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15 include "basic_2/notation/relations/lazyeqalt_4.ma".
16 include "basic_2/substitution/lleq_lleq.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Note: alternative definition of lleq *)
21 inductive lleqa: relation4 ynat term lenv lenv ≝
22 | lleqa_sort: ∀L1,L2,d,k. |L1| = |L2| → lleqa d (⋆k) L1 L2
23 | lleqa_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → lleqa d (#i) L1 L2
24 | lleqa_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
25 ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
26 lleqa (yinj 0) V K1 K2 → lleqa d (#i) L1 L2
27 | lleqa_free: ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → lleqa d (#i) L1 L2
28 | lleqa_gref: ∀L1,L2,d,p. |L1| = |L2| → lleqa d (§p) L1 L2
29 | lleqa_bind: ∀a,I,L1,L2,V,T,d.
30 lleqa d V L1 L2 → lleqa (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
31 lleqa d (ⓑ{a,I}V.T) L1 L2
32 | lleqa_flat: ∀I,L1,L2,V,T,d.
33 lleqa d V L1 L2 → lleqa d T L1 L2 → lleqa d (ⓕ{I}V.T) L1 L2
37 "lazy equivalence (local environment) alternative"
38 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
40 (* Main inversion lemmas ****************************************************)
42 theorem lleqa_inv_lleq: ∀L1,L2,T,d. L1 ⋕⋕[T, d] L2 → L1 ⋕[T, d] L2.
43 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
44 /2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
47 (* Main properties **********************************************************)
49 theorem lleq_lleqa: ∀L1,T,L2,d. L1 ⋕[T, d] L2 → L1 ⋕⋕[T, d] L2.
50 #L1 #T @(f2_ind … rfw … L1 T) -L1 -T
51 #n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
52 [ #i #Hn #L2 #d #H elim (lleq_fwd_lref … H) [ * || * ]
53 /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
54 | #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind … H) -H /3 width=1 by lleqa_bind/
55 | #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat … H) -H /3 width=1 by lleqa_flat/
59 (* Advanced eliminators *****************************************************)
61 lemma lleq_ind_alt: ∀R:relation4 ynat term lenv lenv. (
62 ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
64 ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
66 ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
67 ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
68 K1 ⋕[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
70 ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
72 ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
75 L1 ⋕[V, d]L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
76 R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
79 L1 ⋕[V, d]L2 → L1 ⋕[T, d] L2 →
80 R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
82 ∀d,T,L1,L2. L1 ⋕[T, d] L2 → R d T L1 L2.
83 #R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa … H) -H
84 /3 width=9 by lleqa_inv_lleq/