(* Title: HOL/Finite.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Finite powerset operator
*)
open Finite;
goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";
br lfp_mono 1;
by (REPEAT (ares_tac basic_monos 1));
qed "Fin_mono";
goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)";
by (fast_tac (set_cs addSIs [lfp_lowerbound]) 1);
qed "Fin_subset_Pow";
(* A : Fin(B) ==> A <= B *)
val FinD = Fin_subset_Pow RS subsetD RS PowD;
(*Discharging ~ x:y entails extra work*)
val major::prems = goal Finite.thy
"[| F:Fin(A); P({}); \
\ !!F x. [| x:A; F:Fin(A); x~:F; P(F) |] ==> P(insert x F) \
\ |] ==> P(F)";
by (rtac (major RS Fin.induct) 1);
by (excluded_middle_tac "a:b" 2);
by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*)
by (REPEAT (ares_tac prems 1));
qed "Fin_induct";
(** Simplification for Fin **)
Addsimps Fin.intrs;
(*The union of two finite sets is finite*)
val major::prems = goal Finite.thy
"[| F: Fin(A); G: Fin(A) |] ==> F Un G : Fin(A)";
by (rtac (major RS Fin_induct) 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps (prems @ [Un_insert_left]))));
qed "Fin_UnI";
(*Every subset of a finite set is finite*)
val [subs,fin] = goal Finite.thy "[| A<=B; B: Fin(M) |] ==> A: Fin(M)";
by (EVERY1 [subgoal_tac "ALL C. C<=B --> C: Fin(M)",
rtac mp, etac spec,
rtac subs]);
by (rtac (fin RS Fin_induct) 1);
by (simp_tac (!simpset addsimps [subset_Un_eq]) 1);
by (safe_tac (set_cs addSDs [subset_insert_iff RS iffD1]));
by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2);
by (ALLGOALS Asm_simp_tac);
qed "Fin_subset";
(*The image of a finite set is finite*)
val major::_ = goal Finite.thy
"F: Fin(A) ==> h``F : Fin(h``A)";
by (rtac (major RS Fin_induct) 1);
by (Simp_tac 1);
by (asm_simp_tac
(!simpset addsimps [image_eqI RS Fin.insertI, image_insert]) 1);
qed "Fin_imageI";
val major::prems = goal Finite.thy
"[| c: Fin(A); b: Fin(A); \
\ P(b); \
\ !!(x::'a) 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_insert RS ssubst) 2);
by (ALLGOALS (asm_simp_tac
(!simpset addsimps (prems@[Diff_subset RS Fin_subset]))));
qed "Fin_empty_induct_lemma";
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({})";
by (rtac (Diff_cancel RS subst) 1);
by (rtac (Fin_empty_induct_lemma RS mp) 1);
by (REPEAT (ares_tac (subset_refl::prems) 1));
qed "Fin_empty_induct";