src/ZF/CardinalArith.ML
author paulson
Fri, 11 Aug 2000 13:26:40 +0200
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parent 9491 1a36151ee2fc
child 9842 58d8335cc40c
permissions -rw-r--r--
ZF arith
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(*  Title:      ZF/CardinalArith.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Cardinal arithmetic -- WITHOUT the Axiom of Choice
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Note: Could omit proving the algebraic laws for cardinal addition and
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multiplication.  On finite cardinals these operations coincide with
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addition and multiplication of natural numbers; on infinite cardinals they
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coincide with union (maximum).  Either way we get most laws for free.
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*)
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(*** Cardinal addition ***)
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(** Cardinal addition is commutative **)
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Goalw [eqpoll_def] "A+B eqpoll B+A";
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by (rtac exI 1);
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by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] 
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    lam_bijective 1);
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by Auto_tac;
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qed "sum_commute_eqpoll";
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Goalw [cadd_def] "i |+| j = j |+| i";
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by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
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qed "cadd_commute";
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(** Cardinal addition is associative **)
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Goalw [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
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by (rtac exI 1);
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by (rtac sum_assoc_bij 1);
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qed "sum_assoc_eqpoll";
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(*Unconditional version requires AC*)
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Goalw [cadd_def]
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    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
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\             (i |+| j) |+| k = i |+| (j |+| k)";
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by (rtac cardinal_cong 1);
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by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
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          eqpoll_trans) 1);
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by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
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by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
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          eqpoll_sym) 2);
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by (REPEAT (ares_tac [well_ord_radd] 1));
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qed "well_ord_cadd_assoc";
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(** 0 is the identity for addition **)
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Goalw [eqpoll_def] "0+A eqpoll A";
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by (rtac exI 1);
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by (rtac bij_0_sum 1);
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qed "sum_0_eqpoll";
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Goalw [cadd_def] "Card(K) ==> 0 |+| K = K";
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by (asm_simp_tac (simpset() addsimps [sum_0_eqpoll RS cardinal_cong, 
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				      Card_cardinal_eq]) 1);
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qed "cadd_0";
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Addsimps [cadd_0];
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(** Addition by another cardinal **)
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Goalw [lepoll_def, inj_def] "A lepoll A+B";
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by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1);
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by (Asm_simp_tac 1);
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qed "sum_lepoll_self";
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(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
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Goalw [cadd_def]
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    "[| Card(K);  Ord(L) |] ==> K le (K |+| L)";
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by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
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by (rtac sum_lepoll_self 3);
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1));
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qed "cadd_le_self";
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(** Monotonicity of addition **)
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Goalw [lepoll_def]
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     "[| A lepoll C;  B lepoll D |] ==> A + B  lepoll  C + D";
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by (REPEAT (etac exE 1));
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by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")] 
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    exI 1);
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by (res_inst_tac 
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      [("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")] 
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      lam_injective 1);
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by (typecheck_tac (tcset() addTCs [inj_is_fun]));
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by Auto_tac;
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qed "sum_lepoll_mono";
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Goalw [cadd_def]
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    "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)";
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by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
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by (rtac well_ord_lepoll_imp_Card_le 1);
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by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2));
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
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qed "cadd_le_mono";
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(** Addition of finite cardinals is "ordinary" addition **)
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Goalw [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
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by (rtac exI 1);
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by (res_inst_tac [("c", "%z. if z=Inl(A) then A+B else z"), 
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                  ("d", "%z. if z=A+B then Inl(A) else z")] 
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    lam_bijective 1);
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by (ALLGOALS
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    (asm_simp_tac (simpset() addsimps [succI2, mem_imp_not_eq]
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                             setloop eresolve_tac [sumE,succE])));
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qed "sum_succ_eqpoll";
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(*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
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(*Unconditional version requires AC*)
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Goalw [cadd_def]
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    "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
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by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
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by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
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by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
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qed "cadd_succ_lemma";
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Goal "[| m: nat;  n: nat |] ==> m |+| n = m#+n";
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by (induct_tac "m" 1);
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by (asm_simp_tac (simpset() addsimps [nat_into_Card RS cadd_0]) 1);
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by (asm_simp_tac (simpset() addsimps [cadd_succ_lemma,
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				      nat_into_Card RS Card_cardinal_eq]) 1);
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qed "nat_cadd_eq_add";
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(*** Cardinal multiplication ***)
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(** Cardinal multiplication is commutative **)
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(*Easier to prove the two directions separately*)
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Goalw [eqpoll_def] "A*B eqpoll B*A";
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by (rtac exI 1);
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by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")] 
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    lam_bijective 1);
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by Safe_tac;
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by (ALLGOALS (Asm_simp_tac));
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qed "prod_commute_eqpoll";
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Goalw [cmult_def] "i |*| j = j |*| i";
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by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
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qed "cmult_commute";
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(** Cardinal multiplication is associative **)
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Goalw [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
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by (rtac exI 1);
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6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   150
by (rtac prod_assoc_bij 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   151
qed "prod_assoc_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   152
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   153
(*Unconditional version requires AC*)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   154
Goalw [cmult_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   155
    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   156
\             (i |*| j) |*| k = i |*| (j |*| k)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   157
by (rtac cardinal_cong 1);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   158
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   159
          eqpoll_trans) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   160
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   161
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   162
          eqpoll_sym) 2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   163
by (REPEAT (ares_tac [well_ord_rmult] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   164
qed "well_ord_cmult_assoc";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   165
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   166
(** Cardinal multiplication distributes over addition **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   167
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   168
Goalw [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   169
by (rtac exI 1);
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   170
by (rtac sum_prod_distrib_bij 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   171
qed "sum_prod_distrib_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   172
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   173
Goalw [cadd_def, cmult_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   174
    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   175
\             (i |+| j) |*| k = (i |*| k) |+| (j |*| k)";
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   176
by (rtac cardinal_cong 1);
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   177
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   178
          eqpoll_trans) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   179
by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2);
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   180
by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   181
	  sum_eqpoll_cong RS eqpoll_sym) 2);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   182
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1));
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   183
qed "well_ord_cadd_cmult_distrib";
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   184
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   185
(** Multiplication by 0 yields 0 **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   186
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   187
Goalw [eqpoll_def] "0*A eqpoll 0";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   188
by (rtac exI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   189
by (rtac lam_bijective 1);
4152
451104c223e2 Ran expandshort, especially to introduce Safe_tac
paulson
parents: 4091
diff changeset
   190
by Safe_tac;
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   191
qed "prod_0_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   192
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   193
Goalw [cmult_def] "0 |*| i = 0";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   194
by (asm_simp_tac (simpset() addsimps [prod_0_eqpoll RS cardinal_cong, 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   195
				      Card_0 RS Card_cardinal_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   196
qed "cmult_0";
6070
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   197
Addsimps [cmult_0];
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   198
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   199
(** 1 is the identity for multiplication **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   200
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   201
Goalw [eqpoll_def] "{x}*A eqpoll A";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   202
by (rtac exI 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   203
by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   204
qed "prod_singleton_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   205
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   206
Goalw [cmult_def, succ_def] "Card(K) ==> 1 |*| K = K";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   207
by (asm_simp_tac (simpset() addsimps [prod_singleton_eqpoll RS cardinal_cong, 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   208
				      Card_cardinal_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   209
qed "cmult_1";
6070
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   210
Addsimps [cmult_1];
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   211
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   212
(*** Some inequalities for multiplication ***)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   213
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   214
Goalw [lepoll_def, inj_def] "A lepoll A*A";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   215
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   216
by (Simp_tac 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   217
qed "prod_square_lepoll";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   218
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   219
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   220
Goalw [cmult_def] "Card(K) ==> K le K |*| K";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   221
by (rtac le_trans 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   222
by (rtac well_ord_lepoll_imp_Card_le 2);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   223
by (rtac prod_square_lepoll 3);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   224
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   225
by (asm_simp_tac (simpset() 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   226
		  addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   227
qed "cmult_square_le";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   228
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   229
(** Multiplication by a non-zero cardinal **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   230
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   231
Goalw [lepoll_def, inj_def] "b: B ==> A lepoll A*B";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   232
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   233
by (Asm_simp_tac 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   234
qed "prod_lepoll_self";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   235
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   236
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   237
Goalw [cmult_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   238
    "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   239
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   240
by (rtac prod_lepoll_self 3);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   241
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   242
qed "cmult_le_self";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   243
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   244
(** Monotonicity of multiplication **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   245
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   246
Goalw [lepoll_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   247
     "[| A lepoll C;  B lepoll D |] ==> A * B  lepoll  C * D";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   248
by (REPEAT (etac exE 1));
1090
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   249
by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   250
by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")] 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   251
                  lam_injective 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   252
by (typecheck_tac (tcset() addTCs [inj_is_fun]));
6176
707b6f9859d2 tidied, with left_inverse & right_inverse as default simprules
paulson
parents: 6153
diff changeset
   253
by Auto_tac;
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   254
qed "prod_lepoll_mono";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   255
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   256
Goalw [cmult_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   257
    "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   258
by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   259
by (rtac well_ord_lepoll_imp_Card_le 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   260
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   261
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   262
qed "cmult_le_mono";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   263
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   264
(*** Multiplication of finite cardinals is "ordinary" multiplication ***)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   265
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   266
Goalw [eqpoll_def] "succ(A)*B eqpoll B + A*B";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   267
by (rtac exI 1);
6068
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   268
by (res_inst_tac [("c", "%<x,y>. if x=A then Inl(y) else Inr(<x,y>)"), 
3840
e0baea4d485a fixed dots;
wenzelm
parents: 3736
diff changeset
   269
                  ("d", "case(%y. <A,y>, %z. z)")] 
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   270
    lam_bijective 1);
5488
paulson
parents: 5325
diff changeset
   271
by Safe_tac;
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   272
by (ALLGOALS
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   273
    (asm_simp_tac (simpset() addsimps [succI2, if_type, mem_imp_not_eq])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   274
qed "prod_succ_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   275
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   276
(*Unconditional version requires AC*)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   277
Goalw [cmult_def, cadd_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   278
    "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   279
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   280
by (rtac (cardinal_cong RS sym) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   281
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   282
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   283
qed "cmult_succ_lemma";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   284
6070
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   285
Goal "[| m: nat;  n: nat |] ==> m |*| n = m#*n";
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   286
by (induct_tac "m" 1);
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   287
by (Asm_simp_tac 1);
8201
a81d18b0a9b1 tidied some proofs
paulson
parents: 8127
diff changeset
   288
by (asm_simp_tac (simpset() addsimps [cmult_succ_lemma, nat_cadd_eq_add]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   289
qed "nat_cmult_eq_mult";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   290
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   291
Goal "Card(n) ==> 2 |*| n = n |+| n";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   292
by (asm_simp_tac 
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   293
    (simpset() addsimps [cmult_succ_lemma, Card_is_Ord,
8551
5c22595bc599 tidied using new "inst" rule
paulson
parents: 8201
diff changeset
   294
			 inst "j" "0" cadd_commute]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   295
qed "cmult_2";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   296
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   297
5284
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   298
val sum_lepoll_prod = [sum_eq_2_times RS equalityD1 RS subset_imp_lepoll,
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   299
        asm_rl, lepoll_refl] MRS (prod_lepoll_mono RSN (2, lepoll_trans))
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   300
        |> standard;
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   301
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   302
Goal "[| A lepoll B; 2 lepoll A |] ==> A+B lepoll A*B";
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   303
by (REPEAT (ares_tac [[sum_lepoll_mono, sum_lepoll_prod]
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   304
                MRS lepoll_trans, lepoll_refl] 1));
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   305
qed "lepoll_imp_sum_lepoll_prod";
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   306
c77e9dd9bafc Tidying of AC, especially of AC16_WO4 using a locale
paulson
parents: 5147
diff changeset
   307
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   308
(*** Infinite Cardinals are Limit Ordinals ***)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   309
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   310
(*This proof is modelled upon one assuming nat<=A, with injection
6068
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   311
  lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z 
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   312
  and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   313
  If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   314
Goalw [lepoll_def] "nat lepoll A ==> cons(u,A) lepoll A";
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   315
by (etac exE 1);
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   316
by (res_inst_tac [("x",
6068
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   317
    "lam z:cons(u,A). if z=u then f`0      \
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   318
\                     else if z: range(f) then f`succ(converse(f)`z)  \
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   319
\                     else z")] exI 1);
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   320
by (res_inst_tac [("d", "%y. if y: range(f)     \
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   321
\                            then nat_case(u, %z. f`z, converse(f)`y) \
2d8f3e1f1151 if-then-else syntax for ZF
paulson
parents: 5529
diff changeset
   322
\                            else y")] 
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   323
    lam_injective 1);
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   324
by (fast_tac (claset() addSIs [if_type, apply_type]
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   325
                       addIs  [inj_is_fun, inj_converse_fun]) 1);
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   326
by (asm_simp_tac 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   327
    (simpset() addsimps [inj_is_fun RS apply_rangeI,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   328
			 inj_converse_fun RS apply_rangeI,
6176
707b6f9859d2 tidied, with left_inverse & right_inverse as default simprules
paulson
parents: 6153
diff changeset
   329
			 inj_converse_fun RS apply_funtype]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   330
qed "nat_cons_lepoll";
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   331
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   332
Goal "nat lepoll A ==> cons(u,A) eqpoll A";
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   333
by (etac (nat_cons_lepoll RS eqpollI) 1);
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   334
by (rtac (subset_consI RS subset_imp_lepoll) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   335
qed "nat_cons_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   336
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   337
(*Specialized version required below*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   338
Goalw [succ_def] "nat <= A ==> succ(A) eqpoll A";
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   339
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   340
qed "nat_succ_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   341
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   342
Goalw [InfCard_def] "InfCard(nat)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   343
by (blast_tac (claset() addIs [Card_nat, le_refl, Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   344
qed "InfCard_nat";
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   345
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   346
Goalw [InfCard_def] "InfCard(K) ==> Card(K)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   347
by (etac conjunct1 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   348
qed "InfCard_is_Card";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   349
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   350
Goalw [InfCard_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   351
    "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   352
by (asm_simp_tac (simpset() addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   353
				      Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   354
qed "InfCard_Un";
523
972119df615e ZF/CardinalArith/InfCard_Un: new
lcp
parents: 517
diff changeset
   355
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   356
(*Kunen's Lemma 10.11*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   357
Goalw [InfCard_def] "InfCard(K) ==> Limit(K)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   358
by (etac conjE 1);
7499
23e090051cb8 isatool expandshort;
wenzelm
parents: 6176
diff changeset
   359
by (ftac Card_is_Ord 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   360
by (rtac (ltI RS non_succ_LimitI) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   361
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   362
by (safe_tac (claset() addSDs [Limit_nat RS Limit_le_succD]));
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   363
by (rewtac Card_def);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   364
by (dtac trans 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   365
by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
3016
15763781afb0 Conversion to use blast_tac
paulson
parents: 2925
diff changeset
   366
by (etac (Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   367
by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   368
qed "InfCard_is_Limit";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   369
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   370
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   371
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   372
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   373
(*A general fact about ordermap*)
5325
f7a5e06adea1 Yet more removal of "goal" commands, especially "goal ZF.thy", so ZF.thy
paulson
parents: 5284
diff changeset
   374
Goalw [eqpoll_def]
f7a5e06adea1 Yet more removal of "goal" commands, especially "goal ZF.thy", so ZF.thy
paulson
parents: 5284
diff changeset
   375
    "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   376
by (rtac exI 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   377
by (asm_simp_tac (simpset() addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   378
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   379
by (rtac pred_subset 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   380
qed "ordermap_eqpoll_pred";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   381
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   382
(** Establishing the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   383
5488
paulson
parents: 5325
diff changeset
   384
Goalw [inj_def] "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
paulson
parents: 5325
diff changeset
   385
by (force_tac (claset() addIs [lam_type, Un_least_lt RS ltD, ltI],
paulson
parents: 5325
diff changeset
   386
	       simpset()) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   387
qed "csquare_lam_inj";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   388
5488
paulson
parents: 5325
diff changeset
   389
Goalw [csquare_rel_def] "Ord(K) ==> well_ord(K*K, csquare_rel(K))";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   390
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   391
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   392
qed "well_ord_csquare";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   393
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   394
(** Characterising initial segments of the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   395
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   396
Goalw [csquare_rel_def]
5488
paulson
parents: 5325
diff changeset
   397
 "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z";
paulson
parents: 5325
diff changeset
   398
by (etac rev_mp 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   399
by (REPEAT (etac ltE 1));
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   400
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   401
				      Un_absorb, Un_least_mem_iff, ltD]) 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   402
by (safe_tac (claset() addSEs [mem_irrefl] 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   403
                       addSIs [Un_upper1_le, Un_upper2_le]));
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   404
by (ALLGOALS (asm_simp_tac (simpset() addsimps [lt_def, succI2])));
5488
paulson
parents: 5325
diff changeset
   405
qed "csquareD";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   406
9491
1a36151ee2fc natify, a coercion to reduce the number of type constraints in arithmetic
paulson
parents: 9180
diff changeset
   407
Goalw [Order.pred_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   408
 "z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
9491
1a36151ee2fc natify, a coercion to reduce the number of type constraints in arithmetic
paulson
parents: 9180
diff changeset
   409
by (safe_tac (claset() delrules [SigmaI, succCI]));  
5488
paulson
parents: 5325
diff changeset
   410
by (etac (csquareD RS conjE) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   411
by (rewtac lt_def);
2925
b0ae2e13db93 Using Blast_tac
paulson
parents: 2493
diff changeset
   412
by (ALLGOALS Blast_tac);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   413
qed "pred_csquare_subset";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   414
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   415
Goalw [csquare_rel_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   416
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   417
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   418
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   419
by (REPEAT (etac ltE 1));
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   420
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   421
				      Un_absorb, Un_least_mem_iff, ltD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   422
qed "csquare_ltI";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   423
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   424
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   425
Goalw [csquare_rel_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   426
 "[| x le z;  y le z;  z<K |] ==> \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   427
\      <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   428
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   429
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   430
by (REPEAT (etac ltE 1));
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   431
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   432
				      Un_absorb, Un_least_mem_iff, ltD]) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   433
by (REPEAT_FIRST (etac succE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   434
by (ALLGOALS
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   435
    (asm_simp_tac (simpset() addsimps [subset_Un_iff RS iff_sym, 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   436
				       subset_Un_iff2 RS iff_sym, OrdmemD])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   437
qed "csquare_or_eqI";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   438
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   439
(** The cardinality of initial segments **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   440
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   441
Goal "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   442
\         ordermap(K*K, csquare_rel(K)) ` <x,y> <               \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   443
\         ordermap(K*K, csquare_rel(K)) ` <z,z>";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   444
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   445
by (etac (Limit_is_Ord RS well_ord_csquare) 2);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   446
by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
870
ef6faaa415dc Replaced ordermap_z_lepoll by ordermap_z_lt, which is
lcp
parents: 846
diff changeset
   447
by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   448
by (etac well_ord_is_wf 4);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   449
by (ALLGOALS 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   450
    (blast_tac (claset() addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   451
                         addSEs [ltE])));
870
ef6faaa415dc Replaced ordermap_z_lepoll by ordermap_z_lt, which is
lcp
parents: 846
diff changeset
   452
qed "ordermap_z_lt";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   453
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   454
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   455
Goalw [cmult_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   456
  "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   457
\       | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   458
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   459
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   460
by (subgoals_tac ["z<K"] 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   461
by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   462
by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   463
by (REPEAT_SOME assume_tac);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   464
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   465
by (etac (Limit_is_Ord RS well_ord_csquare) 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   466
by (blast_tac (claset() addIs [ltD]) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   467
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   468
    assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   469
by (REPEAT_FIRST (etac ltE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   470
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   471
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   472
qed "ordermap_csquare_le";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   473
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   474
(*Kunen: "... so the order type <= K" *)
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   475
Goal "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |]  ==>  \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   476
\         ordertype(K*K, csquare_rel(K)) le K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   477
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   478
by (rtac all_lt_imp_le 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   479
by (assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   480
by (etac (well_ord_csquare RS Ord_ordertype) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   481
by (rtac Card_lt_imp_lt 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   482
by (etac InfCard_is_Card 3);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   483
by (etac ltE 2 THEN assume_tac 2);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   484
by (asm_full_simp_tac (simpset() addsimps [ordertype_unfold]) 1);
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   485
by (safe_tac (claset() addSEs [ltE]));
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   486
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   487
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   488
by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1  THEN
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   489
    REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   490
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1  THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   491
    REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   492
(*the finite case: xb Un y < nat *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   493
by (res_inst_tac [("j", "nat")] lt_trans2 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   494
by (asm_full_simp_tac (simpset() addsimps [InfCard_def]) 2);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   495
by (asm_full_simp_tac
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   496
    (simpset() addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   497
			 nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   498
(*case nat le (xb Un y) *)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   499
by (asm_full_simp_tac
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   500
    (simpset() addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   501
			 le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   502
			 Ord_Un, ltI, nat_le_cardinal,
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   503
			 Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   504
qed "ordertype_csquare_le";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   505
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   506
(*Main result: Kunen's Theorem 10.12*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   507
Goal "InfCard(K) ==> K |*| K = K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   508
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   509
by (etac rev_mp 1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   510
by (trans_ind_tac "K" [] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   511
by (rtac impI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   512
by (rtac le_anti_sym 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   513
by (etac (InfCard_is_Card RS cmult_square_le) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   514
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   515
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   516
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   517
by (asm_simp_tac 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   518
    (simpset() addsimps [cmult_def, Ord_cardinal_le,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   519
			 well_ord_csquare RS ordermap_bij RS 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   520
			 bij_imp_eqpoll RS cardinal_cong,
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   521
			 well_ord_csquare RS Ord_ordertype]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   522
qed "InfCard_csquare_eq";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   523
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   524
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   525
Goal "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A eqpoll A";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   526
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   527
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   528
by (rtac well_ord_cardinal_eqE 1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   529
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   530
by (asm_simp_tac (simpset() 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   531
		  addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   532
qed "well_ord_InfCard_square_eq";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   533
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   534
(** Toward's Kunen's Corollary 10.13 (1) **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   535
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   536
Goal "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K";
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   537
by (rtac le_anti_sym 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   538
by (etac ltE 2 THEN
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   539
    REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   540
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   541
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   542
by (asm_full_simp_tac (simpset() addsimps [InfCard_csquare_eq]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   543
qed "InfCard_le_cmult_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   544
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   545
(*Corollary 10.13 (1), for cardinal multiplication*)
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   546
Goal "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   547
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   548
by (typecheck_tac (tcset() addTCs [InfCard_is_Card, Card_is_Ord]));
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   549
by (resolve_tac [cmult_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   550
by (resolve_tac [Un_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   551
by (ALLGOALS
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   552
    (asm_simp_tac 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   553
     (simpset() addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   554
			  subset_Un_iff2 RS iffD1, le_imp_subset])));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   555
qed "InfCard_cmult_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   556
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   557
Goal "InfCard(K) ==> K |+| K = K";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   558
by (asm_simp_tac
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   559
    (simpset() addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   560
by (asm_simp_tac 
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   561
    (simpset() addsimps [InfCard_le_cmult_eq, InfCard_is_Limit, 
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   562
			 Limit_has_0, Limit_has_succ]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   563
qed "InfCard_cdouble_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   564
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   565
(*Corollary 10.13 (1), for cardinal addition*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   566
Goal "[| InfCard(K);  L le K |] ==> K |+| L = K";
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   567
by (rtac le_anti_sym 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   568
by (etac ltE 2 THEN
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   569
    REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   570
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   571
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   572
by (asm_full_simp_tac (simpset() addsimps [InfCard_cdouble_eq]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   573
qed "InfCard_le_cadd_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   574
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   575
Goal "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   576
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
6153
bff90585cce5 new typechecking solver for the simplifier
paulson
parents: 6070
diff changeset
   577
by (typecheck_tac (tcset() addTCs [InfCard_is_Card, Card_is_Ord]));
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   578
by (resolve_tac [cadd_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   579
by (resolve_tac [Un_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   580
by (ALLGOALS
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   581
    (asm_simp_tac 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   582
     (simpset() addsimps [InfCard_le_cadd_eq,
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   583
			  subset_Un_iff2 RS iffD1, le_imp_subset])));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   584
qed "InfCard_cadd_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   585
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   586
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   587
  of all n-tuples of elements of K.  A better version for the Isabelle theory
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   588
  might be  InfCard(K) ==> |list(K)| = K.
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   589
*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   590
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   591
(*** For every cardinal number there exists a greater one
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   592
     [Kunen's Theorem 10.16, which would be trivial using AC] ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   593
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   594
Goalw [jump_cardinal_def] "Ord(jump_cardinal(K))";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   595
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   596
by (blast_tac (claset() addSIs [Ord_ordertype]) 2);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   597
by (rewtac Transset_def);
1075
848bf2e18dff Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents: 989
diff changeset
   598
by (safe_tac subset_cs);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   599
by (asm_full_simp_tac (simpset() addsimps [ordertype_pred_unfold]) 1);
4152
451104c223e2 Ran expandshort, especially to introduce Safe_tac
paulson
parents: 4091
diff changeset
   600
by Safe_tac;
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   601
by (rtac UN_I 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   602
by (rtac ReplaceI 2);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   603
by (ALLGOALS (blast_tac (claset() addIs [well_ord_subset] addSEs [predE])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   604
qed "Ord_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   605
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   606
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   607
Goalw [jump_cardinal_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   608
     "i : jump_cardinal(K) <->   \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   609
\         (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   610
by (fast_tac subset_cs 1);      (*It's vital to avoid reasoning about <=*)
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   611
qed "jump_cardinal_iff";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   612
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   613
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   614
Goal "Ord(K) ==> K < jump_cardinal(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   615
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   616
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   617
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   618
by (rtac subset_refl 2);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   619
by (asm_simp_tac (simpset() addsimps [Memrel_def, subset_iff]) 1);
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   620
by (asm_simp_tac (simpset() addsimps [ordertype_Memrel]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   621
qed "K_lt_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   622
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   623
(*The proof by contradiction: the bijection f yields a wellordering of X
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   624
  whose ordertype is jump_cardinal(K).  *)
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   625
Goal "[| well_ord(X,r);  r <= K * K;  X <= K;       \
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   626
\            f : bij(ordertype(X,r), jump_cardinal(K))  \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   627
\         |] ==> jump_cardinal(K) : jump_cardinal(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   628
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   629
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   630
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   631
by (REPEAT_FIRST (resolve_tac [exI, conjI]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   632
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   633
by (REPEAT (assume_tac 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   634
by (etac (bij_is_inj RS well_ord_rvimage) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   635
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   636
by (asm_simp_tac
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   637
    (simpset() addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   638
			 ordertype_Memrel, Ord_jump_cardinal]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   639
qed "Card_jump_cardinal_lemma";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   640
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   641
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   642
Goal "Card(jump_cardinal(K))";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   643
by (rtac (Ord_jump_cardinal RS CardI) 1);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   644
by (rewtac eqpoll_def);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   645
by (safe_tac (claset() addSDs [ltD, jump_cardinal_iff RS iffD1]));
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   646
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   647
qed "Card_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   648
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   649
(*** Basic properties of successor cardinals ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   650
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   651
Goalw [csucc_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   652
    "Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   653
by (rtac LeastI 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   654
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   655
                      Ord_jump_cardinal] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   656
qed "csucc_basic";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   657
800
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   658
bind_thm ("Card_csucc", csucc_basic RS conjunct1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   659
800
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   660
bind_thm ("lt_csucc", csucc_basic RS conjunct2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   661
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   662
Goal "Ord(K) ==> 0 < csucc(K)";
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   663
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   664
by (REPEAT (assume_tac 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   665
qed "Ord_0_lt_csucc";
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   666
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   667
Goalw [csucc_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   668
    "[| Card(L);  K<L |] ==> csucc(K) le L";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   669
by (rtac Least_le 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   670
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   671
qed "csucc_le";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   672
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   673
Goal "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   674
by (rtac iffI 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   675
by (rtac Card_lt_imp_lt 2);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   676
by (etac lt_trans1 2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   677
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   678
by (resolve_tac [notI RS not_lt_imp_le] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   679
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   680
by (assume_tac 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   681
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   682
by (REPEAT (ares_tac [Ord_cardinal] 1
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   683
     ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   684
qed "lt_csucc_iff";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   685
9180
3bda56c0d70d tidying and unbatchifying
paulson
parents: 8551
diff changeset
   686
Goal "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   687
by (asm_simp_tac 
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   688
    (simpset() addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   689
qed "Card_lt_csucc_iff";
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   690
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   691
Goalw [InfCard_def]
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   692
    "InfCard(K) ==> InfCard(csucc(K))";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   693
by (asm_simp_tac (simpset() addsimps [Card_csucc, Card_is_Ord, 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   694
				      lt_csucc RS leI RSN (2,le_trans)]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   695
qed "InfCard_csucc";
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   696
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   697
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   698
(*** Finite sets ***)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   699
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   700
Goal "n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
6070
032babd0120b ZF: the natural numbers as a datatype
paulson
parents: 6068
diff changeset
   701
by (induct_tac "n" 1);
5529
4a54acae6a15 tidying
paulson
parents: 5488
diff changeset
   702
by (simp_tac (simpset() addsimps eqpoll_0_iff::Fin.intrs) 1);
3736
39ee3d31cfbc Much tidying including step_tac -> clarify_tac or safe_tac; sometimes
paulson
parents: 3016
diff changeset
   703
by (Clarify_tac 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   704
by (subgoal_tac "EX u. u:A" 1);
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   705
by (etac exE 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   706
by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   707
by (assume_tac 2);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   708
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   709
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   710
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   711
by (resolve_tac [Fin.consI] 1);
2925
b0ae2e13db93 Using Blast_tac
paulson
parents: 2493
diff changeset
   712
by (Blast_tac 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   713
by (blast_tac (claset() addIs [subset_consI  RS Fin_mono RS subsetD]) 1); 
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   714
(*Now for the lemma assumed above*)
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   715
by (rewtac eqpoll_def);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   716
by (blast_tac (claset() addIs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   717
val lemma = result();
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   718
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   719
Goalw [Finite_def] "Finite(A) ==> A : Fin(A)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   720
by (blast_tac (claset() addIs [lemma RS spec RS mp]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   721
qed "Finite_into_Fin";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   722
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   723
Goal "A : Fin(U) ==> Finite(A)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   724
by (fast_tac (claset() addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   725
qed "Fin_into_Finite";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   726
5067
62b6288e6005 isatool fixgoal;
wenzelm
parents: 4477
diff changeset
   727
Goal "Finite(A) <-> A : Fin(A)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   728
by (blast_tac (claset() addIs [Finite_into_Fin, Fin_into_Finite]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   729
qed "Finite_Fin_iff";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   730
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   731
Goal "[| Finite(A); Finite(B) |] ==> Finite(A Un B)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   732
by (blast_tac (claset() addSIs [Fin_into_Finite, Fin_UnI] 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   733
                        addSDs [Finite_into_Fin]
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   734
                        addIs  [Un_upper1 RS Fin_mono RS subsetD,
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   735
	 		        Un_upper2 RS Fin_mono RS subsetD]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   736
qed "Finite_Un";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   737
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   738
Goal "[| ALL y:X. Finite(y);  Finite(X) |] ==> Finite(Union(X))";
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   739
by (asm_full_simp_tac (simpset() addsimps [Finite_Fin_iff]) 1);
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   740
by (rtac Fin_UnionI 1);
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   741
by (etac Fin.induct 1);
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   742
by (Simp_tac 1);
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   743
by (blast_tac (claset() addIs [Fin.consI, impOfSubs Fin_mono]) 1);
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   744
qed "Finite_Union";
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   745
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   746
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   747
(** Removing elements from a finite set decreases its cardinality **)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   748
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   749
Goal "A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   750
by (etac Fin_induct 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   751
by (simp_tac (simpset() addsimps [lepoll_0_iff]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   752
by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents: 1622
diff changeset
   753
by (Asm_simp_tac 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   754
by (blast_tac (claset() addSDs [cons_lepoll_consD]) 1);
2925
b0ae2e13db93 Using Blast_tac
paulson
parents: 2493
diff changeset
   755
by (Blast_tac 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   756
qed "Fin_imp_not_cons_lepoll";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   757
5147
825877190618 More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5137
diff changeset
   758
Goal "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)";
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   759
by (rewtac cardinal_def);
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   760
by (rtac Least_equality 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   761
by (fold_tac [cardinal_def]);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   762
by (simp_tac (simpset() addsimps [succ_def]) 1);
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   763
by (blast_tac (claset() addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll] 
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   764
                        addSEs [mem_irrefl]
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   765
                        addSDs [Finite_imp_well_ord]) 1);
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   766
by (blast_tac (claset() addIs [Card_cardinal, Card_is_Ord]) 1);
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   767
by (rtac notI 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   768
by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   769
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   770
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   771
by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   772
by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   773
by (blast_tac (claset() addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll] 
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   774
                    addSDs [Finite_imp_well_ord]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   775
qed "Finite_imp_cardinal_cons";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   776
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   777
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   778
Goal "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|";
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   779
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   780
by (assume_tac 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   781
by (asm_simp_tac (simpset() addsimps [Finite_imp_cardinal_cons,
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   782
                                  Diff_subset RS subset_Finite]) 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3887
diff changeset
   783
by (asm_simp_tac (simpset() addsimps [cons_Diff]) 1);
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   784
qed "Finite_imp_succ_cardinal_Diff";
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   785
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   786
Goal "[| Finite(A);  a:A |] ==> |A-{a}| < |A|";
1622
4b0608ce6150 New theorem Finite_imp_succ_cardinal_Diff
paulson
parents: 1609
diff changeset
   787
by (rtac succ_leE 1);
8127
68c6159440f1 new lemmas for Ntree recursor example; more simprules; more lemmas borrowed
paulson
parents: 7499
diff changeset
   788
by (asm_simp_tac (simpset() addsimps [Finite_imp_succ_cardinal_Diff]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   789
qed "Finite_imp_cardinal_Diff";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   790
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   791
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   792
(** Theorems by Krzysztof Grabczewski, proofs by lcp **)
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   793
3887
04f730731e43 transfer CardinalArith.thy nat_into_Ord;
wenzelm
parents: 3840
diff changeset
   794
val nat_implies_well_ord =
04f730731e43 transfer CardinalArith.thy nat_into_Ord;
wenzelm
parents: 3840
diff changeset
   795
  (transfer CardinalArith.thy nat_into_Ord) RS well_ord_Memrel;
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   796
5137
60205b0de9b9 Huge tidy-up: removal of leading \!\!
paulson
parents: 5116
diff changeset
   797
Goal "[| m:nat; n:nat |] ==> m + n eqpoll m #+ n";
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   798
by (rtac eqpoll_trans 1);
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   799
by (resolve_tac [well_ord_radd RS well_ord_cardinal_eqpoll RS eqpoll_sym] 1);
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4312
diff changeset
   800
by (REPEAT (etac nat_implies_well_ord 1));
4312
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   801
by (asm_simp_tac (simpset() 
63844406913c Tidying, mostly indentation
paulson
parents: 4152
diff changeset
   802
		  addsimps [nat_cadd_eq_add RS sym, cadd_def, eqpoll_refl]) 1);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   803
qed "nat_sum_eqpoll_sum";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   804