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 c08af2080ede3a3be05aa74ab59591658493c654..a0f6f0931217f9d175c2725dea94bbfc4ff2cb13 100644 (file)
--- 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".
 
 
 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))))
  \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 d2802e2ba7e225562aedffac3b7e2b13dacbb117..675a0863b50b00dc510735351e019228402d187d 100644 (file)
--- 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".
 
 
 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).
 
  \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)))
  \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)))))).
 
 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
  \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 1cbfa54149fd3256f96d5130a34c060906527262..dacd5482b5a7790561b96a422db6f3bd2ade9c93 100644 (file)
--- 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".
 
 
 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))) 
  \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).
 
 (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))))))))
  \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))))))).
 
 (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
  \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))))).
 
 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
  \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))))).
 
 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))))).
 
  \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
  \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).
 
 (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 
  \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))).
 
 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))))
  \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 277e12bf925a29de7c2933c916c3dcb256bca588..ae783b82210f1ee9363cb924ba45b746ecf88174 100644 (file)
--- 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".
 
 
 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
  \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).
 
 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) 
  \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)))).
 
 (((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
  \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).
 
 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
  \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).
 
 \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
  \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).
 
 (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))))))
  \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).
 
 \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)))))
  \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 500f4f7585af921bfd4ef6766cb467f9e3928e29..159a244681ea9dd21bc45890fee45fcf3910a2f8 100644 (file)
--- 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".
 
 
 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: 
  \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).
 
 \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
  \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 445f6fc52189b923f185d241d9456d63abb95add..7b62a5a15ed03f87a33fd56e0968900bddbc60b8 100644 (file)
--- 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".
 
 
 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)).
 
  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)).
 
  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)).
 
  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)).
 
  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 (_: 
  \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).
 
 (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))) 
  \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 9eec9c6e83f510aef71bb0d9b342bc4f4ae96ec8..73ec98cfa923752ce46b161df3f76a6556f9dc60 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))))).
 
 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
  \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))).
 
 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
  \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)))).
 
 (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
  \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))))).
 
 (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
  \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)))))).
 
 (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
  \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)).
 
 (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
  \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)).
 
 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
  \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))))))).
 
 (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
  \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)).
 
 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
  \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)).
 
 (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
  \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 2e8391731a8e7430656b4c36656322889f990d5d..43f3bcec5182994addd1ffb6b0b8ac95a14910f0 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 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
  \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)))).
 
 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
  \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 e2fd9ad9490a2143cd264bb573ad940548ad052a..1932956b1299d72b875ab66493ae3cd749265ac5 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)))).
 
 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
  \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 f91be4fff8021339dc43444349cdf438d1f52dca..4426607390107ce0b78a5865f0104415fe6d592d 100644 (file)
--- 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".
 
 
 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)))) 
  \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 797f16001d29e2a23c0dd80fe80378317a0530f6..e9222f5bb5e509f7ee0419dee5fdcf6285848d9c 100644 (file)
--- 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".
 
 
 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
  \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 77eaa993b56634e184558292170250e91edcb98f..58775a829433a68f6efadd8da1a8a243adef5d5a 100644 (file)
--- 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".
 
 
 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 
 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)].
 
 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
  \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))))).
 
 (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)))) 
  \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))))).
 
 (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 (_: 
  \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)))))))).
 
 (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 (_: 
  \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)))))).
 
 (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)))))))))
  \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)))))).
 
 (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)))))))
  \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)))))).
 
 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))))))))
  \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)))).
 
 (\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)))))))))
  \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 72dd176afbab294fa963a2fbda32e94a4ae708f5..7267b7817b707c5655765ac06954c23fec683a61 100644 (file)
--- 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".
 
 
 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))))))))
  \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 68308fb93352096d078eb35d8d091d536d44cbb4..69dfef63d75f625ebd5793d4c7daab8a6fb45b5e 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))).
 
 (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)))))))))
  \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))).
 
 (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
  \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)))).
 
 (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
  \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 f681969a25d7c00319972bba5ad34262f1746766..fb8379af7d3996c3cf486e29cf04f1fd976011e5 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 (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)))))))))
  \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))))).
 
 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: 
  \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))))).
 
 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 
  \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 48d87a2a44b62133e187fe6d543c21673a015406..334505b50f24a5faff51b637f5a8111ac879b3b8 100644 (file)
--- 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".
 
 
 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 
  \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))))).
 
 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))))))))))))
  \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))))))))))).
 
 (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 
  \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))))).
 
 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 
  \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 c25e24f0d0d249c946ae6203e1e9b7fe8270b584..e2edc783c0bf471ff543008a6230fe61f653dee1 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)))).
 
 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))))))))
  \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 4d9b54e26ac8c24af6676ca34f16a0ce7152c417..d26ec13815476970a1e0a7a53306237cc119e82e 100644 (file)
--- 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".
 
 
 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
  \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)))))))).
 
 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
  \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))))))))).
 
 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
  \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)))))))))).
 
 (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 
  \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 f9a28d88aa772a690cdad387bbf9ed88b5cc89f2..4383e2650f73143c4a73514cf9dd34e304574994 100644 (file)
--- 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".
 
 
 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 
  \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).
 
 (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))))))))
  \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)).
 
 (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 
  \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 88de2d2641bec3e6f84e2fa4325d1e2e4e40f0d2..478e65ccaed95fc456e1e2d82bc2e0f2871a0eaa 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 [(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
  \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))))).
 
 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
  \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))))).
 
 (\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
  \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)))))).
 
 (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
  \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).
 
 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 
  \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 1efc0ba5f20ed432ddd0c74a2a6eb03ab2e7a1a3..07e59e05d3be40b8c5510d9075d776f7f9f95293 100644 (file)
--- 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".
 
 
 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 
  \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).
 
 (\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))))))))
  \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 956a7b64e09802a6a69252eda8cfcff38c2aef72..55bad4ad9fb9f8dc0796ba5fb4517fe87aa27fa5 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 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))) 
  \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 dc687c9dd9cbe9037c33ee3144dc9e413721da84..1019462ab94c3d2b3da29c093d76db63e82c5076 100644 (file)
--- 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".
 
 
 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 3bdc08ee14f7977736eeda78b0b0695f20071513..e18a4788ea3109bee128a86e506e0ddce6d1b517 100644 (file)
--- 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".
 
 
 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
  \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 3edfe33a3d7f28702c6b8b34fa851b022b7caad6..7b35a4381bb36b2440172e20b3e2a68ca3c96b78 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 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))))))))
  \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 39936f41f951b08e0f289ba6bf7e90508978f9e3..06b1cc258bddaa67cbde7a9c4cf16709a0bf4033 100644 (file)
--- 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".
 
 
 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))))))))
  \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)))).
 
 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))))))))
  \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 4aba7bc43219e4615ff58ac23c91db2cfa2c6a93..5a92a8ecd928d99cda624829c538776a70454504 100644 (file)
--- 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".
 
 
 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))) 
  \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).
 
 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) 
  \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).
 
 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) 
  \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).
 
 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) 
  \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 e4101c98376f584abecbe4097a8fde3704d3f533..2f082fa2a25787a6b9b7173911197036a6fa0951 100644 (file)
--- 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".
 
 
 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) 
 (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)].
 
 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)))))))
  \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))))).
 
 (\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 (_: 
  \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))))).
 
 (\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 
  \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))))).
 
 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: 
  \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)))))).
 
 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))))) 
  \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)))))).
 
 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: 
  \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))))).
 
 (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)))))))
  \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))))).
 
 (\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 
  \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))))).
 
 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: 
  \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)))))).
 
 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))))) 
  \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 3299f30910f9c30150424fda50de944c8a82ff78..039992420c9c1a151a458ef09a3d49cd9cbe7430 100644 (file)
--- 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".
 
 
 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))) 
  \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))))))).
 
 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) 
  \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))))))).
 
 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))) 
  \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))))))).
 
 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) 
  \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 0dddd9cd8d5e5b5269916b65cf3b90e42b65269f..0e60bfed5f1e803417fd490fefa22061ad51cc5e 100644 (file)
--- 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".
 
 
 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)) 
  \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 079eed63ba19ed29293ecd5f1173c0e9d6be34b1..63593633a83fe42837254ab3565a63fcc93b8b37 100644 (file)
--- 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".
 
 
 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
  \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)))))))).
 
 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))))))))
  \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 bd6677d644947ab7a78e4caa6bae0ea9b89df129..eaef555fd15575f3addb6ea01206cb7fd61428bd 100644 (file)
--- 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".
 
 
 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))))))))
  \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 0df9ed60ce3b92f151a4495b92dc651f3ed2e030..9e1d3014eafb585ea8f679b0ce20e20c2149a594 100644 (file)
--- 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".
 
 
 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
  \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 c096cf25bdf7329f2feda8bf69000601da79db35..a0bb37e961d7171bb606d60a82d49b5c6decff36 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)).
 
 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))))))))))
  \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)).
 
 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))))))))))
  \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 bd9ea33d41e580da472f87af3b815456dc578224..32574ea29f7eb4b590182636c0304e4ac7de69ee 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)).
 
 (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)))))))))
  \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 01312eb8bb6e9a2c69d5e88769557ad6ae28f46d..a63e6baac7e7e821a320a539e2664c486d4575ec 100644 (file)
--- 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".
 
 
 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) 
 (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)].
 
 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
  \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)))).
 
 \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 (_: 
  \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)))))).
 
 (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
  \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)))).
 
 \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 (_: 
  \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 817d2a222e9cd35bb01eef1d837099d39ad96b86..869c843208ecb3b2b47c14112a9399be586b7c19 100644 (file)
--- 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".
 
 
 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)))))))))
  \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 21c4c6e9f5fe534b2201927e23d8344604621d69..a419bad4c7d3fc02d654599b54f53b7e8d60e21b 100644 (file)
--- 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".
 
 
 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)) 
  \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 82f3fd1fd1b542ab5ce68541b1edf41b94e414b0..d70be313e60a9eeb1c03b82db7061b287ec6ecb7 100644 (file)
--- 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".
 
 
 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
  \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).
 
 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
  \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).
 
 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 
  \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).
 
 (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))))))))))
  \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 f4853b5f78fe970dd4e47312cf5dd645e030c4d1..2d759243f71218016677c69743e8f50712c40ef9 100644 (file)
--- 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".
 
 
 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)))))))))
  \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).
 
 (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)))))))))
  \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).
 
 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: 
  \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).
 
 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 (_: 
  \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).
 
 (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: 
  \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).
 
 (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) 
  \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 022e06fa4f74b8ff03a494f80355e4ba0b0eca18..724d7c5a96d8c481e46aded90ece0bd97c4a8b84 100644 (file)
--- 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".
 
 
 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 
 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))].
 
 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
  \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)))))).
 
 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: 
  \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))))))).
 
 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 
  \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 8fb0fcb2c5c7858b08dac20a59b8c18115327732..89493296180c01587ac8dd5396821f39acec98d7 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)))))))))).
 
 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: 
  \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)))))))))).
 
 (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)))))))))
  \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)))))))))).
 
 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 
  \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 e36dea109ea63ad72c172dbb87081fe4efe4a92a..bb427a677eb88e1841d59891604160156c6e0638 100644 (file)
--- 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".
 
 
 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))))))))
  \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))))))))).
 
 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))))))))
  \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 48bc273a6c8929eb35979c6ad650af35f22a7be8..c063401680499c0119ee8c314a55d646c3375601 100644 (file)
--- 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".
 
 
 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))))))))
  \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)])))))))).
 
 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: 
  \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 9724dd2601e134ae2006be0b902ab8d9c0321bca..1d37fd761ffcfa29ea21ebd7078a82d6cf2f9f5d 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))))).
 
 (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: 
  \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))))))).
 
 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)))))))))
  \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))))))).
 
 (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 
  \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 862a804ad29cdeed70a016dbb1df1bc996f9a3ec..d2dc87dfeb6255ae073647c7a2a71211baedaaaf 100644 (file)
--- 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".
 
 
 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))))))))
  \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 b0d4c923b25dfb2e28306cdb5a4b39f2412a384d..0de7f11cd1ac96a17720ee77ea6ab4892d9a421f 100644 (file)
--- 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".
 
 
 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
  \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 c8dc4c0c425211f9e84cba949918bd2ed9c953e9..8d05e4b1491cda699093be8ca651c444e5a89ee3 100644 (file)
--- 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".
 
 
 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 
  \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))).
 
 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 
  \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)).
 
 (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: 
  \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 ea24eb298b2e563ce22cf307049f626cd291f7c3..cd82088fe49610a25085185e97352f1b381f5579 100644 (file)
--- 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".
 
 
 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) 
 (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)].
 
 (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)))))))
  \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))))).
 
 (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 
  \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))))).
 
 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))))))))
  \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)))))).
 
 (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))))) 
  \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 ebd1762a9121d9cdd71342da023caf483428febd..7156f6296e997e5e4932983384dc4ae560b31977 100644 (file)
--- 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".
 
 
 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 
  \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))))))).
 
 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: 
  \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 7929e08da2211f159df6b77b0a0ccda316092a9b..60a4452e3d9756d6590efeb42254263bf51c6a55 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 (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
  \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 86779920a680501b570a69d0c5aaa0dcb29cd750..31221463f22f420d9b24fd23155d21a82537932d 100644 (file)
--- 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".
 
 
 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)) 
  \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 e87c261a674591af3845c0c96f03dfb3e59024bf..7b3094bd1729bf1aa3be49cb1dbf30ce90f0ba96 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 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))))))))
  \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 80cfb2fa86f84286dce8a1ebcd6fbee9540a04bb..4f4a012727acf4b2a72e238463f3335565fbfba2 100644 (file)
--- 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".
 
 
 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 
  \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))).
 
 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 
  \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 12c78237d0966a3db5bf8165b6081c0a0786f466..0fb7bbceb39d14756d6123e486714dc93085bec3 100644 (file)
--- 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".
 
 
 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))))))))
  \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 f1c483123b056164d904a4f0703e843ebc7bcbf9..ebc5ee5136ecbb007bf96990addf937b313de178 100644 (file)
--- 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".
 
 
 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: 
 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 9befbda30463fe8826ad521910fbeead27d52bd1..f648b2323c41b2238df1d279d7472905037bcb37 100644 (file)
--- 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".
 
 
 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 (_: 
  \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))))))))).
 
 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: 
  \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 82dc44057db511c106adb30715433361c923aeed..eb8448a1a348cadeba6c5ca0ae1f3579c8a8c7da 100644 (file)
--- 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".
 
 
 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)))))))
  \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)))).
 
 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: 
  \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 f4dacaf2039c182d5175c51cdb5624963a43a49f..a77911cae174b05991a0bac5c91520a1cfc0c169 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 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 
  \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))).
 
 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
  \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)))))).
 
 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 
  \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))))))).
 
 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 (_: 
  \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))))))).
 
 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
  \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 6ea5bca4cfa60b40a9afdbe77afd40300f8268ee..1d662e8d45b075b64650974e49b51cb9f3021f9d 100644 (file)
--- 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".
 
 
 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))))))))
  \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)))))))).
 
 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))))))))
  \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))))))))).
 
 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))))))))
  \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 1aacc171f50f37f761372c4fee855372e4ca2269..3270fe008c9905a42e76293741fe16ec503390e6 100644 (file)
--- 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".
 
 
 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 
 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))].
 
 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 
  \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))).
 
 \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))))))))
  \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 1a761e160746deaaec552483cf783f2926f0cb49..83889c30aaa663aff785f0a1f1d8aa774ba172cd 100644 (file)
--- 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".
 
 
 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) 
  \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 94e4a50c18ff77fe11df1efd0cd94be42113d465..41b61d8b6ce3683d59ba2237ab15486c3aa9f7a8 100644 (file)
--- 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".
 
 
 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)))))))
  \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))).
 
 (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))))))))
  \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 7c7cc2888f9af97a5919474f8864555aaf11b1cc..31e071b63551ed85c1dbbc1253e217b48c79f862 100644 (file)
--- 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".
 
 
 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 
 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 1aa405d62efd642721680a1105d3040f2d040efd..c115f9d6b0ab040e511245b51affd7016a3333a2 100644 (file)
--- 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".
 
 
 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
  \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).
 
 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
  \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)).
 
 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 
  \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)).
 
 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 
  \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))).
 
 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 
  \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 95b5201a9baadde663e72395ec437708b968c957..91ce1745bcad69dfb68c1ba063bd5250ac6aa922 100644 (file)
--- 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".
 
 
 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)))))).
 
  \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) 
  \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))).
 
 (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
  \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 d5192ec5473c16586b851b8c36781fe3d567c806..42c2de64690174450e8494115675cbb8b967489e 100644 (file)
--- 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".
 
 
 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) 
  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))).
 
 (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
  \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 c7720192a56ef6ddcb565318c8880dd3502b5270..a25e739167fd73ee260734166a89582dada254ad 100644 (file)
--- 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".
 
 
 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 
  \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))))))).
 
 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))))
  \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 61344ece03b5981257939e4a3039d19e4e8a116a..395bb94a79739315d5059a08a29992c7e8a4351c 100644 (file)
--- 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".
 
 
 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
  \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))))))).
 
 (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
  \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))))))).
 
 (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
  \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))))))).
 
 (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)))).
 
  \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)))))))))
  \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))))))))))).
 
 (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)))))))))
  \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))))))))))).
 
 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
  \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 97f8507191fc114ddf5e2b268801fb0e2c6fd436..59ca03b0bd5b69a76d19e2e09d2bd65d3bd91d8e 100644 (file)
--- 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".
 
 
 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 
  \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)]))))))))))).
 
 [(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 
  \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 e091d8b0dc1ff657efdc67826fe33f8aea1ad412..0915f5f57f6dc810d7580faedb3ee24a02efc2c6 100644 (file)
--- 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".
 
 
 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
  \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).
 
 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
  \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))))))).
 
 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))))))))
  \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)).
 
 (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
  \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 dda85e7f30706b714708b4a064366d559b55778a..a8f4df8bdff85989abee7bd7d31e046524b94296 100644 (file)
--- 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".
 
 
 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)))))
  \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 9fba2b7b1b852c309d33ed7a197e175297334c16..6b8d9a268de5d9490228ac363cf777d77409a9f3 100644 (file)
--- 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".
 
 
 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
  \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)).
 
 (\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 
  \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)).
 
 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)))))))))
  \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)))))))))).
 
 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))))))))
  \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)))))))).
 
 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)) 
  \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)))))))))).
 
 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 
  \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)))))))))))).
 
 (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))) 
  \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)))))))))).
 
 (\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)))))))))
  \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)))))))))).
 
 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)))))))
  \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 0151a248f0e270fdb19a79d8f05aef79b7b8c37d..63876304a4af910fb5b5d04a81bc02400fd3fa23 100644 (file)
--- 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".
 
 
 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
  \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 f1f4426c123c822759f2d365b46b3f6524f43774..7390775d31d6dbec7e5c7406af4242b028aa649c 100644 (file)
--- 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".
 
 
 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)))))
  \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)])))))).
 
 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
  \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)))).
 
 (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
  \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)))))).
 
 (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 
  \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)))).
 
 (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
  \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))))))).
 
 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)))))))))
  \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))))).
 
 (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
  \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))))).
 
 (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 
  \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 c3c50a80f98195efcb3d18383e5eedaca0584e28..64e3319056d3485767193b8ae5ddb090b8145832 100644 (file)
--- 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".
 
 
 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))))))))
  \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 c1a4f58f1f05f83916ba8e35a8c280480ee55ad2..56592c5e6ca6bca5d63a7628b111605b0e0aa314 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 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
  \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))))))).
 
 (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
  \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))))))).
 
 (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)))))))))
  \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 3a917c0852be9db7f6798a16907878f1d728ea6d..fc3550a4cfd3602cfb9eedb6ada3e94556b425d6 100644 (file)
--- 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".
 
 
 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: 
  \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)]))))))).
 
 (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
  \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))).
 
 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
  \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))).
 
 _ _) \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)))))))))
  \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))))).
 
 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)))))))
  \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)))))).
 
 \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: 
  \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 f18572e1e1e8204a54ec7ee8a8eb38ab2c3618cc..baef0f83321523681438f4b088596fbead267dfe 100644 (file)
--- 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".
 
 
 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: 
  \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 87db6f5d97350a77cbf0e877d4f6b5bf0a1d70dc..42005d6ba6fb90f8d2f57672b7ed4a32f484f936 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 (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
  \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)).
 
 (\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
  \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)).
 
 (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
  \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)).
 
 (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
  \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 4077638fecbefbee57418fe89335997989003a60..32d8e6f870a9fc897d7cf3c57ac7c5fb8e9a33ee 100644 (file)
--- 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".
 
 
 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: 
 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))].
 
 \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) 
  \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))))).
 
 (\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: 
  \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))))).
 
 (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) 
  \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))))).
 
 (\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: 
  \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))))).
 
 (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
  \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 ee90c179229119738355c69517bda7191145f459..6eec6578a06b79c3d167023197362fa9d08d6ee7 100644 (file)
--- 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".
 
 
 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)) 
  \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)).
 
 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)))).
 
  \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
  \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)))).
 
 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
  \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)).
 
 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
  \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 e2d22dc7cd5c7bfbc3d05035acdc5b33cc9536e4..beb400a3fbef3914eaeb55dcd22e5625116d99f4 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 (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)))))))))
  \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).
 
 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
  \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)))))))).
 
 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)))))))
  \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)))))))))).
 
 (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
  \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)))))))).
 
 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: 
  \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)))).
 
 (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 
  \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))))))))).
 
 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 
  \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))))))))).
 
 (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
  \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).
 
 (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 
  \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).
 
 (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
  \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 6907f694c7d215f5ad4aa6c769b78f11ae7ab22d..610e02018b2285469a6cbe7ab51de0f56f5fff97 100644 (file)
--- 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".
 
 
 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)))).
 
  \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
  \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)))))).
 
 \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
  \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)))))).
 
 (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)))))))
  \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))))))).
 
  \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)))))))
  \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))))))).
 
  \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)))))))
  \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))))))).
 
  \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
  \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)).
 
 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) 
  \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).
 
 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
  \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))))))).
 
 (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: 
  \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).
 
 (\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
  \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)))).
 
 (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))))))))
  \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).
 
 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))))))))
  \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)))))))).
 
 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))))))))
  \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 d66d095ef9004493d2fdaa6c81f3ac19dbbbaa5e..5adcfd6d895e5ff749ce4d90135516520c47a884 100644 (file)
--- 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".
 
 
 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))))))
  \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).
 
 (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
  \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)))))).
 
 (\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 
  \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)).
 
 (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))))))
  \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)))))))).
 
 (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
  \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 8c1489f1973e8fcf0077ddb9628f8882db14e1c8..f5839c798edbe34cfee5bed9b5325df201963e04 100644 (file)
--- 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".
 
 
 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
  \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).
 
 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
  \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).
 
 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
  \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)).
 
 (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
  \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 067edeeef032d344d7ac2011863d069d14eda8b6..286dfbee36c9c19859eaa5417e785aa5705eaf8a 100644 (file)
--- 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".
 
 
 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 
  \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)).
 
 (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
  \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).
 
 (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))))))
  \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)).
 
 (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))))))
  \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)).
 
 (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
  \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)))).
 
 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))))))
  \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)))).
 
 (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) 
  \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).
 
 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
  \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 7cad4fc7a63ca276ddbf8afec411276e85017ff7..8d45d3775af57733596afec64a06f79a4525a4f4 100644 (file)
--- 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".
 
 
 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)))
  \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)))))).
 
 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
  \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 fe0842328619239fab1287e87b48fa45813e015b..0c7cef6283ebb1ab51f50b1e212711a705130a66 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 (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
  \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))))).
 
 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)))).
 
  \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) 
  \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 84dd3f5fe6943e1f8a517aab9dc6d9a7074e11f4..2f42fc60e194d4ba9ce08945248e1ca78dd21f2a 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 (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
  \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)))).
 
 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
  \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 4ee874d1f246a5551164e869b208e9532752d18a..c5b3b0a642ad3b84fe2fc628a6924422ccc4eeee 100644 (file)
--- 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".
 
 
 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))) 
  \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 e42bf9e0072d09a9b1dc273ac5be44e615417781..d7211841f02e9ead56cdd76d9649e3a484764165 100644 (file)
--- 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".
 
 
 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
  \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 1143c6ff3d159d789edf50b73024f5385b0e3c3f..06487b7132a1b634dba5913b43b73246e00100c3 100644 (file)
--- 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".
 
 
 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
  \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))))))).
 
 (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
  \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))))))))).
 
 (\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
  \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))))).
 
 (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
  \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))))))).
 
 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
  \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)))))))).
 
 (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
  \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)).
 
 (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
  \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))))))).
 
 (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
  \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 bd34df9dae7769abdeb14d8cc79896853c796182..39b8275804e983ddbd5e671a12a6eb9d29a1fb88 100644 (file)
--- 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".
 
 
 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))))))
  \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 f2b078af0987b80fc53785cb783c8f7ebf87534b..9e680d7e8150b12c09c12e3894c2f2d7e626eba3 100644 (file)
--- 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".
 
 
 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
  \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 8c602dfe0ec8e1b5d74e14560a87f8b34b8d00ae..5c6a686e908ddad6b7e0317dbcedad85389c2146 100644 (file)
--- 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".
 
 
 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
  \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 43383eee4020e03b46f7e843532b873777ac9e7c..ccaa9eb52f74dfb852210fb6a06751f7ce0b7023 100644 (file)
--- 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".
 
 
 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))))).
 
  \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 
  \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)))))))).
 
 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
  \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))).
 
 (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
  \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))))))).
 
 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
  \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)))))))).
 
 (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
  \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 faaa213cd8828090bb2d07276afc6911c27ba71c..86e794a79c5771b62b45f6495a8b027dd781216a 100644 (file)
--- 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".
 
 
 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)))).
 
  \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)))).
 
  \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)).
 
  \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
  \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)))))).
 
 (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 
  \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)))).
 
 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
  \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)))))).
 
 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
  \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)))))).
 
 (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
  \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 7147efce300e70d423e3e558a9539ff12d2a639d..61df1500c1ec21635fbcadeb70bad49cf14b052e 100644 (file)
--- 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".
 
 
 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 
  \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)))).
 
 (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 
  \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)))).
 
 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 
  \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 db39fa1fd4a634c8ae4e9457a85342d737e6795f..99239c2a7781579c017e9bd9496c3e1882f00627 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 \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) 
  \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))))))).
 
 (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)))))
  \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))))))).
 
 ((\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))))))))
  \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))))))))).
 
 (\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 
  \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).
 
 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 
  \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)))))))))).
 
 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
  \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 28c9587553b0010714c9587fca7c1e1517b2b686..d386b1864166fbc1f0f5307d08edef8983581ccf 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 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
  \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)))).
 
 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
  \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)))).
 
 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
  \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))))).
 
 (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
  \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)))).
 
 (_: (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: 
  \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 0d16aeb79f410690f8eec4cd5ad1d3d5a1113ede..3ce4180a14f7bf06632125510eb61690d32018e7 100644 (file)
--- 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".
 
 
 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
  \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))))))).
 
 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
  \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 abead4fc4fd91c59cbbfe9e876c6f84b2ac17f5a..a629475c6f3ccd5515865b3e538b497922d92ce3 100644 (file)
--- 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".
 
 
 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
  \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 d65a673b8b192f9fe0298c31b7823af9e1ec96cd..8d86935963a631de6bf5b0d75cac9d345f2ed30a 100644 (file)
--- 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".
 
 
 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
  \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))))))).
 
 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
  \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))))).
 
 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
  \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))))).
 
 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
  \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))))).
 
 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
  \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)))).
 
 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
  \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)))))).
 
 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))).
 
  \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
  \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))))).
 
 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)))))))
  \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))))))).
 
 (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
  \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))))))).
 
 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)))))))
  \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))))))).
 
 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
  \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))))))).
 
 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
  \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)))))).
 
 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
  \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))))))))).
 
 (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))))))))
  \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)))))))).
 
 (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))))))))
  \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)))).
 
 (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))))))))
  \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)))))).
 
 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))))))))
  \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)))).
 
 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)))))))))
  \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))))))))).
 
 (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)))))))
  \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 525c45469b95548fdc886bfa59578e26c1aff355..1df35129b7e4e61eea5d4032d87389c2998593a7 100644 (file)
--- 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".
 
 
 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 
  \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 662b6bf2ec7903f74fa8afe0b034f632706c2f7c..a72a7e90dbc89d679f5f1062691687d051279499 100644 (file)
--- 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".
 
 
 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))))))))
  \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)))))))).
 
 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
  \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))).
 
 (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
  \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 96d70c708f8402008a5027f9e8852946ef4910f4..889e820348c86eb07026deb5c63df2e658ca1e42 100644 (file)
--- 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".
 
 
 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))))
  \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 a9898d7ffe06c27064ac2a0ab1cece4a455f8612..972d9217057c5d1416c6d4d45c4bdb5071e8c433 100644 (file)
--- 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".
 
 
 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 
  \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))).
 
 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 
  \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))).
 
 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 (_: 
  \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)))).
 
 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 
  \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)))).
 
 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 
  \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)))).
 
 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)))))))
  \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 57f9498ce34d9d00b920fafccf2c54707d97fb1e..d3a8f78c97f1d1245ef812708682570d673845c5 100644 (file)
--- 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".
 
 
 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) 
  \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))))))))))))))).
 
 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: 
  \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))))))))))))))))))).
 
 (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 
  (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))))))))))))))))))).
 
 (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: 
  \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)))))))))))))))).
 
 (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 
  \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))))))))))))).
 
 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 
  \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))))))))))))))))))).
 
 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) 
  \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))))))))))))))).
 
 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 
  \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 c3558ca724ea945d476d890407faf9da8fddb4a2..d1c31fcc74cbd0a845cfcae819158f6b132c1a9e 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 (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 
  \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)))).
 
 (\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 
  \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 12f17331f94b9af217cd5a7ab5ca65fa30ef60a4..95fb675716784a120aa69337f9a34e009311d0f2 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))).
 
 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)))))))))
  \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 77a73414a5ed4063529291a7c6241da46be6a35a..05a93172a34796a6d5becc8c7844b0643bd67a01 100644 (file)
--- 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".
 
 
 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)))))))))
  \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)))))))).
 
 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)))))))))
  \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))))).
 
 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)))))))))
  \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 1d4dd0cfa5f29b182e7834680b4785ea86104095..58b7e727fac6869a38a6e97aed2411a261f5a593 100644 (file)
--- 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".
 
 
 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 0b088cc7ba623196dab85b1661760a0cc8c2a698..60dbdbfaad161ffb3a74fc68a1f912d2bc678555 100644 (file)
--- 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".
 
 
 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
  \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 c3c58fe31fdac3f6bb620b5ee235bc8839599f9a..97a5e45d8477121403ae1e09dd08f3752f3f5583 100644 (file)
--- 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".
 
 
 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 
  \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))).
 
 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
  \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))))).
 
 (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
  \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))).
 
 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)))))
  \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 fbcbfbf15b38f70f3601577991b25e34fe7b2d87..b518e4360fc92acbc7ab326af7b7a3ea91486495 100644 (file)
--- 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".
 
 
 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 
  \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)))))).
 
 (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)))))))
  \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)))))).
 
 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
  \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 7694da8fbf39d16b6895f5e4844e46370d1f12d0..1c19f4a2f76ea2b083d50977562aa796fcf20b54 100644 (file)
--- 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".
 
 
 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 
  \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)]))))))).
 
 \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
  \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)))).
 
 (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 
  \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)))).
 
 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 
  \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))))).
 
 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 (_: 
  \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))))).
 
 (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
  \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)))).
 
 (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 (_: 
  \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))))).
 
 (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 
  \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 df933985d13bde771f1a1d2fb65930b72ca4b33a..3151fc53cc7dcb6c90e4e03a76c940d2c8335ef9 100644 (file)
--- 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".
 
 
 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))))))))
  \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))))))).
 
 (\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) 
  \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))))))))))))).
 
 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 
  \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 19cb5d83129ea11ab19aa6297b0c070441820f47..5677ca53e81bd12043db48384148628fe08188d8 100644 (file)
--- 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".
 
 
 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)))))))
  \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)))))).
 
 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
  \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)))))).
 
 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)))))))
  \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)))))).
 
 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
  \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)))).
 
 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
  \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)))))).
 
 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
  \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)))))).
 
 (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))))))))
  \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)))))))).
 
 (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)))))))))
  \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)))))))).
 
 (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 (_: 
  \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))))).
 
 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 (_: 
  \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 c27e3c8097c0a10cefe65f5736b777865265993e..8f1063f8e99493d371f2c16330cd389e78336994 100644 (file)
--- 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".
 
 
 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))))))))))
  \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)))))))))).
 
 (\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 
  \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)))))))).
 
 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 
  \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 1da77c869790323585000d1e832be44741e010e3..e3c9fca027c7cb7bcfa21ed6ed72adbcf512627d 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 \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 
  \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))))).
 
 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 (_: 
  \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))))).
 
 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 
  \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)))))).
 
 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: 
  \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 7ff6beaf4bdbf370c98c1a976e2fab153717ea1c..c56ee4c23eac10c64120d730572720271dbc7217 100644 (file)
--- 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".
 
 
 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 
  \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)))))).
 
 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 
  \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)))).
 
 (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 
  \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)))))))))).
 
 (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: 
  \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)))))).
 
 (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 
  \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))).
 
 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 
  \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)))))))).
 
 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 
  \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 77bb9bc5916fbcf12dd3a93e9b31e1ff2a99ce80..04878c2b983bdbd8613dd687023fed9292bb9513 100644 (file)
--- 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".
 
 
 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
  \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 a8ad07418b167c895401fa916f3153164d522c0c..d973e0eb29ceea39a2c37335677ee9704230f010 100644 (file)
--- 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".
 
 
 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))))))))
  \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 299b08b9a21d30de77e693dfd1de7d8e506b691a..c9cf9c301dc34b39fc8678c4463965e5da38b585 100644 (file)
--- 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".
 
 
 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
  \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)))))).
 
 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
  \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)))).
 
 (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)))))))
  \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)))))).
 
 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
  \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)))).
 
 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)))))))
  \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))))))))).
 
 (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
  \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))))))))).
 
 (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))))))))
  \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)))))))).
 
 (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))))))))
  \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)))).
 
 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))))))))
  \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)))).
 
 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)))))))))
  \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)))))))).
 
 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))))))
  \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))))))).
 
 (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 (_: 
  \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))))).
 
 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 (_: 
  \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))))).
 
 (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 (_: 
  \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))))).
 
 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 
  \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 bcecb2777af2fd494db253fa9e12e0217f8aecf9..3c2e56c05b03c614a9ed4b58d25308d5594814fd 100644 (file)
--- 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".
 
 
 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 
  \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)))))))).
 
 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 
  \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 ef4d354959ea2eadc9c85b2b2b354f7e1c09a849..fa49b985541bd15616220e7420b446402df1af87 100644 (file)
--- 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".
 
 
 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
  \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 80ebd666e2e290e36103be9b709e77b6d6c97d52..6dc07a0e181260407f027d8dfc1164f51b1f0a9c 100644 (file)
--- 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".
 
 
 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 
  \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).
 
 (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
  \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).
 
 (\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
  \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).
 
 (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
  \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)))).
 
 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
  \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).
 
 ((((\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).
 
  \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: 
  \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))).
 
 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
  \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)))).
 
 (\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
  \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).
 
 (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
  \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).
 
 (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
  \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).
 
 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
  \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).
 
 (\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
  \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).
 
 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
  \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 48a34e79b3a34c5469f80330780bbe21c5695eae..deac2dbd3e4604f4cc439391ae3655f2321aee13 100644 (file)
--- 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".
 
 
 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
  \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).
 
 (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
  \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).
 
 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
  \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 82d7c11fa459b726680b03d65385339e7e224458..75318f07a90a2c624953e48ce9d9d4ed415948ed 100644 (file)
--- 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".
 
 
 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 
  \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).
 
 (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
  \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).
 
 (\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
  \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).
 
 (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
  \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).
 
 (\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
  \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).
 
 (\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
  \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).
 
 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
  \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).
 
 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)))) 
  \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).
 
 (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))).
 
  \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 
  \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 eb08a1c7fa230938f1b1268c22106fd3b81aaeb7..373c81691d292fa03e970b6ff8b9c0e5257d57f0 100644 (file)
--- 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".
 
 
 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)))))))))))
  \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))))).
 
 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
  \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 8740e6edb1e4631e82d3886b23cfa08acee1ed33..80da46d3cbac77ada65a26bcd7f3b06cb8811dbe 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 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
  \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)).
 
 (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))))))))))
  \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)).
 
 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)))))))))
  \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)))).
 
 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 
  \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)).
 
 (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 
  \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)).
 
 (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
  \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))))))).
 
 \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))))))))))
  \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 7b5d41e5105b83d5838bab8c15ee90f1057ed796..2550405f2bec142ce5f15aee9fe4a6037f770308 100644 (file)
--- 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".
 
 
 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
  \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))))).
 
 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
  \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))))).
 
 (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
  \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).
 
 (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
  \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))))).
 
 \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
  \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 24d0f4ed6eb7b30ff35749db451ef4227c3fa49c..71d9e93fe566cef9074b090726d46fea7b999b88 100644 (file)
--- 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".
 
 
 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
  \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 fc3f7c09ef0b35d1881f7e548d116a1445a3126a..e007361727282c53f5a0ec7933328f6ea4b296a0 100644 (file)
--- 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".
 
 
 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 
  \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)))))))))).
 
 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
  \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 e5d7cf28bfd559f5438c71ab762d1ad95bbc767e..1e140615525dedaa851028994b82c7a8200478bf 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 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
  \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))).
 
 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
  \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))))).
 
 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
  \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))))))).
 
 (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)))))))
  \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))))))).
 
 (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
  \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))))))).
 
 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))))))))
  \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)))))).
 
 (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)))))))
  \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)))).
 
 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
  \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))))).
 
 (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)))))))))
  \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))))))))).
 
 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
  \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)))).
 
 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
  \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))).
 
 (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
  \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))))))))))).
 
 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)))))))))
  \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)))))).
 
 (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
  \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 5dfa365c1ef49d750ad0c418e8d951814ac987dd..5b80b01a2a65f61c82a9223c4afb385bd6b8b60c 100644 (file)
--- 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".
 
 
 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) 
 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))].
 
 ((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
  \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))))).
 
 (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: 
  \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))))).
 
 (_: 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 
  \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))))))).
 
 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)))))))))
  \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)))))).
 
 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 
  \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 69849f3f143614367804b734f7fdc01ca611cdd8..f75c1c75ab0f10748342e9dc81ada0a88d0726ab 100644 (file)
--- 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".
 
 
 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))))))))))
  \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))))).
 
 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
  \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 4a8aa1a3058fe30f13168e7458be3a95e45356cb..54354378cd5c6677f8029556bdb11c4e22af5dd1 100644 (file)
--- 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".
 
 
 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)))))))
  \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 f6680fca4f87307f4dd06b6e613825b6e37b6d13..535f3ff11d9c1f31dd46ce07e9dc50b2007184a9 100644 (file)
--- 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".
 
 
 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)].
 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 46a807164378144319309e68a3b6fd3bd97db14d..0780b7eebed7895f0b9d15fca685f742f1838a61 100644 (file)
--- 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))))))).
 
 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))))))))
  \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))))))).
 
 (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)))))))
  \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)))))).
 
 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))))))))))
  \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))))).
 
 (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
  \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))))).
 
 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)))))))))
  \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)))))))).
 
 (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 
  \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 fe6e5d3d117094d8bf1699645f82a982455f9faa..3ba9fc8e6ccbe24e20841cff90670ce3e4eb0a11 100644 (file)
--- 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".
 
 
 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)))).
 
  \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
  \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))))).
 
 \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 
  \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)))).
 
 [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
  \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)))))).
 
 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)))))))
  \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))))))).
 
  \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
  \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)).
 
 (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
  \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 4a70958c1142832dbce0724de7611e29405fd2ba..93aa0736a340c2d11a253bf2d86a051a4c5be284 100644 (file)
--- 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".
 
 
 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))))))
  \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).
 
 (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
  \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 12a582906a7148e860e64ada6962f2cfde0c7bf8..5af139ab0ac45a3ff2bfb976a458cc05e9142830 100644 (file)
--- 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".
 
 
 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
  \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)))))).
 
 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
  \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))))).
 
 (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 
  \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))))))).
 
 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 
  \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)).
 
 (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)))))))))
  \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).
 
 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: 
  \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)).
 
 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: 
  \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 98238b9cfb48e37c8dd8d7980c90f2ff81ef5d5d..762f857620b7b76de4419d69abaf4ae08330e9cf 100644 (file)
--- 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".
 
 
 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
  \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)).
 
 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)))))))))
  \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))))).
 
 (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)))))))))
  \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)))))).
 
 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))))))))
  \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)))))))))).
 
 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)))))))))
  \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 5fcbf8405d19266e088619817dd7a7963262cdfa..2d32fcad4449ac5136b3ea649e419676a7b8c186 100644 (file)
--- 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".
 
 
 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 
  \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))))).
 
 (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 
  \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))))).
 
 (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
  \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))))))))).
 
 (_: 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
  \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 5689c2721dc5783f707577fb924162b81c32774d..48d11a44e4fd10fd2b0526f9b2c64dc3989d4bf3 100644 (file)
--- 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".
 
 
 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))))))))
  \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)])))))))).
 
 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
  \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))))).
 
 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))))))))
  \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))))).
 
 (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: 
  \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))))))).
 
 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 
  \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))))))).
 
 (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))))))))))
  \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))))))))).
 
 (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: 
  \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 1cfc01a332452d4d2eca17b7c5520279c1b94320..f7bf1d15cdc3c14635c2374e49a90cdaaf6de2ad 100644 (file)
--- 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))))).
 
 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)))))))))
  \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))))).
 
 (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)))))))))
  \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)))))).
 
 (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
  \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)).
 
 (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
  \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 b816890f1cc2a5414f453e4b87529d49616301e8..65b3b604fdaa3b8a49fd553f87b97a55ae1d256d 100644 (file)
--- 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".
 
 
 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)))
  \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)))))).
 
 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)))
  \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)))).
 
 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))))
  \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 f941820bbe7bfe6421042507c7469a5c934ca3e9..267f9d9ad45f18554c0b0180e416666294e00bed 100644 (file)
--- 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".
 
 
 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
  \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)))).
 
 (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))))))
  \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 2ee88a07352526ffaa66f2ac0b22738bd2af649d..f53eb19dcb143f8db2bac44fe0afc6fcb308822c 100644 (file)
--- 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".
 
 
 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)))
  \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)))))).
 
 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
  \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 0daa861c71861348def37ac9def85091ecd5b80b..6df71b39e7fc4926af6a854a83a7cae08c96dd95 100644 (file)
--- 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".
 
 
 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))))))))
  \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))))))).
 
 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))))))))
  \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))))))).
 
 (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).
 
  \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)))))
  \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).
 
 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)))))
  \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)))))))).
 
 (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))).
 
  \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
  \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))))).
 
 (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 
  \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).
 
 (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 
  \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 c90321ff4849101a9cf283d2897400bab7ecea73..4380212393f814964fde300ea3f9e07e650789e2 100644 (file)
--- 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".
 
 
 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))))))))
  \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 7fe662dce3b4adb0e44ef55c8229383562da6deb..fb0df9eb353f9ca5e03e7f5137e12d87dd7c2c68 100644 (file)
--- 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".
 
 
 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)))))))
  \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))))))))))).
 
 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
  \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)))))))))).
 
 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
  \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 c0719c3f15f59e03c60f708ef947212f141cff14..7b2c899d70b58c383ea00ab56f025640a422395f 100644 (file)
--- 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".
 
 
 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
  \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 c2424f2322c0e1f73758f891ee9742b091808b0c..d6d6beb926c530ad90777f157278e259ea93b365 100644 (file)
--- 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".
 
 
 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 
  \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))))).
 
 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 
  \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))))))))))).
 
 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 
  \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 2bbe0e91f2b55ee91dd9e6d7380377d7952d9dc9..13e8ec9ac8d901edc61d345390e2e3def4672673 100644 (file)
--- 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".
 
 
 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: 
 (\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))].
 
 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
  \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))))).
 
 \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 
  \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))))).
 
 (\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 (_: 
  \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))))))).
 
 (_: 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))) 
  \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)))))).
 
 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)) 
  \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)))))).
 
 (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
  \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)))).
 
 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)))))))
  \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 c1581ca573bd58120f1c3cb4ae582cfe7e658506..d98b4020c16547fd3899c9bbf0db9c1f13b9d499 100644 (file)
--- 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".
 
 
 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))) 
  \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))))))).
 
 (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))))))))))))
  \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))))).
 
 _ _) \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))))))))))
  \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)))))))).
 
 (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 
  \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))).
 
 (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)))))))
  \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 e9cf818b747236cac12477489e9939fd745eaa46..f5a2ebb24a6d2e4d25e64866778f4ef10a9e841b 100644 (file)
--- 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)))))))).
 
 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
  \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))))))))).
 
 (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))) 
  \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)))))))).
 
 (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: 
  \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)))))))).
 
 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 
  \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))))))))))))))).
 
 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 (_: 
  \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 63234778874e609f1882ec5e47f734354c40cd47..33f41ebe8f327abcdac98604bc35ffc91c75ff12 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))).
 
 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
  \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))))))).
 
 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
  \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))).
 
 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
  \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)))).
 
 (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
  \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 c7eb51db71a1025d30de2115f30ec452dc70b050..0b897b4f9e84c0b59011421beb71396bfc3b6d35 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))).
 
 (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)))))))))
  \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))))).
 
 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: 
  \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))))))).
 
 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
  \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)))))))))).
 
 (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)))))))))
  \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 8816826ed8b5db59794a96258efdcad94aa1ba25..d9a82d40e8e8f5bb8c1600b44374616d6c3eb978 100644 (file)
--- 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".
 
 
 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))))))))))
  \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))))).
 
 (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
  \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))))).
 
 (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) 
  \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))))))).
 
 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
  \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)))))).
 
 (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 
  \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 390c22bae44ead639fedfe91be6abe93355ae414..e590a65e06294230bce7d0579be90885f0b543a3 100644 (file)
--- 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".
 
 
 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
  \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 6d0f4d55fe1ce46cb3698093998e8109d714105d..43a6d483c3df8d1d37d6155f4813385abd1980a0 100644 (file)
--- 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".
 
 
 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 
  \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))))).
 
 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 
  \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 31c65c5b2d2eb5ac37504e3019b6750cdb0be77a..4f6e1df9f6aa26e88c5ea4ea8d9723ae5ff193dd 100644 (file)
--- 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".
 
 
 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)].
 
 (\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
  \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))).
 
 | (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 (_: 
  \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 7afd013f996fa7774675011c36142dc34289ccee..2b43bb21ada932d75775c0059d32273b231b7b0e 100644 (file)
--- 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".
 
 
 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 
  \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))).
 
 (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 
  \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))).
 
 (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 
  \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))).
 
 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 
  \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 b894fc6789b6bd91dffa1c60215e3c4727f58153..0199af19e72621e6ebfccf7acfd4e27575d07353 100644 (file)
--- 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".
 
 
 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
  \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))).
 
 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))))))))
  \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 c1ade018f1b4c5fcb88f98be6acc9f2496e49b7d..98eb5558cd13f0d380c6911e1e086f103de283bc 100644 (file)
--- 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".
 
 
 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 
 (\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)].
 
 ((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
  \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)))).
 
 [(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 
  \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)))))).
 
 (\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
  \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)))))).
 
 \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
  \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 c8a7aa3116546643316b7be34f6b3ffb23a4da17..9c5fc73f5d55a4b975e3d7139ccd1ae448e47956 100644 (file)
--- 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".
 
 
 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 
  \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)).
 
 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)))))))))
  \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 9d7b3374a15b48a63afc152b235e7d8ec1d53ad5..aec1f02ef99e2c6892d08fc70697b6e6ce42ca75 100644 (file)
--- 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".
 
 
 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: 
  \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)).
 
 (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)))))))
  \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))).
 
 (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
  \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)))).
 
 (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)))))))
  \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 be4d21960d552ed59a1a6c72b5e2e50964632cae..2958902bda14ea3dbe6e078ade9ba8a4f6fca067 100644 (file)
--- 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".
 
 
 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)))))))))
  \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))))).
 
 (_: (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)))))))))
  \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))))).
 
 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)))))))))))
  \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)))))))))))).
 
 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
  \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 327b93fdbc6568894c31889a66e0c7065401e81b..7a6c3f27a12b90c8f972157e8cb1b019fa84fb64 100644 (file)
--- 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".
 
 
 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 
  \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).
 
 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 
  \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).
 
 (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 
  \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).
 
 \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 
  \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 d9c7d049cbd7d3825cb67c211b6d57827248ea7d..724a34747372f80cc491fbc63a85f22a9d59db06 100644 (file)
--- 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".
 
 
 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
  \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).
 
 (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
  \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)))))).
 
 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))).
 
  \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))).
 
  \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
  \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)))).
 
 (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
  \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)))).
 
 (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
  \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).
 
 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)).
 
  \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 
  \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).
 
 (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
  \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)))))).
 
 (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
  \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).
 
 (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))))).
 
  \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 
  \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))).
 
 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 
  \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).
 
 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
  \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)))).
 
 (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
  \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))))))).
 
 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
  \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).
 
 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
  \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))))).
 
 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)))).
 
  \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 
  \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))))).
 
 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
  \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))))).
 
 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)))).
 
  \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
  \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)))))).
 
 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
  \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).
 
 (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))).
 
  \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
  \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)))).
 
 (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))).
 
  \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 
  \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).
 
 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)))))).
 
  \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
  \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)))))).
 
 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
  \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))))))).
 
 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
  \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)))))))).
 
 \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
  \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).
 
 (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)))))).
 
  \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))))).
 
  \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
  \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))))))).
 
 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 
  \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).
 
 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
  \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).
 
 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)))))))))
  \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).
 
 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
  \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))))))).
 
 (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 
  \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 e6e1faf02c1f8af3e9af95cdf704e46c7e2c06b8..c2dff1889f818761797bfb517f05adf4f0d8960a 100644 (file)
--- 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".
 
 
 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))))))
  \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))))))).
 
 (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)))).
 
  \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
  \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 ce17a3c12f47c52679db85e9ac0182490df65953..f80d0b0e9de06fbbd5fb2ddb2c45eea45dad9511 100644 (file)
--- 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".
 
 
 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
  \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 d1139518f29705dea03e882d9b7035eba08c8892..7cb7c3108debd85d9f679114ceb472cf21cc982d 100644 (file)
--- 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".
 
 
 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
  \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)])))))).
 
 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)))).
 
  \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))))))
  \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)]))))))).
 
 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))))))
  \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))))).
 
  \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)))))))
  \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)])))))))).
 
 (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)))))))
  \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)))))).
 
 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)))))))
  \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)])))))))).
 
 \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)))))))))
  \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)])))))))))).
 
 (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 
  \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)])))))))))))).
 
 \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 
  \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)]))))))).
 
 (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 
  \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)])))))))).
 
 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)))))
  \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)])))))).
 
 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 
  \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)]))))))).
 
 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 
  \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)])))))))).
 
 (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: 
  \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)]))))))))).
 
 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 
  \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)]))))))).
 
 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: 
  \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)])))))))).
 
 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: 
  \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)]))))))))).
 
 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: 
  \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)])))))))))).
 
 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: 
  \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)]))))))))))).
 
 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 
  \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)])))))))))).
 
 [(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 
  \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)]))))))))))).
 
 [(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 
  \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)])))))))))))).
 
 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 
  \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)]))))))))))))).
 
 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: 
  \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)]))))))))))))))).
 
 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 
  \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 7eabcfdb5fa947f5baa5b69c7c17eef58cdc5f8d..79919dc763b16dc08dfd994f94168efed4f702c0 100644 (file)
--- 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".
 
 
 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))))))
  \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 518bb5ff064f169a3128a6d89ad55521c865d2c8..8f5ee7307488b980a1b3a4b4fd6ba3efbff83d9a 100644 (file)
--- 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".
 
 
 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 [])).
 
  \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).
 
  \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
  \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)]))))).
 
 \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))).
 
  \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
  \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)])))))).
 
 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
  \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)]))))).
 
 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)))))
  \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)])))))).
 
 \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
  \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])))))).
 
 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))).
 
  \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))].
 
 ((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 4022de81674c0b2bf540c92f43409d909a8ae814..b5069fdf73511272af8581577af1cd7420ac0b2b 100644 (file)
--- 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".
 
 
 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
  \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)))))).
 
 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) 
  \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))))))))).
 
 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: 
  \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)))))))))))).
 
 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)))).
 
  \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
  \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])))))).
 
 (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
  \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)))))).
 
 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
  \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)))))).
 
 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: 
  \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))))).
 
 (\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))).
 
  \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))).
 
  \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))))).
 
  \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))).
 
  \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) 
  \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))).
 
 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
  \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))))).
 
 (\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))).
 
  \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))).
 
  \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).
 
  \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 
  \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))).
 
 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)).
 
  \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).
 
  \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
  \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))).
 
 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)).
 
  \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))))))
  \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)))).
 
 (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.
 
  \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))).
 
  \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)).
 
  \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))).
 
  \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.
 
  \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
  \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))))).
 
 (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)).
 
  \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)).
 
  \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 
  \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))).
 
 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))).
 
  \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
  \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))))).
 
 (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
  \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))))).
 
 (\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))).
 
  \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))).
 
  \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 
  \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))).
 
 (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: 
  \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)).
 
 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).
 
  \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 
  \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)).
 
 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 
  \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)).
 
 (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: 
  \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))).
 
 (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
  \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))).
 
 (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)))).
 
  \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
  \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).
 
 (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).
 
  \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).
 
  \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
  \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))).
 
 (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
  \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))).
 
 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))))).
 
  \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).
 
  \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 
  \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).
 
 ((\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)).
 
  \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
  \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).
 
 (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))))).
 
  \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
  \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))).
 
 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
  \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)))))).
 
 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
  \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))).
 
 (_: (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)))).
 
  \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
  \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))).
 
 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
  \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))).
 
 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
  \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))))).
 
 (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
  \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))))))).
 
 (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
  \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)))))).
 
 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
  \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))))))).
 
 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: 
  \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))))))).
 
 (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).
 
  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
  \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