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4 (* ||A|| A project by Andrea Asperti *)
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15 include "Basic-2/grammar/lenv_length.ma".
17 (* LOCAL ENVIRONMENT EQUALITY ***********************************************)
19 inductive leq: lenv → nat → nat → lenv → Prop ≝
20 | leq_sort: ∀d,e. leq (⋆) d e (⋆)
21 | leq_comp: ∀L1,L2,I1,I2,V1,V2.
22 leq L1 0 0 L2 → leq (L1. 𝕓{I1} V1) 0 0 (L2. 𝕓{I2} V2)
23 | leq_eq: ∀L1,L2,I,V,e. leq L1 0 e L2 → leq (L1. 𝕓{I} V) 0 (e + 1) (L2.𝕓{I} V)
24 | leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
25 leq L1 d e L2 → leq (L1. 𝕓{I1} V1) (d + 1) e (L2. 𝕓{I2} V2)
28 interpretation "local environment equality" 'Eq L1 d e L2 = (leq L1 d e L2).
30 (* Basic properties *********************************************************)
32 lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
34 [ #e elim e -e [ #L elim L -L /2/ | #e #IHe #L elim L -L /2/ ]
35 | #d #IHd #e #L elim L -L /2/
39 lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
40 #L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
43 lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
44 ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
46 #L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
49 lemma leq_fwd_length: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → |L1| = |L2|.
50 #L1 #L2 #d #e #H elim H -H L1 L2 d e; normalize //
53 (* Basic inversion lemmas ***************************************************)
55 lemma leq_inv_sort1_aux: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L1 = ⋆ → L2 = ⋆.
56 #L1 #L2 #d #e #H elim H -H L1 L2 d e
58 | #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #H destruct
59 | #L1 #L2 #I #V #e #_ #_ #H destruct
60 | #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #_ #H destruct
63 lemma leq_inv_sort1: ∀L2,d,e. ⋆ [d, e] ≈ L2 → L2 = ⋆.
66 lemma leq_inv_sort2: ∀L1,d,e. L1 [d, e] ≈ ⋆ → L1 = ⋆.