(* *)
(**************************************************************************)
-include "basic_2/unfold/tpss_ltps.ma".
-include "basic_2/unfold/ltpss.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/ltpss_tps.ma".
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+(* PARALLEL UNFOLD ON LOCAL ENVIRONMENTS ************************************)
(* Properties concerning partial unfold on terms ****************************)
-lemma ltpss_tpss_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
- ∀T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶* U2 →
- d1 + e1 ≤ d2 → L1 ⊢ T2 [d2, e2] ▶* U2.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0 //
-#L #L0 #_ #HL0 #IHL #T2 #U2 #d2 #e2 #HTU2 #Hde1d2
-lapply (ltps_tpss_trans_ge … HL0 HTU2) -HL0 -HTU2 /2 width=1/
+lemma ltpss_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
qed.
-lemma ltpss_tps_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
- ∀T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
- d1 + e1 ≤ d2 → L1 ⊢ T2 [d2, e2] ▶* U2.
-#L1 #L0 #d1 #e1 #HL10 #T2 #U2 #d2 #e2 #HTU2 #Hde1d2
-@(ltpss_tpss_trans_ge … HL10 … Hde1d2) /2 width=1/ (**) (* /3 width=6/ is too slow *)
-qed.
-
-lemma ltpss_tpss_trans_eq: ∀L0,L1,d,e. L0 [d, e] ▶* L1 →
- ∀T2,U2. L1 ⊢ T2 [d, e] ▶* U2 → L0 ⊢ T2 [d, e] ▶* U2.
-#L0 #L1 #d #e #H @(ltpss_ind … H) -L1
-[ /2 width=1/
-| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2
- lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 -HTU2 /2 width=1/
-]
-qed.
-
-lemma ltpss_tps_trans_eq: ∀L0,L1,d,e. L0 [d, e] ▶* L1 →
- ∀T2,U2. L1 ⊢ T2 [d, e] ▶ U2 → L0 ⊢ T2 [d, e] ▶* U2.
-/3 width=3/ qed.
-
-lemma ltpss_tpss_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 →
- L0 ⊢ T2 [d2, e2] ▶* U2 → L0 [d1, e1] ▶* L1 →
- L1 ⊢ T2 [d2, e2] ▶* U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #H @(ltpss_ind … H) -L1
-[ //
-| -HTU2 #L #L1 #_ #HL1 #IHL
- lapply (ltps_tpss_conf_ge … HL1 IHL) -HL1 -IHL //
-]
-qed.
-
-lemma ltpss_tps_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 →
- L0 ⊢ T2 [d2, e2] ▶ U2 → L0 [d1, e1] ▶* L1 →
- L1 ⊢ T2 [d2, e2] ▶* U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #HL01
-@(ltpss_tpss_conf_ge … Hde1d2 … HL01) /2 width=1/ (**) (* /3 width=6/ is too slow *)
-qed.
-
-lemma ltpss_tpss_conf_eq: ∀L0,L1,T2,U2,d,e.
- L0 ⊢ T2 [d, e] ▶* U2 → L0 [d, e] ▶* L1 →
- ∃∃T. L1 ⊢ T2 [d, e] ▶* T & L1 ⊢ U2 [d, e] ▶* T.
-#L0 #L1 #T2 #U2 #d #e #HTU2 #H @(ltpss_ind … H) -L1
+(* Basic_1: was: subst1_subst1_back *)
+lemma ltpss_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
[ /2 width=3/
-| -HTU2 #L #L1 #_ #HL1 * #W2 #HTW2 #HUW2
- elim (ltps_tpss_conf … HL1 HTW2) -HTW2 #T #HT2 #HW2T
- elim (ltps_tpss_conf … HL1 HUW2) -HL1 -HUW2 #U #HU2 #HW2U
- elim (tpss_conf_eq … HW2T … HW2U) -HW2T -HW2U #V #HTV #HUV
- lapply (tpss_trans_eq … HT2 … HTV) -T
- lapply (tpss_trans_eq … HU2 … HUV) -U /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
+ ]
+ ]
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2
+ elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
]
qed.
-
-lemma ltpss_tps_conf_eq: ∀L0,L1,T2,U2,d,e.
- L0 ⊢ T2 [d, e] ▶ U2 → L0 [d, e] ▶* L1 →
- ∃∃T. L1 ⊢ T2 [d, e] ▶* T & L1 ⊢ U2 [d, e] ▶* T.
-/3 width=3/ qed.
-
-lemma ltpss_tpss_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶* T2 →
- ∀L2,ds,es. L1 [ds, es] ▶* L2 →
- ∃∃T. L2 ⊢ T1 [d, e] ▶* T & L1 ⊢ T2 [ds, es] ▶* T.
-#L1 #T1 #T2 #d #e #HT12 #L2 #ds #es #H @(ltpss_ind … H) -L2
-[ /3 width=3/
-| #L #L2 #HL1 #HL2 * #T #HT1 #HT2
- elim (ltps_tpss_conf … HL2 HT1) -HT1 #T0 #HT10 #HT0
- lapply (ltps_tpss_trans_eq … HL2 … HT0) -HL2 -HT0 #HT0
- lapply (ltpss_tpss_trans_eq … HL1 … HT0) -HL1 -HT0 #HT0
- lapply (tpss_trans_eq … HT2 … HT0) -T /2 width=3/
-]
+
+lemma ltpss_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
qed.
-lemma ltpss_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶ T2 →
- ∀L2,ds,es. L1 [ds, es] ▶* L2 →
- ∃∃T. L2 ⊢ T1 [d, e] ▶* T & L1 ⊢ T2 [ds, es] ▶* T.
-/3 width=1/ qed.
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_tps2: ∀L1,L2,I,e.
- L1 [0, e] ▶* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ▶ V2 →
- L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #e #H @(ltpss_ind … H) -L2
-[ /3 width=1/
-| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12
- elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2
- lapply (IHL … HV1) -IHL -HV1 #HL1
- @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
-]
-qed.
-
-lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma ltpss_tps1: ∀L1,L2,I,d,e. L1 [d, e] ▶* L2 →
- ∀V1,V2. L2 ⊢ V1 [d, e] ▶ V2 →
- L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2
-[ /3 width=1/
-| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12
- elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2
- lapply (IHL … HV1) -IHL -HV1 #HL1
- @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+(* Basic_1: was: subst1_subst1 *)
+lemma ltpss_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ T ▶* [d1, e1] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
+ elim (ltpss_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2
+ elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
]
qed.
-
-lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 →
- 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma ltpss_fwd_tpss21: ∀e,K1,I,V1,L2. 0 < e → K1. ⓑ{I} V1 [0, e] ▶* L2 →
- ∃∃K2,V2. K1 [0, e - 1] ▶* K2 & K1 ⊢ V1 [0, e - 1] ▶* V2 &
- L2 = K2. ⓑ{I} V2.
-#e #K1 #I #V1 #L2 #He #H @(ltpss_ind … H) -L2
-[ /2 width=5/
-| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
- elim (ltps_inv_tps21 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H
- lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2
- lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/
-]
-qed-.
-
-lemma ltpss_fwd_tpss11: ∀d,e,I,K1,V1,L2. 0 < d → K1. ⓑ{I} V1 [d, e] ▶* L2 →
- ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
- K1 ⊢ V1 [d - 1, e] ▶* V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #K1 #I #V1 #L2 #Hd #H @(ltpss_ind … H) -L2
-[ /2 width=5/
-| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
- elim (ltps_inv_tps11 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H
- lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2
- lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/
-]
-qed-.
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