS Transset_Pair_D]) 1);
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val Transset_sum_Int_subset = result();
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(** Closure properties **)
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goalw Ord.thy [Transset_def] "Transset(0)";
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by (fast_tac ZF_cs 1);
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val Transset_0 = result();
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goalw Ord.thy [Transset_def]
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    "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Un j)";
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by (fast_tac ZF_cs 1);
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val Transset_Un = result();
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goalw Ord.thy [Transset_def]
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    "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Int j)";
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by (fast_tac ZF_cs 1);
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val Transset_Int = result();
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goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))";
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by (fast_tac ZF_cs 1);
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val Transset_succ = result();
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goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))";
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by (fast_tac ZF_cs 1);
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val Transset_Pow = result();
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goalw Ord.thy [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))";
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by (fast_tac ZF_cs 1);
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val Transset_Union = result();
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val [Transprem] = goalw Ord.thy [Transset_def]
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    "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))";
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by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
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val Transset_Union_family = result();
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val [prem,Transprem] = goalw Ord.thy [Transset_def]
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    "[| j:A;  !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))";
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by (cut_facts_tac [prem] 1);
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by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
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val Transset_Inter_family = result();
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(*** Natural Deduction rules for Ord ***)
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val prems = goalw Ord.thy [Ord_def]
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    "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i) ";
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by (REPEAT (ares_tac (prems@[ballI,conjI]) 1));
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val OrdI = result();
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val [major] = goalw Ord.thy [Ord_def]
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    "Ord(i) ==> Transset(i)";
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by (rtac (major RS conjunct1) 1);
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val Ord_is_Transset = result();
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val [major,minor] = goalw Ord.thy [Ord_def]
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    "[| Ord(i);  j:i |] ==> Transset(j) ";
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by (rtac (minor RS (major RS conjunct2 RS bspec)) 1);
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val Ord_contains_Transset = result();
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(*** Lemmas for ordinals ***)
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goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| Ord(i);  j:i |] ==> Ord(j) ";
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by (fast_tac ZF_cs 1);
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val Ord_in_Ord = result();
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goal Ord.thy "!!i j. [| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)";
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by (REPEAT (ares_tac [OrdI] 1
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     ORELSE eresolve_tac [Ord_contains_Transset, subsetD] 1));
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val Ord_subset_Ord = result();
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goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| j:i;  Ord(i) |] ==> j<=i";
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by (fast_tac ZF_cs 1);
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val OrdmemD = result();
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goal Ord.thy "!!i j k. [| i:j;  j:k;  Ord(k) |] ==> i:k";
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by (REPEAT (ares_tac [OrdmemD RS subsetD] 1));
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val Ord_trans = result();
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goal Ord.thy "!!i j. [| i:j;  Ord(j) |] ==> succ(i) <= j";
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by (REPEAT (ares_tac [OrdmemD RSN (2,succ_subsetI)] 1));
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val Ord_succ_subsetI = result();
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(*** The construction of ordinals: 0, succ, Union ***)
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goal Ord.thy "Ord(0)";
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by (REPEAT (ares_tac [OrdI,Transset_0] 1 ORELSE etac emptyE 1));
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val Ord_0 = result();
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goal Ord.thy "!!i. Ord(i) ==> Ord(succ(i))";
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by (REPEAT (ares_tac [OrdI,Transset_succ] 1
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     ORELSE eresolve_tac [succE,ssubst,Ord_is_Transset,
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			  Ord_contains_Transset] 1));
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val Ord_succ = result();
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val nonempty::prems = goal Ord.thy
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    "[| j:A;  !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))";
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by (rtac (nonempty RS Transset_Inter_family RS OrdI) 1);
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by (rtac Ord_is_Transset 1);
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by (REPEAT (ares_tac ([Ord_contains_Transset,nonempty]@prems) 1
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     ORELSE etac InterD 1));
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val Ord_Inter = result();
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val jmemA::prems = goal Ord.thy
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    "[| j:A;  !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))";
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by (rtac (jmemA RS RepFunI RS Ord_Inter) 1);
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by (etac RepFunE 1);
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by (etac ssubst 1);
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by (eresolve_tac prems 1);
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val Ord_INT = result();
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(*** Natural Deduction rules for Memrel ***)
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goalw Ord.thy [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A";
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by (fast_tac ZF_cs 1);
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val Memrel_iff = result();
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val prems = goal Ord.thy "[| a: b;  a: A;  b: A |]  ==>  <a,b> : Memrel(A)";
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by (REPEAT (resolve_tac (prems@[conjI, Memrel_iff RS iffD2]) 1));
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val MemrelI = result();
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val [major,minor] = goal Ord.thy
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    "[| <a,b> : Memrel(A);  \
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\       [| a: A;  b: A;  a:b |]  ==> P \
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\    |]  ==> P";
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by (rtac (major RS (Memrel_iff RS iffD1) RS conjE) 1);
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by (etac conjE 1);
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by (rtac minor 1);
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by (REPEAT (assume_tac 1));
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val MemrelE = result();
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(*The membership relation (as a set) is well-founded.
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  Proof idea: show A<=B by applying the foundation axiom to A-B *)
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goalw Ord.thy [wf_def] "wf(Memrel(A))";
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by (EVERY1 [rtac (foundation RS disjE RS allI),
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	    etac disjI1,
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	    etac bexE, 
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	    rtac (impI RS allI RS bexI RS disjI2),
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	    etac MemrelE,
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	    etac bspec,
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	    REPEAT o assume_tac]);
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val wf_Memrel = result();
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(*** Transfinite induction ***)
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(*Epsilon induction over a transitive set*)
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val major::prems = goalw Ord.thy [Transset_def]
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    "[| i: k;  Transset(k);                          \
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\       !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) \
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\    |]  ==>  P(i)";
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by (rtac (major RS (wf_Memrel RS wf_induct2)) 1);
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by (fast_tac (ZF_cs addEs [MemrelE]) 1);
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by (resolve_tac prems 1);
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by (assume_tac 1);
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by (cut_facts_tac prems 1);
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by (fast_tac (ZF_cs addIs [MemrelI]) 1);
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val Transset_induct = result();
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(*Induction over an ordinal*)
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val Ord_induct = Ord_is_Transset RSN (2, Transset_induct);
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(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
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val [major,indhyp] = goal Ord.thy
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    "[| Ord(i); \
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\       !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) \
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\    |]  ==>  P(i)";
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by (rtac (major RS Ord_succ RS (succI1 RS Ord_induct)) 1);
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by (rtac indhyp 1);
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by (rtac (major RS Ord_succ RS Ord_in_Ord) 1);
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by (REPEAT (assume_tac 1));
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val trans_induct = result();
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   230
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(*Perform induction on i, then prove the Ord(i) subgoal using prems. *)
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fun trans_ind_tac a prems i = 
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    EVERY [res_inst_tac [("i",a)] trans_induct i,
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	   rename_last_tac a ["1"] (i+1),
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	   ares_tac prems i];
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(*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
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(*Finds contradictions for the following proof*)
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val Ord_trans_tac = EVERY' [etac notE, etac Ord_trans, REPEAT o atac];
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(** Proving that "member" is a linear ordering on the ordinals **)
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val prems = goal Ord.thy
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    "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)";
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by (trans_ind_tac "i" prems 1);
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by (rtac (impI RS allI) 1);
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by (trans_ind_tac "j" [] 1);
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by (DEPTH_SOLVE (swap_res_tac [disjCI,equalityI,subsetI] 1
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     ORELSE ball_tac 1
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     ORELSE eresolve_tac [impE,disjE,allE] 1 
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     ORELSE hyp_subst_tac 1
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     ORELSE Ord_trans_tac 1));
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val Ord_linear_lemma = result();
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   256
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val ordi::ordj::prems = goal Ord.thy
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    "[| Ord(i);  Ord(j);  i:j ==> P;  i=j ==> P;  j:i ==> P |] \
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\    ==> P";
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by (rtac (ordi RS Ord_linear_lemma RS spec RS impE) 1);
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by (rtac ordj 1);
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by (REPEAT (eresolve_tac (prems@[asm_rl,disjE]) 1)); 
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val Ord_linear = result();
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   264
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val prems = goal Ord.thy
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    "[| Ord(i);  Ord(j);  i<=j ==> P;  j<=i ==> P |] \
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\    ==> P";
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by (res_inst_tac [("i","i"),("j","j")] Ord_linear 1);
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by (DEPTH_SOLVE (ares_tac (prems@[subset_refl]) 1
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          ORELSE eresolve_tac [asm_rl,OrdmemD,ssubst] 1));
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val Ord_subset = result();
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   272
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goal Ord.thy "!!i j. [| j<=i;  ~ i<=j;  Ord(i);  Ord(j) |] ==> j:i";
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by (etac Ord_linear 1);
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by (REPEAT (ares_tac [subset_refl] 1
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     ORELSE eresolve_tac [notE,OrdmemD,ssubst] 1));
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val Ord_member = result();
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   278
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val [prem] = goal Ord.thy "Ord(i) ==> 0: succ(i)";
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by (rtac (empty_subsetI RS Ord_member) 1);
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by (fast_tac ZF_cs 1);
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by (rtac (prem RS Ord_succ) 1);
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   283
by (rtac Ord_0 1);
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val Ord_0_mem_succ = result();
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   285
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goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> j:i <-> j<=i & ~(i<=j)";
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by (rtac iffI 1);
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by (rtac conjI 1);
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   289
by (etac OrdmemD 1);
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   290
by (rtac (mem_anti_refl RS notI) 2);
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   291
by (etac subsetD 2);
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   292
by (REPEAT (eresolve_tac [asm_rl, conjE, Ord_member] 1));
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val Ord_member_iff = result();
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   294
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goal Ord.thy "!!i. Ord(i) ==> 0:i  <-> ~ i=0";
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be (Ord_0 RSN (2,Ord_member_iff) RS iff_trans) 1;
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by (fast_tac eq_cs 1);
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val Ord_0_member_iff = result();
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   299
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(** For ordinals, i: succ(j) means 'less-than or equals' **)
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goal Ord.thy "!!i j. [| j<=i;  Ord(i);  Ord(j) |] ==> j : succ(i)";
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by (rtac Ord_member 1);
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by (REPEAT (ares_tac [Ord_succ] 3));
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   305
by (rtac (mem_anti_refl RS notI) 2);
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   306
by (etac subsetD 2);
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   307
by (ALLGOALS (fast_tac ZF_cs));
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val member_succI = result();
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   309
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goalw Ord.thy [Transset_def,Ord_def]
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    "!!i j. [| i : succ(j);  Ord(j) |] ==> i<=j";
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by (fast_tac ZF_cs 1);
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val member_succD = result();
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   314
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   315
goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> j:succ(i) <-> j<=i";
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by (fast_tac (subset_cs addSEs [member_succI, member_succD]) 1);
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val member_succ_iff = result();
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   318
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goal Ord.thy
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    "!!i j. [| Ord(i);  Ord(j) |] ==> i<=succ(j) <-> i<=j | i=succ(j)";
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   321
by (ASM_SIMP_TAC (ZF_ss addrews [member_succ_iff RS iff_sym, Ord_succ]) 1);
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   322
by (fast_tac ZF_cs 1);
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   323
val subset_succ_iff = result();
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   324
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goal Ord.thy "!!i j. [| i:succ(j);  j:k;  Ord(k) |] ==> i:k";
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by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
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val Ord_trans1 = result();
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   328
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   329
goal Ord.thy "!!i j. [| i:j;  j:succ(k);  Ord(k) |] ==> i:k";
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   330
by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
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   331
val Ord_trans2 = result();
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   332
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   333
goal Ord.thy "!!i jk. [| i:j;  j<=k;  Ord(k) |] ==> i:k";
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parents:
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   334
by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
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   335
val Ord_transsub2 = result();
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parents:
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   336
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   337
goal Ord.thy "!!i j. [| i:j;  Ord(j) |] ==> succ(i) : succ(j)";
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parents:
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   338
by (rtac member_succI 1);
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parents:
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   339
by (REPEAT (ares_tac [subsetI,Ord_succ,Ord_in_Ord] 1   
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   340
     ORELSE eresolve_tac [succE,Ord_trans,ssubst] 1));
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   341
val succ_mem_succI = result();
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   342
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   343
goal Ord.thy "!!i j. [| succ(i) : succ(j);  Ord(j) |] ==> i:j";
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   344
by (REPEAT (eresolve_tac [asm_rl, make_elim member_succD, succ_subsetE] 1));
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   345
val succ_mem_succE = result();
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parents:
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   346
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   347
goal Ord.thy "!!i j. Ord(j) ==> succ(i) : succ(j) <-> i:j";
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clasohm
parents:
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   348
by (REPEAT (ares_tac [iffI,succ_mem_succI,succ_mem_succE] 1));
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   349
val succ_mem_succ_iff = result();
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parents:
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   350
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   351
goal Ord.thy "!!i j. [| i<=j;  Ord(i);  Ord(j) |] ==> succ(i) <= succ(j)";
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parents:
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   352
by (rtac (member_succI RS succ_mem_succI RS member_succD) 1);
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clasohm
parents:
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   353
by (REPEAT (ares_tac [Ord_succ] 1));
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clasohm
parents:
diff changeset
   354
val Ord_succ_mono = result();
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clasohm
parents:
diff changeset
   355
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clasohm
parents:
diff changeset
   356
goal Ord.thy "!!i j k. [| i:k;  j:k;  Ord(k) |] ==> i Un j : k";
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clasohm
parents:
diff changeset
   357
by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
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clasohm
parents:
diff changeset
   358
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
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clasohm
parents:
diff changeset
   359
by (ASM_SIMP_TAC (ZF_ss addrews [subset_Un_iff RS iffD1]) 1);
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clasohm
parents:
diff changeset
   360
by (rtac (Un_commute RS ssubst) 1);
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clasohm
parents:
diff changeset
   361
by (ASM_SIMP_TAC (ZF_ss addrews [subset_Un_iff RS iffD1]) 1);
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clasohm
parents:
diff changeset
   362
val Ord_member_UnI = result();
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clasohm
parents:
diff changeset
   363
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clasohm
parents:
diff changeset
   364
goal Ord.thy "!!i j k. [| i:k;  j:k;  Ord(k) |] ==> i Int j : k";
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clasohm
parents:
diff changeset
   365
by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
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clasohm
parents:
diff changeset
   366
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
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clasohm
parents:
diff changeset
   367
by (ASM_SIMP_TAC (ZF_ss addrews [subset_Int_iff RS iffD1]) 1);
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clasohm
parents:
diff changeset
   368
by (rtac (Int_commute RS ssubst) 1);
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clasohm
parents:
diff changeset
   369
by (ASM_SIMP_TAC (ZF_ss addrews [subset_Int_iff RS iffD1]) 1);
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clasohm
parents:
diff changeset
   370
val Ord_member_IntI = result();
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clasohm
parents:
diff changeset
   371
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clasohm
parents:
diff changeset
   372
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clasohm
parents:
diff changeset
   373
(*** Results about limits ***)
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clasohm
parents:
diff changeset
   374
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clasohm
parents:
diff changeset
   375
val prems = goal Ord.thy "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))";
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clasohm
parents:
diff changeset
   376
by (rtac (Ord_is_Transset RS Transset_Union_family RS OrdI) 1);
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clasohm
parents:
diff changeset
   377
by (REPEAT (etac UnionE 1 ORELSE ares_tac ([Ord_contains_Transset]@prems) 1));
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clasohm
parents:
diff changeset
   378
val Ord_Union = result();
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clasohm
parents:
diff changeset
   379
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clasohm
parents:
diff changeset
   380
val prems = goal Ord.thy "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))";
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clasohm
parents:
diff changeset
   381
by (rtac Ord_Union 1);
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clasohm
parents:
diff changeset
   382
by (etac RepFunE 1);
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clasohm
parents:
diff changeset
   383
by (etac ssubst 1);
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clasohm
parents:
diff changeset
   384
by (eresolve_tac prems 1);
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clasohm
parents:
diff changeset
   385
val Ord_UN = result();
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clasohm
parents:
diff changeset
   386
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clasohm
parents:
diff changeset
   387
(*The upper bound must be a successor ordinal --
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clasohm
parents:
diff changeset
   388
  consider that (UN i:nat.i)=nat although nat is an upper bound of each i*)
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clasohm
parents:
diff changeset
   389
val [ordi,limit] = goal Ord.thy
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clasohm
parents:
diff changeset
   390
    "[| Ord(i);  !!y. y:A ==> f(y): succ(i) |] ==> (UN y:A. f(y)) : succ(i)";
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clasohm
parents:
diff changeset
   391
by (rtac (member_succD RS UN_least RS member_succI) 1);
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clasohm
parents:
diff changeset
   392
by (REPEAT (ares_tac [ordi, Ord_UN, ordi RS Ord_succ RS Ord_in_Ord,
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clasohm
parents:
diff changeset
   393
		      limit] 1));
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clasohm
parents:
diff changeset
   394
val sup_least = result();
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clasohm
parents:
diff changeset
   395
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clasohm
parents:
diff changeset
   396
val [jmemi,ordi,limit] = goal Ord.thy
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clasohm
parents:
diff changeset
   397
    "[| j: i;  Ord(i);  !!y. y:A ==> f(y): j |] ==> (UN y:A. succ(f(y))) : i";
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clasohm
parents:
diff changeset
   398
by (cut_facts_tac [jmemi RS (ordi RS Ord_in_Ord)] 1);
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clasohm
parents:
diff changeset
   399
by (rtac (sup_least RS Ord_trans2) 1);
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clasohm
parents:
diff changeset
   400
by (REPEAT (ares_tac [jmemi, ordi, succ_mem_succI, limit] 1));
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clasohm
parents:
diff changeset
   401
val sup_least2 = result();
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clasohm
parents:
diff changeset
   402
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clasohm
parents:
diff changeset
   403
goal Ord.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
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clasohm
parents:
diff changeset
   404
by (fast_tac (eq_cs addSEs [Ord_trans1]) 1);
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clasohm
parents:
diff changeset
   405
val Ord_equality = result();
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clasohm
parents:
diff changeset
   406
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clasohm
parents:
diff changeset
   407
(*Holds for all transitive sets, not just ordinals*)
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clasohm
parents:
diff changeset
   408
goal Ord.thy "!!i. Ord(i) ==> Union(i) <= i";
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clasohm
parents:
diff changeset
   409
by (fast_tac (ZF_cs addSEs [Ord_trans]) 1);
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clasohm
parents:
diff changeset
   410
val Ord_Union_subset = result();