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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "static_2/static/reqx_fqup.ma".
16 include "basic_2/rt_transition/lpx.ma".
17 include "basic_2/rt_transition/rpx.ma".
19 (* EXAMPLES *****************************************************************)
21 (* Lemma "rpx_fwd_lpx_reqx" is not an inversion *****************************)
23 definition L1 (K) (s1) (s0): lenv ≝ K.ⓛ⋆s1.ⓛⓝ#0.⋆s0.
25 definition L (K) (s1) (s0): lenv ≝ K.ⓛ⋆s1.ⓛ⋆s0.
27 definition L2 (K) (i1) (s0): lenv ≝ K.ⓛ#i1.ⓛ⋆s0.
29 definition T: term ≝ #0.
31 (* Basic properties *********************************************************)
33 lemma ex_rpx_fwd_1 (G) (K) (s1) (s0):
34 ❪G,L1 K s1 s0❫ ⊢ ⬈ L K s1 s0.
35 /3 width=1 by lpx_pair, lpx_bind_refl_dx, cpx_eps/ qed.
37 lemma ex_rpx_fwd_2 (K) (s1) (s0) (i1) (i0):
38 L K s1 s0 ≛[T] L2 K i1 i0.
39 /3 width=1 by reqx_pair, reqx_sort/ qed.
41 lemma ex_rpx_fwd_3 (G) (K) (s1) (s0) (i1) (i0):
42 ❪G,L1 K s1 s0❫ ⊢ ⬈[T] L2 K i1 i0 → ⊥.
43 #G #K #s1 #s0 #i1 #i0 #H
44 elim (rpx_inv_zero_pair_sn … H) -H #Y2 #X2 #H #_ normalize #H0 destruct
45 elim (rpx_inv_flat … H) -H #H #_
46 elim (rpx_inv_zero_pair_sn … H) -H #Y2 #X2 #_ #H #H0 destruct
47 elim (cpx_inv_sort1 … H) #s2 #H destruct
50 (* Main properties **********************************************************)
52 theorem ex_rpx_fwd (G) (K) (s1) (s0) (i1) (i0):
53 (❪G,L1 K s1 s0❫ ⊢ ⬈ L K s1 s0 → L K s1 s0 ≛[T] L2 K i1 i0 → ❪G,L1 K s1 s0❫ ⊢ ⬈[T] L2 K i1 i0) → ⊥.
54 /3 width=7 by ex_rpx_fwd_3, ex_rpx_fwd_2, ex_rpx_fwd_1/ qed-.