+++ /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/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 #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 … 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;
- @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- /3 width=6/
-]
-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 #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 … 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;
- @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- /3 width=5/
-]
-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 * #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 #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 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 //
- elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-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 * #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
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
- lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
- 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 [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 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
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
- elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-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
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
- elim (lt_refl_false … H)
- ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
- >IHV12 // >IHT12 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
- >IHV12 // >IHT12 //
-]
-qed.
-(*
- 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) ->
- (EX u1 | (subst0 (minus i h) v u1 u2) &
- t1 = (lift h d u1)
- ).
-
-
- Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (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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