(* Title: HOL/ex/qsort.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
Two verifications of Quicksort
*)
Addsimps ([Qsort.qsort_Nil,Qsort.qsort_Cons]@conj_comms);
goal Qsort.thy "!x. mset (qsort le xs) x = mset xs x";
by(res_inst_tac[("xs","xs"),("p","le")]Qsort.qsort_ind 1);
by(ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
result();
goal Qsort.thy "(Alls x:[x:xs.P(x)].Q(x)) = (Alls x:xs. P(x)-->Q(x))";
by(list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
Addsimps [result()];
goal Qsort.thy
"((Alls x:xs.P(x)) & (Alls x:xs.Q(x))) = (Alls x:xs. P(x)&Q(x))";
by(list.induct_tac "xs" 1);
by(ALLGOALS Asm_simp_tac);
Addsimps [result()];
goal HOL.thy "((~P --> Q) & (P --> Q)) = Q";
by(fast_tac HOL_cs 1);
val lemma = result();
goal Qsort.thy "(Alls x:qsort le xs.P(x)) = (Alls x:xs.P(x))";
by(res_inst_tac[("xs","xs"),("p","le")]Qsort.qsort_ind 1);
by(ALLGOALS(asm_simp_tac (!simpset addsimps [lemma])));
Addsimps [result()];
goal Qsort.thy
"sorted le (xs@ys) = (sorted le xs & sorted le ys & \
\ (Alls x:xs. Alls y:ys. le x y))";
by(list.induct_tac "xs" 1);
by(ALLGOALS Asm_simp_tac);
Addsimps [result()];
goal Qsort.thy
"!!le. [| total(le); transf(le) |] ==> sorted le (qsort le xs)";
by(res_inst_tac[("xs","xs"),("p","le")]Qsort.qsort_ind 1);
by(ALLGOALS(asm_full_simp_tac (!simpset addsimps [list_all_mem_conv]) ));
by(rewrite_goals_tac [Sorting.total_def,Sorting.transf_def]);
by(fast_tac HOL_cs 1);
result();
(* A verification based on predicate calculus rather than combinators *)
val sorted_Cons =
rewrite_rule [list_all_mem_conv RS eq_reflection] Sorting.sorted_Cons;
Addsimps [sorted_Cons];
goal Qsort.thy "x mem qsort le xs = x mem xs";
by(res_inst_tac[("xs","xs"),("p","le")]Qsort.qsort_ind 1);
by(ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by(fast_tac HOL_cs 1);
Addsimps [result()];
goal Qsort.thy
"sorted le (xs@ys) = (sorted le xs & sorted le ys & \
\ (!x. x mem xs --> (!y. y mem ys --> le x y)))";
by(list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by(fast_tac HOL_cs 1);
Addsimps [result()];
goal Qsort.thy
"!!xs. [| total(le); transf(le) |] ==> sorted le (qsort le xs)";
by(res_inst_tac[("xs","xs"),("p","le")]Qsort.qsort_ind 1);
by(Simp_tac 1);
by(asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
by(rewrite_goals_tac [Sorting.total_def,Sorting.transf_def]);
by(fast_tac HOL_cs 1);
result();