--- /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 *********************)
+
+include "LambdaDelta-1/aprem/defs.ma".
+
+theorem aprem_gen_sort:
+ \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem
+i (ASort h n) x) \to False))))
+\def
+ \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n:
+nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
+(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0:
+(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_:
+A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2:
+A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A
+(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a:
+A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2
+(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2)
+(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False
+H4))))))))) i y x H0))) H))))).
+
+theorem aprem_gen_head_O:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x)
+\to (eq A x a1))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O
+(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x))
+(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y
+x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq
+A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
+y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq
+nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0
+a3) (AHead a1 a2))).(let H4 \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 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3)
+in (\lambda (H6: (eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(eq A a a1))
+(refl_equal A a1) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda
+(i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i O) \to ((eq
+A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda (H4: (eq
+nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \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 a3 a0)
+(AHead a1 a2) H5) in ((let H7 \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 a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3
+a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A a4
+(AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0
+(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a
+a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
+
+theorem aprem_gen_head_S:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem
+(S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda
+(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a:
+A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y:
+A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n:
+nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2
+x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
+nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead
+a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda
+(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
+H4 \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 a3)
+(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0
+a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \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 i) H2) in (False_ind
+(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a:
+A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat
+i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3:
+A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0)
+(AHead a1 a2))).(let H6 \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 a3 a0) (AHead a1 a2) H5) in ((let H7 \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 a3 a0) (AHead a1 a2)
+H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4:
+A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3
+a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2
+H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n]))
+(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n
+(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let
+H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in
+H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
+
include "LambdaDelta-1/aprem/props.ma".
+include "LambdaDelta-1/aprem/fwd.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 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))))))))
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b
+i O n H0) in (let H1 \def H_x 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 (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1))))))))
(\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
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)))))))) (\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))))
+(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in
+(eq_ind_r A a1 (\lambda (a0: 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 a0))))))) (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 H_y))) (\lambda (i0:
+nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 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))))
+b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def
+(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6
+\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 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 i0) 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 (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst)
+u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) 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)))) (\lambda (i0: nat).(\lambda
-(_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda
-(_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) (\lambda (d:
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
-b))))))))).(\lambda (H5: (aprem (S i0) (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 (S i0)) \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 i0) 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
-i0))).(\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 i0) 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 i0) 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 i1
-H6 a4) \Rightarrow (\lambda (H7: (eq nat (S i1) (S i0))).(\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 i1 | (S n) \Rightarrow n])) (S i1) (S i0) H7)
-in (eq_ind nat i0 (\lambda (n: nat).((eq A (AHead a4 a0) (AHead a1 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 i0) 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 i0 a0 a3) \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 (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 i0 a5 a3) \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 (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
-i0 a2 a5) \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 (u0:
-T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H16: (aprem
-i0 a2 b)).(let H_x \def (H3 i0 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 i0 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 i0) 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 i0 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 i0) 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 i0 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))) i1 (sym_eq nat i1 i0 H10)))
-H8 H9 H6))))]) in (H6 (refl_equal nat (S i0)) (refl_equal A (AHead a1 a2))
-(refl_equal A b)))))) i 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))).(K_ind (\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)))))))))
-(\lambda (b0: B).(\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)))) (\lambda
-(f: F).(\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 (_:
+b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i
+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))))) (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))))) k H9 (drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7))))))
+(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))).(K_ind (\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))))))))) (\lambda (b0: B).(\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)))) (\lambda (f: F).(\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))))) k 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 (_:
include "LambdaDelta-1/csubt/defs.ma".
+include "LambdaDelta-1/wf3/defs.ma".
+
T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
-theorem ty3_getl_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
-t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
-\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
-(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
-t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
-nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
-T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
-B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
-v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
-nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
-(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
-T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
-H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
-c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
-T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
-w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
-(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
-(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
-(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
-H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl
-n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n
-H0 (CHead d0 (Bind b) v) H7)) 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) u0) (CHead d0 (Bind b)
-v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H10 \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 b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
-(CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abbr
-b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
-T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
-(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
-(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
-d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
-(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr
-H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
-H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
-(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1:
-(ty3 g d u0 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
-T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
-w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
-(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
-(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
-(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
-H5) in (let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl
-n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n
-H0 (CHead d0 (Bind b) v) H7)) 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) u0) (CHead d0 (Bind b)
-v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H10 \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 b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abst])])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
-(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abst
-b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
-T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
-(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
-(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
-d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
-(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst
-H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
-H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0
-t0)).(\lambda (H1: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
-(ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: ((\forall (v0:
-T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to (\forall (b0:
-B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) u0) (CHead d
-(Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda
-(v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead
-(Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: C).(\lambda (v:
-T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind (ex2 T (\lambda
-(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
-u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4:
-T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b)
-i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: (ex2 T
-(\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i
-v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
-(\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda (w: T).(ty3 g d v w)))
-(\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) x t1))).(\lambda (H8:
-(subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) (\lambda (H6: (ex2 T
-(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
-(s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead (Bind
-b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)) (ex T (\lambda
-(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b)
-u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d
-v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda
-(H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
-T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
-T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
-x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
-v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
-u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
-(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
-T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
-B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
-T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
-((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
-(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
-b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
-Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
-(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
-T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
-(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
-t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
-(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
-(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
-(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
-(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
-(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
-(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
-T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
-(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
-x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
-T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
-(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
-(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
-H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
-t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
-nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
-((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
-(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
-v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
-T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
-Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
-(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
-t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
-(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
-T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
-Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
-T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
-Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
-T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
-t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
-(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
-(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
-(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
-(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
-T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
-(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
-(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
-(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
-x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
-H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
-c t u H))))).
-
theorem ty3_gen_appl_nf2:
\forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
(le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6))
H5)) H4))))) H2))))))))).
-inductive wf3 (g: G): C \to (C \to Prop) \def
-| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
-| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
-b) u) (CHead c2 (Bind b) u))))))))
-| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
-| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
-
-theorem wf3_gen_sort1:
- \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
-(eq C x (CSort m)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
-m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
-C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
-(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
-nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
-(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
-\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
-(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
-(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
-(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
-(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
-H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
-T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
-B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
-C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
-True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
-(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
-C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
-c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
-f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
-c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
-
-theorem wf3_gen_bind1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
-B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
-C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
-(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
-C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
-B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
-b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
-(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
-C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
-(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
-(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
-(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
-T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
-(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
-H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
-(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
-c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
-C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
-c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
-(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
-u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
-(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
-c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \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 c0 (Bind
-b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
-(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
-(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
-(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
-T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
-T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
-t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
-C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
-H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
-v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
-\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
-(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
-(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
-C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
-(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
-c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
-(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
-(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
-c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
-b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
-H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
-(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
-in ((let H7 \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 c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
-b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
-T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
-(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
-c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
-b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
-\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
-(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
-(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
-c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
-(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
-Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
-c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
-(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
-H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
-c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
-g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
-(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
-c0 (Flat f) 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 _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
-T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
-g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
-
-theorem wf3_gen_flat1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
-F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
-F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
-f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
-C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
-C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
-nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
-(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
-H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
-(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
-(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
-(CHead c0 (Bind b) 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 _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
-(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
-\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
-u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
-(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) 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 _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
-(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
-v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
-(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
-c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
-C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
-\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
-c1 (Flat f) v) H3) in ((let H6 \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 c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
-(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
-(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
-in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
-H10))))) H5)) H4))))))))) y x H0))) H)))))).
-
-theorem wf3_gen_head2:
- \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
-K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
-K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
-(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
-b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
-C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
-k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
-v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
-B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
-(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
-B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
-g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
-k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
-(Bind b) u) (CHead c k v) H4) in ((let H7 \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 c2 (Bind b) u) (CHead c k v) H4) in
-(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
-(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
-(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
-(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
-(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
-(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
-(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
-(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
-(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
-c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
-k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
-t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
-(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
-((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
-C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
-(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
-k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
-C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
-H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
-H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
-\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
-(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
-b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
-Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
-(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
-Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
-(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
-T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
-(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
-C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
-K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
-(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
-v))))))))))))) x y H0))) H)))))).
-
-theorem wf3_mono:
- \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
-(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
-c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
-c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
-(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
-(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
-H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
-c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
-c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
-t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
-u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
-(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
-b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
-c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
-c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
-(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
-C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
-C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
-u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
-x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
-(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
-b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
-H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
-Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
-(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
-(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
-(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
-u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
-O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
-w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
-C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
-H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
-O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
-C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
-\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
-c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
-(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
-(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
-c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
-C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
-(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
-(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
-(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
-T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
-w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
-(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
-c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
-b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
-C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
-O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
-(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
-H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
-(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
-T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
-(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
-(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
-g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
-(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
-c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
-(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
-(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
-((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
-T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
-u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
-c c1 H)))).
-
-theorem wf3_clear_conf:
- \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
-(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
-\def
- \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
-(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
-c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
-T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
-c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
-c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
-c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
-(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
-
-theorem clear_wf3_trans:
- \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
-(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
-(c2: C).(clear c2 d2))))))))
-\def
- \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
-d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
-d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
-G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
-\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
-(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
-(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
-e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
-\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
-(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
-C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
-C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
-w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
-(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
-(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
-x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
-(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
-C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
-c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
-(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
-(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
-\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
-(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
-T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
-b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
-(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
-x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
-(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
-g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
-(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
-(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
-g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
-((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
-F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
-d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
-(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
-C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
-(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
-(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
-d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
-
-theorem wf3_getl_conf:
- \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
-(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
-C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2)))))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
-C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
-(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
-d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
-(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
-C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
-(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
-(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
-(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
-(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
-g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
-g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
-b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
-(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
-v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
-C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
-(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
-b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
-(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
-x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
-c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
-False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
-g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
-(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
-((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
-Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
-(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
-(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
-((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
-b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
-(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
-nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
-(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
-(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
-n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
-(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
-d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
-(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
-(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
-g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
-k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
-B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
-(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
-b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
-(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
-T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
-(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
-False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
-(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
-w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
-C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
-(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
-b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
-x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
-C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
-n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
-c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
-(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
-C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
-(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
-(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
-O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
-w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
-C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
-(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
-H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
-(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
-n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
-(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
-(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
-(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
-v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
-H1)))))))))))))) c1)))) i)).
-
-theorem wf3_total:
- \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
-C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
-(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
-(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
-H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
-(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
-(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
-(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
-T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
-False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
-(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
-g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
-(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
-(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
-b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
-False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
-(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
-(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
-(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
-
-theorem getl_wf3_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
-d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
-d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
-\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
-in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
-(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
-d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
-d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
-nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
-C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
-(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
-(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
-(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
-(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
-d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
-d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
-(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
-t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
-(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
-(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
-B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
-H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
-(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
-g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
-(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
-(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
-(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
-(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
-(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
-C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
-x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
-(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
-b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
-T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
-c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
-(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
-O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
-d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
-(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
-(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
-n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
-n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
-(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
-k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
-
theorem ty3_shift1:
\forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0)))
H))).
-theorem wf3_pr2_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
-u) \to (pr2 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
-c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
-T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
-C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
-u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
-(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
-(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
-T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
-g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
-(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
-(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
-H5)))))))))))))))))) c1 t1 t2 H))))).
-
-theorem wf3_pr3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
-u) \to (pr3 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
-(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
-t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
-T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
-c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
-(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
-t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
-H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
-
-theorem wf3_pc3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
-u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
-(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
-c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
-(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
-(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
-g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
-H3)))))))))))).
-
-theorem wf3_ty3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
-C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
-T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
-(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
-t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
-T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
-x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
-c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
-C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
-c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
-(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
-c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
-(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
-x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
-t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
-t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
-(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
-x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
-((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
-t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
-(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
-c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
-H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
-(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
-c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
-c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
-theorem wf3_idem:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
-c2 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
-c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
-nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
-c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
-(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
-c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
-(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
-((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
-c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
-(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
-c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
-
-theorem wf3_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
-u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
-u)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
-C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
-(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
-x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
-u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
-
include "LambdaDelta-1/ty3/nf2.ma".
-include "LambdaDelta-1/ty3/dec.ma".
+include "LambdaDelta-1/wf3/props.ma".
(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v
H)))))).
+theorem ty3_getl_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
+t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
+\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
+(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
+t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
+T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
+v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
+H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
+(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
+(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
+(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
+H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl
+n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n
+H0 (CHead d0 (Bind b) v) H7)) 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) u0) (CHead d0 (Bind b)
+v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H10 \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 b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
+(CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abbr
+b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
+T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
+(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
+(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
+d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
+(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr
+H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
+H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
+(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1:
+(ty3 g d u0 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
+(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
+(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
+(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
+H5) in (let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl
+n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n
+H0 (CHead d0 (Bind b) v) H7)) 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) u0) (CHead d0 (Bind b)
+v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H10 \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 b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abst])])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
+((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abst
+b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
+T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
+(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
+(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
+d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
+(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst
+H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
+H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0
+t0)).(\lambda (H1: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: ((\forall (v0:
+T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to (\forall (b0:
+B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) u0) (CHead d
+(Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda
+(v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead
+(Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
+u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b)
+i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: (ex2 T
+(\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i
+v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
+(\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) x t1))).(\lambda (H8:
+(subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
+(s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead (Bind
+b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)) (ex T (\lambda
+(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b)
+u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d
+v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda
+(H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
+x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
+v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
+u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
+T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
+B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
+T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
+b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
+t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
+(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
+(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
+(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
+(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
+T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
+x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
+T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
+(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
+H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
+t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
+nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
+T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
+t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
+T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
+Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
+T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
+Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
+t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
+(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
+(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
+(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
+(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
+x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
+H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
+c t u 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 *********************)
+
+include "LambdaDelta-1/wf3/fwd.ma".
+
+theorem wf3_clear_conf:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
+(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
+\def
+ \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
+(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
+c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
+T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
+c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
+c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
+c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
+(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
+
+theorem clear_wf3_trans:
+ \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
+(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
+(c2: C).(clear c2 d2))))))))
+\def
+ \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
+d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
+d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
+G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
+\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
+(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
+e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
+(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
+w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
+(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
+(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
+C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
+c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
+(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
+(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
+b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
+(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
+(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
+g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
+(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
+g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
+((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
+F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
+d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
+(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
+C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
+(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
+d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 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 *********************)
+
+include "LambdaDelta-1/ty3/defs.ma".
+
+inductive wf3 (g: G): C \to (C \to Prop) \def
+| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
+| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
+b) u) (CHead c2 (Bind b) u))))))))
+| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
+| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
+
--- /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 *********************)
+
+include "LambdaDelta-1/wf3/defs.ma".
+
+theorem wf3_gen_sort1:
+ \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
+(eq C x (CSort m)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
+m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
+C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
+nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
+(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
+\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
+(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
+(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
+(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
+H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
+C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
+(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
+c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
+f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
+c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
+
+theorem wf3_gen_bind1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
+B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
+(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
+C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
+B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
+b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
+(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
+(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
+(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
+H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
+(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
+c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
+C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
+c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
+u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
+(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \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 c0 (Bind
+b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
+(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
+(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
+T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
+T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
+t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
+H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
+v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
+(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
+(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
+(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
+c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
+b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
+H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
+in ((let H7 \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 c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
+b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
+T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
+(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
+c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
+b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
+(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
+Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
+c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
+(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
+H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
+c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
+c0 (Flat f) 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 _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
+T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_flat1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
+F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
+F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
+f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
+C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
+nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
+(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
+H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
+(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
+(CHead c0 (Bind b) 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 _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
+(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
+\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
+u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
+(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) 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 _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
+(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
+v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
+C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
+\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H6 \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 c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
+(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
+(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
+in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
+H10))))) H5)) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_head2:
+ \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
+K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
+K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
+(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
+b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
+C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
+v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
+B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
+B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
+g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
+k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
+(Bind b) u) (CHead c k v) H4) in ((let H7 \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 c2 (Bind b) u) (CHead c k v) H4) in
+(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
+(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
+(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
+(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
+(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
+(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
+(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
+(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
+t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
+((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
+k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
+(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
+k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
+C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
+H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
+H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
+\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
+(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
+b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
+Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
+(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
+Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
+T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
+(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
+C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
+K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
+(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
+v))))))))))))) x y 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 *********************)
+
+include "LambdaDelta-1/wf3/clear.ma".
+
+include "LambdaDelta-1/ty3/dec.ma".
+
+theorem wf3_getl_conf:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
+(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
+C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)))))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
+C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
+(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
+d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
+C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
+(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
+(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
+(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
+(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
+g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
+v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
+(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
+b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
+(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
+x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
+c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
+False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
+(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
+((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
+Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
+(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
+(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
+((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
+(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
+nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
+(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
+(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
+n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
+(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
+d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
+(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
+(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
+g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
+k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
+B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
+(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
+b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
+(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
+T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
+False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
+w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
+(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
+b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
+x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
+C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
+c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
+(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
+C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
+(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
+(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
+w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
+C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
+(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
+H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
+(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
+(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
+(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
+(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
+v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
+H1)))))))))))))) c1)))) i)).
+
+theorem getl_wf3_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
+d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
+d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
+\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
+in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
+(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
+d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
+d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
+C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
+(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
+(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
+(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
+(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
+d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
+d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
+(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
+t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
+(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
+(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
+B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
+H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
+(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
+g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
+(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
+(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
+(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
+(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
+(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
+C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
+x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
+(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
+b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
+T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
+c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
+(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
+O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
+d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
+(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
+(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
+n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
+n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
+(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
+k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
+
--- /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 *********************)
+
+include "LambdaDelta-1/wf3/ty3.ma".
+
+theorem wf3_mono:
+ \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
+(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
+c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
+c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
+(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
+(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
+H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
+c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
+c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
+t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
+u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
+(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
+b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
+c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
+c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
+(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
+C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
+C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
+u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
+x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
+b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
+H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
+Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
+(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
+(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
+u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
+w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
+C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
+H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
+O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
+C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
+\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
+c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
+(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
+(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
+c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
+C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
+(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
+(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
+(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
+T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
+w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
+(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
+c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
+b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
+C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
+O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
+(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
+H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
+(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
+(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
+(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
+g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
+(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
+c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
+(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
+(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
+((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
+T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
+u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
+c c1 H)))).
+
+theorem wf3_total:
+ \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
+C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
+(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
+(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
+H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
+(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
+(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
+(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
+T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
+False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
+g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
+(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
+b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
+False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
+(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
+(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
+(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
+
+theorem wf3_idem:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
+c2 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
+c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
+nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
+c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
+(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
+c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
+(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
+((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
+c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
+c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
+
+theorem wf3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
+u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
+C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
+(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
+x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
+
--- /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 *********************)
+
+include "LambdaDelta-1/wf3/getl.ma".
+
+theorem wf3_pr2_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr2 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
+c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
+T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
+u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
+(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
+(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
+T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
+g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
+(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
+(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
+H5)))))))))))))))))) c1 t1 t2 H))))).
+
+theorem wf3_pr3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr3 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
+(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
+t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
+c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
+(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
+t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
+H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
+
+theorem wf3_pc3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
+u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
+(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
+c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
+(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
+(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
+g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
+H3)))))))))))).
+
+theorem wf3_ty3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
+(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
+t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
+T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
+x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
+c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
+C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
+c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
+c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
+t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
+((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
+t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
+(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
+c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
+H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
+c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
MMAS = $(shell find Base-2 -name "*.mma")
MAS = $(MMAS:%.mma=%.ma)
-XMAS = Base-2/theory.ma
+XMAS = Base-2/theory.ma LambdaDelta-2/theory.ma
$(DIR) all: depends
$(H)$(MAKE) H=$(H) --no-print-directory build
$(H)rm depends
clean:
- $(H)../../matitaclean
+ $(H)../../matitaclean $(MATITAOPTIONS)
$(H)rm -f $(MAS) depends
clean.opt:
- $(H)../../matitaclean.opt
+ $(H)../../matitaclean.opt $(MATITAOPTIONS)
$(H)rm -f $(MAS) depends
clean.ma:
- $(H)../../matitaclean.opt $(MAS)
+ $(H)../../matitaclean.opt $(MATITAOPTIONS) $(MAS)
$(H)rm -f $(MAS) depends
depend:
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* LIFT RELATION
- - Usage: invoke with positive polarity
-*)
-
-include "Unified-Sub/datatypes/Term.ma".
-
-inductive Lift (l: Nat): Nat \to Term \to Term \to Prop \def
- | lift_sort : \forall i,h.
- Lift l i (sort h) (sort h)
- | lift_lref_gt: \forall i,j. i > j \to
- Lift l i (lref j) (lref j)
- | lift_lref_le: \forall i,j1. i <= j1 \to
- \forall j2. (l + j1 == j2) \to
- Lift l i (lref j1) (lref j2)
- | lift_bind : \forall i,u1,u2. Lift l i u1 u2 \to
- \forall t1,t2. Lift l (succ i) t1 t2 \to
- \forall r. Lift l i (intb r u1 t1) (intb r u2 t2)
- | lift_flat : \forall i,u1,u2. Lift l i u1 u2 \to
- \forall t1,t2. Lift l i t1 t2 \to
- \forall r. Lift l i (intf r u1 t1) (intf r u2 t2)
-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "Unified-Sub/Lift/inv.ma".
-
-(* Functional properties ****************************************************)
-
-theorem lift_total: \forall l, t, i. \exists u. Lift l i t u.
- intros 2. elim t; clear t;
- [ autobatch
- | lapply (nle_gt_or_le n i). decompose;
- [ autobatch
- | lapply (nplus_total l n). decompose. autobatch
- ]
- | lapply (H i1). lapply (H1 (succ i1)). decompose. autobatch
- | lapply (H i1). lapply (H1 i1). decompose. autobatch
- ].
-qed.
-
-theorem lift_mono: \forall l,i,t,t1. Lift l i t t1 \to
- \forall t2. Lift l i t t2 \to
- t1 = t2.
- intros 5. elim H; clear H i t t1;
- [ lapply linear lift_inv_sort_1 to H1
- | lapply linear lift_inv_lref_1_gt to H2, H1
- | lapply linear lift_inv_lref_1_le_nplus to H3, H1, H2
- | lapply linear lift_inv_bind_1 to H5. decompose
- | lapply linear lift_inv_flat_1 to H5. decompose
- ]; destruct; autobatch.
-qed.
-
-theorem lift_inj: \forall l,i,t1,t. Lift l i t1 t \to
- \forall t2. Lift l i t2 t \to
- t2 = t1.
- intros 5. elim H; clear H i t1 t;
- [ lapply linear lift_inv_sort_2 to H1
- | lapply linear lift_inv_lref_2_gt to H2, H1
- | lapply nle_nplus to H2 as H.
- lapply linear nle_trans to H1, H as H0.
- lapply lift_inv_lref_2_le_nplus to H3, H0, H2
- | lapply linear lift_inv_bind_2 to H5. decompose
- | lapply linear lift_inv_flat_2 to H5. decompose
- ]; destruct; autobatch.
-qed.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "Unified-Sub/Lift/defs.ma".
-
-(* Inversion properties *****************************************************)
-
-theorem lift_inv_sort_1: \forall l, i, h, x.
- Lift l i (sort h) x \to
- x = sort h.
- intros. inversion H; clear H; intros; destruct. autobatch.
-qed.
-
-theorem lift_inv_lref_1: \forall l, i, j1, x.
- Lift l i (lref j1) x \to
- (i > j1 \land x = lref j1) \lor
- (i <= j1 \land
- \exists j2. (l + j1 == j2) \land x = lref j2
- ).
- intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 7.
-qed.
-
-theorem lift_inv_bind_1: \forall l, i, r, u1, t1, x.
- Lift l i (intb r u1 t1) x \to
- \exists u2, t2.
- Lift l i u1 u2 \land
- Lift l (succ i) t1 t2 \land
- x = intb r u2 t2.
- intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 7.
-qed.
-
-theorem lift_inv_flat_1: \forall l, i, r, u1, t1, x.
- Lift l i (intf r u1 t1) x \to
- \exists u2, t2.
- Lift l i u1 u2 \land
- Lift l i t1 t2 \land
- x = intf r u2 t2.
- intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
-qed.
-
-theorem lift_inv_sort_2: \forall l, i, x, h.
- Lift l i x (sort h) \to
- x = sort h.
- intros. inversion H; clear H; intros; destruct. autobatch.
-qed.
-
-theorem lift_inv_lref_2: \forall l, i, x, j2.
- Lift l i x (lref j2) \to
- (i > j2 \land x = lref j2) \lor
- (i <= j2 \land
- \exists j1. (l + j1 == j2) \land x = lref j1
- ).
- intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 10.
-qed.
-
-theorem lift_inv_bind_2: \forall l, i, r, x, u2, t2.
- Lift l i x (intb r u2 t2) \to
- \exists u1, t1.
- Lift l i u1 u2 \land
- Lift l (succ i) t1 t2 \land
- x = intb r u1 t1.
- intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
-qed.
-
-theorem lift_inv_flat_2: \forall l, i, r, x, u2, t2.
- Lift l i x (intf r u2 t2) \to
- \exists u1, t1.
- Lift l i u1 u2 \land
- Lift l i t1 t2 \land
- x = intf r u1 t1.
- intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
-qed.
-
-(* Corollaries of inversion properties ***************************************)
-
-theorem lift_inv_lref_1_gt: \forall l, i, j1, x.
- Lift l i (lref j1) x \to
- i > j1 \to x = lref j1.
- intros.
- lapply linear lift_inv_lref_1 to H. decompose; destruct;
- [ autobatch
- | lapply linear nle_false to H2, H1. decompose
- ].
-qed.
-
-theorem lift_inv_lref_1_le: \forall l, i, j1, x.
- Lift l i (lref j1) x \to i <= j1 \to
- \exists j2. (l + j1 == j2) \land x = lref j2.
- intros.
- lapply linear lift_inv_lref_1 to H. decompose; destruct;
- [ lapply linear nle_false to H1, H2. decompose
- | autobatch
- ].
-qed.
-
-theorem lift_inv_lref_1_le_nplus: \forall l, i, j1, x.
- Lift l i (lref j1) x \to
- i <= j1 \to \forall j2. (l + j1 == j2) \to
- x = lref j2.
- intros.
- lapply linear lift_inv_lref_1 to H. decompose; destruct;
- [ lapply linear nle_false to H1, H3. decompose
- | lapply linear nplus_mono to H2, H4. destruct. autobatch
- ].
-qed.
-
-theorem lift_inv_lref_2_gt: \forall l, i, x, j2.
- Lift l i x (lref j2) \to
- i > j2 \to x = lref j2.
- intros.
- lapply linear lift_inv_lref_2 to H. decompose; destruct;
- [ autobatch
- | lapply linear nle_false to H2, H1. decompose
- ].
- qed.
-
-theorem lift_inv_lref_2_le: \forall l, i, x, j2.
- Lift l i x (lref j2) \to i <= j2 \to
- \exists j1. (l + j1 == j2) \land x = lref j1.
- intros.
- lapply linear lift_inv_lref_2 to H. decompose; destruct;
- [ lapply linear nle_false to H1, H2. decompose
- | autobatch
- ].
-qed.
-
-theorem lift_inv_lref_2_le_nplus: \forall l, i, x, j2.
- Lift l i x (lref j2) \to
- i <= j2 \to \forall j1. (l + j1 == j2) \to
- x = lref j1.
- intros.
- lapply linear lift_inv_lref_2 to H. decompose; destruct;
- [ lapply linear nle_false to H1, H3. decompose
- | lapply linear nplus_inj_2 to H2, H4. destruct. autobatch
- ].
-qed.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "Unified-Sub/Lift/fun.ma".
-
-(* NOTE: this holds because of nplus_comm_1 *)
-theorem lift_comp: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
- \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
- \forall x. Lift l2 i2 t2 x \to
- \forall i,y. Lift l1 i u1 y \to
- i1 >= i2 \to (l2 + i1 == i) \to x = y.
- intros 5. elim H; clear H i1 t1 t2;
- [ lapply lift_mono to H1, H2. clear H2. destruct.
- lapply linear lift_inv_sort_1 to H1. destruct.
- lapply linear lift_inv_sort_1 to H3. destruct. autobatch
- | lapply lift_mono to H2, H3. clear H3. destruct.
- lapply linear lift_inv_lref_1 to H2.
- decompose; destruct; clear H2 H5;
- lapply linear lift_inv_lref_1_gt to H4; destruct; autobatch width = 4
- | lapply lift_inv_lref_1_le to H3; [ 2: autobatch ]. clear H3.
- lapply lift_inv_lref_1_le to H4; [ 2: autobatch ]. clear H4.
- decompose. destruct. clear H6 i2.
- lapply lift_inv_lref_1_le to H5; [ 2: autobatch depth = 4 width = 4 ].
- decompose. destruct. clear H5 H1 H7 i. autobatch depth = 4 size = 7
- | clear H1 H3.
- lapply linear lift_inv_bind_1 to H5.
- lapply linear lift_inv_bind_1 to H6. decompose. destruct.
- lapply linear lift_inv_bind_1 to H7. decompose. destruct.
- autobatch depth = 4 width = 6 size = 15
- | clear H1 H3.
- lapply linear lift_inv_flat_1 to H5.
- lapply linear lift_inv_flat_1 to H6. decompose. destruct.
- lapply linear lift_inv_flat_1 to H7. decompose. destruct.
- autobatch depth = 4 width = 6 size = 9
- ].
-qed.
-
-theorem lift_comp_rew_dx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
- \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
- \forall u2. Lift l2 i2 t2 u2 \to
- i1 >= i2 \to \forall i. (l2 + i1 == i) \to
- Lift l1 i u1 u2.
- intros.
- lapply (lift_total l1 u1 i). decompose.
- lapply lift_comp to H, H1, H2, H5, H3, H4. destruct. autobatch.
-qed.
-
-theorem lift_comp_rew_sx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
- \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
- \forall i,u2. Lift l2 i t2 u2 \to
- i2 >= i1 \to (l1 + i2 == i) \to
- Lift l1 i1 u1 u2.
- intros.
- lapply (lift_total l1 u1 i1). decompose.
- lapply lift_comp to H1, H, H5, H2, H3, H4. destruct. autobatch.
-qed.
-(*
-theorem lift_trans_le: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
- \forall l2,i2,z. Lift l2 i2 t2 t3 \to
- i1 <= i2 \to
- \forall i. \to i2 <= i \to (l1 + i1 == i) \to
- \forall l. (l1 + l2 == l) \to Lift l i1 t1 t3.
-
-axiom lift_conf_back_ge: \forall l1,i1,u1,u2. Lift l1 i1 u1 u2 \to
- \forall l2,i,t2. Lift l2 i t2 u2 \to
- \forall i2. i2 >= i1 \to (l1 + i2 == i) \to
- \exists t1. | Lift l2 i2 t1 u1 \land
- Lift l1 i1 t1 t2.
-
-*)
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* FLAT CONTEXTS
- - Naming policy:
- - contexts: c d
-*)
-
-include "Unified-Sub/datatypes/Term.ma".
-
-inductive Context: Type \def
- | leaf: Context
- | intb: Context \to IntB \to Term \to Context
-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* POLARIZED TERMS
- - Naming policy:
- - natural numbers : sorts h k, local references i j, lengths l o
- - boolean values : p q
- - generic binding items: r s
- - generic flat items : r s
- - generic head items : m n
- - terms : t u
-*)
-
-include "Unified-Sub/preamble.ma".
-
-inductive Bind: Type \def
- | abbr: Bind
- | abst: Bind
- | excl: Bind
-.
-
-inductive Flat: Type \def
- | appl: Flat
- | cast: Flat
-.
-
-inductive IntB: Type \def
- | bind: Bool \to Bind \to IntB
-.
-
-inductive IntF: Type \def
- | flat: Bool \to Flat \to IntF
-.
-
-inductive Term: Type \def
- | sort: Nat \to Term
- | lref: Nat \to Term
- | intb: IntB \to Term \to Term \to Term
- | intf: IntF \to Term \to Term \to Term
-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* Project started Tue Aug 22, 2006 ***************************************)
-
-(* PREAMBLE
-*)
-
-include "logic/equality.ma".
-include "datatypes/Bool.ma".
-include "NPlus/monoid.ma".
-include "NLE/props.ma".
-include "NLE/nplus.ma".
-
-axiom f_equal_3: \forall (A,B,C,D:Set).
- \forall (f:A \to B \to C \to D).
- \forall (x1,x2:A).
- \forall (y1,y2:B).
- \forall (z1,z2:C).
- x1 = x2 \to y1 = y2 \to z1 = z2 \to
- f x1 y1 z1 = f x2 y2 z2.
baseuri=cic:/matita/LAMBDA-TYPES
-include_paths= ../../legacy ../RELATIONAL
+include_paths= ../../legacy
projects / helm.git / commitdiff