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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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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_atom : ∀L,I,d,e. tps L (𝕒{I}) d e (𝕒{I})
21 | tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
22 ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps L (#i) d e W
23 | tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
24 tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
25 tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
26 | tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
27 tps L V1 d e V2 → tps L T1 d e T2 →
28 tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
31 interpretation "parallel substritution (term)"
32 'PSubst L T1 d e T2 = (tps L T1 d e T2).
34 (* Basic properties *********************************************************)
36 lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
37 ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
38 #L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
40 | #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
41 elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
47 lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
52 lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
53 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
55 #L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
57 | #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
58 lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
59 lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/
65 lemma tps_weak_top: ∀L,T1,T2,d,e.
66 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
67 #L #T1 #T #d #e #H elim H -L T1 T d e
69 | #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
70 lapply (drop_fwd_drop2_length … HLK) #Hi
71 lapply (le_to_lt_to_lt … Hdi Hi) #Hd
72 lapply (plus_minus_m_m_comm (|L|) d ?) /2/
78 lemma tps_weak_all: ∀L,T1,T2,d,e.
79 L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
81 lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
82 lapply (tps_weak_top … HT12) //
85 lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
86 ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
87 #L #T1 #T2 #d #e #H elim H -L T1 T2 d e
89 | #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
92 >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/
94 generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
95 >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/
97 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
98 elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
99 elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
100 -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
101 @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
102 [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
103 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
104 elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
105 -Hdi Hide /3 width=5/
109 (* Basic inversion lemmas ***************************************************)
111 fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} →
113 ∃∃K,V,i. d ≤ i & i < d + e &
114 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
117 #L #T1 #T2 #d #e * -L T1 T2 d e
118 [ #L #I #d #e #J #H destruct -I /2/
119 | #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct -I /3 width=8/
120 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
121 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
125 lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 →
127 ∃∃K,V,i. d ≤ i & i < d + e &
128 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
134 (* Basic-1: was: subst1_gen_sort *)
135 lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k.
137 elim (tps_inv_atom1 … H) -H //
138 * #K #V #i #_ #_ #_ #_ #H destruct
141 (* Basic-1: was: subst1_gen_lref *)
142 lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
144 ∃∃K,V. d ≤ i & i < d + e &
145 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
148 elim (tps_inv_atom1 … H) -H /2/
149 * #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct -i /3/
152 fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
153 ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
154 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
155 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
157 #d #e #L #U1 #U2 * -d e L U1 U2
158 [ #L #k #d #e #I #V1 #T1 #H destruct
159 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
160 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
161 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
165 lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
166 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
167 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
171 fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
172 ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
173 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
175 #d #e #L #U1 #U2 * -d e L U1 U2
176 [ #L #k #d #e #I #V1 #T1 #H destruct
177 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
178 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
179 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
183 lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
184 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
188 fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2.
189 #L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
191 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e;
192 lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide <plus_n_O #Hdd
193 elim (lt_refl_false … Hdd)
199 lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2.
202 (* Basic-1: removed theorems 23:
203 subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
204 subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
205 subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
206 subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
207 subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
208 subst0_confluence_lift subst0_tlt
209 subst1_head subst1_gen_head