1440
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1 |
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|
2 |
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|
3 |
(** monotone functions vs. "&&"- / "||"-semi-morphisms **)
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4 |
|
5069
|
5 |
Goalw [is_mono_def] "is_mono f = (ALL x y. f (x && y) [= f x && f y)";
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4153
|
6 |
by Safe_tac;
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1440
|
7 |
(*==> (level 1)*)
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8 |
by (stac le_inf_eq 1);
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4153
|
9 |
by (rtac conjI 1);
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1899
|
10 |
by (Step_tac 1);
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|
11 |
by (Step_tac 1);
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4153
|
12 |
by (etac mp 1);
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13 |
by (rtac inf_lb1 1);
|
1899
|
14 |
by (Step_tac 1);
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|
15 |
by (Step_tac 1);
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4153
|
16 |
by (etac mp 1);
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17 |
by (rtac inf_lb2 1);
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1440
|
18 |
(*==> (level 11)*)
|
4153
|
19 |
by (rtac (conjI RS (le_trans RS mp)) 1);
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20 |
by (rtac inf_lb2 2);
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1440
|
21 |
by (subgoal_tac "x && y = x" 1);
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4153
|
22 |
by (etac subst 1);
|
1899
|
23 |
by (Fast_tac 1);
|
1440
|
24 |
by (stac inf_connect 1);
|
4153
|
25 |
by (assume_tac 1);
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1440
|
26 |
qed "mono_inf_eq";
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27 |
|
5069
|
28 |
Goalw [is_mono_def] "is_mono f = (ALL x y. f x || f y [= f (x || y))";
|
4153
|
29 |
by Safe_tac;
|
1440
|
30 |
(*==> (level 1)*)
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|
31 |
by (stac ge_sup_eq 1);
|
4153
|
32 |
by (rtac conjI 1);
|
1899
|
33 |
by (Step_tac 1);
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|
34 |
by (Step_tac 1);
|
4153
|
35 |
by (etac mp 1);
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|
36 |
by (rtac sup_ub1 1);
|
1899
|
37 |
by (Step_tac 1);
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|
38 |
by (Step_tac 1);
|
4153
|
39 |
by (etac mp 1);
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|
40 |
by (rtac sup_ub2 1);
|
1440
|
41 |
(*==> (level 11)*)
|
4153
|
42 |
by (rtac (conjI RS (le_trans RS mp)) 1);
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|
43 |
by (rtac sup_ub1 1);
|
1440
|
44 |
by (subgoal_tac "x || y = y" 1);
|
4153
|
45 |
by (etac subst 1);
|
1899
|
46 |
by (Fast_tac 1);
|
1440
|
47 |
by (stac sup_connect 1);
|
4153
|
48 |
by (assume_tac 1);
|
1440
|
49 |
qed "mono_sup_eq";
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50 |
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51 |
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