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14
15
16 include "coq.ma".
17
18 alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
19 alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
20 alias id "trans_equal" = "cic:/Coq/Init/Logic/trans_equal.con".
21 alias id "refl_equal" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1/1)".
22 alias id "Z" = "cic:/Coq/ZArith/BinInt/Z.ind#xpointer(1/1)".
23
24 theorem semicolon: \forall p:Prop.p\to p\land p.
25 intros (p); split; assumption.
26 qed.
27
28 theorem branch:\forall x:nat.x=x.
29 intros (n);
30 elim n
31 [ reflexivity;
32 | reflexivity ].
33 qed.
34
35 theorem pos:\forall x:Z.x=x.
36 intros (n);
37 elim n;
38 [ 3: reflexivity;
39 | 2: reflexivity;
40 | reflexivity ]
41 qed.
42
43 theorem dot:\forall x:Z.x=x.
44 intros (x).
45 elim x.
46 reflexivity. reflexivity. reflexivity.
47 qed.
48
49 theorem dot_slice:\forall x:Z.x=x.
50 intros (x).
51 elim x;
52 [ elim x. reflexivity. reflexivity. reflexivity;
53 | reflexivity
54 | reflexivity ];
55 qed.
56
57 theorem focus:\forall x:Z.x=x.
58 intros (x); elim x.
59 focus 16 17;
60   reflexivity;
61 unfocus.
62 reflexivity.
63 qed.
64
65 theorem skip:\forall x:nat.x=x. 
66 intros (x).
67 apply trans_equal;
68 [ 2: apply (refl_equal nat x);
69 | skip
70 | reflexivity
71 ]
72 qed.
73
74 theorem skip_focus:\forall x:nat.x=x.
75 intros (x).
76 apply trans_equal;
77 [ focus 18; apply (refl_equal nat x); unfocus;
78 | skip  
79 | reflexivity ]
80 qed.