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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/drop.ma".
17 (* PARTIAL SUBSTITUTION ON TERMS ********************************************)
19 inductive tps: lenv → term → nat → nat → term → Prop ≝
20 | tps_sort : ∀L,k,d,e. tps L (⋆k) d e (⋆k)
21 | tps_lref : ∀L,i,d,e. tps L (#i) d e (#i)
22 | tps_subst: ∀L,K,V,U1,U2,i,d,e.
24 ↓[0, i] L ≡ K. 𝕓{Abbr} V → tps K V 0 (d + e - i - 1) U1 →
25 ↑[0, i + 1] U1 ≡ U2 → tps L (#i) d e U2
26 | tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
27 tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
28 tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
29 | tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
30 tps L V1 d e V2 → tps L T1 d e T2 →
31 tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
34 interpretation "partial telescopic substritution"
35 'PSubst L T1 d e T2 = (tps L T1 d e T2).
37 (* Basic properties *********************************************************)
39 lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
40 ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
41 #L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
44 | #L1 #K1 #V #V1 #V2 #i #d #e #Hdi #Hide #HLK1 #_ #HV12 #IHV12 #L2 #HL12
45 elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // #K2 #HK12 #HLK2
46 @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
52 lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
57 lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
58 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
60 #L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
63 | #L #K #V #V1 #V2 #i #d1 #e1 #Hid1 #Hide1 #HLK #_ #HV12 #IHV12 #d2 #e2 #Hd12 #Hde12
64 lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
65 lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 #Hide2
66 @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2/ ] (**) (* /4 width=6/ is too slow *)
72 lemma tps_weak_top: ∀L,T1,T2,d,e.
73 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
74 #L #T1 #T #d #e #H elim H -L T1 T d e
77 | #L #K #V #V1 #V2 #i #d #e #Hdi #_ #HLK #_ #HV12 #IHV12
78 lapply (drop_fwd_drop2_length … HLK) #Hi
79 lapply (le_to_lt_to_lt … Hdi Hi) #Hd
80 lapply (plus_minus_m_m_comm (|L|) d ?) [ /2/ ] -Hd #Hd
81 lapply (drop_fwd_O1_length … HLK) normalize #HKL
82 lapply (tps_weak … IHV12 0 (|L| - i - 1) ? ?) -IHV12 // -HKL /2 width=6/
88 lemma tps_weak_all: ∀L,T1,T2,d,e.
89 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
91 lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
92 lapply (tps_weak_top … HT12) //
95 (* Basic inversion lemmas ***************************************************)
97 lemma tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i →
99 ∃∃K,V1,V2,i. d ≤ i & i < d + e &
100 ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
101 K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
103 #L #T1 #T2 #d #e * -L T1 T2 d e
104 [ #L #k #d #e #i #H destruct
106 | #L #K #V1 #V2 #T2 #i #d #e #Hdi #Hide #HLK #HV12 #HVT2 #j #H destruct -i /3 width=9/
107 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
108 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
112 lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
114 ∃∃K,V1,V2,i. d ≤ i & i < d + e &
115 ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
116 K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
120 lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
121 ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
122 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
123 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
125 #d #e #L #U1 #U2 * -d e L U1 U2
126 [ #L #k #d #e #I #V1 #T1 #H destruct
127 | #L #i #d #e #I #V1 #T1 #H destruct
128 | #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
129 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
130 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
134 lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
135 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
136 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
140 lemma tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
141 ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
142 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
144 #d #e #L #U1 #U2 * -d e L U1 U2
145 [ #L #k #d #e #I #V1 #T1 #H destruct
146 | #L #i #d #e #I #V1 #T1 #H destruct
147 | #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
148 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
149 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
153 lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
154 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &