3 Require Export pr0_defs.
5 Inductive pr1 : T -> T -> Prop :=
6 | pr1_r: (t:?) (pr1 t t)
7 | pr1_u: (t2,t1:?) (pr0 t1 t2) -> (t3:?) (pr1 t2 t3) -> (pr1 t1 t3).
9 Hint pr1 : ltlc := Constructors pr1.
11 Section pr1_props. (******************************************************)
13 Theorem pr1_pr0: (t1,t2:?) (pr0 t1 t2) -> (pr1 t1 t2).
17 Theorem pr1_t: (t2,t1:?) (pr1 t1 t2) ->
18 (t3:?) (pr1 t2 t3) -> (pr1 t1 t3).
19 Intros until 1; XElim H; XEAuto.
22 Theorem pr1_tail_1: (u1,u2:?) (pr1 u1 u2) ->
23 (t:?; k:?) (pr1 (TTail k u1 t) (TTail k u2 t)).
24 Intros; XElim H; XEAuto.
27 Theorem pr1_tail_2: (t1,t2:?) (pr1 t1 t2) ->
28 (u:?; k:?) (pr1 (TTail k u t1) (TTail k u t2)).
29 Intros; XElim H; XEAuto.
34 Hints Resolve pr1_pr0 pr1_t pr1_tail_1 pr1_tail_2 : ltlc.