lemma ltpss_tps2: ∀L1,L2,I,e.
L1 [0, e] ▶* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ▶ V2 →
- L1. 𝕓{I} V1 [0, e + 1] ▶* L2. 𝕓{I} 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
lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
- 0 < e → L1. 𝕓{I} V1 [0, e] ▶* L2. 𝕓{I} 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. 𝕓{I} V1 [d + 1, e] ▶* L2. 𝕓{I} 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
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. 𝕓{I} V1 [d, e] ▶* L2. 𝕓{I} 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. 𝕓{I} V1 [0, e] ▶* L2 →
+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. 𝕓{I} 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
]
qed-.
-lemma ltpss_fwd_tpss11: ∀d,e,I,K1,V1,L2. 0 < d → K1. 𝕓{I} V1 [d, e] ▶* L2 →
+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. 𝕓{I} 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