- transcript: patched to generate CoRN_notation.ma instead of CoRN.ma

[helm.git] / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / sc3 / props.ma

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
index 86a674344b9b4650a25d88c476c66be5126ec3ba..3965938ed02cb29b5d4e0317651535f083cc440b 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
@@ -244,54 +244,58 @@ Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
 T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | 
 (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: 
 T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) 
-(sn3 c (THeads (Flat Appl) vs t)))).((match n in nat return (\lambda (n1: 
-nat).((sc3 g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c 
-(THeads (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) 
-\to (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort 
-n1 n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) with [O 
-\Rightarrow (\lambda (H1: (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) 
-vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort O 
-n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g 
-c (THeads (Flat Appl) vs u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) 
-vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
-(ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) 
-(\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort O (next g 
-n0)))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in 
-(and_ind (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads 
-(Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) 
-u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) 
-(\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda 
-(H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) 
-vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs 
-(THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort O n0) H4 H7) 
-(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))) | (S n1) \Rightarrow 
-(\lambda (H1: (sc3 g (ASort n1 n0) c (THeads (Flat Appl) vs u))).(\lambda 
-(H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c 
-(THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads 
-(Flat Appl) vs u) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs u)) (land 
-(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) 
-(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity 
-g c (THeads (Flat Appl) vs u) (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads 
-(Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat 
-Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity 
-g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c 
-(THeads (Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads 
-(Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat 
-Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat 
-Cast) u t))) (arity_appls_cast g c u t vs (ASort (S n1) n0) H4 H7) 
-(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3))))]) H H0))))))))) (\lambda 
-(a0: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall (u: 
-T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: 
-T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat 
-Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: A).(\lambda (H0: 
-((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a1) c 
-(THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c (THeads (Flat 
-Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead (Flat Cast) u 
-t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda 
-(H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) 
-(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g 
+(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow 
+(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads 
+(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land 
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) 
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: 
+(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: 
+(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat 
+Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs 
+u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads 
+(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat 
+Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs 
+t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat 
+Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat 
+Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs 
+t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort 
+O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) 
+(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t 
+H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with 
+[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c 
+(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) 
+(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 
+n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads 
+(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let 
+H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) 
+(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) 
+(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def 
+H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c 
+(THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort 
+(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g 
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs 
+(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n 
+H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall 
+(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to 
+(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: 
+A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 
+g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c 
+(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc 
+g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
 PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land 
 (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall 
@@ -338,22 +342,6 @@ 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)).
 
-axiom sc3_bind:
- \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
-A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
-T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
-(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
-(THead (Bind b) v t)))))))))))))
-.
-
-axiom sc3_appl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
-TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
-(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
-T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
-(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
-.
-
 theorem 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 
@@ -432,56 +420,56 @@ H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
 I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) 
 (THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 
 (Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil 
-(drop1_nil c)))) in (let H14 \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) 
-x1) (TLRef O) (lift (S x2) O t) H_y1) in (and_ind (sn3 (CHead x0 (Bind Abst) 
-x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O t)) (sn3 c t) 
-(\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef O))).(\lambda (H16: (sn3 
-(CHead x0 (Bind Abst) x1) (lift (S x2) O t))).(sn3_gen_lift (CHead x0 (Bind 
-Abst) x1) t (S x2) O H16 c (drop_drop (Bind Abst) x2 x0 c H12 x1)))) 
-H14))))))))) H11))))) H8)))) H5)))) H2))))) (\lambda (vs: TList).(\lambda (i: 
-nat).(\lambda (c: C).(\lambda (H1: (arity g c (THeads (Flat Appl) vs (TLRef 
-i)) (AHead a0 a1))).(\lambda (H2: (nf2 c (TLRef i))).(\lambda (H3: (sns3 c 
-vs)).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) 
-(\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 (THeads 
-(Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: C).(\lambda (w: T).(\lambda 
-(H4: (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H5: (drop1 is d c)).(let 
-H6 \def H in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to 
-(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: 
-C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef 
-i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef 
-i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
-(TLRef i))))) (\lambda (H7: ((\forall (c0: C).(\forall (t: T).((sc3 g a0 c0 
-t) \to (sn3 c0 t)))))).(\lambda (_: ((\forall (vs0: TList).(\forall (i0: 
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a0) 
-\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat 
-Appl) vs0 (TLRef i0))))))))))).(let H9 \def H0 in (and_ind (\forall (c0: 
-C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
+(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) 
+x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (and_ind (sn3 
+(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S 
+x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef 
+O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O 
+t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop 
+(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) 
+(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g 
+c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c 
+(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) 
+vs (TLRef i)) (AHead a0 a1)) (\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 (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: 
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: 
+PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (and_ind (\forall (c0: 
+C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
 TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
-vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 
+vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 
 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) 
-w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (_: ((\forall (c0: 
-C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))))).(\lambda (H11: 
+w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: 
+C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: 
 ((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 
-(THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 
-c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let 
-H_y \def (H11 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) 
-(lifts1 is vs) (lift1 is (TLRef i))) (\lambda (t: T).(sc3 g a1 d (THead (Flat 
-Appl) w t))) (eq_ind_r T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d 
-(THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is 
-i) d (eq_ind T (lift1 is (TLRef i)) (\lambda (t: T).(arity g d (THead (Flat 
-Appl) w (THeads (Flat Appl) (lifts1 is vs) t)) a1)) (eq_ind T (lift1 is 
-(THeads (Flat Appl) vs (TLRef i))) (\lambda (t: T).(arity g d (THead (Flat 
-Appl) w t) a1)) (arity_appl g d w a0 (sc3_arity_gen g d w a0 H4) (lift1 is 
-(THeads (Flat Appl) vs (TLRef i))) a1 (arity_lift1 g (AHead a0 a1) c is d 
-(THeads (Flat Appl) vs (TLRef i)) H5 H1)) (THeads (Flat Appl) (lifts1 is vs) 
-(lift1 is (TLRef i))) (lifts1_flat Appl is (TLRef i) vs)) (TLRef (trans is 
-i)) (lift1_lref is i)) (eq_ind T (lift1 is (TLRef i)) (\lambda (t: T).(nf2 d 
-t)) (nf2_lift1 c is d (TLRef i) H5 H2) (TLRef (trans is i)) (lift1_lref is 
-i)) (conj (sn3 d w) (sns3 d (lifts1 is 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)).
+(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 
+c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 
+\def H0 in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to 
+(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef 
+i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef 
+i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) 
+\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) 
+\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat 
+Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) 
+in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) 
+(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef 
+(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat 
+Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) 
+(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 
+is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) 
+(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 
+(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 
+(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) 
+(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is 
+(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is 
+(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) 
+(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is 
+vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) 
+(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat 
+Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
 
 theorem sc3_sn3:
  \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c 
@@ -518,3 +506,19 @@ t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def
 ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i 
 c H H0 H1))) H2)))))))))).
 
+axiom sc3_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
+A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
+T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
+(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
+(THead (Bind b) v t)))))))))))))
+.
+
+axiom sc3_appl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
+TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
+T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
+(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
+.
+