Yet more removal of "goal" commands, especially "goal ZF.thy", so ZF.thy
contains fewer theorems than before
(* Title: ZF/CardinalArith.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Cardinal arithmetic -- WITHOUT the Axiom of Choice
Note: Could omit proving the algebraic laws for cardinal addition and
multiplication. On finite cardinals these operations coincide with
addition and multiplication of natural numbers; on infinite cardinals they
coincide with union (maximum). Either way we get most laws for free.
*)
open CardinalArith;
(*** Cardinal addition ***)
(** Cardinal addition is commutative **)
Goalw [eqpoll_def] "A+B eqpoll B+A";
by (rtac exI 1);
by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")]
lam_bijective 1);
by (safe_tac (claset() addSEs [sumE]));
by (ALLGOALS (Asm_simp_tac));
qed "sum_commute_eqpoll";
Goalw [cadd_def] "i |+| j = j |+| i";
by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
qed "cadd_commute";
(** Cardinal addition is associative **)
Goalw [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
by (rtac exI 1);
by (rtac sum_assoc_bij 1);
qed "sum_assoc_eqpoll";
(*Unconditional version requires AC*)
Goalw [cadd_def]
"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
\ (i |+| j) |+| k = i |+| (j |+| k)";
by (rtac cardinal_cong 1);
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
eqpoll_trans) 1);
by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
eqpoll_sym) 2);
by (REPEAT (ares_tac [well_ord_radd] 1));
qed "well_ord_cadd_assoc";
(** 0 is the identity for addition **)
Goalw [eqpoll_def] "0+A eqpoll A";
by (rtac exI 1);
by (rtac bij_0_sum 1);
qed "sum_0_eqpoll";
Goalw [cadd_def] "Card(K) ==> 0 |+| K = K";
by (asm_simp_tac (simpset() addsimps [sum_0_eqpoll RS cardinal_cong,
Card_cardinal_eq]) 1);
qed "cadd_0";
(** Addition by another cardinal **)
Goalw [lepoll_def, inj_def] "A lepoll A+B";
by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1);
by (asm_simp_tac (simpset() addsimps [lam_type]) 1);
qed "sum_lepoll_self";
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
Goalw [cadd_def]
"[| Card(K); Ord(L) |] ==> K le (K |+| L)";
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
by (rtac sum_lepoll_self 3);
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1));
qed "cadd_le_self";
(** Monotonicity of addition **)
Goalw [lepoll_def]
"[| A lepoll C; B lepoll D |] ==> A + B lepoll C + D";
by (REPEAT (etac exE 1));
by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")]
exI 1);
by (res_inst_tac
[("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")]
lam_injective 1);
by (typechk_tac ([inj_is_fun, case_type, InlI, InrI] @ ZF_typechecks));
by (etac sumE 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [left_inverse])));
qed "sum_lepoll_mono";
Goalw [cadd_def]
"[| K' le K; L' le L |] ==> (K' |+| L') le (K |+| L)";
by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
by (rtac well_ord_lepoll_imp_Card_le 1);
by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2));
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
qed "cadd_le_mono";
(** Addition of finite cardinals is "ordinary" addition **)
Goalw [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
by (rtac exI 1);
by (res_inst_tac [("c", "%z. if(z=Inl(A),A+B,z)"),
("d", "%z. if(z=A+B,Inl(A),z)")]
lam_bijective 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [succI2, mem_imp_not_eq]
setloop eresolve_tac [sumE,succE])));
qed "sum_succ_eqpoll";
(*Pulling the succ(...) outside the |...| requires m, n: nat *)
(*Unconditional version requires AC*)
Goalw [cadd_def]
"[| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
qed "cadd_succ_lemma";
val [mnat,nnat] = goal CardinalArith.thy
"[| m: nat; n: nat |] ==> m |+| n = m#+n";
by (cut_facts_tac [nnat] 1);
by (nat_ind_tac "m" [mnat] 1);
by (asm_simp_tac (simpset() addsimps [nat_into_Card RS cadd_0]) 1);
by (asm_simp_tac (simpset() addsimps [nat_into_Ord, cadd_succ_lemma,
nat_into_Card RS Card_cardinal_eq]) 1);
qed "nat_cadd_eq_add";
(*** Cardinal multiplication ***)
(** Cardinal multiplication is commutative **)
(*Easier to prove the two directions separately*)
Goalw [eqpoll_def] "A*B eqpoll B*A";
by (rtac exI 1);
by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")]
lam_bijective 1);
by Safe_tac;
by (ALLGOALS (Asm_simp_tac));
qed "prod_commute_eqpoll";
Goalw [cmult_def] "i |*| j = j |*| i";
by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
qed "cmult_commute";
(** Cardinal multiplication is associative **)
Goalw [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
by (rtac exI 1);
by (rtac prod_assoc_bij 1);
qed "prod_assoc_eqpoll";
(*Unconditional version requires AC*)
Goalw [cmult_def]
"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
\ (i |*| j) |*| k = i |*| (j |*| k)";
by (rtac cardinal_cong 1);
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
eqpoll_trans) 1);
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
eqpoll_sym) 2);
by (REPEAT (ares_tac [well_ord_rmult] 1));
qed "well_ord_cmult_assoc";
(** Cardinal multiplication distributes over addition **)
Goalw [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
by (rtac exI 1);
by (rtac sum_prod_distrib_bij 1);
qed "sum_prod_distrib_eqpoll";
Goalw [cadd_def, cmult_def]
"[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
\ (i |+| j) |*| k = (i |*| k) |+| (j |*| k)";
by (rtac cardinal_cong 1);
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
eqpoll_trans) 1);
by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2);
by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS
sum_eqpoll_cong RS eqpoll_sym) 2);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1));
qed "well_ord_cadd_cmult_distrib";
(** Multiplication by 0 yields 0 **)
Goalw [eqpoll_def] "0*A eqpoll 0";
by (rtac exI 1);
by (rtac lam_bijective 1);
by Safe_tac;
qed "prod_0_eqpoll";
Goalw [cmult_def] "0 |*| i = 0";
by (asm_simp_tac (simpset() addsimps [prod_0_eqpoll RS cardinal_cong,
Card_0 RS Card_cardinal_eq]) 1);
qed "cmult_0";
(** 1 is the identity for multiplication **)
Goalw [eqpoll_def] "{x}*A eqpoll A";
by (rtac exI 1);
by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
qed "prod_singleton_eqpoll";
Goalw [cmult_def, succ_def] "Card(K) ==> 1 |*| K = K";
by (asm_simp_tac (simpset() addsimps [prod_singleton_eqpoll RS cardinal_cong,
Card_cardinal_eq]) 1);
qed "cmult_1";
(*** Some inequalities for multiplication ***)
Goalw [lepoll_def, inj_def] "A lepoll A*A";
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
by (simp_tac (simpset() addsimps [lam_type]) 1);
qed "prod_square_lepoll";
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
Goalw [cmult_def] "Card(K) ==> K le K |*| K";
by (rtac le_trans 1);
by (rtac well_ord_lepoll_imp_Card_le 2);
by (rtac prod_square_lepoll 3);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
by (asm_simp_tac (simpset()
addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
qed "cmult_square_le";
(** Multiplication by a non-zero cardinal **)
Goalw [lepoll_def, inj_def] "b: B ==> A lepoll A*B";
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
by (asm_simp_tac (simpset() addsimps [lam_type]) 1);
qed "prod_lepoll_self";
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
Goalw [cmult_def]
"[| Card(K); Ord(L); 0<L |] ==> K le (K |*| L)";
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
by (rtac prod_lepoll_self 3);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1));
qed "cmult_le_self";
(** Monotonicity of multiplication **)
Goalw [lepoll_def]
"[| A lepoll C; B lepoll D |] ==> A * B lepoll C * D";
by (REPEAT (etac exE 1));
by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")]
lam_injective 1);
by (typechk_tac (inj_is_fun::ZF_typechecks));
by (etac SigmaE 1);
by (asm_simp_tac (simpset() addsimps [left_inverse]) 1);
qed "prod_lepoll_mono";
Goalw [cmult_def]
"[| K' le K; L' le L |] ==> (K' |*| L') le (K |*| L)";
by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
by (rtac well_ord_lepoll_imp_Card_le 1);
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
qed "cmult_le_mono";
(*** Multiplication of finite cardinals is "ordinary" multiplication ***)
Goalw [eqpoll_def] "succ(A)*B eqpoll B + A*B";
by (rtac exI 1);
by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"),
("d", "case(%y. <A,y>, %z. z)")]
lam_bijective 1);
by (safe_tac (claset() addSEs [sumE]));
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [succI2, if_type, mem_imp_not_eq])));
qed "prod_succ_eqpoll";
(*Unconditional version requires AC*)
Goalw [cmult_def, cadd_def]
"[| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
by (rtac (cardinal_cong RS sym) 1);
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
qed "cmult_succ_lemma";
val [mnat,nnat] = goal CardinalArith.thy
"[| m: nat; n: nat |] ==> m |*| n = m#*n";
by (cut_facts_tac [nnat] 1);
by (nat_ind_tac "m" [mnat] 1);
by (asm_simp_tac (simpset() addsimps [cmult_0]) 1);
by (asm_simp_tac (simpset() addsimps [nat_into_Ord, cmult_succ_lemma,
nat_cadd_eq_add]) 1);
qed "nat_cmult_eq_mult";
Goal "Card(n) ==> 2 |*| n = n |+| n";
by (asm_simp_tac
(simpset() addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma,
Card_is_Ord, cadd_0,
read_instantiate [("j","0")] cadd_commute]) 1);
qed "cmult_2";
val sum_lepoll_prod = [sum_eq_2_times RS equalityD1 RS subset_imp_lepoll,
asm_rl, lepoll_refl] MRS (prod_lepoll_mono RSN (2, lepoll_trans))
|> standard;
Goal "[| A lepoll B; 2 lepoll A |] ==> A+B lepoll A*B";
by (REPEAT (ares_tac [[sum_lepoll_mono, sum_lepoll_prod]
MRS lepoll_trans, lepoll_refl] 1));
qed "lepoll_imp_sum_lepoll_prod";
(*** Infinite Cardinals are Limit Ordinals ***)
(*This proof is modelled upon one assuming nat<=A, with injection
lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z)) and inverse
%y. if(y:nat, nat_case(u,%z.z,y), y). If f: inj(nat,A) then
range(f) behaves like the natural numbers.*)
Goalw [lepoll_def]
"nat lepoll A ==> cons(u,A) lepoll A";
by (etac exE 1);
by (res_inst_tac [("x",
"lam z:cons(u,A). if(z=u, f`0, \
\ if(z: range(f), f`succ(converse(f)`z), z))")] exI 1);
by (res_inst_tac [("d", "%y. if(y: range(f), \
\ nat_case(u, %z. f`z, converse(f)`y), y)")]
lam_injective 1);
by (fast_tac (claset() addSIs [if_type, apply_type]
addIs [inj_is_fun, inj_converse_fun]) 1);
by (asm_simp_tac
(simpset() addsimps [inj_is_fun RS apply_rangeI,
inj_converse_fun RS apply_rangeI,
inj_converse_fun RS apply_funtype,
left_inverse, right_inverse, nat_0I, nat_succI,
nat_case_0, nat_case_succ]) 1);
qed "nat_cons_lepoll";
Goal "nat lepoll A ==> cons(u,A) eqpoll A";
by (etac (nat_cons_lepoll RS eqpollI) 1);
by (rtac (subset_consI RS subset_imp_lepoll) 1);
qed "nat_cons_eqpoll";
(*Specialized version required below*)
Goalw [succ_def] "nat <= A ==> succ(A) eqpoll A";
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
qed "nat_succ_eqpoll";
Goalw [InfCard_def] "InfCard(nat)";
by (blast_tac (claset() addIs [Card_nat, le_refl, Card_is_Ord]) 1);
qed "InfCard_nat";
Goalw [InfCard_def] "InfCard(K) ==> Card(K)";
by (etac conjunct1 1);
qed "InfCard_is_Card";
Goalw [InfCard_def]
"[| InfCard(K); Card(L) |] ==> InfCard(K Un L)";
by (asm_simp_tac (simpset() addsimps [Card_Un, Un_upper1_le RSN (2,le_trans),
Card_is_Ord]) 1);
qed "InfCard_Un";
(*Kunen's Lemma 10.11*)
Goalw [InfCard_def] "InfCard(K) ==> Limit(K)";
by (etac conjE 1);
by (forward_tac [Card_is_Ord] 1);
by (rtac (ltI RS non_succ_LimitI) 1);
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
by (safe_tac (claset() addSDs [Limit_nat RS Limit_le_succD]));
by (rewtac Card_def);
by (dtac trans 1);
by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
by (etac (Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1);
by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));
qed "InfCard_is_Limit";
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
(*A general fact about ordermap*)
Goalw [eqpoll_def]
"[| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
by (rtac exI 1);
by (asm_simp_tac (simpset() addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
by (rtac pred_subset 1);
qed "ordermap_eqpoll_pred";
(** Establishing the well-ordering **)
Goalw [inj_def]
"Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
by (fast_tac (claset() addss (simpset())
addIs [lam_type, Un_least_lt RS ltD, ltI]) 1);
qed "csquare_lam_inj";
Goalw [csquare_rel_def]
"Ord(K) ==> well_ord(K*K, csquare_rel(K))";
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
qed "well_ord_csquare";
(** Characterising initial segments of the well-ordering **)
Goalw [csquare_rel_def]
"[| x<K; y<K; z<K |] ==> \
\ <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
by (REPEAT (etac ltE 1));
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
by (safe_tac (claset() addSEs [mem_irrefl]
addSIs [Un_upper1_le, Un_upper2_le]));
by (ALLGOALS (asm_simp_tac (simpset() addsimps [lt_def, succI2, Ord_succ])));
qed_spec_mp "csquareD";
Goalw [pred_def]
"z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
by (safe_tac (claset_of ZF.thy addSEs [SigmaE])); (*avoids using succCI,...*)
by (rtac (csquareD RS conjE) 1);
by (rewtac lt_def);
by (assume_tac 4);
by (ALLGOALS Blast_tac);
qed "pred_csquare_subset";
Goalw [csquare_rel_def]
"[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K)";
by (subgoals_tac ["x<K", "y<K"] 1);
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
by (REPEAT (etac ltE 1));
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
qed "csquare_ltI";
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *)
Goalw [csquare_rel_def]
"[| x le z; y le z; z<K |] ==> \
\ <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
by (subgoals_tac ["x<K", "y<K"] 1);
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
by (REPEAT (etac ltE 1));
by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
by (REPEAT_FIRST (etac succE));
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [subset_Un_iff RS iff_sym,
subset_Un_iff2 RS iff_sym, OrdmemD])));
qed "csquare_or_eqI";
(** The cardinality of initial segments **)
Goal "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \
\ ordermap(K*K, csquare_rel(K)) ` <x,y> < \
\ ordermap(K*K, csquare_rel(K)) ` <z,z>";
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
by (etac (Limit_is_Ord RS well_ord_csquare) 2);
by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
by (etac well_ord_is_wf 4);
by (ALLGOALS
(blast_tac (claset() addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap]
addSEs [ltE])));
qed "ordermap_z_lt";
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
Goalw [cmult_def]
"[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \
\ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|";
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
by (subgoals_tac ["z<K"] 1);
by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
by (REPEAT_SOME assume_tac);
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
by (etac (Limit_is_Ord RS well_ord_csquare) 1);
by (blast_tac (claset() addIs [ltD]) 1);
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
assume_tac 1);
by (REPEAT_FIRST (etac ltE));
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
qed "ordermap_csquare_le";
(*Kunen: "... so the order type <= K" *)
Goal "[| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] ==> \
\ ordertype(K*K, csquare_rel(K)) le K";
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
by (rtac all_lt_imp_le 1);
by (assume_tac 1);
by (etac (well_ord_csquare RS Ord_ordertype) 1);
by (rtac Card_lt_imp_lt 1);
by (etac InfCard_is_Card 3);
by (etac ltE 2 THEN assume_tac 2);
by (asm_full_simp_tac (simpset() addsimps [ordertype_unfold]) 1);
by (safe_tac (claset() addSEs [ltE]));
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1 THEN
REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN
REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
(*the finite case: xb Un y < nat *)
by (res_inst_tac [("j", "nat")] lt_trans2 1);
by (asm_full_simp_tac (simpset() addsimps [InfCard_def]) 2);
by (asm_full_simp_tac
(simpset() addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
(*case nat le (xb Un y) *)
by (asm_full_simp_tac
(simpset() addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt,
Ord_Un, ltI, nat_le_cardinal,
Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
qed "ordertype_csquare_le";
(*Main result: Kunen's Theorem 10.12*)
Goal "InfCard(K) ==> K |*| K = K";
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
by (etac rev_mp 1);
by (trans_ind_tac "K" [] 1);
by (rtac impI 1);
by (rtac le_anti_sym 1);
by (etac (InfCard_is_Card RS cmult_square_le) 2);
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
by (assume_tac 2);
by (assume_tac 2);
by (asm_simp_tac
(simpset() addsimps [cmult_def, Ord_cardinal_le,
well_ord_csquare RS ordermap_bij RS
bij_imp_eqpoll RS cardinal_cong,
well_ord_csquare RS Ord_ordertype]) 1);
qed "InfCard_csquare_eq";
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
Goal "[| well_ord(A,r); InfCard(|A|) |] ==> A*A eqpoll A";
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
by (rtac well_ord_cardinal_eqE 1);
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
by (asm_simp_tac (simpset()
addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
qed "well_ord_InfCard_square_eq";
(** Toward's Kunen's Corollary 10.13 (1) **)
Goal "[| InfCard(K); L le K; 0<L |] ==> K |*| L = K";
by (rtac le_anti_sym 1);
by (etac ltE 2 THEN
REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
by (asm_simp_tac (simpset() addsimps [InfCard_csquare_eq]) 1);
qed "InfCard_le_cmult_eq";
(*Corollary 10.13 (1), for cardinal multiplication*)
Goal "[| InfCard(K); InfCard(L) |] ==> K |*| L = K Un L";
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
by (resolve_tac [cmult_commute RS ssubst] 1);
by (resolve_tac [Un_commute RS ssubst] 1);
by (ALLGOALS
(asm_simp_tac
(simpset() addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
subset_Un_iff2 RS iffD1, le_imp_subset])));
qed "InfCard_cmult_eq";
(*This proof appear to be the simplest!*)
Goal "InfCard(K) ==> K |+| K = K";
by (asm_simp_tac
(simpset() addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
by (rtac InfCard_le_cmult_eq 1);
by (typechk_tac [Ord_0, le_refl, leI]);
by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]);
qed "InfCard_cdouble_eq";
(*Corollary 10.13 (1), for cardinal addition*)
Goal "[| InfCard(K); L le K |] ==> K |+| L = K";
by (rtac le_anti_sym 1);
by (etac ltE 2 THEN
REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
by (asm_simp_tac (simpset() addsimps [InfCard_cdouble_eq]) 1);
qed "InfCard_le_cadd_eq";
Goal "[| InfCard(K); InfCard(L) |] ==> K |+| L = K Un L";
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
by (resolve_tac [cadd_commute RS ssubst] 1);
by (resolve_tac [Un_commute RS ssubst] 1);
by (ALLGOALS
(asm_simp_tac
(simpset() addsimps [InfCard_le_cadd_eq,
subset_Un_iff2 RS iffD1, le_imp_subset])));
qed "InfCard_cadd_eq";
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
of all n-tuples of elements of K. A better version for the Isabelle theory
might be InfCard(K) ==> |list(K)| = K.
*)
(*** For every cardinal number there exists a greater one
[Kunen's Theorem 10.16, which would be trivial using AC] ***)
Goalw [jump_cardinal_def] "Ord(jump_cardinal(K))";
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
by (blast_tac (claset() addSIs [Ord_ordertype]) 2);
by (rewtac Transset_def);
by (safe_tac subset_cs);
by (asm_full_simp_tac (simpset() addsimps [ordertype_pred_unfold]) 1);
by Safe_tac;
by (rtac UN_I 1);
by (rtac ReplaceI 2);
by (ALLGOALS (blast_tac (claset() addIs [well_ord_subset] addSEs [predE])));
qed "Ord_jump_cardinal";
(*Allows selective unfolding. Less work than deriving intro/elim rules*)
Goalw [jump_cardinal_def]
"i : jump_cardinal(K) <-> \
\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*)
qed "jump_cardinal_iff";
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
Goal "Ord(K) ==> K < jump_cardinal(K)";
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
by (rtac subset_refl 2);
by (asm_simp_tac (simpset() addsimps [Memrel_def, subset_iff]) 1);
by (asm_simp_tac (simpset() addsimps [ordertype_Memrel]) 1);
qed "K_lt_jump_cardinal";
(*The proof by contradiction: the bijection f yields a wellordering of X
whose ordertype is jump_cardinal(K). *)
Goal "[| well_ord(X,r); r <= K * K; X <= K; \
\ f : bij(ordertype(X,r), jump_cardinal(K)) \
\ |] ==> jump_cardinal(K) : jump_cardinal(K)";
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
by (REPEAT_FIRST (resolve_tac [exI, conjI]));
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
by (REPEAT (assume_tac 1));
by (etac (bij_is_inj RS well_ord_rvimage) 1);
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
by (asm_simp_tac
(simpset() addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage),
ordertype_Memrel, Ord_jump_cardinal]) 1);
qed "Card_jump_cardinal_lemma";
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
Goal "Card(jump_cardinal(K))";
by (rtac (Ord_jump_cardinal RS CardI) 1);
by (rewtac eqpoll_def);
by (safe_tac (claset() addSDs [ltD, jump_cardinal_iff RS iffD1]));
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
qed "Card_jump_cardinal";
(*** Basic properties of successor cardinals ***)
Goalw [csucc_def]
"Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
by (rtac LeastI 1);
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
Ord_jump_cardinal] 1));
qed "csucc_basic";
bind_thm ("Card_csucc", csucc_basic RS conjunct1);
bind_thm ("lt_csucc", csucc_basic RS conjunct2);
Goal "Ord(K) ==> 0 < csucc(K)";
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
by (REPEAT (assume_tac 1));
qed "Ord_0_lt_csucc";
Goalw [csucc_def]
"[| Card(L); K<L |] ==> csucc(K) le L";
by (rtac Least_le 1);
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
qed "csucc_le";
Goal "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
by (rtac iffI 1);
by (rtac Card_lt_imp_lt 2);
by (etac lt_trans1 2);
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
by (resolve_tac [notI RS not_lt_imp_le] 1);
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
by (assume_tac 1);
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
by (REPEAT (ares_tac [Ord_cardinal] 1
ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
qed "lt_csucc_iff";
Goal "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
by (asm_simp_tac
(simpset() addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
qed "Card_lt_csucc_iff";
Goalw [InfCard_def]
"InfCard(K) ==> InfCard(csucc(K))";
by (asm_simp_tac (simpset() addsimps [Card_csucc, Card_is_Ord,
lt_csucc RS leI RSN (2,le_trans)]) 1);
qed "InfCard_csucc";
(*** Finite sets ***)
Goal "n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
by (etac nat_induct 1);
by (simp_tac (simpset() addsimps (eqpoll_0_iff::Fin.intrs)) 1);
by (Clarify_tac 1);
by (subgoal_tac "EX u. u:A" 1);
by (etac exE 1);
by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
by (assume_tac 2);
by (assume_tac 1);
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
by (assume_tac 1);
by (resolve_tac [Fin.consI] 1);
by (Blast_tac 1);
by (blast_tac (claset() addIs [subset_consI RS Fin_mono RS subsetD]) 1);
(*Now for the lemma assumed above*)
by (rewtac eqpoll_def);
by (blast_tac (claset() addIs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
val lemma = result();
Goalw [Finite_def] "Finite(A) ==> A : Fin(A)";
by (blast_tac (claset() addIs [lemma RS spec RS mp]) 1);
qed "Finite_into_Fin";
Goal "A : Fin(U) ==> Finite(A)";
by (fast_tac (claset() addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
qed "Fin_into_Finite";
Goal "Finite(A) <-> A : Fin(A)";
by (blast_tac (claset() addIs [Finite_into_Fin, Fin_into_Finite]) 1);
qed "Finite_Fin_iff";
Goal "[| Finite(A); Finite(B) |] ==> Finite(A Un B)";
by (blast_tac (claset() addSIs [Fin_into_Finite, Fin_UnI]
addSDs [Finite_into_Fin]
addIs [Un_upper1 RS Fin_mono RS subsetD,
Un_upper2 RS Fin_mono RS subsetD]) 1);
qed "Finite_Un";
(** Removing elements from a finite set decreases its cardinality **)
Goal "A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
by (etac Fin_induct 1);
by (simp_tac (simpset() addsimps [lepoll_0_iff]) 1);
by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
by (Asm_simp_tac 1);
by (blast_tac (claset() addSDs [cons_lepoll_consD]) 1);
by (Blast_tac 1);
qed "Fin_imp_not_cons_lepoll";
Goal "[| Finite(A); a~:A |] ==> |cons(a,A)| = succ(|A|)";
by (rewtac cardinal_def);
by (rtac Least_equality 1);
by (fold_tac [cardinal_def]);
by (simp_tac (simpset() addsimps [succ_def]) 1);
by (blast_tac (claset() addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll]
addSEs [mem_irrefl]
addSDs [Finite_imp_well_ord]) 1);
by (blast_tac (claset() addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1);
by (rtac notI 1);
by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
by (assume_tac 1);
by (assume_tac 1);
by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
by (blast_tac (claset() addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll]
addSDs [Finite_imp_well_ord]) 1);
qed "Finite_imp_cardinal_cons";
Goal "[| Finite(A); a:A |] ==> succ(|A-{a}|) = |A|";
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
by (assume_tac 1);
by (asm_simp_tac (simpset() addsimps [Finite_imp_cardinal_cons,
Diff_subset RS subset_Finite]) 1);
by (asm_simp_tac (simpset() addsimps [cons_Diff]) 1);
qed "Finite_imp_succ_cardinal_Diff";
Goal "[| Finite(A); a:A |] ==> |A-{a}| < |A|";
by (rtac succ_leE 1);
by (asm_simp_tac (simpset() addsimps [Finite_imp_succ_cardinal_Diff,
Ord_cardinal RS le_refl]) 1);
qed "Finite_imp_cardinal_Diff";
(** Theorems by Krzysztof Grabczewski, proofs by lcp **)
val nat_implies_well_ord =
(transfer CardinalArith.thy nat_into_Ord) RS well_ord_Memrel;
Goal "[| m:nat; n:nat |] ==> m + n eqpoll m #+ n";
by (rtac eqpoll_trans 1);
by (resolve_tac [well_ord_radd RS well_ord_cardinal_eqpoll RS eqpoll_sym] 1);
by (REPEAT (etac nat_implies_well_ord 1));
by (asm_simp_tac (simpset()
addsimps [nat_cadd_eq_add RS sym, cadd_def, eqpoll_refl]) 1);
qed "nat_sum_eqpoll_sum";