⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
/2 width=1 by lfxs_sort/ qed.
-lemma lfpr_zero: ∀h,I,G,L1,L2,V.
- ⦃G, L1⦄ ⊢ ➡[h, V] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V.
+lemma lfpr_zero: ∀h,I,G,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
+ ⦃G, L1⦄ ⊢ V1 ➡[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V2.
/2 width=1 by lfxs_zero/ qed.
lemma lfpr_lref: ∀h,I,G,L1,L2,V1,V2,i.
⦃G, L1⦄ ⊢ ➡[h, §l] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2.
/2 width=1 by lfxs_gref/ qed.
+lemma lfpr_pair_repl_dx: ∀h,I,G,L1,L2,T,V,V1.
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V1 →
+ ∀V2. ⦃G, L1⦄ ⊢ V ➡[h] V2 →
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V2.
+/2 width=2 by lfxs_pair_repl_dx/ qed-.
+
(* Basic inversion lemmas ***************************************************)
lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆.
lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
+lemma lfpr_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_sort/ qed-.
+
lemma lfpr_inv_zero: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ➡[h, #0] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
/2 width=1 by lfxs_inv_lref/ qed-.
+lemma lfpr_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_gref/ qed-.
+
lemma lfpr_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 →
⦃G, L1⦄ ⊢ ➡[h, V] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
(* Advanced inversion lemmas ************************************************)
+lemma lfpr_inv_sort_pair_sn: ∀h,I,G,Y2,L1,V1,s. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_sort_pair_sn/ qed-.
+
+lemma lfpr_inv_sort_pair_dx: ∀h,I,G,Y1,L2,V2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_sort_pair_dx/ qed-.
+
lemma lfpr_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #0] Y2 →
∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 &
Y2 = L2.ⓑ{I}V2.
∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 & Y1 = L1.ⓑ{I}V1.
/2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
+lemma lfpr_inv_gref_pair_sn: ∀h,I,G,Y2,L1,V1,l. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_gref_pair_sn/ qed-.
+
+lemma lfpr_inv_gref_pair_dx: ∀h,I,G,Y1,L2,V2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_gref_pair_dx/ qed-.
+
(* Basic forward lemmas *****************************************************)
lemma lfpr_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ➡[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-(* Basic_2A1: removed theorems 11:
+(* Basic_2A1: removed theorems 14:
lpr_inv_atom1 lpr_inv_pair1 lpr_inv_atom2 lpr_inv_pair2
lpr_refl lpr_pair
lpr_fwd_length lpr_lpx
lpr_drop_conf drop_lpr_trans lpr_drop_trans_O1
+ cpr_conf_lpr lpr_cpr_conf_dx lpr_cpr_conf_sn
*)
-(* Basic_1: removed theorems 7: wcpr0_gen_sort wcpr0_gen_head
- wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
- wcpr0_drop wcpr0_drop_back
+(* Basic_1: removed theorems 7:
+ wcpr0_gen_sort wcpr0_gen_head
+ wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+ wcpr0_drop wcpr0_drop_back
*)