include "basic_1/aprem/defs.ma".
-let rec aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall (a1:
-A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: A).(\forall
-(a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to (\forall (a1:
-A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) (a1: aprem n a
-a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) \Rightarrow (f a2
-a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (let TMP_1 \def ((aprem_ind P f
-f0) i a2 a3 a4) in (f0 a2 a3 i a4 TMP_1 a5))].
+implied rec lemma aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall
+(a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2:
+A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to
+(\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A)
+(a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3)
+\Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4
+((aprem_ind P f f0) i a2 a3 a4) a5)].
-theorem aprem_gen_sort:
+lemma 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)).(let TMP_1 \def (ASort h n) in
-(let TMP_2 \def (\lambda (a: A).(aprem i a x)) in (let TMP_3 \def (\lambda
-(_: A).False) in (let TMP_13 \def (\lambda (y: A).(\lambda (H0: (aprem i y
-x)).(let TMP_4 \def (\lambda (_: nat).(\lambda (a: A).(\lambda (_: A).((eq A
-a (ASort h n)) \to False)))) in (let TMP_8 \def (\lambda (a1: A).(\lambda
-(a2: A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let TMP_5 \def
-(AHead a1 a2) in (let TMP_6 \def (\lambda (ee: A).(match ee with [(ASort _ _)
-\Rightarrow False | (AHead _ _) \Rightarrow True])) in (let TMP_7 \def (ASort
-h n) in (let H2 \def (eq_ind A TMP_5 TMP_6 I TMP_7 H1) in (False_ind False
-H2)))))))) in (let TMP_12 \def (\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 TMP_9 \def (AHead a1 a2) in (let TMP_10 \def (\lambda (ee:
-A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) in (let TMP_11 \def (ASort h n) in (let H4 \def (eq_ind A TMP_9
-TMP_10 I TMP_11 H3) in (False_ind False H4)))))))))))) in (aprem_ind TMP_4
-TMP_8 TMP_12 i y x H0)))))) in (insert_eq A TMP_1 TMP_2 TMP_3 TMP_13
-H))))))))).
+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 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 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:
+lemma 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)).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def (\lambda
-(a: A).(aprem O a x)) in (let TMP_3 \def (\lambda (_: A).(eq A x a1)) in (let
-TMP_29 \def (\lambda (y: A).(\lambda (H0: (aprem O y x)).(let TMP_4 \def
-(\lambda (n: nat).(aprem n y x)) in (let TMP_5 \def (\lambda (_: nat).((eq A
-y (AHead a1 a2)) \to (eq A x a1))) in (let TMP_28 \def (\lambda (y0:
-nat).(\lambda (H1: (aprem y0 y x)).(let TMP_6 \def (\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)))))) in (let TMP_14 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda
-(_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
-TMP_7 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead
-a _) \Rightarrow a])) in (let TMP_8 \def (AHead a0 a3) in (let TMP_9 \def
-(AHead a1 a2) in (let H4 \def (f_equal A A TMP_7 TMP_8 TMP_9 H3) in (let
-TMP_10 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 |
-(AHead _ a) \Rightarrow a])) in (let TMP_11 \def (AHead a0 a3) in (let TMP_12
-\def (AHead a1 a2) in (let H5 \def (f_equal A A TMP_10 TMP_11 TMP_12 H3) in
-(let TMP_13 \def (\lambda (H6: (eq A a0 a1)).H6) in (TMP_13 H4))))))))))))))
-in (let TMP_27 \def (\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 TMP_15 \def
-(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _)
-\Rightarrow a4])) in (let TMP_16 \def (AHead a3 a0) in (let TMP_17 \def
-(AHead a1 a2) in (let H6 \def (f_equal A A TMP_15 TMP_16 TMP_17 H5) in (let
-TMP_18 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 |
-(AHead _ a4) \Rightarrow a4])) in (let TMP_19 \def (AHead a3 a0) in (let
-TMP_20 \def (AHead a1 a2) in (let H7 \def (f_equal A A TMP_18 TMP_19 TMP_20
-H5) in (let TMP_26 \def (\lambda (_: (eq A a3 a1)).(let TMP_21 \def (\lambda
-(a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a a1)))) in (let
-H9 \def (eq_ind A a0 TMP_21 H3 a2 H7) in (let TMP_22 \def (\lambda (a4:
-A).(aprem i a4 a)) in (let H10 \def (eq_ind A a0 TMP_22 H2 a2 H7) in (let
-TMP_23 \def (S i) in (let TMP_24 \def (\lambda (ee: nat).(match ee with [O
-\Rightarrow False | (S _) \Rightarrow True])) in (let H11 \def (eq_ind nat
-TMP_23 TMP_24 I O H4) in (let TMP_25 \def (eq A a a1) in (False_ind TMP_25
-H11)))))))))) in (TMP_26 H6)))))))))))))))))) in (aprem_ind TMP_6 TMP_14
-TMP_27 y0 y x H1)))))) in (insert_eq nat O TMP_4 TMP_5 TMP_28 H0)))))) in
-(insert_eq A TMP_1 TMP_2 TMP_3 TMP_29 H)))))))).
+(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 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 with
+[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3)
+(AHead a1 a2) H3) in (\lambda (H6: (eq A a0 a1)).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 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 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
+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:
+lemma 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)).(let TMP_1 \def (AHead a1 a2) in (let
-TMP_3 \def (\lambda (a: A).(let TMP_2 \def (S i) in (aprem TMP_2 a x))) in
-(let TMP_4 \def (\lambda (_: A).(aprem i a2 x)) in (let TMP_38 \def (\lambda
-(y: A).(\lambda (H0: (aprem (S i) y x)).(let TMP_5 \def (S i) in (let TMP_6
-\def (\lambda (n: nat).(aprem n y x)) in (let TMP_7 \def (\lambda (_:
-nat).((eq A y (AHead a1 a2)) \to (aprem i a2 x))) in (let TMP_37 \def
-(\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(let TMP_8 \def (\lambda (n:
+(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)))))) in (let TMP_21 \def (\lambda (a0:
-A).(\lambda (a3: A).(\lambda (H2: (eq nat O (S i))).(\lambda (H3: (eq A
-(AHead a0 a3) (AHead a1 a2))).(let TMP_9 \def (\lambda (e: A).(match e with
-[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) in (let TMP_10
-\def (AHead a0 a3) in (let TMP_11 \def (AHead a1 a2) in (let H4 \def (f_equal
-A A TMP_9 TMP_10 TMP_11 H3) in (let TMP_12 \def (\lambda (e: A).(match e with
-[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) in (let TMP_13
-\def (AHead a0 a3) in (let TMP_14 \def (AHead a1 a2) in (let H5 \def (f_equal
-A A TMP_12 TMP_13 TMP_14 H3) in (let TMP_20 \def (\lambda (H6: (eq A a0
-a1)).(let TMP_15 \def (\lambda (a: A).(aprem i a2 a)) in (let TMP_16 \def
-(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow
-False])) in (let TMP_17 \def (S i) in (let H7 \def (eq_ind nat O TMP_16 I
-TMP_17 H2) in (let TMP_18 \def (aprem i a2 a1) in (let TMP_19 \def (False_ind
-TMP_18 H7) in (eq_ind_r A a1 TMP_15 TMP_19 a0 H6)))))))) in (TMP_20
-H4)))))))))))))) in (let TMP_36 \def (\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 TMP_22 \def (\lambda (e: A).(match e with [(ASort _ _)
-\Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) in (let TMP_23 \def (AHead a3
-a0) in (let TMP_24 \def (AHead a1 a2) in (let H6 \def (f_equal A A TMP_22
-TMP_23 TMP_24 H5) in (let TMP_25 \def (\lambda (e: A).(match e with [(ASort _
-_) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) in (let TMP_26 \def (AHead
-a3 a0) in (let TMP_27 \def (AHead a1 a2) in (let H7 \def (f_equal A A TMP_25
-TMP_26 TMP_27 H5) in (let TMP_35 \def (\lambda (_: (eq A a3 a1)).(let TMP_28
-\def (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to
-(aprem i a2 a)))) in (let H9 \def (eq_ind A a0 TMP_28 H3 a2 H7) in (let
-TMP_29 \def (\lambda (a4: A).(aprem i0 a4 a)) in (let H10 \def (eq_ind A a0
-TMP_29 H2 a2 H7) in (let TMP_30 \def (\lambda (e: nat).(match e with [O
-\Rightarrow i0 | (S n) \Rightarrow n])) in (let TMP_31 \def (S i0) in (let
-TMP_32 \def (S i) in (let H11 \def (f_equal nat nat TMP_30 TMP_31 TMP_32 H4)
-in (let TMP_33 \def (\lambda (n: nat).((eq nat n (S i)) \to ((eq A a2 (AHead
-a1 a2)) \to (aprem i a2 a)))) in (let H12 \def (eq_ind nat i0 TMP_33 H9 i
-H11) in (let TMP_34 \def (\lambda (n: nat).(aprem n a2 a)) in (let H13 \def
-(eq_ind nat i0 TMP_34 H10 i H11) in H13))))))))))))) in (TMP_35
-H6)))))))))))))))))) in (aprem_ind TMP_8 TMP_21 TMP_36 y0 y x H1)))))) in
-(insert_eq nat TMP_5 TMP_6 TMP_7 TMP_37 H0))))))) in (insert_eq A TMP_1 TMP_3
-TMP_4 TMP_38 H))))))))).
+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 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 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 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 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 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 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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