(* Basic_2A1: includes: drop_fwd_length_le4 *)
lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → |L2| ≤ |L1|.
-#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/
+#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by nle_succ_dx, nle_succ_bi/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq1 *)
lemma drops_fwd_fcla: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 →
∃∃n. 𝐂❪f❫ ≘ n & |L1| = |L2| + n.
#f #L1 #L2 #H elim H -f -L1 -L2
-[ /4 width=3 by fcla_isid, ex2_intro/
-| #f #I #L1 #L2 #_ * >length_bind /3 width=3 by fcla_next, ex2_intro, eq_f/
-| #f #I1 #I2 #L1 #L2 #_ #_ * >length_bind >length_bind /3 width=3 by fcla_push, ex2_intro/
+[ /4 width=3 by pr_fcla_isi, ex2_intro/
+| #f #I #L1 #L2 #_ * >length_bind /3 width=3 by pr_fcla_next, ex2_intro, eq_f/
+| #f #I1 #I2 #L1 #L2 #_ #_ * >length_bind >length_bind /3 width=3 by pr_fcla_push, ex2_intro/
]
qed-.
lemma drops_fcla_fwd: ∀f,L1,L2,n. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐂❪f❫ ≘ n →
|L1| = |L2| + n.
#f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+#k #Hm #H <(pr_fcla_mono … Hm … Hn) -f //
qed-.
lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 →
lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐂❪f❫ ≘ n →
n ≤ |L1|.
#f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+#k #Hm #H <(pr_fcla_mono … Hm … Hn) -f //
qed-.
lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⇩*[Ⓣ,f] L1 ≘ K2.ⓘ[I2] →
∃∃n. 𝐂❪f❫ ≘ n & n < |L1|.
#f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H
-#n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/
+#n #Hf #H >H -L1 /3 width=3 by nle_succ_bi, ex2_intro/
qed-.
(* Basic_2A1: includes: drop_fwd_length_lt2 *)
⇩*[Ⓣ,f] L1 ≘ K2.ⓘ[I2] → 𝐂❪f❫ ≘ n →
n < |L1|.
#f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+#k #Hm #H <(pr_fcla_mono … Hm … Hn) -f //
qed-.
(* Basic_2A1: includes: drop_fwd_length_lt4 *)
lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐂❪f❫ ≘ n → 0 < n →
|L2| < |L1|.
#f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f
-/2 width=1 by lt_minus_to_plus_r/ qed-.
+/2 width=1 by nlt_inv_minus_dx/ qed-.
(* Basic_2A1: includes: drop_inv_length_eq *)
lemma drops_inv_length_eq: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → |L1| = |L2| → 𝐈❪f❫.
#f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H
-#n #Hn <HL12 -L2 #H lapply (discr_plus_x_xy … H) -H
-/2 width=3 by fcla_inv_xp/
+#n #Hn <HL12 -L2 #H lapply (nplus_refl_sn … H) -H
+/2 width=3 by pr_fcla_inv_zero/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq2 *)
#f #L1 #L2 #K1 #K2 #HLK1 #HLK2 #HL12
elim (drops_fwd_fcla … HLK1) -HLK1 #n1 #Hn1 #H1 >H1 -L1
elim (drops_fwd_fcla … HLK2) -HLK2 #n2 #Hn2 #H2 >H2 -L2
-<(fcla_mono … Hn2 … Hn1) -f //
+<(pr_fcla_mono … Hn2 … Hn1) -f //
qed-.
theorem drops_conf_div: ∀f1,f2,L1,L2. ⇩*[Ⓣ,f1] L1 ≘ L2 → ⇩*[Ⓣ,f2] L1 ≘ L2 →
#f1 #f2 #L1 #L2 #H1 #H2
elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1
elim (drops_fwd_fcla … H2) -H2 #n2 #Hf2 >H1 -L1 #H
-lapply (injective_plus_r … H) -L2 #H destruct /2 width=3 by ex2_intro/
+lapply (eq_inv_nplus_bi_sn … H) -L2 #H destruct /2 width=3 by ex2_intro/
qed-.
theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2.
#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2
lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1
lapply (drops_fcla_fwd … Hf2 Hn2) -f2 >H1 -L1
-/2 width=1 by injective_plus_r/
+/2 width=1 by eq_inv_nplus_bi_sn/
qed-.