L ⊢ 𝕓{I} V1. T1 ⇒ 𝕓{I} V2. T2.
#I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
elim (tps_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
-lapply (tps_leq_repl … HT0 (⋆. 𝕓{I} V2) ?) -HT0 /2/ #HT0 /3 width=5/
+lapply (tps_leq_repl_dx … HT0 (⋆. 𝕓{I} V2) ?) -HT0 /2/ #HT0
+/3 width=5/
qed.
(* NOTE: new property *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_lift.ma".
+include "Basic-2/substitution/ltps_drop.ma".
+
+(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
+
+lemma ltps_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+ ∀L1,d1,e1. L0 [d1, e1] ≫ L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 [d2, e2] ≫ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltps_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /2/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *)
+| /3/
+]
+qed.
+
+lemma ltps_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+ ∀L1,d1,e1. L0 [d1, e1] ≫ L1 →
+ ∃∃T. L1 ⊢ T2 [d2, e2] ≫ T &
+ L1 ⊢ U2 [d1, e1] ≫ T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+[ /2/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltps_drop_conf_le … HL01 … HLK0 ?) -HL01 HLK0 /2/ #X #H #HLK1
+ elim (ltps_inv_tps11 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+ lapply (drop_fwd_drop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // >arith_a2 /3/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltps_drop_conf_be … HL01 … HLK0 ? ?) -HL01 HLK0 [2,3: /2/ ] #X #H #HLK1
+ elim (ltps_inv_tps21 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+ lapply (drop_fwd_drop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // normalize #HW01
+ lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 //
+ | lapply (ltps_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/
+ ]
+ ]
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 [2: /2/ ] -HL01 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2;
+ elim (IHTU2 … HL01) -IHTU2 HL01 /3 width=5/
+]
+qed.
+(*
+lemma ltps_tps_trans: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst0 i u u1 u2) ->
+ (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
+
+*)
\ No newline at end of file
| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps L (#i) d e W
| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
- tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
+ tps L V1 d e V2 → tps (L. 𝕓{I} V2) T1 (d + 1) e T2 →
tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
tps L V1 d e V2 → tps L T1 d e T2 →
(* Basic properties *********************************************************)
-lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
- ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
+lemma tps_leq_repl_dx: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
+ ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
[ //
| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
-Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
- @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
- [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
+ lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /3 width=5/
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
-Hdi Hide /3 width=5/
fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
- L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 &
U2 = 𝕓{I} V2. T2.
#d #e #L #U1 #U2 * -d e L U1 U2
[ #L #k #d #e #I #V1 #T1 #H destruct
lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
- L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 &
U2 = 𝕓{I} V2. T2.
/2/ qed.
(* Basic-1: removed theorems 23:
subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
subst0_confluence_lift subst0_tlt
subst1_head subst1_gen_head
*)
elim (lift_trans_le … HUV … HVW ?) -HUV HVW V // >arith_a2 // -Hid /3/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- elim (IHV12 … HLK … HWV1 ?) -IHV12 //
- elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK HWV1 Hdetd /3 width=5/ (**) (* just /3 width=5/ is too slow *)
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK Hdetd (**) (* /3 width=5/ is too slow *)
+ /3 width=5/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
lapply (plus_le_weak … Hdetd) #Hedt
- elim (IHV12 … HLK … HWV1 ?) -IHV12 // #W2 #HW12 #HWV2
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
<plus_minus // /3 width=5/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
]
| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V1) ?) -HT02 /2/ #HT02
elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
- lapply (tps_leq_repl … HT1 (L. 𝕓{I} V1) ?) -HT1 /2/
- lapply (tps_leq_repl … HT2 (L. 𝕓{I} V2) ?) -HT2 /3 width=5/
+ lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /2/
+ lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V) ?) -HT2 /3 width=5/
| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
elim (IHV01 … HV02) -IHV01 HV02;
elim (IHV01 … HV02 H) -IHV01 HV02 #V #HV1 #HV2
elim (IHT01 … HT02 ?) -IHT01 HT02
[ -H #T #HT1 #HT2
- lapply (tps_leq_repl … HT1 (L2. 𝕓{I} V1) ?) -HT1 /2/
- lapply (tps_leq_repl … HT2 (L1. 𝕓{I} V2) ?) -HT2 /3 width=5/
+ lapply (tps_leq_repl_dx … HT1 (L2. 𝕓{I} V) ?) -HT1 /2/
+ lapply (tps_leq_repl_dx … HT2 (L1. 𝕓{I} V) ?) -HT2 /3 width=5/
| -HV1 HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %) elim H -H #H
[ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3/ is too slow *)
]
<(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
+ lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V0) ?) -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/
+ lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1;
+ lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V2) ?) -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) ->
- (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
-
- Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
- (u1,u:?; i:?) (subst0 i u u2 u1) ->
- (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
-
-*)
projects / helm.git / commitdiff