(* *)
(**************************************************************************)
-include "Basic_2/substitution/ldrop_ldrop.ma".
-include "Basic_2/substitution/tps.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: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → e = 1 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
+#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
[ //
-| #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 width=1/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
- >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
+| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
+ elim (lt_or_ge i (d+1)) #HiSd
+ [ -Hide1 -HV0
+ lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct
+ | -V -Hdi /2 width=4/
+ ]
+| /4 width=3/
+| /3 width=3/
]
qed.
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ▶ T2 →
+lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
+/2 width=3/ qed-.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-/2 width=8/ qed-.
+#L #T1 #T2 #d #HT12 #K #V #HLK
+lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
+lapply (tps_inv_refl_O2 … HT12) -HT12 //
+qed-.
(* Relocation properties ****************************************************)
(* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+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.
+ L ⊢ U1 ▶ [dt, et] 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 //
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
+| #K #a #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
@tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
]
qed.
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+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.
+ 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 //
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
+| #K #a #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/ ]
qed.
(* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+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.
+ L ⊢ U1 ▶ [dt + e, et] 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 //
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
+| #K #a #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
@tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
qed.
(* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+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.
+ ∃∃T2. K ⊢ T1 ▶ [dt, et] 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/
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
+| #L #a #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
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 *)
]
qed.
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+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.
+ ∃∃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/
#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
+| #L #a #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 *)
qed.
(* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+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.
+ ∃∃T2. K ⊢ T1 ▶ [dt - e, et] 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_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
+| #L #a #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
elim (le_inv_plus_l … Hdetd) #_ #Hedt
elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
(* 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 ▶ [d, e] U2 → ∀T1. ⇧[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
| 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
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
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
(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.
+lemma tps_inv_lift1_ge_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 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
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 →
+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.
+ ∃∃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.
+
+lemma tps_inv_lift1_le_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 → d ≤ dt + et → 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 #Hddet #Hdetde
+elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
+elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
+lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
+lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
+qed.
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