+(* Advanced inversion lemmas ************************************************)
+
+lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
+#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
+elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
+qed-.
+
+lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
+#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
+elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
+#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
+elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
+qed-.
+
+lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.
+