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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "Basic-2/substitution/tps_split.ma".
17 (* PARTIAL SUBSTITUTION ON TERMS ********************************************)
19 (* Main properties **********************************************************)
21 theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
23 #L #T1 #T #d #e #H elim H -L T1 T d e
26 | #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
27 lapply (drop_fwd_drop2 … HLK) #H
28 elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
29 @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
30 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
31 elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
32 @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
33 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
34 elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
39 axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2.
40 ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
41 K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 →
42 ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 →
43 d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
44 ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
46 #L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2
47 #HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
50 theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
51 ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
52 #L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
55 | #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H
56 elim (tps_inv_lref1 … H) -H
57 [ -IHVW1 #HX destruct -T2
58 @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *)
59 | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2
60 elim (lt_or_eq_or_gt i1 i2) #Hi12
61 [ @tps_conf_subst_subst_lt /width=19/
62 | -HVW1; destruct -i2;
63 lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2
64 lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1;
65 elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2
66 lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2
67 elim (lift_total W 0 (i1 + 1)) #T #HWT
68 lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 //
69 lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1
70 lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ]
71 lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ]
73 | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/
76 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
77 elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
78 elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
79 elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
81 [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
83 |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
84 ] // (**) (* /5/ is too slow *)
85 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
86 elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
87 elim (IHV01 … HV02) -IHV01 HV02;
88 elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
93 Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
94 (u1,u:?; i:?) (subst0 i u u1 u2) ->
95 (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
97 Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
98 (u1,u:?; i:?) (subst0 i u u2 u1) ->
99 (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).