set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity".
-include "ceqc/props.ma".
+include "csubc/props.ma".
+include "csubc/getl.ma".
+include "csubc/arity.ma".
+include "lift1/props.ma".
+include "csubc/drop1.ma".
+
+theorem 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)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T
+(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0)))
+(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2
+n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n
+is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
+(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
+a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda
+(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let
+H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2:
+C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1
+(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr)
+(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x
+(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def
+H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2:
+C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
+(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2
+x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u))
+x0)).(let H11 \def (match H10 in csubc return (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (_: (csubc ? c0 c3)).((eq C c0 (CHead x (Bind Abbr) (lift1
+(ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g a0 c2 (lift1 is (TLRef
+i)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H11: (eq C (CSort n)
+(CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H12: (eq C (CSort
+n) x0)).((let H13 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H11) in
+(False_ind ((eq C (CSort n) x0) \to (sc3 g a0 c2 (lift1 is (TLRef i)))) H13))
+H12))) | (csubc_head c0 c3 H11 k v) \Rightarrow (\lambda (H12: (eq C (CHead
+c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H13: (eq C
+(CHead c3 k v) x0)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0)
+\Rightarrow t0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
+u)) H12) in ((let H15 \def (f_equal C K (\lambda (e: C).(match e in C return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
+k0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in
+((let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4]))
+(CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in (eq_ind
+C x (\lambda (c4: C).((eq K k (Bind Abbr)) \to ((eq T v (lift1 (ptrans is i)
+u)) \to ((eq C (CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2
+(lift1 is (TLRef i)))))))) (\lambda (H17: (eq K k (Bind Abbr))).(eq_ind K
+(Bind Abbr) (\lambda (k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C
+(CHead c3 k0 v) x0) \to ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef
+i))))))) (\lambda (H18: (eq T v (lift1 (ptrans is i) u))).(eq_ind T (lift1
+(ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3 (Bind Abbr) t0) x0) \to
+((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))) (\lambda (H19: (eq
+C (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)) x0)).(eq_ind C (CHead c3
+(Bind Abbr) (lift1 (ptrans is i) u)) (\lambda (_: C).((csubc g x c3) \to (sc3
+g a0 c2 (lift1 is (TLRef i))))) (\lambda (_: (csubc g x c3)).(let H21 \def
+(eq_ind_r C x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H9 (CHead c3 (Bind
+Abbr) (lift1 (ptrans is i) u)) H19) in (let H_y \def (sc3_abbr g a0 TNil) in
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y
+(trans is i) c3 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
+u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i)
+O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O)
+(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4)
+(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H21) (lift1 is (TLRef
+i)) (lift1_lref is i))))) x0 H19)) v (sym_eq T v (lift1 (ptrans is i) u)
+H18))) k (sym_eq K k (Bind Abbr) H17))) c0 (sym_eq C c0 x H16))) H15)) H14))
+H13 H11))) | (csubc_abst c0 c3 H11 v a1 H12 w H13) \Rightarrow (\lambda (H14:
+(eq C (CHead c0 (Bind Abst) v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
+u)))).(\lambda (H15: (eq C (CHead c3 (Bind Abbr) w) x0)).((let H16 \def
+(eq_ind C (CHead c0 (Bind Abst) v) (\lambda (e: C).(match e in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H14)
+in (False_ind ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c0 c3) \to
+((sc3 g (asucc g a1) c0 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is
+(TLRef i))))))) H16)) H15 H11 H12 H13)))]) in (H11 (refl_equal C (CHead x
+(Bind Abbr) (lift1 (ptrans is i) u))) (refl_equal C x0)))))) H8))))))
+H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1:
+C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1
+c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda
+(is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4:
+(csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1
+H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1
+(ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind
+Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda
+(x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is
+i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def
+(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is
+i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans
+is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans
+is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda
+(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst)
+(lift1 (ptrans is i) u)) x0)).(let H12 \def (match H11 in csubc return
+(\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C c0
+(CHead x (Bind Abst) (lift1 (ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g
+a0 c2 (lift1 is (TLRef i)))))))) with [(csubc_sort n) \Rightarrow (\lambda
+(H12: (eq C (CSort n) (CHead x (Bind Abst) (lift1 (ptrans is i)
+u)))).(\lambda (H13: (eq C (CSort n) x0)).((let H14 \def (eq_ind C (CSort n)
+(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x (Bind Abst)
+(lift1 (ptrans is i) u)) H12) in (False_ind ((eq C (CSort n) x0) \to (sc3 g
+a0 c2 (lift1 is (TLRef i)))) H14)) H13))) | (csubc_head c0 c3 H12 k v)
+\Rightarrow (\lambda (H13: (eq C (CHead c0 k v) (CHead x (Bind Abst) (lift1
+(ptrans is i) u)))).(\lambda (H14: (eq C (CHead c3 k v) x0)).((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k v)
+(CHead x (Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H16 \def (f_equal
+C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v) (CHead x
+(Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H17 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v) (CHead x
+(Bind Abst) (lift1 (ptrans is i) u)) H13) in (eq_ind C x (\lambda (c4:
+C).((eq K k (Bind Abst)) \to ((eq T v (lift1 (ptrans is i) u)) \to ((eq C
+(CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2 (lift1 is (TLRef
+i)))))))) (\lambda (H18: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda
+(k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C (CHead c3 k0 v) x0) \to
+((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i))))))) (\lambda (H19: (eq
+T v (lift1 (ptrans is i) u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0:
+T).((eq C (CHead c3 (Bind Abst) t0) x0) \to ((csubc g x c3) \to (sc3 g a0 c2
+(lift1 is (TLRef i)))))) (\lambda (H20: (eq C (CHead c3 (Bind Abst) (lift1
+(ptrans is i) u)) x0)).(eq_ind C (CHead c3 (Bind Abst) (lift1 (ptrans is i)
+u)) (\lambda (_: C).((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))
+(\lambda (_: (csubc g x c3)).(let H22 \def (eq_ind_r C x0 (\lambda (c4:
+C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abst) (lift1 (ptrans is i)
+u)) H20) in (let H_y \def (sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans
+is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i)
+(csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T (lift1 is
+(TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c is d1
+(TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i))
+(lift1_lref is i))) (nf2_lref_abst c2 c3 (lift1 (ptrans is i) u) (trans is i)
+H22) I) (lift1 is (TLRef i)) (lift1_lref is i))))) x0 H20)) v (sym_eq T v
+(lift1 (ptrans is i) u) H19))) k (sym_eq K k (Bind Abst) H18))) c0 (sym_eq C
+c0 x H17))) H16)) H15)) H14 H12))) | (csubc_abst c0 c3 H12 v a1 H13 w H14)
+\Rightarrow (\lambda (H15: (eq C (CHead c0 (Bind Abst) v) (CHead x (Bind
+Abst) (lift1 (ptrans is i) u)))).(\lambda (H16: (eq C (CHead c3 (Bind Abbr)
+w) x0)).((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow
+t0])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1 (ptrans is i) u))
+H15) in ((let H18 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _)
+\Rightarrow c4])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1
+(ptrans is i) u)) H15) in (eq_ind C x (\lambda (c4: C).((eq T v (lift1
+(ptrans is i) u)) \to ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c4
+c3) \to ((sc3 g (asucc g a1) c4 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2
+(lift1 is (TLRef i))))))))) (\lambda (H19: (eq T v (lift1 (ptrans is i)
+u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3
+(Bind Abbr) w) x0) \to ((csubc g x c3) \to ((sc3 g (asucc g a1) x t0) \to
+((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))) (\lambda (H20:
+(eq C (CHead c3 (Bind Abbr) w) x0)).(eq_ind C (CHead c3 (Bind Abbr) w)
+(\lambda (_: C).((csubc g x c3) \to ((sc3 g (asucc g a1) x (lift1 (ptrans is
+i) u)) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))
+(\lambda (_: (csubc g x c3)).(\lambda (H22: (sc3 g (asucc g a1) x (lift1
+(ptrans is i) u))).(\lambda (H23: (sc3 g a1 c3 w)).(let H24 \def (eq_ind_r C
+x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abbr) w)
+H20) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is
+i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) c3 w c2 (let H_y0
+\def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def
+(sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g a1) H22) in (sc3_repl g
+a1 c2 (lift (S (trans is i)) O w) (sc3_lift g a1 c3 w H23 c2 (S (trans is i))
+O (getl_drop Abbr c2 c3 w (trans is i) H24)) a0 (asucc_inj g a1 a0
+(arity_mono g x (lift1 (ptrans is i) u) (asucc g a1) H_y1 (asucc g a0)
+H_y0))))) H24) (lift1 is (TLRef i)) (lift1_lref is i))))))) x0 H20)) v
+(sym_eq T v (lift1 (ptrans is i) u) H19))) c0 (sym_eq C c0 x H18))) H17)) H16
+H12 H13 H14)))]) in (H12 (refl_equal C (CHead x (Bind Abst) (lift1 (ptrans is
+i) u))) (refl_equal C x0)))))) H9)))))) H6))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
+d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall
+(d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall
+(c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1:
+C).(\lambda (is: PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2:
+C).(\lambda (H6: (csubc g d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil)
+in (eq_ind_r T (THead (Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1:
+T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1
+(Bind b) (lift1 is u)) (Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2
+(Bind b) (lift1 is u)) (csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1
+is H5 c2 H6)) (lift1 is (THead (Bind b) u t0)) (lift1_bind b is u
+t0))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
+C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
+c2) \to (sc3 g (asucc g a1) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (H2: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3:
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind Abst) u))
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is
+t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1
+c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(eq_ind_r T (THead (Bind
+Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(land (arity g c2 t1
+(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall
+(is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1
+is0 t1)))))))))) (conj (arity g c2 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
+is) t0)) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to
+(\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w
+(lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))))))))))
+(csubc_arity_conf g d1 c2 H5 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is)
+t0)) (AHead a1 a2) (arity_head g d1 (lift1 is u) a1 (arity_lift1 g (asucc g
+a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 (arity_lift1 g a2 (CHead c (Bind
+Abst) u) (Ss is) (CHead d1 (Bind Abst) (lift1 is u)) t0 (drop1_skip_bind Abst
+c is d1 u H4) H2))) (\lambda (d: C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d
+w)).(\lambda (is0: PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead
+(Bind Abst) (lift1 is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0)))
+(\lambda (t1: T).(sc3 g a2 d (THead (Flat Appl) w t1))) (let H8 \def
+(sc3_appl g a1 a2 TNil) in (H8 d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let
+H_y \def (sc3_bind g Abbr (\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9))
+a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def
+(csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C
+(\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g
+a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x:
+C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T
+(lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind
+Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g
+a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1
+(papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x
+u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x
+d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1
+H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0
+is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is)
+t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1)
+is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst)
+(lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1
+(Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u
+t0)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c u a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1
+c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
+(AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
+PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
+d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2
+H5) in (let H6 \def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2))
+(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
+PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
+(lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0)))
+(\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda (H8:
+((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
+PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
+(lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in
+(eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1:
+T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u
+t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1:
+C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2:
+C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in
+(eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1:
+T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0)
+(H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is
+u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall
+(is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
+a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
+a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1
+c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1
+is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
theorem sc3_arity:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
a) \to (sc3 g a c t)))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
-A).(sc3 g a0 c0 t0)))) (\lambda (c0: C).(\lambda (n: nat).(conj (arity g c0
-(TSort n) (ASort O n)) (sn3 c0 (TSort n)) (arity_sort g c0 n) (sn3_nf2 c0
-(TSort n) (nf2_sort c0 n))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: (sc3 g a0
-d u)).(let H_y \def (sc3_abbr g a0 TNil) in (H_y i d u c0 (sc3_lift g a0 d u
-H2 c0 (S i) O (getl_drop Abbr c0 d u i H0)) H0)))))))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
-g a0))).(\lambda (_: (sc3 g (asucc g a0) d u)).(let H3 \def (sc3_abst g a0
-TNil) in (H3 c0 i (arity_abst g c0 d u i H0 a0 H1) (nf2_lref_abst c0 d u i
-H0) I)))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-a1)).(\lambda (H2: (sc3 g a1 c0 u)).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: (sc3 g
-a2 (CHead c0 (Bind b) u) t0)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in
-(H_y c0 u t0 H4 H2))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (H1: (sc3 g
-(asucc g a1) c0 u)).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g
-(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: (sc3 g a2 (CHead c0 (Bind
-Abst) u) t0)).(conj (arity g c0 (THead (Bind Abst) u t0) (AHead a1 a2))
-(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is:
-PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is (THead
-(Bind Abst) u t0))))))))) (arity_head g c0 u a1 H0 t0 a2 H2) (\lambda (d:
-C).(\lambda (w: T).(\lambda (H4: (sc3 g a1 d w)).(\lambda (is:
-PList).(\lambda (H5: (drop1 is d c0)).(let H6 \def (sc3_appl g a1 a2 TNil) in
-(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1:
-T).(sc3 g a2 d (THead (Flat Appl) w t1))) (H6 d w (lift1 (Ss is) t0) (let H_y
-\def (sc3_bind g Abbr (\lambda (H7: (eq B Abbr Abst)).(not_abbr_abst H7)) a1
-a2 TNil) in (H_y d w (lift1 (Ss is) t0) (let H7 \def (sc3_ceqc_trans g a2
-TNil) in (H7 (CHead d (Bind Abst) (lift1 is u)) (lift1 (Ss is) t0) (sc3_lift1
-g (CHead c0 (Bind Abst) u) a2 (Ss is) (CHead d (Bind Abst) (lift1 is u)) t0
-H3 (drop1_skip_bind Abst c0 is d u H5)) (CHead d (Bind Abbr) w) (or_intror
-(csubc g (CHead d (Bind Abbr) w) (CHead d (Bind Abst) (lift1 is u))) (csubc g
-(CHead d (Bind Abst) (lift1 is u)) (CHead d (Bind Abbr) w)) (csubc_abst g d d
-(csubc_refl g d) (lift1 is u) a1 (sc3_lift1 g c0 (asucc g a1) is d u H1 H5) w
-H4)))) H4)) H4 (lift1 is u) (sc3_lift1 g c0 (asucc g a1) is d u H1 H5))
-(lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u
-t0)))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: (sc3 g a1 c0 u)).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
-(H3: (sc3 g (AHead a1 a2) c0 t0)).(let H4 \def H3 in (and_ind (arity g c0 t0
-(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall
-(is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is
-t0)))))))) (sc3 g a2 c0 (THead (Flat Appl) u t0)) (\lambda (_: (arity g c0 t0
-(AHead a1 a2))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d
-w) \to (\forall (is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat
-Appl) w (lift1 is t0)))))))))).(let H_y \def (H6 c0 u H1 PNil) in (H_y
-(drop1_nil c0))))) H4))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (H1: (sc3 g (asucc
-g a0) c0 u)).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3:
-(sc3 g a0 c0 t0)).(let H_y \def (sc3_cast g a0 TNil) in (H_y c0 u H1 t0
-H3)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
-(arity g c0 t0 a1)).(\lambda (H1: (sc3 g a1 c0 t0)).(\lambda (a2: A).(\lambda
-(H2: (leq g a1 a2)).(sc3_repl g a1 c0 t0 H1 a2 H2)))))))) c t a H))))).
+(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y
+(drop1_nil c) c (csubc_refl g c))))))).
projects / helm.git / blobdiff