(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n
-h1)) (aplus g a2 (plus n h2)))).(sym_equal A (asucc g (aplus g a2 (plus n
-h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1
-(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus
-g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g
-g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq
-A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+h1)) (aplus g a2 (plus n h2)))).(sym_eq A (asucc g (aplus g a2 (plus n h2)))
+(asucc g (aplus g a1 (plus n h1))) (sym_eq A (asucc g (aplus g a1 (plus n
+h1))) (asucc g (aplus g a2 (plus n h2))) (sym_eq A (asucc g (aplus g a2 (plus
+n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g g (aplus g
+a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq A (aplus g
+a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
theorem aplus_assoc:
\forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n))
n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g
-(aplus g a n1)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0))))
-(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g
+(aplus g a n1)))) (sym_eq A (asucc g (asucc g (aplus g a (plus n n0))))
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_eq A (asucc g (aplus g
(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0))))
-(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
+(sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a
(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A
(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))
\lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h:
nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))
(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A
-(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
-\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0)
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in
nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
-(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0))
+(ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0))
\to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g
n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0))
-h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g
-(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
+h) (\lambda (a0: A).(eq A a0 (ASort O n0))) H0 (ASort (minus O h) (next_plus
+g (next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
-with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus
-(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S
-i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0)
-(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
-(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n:
-nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in
-(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
-n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
-(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1)
-\Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1)
-n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A
-a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
-(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
-| (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def
-(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m
-with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1
-h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0)
-H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def
-(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n
-i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g
-n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h)
-(S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
-(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda
-(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
-Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
-g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
-nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
-a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
-a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc
-g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with
-[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g
-(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g
-a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec
+next_plus (g0: G) (n1: nat) (i: nat) on i: nat \def (match i with [O
+\Rightarrow n1 | (S i0) \Rightarrow (next g0 (next_plus g0 n1 i0))]) in
+next_plus) g (next g n0) (minus h O))])) (ASort (minus O h) (next_plus g
+(next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def (eq_ind_r nat
+(minus h O) (\lambda (n1: nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h
+(minus_n_O h)) in (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat
+(next_plus g (next g n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0)
+h) n1)) (le_n (next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0)
+P))))) | (S n1) \Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h)
+(ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda
+(a0: A).(eq A a0 (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0
+(minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat
+(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _)
+\Rightarrow n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2:
+(nat \to nat) \def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k)
+\Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k
+l)])])) in minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
+(ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match
+e in A return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead
+_ _) \Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat
+\def (match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus
+g0 n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h)
+(next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat
+(minus n1 h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2:
+nat).(le n2 n1)) (minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda
+(a0: A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0)
+\to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h:
+nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P:
+Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc
+g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g
+(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1
+(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1)))
+in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n:
+nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc
+g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow
+a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3
+P)))))))))) a)).
theorem aplus_inj:
\forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
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