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/
+lapply (ltps_tpss_trans_ge … HL0 HTU2) -HL0 -HTU2 /2 width=1/
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/ (**) (* /3 width=6/ is too slow *)
+@(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/
+[ /2 width=1/
| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2
- lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 HTU2 /2/
+ 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/ qed.
+/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 →
#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 //
+ lapply (ltps_tpss_conf_ge … HL1 IHL) -HL1 -IHL //
]
qed.
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/ (**) (* /3 width=6/ is too slow *)
+@(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
-[ /2/
+[ /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/
+ 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/
]
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/ qed.
+/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/
+[ /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/
+ 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/
]
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/ qed.
+/3 width=1/ qed.
(* Advanced properties ******************************************************)
L1 [0, e] ≫* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ≫ V2 →
L1. 𝕓{I} V1 [0, e + 1] ≫* L2. 𝕓{I} V2.
#L1 #L2 #I #e #H @(ltpss_ind … H) -L2
-[ /3/
+[ /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
+ lapply (IHL … HV1) -IHL -HV1 #HL1
@step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
qed.
L1 [0, e - 1] ≫* L2 → L2 ⊢ V1 [0, e - 1] ≫ V2 →
0 < e → L1. 𝕓{I} V1 [0, e] ≫* L2. 𝕓{I} V2.
#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2/
+>(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. 𝕓{I} V1 [d + 1, e] ≫* L2. 𝕓{I} V2.
#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2
-[ /3/
+[ /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
+ lapply (IHL … HV1) -IHL -HV1 #HL1
@step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
qed.
L1 [d - 1, e] ≫* L2 → L2 ⊢ V1 [d - 1, e] ≫ V2 →
0 < d → L1. 𝕓{I} V1 [d, e] ≫* L2. 𝕓{I} V2.
#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2/
+>(plus_minus_m_m d 1) // /2 width=1/
qed.
(* Advanced forward lemmas **************************************************)
L2 = K2. 𝕓{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 -L;
+| #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/
L2 = K2. 𝕓{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 -L;
+| #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/