src/ZF/Finite.ML
 author lcp Fri, 12 Aug 1994 12:51:34 +0200 changeset 516 1957113f0d7d child 534 cd8bec47e175 permissions -rw-r--r--
installation of new inductive/datatype sections
```
(*  Title: 	ZF/Finite.ML
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

Finite powerset operator

prove X:Fin(A) ==> |X| < nat

prove:  b: Fin(A) ==> inj(b,b)<=surj(b,b)
*)

open Finite;

goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";
by (rtac lfp_mono 1);
by (REPEAT (rtac Fin.bnd_mono 1));
by (REPEAT (ares_tac (Pow_mono::basic_monos) 1));
val Fin_mono = result();

(* A : Fin(B) ==> A <= B *)
val FinD = Fin.dom_subset RS subsetD RS PowD;

(** Induction on finite sets **)

(*Discharging x~:y entails extra work*)
val major::prems = goal Finite.thy
"[| b: Fin(A);  \
\       P(0);        \
\       !!x y. [| x: A;  y: Fin(A);  x~:y;  P(y) |] ==> P(cons(x,y)) \
\    |] ==> P(b)";
by (rtac (major RS Fin.induct) 1);
by (excluded_middle_tac "a:b" 2);
by (etac (cons_absorb RS ssubst) 3 THEN assume_tac 3);	    (*backtracking!*)
by (REPEAT (ares_tac prems 1));
val Fin_induct = result();

(** Simplification for Fin **)
val Fin_ss = arith_ss addsimps Fin.intrs;

(*The union of two finite sets is finite.*)
val major::prems = goal Finite.thy
"[| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)";
by (rtac (major RS Fin_induct) 1);
by (ALLGOALS (asm_simp_tac (Fin_ss addsimps (prems@[Un_0, Un_cons]))));
val Fin_UnI = result();

(*The union of a set of finite sets is finite.*)
val [major] = goal Finite.thy "C : Fin(Fin(A)) ==> Union(C) : Fin(A)";
by (rtac (major RS Fin_induct) 1);
by (ALLGOALS (asm_simp_tac (Fin_ss addsimps [Union_0, Union_cons, Fin_UnI])));
val Fin_UnionI = result();

(*Every subset of a finite set is finite.*)
goal Finite.thy "!!b A. b: Fin(A) ==> ALL z. z<=b --> z: Fin(A)";
by (etac Fin_induct 1);
by (simp_tac (Fin_ss addsimps [subset_empty_iff]) 1);
by (safe_tac (ZF_cs addSDs [subset_cons_iff RS iffD1]));
by (eres_inst_tac [("b","z")] (cons_Diff RS subst) 2);
by (ALLGOALS (asm_simp_tac Fin_ss));
val Fin_subset_lemma = result();

goal Finite.thy "!!c b A. [| c<=b;  b: Fin(A) |] ==> c: Fin(A)";
by (REPEAT (ares_tac [Fin_subset_lemma RS spec RS mp] 1));
val Fin_subset = result();

val major::prems = goal Finite.thy
"[| c: Fin(A);  b: Fin(A);  				\
\       P(b);       						\
\       !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
\    |] ==> c<=b --> P(b-c)";
by (rtac (major RS Fin_induct) 1);
by (rtac (Diff_cons RS ssubst) 2);
by (ALLGOALS (asm_simp_tac (Fin_ss addsimps (prems@[Diff_0, cons_subset_iff,
Diff_subset RS Fin_subset]))));
val Fin_0_induct_lemma = result();

val prems = goal Finite.thy
"[| b: Fin(A);  						\
\       P(b);        						\
\       !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
\    |] ==> P(0)";
by (rtac (Diff_cancel RS subst) 1);
by (rtac (Fin_0_induct_lemma RS mp) 1);
by (REPEAT (ares_tac (subset_refl::prems) 1));
val Fin_0_induct = result();

(*Functions from a finite ordinal*)
val prems = goal Finite.thy "n: nat ==> n->A <= Fin(nat*A)";
by (nat_ind_tac "n" prems 1);
by (simp_tac (ZF_ss addsimps [Pi_empty1, Fin.emptyI, subset_iff, cons_iff]) 1);
by (asm_simp_tac (ZF_ss addsimps [succ_def, mem_not_refl RS cons_fun_eq]) 1);
by (fast_tac (ZF_cs addSIs [Fin.consI]) 1);
val nat_fun_subset_Fin = result();
```