(* *)
(**************************************************************************)
-include "Basic_2/substitution/drop_drop.ma".
+include "Basic_2/substitution/ldrop_ldrop.ma".
include "Basic_2/substitution/tps.ma".
(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
(* Advanced inversion lemmas ************************************************)
-fact tps_inv_refl_SO2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → e = 1 →
- â\88\80K,V. â\86\93[0, d] L â\89¡ K. ð\9d\95\93{Abst} V → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+fact tps_inv_refl_SO2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → e = 1 →
+ â\88\80K,V. â\87©[0, d] L â\89¡ K. â\93\9bV → T1 = T2.
+#L #T1 #T2 #d #e #H elim 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 #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
+ >(le_to_le_to_eq … Hdi ?) /2 width=1/ -d #K #V #HLK
+ lapply (ldrop_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/
+ >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 // /2 width=1/
| #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: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 1] â\89« T2 →
- â\88\80K,V. â\86\93[0, d] L â\89¡ K. ð\9d\95\93{Abst} V → T1 = T2.
-/2 width=8/ qed.
+lemma tps_inv_refl_SO2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 1] â\96¶ T2 →
+ â\88\80K,V. â\87©[0, d] L â\89¡ K. â\93\9bV → T1 = T2.
+/2 width=8/ qed-.
(* Relocation properties ****************************************************)
(* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\89« T2 →
- â\88\80L,U1,U2,d,e. â\86\93[d, e] L ≡ K →
- â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\86\91[d, e] T2 ≡ U2 →
+lemma tps_lift_le: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\96¶ T2 →
+ â\88\80L,U1,U2,d,e. â\87©[d, e] L ≡ K →
+ â\87§[d, e] T1 â\89¡ U1 â\86\92 â\87§[d, e] T2 ≡ U2 →
dt + et ≤ d →
- L â\8a¢ U1 [dt, et] â\89« U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+ L â\8a¢ U1 [dt, et] â\96¶ U2.
+#K #T1 #T2 #dt #et #H elim 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
+ >(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 #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/
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
| #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 *)
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /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/
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+lemma tps_lift_be: ∀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 ≤ d → d ≤ dt + et →
+ L ⊢ U1 [dt, et + e] ▶ U2.
+#K #T1 #T2 #dt #et #H elim 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 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+ | -Hdti
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+ ]
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
+ ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
+| #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 /3 width=6/
]
qed.
(* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\89« T2 →
- â\88\80L,U1,U2,d,e. â\86\93[d, e] L ≡ K →
- â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\86\91[d, e] T2 ≡ U2 →
+lemma tps_lift_ge: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\96¶ T2 →
+ â\88\80L,U1,U2,d,e. â\87©[d, e] L ≡ K →
+ â\87§[d, e] T1 â\89¡ U1 â\86\92 â\87§[d, e] T2 ≡ U2 →
d ≤ dt →
- L â\8a¢ U1 [dt + e, et] â\89« U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+ L â\8a¢ U1 [dt + e, et] â\96¶ U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -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/
+ lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
| #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;
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
@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/
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
]
qed.
(* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
- â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_le: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+ â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
dt + et ≤ d →
- â\88\83â\88\83T2. K â\8a¢ T1 [dt, et] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+ â\88\83â\88\83T2. K â\8a¢ T1 [dt, et] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim 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/
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=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/
+ lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
| #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 *)
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_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/
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
]
qed.
(* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
- â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_ge: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+ â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
d + e ≤ dt →
- â\88\83â\88\83T2. K â\8a¢ T1 [dt - e, et] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+ â\88\83â\88\83T2. K â\8a¢ T1 [dt - e, et] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim 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/
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
+| #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 *)
+ elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+ elim (le_inv_plus_l … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_1_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
| #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/ ]
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
<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/
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
]
qed.
(* Basic_1: was: subst1_gen_lift_eq *)
lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
- L â\8a¢ U1 [d, e] â\89« U2 â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
+ L â\8a¢ U1 [d, e] â\96¶ U2 â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim 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
+ [ 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
+ | 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
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
>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
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
>IHV12 // >IHT12 //
]
qed.
(*
- Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?)
+ 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) &
).
- Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
+ 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 →
+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 →
- â\88\83â\88\83T2. K â\8a¢ T1 [d, dt + et - (d + e)] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+ â\88\83â\88\83T2. K â\8a¢ T1 [d, dt + et - (d + e)] â\96¶ T2 & â\87§[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/
+lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
+qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 [dt, d - dt] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
qed.
projects / helm.git / blobdiff