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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 "ground_2/xoa/ex_5_7.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_atom: ∀G,i. cpce h G (⋆) (#i) (#i)
25 | cpce_zero: ∀G,K,I. (∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
26 cpce h G (K.ⓘ{I}) (#0) (#0)
27 | cpce_eta : ∀n,p,G,K,W,V1,V2,W2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
28 cpce h G K V1 V2 → ⬆*[1] V2 ≘ W2 → cpce h G (K.ⓛW) (#0) (+ⓛW2.ⓐ#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_gref: ∀G,L,l. cpce h G L (§l) (§l)
32 | cpce_bind: ∀p,I,G,K,V1,V2,T1,T2.
33 cpce h G K V1 V2 → cpce h G (K.ⓑ{I}V1) T1 T2 →
34 cpce h G K (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
35 | cpce_flat: ∀I,G,L,V1,V2,T1,T2.
36 cpce h G L V1 V2 → cpce h G L T1 T2 →
37 cpce h G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
41 "context-sensitive parallel eta-conversion (term)"
42 'PConvEta h G L T1 T2 = (cpce h G L T1 T2).
44 (* Basic inversion lemmas ***************************************************)
46 lemma cpce_inv_sort_sn (h) (G) (L) (X2):
47 ∀s. ⦃G,L⦄ ⊢ ⋆s ⬌η[h] X2 → ⋆s = X2.
49 @(insert_eq_0 … (⋆s0)) #X1 * -G -Y -X1 -X2
53 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
54 | #I #G #K #T #U #i #_ #_ #H destruct
56 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
57 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
61 lemma cpce_inv_atom_sn (h) (G) (X2):
62 ∀i. ⦃G,⋆⦄ ⊢ #i ⬌η[h] X2 → #i = X2.
64 @(insert_eq_0 … LAtom) #Y
65 @(insert_eq_0 … (#j)) #X1
69 | #G #K #I #_ #_ #_ //
70 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #_ #H destruct
71 | #I #G #K #T #U #i #_ #_ #_ #H destruct
73 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
74 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
78 lemma cpce_inv_zero_sn (h) (G) (K) (X2):
79 ∀I. ⦃G,K.ⓘ{I}⦄ ⊢ #0 ⬌η[h] X2 →
80 ∨∨ ∧∧ ∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #0 = X2
81 | ∃∃n,p,W,V1,V2,W2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U & ⦃G,K⦄ ⊢ V1 ⬌η[h] V2
82 & ⬆*[1] V2 ≘ W2 & I = BPair Abst W & +ⓛW2.ⓐ#0.#1 = X2.
84 @(insert_eq_0 … (Y0.ⓘ{Z})) #Y
85 @(insert_eq_0 … (#0)) #X1
87 [ #G #L #s #H #_ destruct
88 | #G #i #_ #H destruct
89 | #G #K #I #HI #_ #H destruct /4 width=7 by or_introl, conj/
90 | #n #p #G #K #W #V1 #V2 #W2 #U #HWU #HV12 #HVW2 #_ #H destruct /3 width=12 by or_intror, ex5_7_intro/
91 | #I #G #K #T #U #i #_ #_ #H #_ destruct
92 | #G #L #l #H #_ destruct
93 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
94 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
98 lemma cpce_inv_lref_sn (h) (G) (K) (X2):
99 ∀I,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬌η[h] X2 →
100 ∃∃T2. ⦃G,K⦄ ⊢ #i ⬌η[h] T2 & ⬆*[1] T2 ≘ X2.
102 @(insert_eq_0 … (Y0.ⓘ{Z})) #Y
103 @(insert_eq_0 … (#↑j)) #X1
105 [ #G #L #s #H #_ destruct
106 | #G #i #_ #H destruct
107 | #G #K #I #_ #H #_ destruct
108 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H #_ destruct
109 | #I #G #K #T #U #i #Hi #HTU #H1 #H2 destruct /2 width=3 by ex2_intro/
110 | #G #L #l #H #_ destruct
111 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
112 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
116 lemma cpce_inv_gref_sn (h) (G) (L) (X2):
117 ∀l. ⦃G,L⦄ ⊢ §l ⬌η[h] X2 → §l = X2.
119 @(insert_eq_0 … (§k)) #X1 * -G -Y -X1 -X2
123 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
124 | #I #G #K #T #U #i #_ #_ #H destruct
126 | #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
127 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
131 lemma cpce_inv_bind_sn (h) (G) (K) (X2):
132 ∀p,I,V1,T1. ⦃G,K⦄ ⊢ ⓑ{p,I}V1.T1 ⬌η[h] X2 →
133 ∃∃V2,T2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,K.ⓑ{I}V1⦄ ⊢ T1 ⬌η[h] T2 & ⓑ{p,I}V2.T2 = X2.
134 #h #G #Y #X2 #q #Z #U #X
135 @(insert_eq_0 … (ⓑ{q,Z}U.X)) #X1 * -G -Y -X1 -X2
136 [ #G #L #s #H destruct
138 | #G #K #I #_ #H destruct
139 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
140 | #I #G #K #T #U #i #_ #_ #H destruct
141 | #G #L #l #H destruct
142 | #p #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
143 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
147 lemma cpce_inv_flat_sn (h) (G) (L) (X2):
148 ∀I,V1,T1. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌η[h] X2 →
149 ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,L⦄ ⊢ T1 ⬌η[h] T2 & ⓕ{I}V2.T2 = X2.
150 #h #G #Y #X2 #Z #U #X
151 @(insert_eq_0 … (ⓕ{Z}U.X)) #X1 * -G -Y -X1 -X2
152 [ #G #L #s #H destruct
154 | #G #K #I #_ #H destruct
155 | #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
156 | #I #G #K #T #U #i #_ #_ #H destruct
157 | #G #L #l #H destruct
158 | #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
159 | #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/