-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
-include "Basic-2/substitution/tps_split.ma".
+include "Basic-2/substitution/tps_lift.ma".
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
(* Main properties **********************************************************)
-(*
-theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
- L ⊢ T1 [d, e] ≫ T2.
-#L #T1 #T #d #e #H elim H -L T1 T d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
- lapply (drop_fwd_drop2 … HLK) #H
- elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
- @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
- @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
-]
-qed.
-*)
-
-axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2.
- ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
- K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 →
- ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 →
- d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
- ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
-(*
-#L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2
-#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
-*)
theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
#L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
[ /2/
-| /2/
-| #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H
+| #L #K1 #V1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVT1 #T2 #H
elim (tps_inv_lref1 … H) -H
- [ -IHVW1 #HX destruct -T2
- @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *)
- | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2
- elim (lt_or_eq_or_gt i1 i2) #Hi12
- [ @tps_conf_subst_subst_lt /width=19/
- | -HVW1; destruct -i2;
- lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2
- lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1;
- elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2
- lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2
- elim (lift_total W 0 (i1 + 1)) #T #HWT
- lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 //
- lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1
- lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ]
- lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ]
- /2/
- | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/
- ]
+ [ #HX destruct -T2 /4/
+ | * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (drop_mono … HLK1 … HLK2) -HLK1 #H destruct -V1 K1
+ >(lift_mono … HVT1 … HVT2) -HVT1 /2/
]
| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
+ elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
@ex2_1_intro
[2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
|1: skip
|3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
] // (**) (* /5/ is too slow *)
| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
elim (IHV01 … HV02) -IHV01 HV02;
elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
]
qed.
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
+ ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫ T2.
+#L #T1 #T0 #d1 #e1 #H elim H -L T1 T0 d1 e1
+[ /2/
+| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
+ lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2; normalize /2/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ lapply (tps_leq_repl … HT02 (L. 𝕓{I} V1) ?) -HT02 /2/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 HV02 // #V
+ elim (IHT10 … HT02 ?) -IHT10 HT02 [2: /2/ ] #T #HT1 #HT2
+ lapply (tps_leq_repl … HT2 (L. 𝕓{I} V) ?) -HT2 /3 width=6/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ elim (IHV10 … HV02 ?) -IHV10 HV02 //
+ elim (IHT10 … HT02 ?) -IHT10 HT02 // /3 width=6/
+]
+qed.
(*
Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
(u1,u:?; i:?) (subst0 i u u1 u2) ->
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