elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. 𝕓{Abst} WW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- lapply (tpss_lsubs_conf … HTT2 (L2. 𝕓{Abbr} VV2) ?) -HTT2 /3 width=5/
+ elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ lapply (tpss_lsubs_conf … HTT2 (L2. ⓓVV2) ?) -HTT2 /3 width=5/
| #I #V1 #V2 #T1 #T2 #U2 #HV12 #_ #HTU2 #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
elim (tps_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. 𝕓{I} VV2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (IHT12 … HTT1 (L2. ⓑ{I} VV2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
elim (tpss_strip_neq … HTT2 … HTU2 ?) -T2 /2 width=1/ #T2 #HTT2 #HUT2
- lapply (tps_lsubs_conf … HTT2 (L2. 𝕓{I} V2) ?) -HTT2 /2 width=1/ #HTT2
- elim (ltpss_tps_conf … HTT2 (L2. 𝕓{I} VV2) (d + 1) e ?) -HTT2 /2 width=1/ #W2 #HTTW2 #HTW2
- lapply (tpss_lsubs_conf … HTTW2 (⋆. 𝕓{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
+ lapply (tps_lsubs_conf … HTT2 (L2. ⓑ{I} V2) ?) -HTT2 /2 width=1/ #HTT2
+ elim (ltpss_tps_conf … HTT2 (L2. ⓑ{I} VV2) (d + 1) e ?) -HTT2 /2 width=1/ #W2 #HTTW2 #HTW2
+ lapply (tpss_lsubs_conf … HTTW2 (⋆. ⓑ{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2
- lapply (tpss_lsubs_conf … HTW2 (L2. 𝕓{I} VV2) ?) -HTW2 /2 width=1/ #HTW2
+ lapply (tpss_lsubs_conf … HTW2 (L2. ⓑ{I} VV2) ?) -HTW2 /2 width=1/ #HTW2
lapply (tpss_trans_eq … HUT2 … HTW2) -T2 /3 width=5/
| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct
elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
- elim (IHT12 … HTT1 (L2. 𝕓{Abbr} WW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
elim (lift_total VV2 0 1) #VV #H2VV
- lapply (tpss_lift_ge … HVV2 (L2. 𝕓{Abbr} WW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
+ lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #HVV
@ex2_1_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
| #V1 #TT1 #T1 #T2 #HTT1 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct
qed.
lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
- ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ▶ U1 →
- ∃∃U2. U1 ➡ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ▶ U2.
+ ⋆. ⓑ{I} V1 ⊢ T1 [0, 1] ▶ U1 →
+ ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ U2.
#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
-elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. 𝕓{I} V2) ?) -T1 /2 width=1/ /3 width=3/
+elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ /3 width=3/
qed.
lemma tpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →