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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 "Basic-2/substitution/drop.ma".
17 (* PARALLEL 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,W,i,d,e. d ≤ i → i < d + e →
23 ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps L (#i) d e W
24 | tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
25 tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
26 tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
27 | tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
28 tps L V1 d e V2 → tps L T1 d e T2 →
29 tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
32 interpretation "parallel substritution (term)"
33 'PSubst L T1 d e T2 = (tps L T1 d e T2).
35 (* Basic properties *********************************************************)
37 lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
38 ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
39 #L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
42 | #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
43 elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
49 lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
54 lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
55 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
57 #L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
60 | #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
61 lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
62 lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/
68 lemma tps_weak_top: ∀L,T1,T2,d,e.
69 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
70 #L #T1 #T #d #e #H elim H -L T1 T d e
73 | #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
74 lapply (drop_fwd_drop2_length … HLK) #Hi
75 lapply (le_to_lt_to_lt … Hdi Hi) #Hd
76 lapply (plus_minus_m_m_comm (|L|) d ?) /2/
82 lemma tps_weak_all: ∀L,T1,T2,d,e.
83 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
85 lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
86 lapply (tps_weak_top … HT12) //
89 lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
90 ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
91 #L #T1 #T2 #d #e #H elim H -L T1 T2 d e
94 | #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
97 >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/
99 generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
100 >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/
102 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
103 elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
104 elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
105 -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
106 @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
107 [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
108 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
109 elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
110 -Hdi Hide /3 width=5/
114 (* Basic inversion lemmas ***************************************************)
116 fact tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i →
118 ∃∃K,V. d ≤ i & i < d + e &
119 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
121 #L #T1 #T2 #d #e * -L T1 T2 d e
122 [ #L #k #d #e #i #H destruct
124 | #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #j #H destruct -i /3/
125 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
126 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
130 lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
132 ∃∃K,V. d ≤ i & i < d + e &
133 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
137 fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
138 ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
139 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
140 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
142 #d #e #L #U1 #U2 * -d e L U1 U2
143 [ #L #k #d #e #I #V1 #T1 #H destruct
144 | #L #i #d #e #I #V1 #T1 #H destruct
145 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
146 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
147 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
151 lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
152 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
153 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
157 fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
158 ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
159 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
161 #d #e #L #U1 #U2 * -d e L U1 U2
162 [ #L #k #d #e #I #V1 #T1 #H destruct
163 | #L #i #d #e #I #V1 #T1 #H destruct
164 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
165 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
166 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
170 lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
171 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
175 fact tps_inv_refl0_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2.
176 #L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
179 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e;
180 lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide <plus_n_O #Hdd
181 elim (lt_refl_false … Hdd)
187 lemma tps_inv_refl0: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2.