X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Faplus%2Fprops.ma;h=9eec9c6e83f510aef71bb0d9b342bc4f4ae96ec8;hb=639e798161afea770f41d78673c0fe3be4125beb;hp=7cd40dae7f499bd04052c5037bb324c368ace90c;hpb=15455aa487e001c643b4f46daf82612b8409f1ae;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
index 7cd40dae7..9eec9c6e8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
@@ -27,390 +27,214 @@ theorem aplus_reg_r:
 \def
  \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda 
 (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: 
-nat).(let TMP_5 \def (\lambda (n: nat).(let TMP_1 \def (plus n h1) in (let 
-TMP_2 \def (aplus g a1 TMP_1) in (let TMP_3 \def (plus n h2) in (let TMP_4 
-\def (aplus g a2 TMP_3) in (eq A TMP_2 TMP_4)))))) in (let TMP_11 \def 
-(\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n h1)) (aplus g a2 
-(plus n h2)))).(let TMP_6 \def (plus n h1) in (let TMP_7 \def (aplus g a1 
-TMP_6) in (let TMP_8 \def (plus n h2) in (let TMP_9 \def (aplus g a2 TMP_8) 
-in (let TMP_10 \def (refl_equal G g) in (f_equal2 G A A asucc g g TMP_7 TMP_9 
-TMP_10 H0)))))))) in (nat_ind TMP_5 H TMP_11 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)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n 
+h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
 
 theorem aplus_assoc:
  \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
 (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
 \def
- \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(let TMP_5 \def (\lambda 
-(n: nat).(\forall (h2: nat).(let TMP_1 \def (aplus g a n) in (let TMP_2 \def 
-(aplus g TMP_1 h2) in (let TMP_3 \def (plus n h2) in (let TMP_4 \def (aplus g 
-a TMP_3) in (eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (h2: 
-nat).(let TMP_6 \def (aplus g a h2) in (refl_equal A TMP_6))) in (let TMP_47 
-\def (\lambda (n: nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus 
-g a n) h2) (aplus g a (plus n h2)))))).(\lambda (h2: nat).(let TMP_14 \def 
-(\lambda (n0: nat).(let TMP_8 \def (aplus g a n) in (let TMP_9 \def (asucc g 
-TMP_8) in (let TMP_10 \def (aplus g TMP_9 n0) in (let TMP_11 \def (plus n n0) 
-in (let TMP_12 \def (aplus g a TMP_11) in (let TMP_13 \def (asucc g TMP_12) 
-in (eq A TMP_10 TMP_13)))))))) in (let TMP_19 \def (\lambda (n0: nat).(let 
-TMP_15 \def (aplus g a n) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17 
-\def (aplus g a n0) in (let TMP_18 \def (asucc g TMP_17) in (eq A TMP_16 
-TMP_18)))))) in (let TMP_20 \def (aplus g a n) in (let TMP_21 \def (asucc g 
-TMP_20) in (let TMP_22 \def (refl_equal A TMP_21) in (let TMP_23 \def (plus n 
-O) in (let TMP_24 \def (plus_n_O n) in (let TMP_25 \def (eq_ind nat n TMP_19 
-TMP_22 TMP_23 TMP_24) in (let TMP_46 \def (\lambda (n0: nat).(\lambda (H0: 
-(eq A (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n 
-n0))))).(let TMP_26 \def (plus n n0) in (let TMP_27 \def (S TMP_26) in (let 
-TMP_34 \def (\lambda (n1: nat).(let TMP_28 \def (aplus g a n) in (let TMP_29 
-\def (asucc g TMP_28) in (let TMP_30 \def (aplus g TMP_29 n0) in (let TMP_31 
-\def (asucc g TMP_30) in (let TMP_32 \def (aplus g a n1) in (let TMP_33 \def 
-(asucc g TMP_32) in (eq A TMP_31 TMP_33)))))))) in (let TMP_35 \def (aplus g 
-a n) in (let TMP_36 \def (asucc g TMP_35) in (let TMP_37 \def (aplus g TMP_36 
-n0) in (let TMP_38 \def (plus n n0) in (let TMP_39 \def (aplus g a TMP_38) in 
-(let TMP_40 \def (asucc g TMP_39) in (let TMP_41 \def (refl_equal G g) in 
-(let TMP_42 \def (f_equal2 G A A asucc g g TMP_37 TMP_40 TMP_41 H0) in (let 
-TMP_43 \def (S n0) in (let TMP_44 \def (plus n TMP_43) in (let TMP_45 \def 
-(plus_n_Sm n n0) in (eq_ind nat TMP_27 TMP_34 TMP_42 TMP_44 
-TMP_45))))))))))))))))) in (nat_ind TMP_14 TMP_25 TMP_46 h2))))))))))))) in 
-(nat_ind TMP_5 TMP_7 TMP_47 h1)))))).
+ \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: 
+nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n 
+h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: 
+nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus 
+g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A 
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) 
+(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g 
+(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O 
+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)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) 
+n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) 
+(plus_n_Sm n n0)))) h2)))) h1))).
 
 theorem aplus_asucc:
  \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) 
 h) (asucc g (aplus g a h)))))
 \def
- \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(let TMP_1 \def (S O) in 
-(let TMP_2 \def (plus TMP_1 h) in (let TMP_3 \def (aplus g a TMP_2) in (let 
-TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (aplus g a h) in (let TMP_5 \def 
-(asucc g TMP_4) in (eq A a0 TMP_5)))) in (let TMP_7 \def (aplus g a h) in 
-(let TMP_8 \def (asucc g TMP_7) in (let TMP_9 \def (refl_equal A TMP_8) in 
-(let TMP_10 \def (S O) in (let TMP_11 \def (aplus g a TMP_10) in (let TMP_12 
-\def (aplus g TMP_11 h) in (let TMP_13 \def (S O) in (let TMP_14 \def 
-(aplus_assoc g a TMP_13 h) in (eq_ind_r A TMP_3 TMP_6 TMP_9 TMP_12 
-TMP_14))))))))))))))).
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a 
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) 
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) 
+(aplus_assoc g a (S O) h)))).
 
 theorem aplus_sort_O_S_simpl:
  \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O 
 n) (S k)) (aplus g (ASort O (next g n)) k))))
 \def
- \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(let TMP_1 \def (ASort O 
-n) in (let TMP_2 \def (asucc g TMP_1) in (let TMP_3 \def (aplus g TMP_2 k) in 
-(let TMP_7 \def (\lambda (a: A).(let TMP_4 \def (next g n) in (let TMP_5 \def 
-(ASort O TMP_4) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A a TMP_6))))) in 
-(let TMP_8 \def (next g n) in (let TMP_9 \def (ASort O TMP_8) in (let TMP_10 
-\def (aplus g TMP_9 k) in (let TMP_11 \def (refl_equal A TMP_10) in (let 
-TMP_12 \def (ASort O n) in (let TMP_13 \def (aplus g TMP_12 k) in (let TMP_14 
-\def (asucc g TMP_13) in (let TMP_15 \def (ASort O n) in (let TMP_16 \def 
-(aplus_asucc g k TMP_15) in (eq_ind A TMP_3 TMP_7 TMP_11 TMP_14 
-TMP_16)))))))))))))))).
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc 
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) 
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) 
+k)) (aplus_asucc g k (ASort O n))))).
 
 theorem aplus_sort_S_S_simpl:
  \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A 
 (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
 \def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(let 
-TMP_1 \def (S h) in (let TMP_2 \def (ASort TMP_1 n) in (let TMP_3 \def (asucc 
-g TMP_2) in (let TMP_4 \def (aplus g TMP_3 k) in (let TMP_7 \def (\lambda (a: 
-A).(let TMP_5 \def (ASort h n) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A 
-a TMP_6)))) in (let TMP_8 \def (ASort h n) in (let TMP_9 \def (aplus g TMP_8 
-k) in (let TMP_10 \def (refl_equal A TMP_9) in (let TMP_11 \def (S h) in (let 
-TMP_12 \def (ASort TMP_11 n) in (let TMP_13 \def (aplus g TMP_12 k) in (let 
-TMP_14 \def (asucc g TMP_13) in (let TMP_15 \def (S h) in (let TMP_16 \def 
-(ASort TMP_15 n) in (let TMP_17 \def (aplus_asucc g k TMP_16) in (eq_ind A 
-TMP_4 TMP_7 TMP_10 TMP_14 TMP_17))))))))))))))))))).
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind 
+A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g 
+(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g 
+(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
 
 theorem aplus_asort_O_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O 
 n) h) (ASort O (next_plus g n h)))))
 \def
- \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall 
-(n0: nat).(let TMP_1 \def (ASort O n0) in (let TMP_2 \def (aplus g TMP_1 n) 
-in (let TMP_3 \def (next_plus g n0 n) in (let TMP_4 \def (ASort O TMP_3) in 
-(eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (n: nat).(let TMP_6 \def 
-(ASort O n) in (refl_equal A TMP_6))) in (let TMP_33 \def (\lambda (n: 
-nat).(\lambda (H: ((\forall (n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O 
-(next_plus g n0 n)))))).(\lambda (n0: nat).(let TMP_8 \def (ASort O n0) in 
-(let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let 
-TMP_14 \def (\lambda (a: A).(let TMP_11 \def (next_plus g n0 n) in (let 
-TMP_12 \def (next g TMP_11) in (let TMP_13 \def (ASort O TMP_12) in (eq A a 
-TMP_13))))) in (let TMP_15 \def (next g n0) in (let TMP_16 \def (next_plus g 
-TMP_15 n) in (let TMP_21 \def (\lambda (n1: nat).(let TMP_17 \def (next g n0) 
-in (let TMP_18 \def (ASort O TMP_17) in (let TMP_19 \def (aplus g TMP_18 n) 
-in (let TMP_20 \def (ASort O n1) in (eq A TMP_19 TMP_20)))))) in (let TMP_22 
-\def (next g n0) in (let TMP_23 \def (H TMP_22) in (let TMP_24 \def 
-(next_plus g n0 n) in (let TMP_25 \def (next g TMP_24) in (let TMP_26 \def 
-(next_plus_next g n0 n) in (let TMP_27 \def (eq_ind nat TMP_16 TMP_21 TMP_23 
-TMP_25 TMP_26) in (let TMP_28 \def (ASort O n0) in (let TMP_29 \def (aplus g 
-TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (ASort O 
-n0) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14 
-TMP_27 TMP_30 TMP_32)))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33 
-h))))).
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: 
+nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda 
+(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall 
+(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 
+n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) 
+(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat 
+(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next 
+g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) 
+(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n 
+(ASort O n0)))))) h)).
 
 theorem aplus_asort_le_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h 
 k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
 \def
- \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall 
-(k: nat).(\forall (n0: nat).((le n k) \to (let TMP_1 \def (ASort k n0) in 
-(let TMP_2 \def (aplus g TMP_1 n) in (let TMP_3 \def (minus k n) in (let 
-TMP_4 \def (ASort TMP_3 n0) in (eq A TMP_2 TMP_4))))))))) in (let TMP_13 \def 
-(\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O k)).(let TMP_8 \def 
-(\lambda (n0: nat).(let TMP_6 \def (ASort k n) in (let TMP_7 \def (ASort n0 
-n) in (eq A TMP_6 TMP_7)))) in (let TMP_9 \def (ASort k n) in (let TMP_10 
-\def (refl_equal A TMP_9) in (let TMP_11 \def (minus k O) in (let TMP_12 \def 
-(minus_n_O k) in (eq_ind nat k TMP_8 TMP_10 TMP_11 TMP_12))))))))) in (let 
-TMP_62 \def (\lambda (h0: nat).(\lambda (H: ((\forall (k: nat).(\forall (n: 
-nat).((le h0 k) \to (eq A (aplus g (ASort k n) h0) (ASort (minus k h0) 
-n))))))).(\lambda (k: nat).(let TMP_20 \def (\lambda (n: nat).(\forall (n0: 
-nat).((le (S h0) n) \to (let TMP_14 \def (ASort n n0) in (let TMP_15 \def 
-(aplus g TMP_14 h0) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17 \def 
-(S h0) in (let TMP_18 \def (minus n TMP_17) in (let TMP_19 \def (ASort TMP_18 
-n0) in (eq A TMP_16 TMP_19)))))))))) in (let TMP_42 \def (\lambda (n: 
-nat).(\lambda (H0: (le (S h0) O)).(let TMP_22 \def (\lambda (n0: nat).(let 
-TMP_21 \def (S n0) in (eq nat O TMP_21))) in (let TMP_23 \def (\lambda (n0: 
-nat).(le h0 n0)) in (let TMP_24 \def (ASort O n) in (let TMP_25 \def (aplus g 
-TMP_24 h0) in (let TMP_26 \def (asucc g TMP_25) in (let TMP_27 \def (S h0) in 
-(let TMP_28 \def (minus O TMP_27) in (let TMP_29 \def (ASort TMP_28 n) in 
-(let TMP_30 \def (eq A TMP_26 TMP_29) in (let TMP_40 \def (\lambda (x: 
-nat).(\lambda (H1: (eq nat O (S x))).(\lambda (_: (le h0 x)).(let TMP_31 \def 
-(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow 
-False])) in (let TMP_32 \def (S x) in (let H3 \def (eq_ind nat O TMP_31 I 
-TMP_32 H1) in (let TMP_33 \def (ASort O n) in (let TMP_34 \def (aplus g 
-TMP_33 h0) in (let TMP_35 \def (asucc g TMP_34) in (let TMP_36 \def (S h0) in 
-(let TMP_37 \def (minus O TMP_36) in (let TMP_38 \def (ASort TMP_37 n) in 
-(let TMP_39 \def (eq A TMP_35 TMP_38) in (False_ind TMP_39 H3)))))))))))))) 
-in (let TMP_41 \def (le_gen_S h0 O H0) in (ex2_ind nat TMP_22 TMP_23 TMP_30 
-TMP_40 TMP_41)))))))))))))) in (let TMP_61 \def (\lambda (n: nat).(\lambda 
-(_: ((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n 
-n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: 
-(le (S h0) (S n))).(let TMP_43 \def (S n) in (let TMP_44 \def (ASort TMP_43 
-n0) in (let TMP_45 \def (asucc g TMP_44) in (let TMP_46 \def (aplus g TMP_45 
-h0) in (let TMP_51 \def (\lambda (a: A).(let TMP_47 \def (S n) in (let TMP_48 
-\def (S h0) in (let TMP_49 \def (minus TMP_47 TMP_48) in (let TMP_50 \def 
-(ASort TMP_49 n0) in (eq A a TMP_50)))))) in (let TMP_52 \def (le_S_n h0 n 
-H1) in (let TMP_53 \def (H n n0 TMP_52) in (let TMP_54 \def (S n) in (let 
-TMP_55 \def (ASort TMP_54 n0) in (let TMP_56 \def (aplus g TMP_55 h0) in (let 
-TMP_57 \def (asucc g TMP_56) in (let TMP_58 \def (S n) in (let TMP_59 \def 
-(ASort TMP_58 n0) in (let TMP_60 \def (aplus_asucc g h0 TMP_59) in (eq_ind A 
-TMP_46 TMP_51 TMP_53 TMP_57 TMP_60))))))))))))))))))) in (nat_ind TMP_20 
-TMP_42 TMP_61 k))))))) in (nat_ind TMP_5 TMP_13 TMP_62 h))))).
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: 
+nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort 
+(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O 
+k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) 
+(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: 
+nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A 
+(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: 
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A 
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda 
+(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat 
+O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) 
+h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S 
+x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) 
+H1) in (False_ind (eq A (asucc g (aplus g (ASort O n) h0)) (ASort (minus O (S 
+h0)) n)) H3))))) (le_gen_S h0 O H0)))) (\lambda (n: nat).(\lambda (_: 
+((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n n0) 
+h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: (le (S 
+h0) (S n))).(eq_ind A (aplus g (asucc g (ASort (S n) n0)) h0) (\lambda (a: 
+A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n n0 (le_S_n h0 n H1)) (asucc 
+g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g h0 (ASort (S n) n0))))))) 
+k)))) h)).
 
 theorem aplus_asort_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A 
 (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
 \def
- \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: nat).(let 
-TMP_1 \def (ASort k n) in (let TMP_2 \def (aplus g TMP_1 h) in (let TMP_3 
-\def (minus k h) in (let TMP_4 \def (minus h k) in (let TMP_5 \def (next_plus 
-g n TMP_4) in (let TMP_6 \def (ASort TMP_3 TMP_5) in (let TMP_7 \def (eq A 
-TMP_2 TMP_6) in (let TMP_92 \def (\lambda (H: (lt k h)).(let TMP_8 \def 
-(minus h k) in (let TMP_9 \def (plus k TMP_8) in (let TMP_16 \def (\lambda 
-(n0: nat).(let TMP_10 \def (ASort k n) in (let TMP_11 \def (aplus g TMP_10 
-n0) in (let TMP_12 \def (minus k h) in (let TMP_13 \def (minus h k) in (let 
-TMP_14 \def (next_plus g n TMP_13) in (let TMP_15 \def (ASort TMP_12 TMP_14) 
-in (eq A TMP_11 TMP_15)))))))) in (let TMP_17 \def (ASort k n) in (let TMP_18 
-\def (aplus g TMP_17 k) in (let TMP_19 \def (minus h k) in (let TMP_20 \def 
-(aplus g TMP_18 TMP_19) in (let TMP_25 \def (\lambda (a: A).(let TMP_21 \def 
-(minus k h) in (let TMP_22 \def (minus h k) in (let TMP_23 \def (next_plus g 
-n TMP_22) in (let TMP_24 \def (ASort TMP_21 TMP_23) in (eq A a TMP_24)))))) 
-in (let TMP_26 \def (minus k k) in (let TMP_27 \def (ASort TMP_26 n) in (let 
-TMP_34 \def (\lambda (a: A).(let TMP_28 \def (minus h k) in (let TMP_29 \def 
-(aplus g a TMP_28) in (let TMP_30 \def (minus k h) in (let TMP_31 \def (minus 
-h k) in (let TMP_32 \def (next_plus g n TMP_31) in (let TMP_33 \def (ASort 
-TMP_30 TMP_32) in (eq A TMP_29 TMP_33)))))))) in (let TMP_42 \def (\lambda 
-(n0: nat).(let TMP_35 \def (ASort n0 n) in (let TMP_36 \def (minus h k) in 
-(let TMP_37 \def (aplus g TMP_35 TMP_36) in (let TMP_38 \def (minus k h) in 
-(let TMP_39 \def (minus h k) in (let TMP_40 \def (next_plus g n TMP_39) in 
-(let TMP_41 \def (ASort TMP_38 TMP_40) in (eq A TMP_37 TMP_41))))))))) in 
-(let TMP_49 \def (\lambda (n0: nat).(let TMP_43 \def (ASort O n) in (let 
-TMP_44 \def (minus h k) in (let TMP_45 \def (aplus g TMP_43 TMP_44) in (let 
-TMP_46 \def (minus h k) in (let TMP_47 \def (next_plus g n TMP_46) in (let 
-TMP_48 \def (ASort n0 TMP_47) in (eq A TMP_45 TMP_48)))))))) in (let TMP_50 
-\def (minus h k) in (let TMP_51 \def (aplus_asort_O_simpl g TMP_50 n) in (let 
-TMP_52 \def (minus k h) in (let TMP_53 \def (S k) in (let TMP_54 \def (S h) 
-in (let TMP_55 \def (S k) in (let TMP_56 \def (S TMP_55) in (let TMP_57 \def 
-(S h) in (let TMP_58 \def (S k) in (let TMP_59 \def (le_n_S TMP_58 h H) in 
-(let TMP_60 \def (le_S TMP_56 TMP_57 TMP_59) in (let TMP_61 \def (le_S_n 
-TMP_53 TMP_54 TMP_60) in (let TMP_62 \def (le_S_n k h TMP_61) in (let TMP_63 
-\def (O_minus k h TMP_62) in (let TMP_64 \def (eq_ind_r nat O TMP_49 TMP_51 
-TMP_52 TMP_63) in (let TMP_65 \def (minus k k) in (let TMP_66 \def (minus_n_n 
-k) in (let TMP_67 \def (eq_ind nat O TMP_42 TMP_64 TMP_65 TMP_66) in (let 
-TMP_68 \def (ASort k n) in (let TMP_69 \def (aplus g TMP_68 k) in (let TMP_70 
-\def (le_n k) in (let TMP_71 \def (aplus_asort_le_simpl g k k n TMP_70) in 
-(let TMP_72 \def (eq_ind_r A TMP_27 TMP_34 TMP_67 TMP_69 TMP_71) in (let 
-TMP_73 \def (ASort k n) in (let TMP_74 \def (minus h k) in (let TMP_75 \def 
-(plus k TMP_74) in (let TMP_76 \def (aplus g TMP_73 TMP_75) in (let TMP_77 
-\def (ASort k n) in (let TMP_78 \def (minus h k) in (let TMP_79 \def 
-(aplus_assoc g TMP_77 k TMP_78) in (let TMP_80 \def (eq_ind A TMP_20 TMP_25 
-TMP_72 TMP_76 TMP_79) in (let TMP_81 \def (S k) in (let TMP_82 \def (S h) in 
-(let TMP_83 \def (S k) in (let TMP_84 \def (S TMP_83) in (let TMP_85 \def (S 
-h) in (let TMP_86 \def (S k) in (let TMP_87 \def (le_n_S TMP_86 h H) in (let 
-TMP_88 \def (le_S TMP_84 TMP_85 TMP_87) in (let TMP_89 \def (le_S_n TMP_81 
-TMP_82 TMP_88) in (let TMP_90 \def (le_S_n k h TMP_89) in (let TMP_91 \def 
-(le_plus_minus k h TMP_90) in (eq_ind_r nat TMP_9 TMP_16 TMP_80 h 
-TMP_91))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
-TMP_116 \def (\lambda (H: (le h k)).(let TMP_93 \def (minus k h) in (let 
-TMP_94 \def (ASort TMP_93 n) in (let TMP_99 \def (\lambda (a: A).(let TMP_95 
-\def (minus k h) in (let TMP_96 \def (minus h k) in (let TMP_97 \def 
-(next_plus g n TMP_96) in (let TMP_98 \def (ASort TMP_95 TMP_97) in (eq A a 
-TMP_98)))))) in (let TMP_105 \def (\lambda (n0: nat).(let TMP_100 \def (minus 
-k h) in (let TMP_101 \def (ASort TMP_100 n) in (let TMP_102 \def (minus k h) 
-in (let TMP_103 \def (next_plus g n n0) in (let TMP_104 \def (ASort TMP_102 
-TMP_103) in (eq A TMP_101 TMP_104))))))) in (let TMP_106 \def (minus k h) in 
-(let TMP_107 \def (next_plus g n O) in (let TMP_108 \def (ASort TMP_106 
-TMP_107) in (let TMP_109 \def (refl_equal A TMP_108) in (let TMP_110 \def 
-(minus h k) in (let TMP_111 \def (O_minus h k H) in (let TMP_112 \def 
-(eq_ind_r nat O TMP_105 TMP_109 TMP_110 TMP_111) in (let TMP_113 \def (ASort 
-k n) in (let TMP_114 \def (aplus g TMP_113 h) in (let TMP_115 \def 
-(aplus_asort_le_simpl g h k n H) in (eq_ind_r A TMP_94 TMP_99 TMP_112 TMP_114 
-TMP_115)))))))))))))))) in (lt_le_e k h TMP_7 TMP_92 TMP_116))))))))))))).
+ \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: 
+nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus 
+g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) 
+(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) 
+(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) 
+(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus 
+h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a 
+(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O 
+(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k 
+h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A 
+(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) 
+(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h 
+(le_S_n (S k) (S h) (le_S (S (S k)) (S h) (le_n_S (S k) h H)))))) (minus k k) 
+(minus_n_n k)) (aplus g (ASort k n) k) (aplus_asort_le_simpl g k k n (le_n 
+k))) (aplus g (ASort k n) (plus k (minus h k))) (aplus_assoc g (ASort k n) k 
+(minus h k))) h (le_plus_minus k h (le_S_n k h (le_S_n (S k) (S h) (le_S (S 
+(S k)) (S h) (le_n_S (S k) h H))))))) (\lambda (H: (le h k)).(eq_ind_r A 
+(ASort (minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g 
+n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) 
+n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) 
+(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) 
+(aplus_asort_le_simpl g h k n H))))))).
 
 theorem aplus_ahead_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A 
 (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
 \def
- \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall 
-(a1: A).(\forall (a2: A).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def 
-(aplus g TMP_1 n) in (let TMP_3 \def (aplus g a2 n) in (let TMP_4 \def (AHead 
-a1 TMP_3) in (eq A TMP_2 TMP_4)))))))) in (let TMP_7 \def (\lambda (a1: 
-A).(\lambda (a2: A).(let TMP_6 \def (AHead a1 a2) in (refl_equal A TMP_6)))) 
-in (let TMP_33 \def (\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall 
-(a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 
-n))))))).(\lambda (a1: A).(\lambda (a2: A).(let TMP_8 \def (AHead a1 a2) in 
-(let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let 
-TMP_14 \def (\lambda (a: A).(let TMP_11 \def (aplus g a2 n) in (let TMP_12 
-\def (asucc g TMP_11) in (let TMP_13 \def (AHead a1 TMP_12) in (eq A a 
-TMP_13))))) in (let TMP_15 \def (asucc g a2) in (let TMP_16 \def (aplus g 
-TMP_15 n) in (let TMP_21 \def (\lambda (a: A).(let TMP_17 \def (AHead a1 a2) 
-in (let TMP_18 \def (asucc g TMP_17) in (let TMP_19 \def (aplus g TMP_18 n) 
-in (let TMP_20 \def (AHead a1 a) in (eq A TMP_19 TMP_20)))))) in (let TMP_22 
-\def (asucc g a2) in (let TMP_23 \def (H a1 TMP_22) in (let TMP_24 \def 
-(aplus g a2 n) in (let TMP_25 \def (asucc g TMP_24) in (let TMP_26 \def 
-(aplus_asucc g n a2) in (let TMP_27 \def (eq_ind A TMP_16 TMP_21 TMP_23 
-TMP_25 TMP_26) in (let TMP_28 \def (AHead a1 a2) in (let TMP_29 \def (aplus g 
-TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (AHead a1 
-a2) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14 
-TMP_27 TMP_30 TMP_32))))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33 
-h))))).
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: 
+A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 
+n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A 
+(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: 
+A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda 
+(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g 
+(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) 
+(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n 
+a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) 
+h)).
 
 theorem aplus_asucc_false:
  \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) 
 h) a) \to (\forall (P: Prop).P))))
 \def
- \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).(\forall (h: 
-nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) in (let 
-TMP_70 \def (\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 
-h0) \Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(let 
-TMP_2 \def (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow 
-(ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) 
-\to P)) in (let TMP_36 \def (\lambda (H0: (eq A (aplus g (ASort O (next g 
-n0)) h) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def (ASort 
-O TMP_3) in (let TMP_5 \def (aplus g TMP_4 h) in (let TMP_7 \def (\lambda 
-(a0: A).(let TMP_6 \def (ASort O n0) in (eq A a0 TMP_6))) in (let TMP_8 \def 
-(minus O h) in (let TMP_9 \def (next g n0) in (let TMP_10 \def (minus h O) in 
-(let TMP_11 \def (next_plus g TMP_9 TMP_10) in (let TMP_12 \def (ASort TMP_8 
-TMP_11) in (let TMP_13 \def (next g n0) in (let TMP_14 \def 
-(aplus_asort_simpl g h O TMP_13) in (let H1 \def (eq_ind A TMP_5 TMP_7 H0 
-TMP_12 TMP_14) in (let TMP_18 \def (\lambda (e: A).(match e with [(ASort _ 
-n1) \Rightarrow n1 | (AHead _ _) \Rightarrow (let TMP_16 \def (next g n0) in 
-(let TMP_17 \def (minus h O) in (next_plus g TMP_16 TMP_17)))])) in (let 
-TMP_19 \def (minus O h) in (let TMP_20 \def (next g n0) in (let TMP_21 \def 
-(minus h O) in (let TMP_22 \def (next_plus g TMP_20 TMP_21) in (let TMP_23 
-\def (ASort TMP_19 TMP_22) in (let TMP_24 \def (ASort O n0) in (let H2 \def 
-(f_equal A nat TMP_18 TMP_23 TMP_24 H1) in (let TMP_25 \def (minus h O) in 
-(let TMP_28 \def (\lambda (n1: nat).(let TMP_26 \def (next g n0) in (let 
-TMP_27 \def (next_plus g TMP_26 n1) in (eq nat TMP_27 n0)))) in (let TMP_29 
-\def (minus_n_O h) in (let H3 \def (eq_ind_r nat TMP_25 TMP_28 H2 h TMP_29) 
-in (let TMP_30 \def (le_n n0) in (let TMP_31 \def (next g n0) in (let TMP_32 
-\def (next_plus g TMP_31 h) in (let TMP_33 \def (\lambda (n1: nat).(lt n0 
-n1)) in (let TMP_34 \def (next_plus_lt g h n0) in (let TMP_35 \def (eq_ind 
-nat TMP_32 TMP_33 TMP_34 n0 H3) in (le_lt_false n0 n0 TMP_30 TMP_35 
-P)))))))))))))))))))))))))))))))) in (let TMP_69 \def (\lambda (n1: 
-nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort O 
-(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to 
-P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let 
-TMP_37 \def (ASort n1 n0) in (let TMP_38 \def (aplus g TMP_37 h) in (let 
-TMP_41 \def (\lambda (a0: A).(let TMP_39 \def (S n1) in (let TMP_40 \def 
-(ASort TMP_39 n0) in (eq A a0 TMP_40)))) in (let TMP_42 \def (minus n1 h) in 
-(let TMP_43 \def (minus h n1) in (let TMP_44 \def (next_plus g n0 TMP_43) in 
-(let TMP_45 \def (ASort TMP_42 TMP_44) in (let TMP_46 \def (aplus_asort_simpl 
-g h n1 n0) in (let H1 \def (eq_ind A TMP_38 TMP_41 H0 TMP_45 TMP_46) in (let 
-TMP_47 \def (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
-(AHead _ _) \Rightarrow (minus n1 h)])) in (let TMP_48 \def (minus n1 h) in 
-(let TMP_49 \def (minus h n1) in (let TMP_50 \def (next_plus g n0 TMP_49) in 
-(let TMP_51 \def (ASort TMP_48 TMP_50) in (let TMP_52 \def (S n1) in (let 
-TMP_53 \def (ASort TMP_52 n0) in (let H2 \def (f_equal A nat TMP_47 TMP_51 
-TMP_53 H1) in (let TMP_56 \def (\lambda (e: A).(match e with [(ASort _ n2) 
-\Rightarrow n2 | (AHead _ _) \Rightarrow (let TMP_55 \def (minus h n1) in 
-(next_plus g n0 TMP_55))])) in (let TMP_57 \def (minus n1 h) in (let TMP_58 
-\def (minus h n1) in (let TMP_59 \def (next_plus g n0 TMP_58) in (let TMP_60 
-\def (ASort TMP_57 TMP_59) in (let TMP_61 \def (S n1) in (let TMP_62 \def 
-(ASort TMP_61 n0) in (let H3 \def (f_equal A nat TMP_56 TMP_60 TMP_62 H1) in 
-(let TMP_68 \def (\lambda (H4: (eq nat (minus n1 h) (S n1))).(let TMP_63 \def 
-(minus n1 h) in (let TMP_64 \def (\lambda (n2: nat).(le n2 n1)) in (let 
-TMP_65 \def (minus_le n1 h) in (let TMP_66 \def (S n1) in (let TMP_67 \def 
-(eq_ind nat TMP_63 TMP_64 TMP_65 TMP_66 H4) in (le_Sx_x n1 TMP_67 P))))))) in 
-(TMP_68 H2)))))))))))))))))))))))))))))) in (nat_ind TMP_2 TMP_36 TMP_69 n 
-H))))))))) in (let TMP_88 \def (\lambda (a0: A).(\lambda (_: ((\forall (h: 
-nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: 
+ \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 h0) 
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
+(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O 
+(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) 
+(\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 (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 with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) 
+\Rightarrow (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 n0 n0 (le_n n0) (eq_ind nat 
+(next_plus g (next g n0) h) (\lambda (n1: nat).(lt n0 n1)) (next_plus_lt g h 
+n0) n0 H3) P))))) (\lambda (n1: nat).(\lambda (_: (((eq A (aplus g (match n1 
+with [O \Rightarrow (ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 
+n0)]) h) (ASort n1 n0)) \to P))).(\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 with [(ASort n2 _) \Rightarrow n2 | 
+(AHead _ _) \Rightarrow (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 with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow 
+(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)))))) n 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 TMP_71 \def (asucc g a1) in (let TMP_72 \def 
-(AHead a0 TMP_71) in (let TMP_73 \def (aplus g TMP_72 h) in (let TMP_75 \def 
-(\lambda (a2: A).(let TMP_74 \def (AHead a0 a1) in (eq A a2 TMP_74))) in (let 
-TMP_76 \def (asucc g a1) in (let TMP_77 \def (aplus g TMP_76 h) in (let 
-TMP_78 \def (AHead a0 TMP_77) in (let TMP_79 \def (asucc g a1) in (let TMP_80 
-\def (aplus_ahead_simpl g h a0 TMP_79) in (let H2 \def (eq_ind A TMP_73 
-TMP_75 H1 TMP_78 TMP_80) in (let TMP_83 \def (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow (let TMP_82 \def (asucc g a1) in (aplus g TMP_82 h)) 
-| (AHead _ a2) \Rightarrow a2])) in (let TMP_84 \def (asucc g a1) in (let 
-TMP_85 \def (aplus g TMP_84 h) in (let TMP_86 \def (AHead a0 TMP_85) in (let 
-TMP_87 \def (AHead a0 a1) in (let H3 \def (f_equal A A TMP_83 TMP_86 TMP_87 
-H2) in (H0 h H3 P)))))))))))))))))))))))) in (A_ind TMP_1 TMP_70 TMP_88 
-a))))).
+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 with [(ASort _ _) \Rightarrow (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 
 (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
 \def
- \lambda (g: G).(\lambda (h1: nat).(let TMP_1 \def (\lambda (n: nat).(\forall 
-(h2: nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
-h2))))) in (let TMP_16 \def (\lambda (h2: nat).(let TMP_2 \def (\lambda (n: 
-nat).(\forall (a: A).((eq A (aplus g a O) (aplus g a n)) \to (eq nat O n)))) 
-in (let TMP_3 \def (\lambda (a: A).(\lambda (_: (eq A a a)).(refl_equal nat 
-O))) in (let TMP_15 \def (\lambda (n: nat).(\lambda (_: ((\forall (a: A).((eq 
-A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: (eq A a 
-(asucc g (aplus g a n)))).(let TMP_4 \def (aplus g a n) in (let TMP_5 \def 
-(asucc g TMP_4) in (let TMP_6 \def (\lambda (a0: A).(eq A a a0)) in (let 
-TMP_7 \def (asucc g a) in (let TMP_8 \def (aplus g TMP_7 n) in (let TMP_9 
-\def (aplus_asucc g n a) in (let H1 \def (eq_ind_r A TMP_5 TMP_6 H0 TMP_8 
-TMP_9) in (let TMP_10 \def (asucc g a) in (let TMP_11 \def (aplus g TMP_10 n) 
-in (let TMP_12 \def (sym_eq A a TMP_11 H1) in (let TMP_13 \def (S n) in (let 
-TMP_14 \def (eq nat O TMP_13) in (aplus_asucc_false g a n TMP_12 
-TMP_14))))))))))))))))) in (nat_ind TMP_2 TMP_3 TMP_15 h2))))) in (let TMP_47 
-\def (\lambda (n: nat).(\lambda (H: ((\forall (h2: nat).(\forall (a: A).((eq 
-A (aplus g a n) (aplus g a h2)) \to (eq nat n h2)))))).(\lambda (h2: 
-nat).(let TMP_18 \def (\lambda (n0: nat).(\forall (a: A).((eq A (aplus g a (S 
-n)) (aplus g a n0)) \to (let TMP_17 \def (S n) in (eq nat TMP_17 n0))))) in 
-(let TMP_27 \def (\lambda (a: A).(\lambda (H0: (eq A (asucc g (aplus g a n)) 
-a)).(let TMP_19 \def (aplus g a n) in (let TMP_20 \def (asucc g TMP_19) in 
-(let TMP_21 \def (\lambda (a0: A).(eq A a0 a)) in (let TMP_22 \def (asucc g 
-a) in (let TMP_23 \def (aplus g TMP_22 n) in (let TMP_24 \def (aplus_asucc g 
-n a) in (let H1 \def (eq_ind_r A TMP_20 TMP_21 H0 TMP_23 TMP_24) in (let 
-TMP_25 \def (S n) in (let TMP_26 \def (eq nat TMP_25 O) in (aplus_asucc_false 
-g a n H1 TMP_26)))))))))))) in (let TMP_46 \def (\lambda (n0: nat).(\lambda 
-(_: ((\forall (a: A).((eq A (asucc g (aplus g a n)) (aplus g a n0)) \to (eq 
-nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: (eq A (asucc g (aplus g a n)) 
-(asucc g (aplus g a n0)))).(let TMP_28 \def (aplus g a n) in (let TMP_29 \def 
-(asucc g TMP_28) in (let TMP_32 \def (\lambda (a0: A).(let TMP_30 \def (aplus 
-g a n0) in (let TMP_31 \def (asucc g TMP_30) in (eq A a0 TMP_31)))) in (let 
-TMP_33 \def (asucc g a) in (let TMP_34 \def (aplus g TMP_33 n) in (let TMP_35 
-\def (aplus_asucc g n a) in (let H2 \def (eq_ind_r A TMP_29 TMP_32 H1 TMP_34 
-TMP_35) in (let TMP_36 \def (aplus g a n0) in (let TMP_37 \def (asucc g 
-TMP_36) in (let TMP_40 \def (\lambda (a0: A).(let TMP_38 \def (asucc g a) in 
-(let TMP_39 \def (aplus g TMP_38 n) in (eq A TMP_39 a0)))) in (let TMP_41 
-\def (asucc g a) in (let TMP_42 \def (aplus g TMP_41 n0) in (let TMP_43 \def 
-(aplus_asucc g n0 a) in (let H3 \def (eq_ind_r A TMP_37 TMP_40 H2 TMP_42 
-TMP_43) in (let TMP_44 \def (asucc g a) in (let TMP_45 \def (H n0 TMP_44 H3) 
-in (f_equal nat nat S n n0 TMP_45))))))))))))))))))))) in (nat_ind TMP_18 
-TMP_27 TMP_46 h2))))))) in (nat_ind TMP_1 TMP_16 TMP_47 h1))))).
+ \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: 
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
+h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A 
+(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: 
+(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: 
+A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: 
+(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g 
+a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g 
+n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq 
+nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: 
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
+h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq 
+A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: 
+A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A 
+(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) 
+n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) 
+(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a 
+n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: 
+(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def 
+(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus 
+g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def 
+(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g 
+a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat 
+nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).