--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aprem/defs".
+
+include "A/defs.ma".
+
+inductive aprem: nat \to (A \to (A \to Prop)) \def
+| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
+| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2
+a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aprem/props".
+
+include "aprem/defs.ma".
+
+include "leq/defs.ma".
+
+theorem 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)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
+(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2:
+A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H2 \def (match H1 in aprem
+return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem
+n a a0)).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1)
+b1)))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H2: (eq nat O
+i)).(\lambda (H3: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H4: (eq A a0
+b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to
+((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H5: (eq A (AHead a0 a3) (ASort
+h2 n2))).(let H6 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort h2 n2) H5) in (False_ind ((eq A a0 b2) \to
+(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1
+n1) b1)))) H6))) i H2 H3 H4)))) | (aprem_succ a0 a i0 H2 a3) \Rightarrow
+(\lambda (H3: (eq nat (S i0) i)).(\lambda (H4: (eq A (AHead a3 a0) (ASort h2
+n2))).(\lambda (H5: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A
+(AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1)
+b1))))))) (\lambda (H6: (eq A (AHead a3 a0) (ASort h2 n2))).(let H7 \def
+(eq_ind A (AHead a3 a0) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort h2 n2) H6) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a)
+\to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0)
+(ASort h1 n1) b1))))) H7))) i H3 H4 H5 H2))))]) in (H2 (refl_equal nat i)
+(refl_equal A (ASort h2 n2)) (refl_equal A b2)))))))))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall (i:
+nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1
+b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda (a5:
+A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall (b2:
+A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H4:
+(aprem i (AHead a3 a5) b2)).((match i in nat return (\lambda (n: nat).((aprem
+n (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem n (AHead a0 a4) b1))))) with [O \Rightarrow (\lambda (H5: (aprem O
+(AHead a3 a5) b2)).(let H6 \def (match H5 in aprem return (\lambda (n:
+nat).(\lambda (a: A).(\lambda (a6: A).(\lambda (_: (aprem n a a6)).((eq nat n
+O) \to ((eq A a (AHead a3 a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))))))) with
+[(aprem_zero a6 a7) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H7: (eq
+A (AHead a6 a7) (AHead a3 a5))).(\lambda (H8: (eq A a6 b2)).((let H9 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a6 a7)
+(AHead a3 a5) H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead a _)
+\Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H7) in (eq_ind A a3 (\lambda (a:
+A).((eq A a7 a5) \to ((eq A a b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
+(\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) (\lambda (H11: (eq A a7
+a5)).(eq_ind A a5 (\lambda (_: A).((eq A a3 b2) \to (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))) (\lambda
+(H12: (eq A a3 b2)).(eq_ind A b2 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq
+g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (eq_ind A a3
+(\lambda (a: A).(ex2 A (\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem
+O (AHead a0 a4) b1)))) (ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda
+(b1: A).(aprem O (AHead a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H12) a3
+(sym_eq A a3 b2 H12))) a7 (sym_eq A a7 a5 H11))) a6 (sym_eq A a6 a3 H10)))
+H9)) H8)))) | (aprem_succ a6 a i0 H6 a7) \Rightarrow (\lambda (H7: (eq nat (S
+i0) O)).(\lambda (H8: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H9: (eq A
+a b2)).((let H10 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H7) in (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A
+a b2) \to ((aprem i0 a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
+(\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) H10)) H8 H9 H6))))]) in (H6
+(refl_equal nat O) (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) | (S n)
+\Rightarrow (\lambda (H5: (aprem (S n) (AHead a3 a5) b2)).(let H6 \def (match
+H5 in aprem return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a6:
+A).(\lambda (_: (aprem n0 a a6)).((eq nat n0 (S n)) \to ((eq A a (AHead a3
+a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem (S n) (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7)
+\Rightarrow (\lambda (H6: (eq nat O (S n))).(\lambda (H7: (eq A (AHead a6 a7)
+(AHead a3 a5))).(\lambda (H8: (eq A a6 b2)).((let H9 \def (eq_ind nat O
+(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S n) H6) in (False_ind ((eq
+A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) H9)) H7
+H8)))) | (aprem_succ a6 a i0 H6 a7) \Rightarrow (\lambda (H7: (eq nat (S i0)
+(S n))).(\lambda (H8: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H9: (eq A
+a b2)).((let H10 \def (f_equal nat nat (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n0) \Rightarrow
+n0])) (S i0) (S n) H7) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a7
+a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem n0 a6 a) \to (ex2 A (\lambda
+(b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)))))))
+(\lambda (H11: (eq A (AHead a7 a6) (AHead a3 a5))).(let H12 \def (f_equal A A
+(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a6 | (AHead _ a8) \Rightarrow a8])) (AHead a7 a6) (AHead a3 a5)
+H11) in ((let H13 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a8 _)
+\Rightarrow a8])) (AHead a7 a6) (AHead a3 a5) H11) in (eq_ind A a3 (\lambda
+(_: A).((eq A a6 a5) \to ((eq A a b2) \to ((aprem n a6 a) \to (ex2 A (\lambda
+(b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)))))))
+(\lambda (H14: (eq A a6 a5)).(eq_ind A a5 (\lambda (a8: A).((eq A a b2) \to
+((aprem n a8 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem (S n) (AHead a0 a4) b1)))))) (\lambda (H15: (eq A a b2)).(eq_ind A
+b2 (\lambda (a8: A).((aprem n a5 a8) \to (ex2 A (\lambda (b1: A).(leq g b1
+b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) (\lambda (H16:
+(aprem n a5 b2)).(let H_x \def (H3 n b2 H16) in (let H17 \def H_x in (ex2_ind
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n a4 b1)) (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4)
+b1))) (\lambda (x: A).(\lambda (H18: (leq g x b2)).(\lambda (H19: (aprem n a4
+x)).(ex_intro2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S
+n) (AHead a0 a4) b1)) x H18 (aprem_succ a4 x n H19 a0))))) H17)))) a (sym_eq
+A a b2 H15))) a6 (sym_eq A a6 a5 H14))) a7 (sym_eq A a7 a3 H13))) H12))) i0
+(sym_eq nat i0 n H10))) H8 H9 H6))))]) in (H6 (refl_equal nat (S n))
+(refl_equal A (AHead a3 a5)) (refl_equal A b2))))]) H4)))))))))))) a1 a2
+H)))).
+
+theorem 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
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
+(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
+(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
+A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
+(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
+a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2
+H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/aprem".
+
+include "arity/props.ma".
+
+include "arity/cimp.ma".
+
+include "aprem/props.ma".
+
+theorem 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))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))))
+\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 (_: T).(\lambda (a0:
+A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H1 \def (match H0 in aprem
+return (\lambda (n0: nat).(\lambda (a0: A).(\lambda (a1: A).(\lambda (_:
+(aprem n0 a0 a1)).((eq nat n0 i) \to ((eq A a0 (ASort O n)) \to ((eq A a1 b)
+\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a1 a2)
+\Rightarrow (\lambda (H1: (eq nat O i)).(\lambda (H2: (eq A (AHead a1 a2)
+(ASort O n))).(\lambda (H3: (eq A a1 b)).(eq_ind nat O (\lambda (n0:
+nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a1 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b))))))))) (\lambda (H4: (eq A (AHead a1 a2) (ASort O n))).(let H5 \def
+(eq_ind A (AHead a1 a2) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O n) H4) in (False_ind ((eq A a1 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b))))))) H5))) i H1 H2 H3)))) | (aprem_succ a2 a0 i0 H1 a1) \Rightarrow
+(\lambda (H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a1 a2) (ASort O
+n))).(\lambda (H4: (eq A a0 b)).(eq_ind nat (S i0) (\lambda (n0: nat).((eq A
+(AHead a1 a2) (ASort O n)) \to ((eq A a0 b) \to ((aprem i0 a2 a0) \to (ex2_3
+C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O
+d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u
+(asucc g b)))))))))) (\lambda (H5: (eq A (AHead a1 a2) (ASort O n))).(let H6
+\def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O n) H5) in (False_ind ((eq A a0 b) \to ((aprem i0 a2 a0)
+\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b)))))))) H6))) i H2 H3 H4 H1))))]) in (H1
+(refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A b))))))))
+(\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: ((\forall (i0: nat).(\forall (b: A).((aprem
+i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b:
+A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def
+H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop
+(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0
+\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda
+(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop
+(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus
+i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2)
+H9))))) H7)))))))) H4)))))))))))))) (\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 (_: (arity g d u (asucc g a0))).(\lambda (H2:
+((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem
+i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6:
+(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2)
+x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1:
+C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1
+x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop
+(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x
+x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8
+(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7))))))))
+H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
+u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i:
+nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b)
+u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5:
+(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind
+C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O
+d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0
+(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9
+\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S
+b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g
+a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1)
+b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3:
+((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
+c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b:
+A).(\lambda (H4: (aprem i (AHead a1 a2) b)).((match i in nat return (\lambda
+(n: nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) with
+[O \Rightarrow (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H6 \def (match
+H5 in aprem return (\lambda (n: nat).(\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (aprem n a0 a3)).((eq nat n O) \to ((eq A a0 (AHead a1 a2))
+\to ((eq A a3 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u0:
+T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))))) with [(aprem_zero
+a0 a3) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H7: (eq A (AHead a0
+a3) (AHead a1 a2))).(\lambda (H8: (eq A a0 b)).((let H9 \def (f_equal A A
+(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead _ a4) \Rightarrow a4])) (AHead a0 a3) (AHead a1 a2)
+H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a4 _)
+\Rightarrow a4])) (AHead a0 a3) (AHead a1 a2) H7) in (eq_ind A a1 (\lambda
+(a4: A).((eq A a3 a2) \to ((eq A a4 b) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))))
+(\lambda (H11: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+O j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
+u0 (asucc g b)))))))) (\lambda (H12: (eq A a1 b)).(eq_ind A b (\lambda (_:
+A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus O j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b))))))) (eq_ind A a1 (\lambda (a4: A).(ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g a4))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a1))))) c0 u O (drop_refl
+c0) H0) b H12) a1 (sym_eq A a1 b H12))) a3 (sym_eq A a3 a2 H11))) a0 (sym_eq
+A a0 a1 H10))) H9)) H8)))) | (aprem_succ a0 a3 i0 H6 a4) \Rightarrow (\lambda
+(H7: (eq nat (S i0) O)).(\lambda (H8: (eq A (AHead a4 a0) (AHead a1
+a2))).(\lambda (H9: (eq A a3 b)).((let H10 \def (eq_ind nat (S i0) (\lambda
+(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq A (AHead a4 a0)
+(AHead a1 a2)) \to ((eq A a3 b) \to ((aprem i0 a0 a3) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b))))))))) H10)) H8 H9 H6))))]) in (H6 (refl_equal nat O) (refl_equal A
+(AHead a1 a2)) (refl_equal A b)))) | (S n) \Rightarrow (\lambda (H5: (aprem
+(S n) (AHead a1 a2) b)).(let H6 \def (match H5 in aprem return (\lambda (n0:
+nat).(\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (aprem n0 a0 a3)).((eq
+nat n0 (S n)) \to ((eq A a0 (AHead a1 a2)) \to ((eq A a3 b) \to (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O
+d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H6:
+(eq nat O (S n))).(\lambda (H7: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda
+(H8: (eq A a0 b)).((let H9 \def (eq_ind nat O (\lambda (e: nat).(match e in
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
+\Rightarrow False])) I (S n) H6) in (False_ind ((eq A (AHead a0 a3) (AHead a1
+a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) H9)) H7 H8)))) |
+(aprem_succ a0 a3 i0 H6 a4) \Rightarrow (\lambda (H7: (eq nat (S i0) (S
+n))).(\lambda (H8: (eq A (AHead a4 a0) (AHead a1 a2))).(\lambda (H9: (eq A a3
+b)).((let H10 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
+(\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n0) \Rightarrow n0])) (S
+i0) (S n) H7) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a4 a0) (AHead
+a1 a2)) \to ((eq A a3 b) \to ((aprem n0 a0 a3) \to (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b)))))))))) (\lambda (H11: (eq A (AHead a4 a0) (AHead a1 a2))).(let H12 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a0 | (AHead _ a5) \Rightarrow a5])) (AHead a4 a0)
+(AHead a1 a2) H11) in ((let H13 \def (f_equal A A (\lambda (e: A).(match e in
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a5 _)
+\Rightarrow a5])) (AHead a4 a0) (AHead a1 a2) H11) in (eq_ind A a1 (\lambda
+(_: A).((eq A a0 a2) \to ((eq A a3 b) \to ((aprem n a0 a3) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b)))))))))) (\lambda (H14: (eq A a0 a2)).(eq_ind A a2 (\lambda (a5:
+A).((eq A a3 b) \to ((aprem n a5 a3) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))))
+(\lambda (H15: (eq A a3 b)).(eq_ind A b (\lambda (a5: A).((aprem n a2 a5) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+(S n) j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity
+g d u0 (asucc g b)))))))) (\lambda (H16: (aprem n a2 b)).(let H_x \def (H3 n
+b H16) in (let H17 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus n j) O d (CHead c0 (Bind Abst) u)))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H18: (drop (plus n x2) O x0 (CHead c0 (Bind
+Abst) u))).(\lambda (H19: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus n x2) H18) H19))))))
+H17)))) a3 (sym_eq A a3 b H15))) a0 (sym_eq A a0 a2 H14))) a4 (sym_eq A a4 a1
+H13))) H12))) i0 (sym_eq nat i0 n H10))) H8 H9 H6))))]) in (H6 (refl_equal
+nat (S n)) (refl_equal A (AHead a1 a2)) (refl_equal A b))))]) H4)))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
+u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g c0 t0 (AHead a1 a2))).(\lambda (H3: ((\forall (i: nat).(\forall (b:
+A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i:
+nat).(\lambda (b: A).(\lambda (H4: (aprem i a2 b)).(let H5 \def (H3 (S i) b
+(aprem_succ a2 b i H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6:
+(drop (S (plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g
+b))).(C_ind (\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1
+x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0:
+T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n:
+nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity
+g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i
+x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0:
+T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0
+(CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O
+O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee in
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H11) in (False_ind (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) H13)))))
+(drop_gen_sort n (S (plus i x2)) O c0 H8))))) (\lambda (d: C).(\lambda (IHd:
+(((drop (S (plus i x2)) O d c0) \to ((arity g d x1 (asucc g b)) \to (ex2_3 C
+T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O
+d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b)))))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (H8: (drop (S
+(plus i x2)) O (CHead d k t1) c0)).(\lambda (H9: (arity g (CHead d k t1) x1
+(asucc g b))).((match k in K return (\lambda (k0: K).((arity g (CHead d k0
+t1) x1 (asucc g b)) \to ((drop (r k0 (plus i x2)) O d c0) \to (ex2_3 C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))))))) with [(Bind b0) \Rightarrow (\lambda (H10: (arity g
+(CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Bind b0)
+(plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (CHead d (Bind b0)
+t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O (CHead
+d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d c0 H11 t1) (plus i (S
+x2)) (plus_n_Sm i x2)) H10))) | (Flat f) \Rightarrow (\lambda (H10: (arity g
+(CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Flat f) (plus
+i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g (CHead d (Flat f)
+t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in (ex2_3_ind C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x3: C).(\lambda (x4:
+T).(\lambda (x5: nat).(\lambda (H13: (drop (plus i x5) O x3 c0)).(\lambda
+(H14: (arity g x3 x4 (asucc g b))).(ex2_3_intro C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x3
+x4 x5 H13 H14)))))) H12))))]) H9 (drop_gen_drop k d c0 t1 (plus i x2)
+H8)))))))) x0 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_:
+((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
+a0)).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
+u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4:
+(aprem i a0 b)).(let H_x \def (H3 i b H4) in (let H5 \def H_x in (ex2_3_ind C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H6: (drop (plus i x2) O x0 c0)).(\lambda (H7:
+(arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7))))))
+H5)))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b:
+A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (a2:
+A).(\lambda (H2: (leq g a1 a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda
+(H3: (aprem i a2 b)).(let H_x \def (aprem_repl g a1 a2 H2 i b H3) in (let H4
+\def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem
+i a1 b1)) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x
+b)).(\lambda (H6: (aprem i a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def
+H_x0 in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda
+(H9: (arity g x0 x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8
+(arity_repl g x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5))))))))
+H7)))))) H4))))))))))))) c t a H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/cimp".
+
+include "arity/defs.ma".
+
+include "cimp/props.ma".
+
+theorem 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
+ \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 (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g
+c2 n))))) (\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 (c2: C).((cimp d
+c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c
+c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let
+H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in
+(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7:
+(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def
+(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead
+x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind
+Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _)
+\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono
+c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11
+\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9
+x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x
+H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6)))))
+H4))))))))))))) (\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 (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2:
+C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda
+(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in
+(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2
+(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst)
+u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def
+H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst)
+u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i
+c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i
+H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x |
+(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind
+Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u)
+H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0
+(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0:
+C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12)))))))))
+H6))))) H4))))))))))))) (\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 (c2: C).((cimp c c2) \to (arity g c2 u
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b)
+u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c
+c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
+(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
+((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to
+(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
+(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2:
+C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2
+H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to
+(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0
+a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1
+c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2:
+(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2
+t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/defs".
+
+include "leq/defs.ma".
+
+include "getl/defs.ma".
+
+inductive arity (g:G): C \to (T \to (A \to Prop)) \def
+| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O
+n)))
+| arity_abbr: \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 (arity g c (TLRef i) a)))))))
+| arity_abst: \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 (arity g c (TLRef i) a)))))))
+| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t:
+T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c
+(THead (Bind b) u t) a2)))))))))
+| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind
+Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
+| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to
+(arity g c (THead (Flat Appl) u t) a2)))))))
+| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u
+(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat
+Cast) u t) a))))))
+| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t
+a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/fwd".
+
+include "arity/defs.ma".
+
+include "leq/asucc.ma".
+
+include "leq/fwd.ma".
+
+include "getl/drop.ma".
+
+theorem 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
+ \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
+c t a)) (leq g a (ASort O n)) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: A).((eq T t
+(TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: C).(\lambda (n0:
+nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n1)
+\Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0]))
+(TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort
+O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u
+a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda
+(H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5)))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity
+g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g
+a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0
+(ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
+Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
+g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a1 (ASort O
+n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind
+b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O
+n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 \def
+(eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
+(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
+(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t
+a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda
+(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead
+(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2)
+(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
+g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1
+a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort
+n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g
+a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O
+n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t
+(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat
+Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6)))))))))))
+(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t
+a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O
+n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t
+(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in
+(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1
+(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0:
+T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1
+a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0)))
+H))))).
+
+theorem arity_gen_lref:
+ \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))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
+c t a)) (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)))) (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))) (\lambda (y:
+T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: C).(\lambda (n:
+nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def (eq_ind T (TSort
+n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1:
+(getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g
+d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))))))).(\lambda (H4: (eq T
+(TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e
+in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i0 | (TLRef n)
+\Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in
+(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abbr)
+u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
+T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
+T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
+c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
+u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl
+i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g
+d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g a0))).(\lambda (_:
+(((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
+T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
+T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
+T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
+T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda (H4: (eq T (TLRef
+i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in T
+return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i0 | (TLRef n)
+\Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in
+(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst)
+u))) H1 i H5) in (or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
+T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
+T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
+c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
+u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl
+i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g
+d0 u0 (asucc g a0)))) d u H6 H2))))))))))))) (\lambda (b: B).(\lambda (_:
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr)
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))))))).(\lambda
+(t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t
+a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i (CHead c0 (Bind b) u) (CHead d (Bind Abbr) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i (CHead c0 (Bind b) u) (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda
+(H6: (eq T (THead (Bind b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead
+(Bind b) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda
+(d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
+(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T
+t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+(CHead c0 (Bind Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda
+(u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T
+(THead (Bind Abst) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst)
+u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef
+i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d
+(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1
+a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead
+a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let
+H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
+(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
+d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
+C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g
+a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_:
+(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef
+i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
+H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
+H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
+(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
+\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
+T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
+T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6
+(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
+(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
+(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
+x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
+(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(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:
+ \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 (_:
+A).(arity g (CHead c (Bind b) u) t a2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
+(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
+g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
+T).(arity g c t0 a2)) (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
+A).(arity g (CHead c (Bind b) u) t a2))) (\lambda (y: T).(\lambda (H1: (arity
+g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
+A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) (THead (Bind b) u
+t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t)
+H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0
+a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
+a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
+(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_:
+(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
+a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
+a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
+(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3:
+(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g
+(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u
+t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3))
+(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
+a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u
+t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
+k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
+(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0)
+(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12:
+(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead
+(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u
+a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
+a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g
+(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead
+(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def
+(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u
+H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b)
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0
+(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0
+(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead
+(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def
+(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b
+H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2
+b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9))
+H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u
+t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0:
+A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5:
+(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead
+c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind
+Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0
+t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _)
+\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0
+u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g
+(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0
+(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0
+(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let
+H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda
+(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u
+H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind
+Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1:
+T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
+a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u
+H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g
+a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t
+(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind
+Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u)
+(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda
+(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1))))))
+H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t
+(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False
+return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with
+[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq
+T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0:
+T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_:
+(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
+a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1
+a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u
+t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)))
+H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_:
+(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t))
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_:
+(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b)
+u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b)
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1
+a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T
+(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0
+(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
+a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda
+(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7
+(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11:
+(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
+(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y
+a2 H1))) H0)))))))).
+
+theorem arity_gen_abst:
+ \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 (_:
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) u) t a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
+(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) u) t a2)))) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: A).((eq T t0
+(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq
+A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Bind Abst) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
+a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
+(_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
+(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
+A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
+Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_:
+(((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda
+(a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
+(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
+A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
+Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2:
+(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0
+a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda
+(_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
+t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u
+t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k
+_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 t0) (THead
+(Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0)
+(THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b)
+u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda
+(H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1
+(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
+A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
+(Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t H9) in (let H13
+\def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) u0) t1 a2)) H4
+t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) t1) u
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
+(Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let H15 \def (eq_ind T
+u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) H13 u H10) in (let
+H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H3 u H10) in
+(let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H2 u H10) in
+(let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t (THead (Bind Abst) u t))
+\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b0) u) u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b0)
+u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let H19 \def (eq_ind B b
+(\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) H15 Abst H11) in (let
+H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H11) in
+(let H21 \def (match (H20 (refl_equal B Abst)) in False return (\lambda (_:
+False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))) with []) in
+H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g a1))).(\lambda (H2:
+(((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda
+(a3: A).(eq A (asucc g a1) (AHead a2 a3)))) (\lambda (a2: A).(\lambda (_:
+A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity g
+(CHead c0 (Bind Abst) u) t a3))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (H4:
+(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda
+(a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g
+(CHead c0 (Bind Abst) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4:
+A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind Abst) u) t
+a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) (THead (Bind Abst) u
+t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t1 _) \Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t)
+H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind
+Abst) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0
+(\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
+(_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind Abst) u)
+t a4)))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g
+(CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in (let H11 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
+(_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind Abst) u)
+t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: T).(arity g
+(CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let H13 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H2 u H8) in (let H14
+\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H1 u H8) in
+(ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) a1
+a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: A).(\lambda
+(_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity
+g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (_: (((eq T t0 (THead
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A
+(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
+Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead
+(Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) H6))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0
+u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2
+A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda
+(t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead
+a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u
+t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda
+(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
+(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity
+g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2:
+(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda
+(a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g
+c0 u (asucc g a2)))) (\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0
+(Bind Abst) u) t a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
+a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T
+T (\lambda (e: T).e) t0 (THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T
+t0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u
+t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1
+(THead (Bind Abst) u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind
+Abst) u t))) in (ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10:
+(arity g c0 u (asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u)
+t x1)).(let H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead
+x0 x1) H9) in (let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead
+(Bind Abst) u t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head g
+x0 x1 a2 H12) in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(leq g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (ex3_2 A
+A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda
+(x3: A).(\lambda (H15: (eq A a2 (AHead x2 x3))).(\lambda (H16: (leq g x0
+x2)).(\lambda (H17: (leq g x1 x3)).(eq_ind_r A (AHead x2 x3) (\lambda (a0:
+A).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro
+A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3
+(refl_equal A (AHead x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2)
+(asucc_repl g x0 x2 H16)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3
+H17)) a2 H15)))))) H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem 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)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
+(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (ex2 A (\lambda (a1:
+A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))) (\lambda
+(y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1
+a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0
+u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O n))))) H2)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity
+g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (AHead a1
+a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
+g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
+(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
+d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
+a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3:
+A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T
+(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T
+(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda
+(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc
+g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda
+(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda
+(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
+(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1
+a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
+A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0:
+A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0
+(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5:
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
+\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in
+((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
+t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5)
+in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq
+T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let
+H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t
+H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
+A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0
+(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12
+H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a:
+A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead
+(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda
+(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
+(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat
+Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
+g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T
+t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda
+(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5
+\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let
+H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
+(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in
+(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
+a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0
+t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda
+(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0
+u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t
+(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3))))))
+H8))))))))))))) c y a2 H0))) H)))))).
+
+theorem arity_gen_cast:
+ \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)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
+(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (land (arity g c u (asucc
+g a)) (arity g c t a)) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: A).((eq T t0
+(THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t
+a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u (asucc g (ASort
+O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 a0)).(\lambda (_:
+(((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u (asucc g a0))
+(arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
+t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0:
+A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
+(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t
+(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
+t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T
+t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u
+(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T
+(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T
+(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t
+a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead
+(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0
+t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g
+(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
+u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g
+(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0
+t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
+H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t
+(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
+a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
+c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
+(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land
+(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g
+a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0
+u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0:
+T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t
+a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u
+t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t)
+H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat
+Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0
+(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0
+(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12
+\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in
+(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10)))))))
+H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t))
+\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2:
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u
+t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t)
+H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2
+(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
+T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6
+(refl_equal T (THead (Flat Cast) u t))) in (and_ind (arity g c0 u (asucc g
+a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2))
+(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t
+a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u
+(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1
+H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem 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))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
+(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
+(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
+t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
+(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
+A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
+c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
+c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
+a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
+(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
+a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
+(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
+(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
+A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
+
+theorem arity_gen_lift:
+ \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)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
+(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\forall (c2: C).((drop h d
+c1 c2) \to (arity g c2 t a))) (\lambda (y: T).(\lambda (H0: (arity g c1 y
+a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) \to (\forall (c2:
+C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat d (\lambda (n:
+nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: C).((drop h n
+c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: C).(\lambda (t0:
+T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x
+x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 a0)))))))))
+(\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: T).(\lambda
+(H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda (_: (drop h x
+c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 (ASort O n)))
+(arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) (\lambda (c:
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
+(CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u
+a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x
+x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
+(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
+a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in
+(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
+(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
+(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
+(eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x h) i) (eq T x0 (TLRef
+(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
+(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
+(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
+H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
+(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
+(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
+(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
+(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus
+x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
+(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
+Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
+(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x
+h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
+(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
+(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
+u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
+H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
+(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
+(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
+(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
+nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
+x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
+nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
+(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
+(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
+(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
+(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to
+(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x)
+x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c
+(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift
+h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind
+b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15
+\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
+a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1:
+T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1
+(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal
+T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b
+x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
+(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u
+(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g
+(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c
+(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda
+(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda
+(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0:
+T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S
+x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2
+t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3:
+nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h
+x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x)
+x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c
+(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u
+(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11
+(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3:
+nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall
+(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2)))))))
+H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall
+(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3:
+C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1)
+H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g
+a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T
+(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2))
+(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0
+H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda
+(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4:
+((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall
+(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda
+(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift
+h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
+(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x
+x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1
+a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def
+(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2)
+H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
+(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2
+x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2
+(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0
+x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
+nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
+c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
+(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
+t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
+(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
+c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
+x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
+x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
+a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
+(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
+(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
+(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in
+(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10
+x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast
+u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
+a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1
+(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/lift1".
+
+include "arity/props.ma".
+
+include "drop1/defs.ma".
+
+theorem 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))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
+T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
+(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
+(H0: (arity g c2 t a)).(let H1 \def (match H in drop1 return (\lambda (p:
+PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
+PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a))))))))
+with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda
+(H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0:
+C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C
+c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C
+c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda
+(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda
+(H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
+(False_ind ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1
+hds0 c3 c4) \to (arity g c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1
+(refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1:
+C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1
+(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1
+(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H2 \def (match H0
+in drop1 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0:
+C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c
+c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
+p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def
+(eq_ind PList PNil (\lambda (e: PList).(match e in PList return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
+I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g
+c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4
+hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
+p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def
+(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
+p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n1 d c0 c3) \to ((drop1
+hds0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) (\lambda (H10:
+(eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq
+C c0 c1) \to ((eq C c4 c2) \to ((drop n n1 c0 c3) \to ((drop1 hds0 c3 c4) \to
+(arity g c1 (lift n n0 (lift1 p t)) a))))))) (\lambda (H11: (eq PList hds0
+p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c4 c2) \to
+((drop n n0 c0 c3) \to ((drop1 p0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p
+t)) a)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C
+c4 c2) \to ((drop n n0 c c3) \to ((drop1 p c3 c4) \to (arity g c1 (lift n n0
+(lift1 p t)) a))))) (\lambda (H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c:
+C).((drop n n0 c1 c3) \to ((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p
+t)) a)))) (\lambda (H14: (drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3
+c2)).(arity_lift g c3 (lift1 p t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq
+C c4 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds0 (sym_eq PList hds0 p H11))) d
+(sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))])
+in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C
+c2))))))))))) hds)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/props".
+
+include "arity/fwd.ma".
+
+theorem 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
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0:
+nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n))))))
+(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n))))
+(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda
+(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0
+n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c:
+C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1)
+(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0
+x1) (ASort (S n0) n) H1))))) H0)))) k))).
+
+theorem arity_lift:
+ \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)))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to
+(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop
+h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O
+n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d))))))))
+(\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 (H1:
+(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall
+(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda
+(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1
+c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i
+d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d
+(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity
+g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O
+c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1
+(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n:
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C
+(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i))
+u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h
+(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead
+x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S
+i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0:
+T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i
+c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i))
+(lift_lref_ge i h d0 H4)))))))))))))))) (\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 (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g
+a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0)
+(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1
+t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0
+H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0:
+C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop
+h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d
+(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind
+Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S
+i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0
+(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h
+(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i
+(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11)
+a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i))
+(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus
+i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h)
+(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1)
+(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\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
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u)
+(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind
+g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4
+(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5
+b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
+d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d
+u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1
+a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst)
+d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d)
+(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0))
+(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead
+a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s
+(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1
+(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s
+(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat
+Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift
+h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1
+c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
+t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0
+(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4))
+(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h
+d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0)
+a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1
+c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a
+H))))).
+
+theorem arity_mono:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
+t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
+(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
+A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
+a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
+a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
+a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
+\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
+(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
+Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind
+Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _)
+\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in
+((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
+d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr)
+u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let
+H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr)
+t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0
+t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
+(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
+(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9)))))))
+H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
+(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g
+a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i
+c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i
+H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind
+Abbr) u) (\lambda (ee: C).(match ee 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 True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u
+(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g
+(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i)
+a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2)
+(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead
+d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
+Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
+(ee: C).(match ee 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 x0 (Bind Abbr)
+x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6))
+in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind
+Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind
+C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
+c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead
+d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0
+(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def
+(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u
+H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g
+a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
+(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
+(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2
+(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall
+(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda
+(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4:
+((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3
+a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0)
+a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A
+(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u
+x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8))))
+H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda
+(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0
+u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3:
+((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3
+a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0)
+a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda
+(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda
+(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity
+g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7:
+(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u)
+t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a))
+(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8))
+a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
+A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g
+c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_:
+(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0
+t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity
+g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0
+H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity
+g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g
+c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x
+a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_:
+((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda
+(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
+A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
+(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
+t0 a2 H4) in (and_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g a
+a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
+a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
+A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
+(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
+g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
+
+theorem arity_appls_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs:
+TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to
+((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t)) a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a)
+\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a)))))
+(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity
+g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u)
+(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c
+(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a:
+A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u))
+(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1
+t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda
+(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat
+Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1))
+(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g
+a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat
+Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda
+(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g
+a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))
+a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g
+a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7
+(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc
+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:
+ \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
+(THeads (Flat Appl) vs (TLRef i)) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs:
+TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t
+(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v)
+a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop
+Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0:
+((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to
+(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda
+(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
+v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S
+i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity
+g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x:
+A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat
+Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
+
+theorem arity_appls_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v)
+(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat
+Appl) vs (THead (Bind b) v t)) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
+(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v
+a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind
+b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v)
+t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads
+(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead
+c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts
+(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v)
+(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in
+(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0)
+a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda
+(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity
+g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x
+a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O)
+O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/subst0".
+
+include "arity/props.ma".
+
+include "fsubst0/fwd.ma".
+
+include "csubst0/getl.ma".
+
+include "csubst0/props.ma".
+
+include "subst0/dec.ma".
+
+include "subst0/fwd.ma".
+
+include "getl/getl.ma".
+
+theorem 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 (P: Prop).P))))))))))))
+\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 (_:
+A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
+(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
+(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w
+(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P)))))))))))
+(\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 (_: ((\forall (d0: C).(\forall (u0: T).(\forall
+(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
+(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
+(TLRef i) v)).(\lambda (P: Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O
+w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
+w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
+(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee 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
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d
+(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9))))))
+(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\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 (_: (arity g d u (asucc g a0))).(\lambda
+(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead
+d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v)
+\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
+(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
+Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
+nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
+i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
+H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0
+(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
+d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (ee: C).(match ee 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 d0 (Bind Void)
+u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7))
+in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4))))))))))))))))))
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2:
+((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to
+(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d:
+C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
+(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0)
+v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind
+b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq
+T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0
+t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T
+(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i
+w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v
+(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9
+P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u
+t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda
+(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b)
+i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u
+x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i)
+(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d
+(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u
+x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9
+P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6)))))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl
+i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w:
+T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind
+Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2)))
+(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s
+(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind
+Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8
+P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda
+(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda
+(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1
+d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i
+H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0
+t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall
+(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind
+Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead
+(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T
+(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w
+u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0
+t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2
+t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T
+v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda
+(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0
+i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2:
+T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl)
+i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s
+(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5)))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u
+(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat
+Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5))))))))))))))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2:
+A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d
+u i H3 w v H4 P)))))))))))))))) c t a H))))).
+
+theorem arity_gen_cvoid:
+ \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))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i
+v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i
+v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T
+t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift
+(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3:
+(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u
+(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))
+(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda
+(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i
+x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
+(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x
+(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
+
+theorem arity_fsubst0:
+ \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
+c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
+d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
+t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i
+c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1:
+(fsubst0 i u c (TSort n) c2 t2)).(let H2 \def (fsubst0_gen_base c c2 (TSort
+n) t2 u i H1) in (or3_ind (land (eq C c c2) (subst0 i u (TSort n) t2)) (land
+(eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i u (TSort n) t2)
+(csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: (land (eq C c
+c2) (subst0 i u (TSort n) t2))).(and_ind (eq C c c2) (subst0 i u (TSort n)
+t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c c2)).(\lambda (H5:
+(subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (ASort
+O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 (ASort O n))) c2 H4))) H3))
+(\lambda (H3: (land (eq T (TSort n) t2) (csubst0 i u c c2))).(and_ind (eq T
+(TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n)) (\lambda (H4:
+(eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c c2)).(eq_ind T (TSort n)
+(\lambda (t: T).(arity g c2 t (ASort O n))) (arity_sort g c2 n) t2 H4))) H3))
+(\lambda (H3: (land (subst0 i u (TSort n) t2) (csubst0 i u c c2))).(and_ind
+(subst0 i u (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n))
+(\lambda (H4: (subst0 i u (TSort n) t2)).(\lambda (_: (csubst0 i u c
+c2)).(subst0_gen_sort u t2 i n H4 (arity g c2 t2 (ASort O n))))) H3))
+H2))))))))))) (\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 (H1: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
+(u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) \to
+(\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g c2
+t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda
+(H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2:
+T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5 \def
+(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c c2)
+(subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))
+(land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0)
+(\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind (eq C
+c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c
+c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0:
+C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0))
+(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift
+(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0))
+(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind
+Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c
+(CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind
+Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0
+(CHead d1 (Bind Abbr) u0) H11)) in ((let H14 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr)
+u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11))
+in (\lambda (H15: (eq C d d1)).(let H16 \def (eq_ind_r T u0 (\lambda (t:
+T).(getl i c (CHead d1 (Bind Abbr) t))) H12 u H14) in (eq_ind T u (\lambda
+(t: T).(arity g c (lift (S i) O t) a0)) (let H17 \def (eq_ind_r C d1 (\lambda
+(c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H16 d H15) in (arity_lift g d u
+a0 H1 c (S i) O (getl_drop Abbr c d u i H17))) u0 H14)))) H13)))) t2 H10)))
+(subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda (H6: (land (eq T
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (eq T (TLRef i) t2) (csubst0 i0
+u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8:
+(csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0))
+(lt_le_e i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10
+\def (csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abbr) u) H0) in
+(or4_ind (getl i c2 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
+(arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d (Bind Abbr)
+u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (arity_abbr g c2 d u i H11 a0
+H1))) (\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef
+i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H12: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind x0) x3))).(\lambda (H14:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H18
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H12) in (\lambda (H19: (eq B Abbr
+x0)).(\lambda (H20: (eq C d x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) in (let H22 \def (eq_ind_r C
+x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind x0) x3))) H13 d H20) in (let
+H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead d (Bind b) x3)))
+H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 (H2 d1 u0 (r (Bind Abbr)
+(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) (minus i0 (S i))) u0 d
+u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda (H11: (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H15 \def
+(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u)
+(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3
+(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0
+(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in
+(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def
+(eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) H13 u H18)
+in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0
+c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2
+(CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u i H23 a0 (H2
+d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1
+(Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind Abbr)
+(minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) (\lambda
+(H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
+H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr)
+(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
+(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
+d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0 (TLRef i) t2)
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i i0) (eq T t2
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind
+Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in
+((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17
+\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13
+u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2))
+H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0))
+(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr)
+u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u
+i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0
+H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6))
+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 (H2: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0))
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g
+c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0:
+nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2:
+C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5
+\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c
+c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
+c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2
+a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind
+(eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq
+C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0:
+C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0))
+(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift
+(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0))
+(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind
+Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c
+(CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee 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 d1 (Bind Abbr) u0) (getl_mono c (CHead d
+(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c
+(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2
+H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
+c2))).(and_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0)
+(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c
+c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0
+(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def
+(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind
+(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0)
+(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat
+(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1
+(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d
+(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i)))
+(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2
+(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2
+x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
+H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3))
+H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead
+c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind
+Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11))
+(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq
+C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t)))
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
+Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11))
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
+H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst)
+(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
+(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
+d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0 (TLRef i) t2)
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i i0) (eq T t2
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind
+Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
+(ee: C).(match ee 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 d1 (Bind Abbr)
+u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12))
+in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10)))
+(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall
+(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to
+(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall
+(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b)
+u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead
+(Bind b) u t) c2 t2)).(let H7 \def (fsubst0_gen_base c c2 (THead (Bind b) u
+t) t2 u0 i H6) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead (Bind b) u
+t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (land
+(subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 a2)
+(\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind b) u t)
+t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) (arity g c2
+t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 (THead (Bind b)
+u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t
+a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b
+c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S
+i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u
+x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
+(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
+g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c
+(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
+(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13))
+x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind
+b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both
+(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0)
+(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head
+(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead
+(Bind b) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T (THead (Bind b) u t)
+t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind
+b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u
+t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0
+i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c
+(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u))))
+t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2)
+(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Bind b) u t) t2) (csubst0
+i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead (Bind b) u t)
+t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s
+(Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))))
+(arity g c2 t2 a2) (\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t2 (THead
+(Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))
+(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) x
+t))).(\lambda (H13: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind b) x t)
+(\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x a1 (H2 d1 u0 i
+H5 c2 x (fsubst0_both i u0 c u x H13 c2 H10)) t a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t (fsubst0_fst (S i) u0 (CHead c
+(Bind b) u) t (CHead c2 (Bind b) x) (csubst0_both_bind b i u0 u x H13 c c2
+H10)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c2 t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
+(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
+g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 i H5 c2 u (fsubst0_fst i u0
+c u c2 H10)) x a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u
+(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x
+(fsubst0_both (S i) u0 (CHead c (Bind b) u) t x H13 (CHead c2 (Bind b) u)
+(csubst0_fst_bind b i c c2 u0 H10 u)))) t2 H12)))) H11)) (\lambda (H11:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
+(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c2 t0 a2))
+(arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 (fsubst0_both i u0 c u x0
+H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c
+u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x0)
+x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c2 (Bind b)
+x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 H12)))))) H11))
+(subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) H7))))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
+(asucc g a1))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
+a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: C).(\forall (u0:
+T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind
+Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Bind Abst) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Bind
+Abst) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Bind Abst) u t) t2)) (land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c
+c2)) (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land (eq C c c2) (subst0 i u0
+(THead (Bind Abst) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind
+Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (eq C c
+c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) t2)).(eq_ind C c
+(\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind (ex2 T (\lambda (u2:
+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
+T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u0 t t3)))) (arity g c t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 (AHead a1 a2)) (\lambda
+(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12:
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+(fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst
+(CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i
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+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S
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+Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind
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+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
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+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
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+Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
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+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind
+Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr)
+u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind
+Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2
+H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1
+a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
+x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
+(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head
+g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S
+i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u)
+(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2
+(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2
+(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1
+u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0)
+(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10))
+(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
+t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Appl) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat
+Appl) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c
+c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat
+Appl) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Appl) u t)
+t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0
+(THead (Flat Appl) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda
+(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat
+Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c t2 a2)
+(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2
+(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2
+a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x
+t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t)
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 i H4 c x
+(fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0
+(s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))
+(arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl)
+u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead
+(Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u a1 H0
+x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity
+g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
+t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
+c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
+i u0 c c2))).(and_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead
+(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10:
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
+Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i
+u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda
+(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x)
+(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c
+t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1
+(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head
+(Flat Appl) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
+a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Cast) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat
+Cast) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c
+c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat
+Cast) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Cast) u t)
+t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0
+(THead (Flat Cast) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0))
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda
+(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat
+Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c t2 a0)
+(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2
+(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2
+a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x
+t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t)
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 i H4 c x
+(fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0
+(s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))
+(arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast)
+u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead
+(Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u a0 H0
+x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) (\lambda
+(H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity
+g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
+x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
+H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
+u0 c c2))).(and_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
+(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10:
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
+Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0
+c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda
+(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x)
+(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t
+x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i
+u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t
+t2 i H8)))) H7)) H6))))))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (d1:
+C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u)) \to
+(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity g c2 t2
+a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (d1:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 (Bind
+Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u c t
+c2 t2)).(let H5 \def (fsubst0_gen_base c c2 t t2 u i H4) in (or3_ind (land
+(eq C c c2) (subst0 i u t t2)) (land (eq T t t2) (csubst0 i u c c2)) (land
+(subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 t2 a2) (\lambda (H6: (land
+(eq C c c2) (subst0 i u t t2))).(and_ind (eq C c c2) (subst0 i u t t2) (arity
+g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i u t
+t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (arity_repl g c t2 a1
+(H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) c2 H7))) H6)) (\lambda
+(H6: (land (eq T t t2) (csubst0 i u c c2))).(and_ind (eq T t t2) (csubst0 i u
+c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t t2)).(\lambda (H8: (csubst0 i
+u c c2)).(eq_ind T t (\lambda (t0: T).(arity g c2 t0 a2)) (arity_repl g c2 t
+a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 H8)) a2 H2) t2 H7))) H6))
+(\lambda (H6: (land (subst0 i u t t2) (csubst0 i u c c2))).(and_ind (subst0 i
+u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (subst0 i u t
+t2)).(\lambda (H8: (csubst0 i u c 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:
+ \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
+a)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H:
+(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
+(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u
+c t1 t2 H1)))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/gz/defs".
+
+include "A/defs.ma".
+
+include "G/defs.ma".
+
+definition gz:
+ G
+\def
+ mk_G S lt_n_Sn.
+
+inductive leqz: A \to (A \to Prop) \def
+| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
+(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort
+h2 n2))))))
+| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props".
+
+include "gz/defs.ma".
+
+include "leq/defs.ma".
+
+include "aplus/props.ma".
+
+theorem 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
+ \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0:
+nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h)
+n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y
+\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n)
+(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0:
+nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A
+(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A
+(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O
+\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n:
+nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n))
+k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O
+(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0
+n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n)))
+(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0:
+nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S
+(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0))
+(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz
+(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n:
+nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
+(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
+k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
+(H0: (le (S n) (S k0))).(ex2_ind nat (\lambda (n1: nat).(eq nat (S k0) (S
+n1))) (\lambda (n1: nat).(le n n1)) (eq A (asucc gz (aplus gz (ASort (S n)
+n0) k0)) (ASort O (plus (minus k0 n) n0))) (\lambda (x: nat).(\lambda (H1:
+(eq nat (S k0) (S x))).(\lambda (H2: (le n x)).(let H3 \def (f_equal nat nat
+(\lambda (e: nat).(match e in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow k0 | (S n1) \Rightarrow n1])) (S k0) (S x) H1) in (let H4 \def
+(eq_ind_r nat x (\lambda (n1: nat).(le n n1)) H2 k0 H3) in (eq_ind A (aplus
+gz (ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n)
+n0) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
+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 H4))))))) (le_gen_S n (S
+k0) H0)))))) h)))) k).
+
+theorem 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
+ \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h:
+nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0)
+n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0:
+nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O)
+(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0
+h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda
+(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz
+(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le
+(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0:
+nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
+(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0
+x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O
+n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0)
+n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
+n))))).(\lambda (H0: (le (S k0) (S n0))).(ex2_ind nat (\lambda (n1: nat).(eq
+nat (S n0) (S n1))) (\lambda (n1: nat).(le k0 n1)) (eq A (asucc gz (aplus gz
+(ASort (S n0) n) k0)) (ASort (minus n0 k0) n)) (\lambda (x: nat).(\lambda
+(H1: (eq nat (S n0) (S x))).(\lambda (H2: (le k0 x)).(let H3 \def (f_equal
+nat nat (\lambda (e: nat).(match e in nat return (\lambda (_: nat).nat) with
+[O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x) H1) in (let H4 \def
+(eq_ind_r nat x (\lambda (n1: nat).(le k0 n1)) H2 n0 H3) in (eq_ind A (aplus
+gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0)
+n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) k0) (\lambda (a:
+A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus gz (ASort n0 n)
+k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S
+n0) n))) (ASort (minus n0 k0) n) (IH n0 H4))))))) (le_gen_S k0 (S n0) H0)))))
+h)))) k)).
+
+theorem 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
+(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0:
+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:
+ \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
+(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
+(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k
+h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2
+(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def
+(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort
+h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1
+(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k)
+(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2)
+(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat)
+on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O |
+(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow
+(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2
+k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in
+(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal
+nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1))
+(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2
+k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a
+(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1
+(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort
+h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus
+(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat
+(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2)
+n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind
+A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return
+(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])
+| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in
+(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1
+k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k
+h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A
+a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
+(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2)
+k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort
+(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in
+(let H5 \def (sym_equal A (ASort O (plus (minus k h1) n1)) (ASort (minus h2
+k) n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A
+(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k)))
+(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k)))
+n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort
+n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow
+False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1
+n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A
+(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2)
+k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4
+\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O
+(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le
+k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
+nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
+plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus
+(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in
+(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4:
+A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4
+a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+
+theorem leq_leqz:
+ \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
+(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus
+h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A
+(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1)))
+(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A
+(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2)))
+(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus
+(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus
+h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2))
+(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n:
+nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus
+h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq
+A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz
+n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1
+h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n:
+nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus
+h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal
+nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2
+n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2)
+(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2)
+(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2)
+(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus
+(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2))
+(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1
+h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0
+a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda
+(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/fwd".
+
+include "leq/defs.ma".
+
+theorem leq_gen_sort:
+ \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 (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort
+h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k)
+(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2:
+(eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match
+e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _
+_) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
+(ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to
+((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort
+h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
+h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n:
+nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g
+(ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
+h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2
+n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
+nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))
+(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
+h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq
+nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0
+a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1
+n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead
+a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1
+n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g
+a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal
+A a2))))))).
+
+theorem leq_gen_head:
+ \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 (a4: A).(eq A a
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda
+(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
+(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda
+(a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2))
+\to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a
+(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
+(_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k
+H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda
+(H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda
+(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in
+(False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k)
+(aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
+A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3)))
+(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head
+a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1
+a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A
+(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2)
+H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _)
+\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda
+(a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to
+((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
+(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
+(_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4
+a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3)
+\to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
+(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
+(_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3
+a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g
+a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7
+a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
+A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda
+(H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A
+(AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1
+a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A
+(AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1
+H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A
+a))))))).
+
include "leq/defs.ma".
-theorem leq_gen_sort:
- \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 (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
-h2 n2) k))))))))))
+theorem 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
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return
-(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort
-h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k)
-(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0)
-\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2:
-(eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match
-e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _
-_) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
-(ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to
-((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort
-h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n:
-nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g
-(ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2
-n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
-nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))
-(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq
-nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0
-a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1
-n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead
-a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1
-n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g
-a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal
-A a2))))))).
-
-theorem leq_gen_head:
- \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 (a4: A).(eq A a
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda
-(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda
-(a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2))
-\to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a
-(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k
-H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda
-(H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda
-(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in
-(False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k)
-(aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3)))
-(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head
-a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1
-a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2)
-H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _)
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda
-(a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to
-((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
-(_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4
-a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3)
-\to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
-(_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3
-a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g
-a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7
-a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda
-(H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A
-(AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1
-a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A
-(AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1
-H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A
-a))))))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
+H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
+a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2
+a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
+(ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3
+a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2)
+(AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3:
+(eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
+(AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6)
+(AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A
+(AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g
+a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A
+(AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g
+a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7)
+(AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
+\Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a:
+A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4)))))
+(\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to
+((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12:
+(leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8)))
+a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0
+(refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
theorem leq_refl:
\forall (g: G).(\forall (a: A).(leq g a a))
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/llt/defs".
+
+include "A/defs.ma".
+
+definition lweight:
+ A \to nat
+\def
+ let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow
+O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in
+lweight.
+
+definition llt:
+ A \to (A \to Prop)
+\def
+ \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/llt/props".
+
+include "llt/defs.ma".
+
+include "leq/defs.ma".
+
+theorem lweight_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
+(lweight a1) (lweight a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
+a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
+(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
+a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
+(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
+H)))).
+
+theorem llt_repl:
+ \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
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
+\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
+(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
+
+theorem llt_trans:
+ \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
+a3) \to (llt a1 a3)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
+(lweight a1) (lweight a2) (lweight a3) H H0))))).
+
+theorem llt_head_sx:
+ \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a1)) (S (plus (lweight
+a1) (lweight a2))) (le_n_S (S (lweight a1)) (S (plus (lweight a1) (lweight
+a2))) (le_n_S (lweight a1) (plus (lweight a1) (lweight a2)) (le_plus_l
+(lweight a1) (lweight a2)))))).
+
+theorem llt_head_dx:
+ \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a2)) (S (plus (lweight
+a1) (lweight a2))) (le_n_S (S (lweight a2)) (S (plus (lweight a1) (lweight
+a2))) (le_n_S (lweight a2) (plus (lweight a1) (lweight a2)) (le_plus_r
+(lweight a1) (lweight a2)))))).
+
+theorem 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)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
+n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
+a)))))).
+
+theorem llt_wf_ind:
+ \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
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
+(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
+A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
+(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
+a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
+(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
+a1))))))))))))) a)))).
+
include "leq/defs.ma".
+include "leq/fwd.ma".
+
include "leq/props.ma".
include "leq/asucc.ma".
+include "llt/defs.ma".
+
+include "llt/props.ma".
+
+include "aprem/defs.ma".
+
+include "aprem/props.ma".
+
+include "gz/defs.ma".
+
+include "gz/props.ma".
+
+include "arity/defs.ma".
+
+include "arity/fwd.ma".
+
+include "arity/props.ma".
+
+include "arity/subst0.ma".
+
+include "arity/lift1.ma".
+
+include "arity/cimp.ma".
+
+include "arity/aprem.ma".
+
projects / helm.git / commitdiff