20 by (resolve_tac [notI] 1); |
20 by (resolve_tac [notI] 1); |
21 by (eresolve_tac [Suc_neq_0] 1); |
21 by (eresolve_tac [Suc_neq_0] 1); |
22 by (resolve_tac [notI] 1); |
22 by (resolve_tac [notI] 1); |
23 by (eresolve_tac [notE] 1); |
23 by (eresolve_tac [notE] 1); |
24 by (eresolve_tac [Suc_inject] 1); |
24 by (eresolve_tac [Suc_inject] 1); |
25 val Suc_n_not_n = result(); |
25 qed "Suc_n_not_n"; |
26 |
26 |
27 |
27 |
28 goal Nat.thy "(k+m)+n = k+(m+n)"; |
28 goal Nat.thy "(k+m)+n = k+(m+n)"; |
29 prths ([induct] RL [topthm()]); (*prints all 14 next states!*) |
29 prths ([induct] RL [topthm()]); (*prints all 14 next states!*) |
30 by (resolve_tac [induct] 1); |
30 by (resolve_tac [induct] 1); |
35 back(); |
35 back(); |
36 back(); |
36 back(); |
37 |
37 |
38 goalw Nat.thy [add_def] "0+n = n"; |
38 goalw Nat.thy [add_def] "0+n = n"; |
39 by (resolve_tac [rec_0] 1); |
39 by (resolve_tac [rec_0] 1); |
40 val add_0 = result(); |
40 qed "add_0"; |
41 |
41 |
42 goalw Nat.thy [add_def] "Suc(m)+n = Suc(m+n)"; |
42 goalw Nat.thy [add_def] "Suc(m)+n = Suc(m+n)"; |
43 by (resolve_tac [rec_Suc] 1); |
43 by (resolve_tac [rec_Suc] 1); |
44 val add_Suc = result(); |
44 qed "add_Suc"; |
45 |
45 |
46 val add_ss = FOL_ss addsimps [add_0, add_Suc]; |
46 val add_ss = FOL_ss addsimps [add_0, add_Suc]; |
47 |
47 |
48 goal Nat.thy "(k+m)+n = k+(m+n)"; |
48 goal Nat.thy "(k+m)+n = k+(m+n)"; |
49 by (res_inst_tac [("n","k")] induct 1); |
49 by (res_inst_tac [("n","k")] induct 1); |
50 by (simp_tac add_ss 1); |
50 by (simp_tac add_ss 1); |
51 by (asm_simp_tac add_ss 1); |
51 by (asm_simp_tac add_ss 1); |
52 val add_assoc = result(); |
52 qed "add_assoc"; |
53 |
53 |
54 goal Nat.thy "m+0 = m"; |
54 goal Nat.thy "m+0 = m"; |
55 by (res_inst_tac [("n","m")] induct 1); |
55 by (res_inst_tac [("n","m")] induct 1); |
56 by (simp_tac add_ss 1); |
56 by (simp_tac add_ss 1); |
57 by (asm_simp_tac add_ss 1); |
57 by (asm_simp_tac add_ss 1); |
58 val add_0_right = result(); |
58 qed "add_0_right"; |
59 |
59 |
60 goal Nat.thy "m+Suc(n) = Suc(m+n)"; |
60 goal Nat.thy "m+Suc(n) = Suc(m+n)"; |
61 by (res_inst_tac [("n","m")] induct 1); |
61 by (res_inst_tac [("n","m")] induct 1); |
62 by (ALLGOALS (asm_simp_tac add_ss)); |
62 by (ALLGOALS (asm_simp_tac add_ss)); |
63 val add_Suc_right = result(); |
63 qed "add_Suc_right"; |
64 |
64 |
65 val [prem] = goal Nat.thy "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)"; |
65 val [prem] = goal Nat.thy "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)"; |
66 by (res_inst_tac [("n","i")] induct 1); |
66 by (res_inst_tac [("n","i")] induct 1); |
67 by (simp_tac add_ss 1); |
67 by (simp_tac add_ss 1); |
68 by (asm_simp_tac (add_ss addsimps [prem]) 1); |
68 by (asm_simp_tac (add_ss addsimps [prem]) 1); |