--- /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))))).
+
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