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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/grammar/lenv_length.ma".
17 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **********)
19 inductive lpx_sn (R:lenv→relation term): relation lenv ≝
20 | lpx_sn_stom: lpx_sn R (⋆) (⋆)
21 | lpx_sn_pair: ∀I,K1,K2,V1,V2.
22 lpx_sn R K1 K2 → R K1 V1 V2 → lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
25 (* Basic inversion lemmas ***************************************************)
27 fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
30 | #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
34 lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
35 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
37 fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
38 ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
40 [ #J #K1 #V1 #H destruct
41 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
45 lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
46 ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
47 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
49 fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
52 | #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
56 lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
57 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
59 fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
60 ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
62 [ #J #K2 #V2 #H destruct
63 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
67 lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
68 ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
69 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
71 (* Basic forward lemmas *****************************************************)
73 lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
74 #R #L1 #L2 #H elim H -L1 -L2 normalize //