-(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
-(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7
-a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (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 a6 _) \Rightarrow a6])) (AHead a3 a5)
-(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 a5 | (AHead _ a6)
-\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3
-a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6:
-A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4)
-(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4)))))))))
-H6))))))))))) a y H0))) H))))).
-(* COMMENTS
-Initial nodes: 797
-END *)
+(H: (leq g a (AHead a1 a2))).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def
+(\lambda (a0: A).(leq g a a0)) in (let TMP_7 \def (\lambda (_: A).(let TMP_3
+\def (\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) in (let TMP_4 \def
+(\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) in (let TMP_6 \def (\lambda
+(a3: A).(\lambda (a4: A).(let TMP_5 \def (AHead a3 a4) in (eq A a TMP_5))))
+in (ex3_2 A A TMP_3 TMP_4 TMP_6))))) in (let TMP_50 \def (\lambda (y:
+A).(\lambda (H0: (leq g a y)).(let TMP_12 \def (\lambda (a0: A).(\lambda (a3:
+A).((eq A a3 (AHead a1 a2)) \to (let TMP_8 \def (\lambda (a4: A).(\lambda (_:
+A).(leq g a4 a1))) in (let TMP_9 \def (\lambda (_: A).(\lambda (a5: A).(leq g
+a5 a2))) in (let TMP_11 \def (\lambda (a4: A).(\lambda (a5: A).(let TMP_10
+\def (AHead a4 a5) in (eq A a0 TMP_10)))) in (ex3_2 A A TMP_8 TMP_9
+TMP_11))))))) in (let TMP_22 \def (\lambda (h1: nat).(\lambda (h2:
+nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq
+A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A
+(ASort h2 n2) (AHead a1 a2))).(let TMP_13 \def (ASort h2 n2) in (let TMP_14
+\def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _
+_) \Rightarrow False])) in (let TMP_15 \def (AHead a1 a2) in (let H3 \def
+(eq_ind A TMP_13 TMP_14 I TMP_15 H2) in (let TMP_16 \def (\lambda (a3:
+A).(\lambda (_: A).(leq g a3 a1))) in (let TMP_17 \def (\lambda (_:
+A).(\lambda (a4: A).(leq g a4 a2))) in (let TMP_20 \def (\lambda (a3:
+A).(\lambda (a4: A).(let TMP_18 \def (ASort h1 n1) in (let TMP_19 \def (AHead
+a3 a4) in (eq A TMP_18 TMP_19))))) in (let TMP_21 \def (ex3_2 A A TMP_16
+TMP_17 TMP_20) in (False_ind TMP_21 H3)))))))))))))))) in (let TMP_49 \def
+(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2:
+(((eq A a3 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_:
+A).(leq g a4 a1))) (\lambda (_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda
+(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))))))).(\lambda (a4:
+A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a5
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a6
+a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda (a6:
+A).(\lambda (a7: A).(eq A a4 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead
+a3 a5) (AHead a1 a2))).(let TMP_23 \def (\lambda (e: A).(match e with [(ASort
+_ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) in (let TMP_24 \def
+(AHead a3 a5) in (let TMP_25 \def (AHead a1 a2) in (let H6 \def (f_equal A A
+TMP_23 TMP_24 TMP_25 H5) in (let TMP_26 \def (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) in (let TMP_27
+\def (AHead a3 a5) in (let TMP_28 \def (AHead a1 a2) in (let H7 \def (f_equal
+A A TMP_26 TMP_27 TMP_28 H5) in (let TMP_48 \def (\lambda (H8: (eq A a3
+a1)).(let TMP_33 \def (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (let
+TMP_29 \def (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) in (let TMP_30
+\def (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) in (let TMP_32 \def
+(\lambda (a7: A).(\lambda (a8: A).(let TMP_31 \def (AHead a7 a8) in (eq A a4
+TMP_31)))) in (ex3_2 A A TMP_29 TMP_30 TMP_32)))))) in (let H9 \def (eq_ind A
+a5 TMP_33 H4 a2 H7) in (let TMP_34 \def (\lambda (a6: A).(leq g a4 a6)) in
+(let H10 \def (eq_ind A a5 TMP_34 H3 a2 H7) in (let TMP_39 \def (\lambda (a6:
+A).((eq A a6 (AHead a1 a2)) \to (let TMP_35 \def (\lambda (a7: A).(\lambda
+(_: A).(leq g a7 a1))) in (let TMP_36 \def (\lambda (_: A).(\lambda (a8:
+A).(leq g a8 a2))) in (let TMP_38 \def (\lambda (a7: A).(\lambda (a8: A).(let
+TMP_37 \def (AHead a7 a8) in (eq A a0 TMP_37)))) in (ex3_2 A A TMP_35 TMP_36
+TMP_38)))))) in (let H11 \def (eq_ind A a3 TMP_39 H2 a1 H8) in (let TMP_40
+\def (\lambda (a6: A).(leq g a0 a6)) in (let H12 \def (eq_ind A a3 TMP_40 H1
+a1 H8) in (let TMP_41 \def (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1)))
+in (let TMP_42 \def (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) in (let
+TMP_45 \def (\lambda (a6: A).(\lambda (a7: A).(let TMP_43 \def (AHead a0 a4)
+in (let TMP_44 \def (AHead a6 a7) in (eq A TMP_43 TMP_44))))) in (let TMP_46
+\def (AHead a0 a4) in (let TMP_47 \def (refl_equal A TMP_46) in (ex3_2_intro
+A A TMP_41 TMP_42 TMP_45 a0 a4 H12 H10 TMP_47))))))))))))))) in (TMP_48
+H6))))))))))))))))))) in (leq_ind g TMP_12 TMP_22 TMP_49 a y H0)))))) in
+(insert_eq A TMP_1 TMP_2 TMP_7 TMP_50 H))))))))).
+
+theorem ahead_inj_snd:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
+(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let TMP_1 \def
+(AHead a3 a4) in (let H_x \def (leq_gen_head1 g a1 a2 TMP_1 H) in (let H0
+\def H_x in (let TMP_2 \def (\lambda (a5: A).(\lambda (_: A).(leq g a1 a5)))
+in (let TMP_3 \def (\lambda (_: A).(\lambda (a6: A).(leq g a2 a6))) in (let
+TMP_6 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_4 \def (AHead a3 a4) in
+(let TMP_5 \def (AHead a5 a6) in (eq A TMP_4 TMP_5))))) in (let TMP_7 \def
+(leq g a2 a4) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead
+a3 a4) (AHead x0 x1))).(let TMP_8 \def (\lambda (e: A).(match e with [(ASort
+_ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) in (let TMP_9 \def (AHead
+a3 a4) in (let TMP_10 \def (AHead x0 x1) in (let H4 \def (f_equal A A TMP_8
+TMP_9 TMP_10 H3) in (let TMP_11 \def (\lambda (e: A).(match e with [(ASort _
+_) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) in (let TMP_12 \def (AHead
+a3 a4) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def (f_equal A A TMP_11
+TMP_12 TMP_13 H3) in (let TMP_16 \def (\lambda (H6: (eq A a3 x0)).(let TMP_14
+\def (\lambda (a: A).(leq g a2 a)) in (let H7 \def (eq_ind_r A x1 TMP_14 H2
+a4 H5) in (let TMP_15 \def (\lambda (a: A).(leq g a1 a)) in (let H8 \def
+(eq_ind_r A x0 TMP_15 H1 a3 H6) in H7))))) in (TMP_16 H4))))))))))))))) in
+(ex3_2_ind A A TMP_2 TMP_3 TMP_6 TMP_7 TMP_17 H0)))))))))))))).