19 by (Simp_tac 1); |
19 by (Simp_tac 1); |
20 by (rtac impI 1); |
20 by (rtac impI 1); |
21 by (rtac classical 1); |
21 by (rtac classical 1); |
22 by (dtac not_leE 1); |
22 by (dtac not_leE 1); |
23 by (subgoal_tac "f(na) <= f(f(na))" 1); |
23 by (subgoal_tac "f(na) <= f(f(na))" 1); |
24 by (best_tac (HOL_cs addIs [lessD,Puzzle.f_ax,le_less_trans,le_trans]) 1); |
24 by (best_tac (!claset addIs [lessD,Puzzle.f_ax,le_less_trans,le_trans]) 1); |
25 by (fast_tac (HOL_cs addIs [Puzzle.f_ax]) 1); |
25 by (fast_tac (!claset addIs [Puzzle.f_ax]) 1); |
26 val lemma = result() RS spec RS mp; |
26 val lemma = result() RS spec RS mp; |
27 |
27 |
28 goal Puzzle.thy "n <= f(n)"; |
28 goal Puzzle.thy "n <= f(n)"; |
29 by (fast_tac (HOL_cs addIs [lemma]) 1); |
29 by (fast_tac (!claset addIs [lemma]) 1); |
30 qed "lemma1"; |
30 qed "lemma1"; |
31 |
31 |
32 goal Puzzle.thy "f(n) < f(Suc(n))"; |
32 goal Puzzle.thy "f(n) < f(Suc(n))"; |
33 by (fast_tac (HOL_cs addIs [Puzzle.f_ax,le_less_trans,lemma1]) 1); |
33 by (fast_tac (!claset addIs [Puzzle.f_ax,le_less_trans,lemma1]) 1); |
34 qed "lemma2"; |
34 qed "lemma2"; |
35 |
35 |
36 val prems = goal Puzzle.thy "(!!n.f(n) <= f(Suc(n))) ==> m<n --> f(m) <= f(n)"; |
36 val prems = goal Puzzle.thy "(!!n.f(n) <= f(Suc(n))) ==> m<n --> f(m) <= f(n)"; |
37 by (res_inst_tac[("n","n")]nat_induct 1); |
37 by (res_inst_tac[("n","n")]nat_induct 1); |
38 by (Simp_tac 1); |
38 by (Simp_tac 1); |
39 by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
39 by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
40 by (fast_tac (HOL_cs addIs (le_trans::prems)) 1); |
40 by (fast_tac (!claset addIs (le_trans::prems)) 1); |
41 bind_thm("mono_lemma1", result() RS mp); |
41 bind_thm("mono_lemma1", result() RS mp); |
42 |
42 |
43 val [p1,p2] = goal Puzzle.thy |
43 val [p1,p2] = goal Puzzle.thy |
44 "[| !! n. f(n)<=f(Suc(n)); m<=n |] ==> f(m) <= f(n)"; |
44 "[| !! n. f(n)<=f(Suc(n)); m<=n |] ==> f(m) <= f(n)"; |
45 by (rtac (p2 RS le_imp_less_or_eq RS disjE) 1); |
45 by (rtac (p2 RS le_imp_less_or_eq RS disjE) 1); |
46 by (etac (p1 RS mono_lemma1) 1); |
46 by (etac (p1 RS mono_lemma1) 1); |
47 by (fast_tac (HOL_cs addIs [le_refl]) 1); |
47 by (fast_tac (!claset addIs [le_refl]) 1); |
48 qed "mono_lemma"; |
48 qed "mono_lemma"; |
49 |
49 |
50 val prems = goal Puzzle.thy "m <= n ==> f(m) <= f(n)"; |
50 val prems = goal Puzzle.thy "m <= n ==> f(m) <= f(n)"; |
51 by (fast_tac (HOL_cs addIs ([mono_lemma,less_imp_le,lemma2]@prems)) 1); |
51 by (fast_tac (!claset addIs ([mono_lemma,less_imp_le,lemma2]@prems)) 1); |
52 qed "f_mono"; |
52 qed "f_mono"; |
53 |
53 |
54 goal Puzzle.thy "f(n) = n"; |
54 goal Puzzle.thy "f(n) = n"; |
55 by (rtac le_anti_sym 1); |
55 by (rtac le_anti_sym 1); |
56 by (rtac lemma1 2); |
56 by (rtac lemma1 2); |
57 by (fast_tac (HOL_cs addIs [Puzzle.f_ax,leI] addDs [leD,f_mono,lessD]) 1); |
57 by (fast_tac (!claset addIs [Puzzle.f_ax,leI] addDs [leD,f_mono,lessD]) 1); |
58 result(); |
58 result(); |