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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by (rtac (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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by (rtac (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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(* Ord(succ(j)) ==> Ord(j) *)
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val Ord_succD = succI1 RSN (2, Ord_in_Ord);
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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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goal Ord.thy "Ord(succ(i)) <-> Ord(i)";
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by (fast_tac (ZF_cs addIs [Ord_succ] addDs [Ord_succD]) 1);
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val Ord_succ_iff = 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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(*** < is 'less than' for ordinals ***)
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goalw Ord.thy [lt_def] "!!i j. [| i:j;  Ord(j) |] ==> i<j";
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by (REPEAT (ares_tac [conjI] 1));
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val ltI = result();
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val major::prems = goalw Ord.thy [lt_def]
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    "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P";
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by (rtac (major RS conjE) 1);
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by (REPEAT (ares_tac (prems@[Ord_in_Ord]) 1));
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val ltE = result();
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goal Ord.thy "!!i j. i<j ==> i:j";
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by (etac ltE 1);
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by (assume_tac 1);
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val ltD = result();
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goalw Ord.thy [lt_def] "~ i<0";
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by (fast_tac ZF_cs 1);
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val not_lt0 = result();
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(* i<0 ==> R *)
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val lt0E = standard (not_lt0 RS notE);
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goal Ord.thy "!!i j k. [| i<j;  j<k |] ==> i<k";
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by (fast_tac (ZF_cs addSIs [ltI] addSEs [ltE, Ord_trans]) 1);
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val lt_trans = result();
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goalw Ord.thy [lt_def] "!!i j. [| i<j;  j<i |] ==> P";
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by (REPEAT (eresolve_tac [asm_rl, conjE, mem_anti_sym] 1));
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val lt_anti_sym = result();
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val lt_anti_refl = prove_goal Ord.thy "i<i ==> P"
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 (fn [major]=> [ (rtac (major RS (major RS lt_anti_sym)) 1) ]);
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val lt_not_refl = prove_goal Ord.thy "~ i<i"
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 (fn _=> [ (rtac notI 1), (etac lt_anti_refl 1) ]);
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(** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
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goalw Ord.thy [lt_def] "i le j <-> i<j | (i=j & Ord(j))";
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by (fast_tac (ZF_cs addSIs [Ord_succ] addSDs [Ord_succD]) 1);
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val le_iff = result();
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goal Ord.thy "!!i j. i<j ==> i le j";
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by (asm_simp_tac (ZF_ss addsimps [le_iff]) 1);
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val leI = result();
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goal Ord.thy "!!i. [| i=j;  Ord(j) |] ==> i le j";
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by (asm_simp_tac (ZF_ss addsimps [le_iff]) 1);
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val le_eqI = result();
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val le_refl = refl RS le_eqI;
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val [prem] = goal Ord.thy "(~ (i=j & Ord(j)) ==> i<j) ==> i le j";
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by (rtac (disjCI RS (le_iff RS iffD2)) 1);
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by (etac prem 1);
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val leCI = result();
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val major::prems = goal Ord.thy
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    "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P";
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by (rtac (major RS (le_iff RS iffD1 RS disjE)) 1);
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by (DEPTH_SOLVE (ares_tac prems 1 ORELSE etac conjE 1));
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val leE = result();
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goal Ord.thy "!!i j. [| i le j;  j le i |] ==> i=j";
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by (asm_full_simp_tac (ZF_ss addsimps [le_iff]) 1);
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by (fast_tac (ZF_cs addEs [lt_anti_sym]) 1);
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val le_asym = result();
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goal Ord.thy "i le 0 <-> i=0";
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by (fast_tac (ZF_cs addSIs [Ord_0 RS le_refl] addSEs [leE, lt0E]) 1);
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val le0_iff = result();
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val le0D = standard (le0_iff RS iffD1);
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val lt_cs = 
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    ZF_cs addSIs [le_refl, leCI]
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          addSDs [le0D]
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          addSEs [lt_anti_refl, lt0E, leE];
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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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(*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 < 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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(*The trichotomy law for ordinals!*)
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lcp
parents: 14
diff changeset
   347
val ordi::ordj::prems = goalw Ord.thy [lt_def]
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lcp
parents: 14
diff changeset
   348
    "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P";
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lcp
parents: 14
diff changeset
   349
by (rtac ([ordi,ordj] MRS (Ord_linear_lemma RS spec RS impE)) 1);
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lcp
parents: 14
diff changeset
   350
by (REPEAT (FIRSTGOAL (etac disjE)));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   351
by (DEPTH_SOLVE (ares_tac ([ordi,ordj,conjI] @ prems) 1));
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lcp
parents: 14
diff changeset
   352
val Ord_linear_lt = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   353
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   354
val prems = goal Ord.thy
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   355
    "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   356
by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   357
by (DEPTH_SOLVE (ares_tac ([leI,le_eqI] @ prems) 1));
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lcp
parents: 14
diff changeset
   358
val Ord_linear_le = result();
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lcp
parents: 14
diff changeset
   359
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   360
goal Ord.thy "!!i j. j le i ==> ~ i<j";
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lcp
parents: 14
diff changeset
   361
by (fast_tac (lt_cs addEs [lt_anti_sym]) 1);
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lcp
parents: 14
diff changeset
   362
val le_imp_not_lt = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   363
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   364
goal Ord.thy "!!i j. [| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   365
by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1);
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lcp
parents: 14
diff changeset
   366
by (REPEAT (SOMEGOAL assume_tac));
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lcp
parents: 14
diff changeset
   367
by (fast_tac (lt_cs addEs [lt_anti_sym]) 1);
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lcp
parents: 14
diff changeset
   368
val not_lt_imp_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   369
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   370
goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i";
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lcp
parents: 14
diff changeset
   371
by (REPEAT (ares_tac [iffI, le_imp_not_lt, not_lt_imp_le] 1));
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lcp
parents: 14
diff changeset
   372
val not_lt_iff_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   373
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   374
goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i";
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lcp
parents: 14
diff changeset
   375
by (asm_simp_tac (ZF_ss addsimps [not_lt_iff_le RS iff_sym]) 1);
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lcp
parents: 14
diff changeset
   376
val not_le_iff_lt = result();
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lcp
parents: 14
diff changeset
   377
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   378
goal Ord.thy "!!i. Ord(i) ==> 0 le i";
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lcp
parents: 14
diff changeset
   379
by (etac (not_lt_iff_le RS iffD1) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   380
by (REPEAT (resolve_tac [Ord_0, not_lt0] 1));
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lcp
parents: 14
diff changeset
   381
val Ord_0_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   382
37
cebe01deba80 added ~: for "not in"
lcp
parents: 30
diff changeset
   383
goal Ord.thy "!!i. [| Ord(i);  i~=0 |] ==> 0<i";
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   384
by (etac (not_le_iff_lt RS iffD1) 1);
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lcp
parents: 14
diff changeset
   385
by (rtac Ord_0 1);
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lcp
parents: 14
diff changeset
   386
by (fast_tac lt_cs 1);
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lcp
parents: 14
diff changeset
   387
val Ord_0_lt = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   388
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   389
(*** Results about less-than or equals ***)
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lcp
parents: 14
diff changeset
   390
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lcp
parents: 14
diff changeset
   391
(** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   392
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   393
goal Ord.thy "!!i j. [| j<=i;  Ord(i);  Ord(j) |] ==> j le i";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   394
by (rtac (not_lt_iff_le RS iffD1) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   395
by (assume_tac 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   396
by (assume_tac 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   397
by (fast_tac (ZF_cs addEs [ltE, mem_anti_refl]) 1);
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lcp
parents: 14
diff changeset
   398
val subset_imp_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   399
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   400
goal Ord.thy "!!i j. i le j ==> i<=j";
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lcp
parents: 14
diff changeset
   401
by (etac leE 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   402
by (fast_tac ZF_cs 2);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   403
by (fast_tac (subset_cs addIs [OrdmemD] addEs [ltE]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   404
val le_imp_subset = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   405
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   406
goal Ord.thy "j le i <-> j<=i & Ord(i) & Ord(j)";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   407
by (fast_tac (ZF_cs addSEs [subset_imp_le, le_imp_subset]
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   408
	            addEs [ltE, make_elim Ord_succD]) 1);
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lcp
parents: 14
diff changeset
   409
val le_subset_iff = result();
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   410
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   411
goal Ord.thy "i le succ(j) <-> i le j | i=succ(j) & Ord(i)";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   412
by (simp_tac (ZF_ss addsimps [le_iff]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   413
by (fast_tac (ZF_cs addIs [Ord_succ] addDs [Ord_succD]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   414
val le_succ_iff = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   415
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   416
goal Ord.thy "!!i j. [| i le j;  j<k |] ==> i<k";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   417
by (fast_tac (ZF_cs addEs [leE, lt_trans]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   418
val lt_trans1 = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   419
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   420
goal Ord.thy "!!i j. [| i<j;  j le k |] ==> i<k";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   421
by (fast_tac (ZF_cs addEs [leE, lt_trans]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   422
val lt_trans2 = result();
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   423
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   424
goal Ord.thy "!!i j. [| i le j;  j le k |] ==> i le k";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   425
by (REPEAT (ares_tac [lt_trans1] 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   426
val le_trans = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   427
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   428
goal Ord.thy "!!i j. i<j ==> succ(i) le j";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   429
by (rtac (not_lt_iff_le RS iffD1) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   430
by (fast_tac (lt_cs addEs [lt_anti_sym]) 3);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   431
by (ALLGOALS (fast_tac (ZF_cs addEs [ltE] addIs [Ord_succ])));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   432
val succ_leI = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   433
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   434
goal Ord.thy "!!i j. succ(i) le j ==> i<j";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   435
by (rtac (not_le_iff_lt RS iffD1) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   436
by (fast_tac (lt_cs addEs [lt_anti_sym]) 3);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   437
by (ALLGOALS (fast_tac (ZF_cs addEs [ltE, make_elim Ord_succD])));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   438
val succ_leE = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   439
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   440
goal Ord.thy "succ(i) le j <-> i<j";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   441
by (REPEAT (ares_tac [iffI,succ_leI,succ_leE] 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   442
val succ_le_iff = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   443
14
1c0926788772 ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents: 6
diff changeset
   444
(** Union and Intersection **)
1c0926788772 ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents: 6
diff changeset
   445
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   446
(*Replacing k by succ(k') yields the similar rule for le!*)
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   447
goal Ord.thy "!!i j k. [| i<k;  j<k |] ==> i Un j < k";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   448
by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   449
by (rtac (Un_commute RS ssubst) 4);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   450
by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Un_iff]) 4);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   451
by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Un_iff]) 3);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   452
by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   453
val Un_least_lt = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   454
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   455
(*Replacing k by succ(k') yields the similar rule for le!*)
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   456
goal Ord.thy "!!i j k. [| i<k;  j<k |] ==> i Int j < k";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   457
by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   458
by (rtac (Int_commute RS ssubst) 4);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   459
by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Int_iff]) 4);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   460
by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Int_iff]) 3);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   461
by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   462
val Int_greatest_lt = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   463
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   464
(*** Results about limits ***)
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   465
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   466
val prems = goal Ord.thy "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))";
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   467
by (rtac (Ord_is_Transset RS Transset_Union_family RS OrdI) 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   468
by (REPEAT (etac UnionE 1 ORELSE ares_tac ([Ord_contains_Transset]@prems) 1));
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   469
val Ord_Union = result();
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   470
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   471
val prems = goal Ord.thy "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))";
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   472
by (rtac Ord_Union 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   473
by (etac RepFunE 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   474
by (etac ssubst 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   475
by (eresolve_tac prems 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   476
val Ord_UN = result();
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   477
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   478
(* No < version; consider (UN i:nat.i)=nat *)
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   479
val [ordi,limit] = goal Ord.thy
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   480
    "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (UN x:A. b(x)) le i";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   481
by (rtac (le_imp_subset RS UN_least RS subset_imp_le) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   482
by (REPEAT (ares_tac [ordi, Ord_UN, limit] 1 ORELSE etac (limit RS ltE) 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   483
val UN_least_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   484
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   485
val [jlti,limit] = goal Ord.thy
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   486
    "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (UN x:A. succ(b(x))) < i";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   487
by (rtac (jlti RS ltE) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   488
by (rtac (UN_least_le RS lt_trans2) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   489
by (REPEAT (ares_tac [jlti, succ_leI, limit] 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   490
val UN_succ_least_lt = result();
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   491
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   492
val prems = goal Ord.thy
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   493
    "[| a: A;  i le b(a);  !!x. x:A ==> Ord(b(x)) |] ==> i le (UN x:A. b(x))";
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   494
by (resolve_tac (prems RL [ltE]) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   495
by (rtac (le_imp_subset RS subset_trans RS subset_imp_le) 1);
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   496
by (REPEAT (ares_tac (prems @ [UN_upper, Ord_UN]) 1));
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   497
val UN_upper_le = result();
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   498
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   499
goal Ord.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
30
d49df4181f0d Retrying yet again after network problems
lcp
parents: 14
diff changeset
   500
by (fast_tac (eq_cs addEs [Ord_trans]) 1);
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   501
val Ord_equality = result();
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   502
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   503
(*Holds for all transitive sets, not just ordinals*)
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   504
goal Ord.thy "!!i. Ord(i) ==> Union(i) <= i";
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   505
by (fast_tac (ZF_cs addSEs [Ord_trans]) 1);
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   506
val Ord_Union_subset = result();