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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 "basic_2/grammar/lenv_append.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 →
23 lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
26 (* Basic inversion lemmas ***************************************************)
28 fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
31 | #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
35 lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
36 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
38 fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
39 ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
41 [ #J #K1 #V1 #H destruct
42 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
46 lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
47 ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
48 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
50 fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
53 | #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
57 lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
58 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
60 fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
61 ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
63 [ #J #K2 #V2 #H destruct
64 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
68 lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
69 ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
70 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
72 (* Basic forward lemmas *****************************************************)
74 lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
75 #R #L1 #L2 #H elim H -L1 -L2 normalize //
78 (* Advanced forward lemmas **************************************************)
80 lemma lpx_sn_fwd_append1: ∀R,L1,K1,L. lpx_sn R (K1 @@ L1) L →
81 ∃∃K2,L2. lpx_sn R K1 K2 & L = K2 @@ L2.
84 @(ex2_2_intro … K2 (⋆)) // (* explicit constructor, /2 width=4/ does not work *)
85 | #L1 #I #V1 #IH #K1 #X #H
86 elim (lpx_sn_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
87 elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #H destruct
88 @(ex2_2_intro … (L2.ⓑ{I}V2) HK12) // (* explicit constructor, /2 width=4/ does not work *)
92 lemma lpx_sn_fwd_append2: ∀R,L2,K2,L. lpx_sn R L (K2 @@ L2) →
93 ∃∃K1,L1. lpx_sn R K1 K2 & L = K1 @@ L1.
96 @(ex2_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=4/ does not work *)
97 | #L2 #I #V2 #IH #K2 #X #H
98 elim (lpx_sn_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
99 elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #H destruct
100 @(ex2_2_intro … (L1.ⓑ{I}V1) HK12) // (* explicit constructor, /2 width=4/ does not work *)
104 (* Basic properties *********************************************************)
106 lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
107 #R #HR #L elim L -L // /2 width=1/
110 lemma lpx_sn_append: ∀R. l_appendable_sn R →
111 ∀K1,K2. lpx_sn R K1 K2 → ∀L1,L2. lpx_sn R L1 L2 →
112 lpx_sn R (L1 @@ K1) (L2 @@ K2).
113 #R #HR #K1 #K2 #H elim H -K1 -K2 // /3 width=1/