(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 1 →
+ ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+[ //
+| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct -e;
+ >(le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK
+ lapply (drop_mono … HLK0 … HLK) #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
+ >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 /2/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
+ >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
+]
+qed.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫ T2 →
+ ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
+/2 width=8/ qed.
+
(* Relocation properties ****************************************************)
+(* Basic-1: was: subst1_lift_lt *)
lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
dt + et ≤ d →
L ⊢ U1 [dt, et] ≫ U2.
#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 H2 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
>(lift_mono … H1 … H2) -H1 H2 //
| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
]
qed.
+(* Basic-1: was: subst1_lift_ge *)
lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
d ≤ dt →
L ⊢ U1 [dt + e, et] ≫ U2.
#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 H2 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
>(lift_mono … H1 … H2) -H1 H2 //
| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
]
qed.
+(* Basic-1: was: subst1_gen_lift_lt *)
lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
dt + et ≤ d →
∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
- lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+ ]
| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
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 //
]
qed.
+(* Basic-1: was: subst1_gen_lift_ge *)
lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
d + e ≤ dt →
∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
- lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+ ]
| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
lapply (transitive_le … Hdedt … Hdti) #Hdei
lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
| #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
]
qed.
+(* Basic-1: was: subst1_gen_lift_eq *)
lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
[ //
-| //
| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
elim (lift_inv_lref2 … H) -H * #H
[ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
]
qed.
(*
- Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
- (le (plus d h) i) ->
- (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
-
Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?)
(subst0 i v t1 (lift h d u2)) ->
(le (plus d h) i) ->
lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tps_inv_refl1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 1 →
- ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
-[ //
-| //
-| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct -e;
- >(le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK
- lapply (drop_mono … HLK0 … HLK) #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
- >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 /2/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
- >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
-]
-qed.
-
-lemma tps_inv_refl1: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫ T2 →
- ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
-/2 width=8/ qed.