- elim (IHT12 … HTT1 (L2. 𝕓{I} V2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
-(*lapply (tps_leq_repl … HTT2 (L2. 𝕓{I} VV2) ?) -HTT2 /2/ #HTT2 *)
- elim (tps_conf_neq … HTU2 … HTT2 ?) -HTU2 HTT2 T2 /2/ #T2 #HUT2 #HTT2
- @ex2_1_intro
- [2: @tpr_delta [4: @HVV12 |5: @HTT12 |1,2: skip |6: ] (*|6: ]1,2: skip ]*)
- |1: skip
- |3: @tps_bind [@HVV2 | @HUT2 ]
- ]
-*)
+ elim (IHT12 … HTT1 (L2. 𝕓{I} VV2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
+ elim (tps_conf_neq … HTT2 … HTU2 ?) -HTT2 HTU2 T2 /2/ #T2 #HTT2 #HUT2
+ lapply (tps_leq_repl_dx … HTT2 (L2. 𝕓{I} V2) ?) -HTT2 /2/ #HTT2
+ elim (ltps_tps_conf … HTT2 (L2. 𝕓{I} VV2) (d + 1) 1 ?) -HTT2 /2/ #W2 #HTTW2 #HTW2
+ lapply (tps_leq_repl_dx … HTTW2 (⋆. 𝕓{I} VV2) ?) -HTTW2 /2/ #HTTW2
+ lapply (tps_trans_ge … HUT2 … HTW2 ?) -HUT2 HTW2 // #HUW2
+ /3 width=5/
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X;
+ elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct -Y;
+ elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
+ elim (IHW12 … HWW1 … HL12) -IHW12 HWW1 #WW2 #HWW12 #HWW2
+ elim (IHT12 … HTT1 (L2. 𝕓{Abbr} WW2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
+ elim (lift_total VV2 0 1) #VV #H2VV
+ lapply (tps_lift_ge … HVV2 (L2. 𝕓{Abbr} WW2) … HV2 H2VV ?) -HVV2 HV2 /2/ #HVV
+ @ex2_1_intro [2: @tpr_theta |1: skip |3: @tps_bind [2: @tps_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
+| #V1 #TT1 #T1 #T2 #HT1 #_ #IHT12 #L1 #d #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct -X;
+ elim (tps_inv_lift1_ge … HTT12 L1 … HT1 ?) -HTT12 HT1 /2/ #T2 #HT12 #HTT2
+ elim (IHT12 … HT12 … HL12) -IHT12 HT12 HL12 <minus_plus_m_m /3/
+| #V1 #T1 #T2 #_ #IHT12 #L1 #d #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X;
+ elim (IHT12 … HTT1 … HL12) -IHT12 HTT1 HL12 /3/
+]
+qed.