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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/substitution/ltps.ma".
16 include "Basic_2/unfold/tpss.ma".
18 (* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
20 definition ltpss: nat → nat → relation lenv ≝
21 λd,e. TC … (ltps d e).
23 interpretation "partial unfold (local environment)"
24 'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
26 (* Basic eliminators ********************************************************)
28 lemma ltpss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
29 (∀L,L2. L1 [d, e] ▶* L → L [d, e] ▶ L2 → R L → R L2) →
30 ∀L2. L1 [d, e] ▶* L2 → R L2.
31 #d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
34 (* Basic properties *********************************************************)
36 lemma ltpss_strap: ∀L1,L,L2,d,e.
37 L1 [d, e] ▶ L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
40 lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
43 (* Basic inversion lemmas ***************************************************)
45 lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
46 #d #L1 #L2 #H @(ltpss_ind … H) -L2 //
47 #L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 //
50 lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
51 #d #e #L2 #H @(ltpss_ind … H) -L2 //
52 #L #L2 #_ #HL2 #IHL destruct
53 >(ltps_inv_atom1 … HL2) -HL2 //
56 fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
57 #d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
58 #L2 #L #_ #HL2 #IHL2 #H destruct
59 lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
62 lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
65 fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
66 ∀K2,I,V2. L2 = K2. 𝕓{I} V2 →
67 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
68 K2 ⊢ V1 [0, e - 1] ▶ V2 &
70 #d #e #L1 #L2 * -d e L1 L2
71 [ #d #e #_ #_ #K1 #I #V1 #H destruct
72 | #L1 #I #V #_ #H elim (lt_refl_false … H)
73 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
74 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
78 lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. 𝕓{I} V2 → 0 < e →
79 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
83 fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
84 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
85 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
86 K2 ⊢ V1 [d - 1, e] ▶ V2 &
88 #d #e #L1 #L2 * -d e L1 L2
89 [ #d #e #_ #I #K2 #V2 #H destruct
90 | #L #I #V #H elim (lt_refl_false … H)
91 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
92 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
96 lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. 𝕓{I} V2 → 0 < d →
97 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
98 K2 ⊢ V1 [d - 1, e] ▶ V2 &