-(*
- ||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/drop_drop.ma".
include "Basic-2/substitution/tps.ma".
(* 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 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #_ #HV12 #IHV12 #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+[ #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
lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
- elim (lift_trans_ge … HV12 … HVU2 ?) -HV12 HVU2 V2 // <minus_plus #V2 #HV12 #HVU2
- elim (drop_trans_le … HLK … HKV ?) -HLK HKV K /2/ #X #HLK #H
- elim (drop_inv_skip2 … H ?) -H /2/ -Hid #K #W #HKV #HVW #H destruct -X
- @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2 width=6/ ] (**) (* explicit constructor *)
+ elim (lift_trans_ge … HVW … HVU2 ?) -HVW HVU2 W // <minus_plus #W #HVW #HWU2
+ elim (drop_trans_le … HLK … HKV ?) -HLK HKV K [2: /2/] #X #HLK #H
+ elim (drop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -HVY HVW Y #H destruct -X /2/
| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
]
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 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #HV1 #HV12 #_ #L #U1 #U2 #d #e #HLK #H #HVU2 #Hddt
- <(arith_c1x ? ? ? e) in HV1 #HV1 (**) (* explicit athmetical rewrite and ?'s *)
+[ #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
lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
- lapply (lift_trans_be … HV12 … HVU2 ? ?) -HV12 HVU2 V2 /2/ >plus_plus_comm_23 #HV1U2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid #HLKV
- @tps_subst [4,5: /2/ |6,7,8: // |1,2,3: skip ] (**) (* /3 width=8/ is too slow *)
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -HVW HWU2 W // [ /2/ ] >plus_plus_comm_23 #HVU2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid /3/
| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
]
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 #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #_ #HV12 #IHV12 #K #d #e #HLK #T1 #H #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+[ #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 (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #W #HKL #HKVL #HWV
- elim (IHV12 … HKVL … HWV ?) -HKVL HWV /2/ -Hdetd #W1 #HW1 #HWV1
- elim (lift_trans_le … HWV1 … HV12 ?) -HWV1 HV12 V1 // >arith_a2 /3 width=6/
+ elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #U #HKL #_ #HUV
+ 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 #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #HV1 #HV12 #_ #K #d #e #HLK #T1 #H #Hdedt
+[ #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
- lapply (plus_le_weak … Hdei) #Hei
- <(arith_h1 ? ? ? e ? ?) in HV1 // #HV1
+ lapply (plus_le_weak … Hdei) #Hei
lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
- elim (lift_split … HV12 d (i - e + 1) ? ? ?) -HV12; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
@ex2_1_intro
- [2: @tps_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]
+ [2: @tps_subst [3: /2/ |5,6: // |1,2: skip |4: @arith5 // ]
|1: skip
| //
] (**) (* explicitc constructors *)
| #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 #V1 #V2 #i #d #e #Hdi #Hide #_ #_ #_ #_ #T1 #H
+| #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
elim (lt_refl_false … 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) ->
(le d i) -> (lt i (plus d h)) ->
(EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
*)
+
+lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
+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.
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