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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/unfold/tpss_lift.ma".
17 (* PARALLEL UNFOLD ON TERMS *************************************************)
19 (* alternative definition of tpss *)
20 inductive tpssa: nat → nat → lenv → relation term ≝
21 | tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I})
22 | tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
23 ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
24 ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
25 | tpssa_bind : ∀L,I,V1,V2,T1,T2,d,e.
26 tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
27 tpssa d e L (ⓑ{I} V1. T1) (ⓑ{I} V2. T2)
28 | tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
29 tpssa d e L V1 V2 → tpssa d e L T1 T2 →
30 tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
33 interpretation "parallel unfold (term) alternative"
34 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2).
36 (* Basic properties *********************************************************)
38 lemma tpssa_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
39 ∀L2. L1 ≼ [d, e] L2 → L2 ⊢ T1 ▶▶* [d, e] T2.
40 #L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
42 | #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
43 elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
49 lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
51 #I elim I -I /2 width=1/
54 lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
55 ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
56 #L #T1 #T #d #e #H elim H -L -T1 -T -d -e
58 elim (tps_inv_atom1 … H) -H // * /2 width=6/
59 | #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
60 lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
61 lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
62 elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
63 | #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
64 elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
65 lapply (tps_lsubs_conf … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
66 lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
67 lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
68 lapply (tpssa_lsubs_conf … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
69 | #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
70 elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
74 lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
75 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
78 (* Basic inversion lemmas ***************************************************)
80 lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
81 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
84 lemma tpss_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
85 (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
86 (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
87 ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
88 ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2
90 (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
91 L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
92 R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{I}V1.T1) (ⓑ{I}V2.T2)
94 (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
95 L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
96 R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
98 ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
99 #R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
100 // /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/