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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/xoa/ex_2_4.ma".
16 include "basic_2/notation/relations/pconveta_5.ma".
17 include "basic_2/rt_computation/cpms.ma".
19 (* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
22 inductive cpce (h): relation4 genv lenv term term ≝
23 | cpce_sort: ∀G,L,s. cpce h G L (⋆s) (⋆s)
24 | cpce_ldef: ∀G,K,V. cpce h G (K.ⓓV) (#0) (#0)
25 | cpce_ldec: ∀n,G,K,V,s. ⦃G,K⦄ ⊢ V ➡*[n,h] ⋆s →
26 cpce h G (K.ⓛV) (#0) (#0)
27 | cpce_eta : ∀n,p,G,K,V,W,T. ⦃G,K⦄ ⊢ V ➡*[n,h] ⓛ{p}W.T →
28 cpce h G (K.ⓛV) (#0) (+ⓛW.ⓐ#0.#1)
29 | cpce_lref: ∀I,G,K,T,U,i. cpce h G K (#i) T →
30 ⬆*[1] T ≘ U → cpce h G (K.ⓘ{I}) (#↑i) U
31 | cpce_bind: ∀p,I,G,K,V1,V2,T1,T2.
32 cpce h G K V1 V2 → cpce h G (K.ⓑ{I}V1) T1 T2 →
33 cpce h G K (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
34 | cpce_flat: ∀I,G,L,V1,V2,T1,T2.
35 cpce h G L V1 V2 → cpce h G L T1 T2 →
36 cpce h G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
40 "context-sensitive parallel eta-conversion (term)"
41 'PConvEta h G L T1 T2 = (cpce h G L T1 T2).
43 (* Basic inversion lemmas ***************************************************)
45 lemma cpce_inv_sort_sn (h) (G) (L) (X2):
46 ∀s. ⦃G,L⦄ ⊢ ⋆s ⬌η[h] X2 → ⋆s = X2.
48 @(insert_eq_0 … (⋆s0)) #X1 * -G -Y -X1 -X2
51 | #n #G #K #V #s #_ #_ //
52 | #n #p #G #K #V #W #T #_ #H destruct
53 | #I #G #K #T #U #i #_ #_ #H destruct
54 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
55 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
59 lemma cpce_inv_ldef_sn (h) (G) (K) (X2):
60 ∀V. ⦃G,K.ⓓV⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
62 @(insert_eq_0 … (Y.ⓓX)) #Y1
63 @(insert_eq_0 … (#0)) #X1
67 | #n #G #K #V #s #_ #_ #_ //
68 | #n #p #G #K #V #W #T #_ #_ #H destruct
69 | #I #G #K #T #U #i #_ #_ #H #_ destruct
70 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
71 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
75 lemma cpce_inv_ldec_sn (h) (G) (K) (X2):
76 ∀V. ⦃G,K.ⓛV⦄ ⊢ #0 ⬌η[h] X2 →
77 ∨∨ ∃∃n,s. ⦃G,K⦄ ⊢ V ➡*[n,h] ⋆s & #0 = X2
78 | ∃∃n,p,W,T. ⦃G,K⦄ ⊢ V ➡*[n,h] ⓛ{p}W.T & +ⓛW.ⓐ#0.#1 = X2.
80 @(insert_eq_0 … (Y.ⓛX)) #Y1
81 @(insert_eq_0 … (#0)) #X1
83 [ #G #L #s #H #_ destruct
84 | #G #K #V #_ #H destruct
85 | #n #G #K #V #s #HV #_ #H destruct /3 width=3 by ex2_2_intro, or_introl/
86 | #n #p #G #K #V #W #T #HV #_ #H destruct /3 width=6 by or_intror, ex2_4_intro/
87 | #I #G #K #T #U #i #_ #_ #H #_ destruct
88 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
89 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
93 lemma cpce_inv_lref_sn (h) (G) (K) (X2):
94 ∀I,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬌η[h] X2 →
95 ∃∃T2. ⦃G,K⦄ ⊢ #i ⬌η[h] T2 & ⬆*[1] T2 ≘ X2.
97 @(insert_eq_0 … (Y.ⓘ{Z})) #Y1
98 @(insert_eq_0 … (#↑j)) #X1
100 [ #G #L #s #H #_ destruct
101 | #G #K #V #H #_ destruct
102 | #n #G #K #V #s #_ #H #_ destruct
103 | #n #p #G #K #V #W #T #_ #H #_ destruct
104 | #I #G #K #T #U #i #Hi #HTU #H1 #H2 destruct /2 width=3 by ex2_intro/
105 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
106 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
110 lemma cpce_inv_bind_sn (h) (G) (K) (X2):
111 ∀p,I,V1,T1. ⦃G,K⦄ ⊢ ⓑ{p,I}V1.T1 ⬌η[h] X2 →
112 ∃∃V2,T2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,K.ⓑ{I}V1⦄ ⊢ T1 ⬌η[h] T2 & ⓑ{p,I}V2.T2 = X2.
113 #h #G #Y #X2 #q #Z #U #X
114 @(insert_eq_0 … (ⓑ{q,Z}U.X)) #X1 * -G -Y -X1 -X2
115 [ #G #L #s #H destruct
116 | #G #K #V #H destruct
117 | #n #G #K #V #s #_ #H destruct
118 | #n #p #G #K #V #W #T #_ #H destruct
119 | #I #G #K #T #U #i #_ #_ #H destruct
120 | #p #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
121 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
125 lemma cpce_inv_flat_sn (h) (G) (L) (X2):
126 ∀I,V1,T1. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌η[h] X2 →
127 ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,L⦄ ⊢ T1 ⬌η[h] T2 & ⓕ{I}V2.T2 = X2.
128 #h #G #Y #X2 #Z #U #X
129 @(insert_eq_0 … (ⓕ{Z}U.X)) #X1 * -G -Y -X1 -X2
130 [ #G #L #s #H destruct
131 | #G #K #V #H destruct
132 | #n #G #K #V #s #_ #H destruct
133 | #n #p #G #K #V #W #T #_ #H destruct
134 | #I #G #K #T #U #i #_ #_ #H destruct
135 | #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
136 | #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/