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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground/xoa/ex_4_4.ma".
16 include "basic_2A/notation/relations/lazyeq_4.ma".
17 include "basic_2A/multiple/llpx_sn.ma".
19 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
21 definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2.
23 definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
26 "lazy equivalence (local environment)"
27 'LazyEq T l L1 L2 = (lleq l T L1 L2).
29 definition lleq_transitive: predicate (relation3 lenv term term) ≝
30 λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2.
32 (* Basic inversion lemmas ***************************************************)
34 lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
35 ∀L1,L2,l,k. |L1| = |L2| → R l (⋆k) L1 L2
37 ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2
39 ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
40 ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
41 K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2
43 ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2
45 ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2
48 L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ↑l] L2.ⓑ{I}V →
49 R l V L1 L2 → R (↑l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2
52 L1 ≡[V, l]L2 → L1 ≡[T, l] L2 →
53 R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2
55 ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2.
56 #R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/
59 lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 →
60 L1 ≡[V, l] L2 ∧ L1.ⓑ{I}V ≡[T, ↑l] L2.ⓑ{I}V.
61 /2 width=2 by llpx_sn_inv_bind/ qed-.
63 lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 →
64 L1 ≡[V, l] L2 ∧ L1 ≡[T, l] L2.
65 /2 width=2 by llpx_sn_inv_flat/ qed-.
67 (* Basic forward lemmas *****************************************************)
69 lemma lleq_fwd_length: ∀L1,L2,T,l. L1 ≡[T, l] L2 → |L1| = |L2|.
70 /2 width=4 by llpx_sn_fwd_length/ qed-.
72 lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
73 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
75 | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V &
77 K1 ≡[V, yinj 0] K2 & l ≤ yinj i.
78 #L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/
79 * /3 width=7 by or3_intro2, ex4_4_intro/
82 lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 →
84 /2 width=7 by llpx_sn_fwd_drop_sn/ qed-.
86 lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 →
88 /2 width=7 by llpx_sn_fwd_drop_dx/ qed-.
90 lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,l.
91 L1 ≡[ⓑ{a,I}V.T, l] L2 → L1 ≡[V, l] L2.
92 /2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
94 lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l.
95 L1 ≡[ⓑ{a,I}V.T, l] L2 → L1.ⓑ{I}V ≡[T, ↑l] L2.ⓑ{I}V.
96 /2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
98 lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l.
99 L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[V, l] L2.
100 /2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
102 lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,l.
103 L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[T, l] L2.
104 /2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
106 (* Basic properties *********************************************************)
108 lemma lleq_sort: ∀L1,L2,l,k. |L1| = |L2| → L1 ≡[⋆k, l] L2.
109 /2 width=1 by llpx_sn_sort/ qed.
111 lemma lleq_skip: ∀L1,L2,l,i. yinj i < l → |L1| = |L2| → L1 ≡[#i, l] L2.
112 /2 width=1 by llpx_sn_skip/ qed.
114 lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
115 ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
116 K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2.
117 /2 width=9 by llpx_sn_lref/ qed.
119 lemma lleq_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, l] L2.
120 /2 width=1 by llpx_sn_free/ qed.
122 lemma lleq_gref: ∀L1,L2,l,p. |L1| = |L2| → L1 ≡[§p, l] L2.
123 /2 width=1 by llpx_sn_gref/ qed.
125 lemma lleq_bind: ∀a,I,L1,L2,V,T,l.
126 L1 ≡[V, l] L2 → L1.ⓑ{I}V ≡[T, ↑l] L2.ⓑ{I}V →
127 L1 ≡[ⓑ{a,I}V.T, l] L2.
128 /2 width=1 by llpx_sn_bind/ qed.
130 lemma lleq_flat: ∀I,L1,L2,V,T,l.
131 L1 ≡[V, l] L2 → L1 ≡[T, l] L2 → L1 ≡[ⓕ{I}V.T, l] L2.
132 /2 width=1 by llpx_sn_flat/ qed.
134 lemma lleq_refl: ∀l,T. reflexive … (lleq l T).
135 /2 width=1 by llpx_sn_refl/ qed.
137 lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
138 /2 width=1 by llpx_sn_Y/ qed.
140 lemma lleq_sym: ∀l,T. symmetric … (lleq l T).
141 #l #T #L1 #L2 #H @(lleq_ind … H) -l -T -L1 -L2
142 /2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
145 lemma lleq_ge_up: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
146 ∀T,l,m. ⬆[l, m] T ≡ U →
147 lt ≤ l + m → L1 ≡[U, l] L2.
148 /2 width=6 by llpx_sn_ge_up/ qed-.
150 lemma lleq_ge: ∀L1,L2,T,l1. L1 ≡[T, l1] L2 → ∀l2. l1 ≤ l2 → L1 ≡[T, l2] L2.
151 /2 width=3 by llpx_sn_ge/ qed-.
153 lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
154 L1 ≡[ⓑ{a,I}V.T, 0] L2.
155 /2 width=1 by llpx_sn_bind_O/ qed-.
157 (* Advanceded properties on lazy pointwise extensions ************************)
159 lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) →
160 ∀L1,L2,T,l. L1 ≡[T, l] L2 → llpx_sn R l T L1 L2.
161 /2 width=3 by llpx_sn_co/ qed-.