From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 18 Sep 2006 14:28:30 +0000 (+0000)
Subject: last problem elegantly resolved!
X-Git-Tag: 0.4.95@7852~1021
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=147770d4605b8163f63331f82e800c19a2021d46;p=helm.git

last problem elegantly resolved!
---

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
index fc33cc529..8f4719772 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma
@@ -529,8 +529,8 @@ nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
 nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq 
 nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) 
 in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n 
-z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) 
-(sym_equal nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) 
+z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq 
+nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) 
 H1)))))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall 
 (y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) 
 (plus (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma
index 040b878aa..b4c69f299 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma
@@ -152,8 +152,6 @@ theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
  intros. apply plus_le_reg_l; auto.
 qed.
 
-definition sym_equal \def sym_eq.
-
 default "equality"
  cic:/Coq/Init/Logic/eq.ind
  cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/preamble/sym_eq.con
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
index 7907d8b96..1ca629bff 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma
@@ -29,12 +29,12 @@ theorem aplus_reg_r:
 (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: 
 nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 
 (plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n 
-h1)) (aplus g a2 (plus n h2)))).(sym_equal A (asucc g (aplus g a2 (plus n 
-h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1 
-(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus 
-g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g 
-g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq 
-A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+h1)) (aplus g a2 (plus n h2)))).(sym_eq A (asucc g (aplus g a2 (plus n h2))) 
+(asucc g (aplus g a1 (plus n h1))) (sym_eq A (asucc g (aplus g a1 (plus n 
+h1))) (asucc g (aplus g a2 (plus n h2))) (sym_eq A (asucc g (aplus g a2 (plus 
+n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g g (aplus g 
+a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq A (aplus g 
+a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
 
 theorem aplus_assoc:
  \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
@@ -51,10 +51,10 @@ g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
 n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) 
 n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda 
 (n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g 
-(aplus g a n1)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) 
-(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g 
+(aplus g a n1)))) (sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) 
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_eq A (asucc g (aplus g 
 (asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0)))) 
-(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g 
+(sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g 
 (asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a 
 (plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A 
 (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))) 
@@ -178,53 +178,54 @@ h) a) \to (\forall (P: Prop).P))))
  \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: 
 nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) 
 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A 
-(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in 
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) 
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in 
 nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow 
-(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) 
+(ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) 
 \to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g 
 n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0)) 
-h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g 
-(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2 
+h) (\lambda (a0: A).(eq A a0 (ASort O n0))) H0 (ASort (minus O h) (next_plus 
+g (next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2 
 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
-with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus 
-(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S 
-i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0) 
-(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) 
-(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n: 
-nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in 
-(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g 
-n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n 
-(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1) 
-\Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) 
-n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A 
-a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) 
-(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n 
-| (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def 
-(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m 
-with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1 
-h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) 
-H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def 
-(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n 
-i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g 
-n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) 
-(S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) 
-(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda 
-(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: 
-Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus 
-g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: 
-nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 
-a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g 
-a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc 
-g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A 
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with 
-[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g 
-(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g 
-a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec 
+next_plus (g0: G) (n1: nat) (i: nat) on i: nat \def (match i with [O 
+\Rightarrow n1 | (S i0) \Rightarrow (next g0 (next_plus g0 n1 i0))]) in 
+next_plus) g (next g n0) (minus h O))])) (ASort (minus O h) (next_plus g 
+(next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def (eq_ind_r nat 
+(minus h O) (\lambda (n1: nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h 
+(minus_n_O h)) in (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat 
+(next_plus g (next g n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) 
+h) n1)) (le_n (next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) 
+P))))) | (S n1) \Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) 
+(ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda 
+(a0: A).(eq A a0 (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 
+(minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat 
+(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) 
+\Rightarrow n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: 
+(nat \to nat) \def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) 
+\Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k 
+l)])])) in minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) 
+(ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match 
+e in A return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead 
+_ _) \Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat 
+\def (match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus 
+g0 n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) 
+(next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat 
+(minus n1 h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: 
+nat).(le n2 n1)) (minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda 
+(a0: A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) 
+\to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: 
+nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: 
+Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc 
+g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g 
+(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 
+(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) 
+in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
+(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: 
+nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc 
+g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow 
+a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 
+P)))))))))) a)).
 
 theorem aplus_inj:
  \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma
index d9578d9f3..133d10413 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma
@@ -65,24 +65,24 @@ T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n:
 nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead 
 c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) 
 u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow 
-c])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead 
-e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: C).(match e0 
-in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) 
-\Rightarrow k])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 
-k t) (CHead e (Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0: 
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | 
-(CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead e (Bind b) u) 
-(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in (\lambda (H4: (eq 
-K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 (\lambda (c1: C).(drop 
-(S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) (\lambda (k0: K).(drop (S O) 
-O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) k H4) e 
-H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c0 k t) 
-(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1: 
-(drop (S n) O (CHead c0 k t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e 
-(eq_ind_r nat (S (r k n)) (\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n) 
-(drop_gen_drop k c0 (CHead e (Bind b) u) t n H1)) (r k (S n)) (r_S k n)) 
-t)))) h)))))))) c)).
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead 
+c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) 
+(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def 
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) 
+(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) 
+in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 
+(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) 
+(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 
+c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: 
+(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 
+k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) 
+u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: 
+nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) 
+u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
 
 theorem drop_ctail:
  \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
@@ -126,17 +126,18 @@ c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
 (drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: 
 C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: 
 (eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h n (CHead c2 k t) 
-c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) 
-(CTail k0 u c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k x1) 
-(\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) 
-(let H5 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead 
-c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n 
-(CTail k0 u (CHead c2 k t)) (CTail k0 u (CHead x0 k x1)))))))) H4 (lift h (r 
-k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(drop h (S 
-n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) (drop_skip k h 
-n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) x1) t H2)) c3 
-H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) c1).
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k 
+t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 
+k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k 
+x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u 
+c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n 
+(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop 
+h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 
+(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: 
+T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) 
+(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) 
+x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) 
+c1).
 
 theorem drop_mono:
  \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h 
@@ -152,8 +153,8 @@ n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
 x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 
 (CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r 
 C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h 
-(\lambda (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d 
-(\lambda (n: nat).(eq nat n O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda 
+(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d 
+(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda 
 (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) 
 (drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda 
 (c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: 
@@ -187,23 +188,24 @@ C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
 C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: 
 (eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r 
 C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 
-(\lambda (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall 
-(x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) H0 (CHead x4 k x5) 
-H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 (CHead x0 k 
-x3))) (let H10 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n 
-(CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k 
-t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let 
-H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h 
-(r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0: 
-nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall 
-(x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k 
-t) x2)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda 
-(t0: T).(eq C (CHead x4 k t0) (CHead x0 k x3))) (sym_equal C (CHead x0 k x3) 
-(CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C 
-(CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (H x0 
-(r k n) h H5 x4 H8) (refl_equal K k) (refl_equal T x3))))) x5 (lift_inj x5 x3 
-h (r k n) H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k 
-H1))))))) (drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
+(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to 
+(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 
+(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 
+(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
+nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop 
+h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k 
+n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r 
+k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda 
+(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) 
+(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) 
+t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) 
+H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k 
+x3))) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (sym_eq C (CHead x4 k x3) 
+(CHead x0 k x3) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C 
+CHead x0 x4 k k x3 x3 (H x0 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal 
+T x3))))) x5 (lift_inj x5 x3 h (r k n) H11))))) x1 H6)) x2 H3)))))) 
+(drop_gen_skip_l c0 x1 t h n k H1))))))) (drop_gen_skip_l c0 x2 t h n k 
+H2)))))))) d))))))) c).
 
 theorem drop_conf_lt:
  \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall 
@@ -221,8 +223,8 @@ T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
 T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: 
 T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k 
 u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop 
-h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c: C).(drop h (S 
-(plus O d)) c e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in 
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S 
+(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in 
 (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
 (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) 
 (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C 
@@ -348,115 +350,115 @@ nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
 a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c 
 a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
 d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda 
-(c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1 
-in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop 
-(minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h) 
-O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in 
-(eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r 
-nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq 
-nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h 
-O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in 
-(let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in 
-(eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a 
-e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow 
-(\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0: 
-nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) 
-\to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) 
-(\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O 
-c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) 
-\to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: 
-C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall 
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d 
-h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: 
-nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: 
-(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d 
-O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda 
-(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) 
-(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: 
-(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O 
-O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O 
-H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0))) 
-H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e 
-a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) 
-(eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) 
-c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat 
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) 
-H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) 
-(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S 
-i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
-d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e 
-a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S 
-i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop 
-(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: 
-(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) 
+(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match 
+H1 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to 
+(drop (minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H3: (eq nat 
+(plus d h) O)).(let H4 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d 
+h) O H3) in (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e 
+a)) (eq_ind_r nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq 
+nat d O) (eq nat h O) (drop O O e a) (\lambda (H5: (eq nat d O)).(\lambda 
+(H6: (eq nat h O)).(let H7 \def (eq_ind nat d (\lambda (n: nat).(drop h n a 
+e)) H2 O H5) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) 
+H7 O H6) in (eq_ind C a (\lambda (c0: C).(drop O O c0 a)) (drop_refl a) e 
+(drop_gen_refl a e H8)))))) (plus_O d h H4)) (plus d h) H4) O H4))) | (le_S m 
+H3) \Rightarrow (\lambda (H4: (eq nat (S m) O)).((let H5 \def (eq_ind nat (S 
+m) (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind ((le 
+(plus d h) m) \to (drop (minus O h) O e a)) H5)) H3))]) in (H3 (refl_equal 
+nat O)))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall 
+(c: C).((drop i0 O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c e) \to ((le (plus d h) i0) \to (drop (minus i0 h) O e 
+a))))))))))).(\lambda (a: C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop 
+(S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop 
+h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e 
+a)))))))) (\lambda (n: nat).(\lambda (H0: (drop (S i0) O (CSort n) 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h 
+d (CSort n) e)).(\lambda (H2: (le (plus d h) (S i0))).(and3_ind (eq C e 
+(CSort n)) (eq nat h O) (eq nat d O) (drop (minus (S i0) h) O e a) (\lambda 
+(H3: (eq C e (CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d 
+O)).(and3_ind (eq C a (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (minus 
+(S i0) h) O e a) (\lambda (H6: (eq C a (CSort n))).(\lambda (H7: (eq nat (S 
+i0) O)).(\lambda (_: (eq nat O O)).(let H9 \def (eq_ind nat d (\lambda (n0: 
+nat).(le (plus n0 h) (S i0))) H2 O H5) in (let H10 \def (eq_ind nat h 
+(\lambda (n0: nat).(le (plus O n0) (S i0))) H9 O H4) in (eq_ind_r nat O 
+(\lambda (n0: nat).(drop (minus (S i0) n0) O e a)) (eq_ind_r C (CSort n) 
+(\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) (eq_ind_r C (CSort n) 
+(\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) c0)) (let H11 \def 
+(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: 
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in 
+(False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h 
+H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e 
+H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le 
+(plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: 
+K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) 
+\to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 
+k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e 
+a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
+(CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le 
+(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) 
+t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) 
+(\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O 
+h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) 
+\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda 
+(H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S 
+i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) 
+O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) 
+e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda 
+(_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to 
+(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 
+(Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0 
+(drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t 
+h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda 
+(_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to 
+(drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 
+(Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T 
+(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda 
+(_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0: 
+C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O 
+e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind 
+b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8: 
+(drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda 
+(c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda 
+(n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 
+h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n 
+(plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 
+h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 
+(Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda 
+(H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) 
 e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h 
-n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S 
-i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda 
+n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S 
+i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda 
 (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 
-(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e 
-a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le 
-(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop 
-(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) 
-c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda 
-(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O 
+(Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e 
+a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le 
+(plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop 
+(minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) 
+c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda 
+(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O 
 h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) 
-O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a 
-c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e 
-t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: 
-nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) 
-(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) 
-(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S 
-i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind 
-b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) 
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop 
-(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C 
-e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) 
-x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 
-(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S 
-(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) 
-(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t 
-i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) 
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e 
-H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) 
-(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat 
-f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
-(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S 
-i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le 
-(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h 
-O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind 
-(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S 
-i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 
-(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 
-(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat 
-f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead 
-c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead 
-c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O 
-e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: 
-(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S 
-h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: 
-nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) 
-(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) 
-(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S 
-i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat 
-f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) 
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop 
-(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C 
-e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) 
-x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 
-(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def 
-(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 
-(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) 
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in 
-(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) 
-x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) 
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e 
-H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) 
-k)))) c))))) i).
+O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 
+(drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e 
+t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 
+(CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) 
+h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda 
+(H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda 
+(v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T 
+t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t 
+(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) 
+h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: 
+nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) 
+H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n 
+(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop 
+n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) 
+(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 
+h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 
+H3))))))))) k)))) c))))) i).
 
 theorem drop_conf_rev:
  \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
@@ -645,14 +647,14 @@ nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
 c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) 
 O c1 c)) (let H2 \def (match H1 in le return (\lambda (n: nat).(\lambda (_: 
 (le ? n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n 
-\Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop 
-(plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) 
-H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H 
-O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: 
-(eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e 
+\Rightarrow (\lambda (H2: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop 
+(plus O h) O c1 c2)) (let H3 \def (eq_ind nat d (\lambda (n: nat).(le n O)) 
+H1 O H2) in (let H4 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H 
+O H2) in H4)) d (sym_eq nat d O H2))) | (le_S m H2) \Rightarrow (\lambda (H3: 
+(eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e 
 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 
-c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 
+\Rightarrow True])) I O H3) in (False_ind ((le d m) \to (drop (plus O h) O c1 
+c2)) H4)) H2))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 
 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall 
 (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall 
 (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 
@@ -665,16 +667,16 @@ nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
 n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda 
 (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d 
 O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) 
-e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in 
-(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n) 
-H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S 
-(plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: 
-(eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda 
-(c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0) 
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S 
-(plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O 
-e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: 
+e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) 
+in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort 
+n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop 
+(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda 
+(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) 
+(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind 
+nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind 
+(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n 
+(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: 
 C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: 
 nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d 
 (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda 
@@ -710,21 +712,21 @@ v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
 (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 
 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: 
 (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall 
-(h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c 
-e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) 
-IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S 
-i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t: 
-T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall 
-(e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S 
-(plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in 
-(eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O 
-(CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: 
-K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 
-(plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) 
-d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind 
-b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda 
-(H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 
-e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 
-i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k 
-H))))))))) d))))))) c1)))) i).
+(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O 
+c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) 
+e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: 
+C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t 
+(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k 
+x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S 
+i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k 
+d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S 
+(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind 
+(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to 
+(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r 
+(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 
+(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: 
+F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat 
+f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 
+(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) 
+(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).
 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma
index 9612d298d..463b3de6a 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma
@@ -149,8 +149,8 @@ a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
 (aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) 
 k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort 
 (minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in 
-(let H5 \def (sym_equal A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 
-k) n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A 
+(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) 
+n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A 
 (ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) 
 (minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) 
 n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
index ba33d1441..44ca12dec 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
@@ -32,7 +32,7 @@ theorem lift_lref_lt:
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n 
 d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true 
 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T 
-(TLRef n)) (blt n d) (sym_equal bool (blt n d) true (lt_blt d n H)))))).
+(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
 
 theorem lift_lref_ge:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
@@ -41,7 +41,7 @@ theorem lift_lref_ge:
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d 
 n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true 
 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) 
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_equal bool (blt n d) false 
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false 
 (le_bge d n H)))))).
 
 theorem lift_head:
@@ -78,38 +78,38 @@ theorem lift_gen_sort:
 n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: 
 (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort 
 n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in 
-(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T 
-(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort 
-n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d 
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T 
-(TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def 
-(match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: 
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d H0)) 
+in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with 
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef n0))).(let H3 
+\def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind 
-(eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0 
-h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) 
-(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq 
-T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T 
-(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d 
-(THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d 
-t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 
-in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift 
-h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n))))) with 
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0) 
-(lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k 
-t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h 
-(s k d) t1)))))))))))) t)))).
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H2) in (False_ind (eq T 
+(TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda 
+(H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: 
+T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in 
+(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with 
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef (plus n0 
+h)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) 
+H2) in (False_ind (eq T (TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T 
+(TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: 
+(((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: 
+T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort 
+n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2 
+\def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n) 
+t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) 
+in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead 
+k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T 
+(TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind 
+T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in 
+(False_ind (eq T (THead k t0 t1) (TSort n)) H4)))]) in (H3 (refl_equal T 
+(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
 
 theorem lift_gen_lref:
  \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
@@ -121,7 +121,7 @@ nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
 (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
 h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda 
 (i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def 
-(eq_ind T (lift h d (TSort n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TSort 
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort 
 n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: 
 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
@@ -131,40 +131,40 @@ False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
 nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or 
 (land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T 
 (TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind 
-T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TLRef n) 
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) 
 (lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: 
 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) 
-H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) 
-(TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef 
+n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 
 h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus 
 d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) 
 (TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d 
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef 
-i) t)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal 
-T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort 
-_) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) 
-(TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) (\lambda (n0: 
-nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le (plus d h) 
-n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n (\lambda (n0: 
-nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land 
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) (or_intror (land 
-(lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h) 
-(plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) (plus n h)) (eq 
-T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) (refl_equal T 
-(TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda 
-(k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: nat).(\forall (h: 
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) 
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
-h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: nat).(\forall (h: 
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to (or (land (lt i d) 
-(eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 (TLRef (minus i 
-h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda 
-(H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef i) t)) H1 (THead k 
-(lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def 
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef 
+i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) 
+(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le 
+(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n 
+(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n 
+h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) 
+(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land 
+(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) 
+(plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) 
+(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i 
+H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: 
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to 
+(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 
+(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: 
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to 
+(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 
+(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: 
+nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def 
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 
+(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let 
+H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
 (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) 
 t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) 
 (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) 
@@ -180,35 +180,36 @@ d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0))
 (lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: 
 nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq 
 T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift 
-h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) 
+h d (TLRef n0)) (\lambda (t0: T).(eq T (TLRef n) t0)) H0 (TLRef n0) 
 (lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda 
-(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: 
-T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
-(let H3 \def (match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 
-h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e in T return 
-(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n 
-| (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind 
-nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def 
-(eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 
-(eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus 
-n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq 
-T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 
-t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq 
-T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k 
-t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda 
-(_: (eq ? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq 
-T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: 
-(eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def 
-(eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) 
-t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 
-(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: 
+T).(eq T (TLRef n) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0 
+h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return 
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n1) \Rightarrow 
+n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in 
+(eq_ind nat (plus n0 h) (\lambda (n1: nat).(eq T (TLRef n0) (TLRef n1))) (let 
+H5 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 d)) H (plus n0 h) H4) in 
+(le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d 
+(le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H5)))) n (sym_eq nat n (plus 
+n0 h) H4))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T 
+t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d 
+t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d 
+(THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda 
+(t2: T).(eq T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: 
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) 
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow 
+(\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) 
+t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) 
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+t1)))))))))))) t))))).
 
 theorem lift_gen_lref_false:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
@@ -219,8 +220,8 @@ theorem lift_gen_lref_false:
 n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0: 
 T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: 
 nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: 
-Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? 
-? t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow 
+Prop).(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? 
+? t0)).((eq T t0 (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow 
 (\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T 
 (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
@@ -228,40 +229,41 @@ Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ?
 (H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda 
 (H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d 
 P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) 
-in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3: 
-(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: 
+(\lambda (t0: T).(eq T (TLRef n) t0)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) 
+in (let H4 \def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda 
+(H4: (eq T (TLRef n) (TLRef n0))).(let H5 \def (f_equal T nat (\lambda (e: 
 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) 
-H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 
-(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat 
-n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d 
-n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T 
-(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 
-\def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq 
-T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: 
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
-(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def 
-(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in 
-(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq 
-nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) 
-(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d 
-t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T 
-(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T 
-(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def 
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
+n0) H4) in (eq_ind nat n0 (\lambda (_: nat).P) (let H6 \def (eq_ind_r nat n0 
+(\lambda (n1: nat).(lt n1 d)) H2 n H5) in (le_false d n P H H6)) n (sym_eq 
+nat n n0 H5))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d 
+n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef n) t0)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 
+\def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T 
+t0 (TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H4: 
+(eq T (TLRef n) (TLRef (plus n0 h)))).(let H5 \def (f_equal T nat (\lambda 
+(e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow 
+n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) 
+(TLRef (plus n0 h)) H4) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let 
+H6 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 (plus d h))) H0 (plus n0 h) 
+H5) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H6)))) n 
+(sym_eq nat n (plus n0 h) H5))))]) in (H4 (refl_equal T (TLRef (plus n0 
+h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) 
+(lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: 
+(((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: 
+(eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def 
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef n) t2)) H3 
 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let 
-H5 \def (match H4 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq 
-T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k 
-d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
-t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T 
-(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
+H5 \def (match H4 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with 
+[refl_equal \Rightarrow (\lambda (H5: (eq T (TLRef n) (THead k (lift h d t0) 
+(lift h (s k d) t1)))).(let H6 \def (eq_ind T (TLRef n) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead k (lift h d t0) (lift h (s k d) t1)) H5) in (False_ind P H6)))]) in 
+(H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1))))))))))))) 
+t)))))).
 
 theorem lift_gen_lref_ge:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
@@ -271,7 +273,7 @@ theorem lift_gen_lref_ge:
 n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h 
 d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T 
 (TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 in eq 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort 
+return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort 
 n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow 
 (\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def 
 (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in T return (\lambda 
@@ -280,52 +282,53 @@ n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow
 (eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort 
 n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 
 (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 
-d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef 
-(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def 
-(match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow 
-(\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T 
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: 
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec 
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O 
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef 
-(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T 
-(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n 
-d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H 
-(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in 
-(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def 
-(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) 
-H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 in 
-eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 
-h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda 
-(H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T 
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: 
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec 
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O 
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef 
-(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: 
-nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h 
-n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d 
-t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift 
-h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) 
-(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift 
-h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d 
-t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with 
-[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift 
-h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) 
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq 
-T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d 
-t0) (lift h (s k d) t1)))))))))))) t))))).
+d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef (plus n h)) t0)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 
+\def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T 
+t0 (TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal 
+\Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef n0))).(let H4 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def 
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S 
+(plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | (THead _ _ _) 
+\Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def (\lambda (m: 
+nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
+plus) n h)])) (TLRef (plus n h)) (TLRef n0) H3) in (eq_ind nat (plus n h) 
+(\lambda (n1: nat).(eq T (TLRef n1) (TLRef n))) (let H5 \def (eq_ind_r nat n0 
+(\lambda (n1: nat).(lt n1 d)) H1 (plus n h) H4) in (le_false d n (eq T (TLRef 
+(plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l 
+n h) H5)))) n0 H4)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le 
+d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef (plus n h)) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef (plus 
+n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return 
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow ((let rec plus (n1: nat) on 
+n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O \Rightarrow m | (S 
+p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | 
+(THead _ _ _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def 
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S 
+(plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H3) in 
+(eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) 
+(f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus 
+n0 h) H4))) (plus n0 h) H4)))]) in (H3 (refl_equal T (TLRef (plus n0 
+h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef 
+(plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: 
+T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef 
+n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 
+t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq 
+T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: 
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) 
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow 
+(\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) 
+t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in 
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) 
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+t1)))))))))))) t))))).
 
 theorem lift_gen_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
@@ -363,27 +366,27 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def 
+\Rightarrow (\lambda (H2: (eq T (THead k u t) (TLRef n))).(let H3 \def 
 (eq_ind T (THead k u t) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) 
+T).(eq T t (lift h (s k d) z))))) H3)))]) in (H2 (refl_equal T (TLRef n)))))) 
 (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
 (t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d 
 H0)) in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq 
 ? ? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda 
 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
-h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k 
-u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: 
+h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k 
+u t) (TLRef (plus n h)))).(let H3 \def (eq_ind T (THead k u t) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
+(TLRef (plus n h)) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
 T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u 
 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) 
-z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda 
+z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda 
 (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: 
 nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: 
@@ -394,42 +397,42 @@ T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_:
 T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s 
 k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
 (THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d 
-(THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0 
+(THead k0 t0 t1)) (\lambda (t2: T).(eq T (THead k u t) t2)) H1 (THead k0 
 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def 
 (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 
 (THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
 T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda 
-(H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let 
-H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(H3: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let 
+H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) 
+\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
+H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return 
 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h 
-(s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift 
-h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u 
-(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: 
+(THead _ t2 _) \Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift 
+h (s k0 d) t1)) H3) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in 
+T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 
+(lift h d t0) (lift h (s k0 d) t1)) H3) in (eq_ind K k0 (\lambda (k1: K).((eq 
+T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda 
+(y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k1 y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d 
+T).(eq T t (lift h (s k1 d) z)))))))) (\lambda (H7: (eq T u (lift h d 
 t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1)) 
 \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead 
 k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T 
-t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t: 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H8: (eq T 
+t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t2: 
 T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead 
 k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))) 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k0 d) z)))))) 
 (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) 
 (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h 
 d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s 
 k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d 
 t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1) 
-H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))]) 
+H8))) u (sym_eq T u (lift h d t0) H7))) k (sym_eq K k k0 H6))) H5)) H4)))]) 
 in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) 
 t1)))))))))))))) x)))).
 
@@ -470,13 +473,13 @@ T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def 
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef n))).(let H3 \def 
 (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in T return (\lambda 
 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+| (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d 
 (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n 
 h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda 
@@ -484,13 +487,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n 
-h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in 
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef (plus n 
+h)))).(let H3 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in 
 T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) 
-H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
+H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in 
 (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: 
 T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u 
 t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 
@@ -502,45 +505,45 @@ T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
 h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
 (THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) 
+(lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (THead (Bind b) u t) t2)) 
 H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in 
 (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
 t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in 
+\Rightarrow (\lambda (H3: (eq T (THead (Bind b) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in 
 T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k 
-(lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u 
-| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u 
-t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal 
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) 
-\Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) 
-t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to 
-((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: 
-T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S 
-d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) 
-(\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda 
-(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift 
-h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t: 
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) 
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) 
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) 
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) 
-t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1)) 
-(refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t 
-(lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) 
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
-t1)))))))))))))) x)))).
+\Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Bind b) u t) (THead 
+k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) 
+(THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let 
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | 
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)) H3) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u 
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Bind b) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (S d) z)))))))) (\lambda (H7: (eq T u (lift h d 
+t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b) 
+d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) 
+t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift 
+h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) 
+(\lambda (H8: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind 
+b) d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
+(THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+t2 (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq 
+T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+(lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead 
+(Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) 
+t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H8))) u (sym_eq T u (lift h d 
+t0) H7))) k H6)) H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) 
+(lift h (s k d) t1)))))))))))))) x)))).
 
 theorem lift_gen_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
@@ -578,13 +581,13 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T 
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef n))).(let H3 \def (eq_ind T 
 (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d 
 (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n 
 h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda 
@@ -592,13 +595,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def 
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H3 \def 
 (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda 
 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind 
+| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H2) in (False_ind 
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: 
 ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d 
 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) 
@@ -610,42 +613,42 @@ f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
 (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h: 
 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d 
 (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda 
-(t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s 
+(t2: T).(eq T (THead (Flat f) u t) t2)) H1 (THead k (lift h d t0) (lift h (s 
 k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 in eq return 
 (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) 
 (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
 (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
 u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) 
-with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead 
-k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: 
+with [refl_equal \Rightarrow (\lambda (H3: (eq T (THead (Flat f) u t) (THead 
+k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: 
 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | 
-(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
-(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let 
-H5 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Flat f) u 
+t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal 
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) 
+(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let 
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
 with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) | 
-(THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift 
-h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind 
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat f) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)) H3) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T u 
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))))) (\lambda (H7: (eq T u (lift h d t0))).(eq_ind 
 T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to 
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) 
 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d 
 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda 
-(H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) 
-t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(H8: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) 
+t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift 
+h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
 (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h 
-d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat 
-f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift 
-h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s 
-(Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1)) 
-(refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t 
-(lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) 
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h 
+(s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 
+t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t 
+(lift h (s (Flat f) d) t1) H8))) u (sym_eq T u (lift h d t0) H7))) k H6)) 
+H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
 t1)))))))))))))) x)))).
 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
index 7c205f57a..3e2301883 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
@@ -45,40 +45,41 @@ h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 
 t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K 
 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) 
 (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def 
 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) 
-\Rightarrow t])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) 
-in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow 
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true 
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) 
-(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef 
-i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
-\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) 
-(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 
-d) t1)) | (TLRef _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to 
-nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow 
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true 
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow 
-(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda 
-(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) 
-(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef 
-i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
-\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) 
-(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 
-d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k v (lift h d (THead k0 t0 
-t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k 
-k0)).(let H7 \def (eq_ind K k (\lambda (k: K).(\forall (v: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P: 
-Prop).P)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) 
-(\lambda (t: T).(eq T t t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-(lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) 
-H2)))))))))))) t)).
+[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) 
+\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) 
+H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map 
+(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort 
+n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) 
+with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) 
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in 
+lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat 
+\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) 
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in 
+lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow 
+(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T 
+\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
+(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f 
+i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f 
+(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec 
+lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with 
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i 
+d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) 
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in 
+lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) 
+\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) 
+H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def 
+(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall 
+(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: 
+Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 
+t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 
+d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) 
+H3)) H2)))))))))))) t)).
 
 theorem lift_r:
  \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
@@ -94,11 +95,11 @@ t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
 K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) 
 t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) 
 t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) 
-t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1) 
-(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d 
-t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead 
-k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d 
-t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) 
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) 
+(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) 
+(lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift 
+O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1 
+(lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) 
 (sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1)) 
 (lift_head k t0 t1 O d)))))))) t).
 
@@ -135,17 +136,17 @@ h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
 (lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 
 t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
 T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) 
-(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to 
-(eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: 
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) 
+\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: 
 nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t 
 t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
 T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
-(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall 
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0 
-t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
 (eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
-(lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1 
+(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
 (THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in 
 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
@@ -154,32 +155,31 @@ z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
 (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift 
 h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r 
 T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) 
-(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead 
-(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1) 
-(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 
-(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 
-h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) 
-t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall 
-(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d 
-t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to 
-(eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d 
-t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t: 
-T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0)) 
-(lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq 
-T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d 
-t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift 
-h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift 
-h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d 
-x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat 
-f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0) 
-(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead 
-(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat 
-f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T 
-t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d 
-t0) t1 h d H2)))))))))))) k)) x).
+(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind 
+b) t t0) (THead (Bind b) x0 x1) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind 
+b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K 
+(Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h (S d) 
+H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d 
+H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
+(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
+(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
+(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T 
+(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq 
+T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d 
+x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead 
+(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T 
+(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) 
+(THead (Flat f) x0 x1) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0) 
+(f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f)) 
+(sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) 
+(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
 
 theorem lift_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: 
@@ -242,8 +242,8 @@ H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt
 (plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2) 
 (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 
 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: 
-(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n: 
-nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) 
+(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n0: 
+nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) 
 h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1 
 (le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) 
 (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda 
@@ -283,9 +283,9 @@ t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
 x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 
 t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t 
 t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead 
-(Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead 
-(Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y 
+(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 
+(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 
 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 
 (S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) 
@@ -333,7 +333,7 @@ H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S
 x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S 
 d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift 
 h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind 
-T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus 
+T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus 
 d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t 
 t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead 
 (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift 
@@ -413,12 +413,12 @@ nat T TLRef (plus (plus n h) k) (plus n (plus k h))
 (lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
 (plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1 
 (le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda 
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k: 
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
-(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1: 
-T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
-(lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda 
+(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda 
+(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e 
+(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda 
 (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d 
 h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k 
 d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 
@@ -489,13 +489,13 @@ n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
 (plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1 
 (le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) 
 (lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda 
-(H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: 
-nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h 
-d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k: 
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k 
-d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda 
-(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e 
-d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: 
+(H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: 
+nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift 
+h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall 
+(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h 
+(plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: 
+nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le 
+e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: 
 T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) 
 (eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus 
 k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma
index b67722f58..f03f26618 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma
@@ -141,14 +141,14 @@ d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
 (TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: 
 T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
 (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) 
-(weight_map g t0))) (sym_equal nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef 
+(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef 
 n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d 
 H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: 
 T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
 (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f 
 (TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda 
-(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_equal nat (g (S (plus n h))) 
-(f (plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) 
+(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f 
+(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) 
 (plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift 
 h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda 
 (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat 
@@ -195,32 +195,51 @@ d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
 t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m 
 O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift 
 (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat 
-nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) 
-(sym_equal nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) 
-(H h d f g H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq 
-nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda (m0: 
-nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S 
-(weight_map g (lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) 
-m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x 
-d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g 
-(lift (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x 
-H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: 
-(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: 
-nat).(le d n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) 
-(\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d 
-x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g n) (wadd f (S 
-(weight_map f (lift h d t0))) n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) 
-(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) 
-(lift h (S d) t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd 
-g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h 
-d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) 
-t1)) (weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h 
-(S d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S 
+nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq 
+nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g 
+H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S 
+m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat 
+m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g 
+(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda 
+(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r 
+nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d 
+t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) 
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
+n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: 
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) 
+n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus 
+(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) 
+t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g 
+(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map 
+(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) 
+(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) 
+(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) 
+(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat 
+O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m 
+H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda 
+(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) 
+(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: 
+nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) 
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
+n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S 
+x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g 
+n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat 
+S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) 
+t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S 
+h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) 
+(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S 
+d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S 
 d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) 
 (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda 
 (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) 
 (wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: 
-nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda 
+nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda 
 (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O 
 m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda 
 (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) 
@@ -229,40 +248,21 @@ nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
 n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: 
 nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
 x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) 
-(le_gen_S d m H4))))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) 
-(weight_map (wadd f O) (lift h (S d) t1))) (plus (weight_map g (lift (S h) d 
-t0)) (weight_map (wadd g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat 
-plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) 
-(weight_map (wadd f O) (lift h (S d) t1)) (weight_map (wadd g O) (lift (S h) 
-(S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) (wadd g O) (\lambda 
-(m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) (ex2 nat (\lambda 
-(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O 
-m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: 
-nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m H5)) (\lambda 
-(H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m 
-d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: 
-nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: nat).(\lambda 
-(H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda 
-(n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) H5)) 
-(lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
-n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S 
-x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g 
-n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) b) (lift (S h) d 
-(THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 (S h) d)) (lift h d (THead 
-(Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r 
-T (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2: 
-T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead (Flat f0) t0 
-t1))))) (eq_ind_r T (THead (Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat 
-f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Flat f0) (lift h d 
-t0) (lift h (s (Flat f0) d) t1))) (weight_map g t2))) (f_equal nat nat S 
-(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus 
-(weight_map g (lift (S h) d t0)) (weight_map g (lift (S h) d t1))) (f_equal2 
-nat nat nat plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d 
-t0)) (weight_map f (lift h d t1)) (weight_map g (lift (S h) d t1)) (H h d f g 
-H1 H2 H3) (H0 h d f g H1 H2 H3))) (lift (S h) d (THead (Flat f0) t0 t1)) 
-(lift_head (Flat f0) t0 t1 (S h) d)) (lift h d (THead (Flat f0) t0 t1)) 
-(lift_head (Flat f0) t0 t1 h d))) k))))))))))))) t)).
+(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head 
+(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind 
+b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) 
+(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) 
+(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead 
+(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: 
+T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) 
+d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d 
+t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) 
+(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f 
+(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) 
+(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) 
+(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) 
+(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
+k))))))))))))) t)).
 
 theorem lift_weight_add_O:
  \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma
index 761030b2a..377c4ea39 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma
@@ -145,7 +145,7 @@ t0)) \to (\forall (P0: Prop).P0)))) H10 (THead (Bind Abst) x5 x6) H24) in
 x6) H24) in (let H29 \def (nf2_gen_abst c x5 x6 H28) in (and_ind (nf2 c x5) 
 (nf2 (CHead c (Bind Abst) x5) x6) P (\lambda (H30: (nf2 c x5)).(\lambda (_: 
 (nf2 (CHead c (Bind Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 
-x5 H30 u2 H7) in (H27 x6 (sym_equal T (THead (Bind Abst) x0 x6) (THead (Bind 
+x5 H30 u2 H7) in (H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind 
 Abst) x5 x6) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 
 (refl_equal K (Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T 
 x6))) P)))) H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma
index 012d5b0ee..db6cf7df6 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma
@@ -20,11 +20,15 @@ include "pr2/defs.ma".
 
 include "pr0/subst1.ma".
 
-include "csubst1/defs.ma".
+include "pr0/fwd.ma".
+
+include "csubst1/getl.ma".
+
+include "csubst1/fwd.ma".
 
 include "subst1/subst1.ma".
 
-include "getl/props.ma".
+include "getl/drop.ma".
 
 theorem pr2_delta1:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
@@ -121,12 +125,166 @@ H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0
 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) 
 (refl_equal T t1) (refl_equal T t2)))))))))).
 
-axiom pr2_gen_cabbr:
+theorem pr2_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
 \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
 a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
 (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
 x1 x2))))))))))))))))
-.
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: 
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to 
+(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 
+a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda 
+(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 
+x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
+d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d 
+x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: 
+T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 
+(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda 
+(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T 
+(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
+x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda 
+(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) 
+H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 
+(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 
+H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S 
+O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: 
+T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda 
+(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) 
+(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: 
+T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) 
+x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x 
+(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: 
+T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) 
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: 
+(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) 
+H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 
+d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: 
+(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: 
+T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 
+t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: 
+T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) 
+d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d 
+(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: 
+T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) 
+(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) 
+u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) 
+(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 
+(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) 
+d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
+(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 
+t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: 
+T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) 
+x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 
+(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl 
+i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 
+(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) 
+(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 
+u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: 
+T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) 
+x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop 
+(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: 
+T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) 
+H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) 
+x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S 
+i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 
+x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: 
+(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda 
+(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind 
+nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S 
+O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in 
+(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S 
+i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: 
+T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) 
+(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S 
+i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 
+(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift 
+(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat 
+(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S 
+i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let 
+H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: 
+nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) 
+in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) 
+(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) 
+d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) 
+(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) 
+(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S 
+i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i 
+H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 
+c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i 
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) 
+(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: 
+nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def 
+(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 
+\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in 
+(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind 
+Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: 
+T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let 
+H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) 
+H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead 
+e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ 
+_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) 
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in 
+((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind 
+Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d 
+e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind 
+Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: 
+T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r 
+T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u 
+(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) 
+(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: 
+C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda 
+(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) 
+t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: 
+T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t 
+x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind 
+T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) 
+(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) 
+(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i 
+(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i 
+x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) 
+(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) 
+H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T 
+(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S 
+O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) 
+(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 
+u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T 
+(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 
+(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) 
+d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq 
+T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 
+x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 
+(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t 
+(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u 
+(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 
+(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d 
+(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: 
+nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3 
+H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S 
+O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S 
+O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift 
+(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) 
+(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 
+x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma
index c09ce6c95..8e1e4e5b9 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma
@@ -38,8 +38,8 @@ T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t
 u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) 
 (plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: 
 T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) 
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
 (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: 
 T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: 
 nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i 
@@ -68,13 +68,13 @@ u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T
 (plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 
 u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 
 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u2: 
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda 
-(t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t 
+(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: 
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda 
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t 
 u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: 
-(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: 
-T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda (t: T).(subst0 
-(s k i) u1 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t 
+(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: 
+T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 
+(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t 
 t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: 
 (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
 (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: 
@@ -114,8 +114,8 @@ T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift
 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 
 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda 
 (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: 
-((\forall (u2: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u2) \to 
-(ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus 
+((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to 
+(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus 
 i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: 
 T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v 
 u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: 
@@ -144,12 +144,12 @@ t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead
 k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 
 u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: 
 T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u2) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) 
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
 (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: 
 K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 
-t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 t0 t)) 
+t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
 (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: 
 T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v 
 u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: 
@@ -191,21 +191,21 @@ u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t)))
 (\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
 T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
 T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda 
-(H3: (ex2 T (\lambda (u2: T).(eq T t3 (THead k u2 t))) (\lambda (u3: 
+(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: 
 T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 
 t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) 
 (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 
 i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead 
 k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda 
-(H3: (ex2 T (\lambda (t2: T).(eq T t3 (THead k u2 t2))) (\lambda (t2: 
-T).(subst0 (s k i0) v0 t t2)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k 
+(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: 
+T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k 
 u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k 
 u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda 
 (H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: 
 T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 
-H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-t3 (THead k u2 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t t2))))).(ex3_2_ind T T 
+H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T 
+t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) 
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T 
 (\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
 T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
 T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: 
@@ -229,16 +229,16 @@ T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0
 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x 
 t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda 
 (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) 
-t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u 
-t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t0 t2)))).(ex2_ind T (\lambda (t5: 
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u 
+t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: 
 T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) 
 (subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 
 (THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k 
 u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 
 i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 
-t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k 
+T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 
+t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k 
 u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) 
 t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 
@@ -257,22 +257,22 @@ v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4:
 t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 
-t0) t4) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) 
+t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) 
 (\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 
 (THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead 
 k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda 
 (H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 
 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 
-H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u2 t2))) 
-(\lambda (t2: T).(subst0 (s k i0) v0 t3 t2)))).(ex2_ind T (\lambda (t5: 
+H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) 
+(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: 
 T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) 
 (subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 
 (THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead 
 k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 
 u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i0) v0 t3 t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda 
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda 
 (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 
 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) 
 (subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
@@ -318,8 +318,8 @@ nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t)))
 T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: 
 T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead 
 k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead k u2 t))) (\lambda (u2: T).(subst0 i2 u3 u0 
-u2)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
+(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 
+u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
 T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) 
 t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq 
 T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k 
@@ -332,7 +332,7 @@ k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda
 u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) 
 (subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 
 H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead 
-k u0 t3))) (\lambda (t2: T).(subst0 (s k i2) u3 t t2)))).(ex2_ind T (\lambda 
+k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda 
 (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t 
 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
 T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 
@@ -341,10 +341,10 @@ x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x)
 (\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 
 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) 
 (THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) 
-t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i2) u3 t 
-t2))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
+t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq 
+T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 
+u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t 
+t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: 
 T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) 
@@ -373,7 +373,7 @@ t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: 
 T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (H4: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) (\lambda 
+(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda 
 (u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k 
 u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 
 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
@@ -383,8 +383,8 @@ T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
 (ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: 
 T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 
 k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T 
-(\lambda (t2: T).(eq T t4 (THead k u t2))) (\lambda (t2: T).(subst0 (s k i2) 
-u2 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda 
+(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) 
+u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda 
 (t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 
 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
 T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 
@@ -398,9 +398,9 @@ T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
 v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) 
 (subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (\lambda (H7: (eq nat 
 (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))) t4 H5)))) H4)) (\lambda (H4: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) 
 (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i2) u2 t3 t2))))).(ex3_2_ind T T (\lambda 
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda 
 (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
 T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: 
 T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k 
@@ -423,8 +423,8 @@ T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq
 nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: 
 T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: 
 T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: ((\forall (t4: 
-T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq 
-nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t3 t)) (\lambda (t: 
+T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) \to ((not (eq 
+nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 t)) (\lambda (t: 
 T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: T).(\lambda (u3: 
 T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k u0 t2) 
 t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq 
@@ -434,8 +434,8 @@ u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: 
 T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda 
-(u2: T).(subst0 i2 u3 u0 u2)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k 
+(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda 
+(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k 
 u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: 
 T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
 (\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 
@@ -448,8 +448,8 @@ u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t:
 T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x 
 t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 
 i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T 
-(\lambda (t2: T).(eq T t4 (THead k u0 t2))) (\lambda (t3: T).(subst0 (s k i2) 
-u3 t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda 
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) 
+u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda 
 (t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 
 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
 T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 
@@ -463,9 +463,9 @@ x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
 (t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 
 i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 
 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H9))))) t4 H7)))) 
-H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 
-(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t2 t3))))).(ex3_2_ind T 
+H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 
+(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))))).(ex3_2_ind T 
 T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: 
 T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: 
 T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k 
@@ -521,8 +521,8 @@ t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
 T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) 
 (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
 T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
-(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 
-t))) (\lambda (u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq 
+(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 
+t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
 T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead 
 k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
 (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
@@ -543,77 +543,78 @@ t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x
 t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) 
 (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) 
 (THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead 
-k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) 
-(\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v 
-u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) (THead 
-k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k 
-x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x x0)).(or4_intro1 (eq T 
-(THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k 
-u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v 
-(THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) 
-(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) (subst0_fst v x0 u2 i0 H7 t 
-k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) (\lambda (H6: (subst0 i0 v u2 
-x)).(or4_intro2 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: 
+k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) 
+(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 
+v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) 
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: 
+T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x 
+x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: 
 T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x 
 t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x 
-t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t k))) (\lambda (H6: (subst0 i0 v 
-x u2)).(or4_intro3 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: 
-T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x 
-t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x 
-t) (THead k u2 t)) (subst0_fst v u2 x i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) 
-(\lambda (H3: (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda 
-(t2: T).(subst0 (s k i0) v t t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t t3)) (or4 (eq T 
-(THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
-(\lambda (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) 
-(subst0 i0 v t2 (THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 
-(THead k u1 x))).(\lambda (H5: (subst0 (s k i0) v t x)).(eq_ind_r T (THead k 
-u1 x) (\lambda (t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: 
-T).(subst0 i0 v (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) 
-(subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind 
-(eq T u2 u2) (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: 
-T).(subst0 i0 v u2 t3))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T 
-(THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k 
-u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v 
-(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))) 
-(\lambda (_: (eq T u2 u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) 
+t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) 
+t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) 
+(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) 
+(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x 
+t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x 
+t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t 
+k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) 
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x 
+i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: 
+T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda 
+(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 
+t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) 
+(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 
+(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T 
+(THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) 
+(\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) 
+(subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 u2) (ex2 T (\lambda (t3: 
+T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3))) (subst0 i0 v 
+u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T 
+(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 
+v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 
+v (THead k u1 x) (THead k u2 t))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq 
+T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead 
+k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v 
+(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) 
+(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 
+u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: 
+T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T 
+(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) 
+(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 
+v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) 
+(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) 
+(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda 
+(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) 
 (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
 T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 
 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: 
 T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 
 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 
-H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda 
-(t: T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v u2 t3)) 
-(\lambda (t3: T).(subst0 i0 v u2 t3)) (or4 (eq T (THead k u2 t) (THead k u1 
-x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 
-x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))) (\lambda (x0: T).(\lambda 
-(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T 
-(THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k 
-u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v 
-(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) 
-(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 
-u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 
+H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead 
+k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) 
+t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k 
+u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 
+T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 
+i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) 
+(subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: (subst0 i0 v u2 
 u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: 
 T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 
 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 
 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) 
 t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) 
-(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: 
-(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T 
-(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 
-v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 
-v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v 
-(THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead 
-k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 
-u2 H0)) t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 
-i0 v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v t 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) 
+t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
+T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 
+u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t 
+t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) 
 (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
@@ -636,9 +637,9 @@ x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1))
 (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: 
 T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 
 x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 
-x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) 
-(\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 
-v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) 
+x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) 
+(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 
+i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) 
 (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
 (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 
 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda 
@@ -721,8 +722,8 @@ t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x
 t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 
 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) 
 (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0)) 
-t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u0 
-t2))) (\lambda (t2: T).(subst0 (s k i0) v t3 t2)))).(ex2_ind T (\lambda (t5: 
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u0 
+t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))).(ex2_ind T (\lambda (t5: 
 T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)) 
 (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k 
 u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) 
@@ -766,9 +767,9 @@ x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T
 u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v 
 (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) 
 (subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) 
 (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i0) v t3 t2))))).(ex3_2_ind T T (\lambda 
+T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))))).(ex3_2_ind T T (\lambda 
 (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: 
 T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5: 
 T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda 
@@ -834,8 +835,8 @@ T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5:
 T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T (THead k u2 t3) t4) (ex2 T 
 (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v 
 t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 (THead k u2 t3))) 
-(\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda 
-(u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k 
+(\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t2))) (\lambda 
+(u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k 
 u3 t2))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t3) t4) 
 (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
 T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 
@@ -1002,8 +1003,8 @@ k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T
 u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v 
 (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) 
 (subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6)))) 
-H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u1 t2))) 
-(\lambda (t3: T).(subst0 (s k i0) v t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq 
+H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) 
+(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq 
 T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T 
 (THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) 
 (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 
@@ -1159,10 +1160,10 @@ u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
 T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 
 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 
 x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 
-x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t2 
-t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
+x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: 
+T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v 
+u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) 
 t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
@@ -1353,12 +1354,12 @@ t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t:
 T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S 
 O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) 
 (\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def 
-(sym_equal T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) 
-i H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) 
-t)) (\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda 
-(t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S 
-O) i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S 
-O) i t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) 
+(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i 
+H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) 
+(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: 
+T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) 
+i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i 
+t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) 
 x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S 
 O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) 
 (plus_comm i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma
index 182118963..0c54f9156 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma
@@ -176,11 +176,11 @@ y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i t1)) \to
 (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H2: 
 (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda (t: 
 T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 \def 
-(sym_equal T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u 
-(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda 
-(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) 
-H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: 
-(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: 
+(sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u (lift 
+(S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: 
+nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H3)) 
+in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: (subst0 i 
+u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: 
 T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 
 (\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T 
 (lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (eq T t1 t3) (\lambda 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma
index 23f81bacb..73069bcd5 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma
@@ -161,12 +161,12 @@ A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda
 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1: 
 A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1 
 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g 
-(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0) 
-(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in 
-(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A 
-x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) 
-a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u 
-t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 
+(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) (AHead 
+x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A 
+(\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g 
+a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) 
+(asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 
 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 
 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) 
 in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma
index eb09bb829..094ce7ecb 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma
@@ -498,7 +498,7 @@ Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0
 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
 T).(\lambda (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 
 x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind 
-Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind 
+Abst) d) u0 t x5)).(let H18 \def (sym_eq T (lift (S O) d x1) (THead (Bind 
 Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 
 (THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S 
 O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z)))) 
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma
deleted file mode 100644
index 32e613ebb..000000000
--- a/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma
+++ /dev/null
@@ -1,183 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* Problematic objects for disambiguation/typechecking ********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
-
-include "LambdaDelta/theory.ma".
-
-theorem pr2_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
-(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
-x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: 
-C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to 
-(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 
-a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda 
-(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 
-x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
-C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
-d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d 
-x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: 
-T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
-x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 
-(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda 
-(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T 
-(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
-x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda 
-(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) 
-H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 
-(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 
-H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S 
-O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
-(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
-a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: 
-T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda 
-(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) 
-(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: 
-T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) 
-x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x 
-(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: 
-T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) 
-(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: 
-(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) 
-H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 
-d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: 
-(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: 
-T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 
-t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: 
-T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) 
-d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d 
-(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: 
-T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) 
-(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) 
-u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) 
-(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 
-(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) 
-d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
-(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
-C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 
-t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: 
-T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) 
-x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 
-(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl 
-i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 
-(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) 
-(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
-C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 
-u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: 
-T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) 
-x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop 
-(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: 
-T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) 
-H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) 
-x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S 
-i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 
-x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: 
-(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda 
-(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind 
-nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S 
-O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in 
-(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S 
-i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: 
-T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) 
-(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S 
-i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 
-(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift 
-(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat 
-(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S 
-i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let 
-H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: 
-nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) 
-in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) 
-(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) 
-d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) 
-(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) 
-(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S 
-i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i 
-H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 
-c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i 
-(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) 
-(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: 
-nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def 
-(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 
-\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in 
-(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind 
-Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: 
-T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let 
-H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) 
-H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead 
-e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match 
-e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ 
-_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) 
-(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in 
-((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda 
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
-(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind 
-Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d 
-e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind 
-Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: 
-T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r 
-T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u 
-(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) 
-(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: 
-C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda 
-(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) 
-t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: 
-T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t 
-x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind 
-T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) 
-(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) 
-(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i 
-(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i 
-x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) 
-(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) 
-H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T 
-(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S 
-O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) 
-(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 
-u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T 
-(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 
-(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) 
-d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq 
-T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 
-x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 
-(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t 
-(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u 
-(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 
-(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d 
-(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: 
-nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3 
-H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S 
-O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S 
-O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift 
-(S O) d0 x0) u0 d0 H11 (sym_not_equal nat d0 i (lt_neq d0 i H12)))))))))) 
-(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 
-x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
-