From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 6 Sep 2015 17:42:29 +0000 (+0000)
Subject: flavour and source information exported for the objects of lambdadelta version 1
X-Git-Tag: make_still_working~698
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=8de8cf8adfa6fcda91047eb2c25535893ede046a;p=helm.git

flavour and source information exported for the objects of lambdadelta version 1
---

diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma
index c08af2080..a0f6f0931 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma
@@ -16,13 +16,13 @@
 
 include "basic_1/A/defs.ma".
 
-let rec A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall (n0: 
-nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: A).((P 
-a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with [(ASort n 
-n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (f0 a0 ((A_rect P f f0) 
-a0) a1 ((A_rect P f f0) a1))].
+implied let rec A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall 
+(n0: nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: 
+A).((P a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with 
+[(ASort n n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (f0 a0 
+((A_rect P f f0) a0) a1 ((A_rect P f f0) a1))].
 
-theorem A_ind:
+implied lemma A_ind:
  \forall (P: ((A \to Prop))).(((\forall (n: nat).(\forall (n0: nat).(P (ASort 
 n n0))))) \to (((\forall (a: A).((P a) \to (\forall (a0: A).((P a0) \to (P 
 (AHead a a0))))))) \to (\forall (a: A).(P a))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma
index d2802e2ba..675a0863b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma
@@ -16,19 +16,19 @@
 
 include "basic_1/C/defs.ma".
 
-let rec C_rect (P: (C \to Type[0])) (f: (\forall (n: nat).(P (CSort n)))) 
-(f0: (\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k 
-t))))))) (c: C) on c: P c \def match c with [(CSort n) \Rightarrow (f n) | 
-(CHead c0 k t) \Rightarrow (f0 c0 ((C_rect P f f0) c0) k t)].
+implied let rec C_rect (P: (C \to Type[0])) (f: (\forall (n: nat).(P (CSort 
+n)))) (f0: (\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P 
+(CHead c k t))))))) (c: C) on c: P c \def match c with [(CSort n) \Rightarrow 
+(f n) | (CHead c0 k t) \Rightarrow (f0 c0 ((C_rect P f f0) c0) k t)].
 
-theorem C_ind:
+implied lemma C_ind:
  \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
 (((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k 
 t))))))) \to (\forall (c: C).(P c))))
 \def
  \lambda (P: ((C \to Prop))).(C_rect P).
 
-theorem clt_wf__q_ind:
+fact clt_wf__q_ind:
  \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to 
 Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 
 c))))) P n))) \to (\forall (c: C).(P c)))
@@ -39,7 +39,7 @@ Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
 n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight 
 c)))))).
 
-theorem clt_wf_ind:
+lemma clt_wf_ind:
  \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) 
 \to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/props.ma b/matita/matita/contribs/lambdadelta/basic_1/C/props.ma
index 1cbfa5414..dacd5482b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/C/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/C/props.ma
@@ -18,7 +18,7 @@ include "basic_1/C/fwd.ma".
 
 include "basic_1/T/props.ma".
 
-theorem cle_r:
+lemma cle_r:
  \forall (c: C).(cle c c)
 \def
  \lambda (c: C).(C_ind (\lambda (c0: C).(le (cweight c0) (cweight c0))) 
@@ -26,7 +26,7 @@ theorem cle_r:
 (cweight c0))).(\lambda (_: K).(\lambda (t: T).(le_n (plus (cweight c0) 
 (tweight t))))))) c).
 
-theorem cle_head:
+lemma cle_head:
  \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (u1: T).(\forall 
 (u2: T).((tle u1 u2) \to (\forall (k: K).(cle (CHead c1 k u1) (CHead c2 k 
 u2))))))))
@@ -36,7 +36,7 @@ c2))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (le (tweight u1)
 (tweight u2))).(\lambda (_: K).(le_plus_plus (cweight c1) (cweight c2) 
 (tweight u1) (tweight u2) H H0))))))).
 
-theorem cle_trans_head:
+lemma cle_trans_head:
  \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (k: K).(\forall 
 (u: T).(cle c1 (CHead c2 k u))))))
 \def
@@ -44,7 +44,7 @@ theorem cle_trans_head:
 c2))).(\lambda (_: K).(\lambda (u: T).(le_plus_trans (cweight c1) (cweight 
 c2) (tweight u) H))))).
 
-theorem clt_cong:
+lemma clt_cong:
  \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: 
 T).(clt (CHead c k t) (CHead d k t))))))
 \def
@@ -52,14 +52,14 @@ T).(clt (CHead c k t) (CHead d k t))))))
 d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) 
 (tweight t) H))))).
 
-theorem clt_head:
+lemma clt_head:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
 \def
  \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight 
 c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_reg_l O 
 (tweight u) (cweight c) (tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
 
-theorem chead_ctail:
+lemma chead_ctail:
  \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: 
 K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
 \def
@@ -83,7 +83,7 @@ C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
 (CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 
 k t) H1))))) H0))))))))) c).
 
-theorem clt_thead:
+lemma clt_thead:
  \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
 \def
  \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt 
@@ -91,7 +91,7 @@ c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
 C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: 
 T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
 
-theorem c_tail_ind:
+lemma c_tail_ind:
  \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
 (((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t 
 c))))))) \to (\forall (c: C).(P c))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
index 277e12bf9..ae783b822 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/T/fwd.ma".
 
-theorem terms_props__bind_dec:
+fact terms_props__bind_dec:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -55,7 +55,7 @@ Void \Rightarrow True])) I Abst H) in (False_ind P H0))))) (or_introl (eq B
 Void Void) ((eq B Void Void) \to (\forall (P: Prop).P)) (refl_equal B Void)) 
 b2)) b1).
 
-theorem bind_dec_not:
+lemma bind_dec_not:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
 \def
  \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
@@ -65,7 +65,7 @@ b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
 (((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
 b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
 
-theorem terms_props__flat_dec:
+fact terms_props__flat_dec:
  \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -85,7 +85,7 @@ Prop).P)) (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let H0 \def
 Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) 
 f1).
 
-theorem terms_props__kind_dec:
+fact terms_props__kind_dec:
  \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -130,7 +130,7 @@ f) (Flat f0) H1) in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1)
 \to (\forall (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) 
 k2))) k1).
 
-theorem term_dec:
+lemma term_dec:
  \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -272,7 +272,7 @@ H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
 (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 
 (refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
 
-theorem binder_dec:
+lemma binder_dec:
  \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
 T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall 
 (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
@@ -336,7 +336,7 @@ t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) 
 in (False_ind P H2))))))))))))) k)) t).
 
-theorem abst_dec:
+lemma abst_dec:
  \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead 
 (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to 
 (\forall (P: Prop).P)))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma
index 500f4f758..159a24468 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma
@@ -16,14 +16,14 @@
 
 include "basic_1/T/defs.ma".
 
-let rec T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort n)))) 
-(f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall (t: 
-T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t: T) on 
-t: P t \def match t with [(TSort n) \Rightarrow (f n) | (TLRef n) \Rightarrow 
-(f0 n) | (THead k t0 t1) \Rightarrow (f1 k t0 ((T_rect P f f0 f1) t0) t1 
-((T_rect P f f0 f1) t1))].
+implied let rec T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort 
+n)))) (f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall 
+(t: T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t: 
+T) on t: P t \def match t with [(TSort n) \Rightarrow (f n) | (TLRef n) 
+\Rightarrow (f0 n) | (THead k t0 t1) \Rightarrow (f1 k t0 ((T_rect P f f0 f1) 
+t0) t1 ((T_rect P f f0 f1) t1))].
 
-theorem T_ind:
+implied lemma T_ind:
  \forall (P: ((T \to Prop))).(((\forall (n: nat).(P (TSort n)))) \to 
 (((\forall (n: nat).(P (TLRef n)))) \to (((\forall (k: K).(\forall (t: T).((P 
 t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) \to (\forall (t: 
@@ -31,7 +31,7 @@ T).(P t)))))
 \def
  \lambda (P: ((T \to Prop))).(T_rect P).
 
-theorem thead_x_y_y:
+lemma thead_x_y_y:
  \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to 
 (\forall (P: Prop).P))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/props.ma b/matita/matita/contribs/lambdadelta/basic_1/T/props.ma
index 445f6fc52..7b62a5a15 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/T/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/T/props.ma
@@ -16,35 +16,35 @@
 
 include "basic_1/T/fwd.ma".
 
-theorem not_abbr_abst:
+lemma not_abbr_abst:
  not (eq B Abbr Abst)
 \def
  \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
 B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
 \Rightarrow False])) I Abst H) in (False_ind False H0)).
 
-theorem not_void_abst:
+lemma not_void_abst:
  not (eq B Void Abst)
 \def
  \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: 
 B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
 \Rightarrow True])) I Abst H) in (False_ind False H0)).
 
-theorem not_abbr_void:
+lemma not_abbr_void:
  not (eq B Abbr Void)
 \def
  \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
 B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
 \Rightarrow False])) I Void H) in (False_ind False H0)).
 
-theorem not_abst_void:
+lemma not_abst_void:
  not (eq B Abst Void)
 \def
  \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: 
 B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void 
 \Rightarrow False])) I Void H) in (False_ind False H0)).
 
-theorem tweight_lt:
+lemma tweight_lt:
  \forall (t: T).(lt O (tweight t))
 \def
  \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: 
@@ -53,7 +53,7 @@ nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda
 (tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S 
 O) (tweight t0) (tweight t1) H))))))) t).
 
-theorem tle_r:
+lemma tle_r:
  \forall (t: T).(tle t t)
 \def
  \lambda (t: T).(T_ind (\lambda (t0: T).(le (tweight t0) (tweight t0))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
index 9eec9c6e8..73ec98cfa 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
@@ -20,7 +20,7 @@ include "basic_1/A/fwd.ma".
 
 include "basic_1/next_plus/props.ma".
 
-theorem aplus_reg_r:
+lemma aplus_reg_r:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall 
 (h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A 
 (aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
@@ -32,7 +32,7 @@ nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
 h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n 
 h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
 
-theorem aplus_assoc:
+lemma aplus_assoc:
  \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
 (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
 \def
@@ -51,7 +51,7 @@ n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
 n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) 
 (plus_n_Sm n n0)))) h2)))) h1))).
 
-theorem aplus_asucc:
+lemma aplus_asucc:
  \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) 
 h) (asucc g (aplus g a h)))))
 \def
@@ -60,7 +60,7 @@ h) (asucc g (aplus g a h)))))
 (refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) 
 (aplus_assoc g a (S O) h)))).
 
-theorem aplus_sort_O_S_simpl:
+lemma aplus_sort_O_S_simpl:
  \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O 
 n) (S k)) (aplus g (ASort O (next g n)) k))))
 \def
@@ -69,7 +69,7 @@ g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
 (refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) 
 k)) (aplus_asucc g k (ASort O n))))).
 
-theorem aplus_sort_S_S_simpl:
+lemma aplus_sort_S_S_simpl:
  \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A 
 (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
 \def
@@ -78,7 +78,7 @@ A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
 (ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g 
 (ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
 
-theorem aplus_asort_O_simpl:
+lemma aplus_asort_O_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O 
 n) h) (ASort O (next_plus g n h)))))
 \def
@@ -93,7 +93,7 @@ g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
 (next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n 
 (ASort O n0)))))) h)).
 
-theorem aplus_asort_le_simpl:
+lemma aplus_asort_le_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h 
 k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
 \def
@@ -120,7 +120,7 @@ A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n n0 (le_S_n h0 n H1)) (asucc
 g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g h0 (ASort (S n) n0))))))) 
 k)))) h)).
 
-theorem aplus_asort_simpl:
+lemma aplus_asort_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A 
 (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
 \def
@@ -147,7 +147,7 @@ n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
 (next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) 
 (aplus_asort_le_simpl g h k n H))))))).
 
-theorem aplus_ahead_simpl:
+lemma aplus_ahead_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A 
 (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
 \def
@@ -163,7 +163,7 @@ A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda
 a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) 
 h)).
 
-theorem aplus_asucc_false:
+lemma aplus_asucc_false:
  \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) 
 h) a) \to (\forall (P: Prop).P))))
 \def
@@ -209,7 +209,7 @@ a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 (AHead a0 (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:
+lemma aplus_inj:
  \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A 
 (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
index 2e8391731..43f3bcec5 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
@@ -16,15 +16,15 @@
 
 include "basic_1/aprem/defs.ma".
 
-let rec aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall (a1: 
-A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: A).(\forall 
-(a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to (\forall (a1: 
-A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) (a1: aprem n a 
-a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) \Rightarrow (f a2 
-a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4 ((aprem_ind P f 
-f0) i a2 a3 a4) a5)].
+implied let rec aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall 
+(a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: 
+A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to 
+(\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) 
+(a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) 
+\Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4 
+((aprem_ind P f f0) i a2 a3 a4) a5)].
 
-theorem aprem_gen_sort:
+lemma aprem_gen_sort:
  \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem 
 i (ASort h n) x) \to False))))
 \def
@@ -43,7 +43,7 @@ A (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow
 False | (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False 
 H4))))))))) i y x H0))) H))))).
 
-theorem aprem_gen_head_O:
+lemma aprem_gen_head_O:
  \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) 
 \to (eq A x a1))))
 \def
@@ -74,7 +74,7 @@ a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee
 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
 (eq A a a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
 
-theorem aprem_gen_head_S:
+lemma aprem_gen_head_S:
  \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem 
 (S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma
index e2fd9ad94..1932956b1 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma
@@ -18,7 +18,7 @@ include "basic_1/aprem/fwd.ma".
 
 include "basic_1/leq/fwd.ma".
 
-theorem aprem_repl:
+lemma aprem_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
 (i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g 
 b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
@@ -56,7 +56,7 @@ A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0
 a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 
 H)))).
 
-theorem aprem_asucc:
+lemma aprem_asucc:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i 
 a1 a2) \to (aprem i (asucc g a1) a2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma
index f91be4fff..442660739 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma
@@ -20,7 +20,7 @@ include "basic_1/arity/cimp.ma".
 
 include "basic_1/aprem/props.ma".
 
-theorem arity_aprem:
+lemma arity_aprem:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat 
 (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma
index 797f16001..e9222f5bb 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma
@@ -18,7 +18,7 @@ include "basic_1/arity/fwd.ma".
 
 include "basic_1/cimp/props.ma".
 
-theorem arity_cimp_conf:
+lemma arity_cimp_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
 t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma
index 77eaa993b..58775a829 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma
@@ -20,12 +20,12 @@ include "basic_1/leq/asucc.ma".
 
 include "basic_1/getl/drop.ma".
 
-let rec arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f: (\forall (c: 
-C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0: (\forall (c: 
-C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
-(Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a) \to (P c 
-(TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall (u: 
-T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: 
+implied let rec arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f: 
+(\forall (c: C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0: 
+(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a) 
+\to (P c (TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall 
+(u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: 
 A).((arity g d u (asucc g a)) \to ((P d u (asucc g a)) \to (P c (TLRef i) 
 a)))))))))) (f2: (\forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
 C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to 
@@ -61,7 +61,7 @@ a3) a4)) | (arity_cast c0 u a1 a2 t0 a3) \Rightarrow (f5 c0 u a1 a2
 a2 a3 l) \Rightarrow (f6 c0 t0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) 
 c0 t0 a1 a2) a3 l)].
 
-theorem arity_gen_sort:
+lemma arity_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c 
 (TSort n) a) \to (leq g a (ASort O n))))))
 \def
@@ -130,7 +130,7 @@ H7 \def (eq_ind T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TSort n) H5) in
 (leq_trans g a2 a1 (leq_sym g a1 a2 H3) (ASort O n) (H6 (refl_equal T (TSort 
 n))))))))))))))) c y a H0))) H))))).
 
-theorem arity_gen_lref:
+lemma arity_gen_lref:
  \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c 
 (TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c 
 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) 
@@ -324,7 +324,7 @@ T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
 (asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) 
 (asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
 
-theorem arity_gen_bind:
+lemma arity_gen_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: 
 C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind 
 b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: 
@@ -485,7 +485,7 @@ c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
 (arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y 
 a2 H1))) H0)))))))).
 
-theorem arity_gen_abst:
+lemma arity_gen_abst:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
 A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: 
 A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
@@ -698,7 +698,7 @@ A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
 (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) 
 H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
 
-theorem arity_gen_appl:
+lemma arity_gen_appl:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: 
 A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity 
 g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
@@ -815,7 +815,7 @@ A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10
 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c 
 y a2 H0))) H)))))).
 
-theorem arity_gen_cast:
+lemma arity_gen_cast:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
 A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) 
 (arity g c t a)))))))
@@ -920,7 +920,7 @@ a1))).(\lambda (H10: (arity g c0 t a1)).(conj (arity g c0 u (asucc g a2))
 g a1 a2 H3)) (arity_repl g c0 t a1 H10 a2 H3)))) H8))))))))))))) c y a H0))) 
 H)))))).
 
-theorem arity_gen_appls:
+lemma arity_gen_appls:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall 
 (a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: 
 A).(arity g c t a))))))))
@@ -942,7 +942,7 @@ a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
 (\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: 
 A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
 
-theorem arity_gen_lift:
+lemma arity_gen_lift:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: 
 nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: 
 C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma
index 72dd176af..7267b7817 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma
@@ -18,7 +18,7 @@ include "basic_1/arity/props.ma".
 
 include "basic_1/drop1/fwd.ma".
 
-theorem arity_lift1:
+lemma arity_lift1:
  \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: 
 PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t 
 a) \to (arity g c1 (lift1 hds t) a))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma
index 68308fb93..69dfef63d 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma
@@ -26,7 +26,7 @@ include "basic_1/pr0/props.ma".
 
 include "basic_1/arity/subst0.ma".
 
-theorem arity_sred_wcpr0_pr0:
+lemma arity_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
 c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 
 t2) \to (arity g c2 t2 a)))))))))
@@ -527,7 +527,7 @@ a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda
 (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 H3 t2 H4) a2 H2)))))))))))) c1 
 t1 a H))))).
 
-theorem arity_sred_wcpr0_pr1:
+lemma arity_sred_wcpr0_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
 (c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 
 c2) \to (arity g c2 t2 a)))))))))
@@ -546,7 +546,7 @@ a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3:
 (arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl 
 c2)))))))))))))) t1 t2 H))).
 
-theorem arity_sred_pr2:
+lemma arity_sred_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
 \def
@@ -563,7 +563,7 @@ G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a
 (arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
 H2)))))))))))))) c t1 t2 H)))).
 
-theorem arity_sred_pr3:
+lemma arity_sred_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma
index f681969a2..fb8379af7 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/arity/fwd.ma".
 
-theorem node_inh:
+lemma node_inh:
  \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: 
 C).(\lambda (t: T).(arity g c t (ASort k n)))))))
 \def
@@ -33,7 +33,7 @@ C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1)
 (TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 
 x1) (ASort (S n0) n) H1))))) H0)))) k))).
 
-theorem arity_lift:
+lemma arity_lift:
  \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 
 t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 
 c2) \to (arity g c1 (lift h d t) a)))))))))
@@ -155,7 +155,7 @@ C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1
 c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a 
 H))))).
 
-theorem arity_repellent:
+lemma arity_repellent:
  \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: 
 A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c 
 (THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: 
@@ -210,7 +210,7 @@ a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7
 g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g 
 (asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
 
-theorem arity_appls_abbr:
+lemma arity_appls_abbr:
  \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
 nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall 
 (a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma
index 48d87a2a4..334505b50 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma
@@ -26,7 +26,7 @@ include "basic_1/subst0/fwd.ma".
 
 include "basic_1/getl/getl.ma".
 
-theorem arity_gen_cvoid_subst0:
+lemma arity_gen_cvoid_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
 (Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to 
@@ -232,7 +232,7 @@ nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
 T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d 
 u i H3 w v H4 P)))))))))))))))) c t a H))))).
 
-theorem arity_gen_cvoid:
+lemma arity_gen_cvoid:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
 (Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
@@ -253,7 +253,7 @@ x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
 (ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x 
 (refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
 
-theorem arity_fsubst0:
+lemma arity_fsubst0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
 c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u 
@@ -1101,7 +1101,7 @@ c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2)
 c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 
 c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
 
-theorem arity_subst0:
+lemma arity_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c 
 t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead 
 d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma
index c25e24f0d..e2edc783c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma
@@ -18,7 +18,7 @@ include "basic_1/asucc/defs.ma".
 
 include "basic_1/A/fwd.ma".
 
-theorem asucc_gen_sort:
+lemma asucc_gen_sort:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A 
 (ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: 
 nat).(eq A a (ASort h0 n0)))))))))
@@ -41,7 +41,7 @@ n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2
 H1) in (False_ind (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A 
 (AHead a0 a1) (ASort h0 n0))))) H2))))))) a)))).
 
-theorem asucc_gen_head:
+lemma asucc_gen_head:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A 
 (AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 
 a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma
index 4d9b54e26..d26ec1381 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma
@@ -18,7 +18,7 @@ include "basic_1/cimp/defs.ma".
 
 include "basic_1/getl/getl.ma".
 
-theorem cimp_flat_sx:
+lemma cimp_flat_sx:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) 
 c)))
 \def
@@ -36,7 +36,7 @@ b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind
 b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) 
 d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
 
-theorem cimp_flat_dx:
+lemma cimp_flat_dx:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) 
 v))))
 \def
@@ -45,7 +45,7 @@ C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
 b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 
 (Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
 
-theorem cimp_bind:
+lemma cimp_bind:
  \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
 (v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
 \def
@@ -86,7 +86,7 @@ C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C (\lambda (d2: C).(getl (S h0)
 (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) h0 c2 
 (CHead x (Bind b0) w) H3 v)))) H2)))))) h H0)))))))))).
 
-theorem cimp_getl_conf:
+lemma cimp_getl_conf:
  \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
 (d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) 
 \to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma
index f9a28d88a..4383e2650 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/clear/fwd.ma".
 
 include "basic_1/drop/fwd.ma".
 
-theorem drop_clear:
+lemma drop_clear:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to 
 (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead 
 e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
@@ -63,7 +63,7 @@ B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c
 (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) k (drop_gen_drop k c c2 t i 
 H0))))))))) c1).
 
-theorem drop_clear_O:
+lemma drop_clear_O:
  \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c 
 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 
 e2) \to (drop (S i) O c e2))))))))
@@ -99,7 +99,7 @@ H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1
 (Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead 
 e1 (Bind b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
 
-theorem drop_clear_S:
+lemma drop_clear_S:
  \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop 
 h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear 
 x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
index 88de2d264..478e65cca 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
@@ -18,15 +18,15 @@ include "basic_1/clear/defs.ma".
 
 include "basic_1/C/fwd.ma".
 
-let rec clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: B).(\forall (e: 
-C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) u)))))) (f0: 
-(\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to (\forall (f0: 
-F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) (c0: C) (c1: 
-clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) \Rightarrow 
-(f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 ((clear_ind P f 
-f0) e c2 c3) f1 u)].
+implied let rec clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: 
+B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) 
+u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to 
+(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) 
+(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) 
+\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 
+((clear_ind P f f0) e c2 c3) f1 u)].
 
-theorem clear_gen_sort:
+lemma clear_gen_sort:
  \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: 
 Prop).P)))
 \def
@@ -44,7 +44,7 @@ H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda (ee: C).(match ee with
 [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
 H3) in (False_ind P H4))))))))) y x H0))) H)))).
 
-theorem clear_gen_bind:
+lemma clear_gen_bind:
  \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
 (CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
 \def
@@ -76,7 +76,7 @@ with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with
 b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0))) 
 H))))).
 
-theorem clear_gen_flat:
+lemma clear_gen_flat:
  \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
 (CHead e (Flat f) u) x) \to (clear e x)))))
 \def
@@ -106,7 +106,7 @@ e (Flat f) u)) \to (clear e c))) H2 e H8) in (let H10 \def (eq_ind C e0
 (\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x 
 H0))) H))))).
 
-theorem clear_gen_flat_r:
+lemma clear_gen_flat_r:
  \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x 
 (CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
 \def
@@ -127,7 +127,7 @@ u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
 (c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C 
 (CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
 
-theorem clear_gen_all:
+lemma clear_gen_all:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: 
 B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
 \def
@@ -173,7 +173,7 @@ H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t)
 c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f 
 c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
 
-theorem clear_cle:
+lemma clear_cle:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
 \def
  \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma
index 1efc0ba5f..07e59e05d 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/clear/fwd.ma".
 
-theorem clear_clear:
+lemma clear_clear:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
 \def
  \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
@@ -51,7 +51,7 @@ c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))))
 (\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c 
 c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
 
-theorem clear_ctail:
+lemma clear_ctail:
  \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 
 (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k 
 u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma
index 956a7b64e..55bad4ad9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/clen/defs.ma".
 
 include "basic_1/getl/props.ma".
 
-theorem getl_ctail_clen:
+lemma getl_ctail_clen:
  \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: 
 nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
 \def
@@ -42,7 +42,7 @@ F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t
 c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) 
 t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
 
-theorem getl_gen_tail:
+lemma getl_gen_tail:
  \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall 
 (c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 
 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma
index dc687c9dd..1019462ab 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/cnt/defs.ma".
 
-let rec cnt_ind (P: (T \to Prop)) (f: (\forall (n: nat).(P (TSort n)))) (f0: 
-(\forall (t: T).((cnt t) \to ((P t) \to (\forall (k: K).(\forall (v: T).(P 
-(THead k v t)))))))) (t: T) (c: cnt t) on c: P t \def match c with [(cnt_sort 
-n) \Rightarrow (f n) | (cnt_head t0 c0 k v) \Rightarrow (f0 t0 c0 ((cnt_ind P 
-f f0) t0 c0) k v)].
+implied let rec cnt_ind (P: (T \to Prop)) (f: (\forall (n: nat).(P (TSort 
+n)))) (f0: (\forall (t: T).((cnt t) \to ((P t) \to (\forall (k: K).(\forall 
+(v: T).(P (THead k v t)))))))) (t: T) (c: cnt t) on c: P t \def match c with 
+[(cnt_sort n) \Rightarrow (f n) | (cnt_head t0 c0 k v) \Rightarrow (f0 t0 c0 
+((cnt_ind P f f0) t0 c0) k v)].
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma
index 3bdc08ee1..e18a4788e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma
@@ -18,7 +18,7 @@ include "basic_1/cnt/fwd.ma".
 
 include "basic_1/lift/props.ma".
 
-theorem cnt_lift:
+lemma cnt_lift:
  \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i 
 d t)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma
index 3edfe33a3..7b35a4381 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma
@@ -22,7 +22,7 @@ include "basic_1/arity/fwd.ma".
 
 include "basic_1/csubv/getl.ma".
 
-theorem csuba_arity:
+lemma csuba_arity:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
 t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
 \def
@@ -101,7 +101,7 @@ g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
 C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
 H2)))))))))) c1 t a H))))).
 
-theorem csuba_arity_rev:
+lemma csuba_arity_rev:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
 t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 
 t a))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma
index 39936f41f..06b1cc258 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/csuba/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem csuba_clear_conf:
+lemma csuba_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to 
 (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
 (\lambda (e2: C).(clear c2 e2))))))))
@@ -69,7 +69,7 @@ e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2))
 u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 
 (clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
 
-theorem csuba_clear_trans:
+lemma csuba_clear_trans:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to 
 (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
 (\lambda (e2: C).(clear c2 e2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma
index 4aba7bc43..5a92a8ecd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/csuba/fwd.ma".
 
 include "basic_1/drop/fwd.ma".
 
-theorem csuba_drop_abbr:
+lemma csuba_drop_abbr:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
 O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
 c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
@@ -184,7 +184,7 @@ d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10))))
 H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n 
 H1)))))))))))) c1)))) i).
 
-theorem csuba_drop_abst:
+lemma csuba_drop_abst:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
 O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
@@ -798,7 +798,7 @@ A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
 Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
 (drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
 
-theorem csuba_drop_abst_rev:
+lemma csuba_drop_abst_rev:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
 O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
 c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
@@ -1304,7 +1304,7 @@ Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
 H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n 
 H1)))))))))))) c1)))) i).
 
-theorem csuba_drop_abbr_rev:
+lemma csuba_drop_abbr_rev:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
 O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
 g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma
index e4101c983..2f082fa2a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/csuba/defs.ma".
 
-let rec csuba_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: nat).(P 
-(CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csuba g c1 
-c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
+implied let rec csuba_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csuba 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
 (CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csuba g c1 
 c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
 T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
@@ -32,7 +32,7 @@ u) \Rightarrow (f0 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) k u) |
 f1 f2) c2 c3 c4) b n u1 u2) | (csuba_abst c2 c3 c4 t a a0 u a1) \Rightarrow 
 (f2 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) t a a0 u a1)].
 
-theorem csuba_gen_abbr:
+lemma csuba_gen_abbr:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
 (CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
@@ -95,7 +95,7 @@ _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H5) in (False_ind (ex2 C
 (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) 
 (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) y c H0))) H))))).
 
-theorem csuba_gen_void:
+lemma csuba_gen_void:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
 (CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
 C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: 
@@ -178,7 +178,7 @@ C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2)))))
 (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) 
 H6)))))))))))) y c H0))) H))))).
 
-theorem csuba_gen_abst:
+lemma csuba_gen_abst:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
 (CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead 
 d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
@@ -318,7 +318,7 @@ g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
 A).(arity g d2 u2 a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 
 H10 H4)))))))) H6)))))))))))) y c H0))) H))))).
 
-theorem csuba_gen_flat:
+lemma csuba_gen_flat:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
 (f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: 
 C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
@@ -386,7 +386,7 @@ H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead
 c2 (Bind Abbr) u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
 T).(csuba g d1 d2)))) H6)))))))))))) y c H0))) H)))))).
 
-theorem csuba_gen_bind:
+lemma csuba_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
 (v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
@@ -492,7 +492,7 @@ v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))
 Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) 
 H6)))))))))))) y c2 H0))) H)))))).
 
-theorem csuba_gen_abst_rev:
+lemma csuba_gen_abst_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
@@ -596,7 +596,7 @@ u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
 (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda 
 (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H6)))))))))))) c y H0))) H))))).
 
-theorem csuba_gen_void_rev:
+lemma csuba_gen_void_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
 Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
@@ -667,7 +667,7 @@ _) \Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind (ex2 C
 (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) 
 (\lambda (d2: C).(csuba g d2 d1))) H6)))))))))))) c y H0))) H))))).
 
-theorem csuba_gen_abbr_rev:
+lemma csuba_gen_abbr_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
@@ -851,7 +851,7 @@ d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
 u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) 
 H6)))))))))))) c y H0))) H))))).
 
-theorem csuba_gen_flat_rev:
+lemma csuba_gen_flat_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
 (f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: 
 C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
@@ -919,7 +919,7 @@ H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead
 c1 (Bind Abst) t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
 T).(csuba g d2 d1)))) H6)))))))))))) c y H0))) H)))))).
 
-theorem csuba_gen_bind_rev:
+lemma csuba_gen_bind_rev:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
 (v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma
index 3299f3091..039992420 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma
@@ -20,7 +20,7 @@ include "basic_1/csuba/clear.ma".
 
 include "basic_1/getl/clear.ma".
 
-theorem csuba_getl_abbr:
+lemma csuba_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
 (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g 
 c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
@@ -134,7 +134,7 @@ x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u)))
 x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) 
 i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
 
-theorem csuba_getl_abst:
+lemma csuba_getl_abst:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
 (i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba 
 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) 
@@ -468,7 +468,7 @@ c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22))
 H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 
 H2)))) H0))))))).
 
-theorem csuba_getl_abst_rev:
+lemma csuba_getl_abst_rev:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
 (i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g 
 c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
@@ -694,7 +694,7 @@ T).(csuba g d2 d1))) x9 x10 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind
 Void) x10) n H23) H24)))))) H22)) H21)))))))) H17)))))) H14))))))) 
 H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
 
-theorem csuba_getl_abbr_rev:
+lemma csuba_getl_abbr_rev:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
 (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba 
 g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma
index 0dddd9cd8..0e60bfed5 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma
@@ -18,7 +18,7 @@ include "basic_1/csuba/defs.ma".
 
 include "basic_1/C/fwd.ma".
 
-theorem csuba_refl:
+lemma csuba_refl:
  \forall (g: G).(\forall (c: C).(csuba g c c))
 \def
  \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma
index 079eed63b..63593633a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/csubc/csuba.ma".
 
-theorem csubc_arity_conf:
+lemma csubc_arity_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
 (\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
 \def
@@ -24,7 +24,7 @@ theorem csubc_arity_conf:
 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t 
 a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
 
-theorem csubc_arity_trans:
+lemma csubc_arity_trans:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
 ((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to 
 (arity g c1 t a))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma
index bd6677d64..eaef555fd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/csubc/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem csubc_clear_conf:
+lemma csubc_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall 
 (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda 
 (e2: C).(csubc g e1 e2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma
index 0df9ed60c..9e1d3014e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma
@@ -18,7 +18,7 @@ include "basic_1/csubc/fwd.ma".
 
 include "basic_1/sc3/props.ma".
 
-theorem csubc_csuba:
+lemma csubc_csuba:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
 g c1 c2))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma
index c096cf25b..a0bb37e96 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/csubc/fwd.ma".
 
 include "basic_1/sc3/props.ma".
 
-theorem csubc_drop_conf_O:
+lemma csubc_drop_conf_O:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h 
 O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
 C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
@@ -133,7 +133,7 @@ e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2)
 e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) 
 H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)).
 
-theorem drop_csubc_trans:
+lemma drop_csubc_trans:
  \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
 (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
@@ -299,7 +299,7 @@ x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n
 x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) 
 (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
 
-theorem csubc_drop_conf_rev:
+lemma csubc_drop_conf_rev:
  \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
 (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma
index bd9ea33d4..32574ea29 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/csubc/drop.ma".
 
-theorem drop1_csubc_trans:
+lemma drop1_csubc_trans:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
 C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
 (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
@@ -50,7 +50,7 @@ x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1
 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 
 n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
 
-theorem csubc_drop1_conf_rev:
+lemma csubc_drop1_conf_rev:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
 C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
 (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma
index 01312eb8b..a63e6baac 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/csubc/defs.ma".
 
-let rec csubc_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: nat).(P 
-(CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubc g c1 
-c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (v: T).(P (CHead c1 k v) 
+implied let rec csubc_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubc 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (v: T).(P (CHead c1 k v) 
 (CHead c2 k v))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubc g c1 
 c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
 T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
@@ -32,7 +32,7 @@ v) \Rightarrow (f0 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) k v) |
 f1 f2) c2 c3 c4) b n u1 u2) | (csubc_abst c2 c3 c4 v a s0 w s1) \Rightarrow 
 (f2 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) v a s0 w s1)].
 
-theorem csubc_gen_sort_l:
+lemma csubc_gen_sort_l:
  \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to 
 (eq C x (CSort n)))))
 \def
@@ -65,7 +65,7 @@ c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v)
 \Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) 
 w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
 
-theorem csubc_gen_head_l:
+lemma csubc_gen_head_l:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: 
 K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x 
 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
@@ -348,7 +348,7 @@ g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2
 (Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) 
 H)))))).
 
-theorem csubc_gen_sort_r:
+lemma csubc_gen_sort_r:
  \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to 
 (eq C x (CSort n)))))
 \def
@@ -381,7 +381,7 @@ c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w)
 \Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) 
 v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
 
-theorem csubc_gen_head_r:
+lemma csubc_gen_head_r:
  \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: 
 K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x 
 (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma
index 817d2a222..869c84320 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/csubc/drop.ma".
 
 include "basic_1/csubc/clear.ma".
 
-theorem csubc_getl_conf:
+lemma csubc_getl_conf:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i 
 c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
 C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma
index 21c4c6e9f..a419bad4c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma
@@ -18,7 +18,7 @@ include "basic_1/csubc/defs.ma".
 
 include "basic_1/sc3/props.ma".
 
-theorem csubc_refl:
+lemma csubc_refl:
  \forall (g: G).(\forall (c: C).(csubc g c c))
 \def
  \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma
index 82f3fd1fd..d70be313e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma
@@ -20,7 +20,7 @@ include "basic_1/csubst0/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem csubst0_clear_O:
+lemma csubst0_clear_O:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
 (\forall (c: C).((clear c1 c) \to (clear c2 c))))))
 \def
@@ -101,7 +101,7 @@ O H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0))
 H6 O H9) in (clear_flat x1 c0 (H x1 v H10 c0 (clear_gen_flat f c c0 t H8)) f 
 x0)))))) k H1 H4) c2 H5)))))))) H3)) H2))))))))))) c1).
 
-theorem csubst0_clear_O_back:
+lemma csubst0_clear_O_back:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
 (\forall (c: C).((clear c2 c) \to (clear c1 c))))))
 \def
@@ -183,7 +183,7 @@ v c x1)) H7 O H9) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0
 n v t x0)) H6 O H9) in (clear_flat c c0 (H x1 v H11 c0 (clear_gen_flat f x1 
 c0 x0 H10)) f t)))))) k H4 H8))))))))) H3)) H2))))))))))) c1).
 
-theorem csubst0_clear_S:
+lemma csubst0_clear_S:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
 (S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 
 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq 
@@ -1022,7 +1022,7 @@ T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H15 (clear_flat x1
 (CHead x5 (Bind x3) x7) H16 f x0) H17 H18))))))))))) H14)) H13)))))))) k H1 
 H4) c2 H5)))))))) H3)) H2)))))))))))) c1).
 
-theorem csubst0_clear_trans:
+lemma csubst0_clear_trans:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
 i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C 
 (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma
index f4853b5f7..2d759243f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/csubst0/fwd.ma".
 
 include "basic_1/drop/fwd.ma".
 
-theorem csubst0_drop_gt:
+lemma csubst0_drop_gt:
  \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
 c1 e) \to (drop n O c2 e)))))))))
@@ -160,7 +160,7 @@ f) n0 x1 e (H13 x1 v H9 e H12) x0)))) H15)) (lt_gen_xS x2 n0 H14)))))) k
 (drop_gen_drop k c e t n0 H3) H10 H11))) c2 H7)))))))) H5)) H4))))))))))) 
 c1)))))) n).
 
-theorem csubst0_drop_gt_back:
+lemma csubst0_drop_gt_back:
  \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
 c2 e) \to (drop n O c1 e)))))))))
@@ -297,7 +297,7 @@ x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H13 x1 v H9 e H15)
 t)))) H16)) (lt_gen_xS x2 n0 H14)))))) k H11 H12 (drop_gen_drop k x1 e x0 n0 
 H10)))))))))))) H5)) H4))))))))))) c1)))))) n).
 
-theorem csubst0_drop_lt:
+lemma csubst0_drop_lt:
  \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
 c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
@@ -2291,7 +2291,7 @@ x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3
 x7) H17 x0) H18 H19)) e H16)))))))))) H15)) H14)))))) k (drop_gen_drop k c e 
 t n0 H2) H9 H10) i H5))) c2 H6)))))))) H4)) H3))))))))))) c1)))))) n).
 
-theorem csubst0_drop_eq:
+lemma csubst0_drop_eq:
  \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
 n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) 
 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
@@ -4191,7 +4191,7 @@ C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
 (Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))))) k 
 (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))))) H3)) H2))))))))))) c1)))) n).
 
-theorem csubst0_drop_eq_back:
+lemma csubst0_drop_eq_back:
  \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
 n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) 
 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
@@ -6039,7 +6039,7 @@ C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
 (Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) k H4 
 (drop_gen_drop k x1 e x0 n0 H8)))))))))) H3)) H2))))))))))) c1)))) n).
 
-theorem csubst0_drop_lt_back:
+lemma csubst0_drop_lt_back:
  \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O 
 c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
index 022e06fa4..724d7c5a9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
@@ -18,10 +18,10 @@ include "basic_1/csubst0/defs.ma".
 
 include "basic_1/C/fwd.ma".
 
-let rec csubst0_ind (P: (nat \to (T \to (C \to (C \to Prop))))) (f: (\forall 
-(k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: 
-T).((subst0 i v u1 u2) \to (\forall (c: C).(P (s k i) v (CHead c k u1) (CHead 
-c k u2)))))))))) (f0: (\forall (k: K).(\forall (i: nat).(\forall (c1: 
+implied let rec csubst0_ind (P: (nat \to (T \to (C \to (C \to Prop))))) (f: 
+(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(P (s k i) v (CHead c k u1) 
+(CHead c k u2)))))))))) (f0: (\forall (k: K).(\forall (i: nat).(\forall (c1: 
 C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to ((P i v c1 c2) 
 \to (\forall (u: T).(P (s k i) v (CHead c1 k u) (CHead c2 k u))))))))))) (f1: 
 (\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
@@ -34,7 +34,7 @@ v c4 ((csubst0_ind P f f0 f1) i v c2 c3 c4) u) | (csubst0_both k i v u1 u2 s0
 c2 c3 c4) \Rightarrow (f1 k i v u1 u2 s0 c2 c3 c4 ((csubst0_ind P f f0 f1) i 
 v c2 c3 c4))].
 
-theorem csubst0_gen_sort:
+lemma csubst0_gen_sort:
  \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 
 i v (CSort n) x) \to (\forall (P: Prop).P)))))
 \def
@@ -61,7 +61,7 @@ c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee:
 C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
 True])) I (CSort n) H4) in (False_ind P H5))))))))))))) i v y x H0))) H)))))).
 
-theorem csubst0_gen_head:
+lemma csubst0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
 (v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T 
 nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
@@ -274,7 +274,7 @@ C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) u2 c2 i0 (refl_equal nat (s k
 i0)) (refl_equal C (CHead c2 k u2)) H12 H11)) k0 H8))))))) H6)) 
 H5))))))))))))) i v y x H0))) H))))))).
 
-theorem csubst0_gen_S_bind_2:
+lemma csubst0_gen_S_bind_2:
  \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall 
 (v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to 
 (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma
index 8fb0fcb2c..894932961 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma
@@ -20,7 +20,7 @@ include "basic_1/csubst0/drop.ma".
 
 include "basic_1/getl/fwd.ma".
 
-theorem csubst0_getl_ge:
+lemma csubst0_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
 e) \to (getl n c2 e)))))))))
@@ -103,7 +103,7 @@ H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e
 x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
 i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
 
-theorem csubst0_getl_lt:
+lemma csubst0_getl_lt:
  \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
 e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
@@ -1018,7 +1018,7 @@ v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8))
 (clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) 
 H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
 
-theorem csubst0_getl_ge_back:
+lemma csubst0_getl_ge_back:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 
 e) \to (getl n c1 e)))))))))
@@ -1101,7 +1101,7 @@ H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0
 x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
 i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
 
-theorem csubst0_getl_lt_back:
+lemma csubst0_getl_lt_back:
  \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 
 e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma
index e36dea109..bb427a677 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/csubst0/defs.ma".
 
-theorem csubst0_snd_bind:
+lemma csubst0_snd_bind:
  \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
 (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c 
 (Bind b) u1) (CHead c (Bind b) u2))))))))
@@ -27,7 +27,7 @@ b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b)
 u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S 
 i))))))))).
 
-theorem csubst0_fst_bind:
+lemma csubst0_fst_bind:
  \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall 
 (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 
 (Bind b) u) (CHead c2 (Bind b) u))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma
index 48bc273a6..c06340168 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma
@@ -22,7 +22,7 @@ include "basic_1/subst1/defs.ma".
 
 include "basic_1/s/fwd.ma".
 
-theorem csubst1_ind:
+implied lemma csubst1_ind:
  \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (P: ((C \to 
 Prop))).((P c1) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2)))) \to 
 (\forall (c: C).((csubst1 i v c1 c) \to (P c))))))))
@@ -33,7 +33,7 @@ c2) \to (P c2))))).(\lambda (c: C).(\lambda (c0: (csubst1 i v c1 c)).(match
 c0 with [csubst1_refl \Rightarrow f | (csubst1_sing x x0) \Rightarrow (f0 x 
 x0)])))))))).
 
-theorem csubst1_gen_head:
+lemma csubst1_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
 (v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T 
 C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma
index 9724dd260..1d37fd761 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma
@@ -22,7 +22,7 @@ include "basic_1/subst1/props.ma".
 
 include "basic_1/drop/props.ma".
 
-theorem csubst1_getl_ge:
+lemma csubst1_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 
 e) \to (getl n c2 e)))))))))
@@ -34,7 +34,7 @@ c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to
 (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
 (getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
 
-theorem csubst1_getl_lt:
+lemma csubst1_getl_lt:
  \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 
 e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: 
@@ -137,7 +137,7 @@ x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4)
 x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 
 H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
 
-theorem csubst1_getl_ge_back:
+lemma csubst1_getl_ge_back:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 
 e) \to (getl n c1 e)))))))))
@@ -149,7 +149,7 @@ c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to
 (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
 (getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
 
-theorem getl_csubst1:
+lemma getl_csubst1:
  \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c 
 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: 
 C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma
index 862a804ad..d2dc87dfe 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/csubt/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem csubt_clear_conf:
+lemma csubt_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to 
 (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
 (\lambda (e2: C).(clear c2 e2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma
index b0d4c923b..0de7f11cd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/ty3/arity.ma".
 
-theorem csubt_csuba:
+lemma csubt_csuba:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba 
 g c1 c2))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma
index c8dc4c0c4..8d05e4b14 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/csubt/fwd.ma".
 
 include "basic_1/drop/fwd.ma".
 
-theorem csubt_drop_flat:
+lemma csubt_drop_flat:
  \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
 (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
@@ -116,7 +116,7 @@ u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0
 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 
 H5)))))))))))))) c1 c2 H0)))))) n))).
 
-theorem csubt_drop_abbr:
+lemma csubt_drop_abbr:
  \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
 c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
 Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
@@ -215,7 +215,7 @@ d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C
 (CHead x (Bind Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind 
 Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
 
-theorem csubt_drop_abst:
+lemma csubt_drop_abst:
  \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
 c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
 Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
index ea24eb298..cd82088fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/csubt/defs.ma".
 
-let rec csubt_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: nat).(P 
-(CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubt g c1 
-c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
+implied let rec csubt_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubt 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
 (CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubt g c1 
 c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
 T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
@@ -32,7 +32,7 @@ t) \to (P (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))))) (c: C)
 (csubt_abst c2 c3 c4 u t t0 t1) \Rightarrow (f2 c2 c3 c4 ((csubt_ind g P f f0 
 f1 f2) c2 c3 c4) u t t0 t1)].
 
-theorem csubt_gen_abbr:
+lemma csubt_gen_abbr:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
 (CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
@@ -95,7 +95,7 @@ I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C
 (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
 e1 e2))) H6))))))))))) y c2 H0))) H))))).
 
-theorem csubt_gen_abst:
+lemma csubt_gen_abst:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
 (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
 e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda 
@@ -218,7 +218,7 @@ C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3
 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13 H11 
 H9))))))))) H6))))))))))) y c2 H0))) H))))).
 
-theorem csubt_gen_flat:
+lemma csubt_gen_flat:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall 
 (f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C 
 c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
@@ -278,7 +278,7 @@ H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u)
 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 
 H0))) H)))))).
 
-theorem csubt_gen_bind:
+lemma csubt_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
 (v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma
index ebd1762a9..7156f6296 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma
@@ -20,7 +20,7 @@ include "basic_1/csubt/drop.ma".
 
 include "basic_1/getl/clear.ma".
 
-theorem csubt_getl_abbr:
+lemma csubt_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
 (n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g 
 c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
@@ -134,7 +134,7 @@ d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22
 H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 
 H2)))) H0))))))).
 
-theorem csubt_getl_abst:
+lemma csubt_getl_abst:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
 (n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g 
 c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma
index 7929e08da..60a4452e3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma
@@ -18,7 +18,7 @@ include "basic_1/csubt/getl.ma".
 
 include "basic_1/pc3/left.ma".
 
-theorem csubt_pr2:
+lemma csubt_pr2:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
 t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
 \def
@@ -36,7 +36,7 @@ i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_:
 (csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
 c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
 
-theorem csubt_pc3:
+lemma csubt_pc3:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
 t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma
index 86779920a..31221463f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma
@@ -18,7 +18,7 @@ include "basic_1/csubt/defs.ma".
 
 include "basic_1/C/fwd.ma".
 
-theorem csubt_refl:
+lemma csubt_refl:
  \forall (g: G).(\forall (c: C).(csubt g c c))
 \def
  \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma
index e87c261a6..7b3094bd1 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma
@@ -20,7 +20,7 @@ include "basic_1/csubt/props.ma".
 
 include "basic_1/ty3/fwd.ma".
 
-theorem csubt_ty3:
+lemma csubt_ty3:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
 t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
 \def
@@ -86,7 +86,7 @@ T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c
 c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
 c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
 
-theorem csubt_ty3_ld:
+lemma csubt_ty3_ld:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
 v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
 t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma
index 80cfb2fa8..4f4a01272 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/csubv/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem csubv_clear_conf:
+lemma csubv_clear_conf:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
 B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to 
 (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 
@@ -105,7 +105,7 @@ B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))))
 v2) (CHead d2 (Bind b2) v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind 
 x0) x2) H5 f2 v2))))))) H3))))))))))))))) c1 c2 H))).
 
-theorem csubv_clear_conf_void:
+lemma csubv_clear_conf_void:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
 C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T 
 (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma
index 12c78237d..0fb7bbceb 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma
@@ -20,7 +20,7 @@ include "basic_1/csubv/fwd.ma".
 
 include "basic_1/drop/fwd.ma".
 
-theorem csubv_drop_conf:
+lemma csubv_drop_conf:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: 
 C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 
 e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma
index f1c483123..ebc5ee513 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma
@@ -16,10 +16,10 @@
 
 include "basic_1/csubv/defs.ma".
 
-let rec csubv_ind (P: (C \to (C \to Prop))) (f: (\forall (n: nat).(P (CSort 
-n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to ((P 
-c1 c2) \to (\forall (v1: T).(\forall (v2: T).(P (CHead c1 (Bind Void) v1) 
-(CHead c2 (Bind Void) v2))))))))) (f1: (\forall (c1: C).(\forall (c2: 
+implied let rec csubv_ind (P: (C \to (C \to Prop))) (f: (\forall (n: nat).(P 
+(CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubv c1 c2) 
+\to ((P c1 c2) \to (\forall (v1: T).(\forall (v2: T).(P (CHead c1 (Bind Void) 
+v1) (CHead c2 (Bind Void) v2))))))))) (f1: (\forall (c1: C).(\forall (c2: 
 C).((csubv c1 c2) \to ((P c1 c2) \to (\forall (b1: B).((not (eq B b1 Void)) 
 \to (\forall (b2: B).(\forall (v1: T).(\forall (v2: T).(P (CHead c1 (Bind b1) 
 v1) (CHead c2 (Bind b2) v2)))))))))))) (f2: (\forall (c1: C).(\forall (c2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma
index 9befbda30..f648b2323 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma
@@ -20,7 +20,7 @@ include "basic_1/csubv/drop.ma".
 
 include "basic_1/getl/fwd.ma".
 
-theorem csubv_getl_conf:
+lemma csubv_getl_conf:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
 B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 
 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
@@ -53,7 +53,7 @@ B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))))
 b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 
 H9))))))) H7)))))) H4)))))) H1))))))))).
 
-theorem csubv_getl_conf_void:
+lemma csubv_getl_conf_void:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
 C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) 
 \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma
index 82dc44057..eb8448a1a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma
@@ -20,7 +20,7 @@ include "basic_1/C/fwd.ma".
 
 include "basic_1/T/props.ma".
 
-theorem csubv_bind_same:
+lemma csubv_bind_same:
  \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall 
 (v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) 
 v2)))))))
@@ -32,7 +32,7 @@ T).(csubv_bind c1 c2 H Abbr not_abbr_void Abbr v1 v2))) (\lambda (v1:
 T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst not_abst_void Abst v1 v2))) 
 (\lambda (v1: T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))).
 
-theorem csubv_refl:
+lemma csubv_refl:
  \forall (c: C).(csubv c c)
 \def
  \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma
index f4dacaf20..a77911cae 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma
@@ -22,19 +22,19 @@ include "basic_1/r/props.ma".
 
 include "basic_1/C/fwd.ma".
 
-let rec drop_ind (P: (nat \to (nat \to (C \to (C \to Prop))))) (f: (\forall 
-(c: C).(P O O c c))) (f0: (\forall (k: K).(\forall (h: nat).(\forall (c: 
-C).(\forall (e: C).((drop (r k h) O c e) \to ((P (r k h) O c e) \to (\forall 
-(u: T).(P (S h) O (CHead c k u) e))))))))) (f1: (\forall (k: K).(\forall (h: 
-nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h (r k d) c e) 
-\to ((P h (r k d) c e) \to (\forall (u: T).(P h (S d) (CHead c k (lift h (r k 
-d) u)) (CHead e k u))))))))))) (n: nat) (n0: nat) (c: C) (c0: C) (d: drop n 
-n0 c c0) on d: P n n0 c c0 \def match d with [(drop_refl c1) \Rightarrow (f 
-c1) | (drop_drop k h c1 e d0 u) \Rightarrow (f0 k h c1 e d0 ((drop_ind P f f0 
-f1) (r k h) O c1 e d0) u) | (drop_skip k h d0 c1 e d1 u) \Rightarrow (f1 k h 
-d0 c1 e d1 ((drop_ind P f f0 f1) h (r k d0) c1 e d1) u)].
+implied let rec drop_ind (P: (nat \to (nat \to (C \to (C \to Prop))))) (f: 
+(\forall (c: C).(P O O c c))) (f0: (\forall (k: K).(\forall (h: nat).(\forall 
+(c: C).(\forall (e: C).((drop (r k h) O c e) \to ((P (r k h) O c e) \to 
+(\forall (u: T).(P (S h) O (CHead c k u) e))))))))) (f1: (\forall (k: 
+K).(\forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop 
+h (r k d) c e) \to ((P h (r k d) c e) \to (\forall (u: T).(P h (S d) (CHead c 
+k (lift h (r k d) u)) (CHead e k u))))))))))) (n: nat) (n0: nat) (c: C) (c0: 
+C) (d: drop n n0 c c0) on d: P n n0 c c0 \def match d with [(drop_refl c1) 
+\Rightarrow (f c1) | (drop_drop k h c1 e d0 u) \Rightarrow (f0 k h c1 e d0 
+((drop_ind P f f0 f1) (r k h) O c1 e d0) u) | (drop_skip k h d0 c1 e d1 u) 
+\Rightarrow (f1 k h d0 c1 e d1 ((drop_ind P f f0 f1) h (r k d0) c1 e d1) u)].
 
-theorem drop_gen_sort:
+lemma drop_gen_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop 
 h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
 \def
@@ -65,7 +65,7 @@ with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
 k d0) u))) (eq nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) 
 H))))).
 
-theorem drop_gen_refl:
+lemma drop_gen_refl:
  \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
 \def
  \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O 
@@ -94,7 +94,7 @@ nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4)
 in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h 
 H6)))))))))))))) y y0 x e H1))) H0))) H))).
 
-theorem drop_gen_drop:
+lemma drop_gen_drop:
  \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
 nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
 \def
@@ -173,7 +173,7 @@ H18) in (let H22 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee with [O
 k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) 
 y1 y0 y x H2))) H1))) H0))) H)))))).
 
-theorem drop_gen_skip_r:
+lemma drop_gen_skip_r:
  \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
 (d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda 
 (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k 
@@ -250,7 +250,7 @@ C (CHead c0 k (lift h0 (r k d) u)) (CHead e0 k (lift h0 (r k d) u))))
 h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) u0 H8)))) H7)) H6)))))))))))) h y0 
 x y H1))) H0))) H))))))).
 
-theorem drop_gen_skip_l:
+lemma drop_gen_skip_l:
  \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
 (d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T 
 (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: 
@@ -366,7 +366,7 @@ u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h0 (r k d) u0)) H19) d0
 H17)))) u H13)) k0 H9))))))))) H7)) H6)))))))))))) h y0 y x H1))) H0))) 
 H))))))).
 
-theorem drop_S:
+lemma drop_S:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
 nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
index 6ea5bca4c..1d662e8d4 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/drop/fwd.ma".
 
-theorem drop_skip_bind:
+lemma drop_skip_bind:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
 d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) 
 (lift h d u)) (CHead e (Bind b) u))))))))
@@ -26,7 +26,7 @@ d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
 d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e 
 (Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
 
-theorem drop_skip_flat:
+lemma drop_skip_flat:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
 (S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat 
 f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
@@ -37,7 +37,7 @@ f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
 e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S 
 d))))))))).
 
-theorem drop_ctail:
+lemma drop_ctail:
  \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
 h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) 
 (CTail k u c2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma
index 1aacc171f..3270fe008 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma
@@ -16,8 +16,8 @@
 
 include "basic_1/drop1/defs.ma".
 
-let rec drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: (\forall (c: 
-C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: C).(\forall (h: 
+implied let rec drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: (\forall 
+(c: C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: C).(\forall (h: 
 nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: 
 PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons h d hds) c1 
 c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P p c c0 \def 
@@ -25,7 +25,7 @@ match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1 c2 h d0 d1
 c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f f0) hds c2 
 c3 d2))].
 
-theorem drop1_gen_pnil:
+lemma drop1_gen_pnil:
  \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
 \def
  \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq 
@@ -41,7 +41,7 @@ PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to
 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H4) in 
 (False_ind (eq C c3 c5) H5)))))))))))) y c1 c2 H0))) H))).
 
-theorem drop1_gen_pcons:
+lemma drop1_gen_pcons:
  \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: 
 nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda 
 (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma
index 1a761e160..83889c30a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/drop1/fwd.ma".
 
 include "basic_1/getl/drop.ma".
 
-theorem drop1_getl_trans:
+lemma drop1_getl_trans:
  \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
 \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl 
 i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma
index 94e4a50c1..41b61d8b6 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma
@@ -20,7 +20,7 @@ include "basic_1/drop/props.ma".
 
 include "basic_1/getl/defs.ma".
 
-theorem drop1_skip_bind:
+lemma drop1_skip_bind:
  \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: 
 C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) 
 (lift1 hds u)) (CHead e (Bind b) u)))))))
@@ -43,7 +43,7 @@ e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b)
 (lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e 
 (Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
 
-theorem drop1_cons_tail:
+lemma drop1_cons_tail:
  \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop 
 h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to 
 (drop1 (PConsTail hds h d) c1 c3))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma
index 7c7cc2888..31e071b63 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma
@@ -16,10 +16,10 @@
 
 include "basic_1/ex0/defs.ma".
 
-let rec leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: nat).(\forall 
-(h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus h1 n2) (plus 
-h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall (a1: 
-A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3: 
+implied let rec leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: 
+nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus 
+h1 n2) (plus h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall 
+(a1: A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3: 
 A).(\forall (a4: A).((leqz a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead 
 a2 a4))))))))))) (a: A) (a0: A) (l: leqz a a0) on l: P a a0 \def match l with 
 [(leqz_sort h1 h2 n1 n2 e) \Rightarrow (f h1 h2 n1 n2 e) | (leqz_head a1 a2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma
index 1aa405d62..c115f9d6b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma
@@ -20,7 +20,7 @@ include "basic_1/leq/fwd.ma".
 
 include "basic_1/aplus/props.ma".
 
-theorem aplus_gz_le:
+lemma aplus_gz_le:
  \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A 
 (aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
 \def
@@ -53,7 +53,7 @@ A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0)
 k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S 
 n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
 
-theorem aplus_gz_ge:
+lemma aplus_gz_ge:
  \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A 
 (aplus gz (ASort h n) k) (ASort (minus h k) n)))))
 \def
@@ -80,7 +80,7 @@ k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0) n) k0)) a))
 gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S n0) n))) 
 (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
 
-theorem next_plus_gz:
+lemma next_plus_gz:
  \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
 \def
  \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat 
@@ -88,7 +88,7 @@ theorem next_plus_gz:
 nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat 
 S (next_plus gz n n0) (plus n0 n) H))) h)).
 
-theorem leqz_leq:
+lemma leqz_leq:
  \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
 \def
  \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz 
@@ -153,7 +153,7 @@ A).(\lambda (a3: A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0
 a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda 
 (H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
 
-theorem leq_leqz:
+lemma leq_leqz:
  \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
 \def
  \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma
index 95b5201a9..91ce1745b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma
@@ -28,14 +28,14 @@ include "basic_1/arity/defs.ma".
 
 include "basic_1/leq/props.ma".
 
-theorem ex1__leq_sort_SS:
+fact ex1__leq_sort_SS:
  \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc 
 g (asucc g (ASort (S (S k)) n))))))
 \def
  \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g 
 (asucc g (ASort (S (S k)) n)))))).
 
-theorem ex1_arity:
+lemma ex1_arity:
  \forall (g: G).(arity g ex1_c ex1_t (ASort O O))
 \def
  \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) 
@@ -67,7 +67,7 @@ O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead
 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
 (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))).
 
-theorem ex1_ty3:
+lemma ex1_ty3:
  \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: 
 Prop).P)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
index d5192ec54..42c2de646 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
@@ -22,7 +22,7 @@ include "basic_1/pr2/fwd.ma".
 
 include "basic_1/arity/fwd.ma".
 
-theorem ex2_nf2:
+lemma ex2_nf2:
  nf2 ex2_c ex2_t
 \def
  \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) 
@@ -130,7 +130,7 @@ False])) I (THead (Bind x0) x1 x2) H3) in (False_ind (eq T (THead (Flat Appl)
 (TSort O) (TSort O)) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
 (TSort O)) x3))) H9)) t2 H8))))))))))))))) H1)) H0))).
 
-theorem ex2_arity:
+lemma ex2_arity:
  \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: 
 Prop).P)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma
index c7720192a..a25e73916 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/flt/defs.ma".
 
-theorem flt_wf__q_ind:
+fact flt_wf__q_ind:
  \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C 
 \to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq 
 nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall 
@@ -28,7 +28,7 @@ nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
 C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: 
 C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
 
-theorem flt_wf_ind:
+lemma flt_wf_ind:
  \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: 
 T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) 
 \to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma
index 61344ece0..395bb94a7 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma
@@ -18,7 +18,7 @@ include "basic_1/flt/defs.ma".
 
 include "basic_1/C/props.ma".
 
-theorem flt_thead_sx:
+lemma flt_thead_sx:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c 
 (THead k u t)))))
 \def
@@ -26,7 +26,7 @@ theorem flt_thead_sx:
 (tweight u) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
 u) (plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t))))))).
 
-theorem flt_thead_dx:
+lemma flt_thead_dx:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c 
 (THead k u t)))))
 \def
@@ -34,7 +34,7 @@ theorem flt_thead_dx:
 (tweight t) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
 t) (plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t))))))).
 
-theorem flt_shift:
+lemma flt_shift:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c 
 k u) t c (THead k u t)))))
 \def
@@ -48,14 +48,14 @@ t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
 (plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) 
 (tweight t))))))).
 
-theorem flt_arith0:
+lemma flt_arith0:
  \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t 
 (CHead c k t) (TLRef i)))))
 \def
  \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: 
 nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
 
-theorem flt_arith1:
+lemma flt_arith1:
  \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle 
 (CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: 
 nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
@@ -70,7 +70,7 @@ nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
 (tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S 
 O))))))))))).
 
-theorem flt_arith2:
+lemma flt_arith2:
  \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 
 t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt 
 c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
@@ -82,7 +82,7 @@ c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
 t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S 
 O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
 
-theorem cle_flt_trans:
+lemma cle_flt_trans:
  \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (c3: C).(\forall 
 (u2: T).(\forall (u3: T).((flt c2 u2 c3 u3) \to (flt c1 u2 c3 u3)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma
index 97f850719..59ca03b0b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/fsubst0/defs.ma".
 
-theorem fsubst0_ind:
+implied lemma fsubst0_ind:
  \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (t1: T).(\forall 
 (P: ((C \to (T \to Prop)))).(((\forall (t2: T).((subst0 i v t1 t2) \to (P c1 
 t2)))) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2 t1)))) \to 
@@ -33,7 +33,7 @@ C).(\lambda (t: T).(\lambda (f2: (fsubst0 i v c1 t1 c t)).(match f2 with
 [(fsubst0_snd x x0) \Rightarrow (f x x0) | (fsubst0_fst x x0) \Rightarrow (f0 
 x x0) | (fsubst0_both x x0 x1 x2) \Rightarrow (f1 x x0 x1 x2)]))))))))))).
 
-theorem fsubst0_gen_base:
+lemma fsubst0_gen_base:
  \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall 
 (v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 
 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma
index e091d8b0d..0915f5f57 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/getl/props.ma".
 
 include "basic_1/clear/drop.ma".
 
-theorem clear_getl_trans:
+lemma clear_getl_trans:
  \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to 
 (\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
 \def
@@ -48,7 +48,7 @@ H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f)
 t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 
 c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i).
 
-theorem getl_clear_trans:
+lemma getl_clear_trans:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to 
 (\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
 \def
@@ -66,7 +66,7 @@ x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1
 c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 
 x1 c3 x2 H7)))))))) H4))))) H1))))))).
 
-theorem getl_clear_bind:
+lemma getl_clear_bind:
  \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c 
 (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) 
 \to (getl (S n) c e2))))))))
@@ -102,7 +102,7 @@ H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1
 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 
 (Bind b) v) t H2) e2 n H1) f t))) k H0))))))))))) c)).
 
-theorem getl_clear_conf:
+lemma getl_clear_conf:
  \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to 
 (\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma
index dda85e7f3..a8f4df8bd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/getl/props.ma".
 
-theorem getl_dec:
+lemma getl_dec:
  \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda 
 (b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: 
 C).((getl i c d) \to (\forall (P: Prop).P)))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma
index 9fba2b7b1..6b8d9a268 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma
@@ -18,7 +18,7 @@ include "basic_1/getl/props.ma".
 
 include "basic_1/clear/drop.ma".
 
-theorem getl_drop:
+lemma getl_drop:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
 nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
 \def
@@ -58,7 +58,7 @@ 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) (getl_gen_S k c0 (CHead e 
 (Bind b) u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
 
-theorem getl_drop_conf_lt:
+lemma getl_drop_conf_lt:
  \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: 
 nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
 nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
@@ -325,7 +325,7 @@ h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22))))))))
 H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k 
 H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
 
-theorem getl_drop_conf_ge:
+lemma getl_drop_conf_ge:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
 (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d 
 h) i) \to (getl (minus i h) e a)))))))))
@@ -338,7 +338,7 @@ a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
 x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i 
 x c H3 e h d H0 H1) H4)))) H2)))))))))).
 
-theorem getl_conf_ge_drop:
+lemma getl_conf_ge_drop:
  \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
 nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 
 c2) \to (drop i O c2 e))))))))
@@ -351,7 +351,7 @@ u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
 i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in 
 H3)))))))).
 
-theorem getl_drop_conf_rev:
+lemma getl_drop_conf_rev:
  \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
 (\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i 
 c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) 
@@ -362,7 +362,7 @@ e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
 nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 
 H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
 
-theorem drop_getl_trans_lt:
+lemma drop_getl_trans_lt:
  \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: 
 C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda 
@@ -396,7 +396,7 @@ d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d
 (le_S_n (S i) (S d) (le_S (S (S i)) (S d) (le_n_S (S i) d H)))) c1 c2 h H0 x 
 H3))))) H2)))))))))))).
 
-theorem drop_getl_trans_le:
+lemma drop_getl_trans_le:
  \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 
 e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) 
@@ -421,7 +421,7 @@ O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
 (\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) 
 H2)))))))))).
 
-theorem drop_getl_trans_ge:
+lemma drop_getl_trans_ge:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
 nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) 
 \to ((le d i) \to (getl (plus i h) c1 e2)))))))))
@@ -434,7 +434,7 @@ C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
 C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro 
 (plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
 
-theorem getl_drop_trans:
+lemma getl_drop_trans:
  \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to 
 (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i 
 h)) O c1 e2)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma
index 0151a248f..63876304a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma
@@ -18,7 +18,7 @@ include "basic_1/getl/fwd.ma".
 
 include "basic_1/flt/props.ma".
 
-theorem getl_flt:
+lemma getl_flt:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
 nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma
index f1f4426c1..7390775d3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma
@@ -20,7 +20,7 @@ include "basic_1/drop/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem getl_ind:
+implied lemma getl_ind:
  \forall (h: nat).(\forall (c1: C).(\forall (c2: C).(\forall (P: 
 Prop).(((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to P)))) \to 
 ((getl h c1 c2) \to P)))))
@@ -30,7 +30,7 @@ Prop).(\lambda (f: ((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to
 P))))).(\lambda (g: (getl h c1 c2)).(match g with [(getl_intro x x0 x1) 
 \Rightarrow (f x x0 x1)])))))).
 
-theorem getl_gen_all:
+lemma getl_gen_all:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 
 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
 \def
@@ -40,7 +40,7 @@ C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda
 (H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda 
 (e0: C).(clear e0 c2)) e H0 H1)))) H)))).
 
-theorem getl_gen_sort:
+lemma getl_gen_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to 
 (\forall (P: Prop).P))))
 \def
@@ -54,7 +54,7 @@ e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2:
 (CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 
 H1))))) H0)))))).
 
-theorem getl_gen_O:
+lemma getl_gen_O:
  \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
 \def
  \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def 
@@ -63,7 +63,7 @@ theorem getl_gen_O:
 (drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 
 (\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
 
-theorem getl_gen_S:
+lemma getl_gen_S:
  \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
 nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
 \def
@@ -74,7 +74,7 @@ k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0:
 C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 
 x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
 
-theorem getl_gen_2:
+lemma getl_gen_2:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 
 B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind 
 b) v)))))))))
@@ -96,7 +96,7 @@ B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c
 (Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) 
 H3))))) H0))))).
 
-theorem getl_gen_flat:
+lemma getl_gen_flat:
  \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
 nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
 \def
@@ -108,7 +108,7 @@ H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to
 (getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) 
 d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
 
-theorem getl_gen_bind:
+lemma getl_gen_bind:
  \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
 nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d 
 (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma
index c3c50a80f..64e331905 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/getl/drop.ma".
 
 include "basic_1/getl/clear.ma".
 
-theorem getl_conf_le:
+lemma getl_conf_le:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
 (e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e 
 a))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma
index c1a4f58f1..56592c5e6 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma
@@ -20,7 +20,7 @@ include "basic_1/clear/props.ma".
 
 include "basic_1/drop/props.ma".
 
-theorem getl_refl:
+lemma getl_refl:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) 
 (CHead c (Bind b) u))))
 \def
@@ -28,7 +28,7 @@ theorem getl_refl:
 b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) 
 u)) (clear_bind b c u)))).
 
-theorem getl_head:
+lemma getl_head:
  \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k 
 h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
 \def
@@ -39,7 +39,7 @@ C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1:
 (drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k 
 u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
 
-theorem getl_flat:
+lemma getl_flat:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall 
 (f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
 \def
@@ -56,7 +56,7 @@ x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to
 (S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat 
 f) h0 c x H3 u) H2)))) h H1)))) H0))))))).
 
-theorem getl_ctail:
+lemma getl_ctail:
  \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
 nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: 
 T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma
index 3a917c085..fc3550a4c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma
@@ -18,7 +18,7 @@ include "basic_1/iso/defs.ma".
 
 include "basic_1/tlist/defs.ma".
 
-theorem iso_ind:
+implied lemma iso_ind:
  \forall (P: ((T \to (T \to Prop)))).(((\forall (n1: nat).(\forall (n2: 
 nat).(P (TSort n1) (TSort n2))))) \to (((\forall (i1: nat).(\forall (i2: 
 nat).(P (TLRef i1) (TLRef i2))))) \to (((\forall (v1: T).(\forall (v2: 
@@ -35,7 +35,7 @@ nat).(\forall (n2: nat).(P (TSort n1) (TSort n2)))))).(\lambda (f0: ((\forall
 (f x x0) | (iso_lref x x0) \Rightarrow (f0 x x0) | (iso_head x x0 x1 x2 x3) 
 \Rightarrow (f1 x x0 x1 x2 x3)]))))))).
 
-theorem iso_gen_sort:
+lemma iso_gen_sort:
  \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda 
 (n2: nat).(eq T u2 (TSort n2))))))
 \def
@@ -60,7 +60,7 @@ K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T
 n1) H1) in (False_ind (ex nat (\lambda (n2: nat).(eq T (THead k v2 t2) (TSort 
 n2)))) H2)))))))) y u2 H0))) H))).
 
-theorem iso_gen_lref:
+lemma iso_gen_lref:
  \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda 
 (n2: nat).(eq T u2 (TLRef n2))))))
 \def
@@ -85,7 +85,7 @@ ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _
 _ _) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: 
 nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
 
-theorem iso_gen_head:
+lemma iso_gen_head:
  \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso 
 (THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
 (THead k v2 t2)))))))))
@@ -121,7 +121,7 @@ k1 v2 t2) (THead k v3 t3)))))) (ex_2_intro T T (\lambda (v3: T).(\lambda (t3:
 T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 
 t2))) k0 H6)))) H3)) H2)))))))) y u2 H0))) H))))).
 
-theorem iso_flats_lref_bind_false:
+lemma iso_flats_lref_bind_false:
  \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall 
 (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind 
 b) v t)) \to (\forall (P: Prop).P)))))))
@@ -149,7 +149,7 @@ v t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat f) x0 x1) 
 H2) in (False_ind P H3))))) H1)))))))) vs)))))).
 
-theorem iso_flats_flat_bind_false:
+lemma iso_flats_flat_bind_false:
  \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall 
 (v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads 
 (Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma
index f18572e1e..baef0f833 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma
@@ -18,7 +18,7 @@ include "basic_1/T/fwd.ma".
 
 include "basic_1/iso/fwd.ma".
 
-theorem iso_refl:
+lemma iso_refl:
  \forall (t: T).(iso t t)
 \def
  \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
index 87db6f5d9..42005d6ba 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/leq/props.ma".
 
-theorem asucc_repl:
+lemma asucc_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
 (asucc g a1) (asucc g a2)))))
 \def
@@ -96,7 +96,7 @@ h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g 
 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
 
-theorem asucc_inj:
+lemma asucc_inj:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc 
 g a2)) \to (leq g a1 a2))))
 \def
@@ -306,7 +306,7 @@ _ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda
 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0 
 a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
 
-theorem leq_asucc:
+lemma leq_asucc:
  \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
 a0)))))
 \def
@@ -322,7 +322,7 @@ g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) 
 a)).
 
-theorem leq_ahead_asucc_false:
+lemma leq_ahead_asucc_false:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 
 (asucc g a1)) \to (\forall (P: Prop).P))))
 \def
@@ -374,7 +374,7 @@ x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def
 (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in 
 (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
 
-theorem leq_asucc_false:
+lemma leq_asucc_false:
  \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
 Prop).P)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma
index 4077638fe..32d8e6f87 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/leq/defs.ma".
 
-let rec leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1: 
+implied let rec leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1: 
 nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k: 
 nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (P 
 (ASort h1 n1) (ASort h2 n2))))))))) (f0: (\forall (a1: A).(\forall (a2: 
@@ -27,7 +27,7 @@ k e) \Rightarrow (f h1 h2 n1 n2 k e) | (leq_head a1 a2 l0 a3 a4 l1)
 \Rightarrow (f0 a1 a2 l0 ((leq_ind g P f f0) a1 a2 l0) a3 a4 l1 ((leq_ind g P 
 f f0) a3 a4 l1))].
 
-theorem leq_gen_sort1:
+lemma leq_gen_sort1:
  \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
 g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
 nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
@@ -82,7 +82,7 @@ n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a3 a5) 
 (ASort h2 n2)))))) H6))))))))))) y a2 H0))) H))))).
 
-theorem leq_gen_head1:
+lemma leq_gen_head1:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g 
 (AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
@@ -130,7 +130,7 @@ A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda
 (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) a3 a5 H12 H10 
 (refl_equal A (AHead a3 a5))))))))) H6))))))))))) y a H0))) H))))).
 
-theorem leq_gen_sort2:
+lemma leq_gen_sort2:
  \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
 g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
 nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) 
@@ -185,7 +185,7 @@ n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a1 a4) 
 (ASort h2 n2)))))) H6))))))))))) a2 y H0))) H))))).
 
-theorem leq_gen_head2:
+lemma leq_gen_head2:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a 
 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 
 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
@@ -233,7 +233,7 @@ A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda
 (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) (AHead a6 a7)))) a0 a4 H12 H10 
 (refl_equal A (AHead a0 a4))))))))) H6))))))))))) a y H0))) H))))).
 
-theorem ahead_inj_snd:
+lemma ahead_inj_snd:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma
index ee90c1792..6eec6578a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma
@@ -18,7 +18,7 @@ include "basic_1/leq/fwd.ma".
 
 include "basic_1/aplus/props.ma".
 
-theorem leq_refl:
+lemma leq_refl:
  \forall (g: G).(\forall (a: A).(leq g a a))
 \def
  \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) 
@@ -27,14 +27,14 @@ theorem leq_refl:
 a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 
 H0))))) a)).
 
-theorem leq_eq:
+lemma leq_eq:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 
 a2))))
 \def
  \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 
 a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
 
-theorem leq_sym:
+lemma leq_sym:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
 a2 a1))))
 \def
@@ -92,7 +92,7 @@ A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8:
 a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) 
 H5))))))))))))) a1 a2 H)))).
 
-theorem leq_ahead_false_1:
+lemma leq_ahead_false_1:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 
 \to (\forall (P: Prop).P))))
 \def
@@ -139,7 +139,7 @@ g a2 x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def
 H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a 
 a0) a3)) H3 a H8) in (H a0 H10 P))))) H6))))))) H2)))))))))) a1)).
 
-theorem leq_ahead_false_2:
+lemma leq_ahead_false_2:
  \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) 
 \to (\forall (P: Prop).P))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma
index e2d22dc7c..beb400a3f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/lift/props.ma".
 
-theorem lift_gen_sort:
+lemma lift_gen_sort:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T 
 (TSort n) (lift h d t)) \to (eq T t (TSort n))))))
 \def
@@ -52,7 +52,7 @@ t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T
 (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k t0 t1) 
 (TSort n)) H3))))))))) t)))).
 
-theorem lift_gen_lref:
+lemma lift_gen_lref:
  \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
 (TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le 
 (plus d h) i) (eq T t (TLRef (minus i h)))))))))
@@ -109,7 +109,7 @@ 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))))) 
 H3)))))))))))) t).
 
-theorem lift_gen_lref_lt:
+lemma lift_gen_lref_lt:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall 
 (t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
 \def
@@ -128,7 +128,7 @@ d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
 T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S 
 n) d h H))) t H4))) H2)) H1)))))))).
 
-theorem lift_gen_lref_false:
+lemma lift_gen_lref_false:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
 (plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall 
 (P: Prop).P)))))))
@@ -145,7 +145,7 @@ h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
 (H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false 
 (plus d h) n P H4 H0))) H3)) H2)))))))))).
 
-theorem lift_gen_lref_ge:
+lemma lift_gen_lref_ge:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
 (t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
 \def
@@ -167,7 +167,7 @@ h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
 h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus 
 (plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
 
-theorem lift_gen_head:
+lemma lift_gen_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
 nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: 
@@ -265,7 +265,7 @@ t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0)
 (lift h d t0)) (refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) 
 H4)) H3))))))))))) x)))).
 
-theorem lift_gen_bind:
+lemma lift_gen_bind:
  \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
 nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T 
 T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda 
@@ -299,7 +299,7 @@ y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
 z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d 
 x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
 
-theorem lift_gen_flat:
+lemma lift_gen_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
 nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T 
 T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda 
@@ -333,7 +333,7 @@ T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
 (refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x 
 H1)))))) H0))))))))).
 
-theorem lift_inj:
+lemma lift_inj:
  \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T 
 (lift h d x) (lift h d t)) \to (eq T x t)))))
 \def
@@ -396,7 +396,7 @@ x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead
 (refl_equal K (Flat f)) (H x0 h d H4) (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:
+lemma lift_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: 
 nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 
 t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 
@@ -592,7 +592,7 @@ H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
 (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2))))))))))))) 
 t1).
 
-theorem lifts_inj:
+lemma lifts_inj:
  \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: 
 nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
index 6907f694c..610e02018 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
@@ -20,14 +20,14 @@ include "basic_1/s/props.ma".
 
 include "basic_1/T/fwd.ma".
 
-theorem lift_sort:
+lemma lift_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
 n)) (TSort n))))
 \def
  \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort 
 n)))).
 
-theorem lift_lref_lt:
+lemma lift_lref_lt:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T 
 (lift h d (TLRef n)) (TLRef n)))))
 \def
@@ -36,7 +36,7 @@ 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_eq bool (blt n d) true (lt_blt d n H)))))).
 
-theorem lift_lref_ge:
+lemma lift_lref_ge:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
 (lift h d (TLRef n)) (TLRef (plus n h))))))
 \def
@@ -46,7 +46,7 @@ n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
 (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:
+lemma lift_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) 
 t)))))))
@@ -54,7 +54,7 @@ t)))))))
  \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
 (d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
 
-theorem lift_bind:
+lemma lift_bind:
  \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) 
 (lift h (S d) t)))))))
@@ -62,7 +62,7 @@ theorem lift_bind:
  \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
 (d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
 
-theorem lift_flat:
+lemma lift_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) 
 (lift h d t)))))))
@@ -70,7 +70,7 @@ theorem lift_flat:
  \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
 (d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
 
-theorem thead_x_lift_y_y:
+lemma thead_x_lift_y_y:
  \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall 
 (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
 \def
@@ -112,7 +112,7 @@ 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:
+lemma lift_r:
  \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
 \def
  \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) 
@@ -132,7 +132,7 @@ t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1)
 t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))))) (lift O 
 d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
 
-theorem lift_lref_gt:
+lemma lift_lref_gt:
  \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef 
 (pred n))) (TLRef n))))
 \def
@@ -145,7 +145,7 @@ theorem lift_lref_gt:
 (pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) 
 (S_pred n d H))))))).
 
-theorem lift_tle:
+lemma lift_tle:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(tle t (lift h d t))))
 \def
  \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
@@ -162,7 +162,7 @@ nat).(plus x h)) (s k d) t1))) (le_plus_plus (tweight t0) (tweight (lref_map
 (\lambda (x: nat).(plus x h)) d t0)) (tweight t1) (tweight (lref_map (\lambda 
 (x: nat).(plus x h)) (s k d) t1)) H_y H_y0))))))))))) t).
 
-theorem lifts_tapp:
+lemma lifts_tapp:
  \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
 TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
 \def
@@ -176,7 +176,7 @@ t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
 (TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
 v)) H)))) vs)))).
 
-theorem lift_free:
+lemma lift_free:
  \forall (t: T).(\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 t)) (lift (plus k h) d t))))))))
@@ -233,7 +233,7 @@ k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
 h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h 
 d))))))))))))) t).
 
-theorem lift_free_sym:
+lemma lift_free_sym:
  \forall (t: T).(\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 t)) (lift (plus h k) d t))))))))
@@ -243,7 +243,7 @@ nat).(\lambda (e: nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d
 e)).(eq_ind_r nat (plus k h) (\lambda (n: nat).(eq T (lift k e (lift h d t)) 
 (lift n d t))) (lift_free t h k d e H H0) (plus h k) (plus_sym h k)))))))).
 
-theorem lift_d:
+lemma lift_d:
  \forall (t: T).(\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 t)) 
 (lift k e (lift h d t))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma
index d66d095ef..5adcfd6d8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma
@@ -18,7 +18,7 @@ include "basic_1/lift/props.ma".
 
 include "basic_1/tlt/props.ma".
 
-theorem lift_weight_map:
+lemma lift_weight_map:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to 
 nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat 
 (weight_map f (lift h d t)) (weight_map f t))))))
@@ -106,7 +106,7 @@ t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1))
 (lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
 k)))))))))) t).
 
-theorem lift_weight:
+lemma lift_weight:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d 
 t)) (weight t))))
 \def
@@ -114,7 +114,7 @@ t)) (weight t))))
 (\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat 
 O)))))).
 
-theorem lift_weight_add:
+lemma lift_weight_add:
  \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: 
 nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
 (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to 
@@ -262,7 +262,7 @@ t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0))
 (lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
 k))))))))))))) t)).
 
-theorem lift_weight_add_O:
+lemma lift_weight_add_O:
  \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to 
 nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) 
 O t))))))
@@ -273,7 +273,7 @@ nat))).(lift_weight_add (minus (wadd f w O) O) t h O f (wadd f w) (\lambda
 (minus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal 
 nat (f m)))))))).
 
-theorem lift_tlt_dx:
+lemma lift_tlt_dx:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(tlt t (THead k u (lift h d t)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma
index 8c1489f19..f5839c798 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma
@@ -18,7 +18,7 @@ include "basic_1/lift/props.ma".
 
 include "basic_1/drop1/defs.ma".
 
-theorem lift1_lift1:
+lemma lift1_lift1:
  \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
 (lift1 is2 t)) (lift1 (papp is1 is2) t))))
 \def
@@ -31,7 +31,7 @@ t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t))))
 T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal 
 nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
 
-theorem lift1_xhg:
+lemma lift1_xhg:
  \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
 (lift (S O) O (lift1 hds t))))
 \def
@@ -49,7 +49,7 @@ nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
 p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S 
 d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
 
-theorem lifts1_xhg:
+lemma lifts1_xhg:
  \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts 
 (S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
 \def
@@ -66,7 +66,7 @@ t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
 (lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) 
 (lift (S O) O t)) (lift1_xhg hds t))))) ts)).
 
-theorem lift1_free:
+lemma lift1_free:
  \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
 (lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
index 067edeeef..286dfbee3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
@@ -18,7 +18,7 @@ include "basic_1/lift1/defs.ma".
 
 include "basic_1/lift/props.ma".
 
-theorem lift1_sort:
+lemma lift1_sort:
  \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
 \def
  \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T 
@@ -27,7 +27,7 @@ nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p
 (TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 
 n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
 
-theorem lift1_lref:
+lemma lift1_lref:
  \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
 (trans hds i))))
 \def
@@ -41,7 +41,7 @@ T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow
 (TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false 
 \Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
 
-theorem lift1_bind:
+lemma lift1_bind:
  \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
 (lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss 
 hds) t))))))
@@ -62,7 +62,7 @@ n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0
 (lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u 
 t)) (H u t)))))))) hds)).
 
-theorem lift1_flat:
+lemma lift1_flat:
  \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
 (lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds 
 t))))))
@@ -82,7 +82,7 @@ n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f)
 (lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 
 p (THead (Flat f) u t)) (H u t)))))))) hds)).
 
-theorem lift1_cons_tail:
+lemma lift1_cons_tail:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq 
 T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
 \def
@@ -95,7 +95,7 @@ t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
 t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p 
 h d) t) H))))) hds)))).
 
-theorem lifts1_flat:
+lemma lifts1_flat:
  \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: 
 TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds 
 ts) (lift1 hds t))))))
@@ -115,7 +115,7 @@ f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
 (lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 
 (THeads (Flat f) t1 t)))))) ts)))).
 
-theorem lifts1_nil:
+lemma lifts1_nil:
  \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
 \def
  \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) 
@@ -124,7 +124,7 @@ t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
 TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 
 PNil t0) H)))) ts).
 
-theorem lifts1_cons:
+lemma lifts1_cons:
  \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: 
 TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma
index 7cad4fc7a..8d45d3775 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/llt/defs.ma".
 
-theorem llt_wf__q_ind:
+fact llt_wf__q_ind:
  \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to 
 Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 
 a))))) P n))) \to (\forall (a: A).(P a)))
@@ -27,7 +27,7 @@ Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
 n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight 
 a)))))).
 
-theorem llt_wf_ind:
+lemma llt_wf_ind:
  \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 
 a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma
index fe0842328..0c7cef628 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma
@@ -18,7 +18,7 @@ include "basic_1/llt/defs.ma".
 
 include "basic_1/leq/fwd.ma".
 
-theorem lweight_repl:
+lemma lweight_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat 
 (lweight a1) (lweight a2)))))
 \def
@@ -34,7 +34,7 @@ a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
 (lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 
 H)))).
 
-theorem llt_repl:
+lemma llt_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
 (a3: A).((llt a1 a3) \to (llt a2 a3))))))
 \def
@@ -51,13 +51,13 @@ a3) \to (llt a1 a3)))))
 a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans 
 (lweight a1) (lweight a2) (lweight a3) H H0))))).
 
-theorem llt_head_sx:
+lemma llt_head_sx:
  \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
 \def
  \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) 
 (lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
 
-theorem llt_head_dx:
+lemma llt_head_dx:
  \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
 \def
  \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma
index 84dd3f5fe..2f42fc60e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/next_plus/defs.ma".
 
-theorem next_plus_assoc:
+lemma next_plus_assoc:
  \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq 
 nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2))))))
 \def
@@ -36,7 +36,7 @@ n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g
 (next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) 
 (plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))).
 
-theorem next_plus_next:
+lemma next_plus_next:
  \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g 
 (next g n) h) (next g (next_plus g n h)))))
 \def
@@ -45,7 +45,7 @@ g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n
 h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n 
 (S O)) h) (next_plus_assoc g n (S O) h)))).
 
-theorem next_plus_lt:
+lemma next_plus_lt:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next 
 g n) h))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
index 4ee874d1f..c5b3b0a64 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
@@ -18,7 +18,7 @@ include "basic_1/nf2/fwd.ma".
 
 include "basic_1/arity/subst0.ma".
 
-theorem arity_nf2_inv_all:
+lemma arity_nf2_inv_all:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
index e42bf9e00..d7211841f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
@@ -22,7 +22,7 @@ include "basic_1/pr0/dec.ma".
 
 include "basic_1/C/props.ma".
 
-theorem nf2_dec:
+lemma nf2_dec:
  \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
 T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
index 1143c6ff3..06487b713 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
@@ -22,7 +22,7 @@ include "basic_1/subst0/dec.ma".
 
 include "basic_1/T/props.ma".
 
-theorem nf2_gen_lref:
+lemma nf2_gen_lref:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
 \def
@@ -33,7 +33,7 @@ Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
 (lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef 
 i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
 
-theorem nf2_gen_abst:
+lemma nf2_gen_abst:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u 
 t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
 \def
@@ -55,7 +55,7 @@ T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _)
 (\lambda (t0: T).(pr2 (CHead c (Bind Abst) u) t t0)) H0 t H1) in (eq_ind T t 
 (\lambda (t0: T).(eq T t t0)) (refl_equal T t) t2 H1))))))))).
 
-theorem nf2_gen_cast:
+lemma nf2_gen_cast:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u 
 t)) \to (\forall (P: Prop).P))))
 \def
@@ -63,7 +63,7 @@ t)) \to (\forall (P: Prop).P))))
 (Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t 
 (pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
 
-theorem nf2_gen_beta:
+lemma nf2_gen_beta:
  \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c 
 (THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
 \def
@@ -78,7 +78,7 @@ Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
 Abst) v t)) (THead (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t 
 (pr0_refl t))))) in (False_ind P H0))))))).
 
-theorem nf2_gen_flat:
+lemma nf2_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c 
 (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
 \def
@@ -95,7 +95,7 @@ _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
 (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) 
 (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
 
-theorem nf2_gen__nf2_gen_aux:
+fact nf2_gen__nf2_gen_aux:
  \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T 
 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
 \def
@@ -136,7 +136,7 @@ t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
 (S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 
 P)))))) H3)) H2))))))))))) x)).
 
-theorem nf2_gen_abbr:
+lemma nf2_gen_abbr:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u 
 t)) \to (\forall (P: Prop).P))))
 \def
@@ -161,7 +161,7 @@ t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H (lift (S O) O x) H2) in
 (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) u))) P))) H1))) 
 H0))))))).
 
-theorem nf2_gen_void:
+lemma nf2_gen_void:
  \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u 
 (lift (S O) O t))) \to (\forall (P: Prop).P))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
index bd34df9da..39b827580 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
@@ -18,7 +18,7 @@ include "basic_1/nf2/pr3.ma".
 
 include "basic_1/iso/props.ma".
 
-theorem nf2_iso_appls_lref:
+lemma nf2_iso_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
 TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso 
 (THeads (Flat Appl) vs (TLRef i)) u))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
index f2b078af0..9e680d7e8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
@@ -18,7 +18,7 @@ include "basic_1/nf2/props.ma".
 
 include "basic_1/drop1/fwd.ma".
 
-theorem nf2_lift1:
+lemma nf2_lift1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
 hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
index 8c602dfe0..5c6a686e9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
@@ -18,7 +18,7 @@ include "basic_1/nf2/defs.ma".
 
 include "basic_1/pr3/pr3.ma".
 
-theorem nf2_pr3_unfold:
+lemma nf2_pr3_unfold:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
 t1) \to (eq T t1 t2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
index 43383eee4..ccaa9eb52 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
@@ -18,14 +18,14 @@ include "basic_1/nf2/defs.ma".
 
 include "basic_1/pr2/fwd.ma".
 
-theorem nf2_sort:
+lemma nf2_sort:
  \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
 \def
  \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort 
 n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal 
 T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
 
-theorem nf2_csort_lref:
+lemma nf2_csort_lref:
  \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
 \def
  \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort 
@@ -84,7 +84,7 @@ b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T
 u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 
 H3)))))) H2)))))))).
 
-theorem nfs2_tapp:
+lemma nfs2_tapp:
  \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) 
 \to (land (nfs2 c ts) (nf2 c t)))))
 \def
@@ -102,7 +102,7 @@ t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
 (land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) 
 H6))) H4))))) H1)))))) ts))).
 
-theorem nf2_appls_lref:
+lemma nf2_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
 TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
 \def
@@ -263,7 +263,7 @@ theorem nf2_appl_lref:
 nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 
 (TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
 
-theorem nf2_lref_abst:
+lemma nf2_lref_abst:
  \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c 
 (CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
 \def
@@ -291,7 +291,7 @@ _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e
 (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) H3)) in (False_ind (eq T (TLRef 
 i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) H1)))))))).
 
-theorem nf2_lift:
+lemma nf2_lift:
  \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: 
 nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
index faaa213cd..86e794a79 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
@@ -18,27 +18,27 @@ include "basic_1/pc1/defs.ma".
 
 include "basic_1/pr1/pr1.ma".
 
-theorem pc1_pr0_r:
+lemma pc1_pr0_r:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T 
 (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) 
 (pr1_refl t2)))).
 
-theorem pc1_pr0_x:
+lemma pc1_pr0_x:
  \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T 
 (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) 
 (pr1_pr0 t2 t1 H)))).
 
-theorem pc1_refl:
+lemma pc1_refl:
  \forall (t: T).(pc1 t t)
 \def
  \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: 
 T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
 
-theorem pc1_pr0_u:
+lemma pc1_pr0_u:
  \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
 t3) \to (pc1 t1 t3)))))
 \def
@@ -49,7 +49,7 @@ T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda
 (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) 
 H3)))) H1)))))).
 
-theorem pc1_s:
+lemma pc1_s:
  \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
 \def
  \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in 
@@ -58,7 +58,7 @@ t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2
 x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 
 H1)))) H0)))).
 
-theorem pc1_head_1:
+lemma pc1_head_1:
  \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall 
 (k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
 \def
@@ -70,7 +70,7 @@ T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0))
 t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) 
 H0)))))).
 
-theorem pc1_head_2:
+lemma pc1_head_2:
  \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall 
 (k: K).(pc1 (THead k u t1) (THead k u t2))))))
 \def
@@ -97,7 +97,7 @@ x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1
 (pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x 
 H2))))) H4))))) H1)))))).
 
-theorem pc1_pr0_u2:
+lemma pc1_pr0_u2:
  \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 
 t2) \to (pc1 t1 t2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma
index 7147efce3..61df1500c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma
@@ -20,7 +20,7 @@ include "basic_1/fsubst0/fwd.ma".
 
 include "basic_1/csubst0/getl.ma".
 
-theorem pc3_pr2_fsubst0:
+lemma pc3_pr2_fsubst0:
  \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall 
 (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
@@ -332,7 +332,7 @@ t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
 (csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
 H2))))))))))) c2 t4 H3)))))))))))))))) c1 t1 t H)))).
 
-theorem pc3_pr2_fsubst0_back:
+lemma pc3_pr2_fsubst0_back:
  \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall 
 (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
@@ -633,7 +633,7 @@ H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n
 i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 
 H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
 
-theorem pc3_fsubst0:
+lemma pc3_fsubst0:
  \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall 
 (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma
index db39fa1fd..99239c2a7 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/pc3/props.ma".
 
-theorem pc3_gen_sort:
+lemma pc3_gen_sort:
  \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort 
 n)) \to (eq nat m n))))
 \def
@@ -30,7 +30,7 @@ H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
 \Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H3) in 
 H4))))) H0))))).
 
-theorem pc3_gen_abst:
+lemma pc3_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall 
 (t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to 
 (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
@@ -73,7 +73,7 @@ u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))
 (CHead c (Bind b) u) t1 x1 (H15 b u) t2 (H6 b u))))))))) H12)))))))) 
 H7))))))) H3))))) H0))))))).
 
-theorem pc3_gen_abst_shift:
+lemma pc3_gen_abst_shift:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c 
 (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind 
 Abst) u) t1 t2)))))
@@ -86,7 +86,7 @@ Abst) u) t1 t2)))))
 ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 
 Abst u))) H0))))))).
 
-theorem pc3_gen_lift:
+lemma pc3_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall 
 (d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d 
 c e) \to (pc3 e t1 t2))))))))
@@ -107,7 +107,7 @@ T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
 (\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e 
 t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
 
-theorem pc3_gen_not_abst:
+lemma pc3_gen_not_abst:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: 
 T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) 
 u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S 
@@ -226,7 +226,7 @@ u1) t1 (lift (S O) O t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t
 t2) (THead (Bind Abst) x0 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 
 Abst x0)))))))))) H6))) H4))))) H1))))))))) b).
 
-theorem pc3_gen_lift_abst:
+lemma pc3_gen_lift_abst:
  \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall 
 (h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) 
 \to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda 
@@ -281,7 +281,7 @@ T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
 t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 
 x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
 
-theorem pc3_gen_sort_abst:
+lemma pc3_gen_sort_abst:
  \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c 
 (TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
index 28c958755..d386b1864 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
@@ -16,17 +16,17 @@
 
 include "basic_1/pc3/props.ma".
 
-let rec pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P 
-t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: (\forall 
-(t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3_left c t1 
-t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: pc3_left c t t0) 
-on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow (f t1) | 
-(pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1 ((pc3_left_ind c 
-P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1) \Rightarrow (f1 t1 t2 p0 
-t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
+implied let rec pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall 
+(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to 
+(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: 
+(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: 
+pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow 
+(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1 
+((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1) 
+\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
 
-theorem pc3_ind_left__pc3_left_pr3:
+fact pc3_ind_left__pc3_left_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
 (pc3_left c t1 t2))))
 \def
@@ -36,7 +36,7 @@ t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: 
 (pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_trans:
+fact pc3_ind_left__pc3_left_trans:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
 \def
@@ -53,7 +53,7 @@ t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
 t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 
 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_sym:
+fact pc3_ind_left__pc3_left_sym:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (pc3_left c t2 t1))))
 \def
@@ -68,7 +68,7 @@ T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
 t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 
 (pc3_left_r c t3))))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_pc3:
+fact pc3_ind_left__pc3_left_pc3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to 
 (pc3_left c t1 t2))))
 \def
@@ -79,7 +79,7 @@ x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
 (pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x 
 (pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
 
-theorem pc3_ind_left__pc3_pc3_left:
+fact pc3_ind_left__pc3_pc3_left:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (pc3 c t1 t2))))
 \def
@@ -92,7 +92,7 @@ T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
 (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 
 (pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
 
-theorem pc3_ind_left:
+lemma pc3_ind_left:
  \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 
 t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
 T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma
index 0d16aeb79..3ce4180a1 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma
@@ -18,7 +18,7 @@ include "basic_1/pc3/defs.ma".
 
 include "basic_1/nf2/pr3.ma".
 
-theorem pc3_nf2:
+lemma pc3_nf2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
 t1) \to ((nf2 c t2) \to (eq T t1 t2))))))
 \def
@@ -33,7 +33,7 @@ t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1
 H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 
 H_y0))))))))) H2))))))).
 
-theorem pc3_nf2_unfold:
+lemma pc3_nf2_unfold:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
 t2) \to (pr3 c t1 t2)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma
index abead4fc4..a629475c6 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma
@@ -20,7 +20,7 @@ include "basic_1/pc1/defs.ma".
 
 include "basic_1/pr3/pr1.ma".
 
-theorem pc3_pc1:
+lemma pc3_pc1:
  \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 
 t2))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma
index d65a673b8..8d8693596 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma
@@ -18,7 +18,7 @@ include "basic_1/pc3/defs.ma".
 
 include "basic_1/pr3/pr3.ma".
 
-theorem clear_pc3_trans:
+lemma clear_pc3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
 \def
@@ -30,7 +30,7 @@ x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
 t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 
 H0))))) H1))))))).
 
-theorem pc3_pr2_r:
+lemma pc3_pr2_r:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c 
 t1 t2))))
 \def
@@ -38,7 +38,7 @@ t1 t2))))
 t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
 t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
 
-theorem pc3_pr2_x:
+lemma pc3_pr2_x:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c 
 t1 t2))))
 \def
@@ -46,7 +46,7 @@ t1 t2))))
 t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
 t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
 
-theorem pc3_pr3_r:
+lemma pc3_pr3_r:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c 
 t1 t2))))
 \def
@@ -54,7 +54,7 @@ t1 t2))))
 t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
 t2 H (pr3_refl c t2))))).
 
-theorem pc3_pr3_x:
+lemma pc3_pr3_x:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c 
 t1 t2))))
 \def
@@ -62,7 +62,7 @@ t1 t2))))
 t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
 t1 (pr3_refl c t1) H)))).
 
-theorem pc3_pr3_t:
+lemma pc3_pr3_t:
  \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall 
 (t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
 \def
@@ -70,13 +70,13 @@ theorem pc3_pr3_t:
 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: 
 T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
 
-theorem pc3_refl:
+lemma pc3_refl:
  \forall (c: C).(\forall (t: T).(pc3 c t t))
 \def
  \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) 
 (\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
 
-theorem pc3_s:
+lemma pc3_s:
  \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c 
 t2 t1))))
 \def
@@ -86,7 +86,7 @@ T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
 x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) 
 (\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
 
-theorem pc3_thin_dx:
+lemma pc3_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
 (u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u 
 t2)))))))
@@ -100,7 +100,7 @@ x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
 (Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) 
 H0))))))).
 
-theorem pc3_head_1:
+lemma pc3_head_1:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
 (k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
 \def
@@ -113,7 +113,7 @@ u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
 H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl 
 (CHead c k x) t)))))) H0))))))).
 
-theorem pc3_head_2:
+lemma pc3_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u 
 t2)))))))
@@ -127,7 +127,7 @@ T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
 t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) 
 (pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
 
-theorem pc3_pr2_u:
+lemma pc3_pr2_u:
  \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall 
 (t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
 \def
@@ -154,7 +154,7 @@ x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
 H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) 
 H4))))) H1))))))).
 
-theorem pc3_pr2_u2:
+lemma pc3_pr2_u2:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
 (t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
 \def
@@ -162,7 +162,7 @@ theorem pc3_pr2_u2:
 t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x 
 c t1 t0 H) t2 H0)))))).
 
-theorem pc3_pr3_conf:
+lemma pc3_pr3_conf:
  \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall 
 (t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
 \def
@@ -190,7 +190,7 @@ u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
 (CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c 
 u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
 
-theorem pc3_pr0_pr2_t:
+lemma pc3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
 (CHead c k u1) t1 t2))))))))
@@ -259,7 +259,7 @@ u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u
 (r (Flat f) i0) H10 t3 t4 H3 t H9) f u1))))) k IHi (getl_gen_S k c (CHead d 
 (Bind Abbr) u) u2 i0 H8)))))) i H7 H4)))))))))))))) y t1 t2 H1))) H0)))))))).
 
-theorem pc3_pr2_pr2_t:
+lemma pc3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
 (CHead c k u1) t1 t2))))))))
@@ -338,7 +338,7 @@ t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) H12 t4 t5 H6 t6 H11) f t)))) k
 (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 i1 H10)))))) i0 H9 
 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 H)))).
 
-theorem pc3_pr2_pr3_t:
+lemma pc3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to 
 (pc3 (CHead c k u1) t1 t2))))))))
@@ -354,7 +354,7 @@ T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
 u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
 u1 H3)))))))))) t1 t2 H)))))).
 
-theorem pc3_pr3_pc3_t:
+lemma pc3_pr3_pc3_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 
 (CHead c k u1) t1 t2))))))))
@@ -375,7 +375,7 @@ t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
 x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 
 H0)))))) H4))))))))))))) u2 u1 H)))).
 
-theorem pc3_lift:
+lemma pc3_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
 d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift 
 h d t1) (lift h d t2)))))))))
@@ -388,7 +388,7 @@ T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
 (lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H 
 t2 x H3))))) H1))))))))).
 
-theorem pc3_eta:
+lemma pc3_eta:
  \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t 
 (THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead 
 (Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma
index 525c45469..1df35129b 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma
@@ -18,7 +18,7 @@ include "basic_1/pc3/props.ma".
 
 include "basic_1/pr3/subst1.ma".
 
-theorem pc3_gen_cabbr:
+lemma pc3_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 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 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma
index 662b6bf2e..a72a7e90d 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma
@@ -18,7 +18,7 @@ include "basic_1/pc3/props.ma".
 
 include "basic_1/wcpr0/getl.ma".
 
-theorem pc3_wcpr0__pc3_wcpr0_t_aux:
+fact pc3_wcpr0__pc3_wcpr0_t_aux:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall 
 (u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 
 (CHead c2 k u) t1 t2))))))))
@@ -54,7 +54,7 @@ H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k
 u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) 
 H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
 
-theorem pc3_wcpr0_t:
+lemma pc3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
 T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
 \def
@@ -74,7 +74,7 @@ u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3
 (pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x 
 H6)))))) H4))))))))))))) c1 c2 H))).
 
-theorem pc3_wcpr0:
+lemma pc3_wcpr0:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
 T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
index 96d70c708..889e82034 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
@@ -22,7 +22,7 @@ include "basic_1/T/dec.ma".
 
 include "basic_1/T/props.ma".
 
-theorem nf0_dec:
+lemma nf0_dec:
  \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 
 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
 T).(pr0 t1 t2))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
index a9898d7ff..972d92170 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
@@ -18,40 +18,41 @@ include "basic_1/pr0/defs.ma".
 
 include "basic_1/subst0/fwd.ma".
 
-let rec pr0_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0: 
-(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall 
-(t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (k: K).(P 
-(THead k u1 t1) (THead k u2 t2)))))))))))) (f1: (\forall (u: T).(\forall (v1: 
-T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall (t1: T).(\forall 
-(t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))))))) (f2: (\forall (b: B).((not 
-(eq B b Abst)) \to (\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 
-v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to 
-(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)))))))))))))))))) (f3: (\forall (u1: T).(\forall (u2: 
-T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 
-t2) \to ((P t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to (P (THead 
-(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))))))) (f4: (\forall (b: 
-B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) 
-\to ((P t1 t2) \to (\forall (u: T).(P (THead (Bind b) u (lift (S O) O t1)) 
-t2))))))))) (f5: (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 
-t2) \to (\forall (u: T).(P (THead (Flat Cast) u t1) t2))))))) (t: T) (t0: T) 
-(p: pr0 t t0) on p: P t t0 \def match p with [(pr0_refl t1) \Rightarrow (f 
-t1) | (pr0_comp u1 u2 p0 t1 t2 p1 k) \Rightarrow (f0 u1 u2 p0 ((pr0_ind P f 
-f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 
-p1) k) | (pr0_beta u v1 v2 p0 t1 t2 p1) \Rightarrow (f1 u v1 v2 p0 ((pr0_ind 
-P f f0 f1 f2 f3 f4 f5) v1 v2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 
-t2 p1)) | (pr0_upsilon b n v1 v2 p0 u1 u2 p1 t1 t2 p2) \Rightarrow (f2 b n v1 
-v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) v1 v2 p0) u1 u2 p1 ((pr0_ind P f f0 f1 
-f2 f3 f4 f5) u1 u2 p1) t1 t2 p2 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p2)) | 
-(pr0_delta u1 u2 p0 t1 t2 p1 w s0) \Rightarrow (f3 u1 u2 p0 ((pr0_ind P f f0 
-f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) 
-w s0) | (pr0_zeta b n t1 t2 p0 u) \Rightarrow (f4 b n t1 t2 p0 ((pr0_ind P f 
-f0 f1 f2 f3 f4 f5) t1 t2 p0) u) | (pr0_tau t1 t2 p0 u) \Rightarrow (f5 t1 t2 
-p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p0) u)].
+implied let rec pr0_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
+t))) (f0: (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to 
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall 
+(k: K).(P (THead k u1 t1) (THead k u2 t2)))))))))))) (f1: (\forall (u: 
+T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall 
+(t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))))))) (f2: 
+(\forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
+T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 
+u2) \to ((P u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P 
+t1 t2) \to (P (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))))))))))))))) (f3: (\forall 
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (w: T).((subst0 
+O u2 t2 w) \to (P (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 
+w))))))))))))) (f4: (\forall (b: B).((not (eq B b Abst)) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead 
+(Bind b) u (lift (S O) O t1)) t2))))))))) (f5: (\forall (t1: T).(\forall (t2: 
+T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead (Flat Cast) u 
+t1) t2))))))) (t: T) (t0: T) (p: pr0 t t0) on p: P t t0 \def match p with 
+[(pr0_refl t1) \Rightarrow (f t1) | (pr0_comp u1 u2 p0 t1 t2 p1 k) 
+\Rightarrow (f0 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 
+((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) k) | (pr0_beta u v1 v2 p0 t1 t2 
+p1) \Rightarrow (f1 u v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) v1 v2 p0) t1 
+t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1)) | (pr0_upsilon b n v1 v2 p0 
+u1 u2 p1 t1 t2 p2) \Rightarrow (f2 b n v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
+f5) v1 v2 p0) u1 u2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p1) t1 t2 p2 
+((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p2)) | (pr0_delta u1 u2 p0 t1 t2 p1 w 
+s0) \Rightarrow (f3 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 
+p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) w s0) | (pr0_zeta b n t1 t2 p0 
+u) \Rightarrow (f4 b n t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p0) u) 
+| (pr0_tau t1 t2 p0 u) \Rightarrow (f5 t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
+f5) t1 t2 p0) u)].
 
-theorem pr0_gen_sort:
+lemma pr0_gen_sort:
  \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
 \def
  \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq 
@@ -110,7 +111,7 @@ T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TSort n))).(let H4 \def
 True])) I (TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
 H4)))))))) y x H0))) H))).
 
-theorem pr0_gen_lref:
+lemma pr0_gen_lref:
  \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
 \def
  \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq 
@@ -169,7 +170,7 @@ T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TLRef n))).(let H4 \def
 True])) I (TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
 H4)))))))) y x H0))) H))).
 
-theorem pr0_gen_abst:
+lemma pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
 Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
@@ -323,7 +324,7 @@ T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
 t3)))) H4)))))))) y x H0))) H)))).
 
-theorem pr0_gen_appl:
+lemma pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
@@ -1088,7 +1089,7 @@ T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
 
-theorem pr0_gen_cast:
+lemma pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
@@ -1248,7 +1249,7 @@ T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) 
 H8))))) H4)))))))) y x H0))) H)))).
 
-theorem pr0_gen_lift:
+lemma pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
 (lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
 (t2: T).(pr0 t1 t2)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
index 57f9498ce..d3a8f78c9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
@@ -20,7 +20,7 @@ include "basic_1/lift/tlt.ma".
 
 include "basic_1/tlt/fwd.ma".
 
-theorem pr0_confluence__pr0_cong_upsilon_refl:
+fact pr0_confluence__pr0_cong_upsilon_refl:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to 
 (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) 
@@ -40,7 +40,7 @@ t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5)
 O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind 
 b))))))))))))))).
 
-theorem pr0_confluence__pr0_cong_upsilon_cong:
+fact pr0_confluence__pr0_cong_upsilon_cong:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
 T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall 
 (t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: 
@@ -62,7 +62,7 @@ Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp
 (lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat 
 Appl)) (Bind b))))))))))))))))))).
 
-theorem pr0_confluence__pr0_cong_upsilon_delta:
+fact pr0_confluence__pr0_cong_upsilon_delta:
  (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
 T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: 
 T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 
@@ -103,7 +103,7 @@ O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl))
 (lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 
 H5))))))))))))))))))).
 
-theorem pr0_confluence__pr0_cong_upsilon_zeta:
+fact pr0_confluence__pr0_cong_upsilon_zeta:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
 T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 
 u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: 
@@ -125,7 +125,7 @@ Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl)
 (lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) 
 O)))))))))))))))).
 
-theorem pr0_confluence__pr0_cong_delta:
+fact pr0_confluence__pr0_cong_delta:
  \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
 (\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall 
 (t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda 
@@ -151,7 +151,7 @@ x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda
 u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) 
 (pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
 
-theorem pr0_confluence__pr0_upsilon_upsilon:
+fact pr0_confluence__pr0_upsilon_upsilon:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
 T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: 
 T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to 
@@ -175,7 +175,7 @@ H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O
 x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S 
 O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
 
-theorem pr0_confluence__pr0_delta_delta:
+fact pr0_confluence__pr0_delta_delta:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to 
 (\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) 
@@ -244,7 +244,7 @@ w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T
 x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) 
 (pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
 
-theorem pr0_confluence__pr0_delta_tau:
+fact pr0_confluence__pr0_delta_tau:
  \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
 (\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T 
 (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
index c3558ca72..d1c31fcc7 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
@@ -18,7 +18,7 @@ include "basic_1/pr0/fwd.ma".
 
 include "basic_1/subst0/props.ma".
 
-theorem pr0_lift:
+lemma pr0_lift:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
 (d: nat).(pr0 (lift h d t1) (lift h d t2))))))
 \def
@@ -128,7 +128,7 @@ d u) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(pr0 t (lift h d t4)))
 (lift h d (THead (Flat Cast) u t3)) (lift_head (Flat Cast) u t3 h d))))))))) 
 t1 t2 H))).
 
-theorem pr0_gen_abbr:
+lemma pr0_gen_abbr:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
@@ -358,7 +358,7 @@ T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
 (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
 H4)))))))) y x H0))) H)))).
 
-theorem pr0_gen_void:
+lemma pr0_gen_void:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
index 12f17331f..95fb67571 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
@@ -18,7 +18,7 @@ include "basic_1/pr0/props.ma".
 
 include "basic_1/subst0/subst0.ma".
 
-theorem pr0_subst0_back:
+lemma pr0_subst0_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
 T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
@@ -69,7 +69,7 @@ t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda
 t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
 H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
 
-theorem pr0_subst0_fwd:
+lemma pr0_subst0_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
 T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
index 77a73414a..05a93172a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
@@ -18,7 +18,7 @@ include "basic_1/pr0/subst0.ma".
 
 include "basic_1/subst1/fwd.ma".
 
-theorem pr0_delta1:
+lemma pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
 (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead 
 (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
@@ -30,7 +30,7 @@ Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind
 Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H 
 t1 t2 H0 t0 H2))) w H1)))))))).
 
-theorem pr0_subst1_back:
+lemma pr0_subst1_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
@@ -48,7 +48,7 @@ T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x 
 H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
 
-theorem pr0_subst1_fwd:
+lemma pr0_subst1_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma
index 1d4dd0cfa..58b7e727f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma
@@ -16,9 +16,10 @@
 
 include "basic_1/pr1/defs.ma".
 
-let rec pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0: 
-(\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pr1 t2 
-t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t t0) on p: P 
-t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | (pr1_sing t2 t1 p0 
-t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr1_ind P f f0) t2 t3 p1))].
+implied let rec pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
+t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
+T).((pr1 t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t 
+t0) on p: P t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | 
+(pr1_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr1_ind P f f0) t2 
+t3 p1))].
 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma
index 0b088cc7b..60dbdbfaa 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma
@@ -18,7 +18,7 @@ include "basic_1/pr1/props.ma".
 
 include "basic_1/pr0/pr0.ma".
 
-theorem pr1_strip:
+lemma pr1_strip:
  \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 
 t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma
index c3c58fe31..97a5e45d8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma
@@ -22,7 +22,7 @@ include "basic_1/subst1/props.ma".
 
 include "basic_1/T/props.ma".
 
-theorem pr1_pr0:
+lemma pr1_pr0:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H 
@@ -40,7 +40,7 @@ t3) \to (pr1 t1 t3)))))
 t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 
 t5 H3)))))))))) t1 t2 H))).
 
-theorem pr1_head_1:
+lemma pr1_head_1:
  \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall 
 (k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
 \def
@@ -52,7 +52,7 @@ t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda
 (THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k 
 t3 t) H2))))))) u1 u2 H))))).
 
-theorem pr1_head_2:
+lemma pr1_head_2:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall 
 (k: K).(pr1 (THead k u t1) (THead k u t2))))))
 \def
@@ -84,7 +84,7 @@ t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp
 t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v 
 w H))).
 
-theorem pr1_eta:
+lemma pr1_eta:
  \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in 
 (\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) 
 (TLRef O) (lift (S O) O t))) t)))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
index fbcbfbf15..b518e4360 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
@@ -18,7 +18,7 @@ include "basic_1/pr2/props.ma".
 
 include "basic_1/clen/getl.ma".
 
-theorem pr2_gen_ctail:
+lemma pr2_gen_ctail:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: 
 T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 
@@ -76,7 +76,7 @@ t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
 (refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 
 t2 H0))) H)))))).
 
-theorem pr2_gen_cbind:
+lemma pr2_gen_cbind:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) 
 (THead (Bind b) v t2)))))))
@@ -129,7 +129,7 @@ nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c
 Abbr) u) x H9) t3 t4 H2 t H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) 
 H)))))).
 
-theorem pr2_gen_cflat:
+lemma pr2_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
index 7694da8fb..1c19f4a2f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
@@ -22,7 +22,7 @@ include "basic_1/getl/clear.ma".
 
 include "basic_1/getl/drop.ma".
 
-theorem pr2_ind:
+implied lemma pr2_ind:
  \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1: 
 T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c: 
 C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
@@ -40,7 +40,7 @@ T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2)
 \Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8) 
 \Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))).
 
-theorem pr2_gen_sort:
+lemma pr2_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to 
 (eq T x (TSort n)))))
 \def
@@ -62,7 +62,7 @@ t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
 (pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) 
 t1 H4))))))))))))) c y x H0))) H)))).
 
-theorem pr2_gen_lref:
+lemma pr2_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to 
 (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c 
 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S 
@@ -112,7 +112,7 @@ Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
 n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) 
 (subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
 
-theorem pr2_gen_abst:
+lemma pr2_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
@@ -234,7 +234,7 @@ H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
 H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) 
 (pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
-theorem pr2_gen_cast:
+lemma pr2_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -366,7 +366,7 @@ u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
 (pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 
 H5)))))))))))))) c y x H0))) H))))).
 
-theorem pr2_gen_csort:
+lemma pr2_gen_csort:
  \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) 
 \to (pr0 t1 t2))))
 \def
@@ -383,7 +383,7 @@ t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
 (Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) 
 u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
 
-theorem pr2_gen_appl:
+lemma pr2_gen_appl:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -2728,7 +2728,7 @@ H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s
 (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
 (pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
-theorem pr2_gen_lift:
+lemma pr2_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
index df933985d..3151fc53c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
@@ -20,7 +20,7 @@ include "basic_1/pr0/pr0.ma".
 
 include "basic_1/getl/fwd.ma".
 
-theorem pr2_confluence__pr2_free_free:
+fact pr2_confluence__pr2_free_free:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 
 t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
 T).(pr2 c t2 t))))))))
@@ -33,7 +33,7 @@ x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
 (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) 
 (pr0_confluence t0 t2 H0 t1 H))))))).
 
-theorem pr2_confluence__pr2_free_delta:
+fact pr2_confluence__pr2_free_delta:
  \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall 
 (t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to 
 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) 
@@ -59,7 +59,7 @@ T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0
 H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) 
 (pr0_confluence t0 t4 H1 t1 H))))))))))))).
 
-theorem pr2_confluence__pr2_delta_delta:
+fact pr2_confluence__pr2_delta_delta:
  \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall 
 (t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: 
 T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma
index 19cb5d831..5677ca53e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma
@@ -18,7 +18,7 @@ include "basic_1/pr2/fwd.ma".
 
 include "basic_1/pr0/subst0.ma".
 
-theorem pr2_thin_dx:
+lemma pr2_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
 t2)))))))
@@ -36,7 +36,7 @@ H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
 t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 
 u)))))))))))) c t1 t2 H)))))).
 
-theorem pr2_head_1:
+lemma pr2_head_1:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
 (k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
 \def
@@ -52,7 +52,7 @@ t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
 t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c 
 u1 u2 H)))))).
 
-theorem pr2_head_2:
+lemma pr2_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u 
 t2)))))))
@@ -141,7 +141,7 @@ c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6)
 H3 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) 
 H4 u))))))))))))) i)))))) k) y t1 t2 H0))) H)))))).
 
-theorem clear_pr2_trans:
+lemma clear_pr2_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
 \def
@@ -156,7 +156,7 @@ t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
 C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c 
 (CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
 
-theorem pr2_cflat:
+lemma pr2_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
 \def
@@ -171,7 +171,7 @@ u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
 i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c 
 t1 t2 H)))))).
 
-theorem pr2_ctail:
+lemma pr2_ctail:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
 \def
@@ -185,7 +185,7 @@ T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
 (subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail 
 Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
 
-theorem pr2_change:
+lemma pr2_change:
  \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
 T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to 
 (\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
@@ -233,7 +233,7 @@ nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))
 (CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) 
 y t1 t2 H1))) H0)))))))).
 
-theorem pr2_lift:
+lemma pr2_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift 
 h d t1) (lift h d t2)))))))))
@@ -274,7 +274,7 @@ h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
 (lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h 
 H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
 
-theorem pr2_gen_abbr:
+lemma pr2_gen_abbr:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -824,7 +824,7 @@ B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
 H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) 
 (pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
-theorem pr2_gen_void:
+lemma pr2_gen_void:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma
index c27e3c809..8f1063f8e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma
@@ -22,7 +22,7 @@ include "basic_1/csubst1/getl.ma".
 
 include "basic_1/subst1/subst1.ma".
 
-theorem pr2_delta1:
+lemma pr2_delta1:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) 
 \to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
@@ -34,7 +34,7 @@ t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
 (\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 
 H0 t0 H2))) t H1)))))))))).
 
-theorem pr2_subst1:
+lemma pr2_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c 
@@ -100,7 +100,7 @@ T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20))))
 H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 
 H5))))))))))))))) y t1 t2 H1))) H0)))))))).
 
-theorem pr2_gen_cabbr:
+lemma 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 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
index 1da77c869..e3c9fca02 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
@@ -18,14 +18,14 @@ include "basic_1/pr3/defs.ma".
 
 include "basic_1/pr2/fwd.ma".
 
-let rec pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
-t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: 
-T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr3 
-c t t0) on p: P t t0 \def match p with [(pr3_refl t1) \Rightarrow (f t1) | 
-(pr3_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr3_ind c P f f0) 
-t2 t3 p1))].
+implied let rec pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: 
+T).(P t t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to 
+(\forall (t3: T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) 
+(t0: T) (p: pr3 c t t0) on p: P t t0 \def match p with [(pr3_refl t1) 
+\Rightarrow (f t1) | (pr3_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 
+((pr3_ind c P f f0) t2 t3 p1))].
 
-theorem pr3_gen_sort:
+lemma pr3_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
 (eq T x (TSort n)))))
 \def
@@ -42,7 +42,7 @@ T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n))
 \to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T 
 (TSort n)))) t1 H4))))))))) y x H0))) H)))).
 
-theorem pr3_gen_abst:
+lemma pr3_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
@@ -115,7 +115,7 @@ x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u:
 T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) 
 H11)))))))) H6)))))))))))) y x H0))))) H))))).
 
-theorem pr3_gen_cast:
+lemma pr3_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -217,7 +217,7 @@ t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
 T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 
 H2))) H6)))))))))))) y x H0))))) H))))).
 
-theorem pr3_gen_lift:
+lemma pr3_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 
@@ -254,7 +254,7 @@ h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4
 H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x 
 H0)))) H)))))).
 
-theorem pr3_gen_lref:
+lemma pr3_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to 
 (or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda 
 (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
index 7ff6beaf4..c56ee4c23 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
@@ -20,7 +20,7 @@ include "basic_1/iso/props.ma".
 
 include "basic_1/tlist/fwd.ma".
 
-theorem pr3_iso_appls_abbr:
+lemma pr3_iso_appls_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
 Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
@@ -199,7 +199,7 @@ c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat
 Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
 (lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
 
-theorem pr3_iso_appls_cast:
+lemma pr3_iso_appls_cast:
  \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
 \def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: 
 T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
@@ -359,7 +359,7 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
 (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
 x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
 
-theorem pr3_iso_appl_bind:
+lemma pr3_iso_appl_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
 T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
 in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
@@ -576,7 +576,7 @@ O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
 (Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
 H10))) u2 H6))))))))))))) H3)) H2)))))))))).
 
-theorem pr3_iso_appls_appl_bind:
+lemma pr3_iso_appls_appl_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
 T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
 (THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
@@ -744,7 +744,7 @@ Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3)
 (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
 H7)))))))))))))) H4)) H3))))))))) vs)))))).
 
-theorem pr3_iso_appls_bind:
+lemma pr3_iso_appls_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
 T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
 in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
@@ -824,7 +824,7 @@ t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
 Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) 
 (lifts_tapp (S O) O t ts))))))))))) vs))).
 
-theorem pr3_iso_beta:
+lemma pr3_iso_beta:
  \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
 Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
 u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
@@ -962,7 +962,7 @@ Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in
 False with []) in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) 
 H1)))))))).
 
-theorem pr3_iso_appls_beta:
+lemma pr3_iso_appls_beta:
  \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
 \def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
 (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
index 77bb9bc59..04878c2b9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
@@ -18,7 +18,7 @@ include "basic_1/pr3/defs.ma".
 
 include "basic_1/pr1/fwd.ma".
 
-theorem pr3_pr1:
+lemma pr3_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
 t2))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
index a8ad07418..d973e0eb2 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
@@ -18,7 +18,7 @@ include "basic_1/pr3/props.ma".
 
 include "basic_1/pr2/pr2.ma".
 
-theorem pr3_strip:
+lemma pr3_strip:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
 (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
 T).(pr3 c t2 t))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
index 299b08b9a..c9cf9c301 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
@@ -22,7 +22,7 @@ include "basic_1/pr2/props.ma".
 
 include "basic_1/pr1/props.ma".
 
-theorem clear_pr3_trans:
+lemma clear_pr3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
 \def
@@ -33,7 +33,7 @@ T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
 T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
 
-theorem pr3_pr2:
+lemma pr3_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
 t1 t2))))
 \def
@@ -52,7 +52,7 @@ t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
 (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
 (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
 
-theorem pr3_thin_dx:
+lemma pr3_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u 
 t2)))))))
@@ -66,7 +66,7 @@ t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
 t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
 t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
 
-theorem pr3_head_1:
+lemma pr3_head_1:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
 (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
 \def
@@ -80,7 +80,7 @@ T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
 c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
 (H2 k t)))))))))) u1 u2 H)))).
 
-theorem pr3_head_2:
+lemma pr3_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u 
 t2)))))))
@@ -114,7 +114,7 @@ u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
 u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
 
-theorem pr3_cflat:
+lemma pr3_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
 \def
@@ -137,7 +137,7 @@ u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
 (f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
 u2))))))))).
 
-theorem pr3_pr0_pr2_t:
+lemma pr3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
 (CHead c k u1) t1 t2))))))))
@@ -206,7 +206,7 @@ u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
 (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
 u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
 
-theorem pr3_pr2_pr2_t:
+lemma pr3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
 (CHead c k u1) t1 t2))))))))
@@ -286,7 +286,7 @@ i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0
 f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c 
 u1 u2 H)))).
 
-theorem pr3_pr2_pr3_t:
+lemma pr3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to 
 (pr3 (CHead c k u1) t1 t2))))))))
@@ -318,7 +318,7 @@ t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
 
-theorem pr3_lift:
+lemma pr3_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift 
 h d t1) (lift h d t2)))))))))
@@ -332,7 +332,7 @@ t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
 t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
 H0)))))))).
 
-theorem pr3_eta:
+lemma pr3_eta:
  \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind 
 Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v 
 (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
@@ -356,7 +356,7 @@ Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u
 (pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead 
 (Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
 
-theorem pr3_gen_void:
+lemma pr3_gen_void:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -486,7 +486,7 @@ c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))
 O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
 O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
 
-theorem pr3_gen_abbr:
+lemma pr3_gen_abbr:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -753,7 +753,7 @@ x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing
 (pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
 (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
 
-theorem pr3_gen_appl:
+lemma pr3_gen_appl:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -1518,7 +1518,7 @@ y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4))
 H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
 x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
 
-theorem pr3_gen_bind:
+lemma pr3_gen_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
 T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
index bcecb2777..3c2e56c05 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
@@ -18,7 +18,7 @@ include "basic_1/pr3/fwd.ma".
 
 include "basic_1/pr2/subst1.ma".
 
-theorem pr3_subst1:
+lemma pr3_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) 
 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c 
@@ -45,7 +45,7 @@ t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
 i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
 e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
 
-theorem pr3_gen_cabbr:
+lemma pr3_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 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 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
index ef4d35495..fa49b9855 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
@@ -18,7 +18,7 @@ include "basic_1/pr3/props.ma".
 
 include "basic_1/wcpr0/getl.ma".
 
-theorem pr3_wcpr0_t:
+lemma pr3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
 T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
index 80ebd666e..6dc07a0e1 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
@@ -18,7 +18,7 @@ include "basic_1/r/defs.ma".
 
 include "basic_1/s/defs.ma".
 
-theorem r_S:
+lemma r_S:
  \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S 
@@ -26,7 +26,7 @@ i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
 (Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat 
 f) i))))) k).
 
-theorem r_plus:
+lemma r_plus:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
 (plus (r k i) j))))
 \def
@@ -36,7 +36,7 @@ nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r 
 (Flat f) i) j))))) k).
 
-theorem r_plus_sym:
+lemma r_plus_sym:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
 (plus i (r k j)))))
 \def
@@ -45,7 +45,7 @@ nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
 (i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: 
 F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
 
-theorem r_minus:
+lemma r_minus:
  \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat 
 (minus (r k i) (S n)) (r k (minus i (S n)))))))
 \def
@@ -54,7 +54,7 @@ K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
 n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: 
 F).(minus_x_Sy i n H)) k)))).
 
-theorem r_dis:
+lemma r_dis:
  \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) 
 \to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
 \def
@@ -68,14 +68,14 @@ nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
 ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda 
 (i: nat).(refl_equal nat (S i)))))))) k).
 
-theorem s_r:
+lemma s_r:
  \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 
 i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) 
 (\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
 
-theorem r_arith0:
+lemma r_arith0:
  \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
 \def
  \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: 
@@ -83,7 +83,7 @@ nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
 nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) 
 (minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
 
-theorem r_arith1:
+lemma r_arith1:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S 
 i)) (S j)) (minus (r k i) j))))
 \def
@@ -91,7 +91,7 @@ i)) (S j)) (minus (r k i) j))))
 (\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat 
 (minus (r k i) j)) (r k (S i)) (r_S k i)))).
 
-theorem r_arith2:
+lemma r_arith2:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to 
 (le (r k i) j))))
 \def
@@ -101,7 +101,7 @@ nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda
 (le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
 nat).(\lambda (H: (le (S i) j)).H)))) k).
 
-theorem r_arith3:
+lemma r_arith3:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to 
 (le j (r k i)))))
 \def
@@ -111,7 +111,7 @@ nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda
 (le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
 nat).(\lambda (H: (le j (S i))).H)))) k).
 
-theorem r_arith4:
+lemma r_arith4:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k 
 j)) (minus (r k i) j))))
 \def
@@ -121,7 +121,7 @@ B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i)
 j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
 (minus (r (Flat f) i) j))))) k).
 
-theorem r_arith5:
+lemma r_arith5:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to 
 (lt j (r k i)))))
 \def
@@ -131,7 +131,7 @@ nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda
 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S 
 i))).H)))) k).
 
-theorem r_arith6:
+lemma r_arith6:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S 
 j)) (minus i (s k j)))))
 \def
@@ -141,7 +141,7 @@ B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b)
 j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
 (minus i (s (Flat f) j)))))) k).
 
-theorem r_arith7:
+lemma r_arith7:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j)) 
 \to (eq nat (r k i) j))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma
index 48a34e79b..deac2dbd3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/s/defs.ma".
 
-theorem s_inj:
+lemma s_inj:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) 
 \to (eq nat i j))))
 \def
@@ -26,7 +26,7 @@ B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
 (Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda 
 (j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
 
-theorem s_le_gen:
+lemma s_le_gen:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k i) (s k j)) \to 
 (le i j))))
 \def
@@ -36,7 +36,7 @@ nat).(\lambda (j: nat).(\lambda (H: (le (s (Bind b) i) (s (Bind b)
 j))).(le_S_n i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
 nat).(\lambda (H: (le (s (Flat f) i) (s (Flat f) j))).H)))) k).
 
-theorem s_lt_gen:
+lemma s_lt_gen:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k i) (s k j)) \to 
 (lt i j))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/props.ma b/matita/matita/contribs/lambdadelta/basic_1/s/props.ma
index 82d7c11fa..75318f07a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/s/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/s/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/s/defs.ma".
 
-theorem s_S:
+lemma s_S:
  \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S 
@@ -24,7 +24,7 @@ i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
 (Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat 
 f) i))))) k).
 
-theorem s_plus:
+lemma s_plus:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
 (plus (s k i) j))))
 \def
@@ -34,7 +34,7 @@ nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s 
 (Flat f) i) j))))) k).
 
-theorem s_plus_sym:
+lemma s_plus_sym:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
 (plus i (s k j)))))
 \def
@@ -45,7 +45,7 @@ nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
 (plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
 nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
 
-theorem s_minus:
+lemma s_minus:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s 
 k (minus i j)) (minus (s k i) j)))))
 \def
@@ -57,7 +57,7 @@ j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j 
 i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
 
-theorem minus_s_s:
+lemma minus_s_s:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s 
 k j)) (minus i j))))
 \def
@@ -67,7 +67,7 @@ B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i 
 j))))) k).
 
-theorem s_le:
+lemma s_le:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) 
 (s k j)))))
 \def
@@ -76,7 +76,7 @@ nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
 nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: 
 F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
 
-theorem s_lt:
+lemma s_lt:
  \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) 
 (s k j)))))
 \def
@@ -85,7 +85,7 @@ nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
 nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(lt_n_S i j H))))) (\lambda (_: 
 F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
 
-theorem s_inc:
+lemma s_inc:
  \forall (k: K).(\forall (i: nat).(le i (s k i)))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) 
@@ -94,14 +94,14 @@ theorem s_inc:
 (S (S (s (Bind b) i))) (le_n (S (S (s (Bind b) i)))))))))) (\lambda (f: 
 F).(\lambda (i: nat).(le_n (s (Flat f) i)))) k).
 
-theorem s_arith0:
+lemma s_arith0:
  \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
 \def
  \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: 
 nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal 
 nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
 
-theorem s_arith1:
+lemma s_arith1:
  \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
 \def
  \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma
index eb08a1c7f..373c81691 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma
@@ -22,7 +22,7 @@ include "basic_1/csubc/drop1.ma".
 
 include "basic_1/csubc/props.ma".
 
-theorem sc3_arity_csubc:
+lemma sc3_arity_csubc:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
 t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall 
 (c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
@@ -303,7 +303,7 @@ a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1
 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1 
 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
 
-theorem sc3_arity:
+lemma sc3_arity:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to (sc3 g a c t)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma
index 8740e6edb..80da46d3c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma
@@ -36,7 +36,7 @@ include "basic_1/drop1/props.ma".
 
 include "basic_1/lift1/drop1.ma".
 
-theorem sc3_arity_gen:
+lemma sc3_arity_gen:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c 
 t) \to (arity g c t a)))))
 \def
@@ -57,7 +57,7 @@ g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
 a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
 Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
 
-theorem sc3_repl:
+lemma sc3_repl:
  \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c 
 t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
 \def
@@ -124,7 +124,7 @@ Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0)
 (llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 
 H11))))))) H7))))) H4)))))))))))) a2)) a1)).
 
-theorem sc3_lift:
+lemma sc3_lift:
  \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e 
 t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) 
 \to (sc3 g a c (lift h d t))))))))))
@@ -167,7 +167,7 @@ is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
 t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) 
 (lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
 
-theorem sc3_lift1:
+lemma sc3_lift1:
  \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: 
 PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) 
 \to (sc3 g a c (lift1 hds t)))))))))
@@ -187,7 +187,7 @@ e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n
 n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 
 H4) c n n0 H3)))) H2))))))))))) hds)))).
 
-theorem sc3_abbr:
+lemma sc3_abbr:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: 
 nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads 
 (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to 
@@ -371,7 +371,7 @@ is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
 (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl 
 is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
 
-theorem sc3_props__sc3_sn3_abst:
+fact sc3_props__sc3_sn3_abst:
  \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g 
 a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def 
 (THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to 
@@ -500,7 +500,7 @@ vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
 (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat 
 Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
 
-theorem sc3_sn3:
+lemma sc3_sn3:
  \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c 
 t) \to (sn3 c t)))))
 \def
@@ -516,7 +516,7 @@ C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i))
 \to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef 
 i))))))))))).(H1 c t H))) H0))))))).
 
-theorem sc3_abst:
+lemma sc3_abst:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
 (i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef 
 i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
index 7b5d41e51..2550405f2 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
@@ -18,16 +18,16 @@ include "basic_1/sn3/defs.ma".
 
 include "basic_1/pr3/props.ma".
 
-let rec sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: T).(((\forall (t2: 
-T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (sn3 
-c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: 
-Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) (t: T) (s0: sn3 
-c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) \Rightarrow (f t1 s1 
-(\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to (\forall (P0: 
-Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).((sn3_ind c P f) t2 (s1 t2 p 
-p0))))))].
+implied let rec sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: 
+T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 
+c t1 t2) \to (sn3 c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) 
+(t: T) (s0: sn3 c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) 
+\Rightarrow (f t1 s1 (\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to 
+(\forall (P0: Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).((sn3_ind c P f) t2 
+(s1 t2 p p0))))))].
 
-theorem sn3_gen_bind:
+lemma sn3_gen_bind:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
 \def
@@ -79,7 +79,7 @@ in (land_ind (sn3 c x) (sn3 (CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b)
 x) t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) 
 t2)).H10)) H8))))))))))))))) y H0))))) H))))).
 
-theorem sn3_gen_flat:
+lemma sn3_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
 \def
@@ -127,7 +127,7 @@ H7 x f) x t2 (refl_equal T (THead (Flat f) x t2))) in (land_ind (sn3 c x)
 (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 c 
 t2)).H10)) H8))))))))))))))) y H0))))) H))))).
 
-theorem sn3_gen_head:
+lemma sn3_gen_head:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead k u t)) \to (sn3 c u)))))
 \def
@@ -142,7 +142,7 @@ F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
 (land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
 (sn3 c t)).H1)) H0)))))))) k).
 
-theorem sn3_gen_cflat:
+lemma sn3_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
 c (Flat f) u) t) \to (sn3 c t)))))
 \def
@@ -156,7 +156,7 @@ t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
 \to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
 (pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
 
-theorem sn3_gen_lift:
+lemma sn3_gen_lift:
  \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
index 24d0f4ed6..71d9e93fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
@@ -20,7 +20,7 @@ include "basic_1/drop1/fwd.ma".
 
 include "basic_1/lift1/props.ma".
 
-theorem sns3_lifts1:
+lemma sns3_lifts1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
 (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
index fc3f7c09e..e00736172 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
@@ -20,7 +20,7 @@ include "basic_1/nf2/dec.ma".
 
 include "basic_1/nf2/pr3.ma".
 
-theorem sn3_nf2:
+lemma sn3_nf2:
  \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
 \def
  \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
@@ -31,7 +31,7 @@ in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P:
 Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
 (refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
 
-theorem nf2_sn3:
+lemma nf2_sn3:
  \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
 t u)) (\lambda (u: T).(nf2 c u)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
index e5d7cf28b..1e1406155 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
@@ -20,7 +20,7 @@ include "basic_1/nf2/iso.ma".
 
 include "basic_1/pr3/iso.ma".
 
-theorem sn3_pr3_trans:
+lemma sn3_pr3_trans:
  \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
 t2) \to (sn3 c t2)))))
 \def
@@ -40,7 +40,7 @@ H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
 H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: 
 Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
 
-theorem sn3_pr2_intro:
+lemma sn3_pr2_intro:
  \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to 
 (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
 \def
@@ -135,7 +135,7 @@ H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
 H7))))))))) t H2)))))) u H))).
 
-theorem sn3_cflat:
+lemma sn3_cflat:
  \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
 T).(sn3 (CHead c (Flat f) u) t)))))
 \def
@@ -149,7 +149,7 @@ F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
 Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
 (pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
 
-theorem sn3_shift:
+lemma sn3_shift:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
 (THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
 \def
@@ -159,7 +159,7 @@ H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
 (Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
 v) t)).H2)) H0))))))).
 
-theorem sn3_change:
+lemma sn3_change:
  \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
 T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 
 (CHead c (Bind b) v2) t)))))))
@@ -177,7 +177,7 @@ Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
 (pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
 v1)))))))))) t H0))))))).
 
-theorem sn3_gen_def:
+lemma sn3_gen_def:
  \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
 \def
@@ -188,7 +188,7 @@ i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
 i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
 Abbr c d v i H))))))).
 
-theorem sn3_cdelta:
+lemma sn3_cdelta:
  \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
 (\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: 
 C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
@@ -242,7 +242,7 @@ v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
 (sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
 H0)))))).
 
-theorem sn3_cpr3_trans:
+lemma sn3_cpr3_trans:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
 (k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
 t)))))))
@@ -661,7 +661,7 @@ x0) (THead (Bind x1) x2 x3) H14) in (\lambda (H23: (eq T t2 x2)).(\lambda
 False with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
 H11))))))))) w H4))))))))))) y H0))))) H)))).
 
-theorem sn3_appl_lref:
+lemma sn3_appl_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
 T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
 \def
@@ -780,7 +780,7 @@ T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
 (False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
 x3))) H14)) t2 H9)))))))))))))) H6)) H5))))))))) v H0))))).
 
-theorem sn3_appl_abbr:
+lemma sn3_appl_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
 (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
@@ -2099,7 +2099,7 @@ theorem sn3_appl_appls:
 Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 
 H1))))))))).
 
-theorem sn3_appls_lref:
+lemma sn3_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: 
 TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
 \def
@@ -2284,7 +2284,7 @@ Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
 t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
 t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
 
-theorem sn3_lift:
+lemma sn3_lift:
  \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
 nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
 \def
@@ -2310,7 +2310,7 @@ H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
 (refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) 
 H5))))))))))))) t H))).
 
-theorem sn3_abbr:
+lemma sn3_abbr:
  \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
 \def
@@ -2348,7 +2348,7 @@ Abbr) t))) H8 v H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t)))
 v))) H12 d H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) 
 x1 H10)))) H9))) t2 H6)))))) H4)) H3))))))))))).
 
-theorem sn3_appls_abbr:
+lemma sn3_appls_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
 (CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
 vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
@@ -2387,7 +2387,7 @@ Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
 (pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) 
 H3)))))))) vs0))) vs)))))).
 
-theorem sns3_lifts:
+lemma sns3_lifts:
  \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
 i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
index 5dfa365c1..5b80b01a2 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
@@ -16,20 +16,20 @@
 
 include "basic_1/sty0/defs.ma".
 
-let rec sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: (\forall (c: 
-C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0: (\forall (c: 
-C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d 
-(Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c 
-(TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall (d: 
-C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) \to 
-(\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift (S i) 
-O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall 
-(t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to ((P (CHead 
-c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind b) v 
-t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
-(t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat Appl) v t1) 
-(THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall (v1: 
-T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1: 
+implied let rec sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
+(\forall (c: C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0: 
+(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w) 
+\to (P c (TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall 
+(d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) 
+\to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift 
+(S i) O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v: 
+T).(\forall (t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to 
+((P (CHead c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind 
+b) v t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1: 
+T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
+Appl) v t1) (THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall 
+(v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1: 
 T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
 Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))))))) (c: C) (t: T) (t0: T) (s0: 
 sty0 g c t t0) on s0: P c t t0 \def match s0 with [(sty0_sort c0 n) 
@@ -43,7 +43,7 @@ t2 s1)) | (sty0_cast c0 v1 v2 s1 t1 t2 s2) \Rightarrow (f4 c0 v1 v2 s1
 ((sty0_ind g P f f0 f1 f2 f3 f4) c0 v1 v2 s1) t1 t2 s2 ((sty0_ind g P f f0 f1 
 f2 f3 f4) c0 t1 t2 s2))].
 
-theorem sty0_gen_sort:
+lemma sty0_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
 (TSort n) x) \to (eq T x (TSort (next g n)))))))
 \def
@@ -95,7 +95,7 @@ T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
 (THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) c y x H0))) 
 H))))).
 
-theorem sty0_gen_lref:
+lemma sty0_gen_lref:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
 (TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
 (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
@@ -266,7 +266,7 @@ T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
 (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) H6)))))))))))) 
 c y x H0))) H))))).
 
-theorem sty0_gen_bind:
+lemma sty0_gen_bind:
  \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
 T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda 
 (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead 
@@ -365,7 +365,7 @@ H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1
 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Bind b) u 
 t3)))) H6)))))))))))) c y x H0))) H))))))).
 
-theorem sty0_gen_appl:
+lemma sty0_gen_appl:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: 
 T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g 
 c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
@@ -446,7 +446,7 @@ True])])])) I (THead (Flat Appl) u t1) H5) in (False_ind (ex2 T (\lambda (t3:
 T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead 
 (Flat Appl) u t3)))) H6)))))))))))) c y x H0))) H)))))).
 
-theorem sty0_gen_cast:
+lemma sty0_gen_cast:
  \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall 
 (x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: 
 T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma
index 69849f3f1..f75c1c75a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma
@@ -18,7 +18,7 @@ include "basic_1/sty0/fwd.ma".
 
 include "basic_1/getl/drop.ma".
 
-theorem sty0_lift:
+lemma sty0_lift:
  \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e 
 t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
 e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
@@ -163,7 +163,7 @@ h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
 Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) 
 v1 t3 h d))))))))))))))) e t1 t2 H))))).
 
-theorem sty0_correct:
+lemma sty0_correct:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
 t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma
index 4a8aa1a30..54354378c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma
@@ -18,7 +18,7 @@ include "basic_1/sty1/props.ma".
 
 include "basic_1/cnt/props.ma".
 
-theorem sty1_cnt:
+lemma sty1_cnt:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
 t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt 
 t2)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma
index f6680fca4..535f3ff11 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/sty1/defs.ma".
 
-let rec sty1_ind (g: G) (c: C) (t1: T) (P: (T \to Prop)) (f: (\forall (t2: 
-T).((sty0 g c t1 t2) \to (P t2)))) (f0: (\forall (t: T).((sty1 g c t1 t) \to 
-((P t) \to (\forall (t2: T).((sty0 g c t t2) \to (P t2))))))) (t: T) (s0: 
+implied let rec sty1_ind (g: G) (c: C) (t1: T) (P: (T \to Prop)) (f: (\forall 
+(t2: T).((sty0 g c t1 t2) \to (P t2)))) (f0: (\forall (t: T).((sty1 g c t1 t) 
+\to ((P t) \to (\forall (t2: T).((sty0 g c t t2) \to (P t2))))))) (t: T) (s0: 
 sty1 g c t1 t) on s0: P t \def match s0 with [(sty1_sty0 t2 s1) \Rightarrow 
 (f t2 s1) | (sty1_sing t0 s1 t2 s2) \Rightarrow (f0 t0 s1 ((sty1_ind g c t1 P 
 f f0) t0 s1) t2 s2)].
diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma
index 46a807164..0780b7eeb 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma
@@ -29,7 +29,7 @@ c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g
 c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 
 g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
 
-theorem sty1_bind:
+lemma sty1_bind:
  \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
 T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead 
 (Bind b) v t1) (THead (Bind b) v t2))))))))
@@ -45,7 +45,7 @@ t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c
 (Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) 
 H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
 
-theorem sty1_appl:
+lemma sty1_appl:
  \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat 
 Appl) v t2)))))))
@@ -60,7 +60,7 @@ t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t
 t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 
 (THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
 
-theorem sty1_lift:
+lemma sty1_lift:
  \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e 
 t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
 e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
@@ -78,7 +78,7 @@ c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1:
 (H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) 
 (lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
 
-theorem sty1_correct:
+lemma sty1_correct:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c 
 t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
 \def
@@ -89,7 +89,7 @@ t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1
 t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: 
 T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
 
-theorem sty1_abbr:
+lemma sty1_abbr:
  \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
 nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) 
 \to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
@@ -105,7 +105,7 @@ t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
 (sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w 
 H0)))))))).
 
-theorem sty1_cast2:
+lemma sty1_cast2:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c 
 t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T 
 (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma
index fe6e5d3d1..3ba9fc8e6 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma
@@ -18,14 +18,14 @@ include "basic_1/subst/defs.ma".
 
 include "basic_1/subst0/fwd.ma".
 
-theorem subst_sort:
+lemma subst_sort:
  \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort 
 k)) (TSort k))))
 \def
  \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort 
 k)))).
 
-theorem subst_lref_lt:
+lemma subst_lref_lt:
  \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T 
 (subst d v (TLRef i)) (TLRef i)))))
 \def
@@ -35,7 +35,7 @@ d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
 \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) 
 (refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
 
-theorem subst_lref_eq:
+lemma subst_lref_eq:
  \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
 \def
  \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq 
@@ -43,7 +43,7 @@ T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with
 [true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift 
 i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
 
-theorem subst_lref_gt:
+lemma subst_lref_gt:
  \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T 
 (subst d v (TLRef i)) (TLRef (pred i))))))
 \def
@@ -56,7 +56,7 @@ i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true
 i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) 
 (le_bge d i (lt_le_weak d i H)))))).
 
-theorem subst_head:
+lemma subst_head:
  \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: 
 nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w 
 t)))))))
@@ -64,7 +64,7 @@ t)))))))
  \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: 
 nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).
 
-theorem subst_lift_SO:
+lemma subst_lift_SO:
  \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S 
 O) d t)) t)))
 \def
@@ -103,7 +103,7 @@ d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))))) (subst d v (THead k (lift
 (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) (lift_head k t0 t1 (S O) 
 d)))))))) t)).
 
-theorem subst_subst0:
+lemma subst_subst0:
  \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 
 d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
index 4a70958c1..93aa0736a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
@@ -18,7 +18,7 @@ include "basic_1/subst0/defs.ma".
 
 include "basic_1/lift/props.ma".
 
-theorem dnf_dec2:
+lemma dnf_dec2:
  \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: 
 T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
 O) d v))))))
@@ -148,7 +148,7 @@ x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x)
 (S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) 
 d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
 
-theorem dnf_dec:
+lemma dnf_dec:
  \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or 
 (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
index 12a582906..5af139ab0 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
@@ -18,26 +18,26 @@ include "basic_1/subst0/defs.ma".
 
 include "basic_1/lift/fwd.ma".
 
-let rec subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f: (\forall 
-(v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0: (\forall 
-(v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 i v u1 
-u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v (THead k u1 
-t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v: T).(\forall 
-(t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to ((P 
-(s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead k u 
-t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall 
-(i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k: K).(\forall 
-(t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k i) v t1 t2) 
-\to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat) (t: T) (t0: 
-T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match s0 with 
-[(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2 k) 
-\Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2 k) | 
-(subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1 ((subst0_ind P f 
-f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1 k t2 t3 s2) 
-\Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) k t2 t3 
-s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
+implied let rec subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f: 
+(\forall (v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0: 
+(\forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 
+i v u1 u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v 
+(THead k u1 t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 
+t2) \to ((P (s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead 
+k u t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2: 
+T).(\forall (i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k: 
+K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k 
+i) v t1 t2) \to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat) 
+(t: T) (t0: T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match 
+s0 with [(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2 
+k) \Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2 
+k) | (subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1 
+((subst0_ind P f f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1 
+k t2 t3 s2) \Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 
+s1) k t2 t3 s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
 
-theorem subst0_gen_sort:
+lemma subst0_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
 i v (TSort n) x) \to (\forall (P: Prop).P)))))
 \def
@@ -71,7 +71,7 @@ n)) \to P))).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let H6 \def
 True])) I (TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) 
 H)))))).
 
-theorem subst0_gen_lref:
+lemma subst0_gen_lref:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
 \def
@@ -113,7 +113,7 @@ False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
 (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 t2) (lift 
 (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
 
-theorem subst0_gen_head:
+lemma subst0_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
 (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T 
 (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
@@ -309,7 +309,7 @@ T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k
 u2 t2)) H16 H14)))) k0 H10)))))))) H7)) H6)))))))))))))) i v y x H0))) 
 H))))))).
 
-theorem subst0_gen_lift_lt:
+lemma subst0_gen_lift_lt:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) 
 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
@@ -492,7 +492,7 @@ i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
 (lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
 H2))))))))))))) t1)).
 
-theorem subst0_gen_lift_false:
+lemma subst0_gen_lift_false:
  \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u 
 (lift h d t) x) \to (\forall (P: Prop).P)))))))))
@@ -563,7 +563,7 @@ t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) t1) x1)).(H u x0 h d
 i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d t0) (lift h (s k d) 
 t1) x i H4))))))))))))))))) t).
 
-theorem subst0_gen_lift_ge:
+lemma subst0_gen_lift_ge:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) 
 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
@@ -719,7 +719,7 @@ nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i
 H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u 
 (lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
 
-theorem subst0_gen_lift_rev_ge:
+lemma subst0_gen_lift_rev_ge:
  \forall (t1: T).(\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst0 i v t1 (lift h d u2)) \to ((le (plus d h) 
 i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
index 98238b9cf..762f85762 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/subst0/fwd.ma".
 
-theorem subst0_refl:
+lemma subst0_refl:
  \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to 
 (\forall (P: Prop).P))))
 \def
@@ -73,7 +73,7 @@ t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u
 t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 
 t1 (THead k t0 t1) d H1)))))))))) t)).
 
-theorem subst0_lift_lt:
+lemma subst0_lift_lt:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
 i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i 
 (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
@@ -143,7 +143,7 @@ k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d)
 (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) 
 (lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
 
-theorem subst0_lift_ge:
+lemma subst0_lift_ge:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
 (h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 
 (plus i h) u (lift h d t1) (lift h d t2)))))))))
@@ -201,7 +201,7 @@ h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d
 i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead 
 k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
 
-theorem subst0_lift_ge_S:
+lemma subst0_lift_ge_S:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d 
 t1) (lift (S O) d t2))))))))
@@ -213,7 +213,7 @@ t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
 i) (\lambda (n: nat).(eq nat n (S i))) (le_antisym (plus (S O) i) (S i) (le_n 
 (S i)) (le_n (plus (S O) i))) (plus i (S O)) (plus_sym i (S O)))))))))).
 
-theorem subst0_lift_ge_s:
+lemma subst0_lift_ge_s:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
 i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s 
 (Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
index 5fcbf8405..2d32fcad4 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
@@ -18,7 +18,7 @@ include "basic_1/subst0/fwd.ma".
 
 include "basic_1/lift/tlt.ma".
 
-theorem subst0_weight_le:
+lemma subst0_weight_le:
  \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
 u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
@@ -220,7 +220,7 @@ nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5:
 (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 
 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
 
-theorem subst0_weight_lt:
+lemma subst0_weight_lt:
  \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
 u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
@@ -423,7 +423,7 @@ f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
 (weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t 
 z H))))).
 
-theorem subst0_tlt_head:
+lemma subst0_tlt_head:
  \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt 
 (THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
 \def
@@ -447,7 +447,7 @@ nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
 (_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda 
 (_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
 
-theorem subst0_tlt:
+lemma subst0_tlt:
  \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z 
 (THead (Bind Abbr) u t)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma
index 5689c2721..48d11a44e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma
@@ -18,7 +18,7 @@ include "basic_1/subst1/defs.ma".
 
 include "basic_1/subst0/fwd.ma".
 
-theorem subst1_ind:
+implied lemma subst1_ind:
  \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (P: ((T \to 
 Prop))).((P t1) \to (((\forall (t2: T).((subst0 i v t1 t2) \to (P t2)))) \to 
 (\forall (t: T).((subst1 i v t1 t) \to (P t))))))))
@@ -29,7 +29,7 @@ t2) \to (P t2))))).(\lambda (t: T).(\lambda (s0: (subst1 i v t1 t)).(match s0
 with [subst1_refl \Rightarrow f | (subst1_single x x0) \Rightarrow (f0 x 
 x0)])))))))).
 
-theorem subst1_gen_sort:
+lemma subst1_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
 i v (TSort n) x) \to (eq T x (TSort n))))))
 \def
@@ -39,7 +39,7 @@ t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
 i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x 
 H))))).
 
-theorem subst1_gen_lref:
+lemma subst1_gen_lref:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
 i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift 
 (S n) O v))))))))
@@ -56,7 +56,7 @@ nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
 (eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x 
 H))))).
 
-theorem subst1_gen_head:
+lemma subst1_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
 (x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
@@ -116,7 +116,7 @@ i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
 x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) 
 H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
 
-theorem subst1_gen_lift_lt:
+lemma subst1_gen_lift_lt:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) 
 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
@@ -139,7 +139,7 @@ t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
 (subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x 
 H))))))).
 
-theorem subst1_gen_lift_eq:
+lemma subst1_gen_lift_eq:
  \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u 
 (lift h d t) x) \to (eq T x (lift h d t))))))))))
@@ -151,7 +151,7 @@ h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
 (t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t 
 u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
 
-theorem subst1_gen_lift_ge:
+lemma subst1_gen_lift_ge:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) 
 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma
index 1cfc01a33..f7bf1d15c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma
@@ -39,7 +39,7 @@ k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
 i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both 
 v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
 
-theorem subst1_lift_lt:
+lemma subst1_lift_lt:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 
 i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i 
 (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
@@ -54,7 +54,7 @@ nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
 (minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d 
 H1 h))))))) t2 H))))).
 
-theorem subst1_lift_ge:
+lemma subst1_lift_ge:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
 (h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 
 (plus i h) u (lift h d t1) (lift h d t2)))))))))
@@ -67,7 +67,7 @@ d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
 (d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) 
 (lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
 
-theorem subst1_ex:
+lemma subst1_ex:
  \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: 
 T).(subst1 d u t1 (lift (S O) d t2))))))
 \def
@@ -107,7 +107,7 @@ d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
 (lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k 
 x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
 
-theorem subst1_lift_S:
+lemma subst1_lift_S:
  \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i 
 (TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma
index b816890f1..65b3b604f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/tlist/props.ma".
 
-theorem tslt_wf__q_ind:
+fact tslt_wf__q_ind:
  \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
 \to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
 \to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
@@ -27,7 +27,7 @@ Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
 ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
 (tslen ts)))))).
 
-theorem tslt_wf_ind:
+lemma tslt_wf_ind:
  \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
 TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
 TList).(P ts)))
@@ -45,7 +45,7 @@ m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
 H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
 ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
 
-theorem tlist_ind_rev:
+lemma tlist_ind_rev:
  \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
 TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
 TList).(P ts))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma
index f941820bb..267f9d9ad 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/tlist/defs.ma".
 
-theorem theads_tapp:
+lemma theads_tapp:
  \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T 
 (THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
 \def
@@ -28,7 +28,7 @@ v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
 (THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads 
 k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
 
-theorem tcons_tapp_ex:
+lemma tcons_tapp_ex:
  \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
 TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
 (ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma
index 2ee88a073..f53eb19dc 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/tlt/defs.ma".
 
-theorem tlt_wf__q_ind:
+fact tlt_wf__q_ind:
  \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to 
 Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 
 t))))) P n))) \to (\forall (t: T).(P t)))
@@ -27,7 +27,7 @@ Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t)
 n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight 
 t)))))).
 
-theorem tlt_wf_ind:
+lemma tlt_wf_ind:
  \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) 
 \to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma
index 0daa861c7..6df71b39e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma
@@ -18,7 +18,7 @@ include "basic_1/T/fwd.ma".
 
 include "basic_1/tlt/defs.ma".
 
-theorem wadd_le:
+lemma wadd_le:
  \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
 nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to 
 (\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
@@ -29,7 +29,7 @@ nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
 nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le 
 (wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
 
-theorem wadd_lt:
+lemma wadd_lt:
  \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
 nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to 
 (\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
@@ -41,14 +41,14 @@ nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S_n (S v) (S w) (le_S
 (S (S v)) (S w) (le_n_S (S v) w H0)))) (\lambda (n0: nat).(\lambda (_: (le 
 (wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
 
-theorem wadd_O:
+lemma wadd_O:
  \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: 
 nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat 
 (wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
 
-theorem weight_le:
+lemma weight_le:
  \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) 
 (weight_map g t)))))
@@ -127,7 +127,7 @@ nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus
 t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) 
 (weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
 
-theorem weight_eq:
+lemma weight_eq:
  \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f 
 t) (weight_map g t)))))
@@ -139,14 +139,14 @@ nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n)
 (H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: 
 nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
 
-theorem weight_add_O:
+lemma weight_add_O:
  \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) 
 (weight_map (\lambda (_: nat).O) t))
 \def
  \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: 
 nat).O) (\lambda (n: nat).(wadd_O n))).
 
-theorem weight_add_S:
+lemma weight_add_S:
  \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) 
 O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
 \def
@@ -163,7 +163,7 @@ theorem tlt_trans:
 (weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) 
 (weight v) (weight t) H H0))))).
 
-theorem tlt_head_sx:
+lemma tlt_head_sx:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
@@ -191,7 +191,7 @@ t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S
 (weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: 
 nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
 
-theorem tlt_head_dx:
+lemma tlt_head_dx:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
 \def
  \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma
index c90321ff4..438021239 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma
@@ -20,7 +20,7 @@ include "basic_1/arity/pr3.ma".
 
 include "basic_1/asucc/fwd.ma".
 
-theorem ty3_arity:
+lemma ty3_arity:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
 t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity 
 g c t2 (asucc g a1))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma
index 7fe662dce..fb0df9eb3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma
@@ -18,7 +18,7 @@ include "basic_1/ty3/arity.ma".
 
 include "basic_1/sc3/arity.ma".
 
-theorem ty3_predicative:
+lemma ty3_predicative:
  \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: 
 T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: 
 Prop).P)))))))
@@ -80,7 +80,7 @@ c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
 H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) 
 t (lift (S O) O u2) H0))))))))))).
 
-theorem ty3_acyclic:
+lemma ty3_acyclic:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
 u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
 \def
@@ -92,7 +92,7 @@ u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
 c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x 
 (arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
 
-theorem ty3_sn3:
+lemma ty3_sn3:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
 u) \to (sn3 c t)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma
index c0719c3f1..7b2c899d7 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma
@@ -22,7 +22,7 @@ include "basic_1/getl/dec.ma".
 
 include "basic_1/flt/fwd.ma".
 
-theorem ty3_inference:
+lemma ty3_inference:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
 T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma
index c2424f232..d6d6beb92 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma
@@ -20,7 +20,7 @@ include "basic_1/pc3/fsubst0.ma".
 
 include "basic_1/getl/getl.ma".
 
-theorem ty3_fsubst0:
+lemma ty3_fsubst0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
 t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: 
 T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
@@ -949,7 +949,7 @@ i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst
 i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 
 t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
 
-theorem ty3_csubst0:
+lemma ty3_csubst0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
 (CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g 
@@ -961,7 +961,7 @@ nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2:
 C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 
 (fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
 
-theorem ty3_subst0:
+lemma ty3_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 
 t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e 
 (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
index 2bbe0e91f..13e8ec9ac 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
@@ -18,8 +18,8 @@ include "basic_1/ty3/defs.ma".
 
 include "basic_1/pc3/props.ma".
 
-let rec ty3_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: (\forall (c: 
-C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to ((P c t2 t) \to 
+implied let rec ty3_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: (\forall 
+(c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to ((P c t2 t) \to 
 (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((P c u t1) \to ((pc3 c 
 t1 t2) \to (P c u t2)))))))))))) (f0: (\forall (c: C).(\forall (m: nat).(P c 
 (TSort m) (TSort (next g m)))))) (f1: (\forall (n: nat).(\forall (c: 
@@ -54,15 +54,15 @@ t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 v (THead (Bind Abst) u t3) t4)) |
 f1 f2 f3 f4 f5) c0 t2 t3 t4) t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t3 
 t5 t6))].
 
-let rec tys3_ind (g: G) (c: C) (P: (TList \to (T \to Prop))) (f: (\forall (u: 
-T).(\forall (u0: T).((ty3 g c u u0) \to (P TNil u))))) (f0: (\forall (t: 
-T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: TList).((tys3 g c ts u) 
-\to ((P ts u) \to (P (TCons t ts) u)))))))) (t: TList) (t0: T) (t1: tys3 g c 
-t t0) on t1: P t t0 \def match t1 with [(tys3_nil u u0 t2) \Rightarrow (f u 
-u0 t2) | (tys3_cons t2 u t3 ts t4) \Rightarrow (f0 t2 u t3 ts t4 ((tys3_ind g 
-c P f f0) ts u t4))].
+implied let rec tys3_ind (g: G) (c: C) (P: (TList \to (T \to Prop))) (f: 
+(\forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (P TNil u))))) (f0: 
+(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
+TList).((tys3 g c ts u) \to ((P ts u) \to (P (TCons t ts) u)))))))) (t: 
+TList) (t0: T) (t1: tys3 g c t t0) on t1: P t t0 \def match t1 with 
+[(tys3_nil u u0 t2) \Rightarrow (f u u0 t2) | (tys3_cons t2 u t3 ts t4) 
+\Rightarrow (f0 t2 u t3 ts t4 ((tys3_ind g c P f f0) ts u t4))].
 
-theorem ty3_gen_sort:
+lemma ty3_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
 (TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
 \def
@@ -127,7 +127,7 @@ _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
 \Rightarrow True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) 
 (THead (Flat Cast) t0 t2)) H6))))))))))) c y x H0))) H))))).
 
-theorem ty3_gen_lref:
+lemma ty3_gen_lref:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
 (TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
 (t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
@@ -417,7 +417,7 @@ C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
 (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) 
 H6))))))))))) c y x H0))) H))))).
 
-theorem ty3_gen_bind:
+lemma ty3_gen_bind:
  \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
 T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda 
 (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: 
@@ -584,7 +584,7 @@ T (THead (Flat Cast) t2 t0) (\lambda (ee: T).(match ee with [(TSort _)
 (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 
 g (CHead c0 (Bind b) u) t1 t4)))) H6))))))))))) c y x H0))) H))))))).
 
-theorem ty3_gen_appl:
+lemma ty3_gen_appl:
  \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
 T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: 
 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
@@ -744,7 +744,7 @@ t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u
 t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) c y x 
 H0))) H)))))).
 
-theorem ty3_gen_cast:
+lemma ty3_gen_cast:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall 
 (x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: 
 T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
@@ -873,7 +873,7 @@ T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0
 (THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) 
 H)))))).
 
-theorem tys3_gen_nil:
+lemma tys3_gen_nil:
  \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
 (\lambda (u0: T).(ty3 g c u u0))))))
 \def
@@ -892,7 +892,7 @@ TList).(match ee with [TNil \Rightarrow False | (TCons _ _) \Rightarrow
 True])) I TNil H4) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u0 u1))) 
 H5))))))))) y u H0))) H)))).
 
-theorem tys3_gen_cons:
+lemma tys3_gen_cons:
  \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
 (u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts 
 u)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma
index c1581ca57..d98b4020c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma
@@ -20,7 +20,7 @@ include "basic_1/pc3/nf2.ma".
 
 include "basic_1/nf2/fwd.ma".
 
-theorem ty3_gen_appl_nf2:
+lemma ty3_gen_appl_nf2:
  \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
 T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: 
 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
@@ -86,7 +86,7 @@ x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
 (pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) 
 (ty3_gen_appl g c w v x H))))))).
 
-theorem ty3_inv_lref_nf2_pc3:
+lemma ty3_inv_lref_nf2_pc3:
  \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c 
 (TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to 
 ((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
@@ -195,7 +195,7 @@ ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _
 _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: 
 T).(eq T u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
 
-theorem ty3_inv_lref_nf2:
+lemma ty3_inv_lref_nf2:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c 
 (TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: 
 T).(eq T u (lift (S i) O u0))))))))))
@@ -204,7 +204,7 @@ T).(eq T u (lift (S i) O u0))))))))))
 (H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: 
 (nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
 
-theorem ty3_inv_appls_lref_nf2:
+lemma ty3_inv_appls_lref_nf2:
  \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: 
 T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to 
 ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S 
@@ -262,7 +262,7 @@ i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
 (THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t 
 Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
 
-theorem ty3_inv_lref_lref_nf2:
+lemma ty3_inv_lref_lref_nf2:
  \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c 
 (TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i 
 j)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma
index e9cf818b7..f5a2ebb24 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma
@@ -28,7 +28,7 @@ T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs:
 TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w 
 u)) \to False)))))))).
 
-theorem ty3_nf2_inv_abst_premise_csort:
+lemma ty3_nf2_inv_abst_premise_csort:
  \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise 
 (CSort m) w u)))
 \def
@@ -38,7 +38,7 @@ wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs
 (lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d 
 (Bind Abst) wi) H False))))))))).
 
-theorem ty3_nf2_inv_all:
+lemma ty3_nf2_inv_all:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
 u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
 t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
@@ -61,7 +61,7 @@ TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2:
 (arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g 
 c t x H2 H0)))) H1)))))))).
 
-theorem ty3_nf2_inv_sort:
+lemma ty3_nf2_inv_sort:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t 
 (TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort 
 n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
@@ -173,7 +173,7 @@ nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))
 x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) 
 H2)) H1)))))))).
 
-theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux:
+fact ty3_nf2_gen__ty3_nf2_inv_abst_aux:
  \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise 
 c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead 
 (Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to 
@@ -193,7 +193,7 @@ wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift
 vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) 
 w1 u1) H0))))))))))))))).
 
-theorem ty3_nf2_inv_abst:
+lemma ty3_nf2_inv_abst:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: 
 T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to 
 ((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
index 632347788..33f41ebe8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
@@ -26,7 +26,7 @@ include "basic_1/pc3/wcpr0.ma".
 
 include "basic_1/pc1/props.ma".
 
-theorem ty3_sred_wcpr0_pr0:
+lemma ty3_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
 t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) 
 \to (ty3 g c2 t2 t)))))))))
@@ -621,7 +621,7 @@ t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10)))
 H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
 t4))))))))))))))) c1 t1 t H))))).
 
-theorem ty3_sred_pr0:
+lemma ty3_sred_pr0:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall 
 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -629,7 +629,7 @@ theorem ty3_sred_pr0:
 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 
 t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
 
-theorem ty3_sred_pr1:
+lemma ty3_sred_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -643,7 +643,7 @@ C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g:
 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t 
 (ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
 
-theorem ty3_sred_pr2:
+lemma ty3_sred_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
@@ -660,7 +660,7 @@ G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0
 (ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
 H2)))))))))))))) c t1 t2 H)))).
 
-theorem ty3_sred_pr3:
+lemma ty3_sred_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma
index c7eb51db7..0b897b4f9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/ty3/pr3.ma".
 
-theorem ty3_cred_pr2:
+lemma ty3_cred_pr2:
  \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 
 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
 (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
@@ -41,7 +41,7 @@ c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0)
 (CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 
 v2 H))))).
 
-theorem ty3_cred_pr3:
+lemma ty3_cred_pr3:
  \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 
 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
 (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
@@ -58,7 +58,7 @@ B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to
 T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b 
 t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
 
-theorem ty3_gen_lift:
+lemma ty3_gen_lift:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: 
 nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop 
 h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: 
@@ -443,7 +443,7 @@ x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21)))))
 H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 
 H5))))))))))))))) c y x H0))))) H))))))).
 
-theorem ty3_tred:
+lemma ty3_tred:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
 t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
 \def
@@ -466,7 +466,7 @@ T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def
 (ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g 
 t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
 
-theorem ty3_sred_back:
+lemma ty3_sred_back:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c 
 t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 
 t) \to (ty3 g c t1 t)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma
index 8816826ed..d9a82d40e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma
@@ -18,7 +18,7 @@ include "basic_1/ty3/fwd.ma".
 
 include "basic_1/pc3/fwd.ma".
 
-theorem ty3_lift:
+lemma ty3_lift:
  \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e 
 t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c 
 e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
@@ -185,7 +185,7 @@ H4)) (lift h d (THead (Flat Cast) t4 t3)) (lift_head (Flat Cast) t4 t3 h d))
 (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) t3 t0 h 
 d)))))))))))))) e t1 t2 H))))).
 
-theorem ty3_correct:
+lemma ty3_correct:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
 t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
 \def
@@ -415,7 +415,7 @@ t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t (THead (Flat Cast) x0 t2) c0
 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 (H3 x0 H7) (Flat Cast) t2) t4 
 H5))))) (ty3_gen_cast g c0 t0 t2 t4 H4)))))))))))) c u t1 H))))).
 
-theorem ty3_gen_abst_abst:
+lemma ty3_gen_abst_abst:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
 (t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 
 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) 
@@ -446,7 +446,7 @@ c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5
 Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind 
 Abst) u t2) H))))))).
 
-theorem ty3_typecheck:
+lemma ty3_typecheck:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t 
 v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
 \def
@@ -457,7 +457,7 @@ c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))
 (THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v 
 H)))))).
 
-theorem ty3_getl_subst0:
+lemma ty3_getl_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
 u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t 
 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma
index 390c22bae..e590a65e0 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma
@@ -18,7 +18,7 @@ include "basic_1/ty3/pr3_props.ma".
 
 include "basic_1/sty0/fwd.ma".
 
-theorem ty3_sty0:
+lemma ty3_sty0:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
 t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma
index 6d0f4d55f..43a6d483c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma
@@ -20,7 +20,7 @@ include "basic_1/pc3/subst1.ma".
 
 include "basic_1/getl/getl.ma".
 
-theorem ty3_gen_cabbr:
+lemma ty3_gen_cabbr:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g 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 
@@ -554,7 +554,7 @@ O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat
 Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 
 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
 
-theorem ty3_gen_cvoid:
+lemma ty3_gen_cvoid:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
 (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
index 31c65c5b2..4f6e1df9f 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
@@ -16,15 +16,15 @@
 
 include "basic_1/wcpr0/defs.ma".
 
-let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c c))) 
-(f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to 
+implied let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c 
+c))) (f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to 
 (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P (CHead 
 c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on w: P c 
 c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp c1 c2 
 w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1 u2 p 
 k)].
 
-theorem wcpr0_gen_sort:
+lemma wcpr0_gen_sort:
  \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
 n))))
 \def
@@ -42,7 +42,7 @@ c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
 | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C 
 (CHead c2 k u2) (CHead c1 k u1)) H5))))))))))) y x H0))) H))).
 
-theorem wcpr0_gen_head:
+lemma wcpr0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma
index 7afd013f9..2b43bb21a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/wcpr0/fwd.ma".
 
 include "basic_1/getl/props.ma".
 
-theorem wcpr0_drop:
+lemma wcpr0_drop:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
 e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
@@ -116,7 +116,7 @@ e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
 (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) 
 c1 c2 H))).
 
-theorem wcpr0_drop_back:
+lemma wcpr0_drop_back:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
 e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
@@ -214,7 +214,7 @@ e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
 (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) 
 c2 c1 H))).
 
-theorem wcpr0_getl:
+lemma wcpr0_getl:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
@@ -330,7 +330,7 @@ e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat
 f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 
 H))).
 
-theorem wcpr0_getl_back:
+lemma wcpr0_getl_back:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
 nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma
index b894fc678..0199af19e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma
@@ -18,7 +18,7 @@ include "basic_1/wf3/fwd.ma".
 
 include "basic_1/clear/fwd.ma".
 
-theorem wf3_clear_conf:
+lemma wf3_clear_conf:
  \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall 
 (c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
 \def
@@ -32,7 +32,7 @@ c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
 C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def 
 (wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
 
-theorem clear_wf3_trans:
+lemma clear_wf3_trans:
  \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall 
 (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda 
 (c2: C).(clear c2 d2))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
index c1ade018f..98eb5558c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/wf3/defs.ma".
 
-let rec wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: nat).(P 
-(CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) 
-\to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
+implied let rec wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: 
+nat).(P (CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g 
+c1 c2) \to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
 (\forall (b: B).(P (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))))))))))) (f1: 
 (\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall 
 (u: T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(P 
@@ -31,7 +31,7 @@ f2) c1 c2 w0) u t t0 b) | (wf3_void c1 c2 w0 u f3 b) \Rightarrow (f1 c1 c2 w0
 ((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3 b) | (wf3_flat c1 c2 w0 u f3) 
 \Rightarrow (f2 c1 c2 w0 ((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3)].
 
-theorem wf3_gen_sort1:
+lemma wf3_gen_sort1:
  \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to 
 (eq C x (CSort m)))))
 \def
@@ -63,7 +63,7 @@ H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: C).(match ee with
 [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) 
 H3) in (False_ind (eq C c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
 
-theorem wf3_gen_bind1:
+lemma wf3_gen_bind1:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: 
 B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: 
 C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda 
@@ -191,7 +191,7 @@ C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3))
 (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H4))))))))) y 
 x H0))) H)))))).
 
-theorem wf3_gen_flat1:
+lemma wf3_gen_flat1:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: 
 F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
 \def
@@ -235,7 +235,7 @@ C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) in (let H10
 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in H10))))) H5)) 
 H4))))))))) y x H0))) H)))))).
 
-theorem wf3_gen_head2:
+lemma wf3_gen_head2:
  \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: 
 K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma
index c8a7aa311..9c5fc73f5 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma
@@ -18,7 +18,7 @@ include "basic_1/wf3/clear.ma".
 
 include "basic_1/ty3/dec.ma".
 
-theorem wf3_getl_conf:
+lemma wf3_getl_conf:
  \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall 
 (v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: 
 C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda 
@@ -139,7 +139,7 @@ C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
 v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n 
 H1)))))))))))))) c1)))) i)).
 
-theorem getl_wf3_trans:
+lemma getl_wf3_trans:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to 
 (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: 
 C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma
index 9d7b3374a..aec1f02ef 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma
@@ -18,7 +18,7 @@ include "basic_1/wf3/ty3.ma".
 
 include "basic_1/app/defs.ma".
 
-theorem wf3_total:
+lemma wf3_total:
  \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
 \def
  \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: 
@@ -41,7 +41,7 @@ False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
 (f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x 
 (wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
 
-theorem ty3_shift1:
+lemma ty3_shift1:
  \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall 
 (t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c 
 t2)))))))
@@ -123,7 +123,7 @@ False | (Flat _) \Rightarrow True])) I (Bind x) H9) in (False_ind (ty3 g
 (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
 t2))) H10)))) H8)))))))))))))))) y c H0))) H))).
 
-theorem wf3_idem:
+lemma wf3_idem:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g 
 c2 c2))))
 \def
@@ -139,7 +139,7 @@ c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
 (c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 
 c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
 
-theorem wf3_ty3:
+lemma wf3_ty3:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t 
 u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t 
 u)))))))
diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma
index be4d21960..2958902bd 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma
@@ -16,7 +16,7 @@
 
 include "basic_1/wf3/getl.ma".
 
-theorem wf3_pr2_conf:
+lemma wf3_pr2_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
 u) \to (pr2 c2 t1 t2)))))))))
@@ -41,7 +41,7 @@ g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
 (_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) 
 H5)))))))))))))))))) c1 t1 t2 H))))).
 
-theorem wf3_pr3_conf:
+lemma wf3_pr3_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 
 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
 u) \to (pr3 c2 t1 t2)))))))))
@@ -58,7 +58,7 @@ c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
 t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 
 H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
 
-theorem wf3_pc3_conf:
+lemma wf3_pc3_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 
 u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
@@ -72,7 +72,7 @@ c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
 g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) 
 H3)))))))))))).
 
-theorem wf3_ty3_conf:
+lemma wf3_ty3_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
index 327b93fdb..7a6c3f27a 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/blt/defs.ma".
 
-theorem lt_blt:
+lemma lt_blt:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
@@ -35,7 +35,7 @@ true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0
 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda 
 (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
 
-theorem le_bge:
+lemma le_bge:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
@@ -54,7 +54,7 @@ in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in
 (eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 
 (le_S_n n n0 H1))))) y)))) x).
 
-theorem blt_lt:
+lemma blt_lt:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
@@ -71,7 +71,7 @@ nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m)
 \Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool 
 (blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
 
-theorem bge_le:
+lemma bge_le:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
index d9c7d049c..724a34747 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/preamble.ma".
 
-theorem nat_dec:
+lemma nat_dec:
  \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to 
 (\forall (P: Prop).P))))
 \def
@@ -54,7 +54,7 @@ Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or
 (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3) 
 in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
 
-theorem simpl_plus_r:
+lemma simpl_plus_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) 
 (plus p n)) \to (eq nat m p))))
 \def
@@ -64,18 +64,18 @@ theorem simpl_plus_r:
 nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m) 
 (plus_sym n m)))))).
 
-theorem minus_Sx_Sy:
+lemma minus_Sx_Sy:
  \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
 \def
  \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
 
-theorem minus_plus_r:
+lemma minus_plus_r:
  \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
 \def
  \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: 
 nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
 
-theorem plus_permute_2_in_3:
+lemma plus_permute_2_in_3:
  \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x 
 y) z) (plus (plus x z) y))))
 \def
@@ -86,7 +86,7 @@ nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
 (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z 
 y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
 
-theorem plus_permute_2_in_3_assoc:
+lemma plus_permute_2_in_3_assoc:
  \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n 
 h) k) (plus n (plus k h)))))
 \def
@@ -96,7 +96,7 @@ nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
 (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k 
 h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
 
-theorem plus_O:
+lemma plus_O:
  \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat 
 x O) (eq nat y O))))
 \def
@@ -111,13 +111,13 @@ y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
 True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in 
 (H1 (refl_equal nat O))))))) x).
 
-theorem minus_Sx_SO:
+lemma minus_Sx_SO:
  \forall (x: nat).(eq nat (minus (S x) (S O)) x)
 \def
  \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal 
 nat x) (minus x O) (minus_n_O x)).
 
-theorem nat_dec_neg:
+lemma nat_dec_neg:
  \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
 \def
  \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq 
@@ -136,7 +136,7 @@ n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
 (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H 
 n0)))) j)))) i).
 
-theorem neq_eq_e:
+lemma neq_eq_e:
  \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) 
 \to P)) \to ((((eq nat i j) \to P)) \to P))))
 \def
@@ -144,7 +144,7 @@ theorem neq_eq_e:
 (eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def 
 (nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
 
-theorem le_false:
+lemma le_false:
  \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S 
 n) m) \to P))))
 \def
@@ -173,13 +173,13 @@ O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1)
 (S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) 
 (le_S_n (S n1) n H2))))))) n0)))) m).
 
-theorem le_Sx_x:
+lemma le_Sx_x:
  \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
 \def
  \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def 
 le_Sn_n in (False_ind P (H0 x H))))).
 
-theorem le_n_pred:
+lemma le_n_pred:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
@@ -187,7 +187,7 @@ theorem le_n_pred:
 nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans 
 (pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
 
-theorem minus_le:
+lemma minus_le:
  \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n 
@@ -197,7 +197,7 @@ y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H:
 nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow 
 (minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
 
-theorem le_plus_minus_sym:
+lemma le_plus_minus_sym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) 
 n))))
 \def
@@ -205,7 +205,7 @@ n))))
 (plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) 
 (plus (minus m n) n) (plus_sym (minus m n) n)))).
 
-theorem le_minus_minus:
+lemma le_minus_minus:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) 
 \to (le (minus y x) (minus z x))))))
 \def
@@ -215,7 +215,7 @@ nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
 z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z 
 (le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
 
-theorem le_minus_plus:
+lemma le_minus_plus:
  \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat 
 (minus (plus x y) z) (plus (minus x z) y)))))
 \def
@@ -246,7 +246,7 @@ nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus
 n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S 
 n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
 
-theorem le_minus:
+lemma le_minus:
  \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to 
 (le x (minus z y)))))
 \def
@@ -255,14 +255,14 @@ x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
 y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x 
 y))))).
 
-theorem le_trans_plus_r:
+lemma le_trans_plus_r:
  \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to 
 (le y z))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus 
 x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
 
-theorem lt_x_O:
+lemma lt_x_O:
  \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
 \def
  \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def 
@@ -270,7 +270,7 @@ theorem lt_x_O:
 ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in 
 (False_ind P H0))))).
 
-theorem le_gen_S:
+lemma le_gen_S:
  \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: 
 nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
 \def
@@ -286,14 +286,14 @@ m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
 nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) 
 x H1 H0))]) in (H0 (refl_equal nat x))))).
 
-theorem lt_x_plus_x_Sy:
+lemma lt_x_plus_x_Sy:
  \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: 
 nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) 
 (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
 
-theorem simpl_lt_plus_r:
+lemma simpl_lt_plus_r:
  \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m 
 p)) \to (lt n m))))
 \def
@@ -303,7 +303,7 @@ n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
 H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 
 (plus p m) (plus_sym m p)) in H1)))))).
 
-theorem minus_x_Sy:
+lemma minus_x_Sy:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S 
 (minus x (S y))))))
 \def
@@ -326,14 +326,14 @@ n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat
 (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) 
 y)))) x).
 
-theorem lt_plus_minus:
+lemma lt_plus_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus 
 y (S x)))))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S 
 x) y H))).
 
-theorem lt_plus_minus_r:
+lemma lt_plus_minus_r:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y 
 (S x)) x)))))
 \def
@@ -341,14 +341,14 @@ theorem lt_plus_minus_r:
 (plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x 
 y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
 
-theorem minus_x_SO:
+lemma minus_x_SO:
  \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
 \def
  \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: 
 nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal 
 nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
 
-theorem le_x_pred_y:
+lemma le_x_pred_y:
  \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
 \def
  \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to 
@@ -363,14 +363,14 @@ True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
 x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S 
 n))).(le_S_n x n H0))))) y).
 
-theorem lt_le_minus:
+lemma lt_le_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S 
 O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) 
 (plus_sym x (S O)))))).
 
-theorem lt_le_e:
+lemma lt_le_e:
  \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) 
 \to ((((le d n) \to P)) \to P))))
 \def
@@ -378,7 +378,7 @@ theorem lt_le_e:
 d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in 
 (or_ind (le d n) (lt n d) P H0 H H1)))))).
 
-theorem lt_eq_e:
+lemma lt_eq_e:
  \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
 \to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
 \def
@@ -386,7 +386,7 @@ theorem lt_eq_e:
 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x 
 y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
 
-theorem lt_eq_gt_e:
+lemma lt_eq_gt_e:
  \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
 \to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
 \def
@@ -395,7 +395,7 @@ y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
 \to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda 
 (H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
 
-theorem lt_gen_xS:
+lemma lt_gen_xS:
  \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 
 nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
 \def
@@ -411,21 +411,21 @@ nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
 (ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt 
 m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
 
-theorem le_lt_false:
+lemma le_lt_false:
  \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: 
 Prop).P))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt 
 y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
 
-theorem lt_neq:
+lemma lt_neq:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq 
 nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in 
 (lt_n_n y H1))))).
 
-theorem arith0:
+lemma arith0:
  \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) 
 \to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
 \def
@@ -440,7 +440,7 @@ h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
 d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) 
 (minus_plus h2 (plus d2 h1))))))).
 
-theorem O_minus:
+lemma O_minus:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
@@ -458,7 +458,7 @@ nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
 x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S 
 n))).(H n (le_S_n x0 n H1))))) y)))) x).
 
-theorem minus_minus:
+lemma minus_minus:
  \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) 
 \to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
 \def
@@ -497,7 +497,7 @@ H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) (\lambda
 nat (minus (S x0) (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 
 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
 
-theorem plus_plus:
+lemma plus_plus:
  \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: 
 nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z 
 x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
@@ -572,7 +572,7 @@ x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) 
 x1)))) z).
 
-theorem le_S_minus:
+lemma le_S_minus:
  \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to 
 (le d (S (minus n h))))))
 \def
@@ -582,7 +582,7 @@ d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
 (le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S 
 d (minus n h) (le_minus d n h H))))))).
 
-theorem lt_x_pred_y:
+lemma lt_x_pred_y:
  \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
 \def
  \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred 
diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma
index e6e1faf02..c2dff1889 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/preamble.ma".
 
-theorem insert_eq:
+lemma insert_eq:
  \forall (S: Type[0]).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall 
 (G: ((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) 
 \to ((P x) \to (G x))))))
@@ -25,14 +25,14 @@ theorem insert_eq:
 (G: ((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to 
 (G y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
 
-theorem unintro:
+lemma unintro:
  \forall (A: Type[0]).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall 
 (x: A).(P x))) \to (P a))))
 \def
  \lambda (A: Type[0]).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda 
 (H: ((\forall (x: A).(P x)))).(H a)))).
 
-theorem xinduction:
+lemma xinduction:
  \forall (A: Type[0]).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall 
 (x: A).((eq A t x) \to (P x)))) \to (P t))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma b/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma
index ce17a3c12..f80d0b0e9 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/plist/defs.ma".
 
-theorem papp_ss:
+lemma papp_ss:
  \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss 
 is2)) (Ss (papp is1 is2))))
 \def
diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma b/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma
index d1139518f..7cb7c3108 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/types/defs.ma".
 
-theorem and3_rect:
+implied lemma and3_rect:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
 Type[0]).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
 \def
@@ -24,14 +24,14 @@ Type[0]).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
 Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to P))))).(\lambda (a: (and3 P0 
 P1 P2)).(match a with [(and3_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
 
-theorem and3_ind:
+implied lemma and3_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
 Prop).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
 \def
  \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
 Prop).(and3_rect P0 P1 P2 P)))).
 
-theorem and4_rect:
+implied lemma and4_rect:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to 
 ((and4 P0 P1 P2 P3) \to P))))))
@@ -41,7 +41,7 @@ Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to (P3 \to
 P)))))).(\lambda (a: (and4 P0 P1 P2 P3)).(match a with [(and4_intro x x0 x1 
 x2) \Rightarrow (f x x0 x1 x2)]))))))).
 
-theorem and4_ind:
+implied lemma and4_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to ((and4 
 P0 P1 P2 P3) \to P))))))
@@ -49,7 +49,7 @@ P0 P1 P2 P3) \to P))))))
  \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
 Prop).(\lambda (P: Prop).(and4_rect P0 P1 P2 P3 P))))).
 
-theorem and5_rect:
+implied lemma and5_rect:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P4: Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 
 \to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
@@ -59,7 +59,7 @@ Prop).(\lambda (P4: Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to
 (P2 \to (P3 \to (P4 \to P))))))).(\lambda (a: (and5 P0 P1 P2 P3 P4)).(match a 
 with [(and5_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
 
-theorem and5_ind:
+implied lemma and5_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 
 \to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
@@ -68,7 +68,7 @@ Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3
 Prop).(\lambda (P4: Prop).(\lambda (P: Prop).(and5_rect P0 P1 P2 P3 P4 
 P)))))).
 
-theorem or3_ind:
+implied lemma or3_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
 Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) \to ((or3 P0 P1 P2) 
 \to P)))))))
@@ -79,7 +79,7 @@ Prop).(\lambda (f: ((P0 \to P))).(\lambda (f0: ((P1 \to P))).(\lambda (f1:
 \Rightarrow (f x) | (or3_intro1 x) \Rightarrow (f0 x) | (or3_intro2 x) 
 \Rightarrow (f1 x)])))))))).
 
-theorem or4_ind:
+implied lemma or4_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) 
 \to (((P3 \to P)) \to ((or4 P0 P1 P2 P3) \to P)))))))))
@@ -91,7 +91,7 @@ P))).(\lambda (f1: ((P2 \to P))).(\lambda (f2: ((P3 \to P))).(\lambda (o:
 (or4_intro1 x) \Rightarrow (f0 x) | (or4_intro2 x) \Rightarrow (f1 x) | 
 (or4_intro3 x) \Rightarrow (f2 x)])))))))))).
 
-theorem or5_ind:
+implied lemma or5_ind:
  \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
 Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) 
 \to (((P2 \to P)) \to (((P3 \to P)) \to (((P4 \to P)) \to ((or5 P0 P1 P2 P3 
@@ -105,7 +105,7 @@ P4)).(match o with [(or5_intro0 x) \Rightarrow (f x) | (or5_intro1 x)
 \Rightarrow (f0 x) | (or5_intro2 x) \Rightarrow (f1 x) | (or5_intro3 x) 
 \Rightarrow (f2 x) | (or5_intro4 x) \Rightarrow (f3 x)])))))))))))).
 
-theorem ex3_ind:
+implied lemma ex3_ind:
  \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
 Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P: Prop).(((\forall (x0: 
 A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to P))))) \to ((ex3 A0 P0 P1 P2) \to 
@@ -117,7 +117,7 @@ Prop))).(\lambda (P2: ((A0 \to Prop))).(\lambda (P: Prop).(\lambda (f:
 (e: (ex3 A0 P0 P1 P2)).(match e with [(ex3_intro x x0 x1 x2) \Rightarrow (f x 
 x0 x1 x2)]))))))).
 
-theorem ex4_ind:
+implied lemma ex4_ind:
  \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
 Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P3: ((A0 \to 
 Prop))).(\forall (P: Prop).(((\forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 
@@ -130,7 +130,7 @@ x0) \to ((P2 x0) \to ((P3 x0) \to P))))))).(\lambda (e: (ex4 A0 P0 P1 P2
 P3)).(match e with [(ex4_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 
 x3)])))))))).
 
-theorem ex_2_ind:
+implied lemma ex_2_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
 Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) 
 \to P)))) \to ((ex_2 A0 A1 P0) \to P)))))
@@ -140,7 +140,7 @@ Prop)))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1:
 A1).((P0 x0 x1) \to P))))).(\lambda (e: (ex_2 A0 A1 P0)).(match e with 
 [(ex_2_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
 
-theorem ex2_2_ind:
+implied lemma ex2_2_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
 Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P: 
 Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
@@ -152,7 +152,7 @@ Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P: Prop).(\lambda
 P)))))).(\lambda (e: (ex2_2 A0 A1 P0 P1)).(match e with [(ex2_2_intro x x0 x1 
 x2) \Rightarrow (f x x0 x1 x2)]))))))).
 
-theorem ex3_2_ind:
+implied lemma ex3_2_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
 Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
 \to Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 
@@ -166,7 +166,7 @@ A1).((P0 x0 x1) \to ((P1 x0 x1) \to ((P2 x0 x1) \to P))))))).(\lambda (e:
 (ex3_2 A0 A1 P0 P1 P2)).(match e with [(ex3_2_intro x x0 x1 x2 x3) 
 \Rightarrow (f x x0 x1 x2 x3)])))))))).
 
-theorem ex4_2_ind:
+implied lemma ex4_2_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
 Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
 \to Prop)))).(\forall (P3: ((A0 \to (A1 \to Prop)))).(\forall (P: 
@@ -182,7 +182,7 @@ x0 x1) \to ((P2 x0 x1) \to ((P3 x0 x1) \to P)))))))).(\lambda (e: (ex4_2 A0
 A1 P0 P1 P2 P3)).(match e with [(ex4_2_intro x x0 x1 x2 x3 x4) \Rightarrow (f 
 x x0 x1 x2 x3 x4)]))))))))).
 
-theorem ex_3_ind:
+implied lemma ex_3_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: Prop).(((\forall (x0: 
 A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to P))))) \to ((ex_3 
@@ -194,7 +194,7 @@ A0 A1 A2 P0) \to P))))))
 P)))))).(\lambda (e: (ex_3 A0 A1 A2 P0)).(match e with [(ex_3_intro x x0 x1 
 x2) \Rightarrow (f x x0 x1 x2)]))))))).
 
-theorem ex2_3_ind:
+implied lemma ex2_3_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
 \to Prop))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: 
@@ -208,7 +208,7 @@ A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to P)))))) \to
 P))))))).(\lambda (e: (ex2_3 A0 A1 A2 P0 P1)).(match e with [(ex2_3_intro x 
 x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
 
-theorem ex3_3_ind:
+implied lemma ex3_3_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
 \to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: 
@@ -224,7 +224,7 @@ A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to
 P)))))))).(\lambda (e: (ex3_3 A0 A1 A2 P0 P1 P2)).(match e with [(ex3_3_intro 
 x x0 x1 x2 x3 x4) \Rightarrow (f x x0 x1 x2 x3 x4)]))))))))).
 
-theorem ex4_3_ind:
+implied lemma ex4_3_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
 \to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
@@ -242,7 +242,7 @@ x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to P))))))))).(\lambda (e: (ex4_3
 A0 A1 A2 P0 P1 P2 P3)).(match e with [(ex4_3_intro x x0 x1 x2 x3 x4 x5) 
 \Rightarrow (f x x0 x1 x2 x3 x4 x5)])))))))))).
 
-theorem ex5_3_ind:
+implied lemma ex5_3_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
 \to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
@@ -262,7 +262,7 @@ A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2)
 A1 A2 P0 P1 P2 P3 P4)).(match e with [(ex5_3_intro x x0 x1 x2 x3 x4 x5 x6) 
 \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6)]))))))))))).
 
-theorem ex3_4_ind:
+implied lemma ex3_4_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
 Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
@@ -281,7 +281,7 @@ P))))))))).(\lambda (e: (ex3_4 A0 A1 A2 A3 P0 P1 P2)).(match e with
 [(ex3_4_intro x x0 x1 x2 x3 x4 x5) \Rightarrow (f x x0 x1 x2 x3 x4 
 x5)])))))))))).
 
-theorem ex4_4_ind:
+implied lemma ex4_4_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
 Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
@@ -302,7 +302,7 @@ P)))))))))).(\lambda (e: (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)).(match e with
 [(ex4_4_intro x x0 x1 x2 x3 x4 x5 x6) \Rightarrow (f x x0 x1 x2 x3 x4 x5 
 x6)]))))))))))).
 
-theorem ex4_5_ind:
+implied lemma ex4_5_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
 (A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
@@ -325,7 +325,7 @@ x4) \to P))))))))))).(\lambda (e: (ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3)).(match
 e with [(ex4_5_intro x x0 x1 x2 x3 x4 x5 x6 x7) \Rightarrow (f x x0 x1 x2 x3 
 x4 x5 x6 x7)])))))))))))).
 
-theorem ex5_5_ind:
+implied lemma ex5_5_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
 (A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
@@ -351,7 +351,7 @@ x4) \to ((P4 x0 x1 x2 x3 x4) \to P)))))))))))).(\lambda (e: (ex5_5 A0 A1 A2
 A3 A4 P0 P1 P2 P3 P4)).(match e with [(ex5_5_intro x x0 x1 x2 x3 x4 x5 x6 x7 
 x8) \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))))))))).
 
-theorem ex6_6_ind:
+implied lemma ex6_6_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (P0: 
 ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P1: 
@@ -382,7 +382,7 @@ P)))))))))))))).(\lambda (e: (ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4
 P5)).(match e with [(ex6_6_intro x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) 
 \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)]))))))))))))))).
 
-theorem ex6_7_ind:
+implied lemma ex6_7_ind:
  \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
 (A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (A6: 
 Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/props.ma b/matita/matita/contribs/lambdadelta/ground_1/types/props.ma
index 7eabcfdb5..79919dc76 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/types/props.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/types/props.ma
@@ -16,7 +16,7 @@
 
 include "ground_1/types/defs.ma".
 
-theorem ex2_sym:
+lemma ex2_sym:
  \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
 Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A 
 (\lambda (x: A).(Q x)) (\lambda (x: A).(P x))))))
diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma
index 518bb5ff0..8f5ee7307 100644
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma
@@ -16,17 +16,17 @@
 
 include "legacy_1/coq/defs.ma".
 
-theorem False_rect:
+implied lemma False_rect:
  \forall (P: Type[0]).(False \to P)
 \def
  \lambda (P: Type[0]).(\lambda (f: False).(match f in False with [])).
 
-theorem False_ind:
+implied lemma False_ind:
  \forall (P: Prop).(False \to P)
 \def
  \lambda (P: Prop).(False_rect P).
 
-theorem land_rect:
+implied lemma land_rect:
  \forall (A: Prop).(\forall (B: Prop).(\forall (P: Type[0]).(((A \to (B \to 
 P))) \to ((land A B) \to P))))
 \def
@@ -34,13 +34,13 @@ P))) \to ((land A B) \to P))))
 \to (B \to P)))).(\lambda (a: (land A B)).(match a with [(conj x x0) 
 \Rightarrow (f x x0)]))))).
 
-theorem land_ind:
+implied lemma land_ind:
  \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to (B \to P))) 
 \to ((land A B) \to P))))
 \def
  \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Prop).(land_rect A B P))).
 
-theorem or_ind:
+implied lemma or_ind:
  \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to P)) \to 
 (((B \to P)) \to ((or A B) \to P)))))
 \def
@@ -48,7 +48,7 @@ theorem or_ind:
 P))).(\lambda (f0: ((B \to P))).(\lambda (o: (or A B)).(match o with 
 [(or_introl x) \Rightarrow (f x) | (or_intror x) \Rightarrow (f0 x)])))))).
 
-theorem ex_ind:
+implied lemma ex_ind:
  \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (P0: 
 Prop).(((\forall (x: A).((P x) \to P0))) \to ((ex A P) \to P0))))
 \def
@@ -56,7 +56,7 @@ Prop).(((\forall (x: A).((P x) \to P0))) \to ((ex A P) \to P0))))
 Prop).(\lambda (f: ((\forall (x: A).((P x) \to P0)))).(\lambda (e: (ex A 
 P)).(match e with [(ex_intro x x0) \Rightarrow (f x x0)]))))).
 
-theorem ex2_ind:
+implied lemma ex2_ind:
  \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
 Prop))).(\forall (P0: Prop).(((\forall (x: A).((P x) \to ((Q x) \to P0)))) 
 \to ((ex2 A P Q) \to P0)))))
@@ -66,7 +66,7 @@ Prop))).(\lambda (P0: Prop).(\lambda (f: ((\forall (x: A).((P x) \to ((Q x)
 \to P0))))).(\lambda (e: (ex2 A P Q)).(match e with [(ex_intro2 x x0 x1) 
 \Rightarrow (f x x0 x1)])))))).
 
-theorem eq_rect:
+implied lemma eq_rect:
  \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Type[0]))).((P x) 
 \to (\forall (y: A).((eq A x y) \to (P y))))))
 \def
@@ -74,21 +74,21 @@ theorem eq_rect:
 Type[0]))).(\lambda (f: (P x)).(\lambda (y: A).(\lambda (e: (eq A x 
 y)).(match e with [refl_equal \Rightarrow f])))))).
 
-theorem eq_ind:
+implied lemma eq_ind:
  \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
 (\forall (y: A).((eq A x y) \to (P y))))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(eq_rect A 
 x P))).
 
-let rec le_ind (n: nat) (P: (nat \to Prop)) (f: P n) (f0: (\forall (m: 
-nat).((le n m) \to ((P m) \to (P (S m)))))) (n0: nat) (l: le n n0) on l: P n0 
-\def match l with [le_n \Rightarrow f | (le_S m l0) \Rightarrow (f0 m l0 
+implied let rec le_ind (n: nat) (P: (nat \to Prop)) (f: P n) (f0: (\forall 
+(m: nat).((le n m) \to ((P m) \to (P (S m)))))) (n0: nat) (l: le n n0) on l: 
+P n0 \def match l with [le_n \Rightarrow f | (le_S m l0) \Rightarrow (f0 m l0 
 ((le_ind n P f f0) m l0))].
 
-let rec Acc_ind (A: Type[0]) (R: (A \to (A \to Prop))) (P: (A \to Prop)) (f: 
-(\forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to (((\forall 
-(y: A).((R y x) \to (P y)))) \to (P x))))) (a: A) (a0: Acc A R a) on a0: P a 
-\def match a0 with [(Acc_intro x a1) \Rightarrow (f x a1 (\lambda (y: 
-A).(\lambda (r0: (R y x)).((Acc_ind A R P f) y (a1 y r0)))))].
+implied let rec Acc_ind (A: Type[0]) (R: (A \to (A \to Prop))) (P: (A \to 
+Prop)) (f: (\forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to 
+(((\forall (y: A).((R y x) \to (P y)))) \to (P x))))) (a: A) (a0: Acc A R a) 
+on a0: P a \def match a0 with [(Acc_intro x a1) \Rightarrow (f x a1 (\lambda 
+(y: A).(\lambda (r0: (R y x)).((Acc_ind A R P f) y (a1 y r0)))))].
 
diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
index 4022de816..b5069fdf7 100644
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
+++ b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
@@ -16,7 +16,7 @@
 
 include "legacy_1/coq/fwd.ma".
 
-theorem f_equal:
+lemma f_equal:
  \forall (A: Type[0]).(\forall (B: Type[0]).(\forall (f: ((A \to 
 B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
 \def
@@ -24,7 +24,7 @@ B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
 B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x 
 (\lambda (a: A).(eq B (f x) (f a))) (refl_equal B (f x)) y H)))))).
 
-theorem f_equal2:
+lemma f_equal2:
  \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (B: Type[0]).(\forall 
 (f: ((A1 \to (A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: 
 A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) 
@@ -37,7 +37,7 @@ A1).((eq A2 x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2
 y2)).(eq_ind A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B 
 (f x1 x2)) y2 H0)) y1 H))))))))).
 
-theorem f_equal3:
+lemma f_equal3:
  \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (A3: Type[0]).(\forall 
 (B: Type[0]).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: 
 A1).(\forall (y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: 
@@ -54,14 +54,14 @@ A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3)))))
 x3 (\lambda (a: A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 
 x3)) y3 H1)) y2 H0)) y1 H)))))))))))).
 
-theorem sym_eq:
+lemma sym_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y 
 x))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x 
 y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
 
-theorem eq_ind_r:
+lemma eq_ind_r:
  \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
 (\forall (y: A).((eq A y x) \to (P y))))))
 \def
@@ -69,7 +69,7 @@ theorem eq_ind_r:
 (H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) 
 with [refl_equal \Rightarrow H])))))).
 
-theorem trans_eq:
+lemma trans_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A 
 x y) \to ((eq A y z) \to (eq A x z))))))
 \def
@@ -77,7 +77,7 @@ x y) \to ((eq A y z) \to (eq A x z))))))
 A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda 
 (a: A).(eq A x a)) H z H0)))))).
 
-theorem sym_not_eq:
+lemma sym_not_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to 
 (not (eq A y x)))))
 \def
@@ -85,7 +85,7 @@ theorem sym_not_eq:
 A x y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a 
 y)) (refl_equal A y) x h2)))))).
 
-theorem nat_double_ind:
+lemma nat_double_ind:
  \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to 
 (((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: 
 nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: 
@@ -99,31 +99,31 @@ nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind
 (\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R 
 (S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
 
-theorem eq_add_S:
+lemma eq_add_S:
  \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S 
 m))).(f_equal nat nat pred (S n) (S m) H))).
 
-theorem O_S:
+lemma O_S:
  \forall (n: nat).(not (eq nat O (S n)))
 \def
  \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda 
 (n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
 
-theorem not_eq_S:
+lemma not_eq_S:
  \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S 
 n) (S m)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda 
 (H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
 
-theorem pred_Sn:
+lemma pred_Sn:
  \forall (m: nat).(eq nat m (pred (S m)))
 \def
  \lambda (m: nat).(refl_equal nat (pred (S m))).
 
-theorem S_pred:
+lemma S_pred:
  \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) 
@@ -131,7 +131,7 @@ theorem S_pred:
 m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 
 (S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
 
-theorem le_trans:
+lemma le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) 
 \to (le n p)))))
 \def
@@ -140,26 +140,26 @@ m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H
 (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n 
 m0 IHle)))) p H0))))).
 
-theorem le_trans_S:
+lemma le_trans_S:
  \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S 
 n) m (le_S n n (le_n n)) H))).
 
-theorem le_n_S:
+lemma le_n_S:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
 (n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le 
 n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
 
-theorem le_O_n:
+lemma le_O_n:
  \forall (n: nat).(le O n)
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda 
 (n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
 
-theorem le_S_n:
+lemma le_S_n:
  \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S 
@@ -167,21 +167,21 @@ n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0:
 nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S 
 n m0 H0)))) (S m) H))).
 
-theorem le_Sn_O:
+lemma le_Sn_O:
  \forall (n: nat).(not (le (S n) O))
 \def
  \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: 
 nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: 
 (IsSucc m)).I))) O H)).
 
-theorem le_Sn_n:
+lemma le_Sn_n:
  \forall (n: nat).(not (le (S n) n))
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O 
 O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le 
 (S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
 
-theorem le_antisym:
+lemma le_antisym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n 
 m))))
 \def
@@ -192,12 +192,12 @@ nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n)
 m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) 
 in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
 
-theorem le_n_O_eq:
+lemma le_n_O_eq:
  \forall (n: nat).((le n O) \to (eq nat O n))
 \def
  \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
 
-theorem le_elim_rel:
+lemma le_elim_rel:
  \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to 
 (((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S 
 q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
@@ -213,34 +213,34 @@ n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_:
 (P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 
 H1)))))) m Le))))) n)))).
 
-theorem lt_n_n:
+lemma lt_n_n:
  \forall (n: nat).(not (lt n n))
 \def
  le_Sn_n.
 
-theorem lt_n_S:
+lemma lt_n_S:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m 
 H))).
 
-theorem lt_n_Sn:
+lemma lt_n_Sn:
  \forall (n: nat).(lt n (S n))
 \def
  \lambda (n: nat).(le_n (S n)).
 
-theorem lt_S_n:
+lemma lt_S_n:
  \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S 
 n) m H))).
 
-theorem lt_n_O:
+lemma lt_n_O:
  \forall (n: nat).(not (lt n O))
 \def
  le_Sn_O.
 
-theorem lt_trans:
+lemma lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) 
 \to (lt n p)))))
 \def
@@ -249,17 +249,17 @@ m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S
 (S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt 
 n m0)).(le_S (S n) m0 IHle)))) p H0))))).
 
-theorem lt_O_Sn:
+lemma lt_O_Sn:
  \forall (n: nat).(lt O (S n))
 \def
  \lambda (n: nat).(le_n_S O n (le_O_n n)).
 
-theorem lt_le_S:
+lemma lt_le_S:
  \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
 \def
  \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
 
-theorem le_not_lt:
+lemma le_not_lt:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
@@ -267,12 +267,12 @@ theorem le_not_lt:
 m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle 
 (le_trans_S (S m0) n H1)))))) m H))).
 
-theorem le_lt_n_Sm:
+lemma le_lt_n_Sm:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
 
-theorem le_lt_trans:
+lemma le_lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) 
 \to (lt n p)))))
 \def
@@ -281,7 +281,7 @@ m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0))
 (le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
 (lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
 
-theorem lt_le_trans:
+lemma lt_le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) 
 \to (lt n p)))))
 \def
@@ -290,19 +290,19 @@ m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H
 (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S 
 (S n) m0 IHle)))) p H0))))).
 
-theorem lt_le_weak:
+lemma lt_le_weak:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m 
 H))).
 
-theorem lt_n_Sm_le:
+lemma lt_n_Sm_le:
  \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m 
 H))).
 
-theorem le_lt_or_eq:
+lemma le_lt_or_eq:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
@@ -311,7 +311,7 @@ theorem le_lt_or_eq:
 (or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) 
 (le_n_S n m0 H0))))) m H))).
 
-theorem le_or_lt:
+lemma le_or_lt:
  \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: 
@@ -324,14 +324,14 @@ n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S
 n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S 
 m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
 
-theorem plus_n_O:
+lemma plus_n_O:
  \forall (n: nat).(eq nat n (plus n O))
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
 O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
 
-theorem plus_n_Sm:
+lemma plus_n_Sm:
  \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
 \def
  \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S 
@@ -339,7 +339,7 @@ theorem plus_n_Sm:
 nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat 
 S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
 
-theorem plus_sym:
+lemma plus_sym:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus 
@@ -348,7 +348,7 @@ y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S
 (plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) 
 (plus_n_Sm m y)))) n)).
 
-theorem plus_Snm_nSm:
+lemma plus_Snm_nSm:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: 
@@ -356,7 +356,7 @@ nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda
 (n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n 
 (S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
 
-theorem plus_assoc_l:
+lemma plus_assoc_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m 
 p)) (plus (plus n m) p))))
 \def
@@ -366,14 +366,14 @@ nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat
 (plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 
 m) p) H))) n))).
 
-theorem plus_assoc_r:
+lemma plus_assoc_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n 
 m) p) (plus n (plus m p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n 
 (plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
 
-theorem simpl_plus_l:
+lemma simpl_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) 
 (plus n p)) \to (eq nat m p))))
 \def
@@ -386,21 +386,21 @@ nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn
 (plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) 
 (eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
 
-theorem minus_n_O:
+lemma minus_n_O:
  \forall (n: nat).(eq nat n (minus n O))
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
 O))).(refl_equal nat (S n0)))) n).
 
-theorem minus_n_n:
+lemma minus_n_n:
  \forall (n: nat).(eq nat O (minus n n))
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
 n0))).IHn)) n).
 
-theorem minus_Sn_m:
+lemma minus_Sn_m:
  \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) 
 (minus (S n) m))))
 \def
@@ -411,7 +411,7 @@ n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p
 (le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow 
 (S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
 
-theorem plus_minus:
+lemma plus_minus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) 
 \to (eq nat p (minus n m)))))
 \def
@@ -426,20 +426,20 @@ p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda
 nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H 
 (eq_add_S m0 (plus n0 p) H0)))))) m n))).
 
-theorem minus_plus:
+lemma minus_plus:
  \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
 \def
  \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) 
 (plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
 
-theorem le_pred_n:
+lemma le_pred_n:
  \forall (n: nat).(le (pred n) n)
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) 
 (\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 
 (le_n n0)))) n).
 
-theorem le_plus_l:
+lemma le_plus_l:
  \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus 
@@ -447,14 +447,14 @@ n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn:
 ((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus 
 n0 m) (IHn m))))) n).
 
-theorem le_plus_r:
+lemma le_plus_r:
  \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus 
 n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m 
 (plus n0 m) H))) n)).
 
-theorem simpl_le_plus_l:
+lemma simpl_le_plus_l:
  \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p 
 m)) \to (le n m))))
 \def
@@ -466,14 +466,14 @@ nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n)
 (H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus 
 p0 m) H))))))) p).
 
-theorem le_plus_trans:
+lemma le_plus_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n 
 (plus m p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
 m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
 
-theorem le_reg_l:
+lemma le_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus 
 p n) (plus p m)))))
 \def
@@ -483,7 +483,7 @@ nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H)
 m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
 p))).
 
-theorem le_plus_plus:
+lemma le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
 n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
 \def
@@ -493,7 +493,7 @@ nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0:
 nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S 
 (plus n p) (plus m0 q) H2)))) m H)))))).
 
-theorem le_plus_minus:
+lemma le_plus_minus:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m 
 n)))))
 \def
@@ -503,14 +503,14 @@ n)))))
 (_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat 
 S q (plus p (minus q p)) H0))))) n m Le))).
 
-theorem le_plus_minus_r:
+lemma le_plus_minus_r:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m 
 n)) m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m 
 (plus n (minus m n)) (le_plus_minus n m H)))).
 
-theorem simpl_lt_plus_l:
+lemma simpl_lt_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p 
 m)) \to (lt n m))))
 \def
@@ -520,7 +520,7 @@ nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H)
 m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S 
 (plus p0 n)) (plus p0 m) H))))) p))).
 
-theorem lt_reg_l:
+lemma lt_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
 p n) (plus p m)))))
 \def
@@ -530,7 +530,7 @@ nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H)
 m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
 p))).
 
-theorem lt_reg_r:
+lemma lt_reg_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
 n p) (plus m p)))))
 \def
@@ -541,7 +541,7 @@ nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt
 (plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m 
 p)) (plus n p) (plus_sym n p))))).
 
-theorem le_lt_plus_plus:
+lemma le_lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 \def
@@ -550,7 +550,7 @@ nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n
 (S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) 
 (plus (S n) p) (plus_Snm_nSm n p))))))).
 
-theorem lt_le_plus_plus:
+lemma lt_le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
 n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
 \def
@@ -558,7 +558,7 @@ n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
 nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m 
 p q H H0)))))).
 
-theorem lt_plus_plus:
+lemma lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 \def
@@ -566,7 +566,7 @@ n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q 
 H (lt_le_weak p q H0))))))).
 
-theorem well_founded_ltof:
+lemma well_founded_ltof:
  \forall (A: Type[0]).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
 \def
  \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: 
@@ -580,12 +580,12 @@ nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A
 (lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a 
 (le_n (S (f a))))))).
 
-theorem lt_wf:
+lemma lt_wf:
  well_founded nat lt
 \def
  well_founded_ltof nat (\lambda (m: nat).m).
 
-theorem lt_wf_ind:
+lemma lt_wf_ind:
  \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: 
 nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
 \def