(* This file was automatically generated: do not edit *********************)
-include "Basic-1/preamble.ma".
+include "basic_1/preamble.ma".
-inductive A: Set \def
+inductive A: Type[0] \def
| ASort: nat \to (nat \to A)
| AHead: A \to (A \to A).
--- /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 "basic_1/A/defs.ma".
+
+let rec A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall (n0:
+nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: A).((P
+a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with [(ASort n
+n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (let TMP_1 \def ((A_rect
+P f f0) a0) in (let TMP_2 \def ((A_rect P f f0) a1) in (f0 a0 TMP_1 a1
+TMP_2)))].
+
+theorem A_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).(\forall (n0: nat).(P (ASort
+n n0))))) \to (((\forall (a: A).((P a) \to (\forall (a0: A).((P a0) \to (P
+(AHead a a0))))))) \to (\forall (a: A).(P a))))
+\def
+ \lambda (P: ((A \to Prop))).(A_rect P).
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/asucc/defs.ma".
+include "basic_1/asucc/defs.ma".
-definition aplus:
- G \to (A \to (nat \to A))
-\def
- 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.
+let rec aplus (g: G) (a: A) (n: nat) on n: A \def match n with [O \Rightarrow
+a | (S n0) \Rightarrow (let TMP_1 \def (aplus g a n0) in (asucc g TMP_1))].
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/aplus/defs.ma".
+include "basic_1/aplus/defs.ma".
-include "Basic-1/next_plus/props.ma".
+include "basic_1/A/fwd.ma".
+
+include "basic_1/next_plus/props.ma".
theorem aplus_reg_r:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
\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).(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))))))).
-(* COMMENTS
-Initial nodes: 143
-END *)
+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))))))))).
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).(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))).
-(* COMMENTS
-Initial nodes: 361
-END *)
+ \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)))))).
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).(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)))).
-(* COMMENTS
-Initial nodes: 87
-END *)
+ \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))))))))))))))).
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).(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))))).
-(* COMMENTS
-Initial nodes: 97
-END *)
+ \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)))))))))))))))).
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).(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)))))).
-(* COMMENTS
-Initial nodes: 97
-END *)
+ \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))))))))))))))))))).
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).(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)).
-(* COMMENTS
-Initial nodes: 229
-END *)
+ \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))))).
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).(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 in nat return (\lambda (_: nat).Prop) 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)).
-(* COMMENTS
-Initial nodes: 484
-END *)
+ \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))))).
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).(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 (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 (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))))))).
-(* COMMENTS
-Initial nodes: 587
-END *)
+ \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))))))))))))).
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).(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)).
-(* COMMENTS
-Initial nodes: 239
-END *)
+ \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))))).
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).(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 in A return (\lambda (_: A).nat) 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))))) (\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
+ \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
-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)))))) 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 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)).
-(* COMMENTS
-Initial nodes: 977
-END *)
+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:
+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))))).
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).(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)).
-(* COMMENTS
-Initial nodes: 599
-END *)
+ \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))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/A/defs.ma".
+include "basic_1/A/defs.ma".
inductive aprem: nat \to (A \to (A \to Prop)) \def
| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/aprem/defs.ma".
+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))].
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))))).
-(* COMMENTS
-Initial nodes: 227
-END *)
+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))))))))).
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)).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)))).
-(* COMMENTS
-Initial nodes: 500
-END *)
+(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)))))))).
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:
+(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:
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:
+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 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))))).
-(* COMMENTS
-Initial nodes: 631
-END *)
+(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))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/aprem/fwd.ma".
+include "basic_1/aprem/fwd.ma".
-include "Basic-1/leq/defs.ma".
+include "basic_1/leq/fwd.ma".
theorem aprem_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
-(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
-(b1: A).(aprem i a b1)))))))) (\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 (i: nat).(\lambda (b2:
-A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2
-i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g
-b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda
-(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall
-(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g
-b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda
-(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall
-(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
-(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda
-(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead
-a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem
-n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y
-\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A
-(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))
-(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead
-a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda
-(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
-(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0)
-(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let
-H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1:
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1:
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda
-(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2
-A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0
-a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2
-H)))).
-(* COMMENTS
-Initial nodes: 621
-END *)
+a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(\forall (i:
+nat).(\forall (b2: A).((aprem i a0 b2) \to (let TMP_1 \def (\lambda (b1:
+A).(leq g b1 b2)) in (let TMP_2 \def (\lambda (b1: A).(aprem i a b1)) in (ex2
+A TMP_1 TMP_2)))))))) in (let TMP_8 \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 (i:
+nat).(\lambda (b2: A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def
+(aprem_gen_sort b2 i h2 n2 H1) in (let H2 \def H_x in (let TMP_4 \def
+(\lambda (b1: A).(leq g b1 b2)) in (let TMP_6 \def (\lambda (b1: A).(let
+TMP_5 \def (ASort h1 n1) in (aprem i TMP_5 b1))) in (let TMP_7 \def (ex2 A
+TMP_4 TMP_6) in (False_ind TMP_7 H2))))))))))))))) in (let TMP_37 \def
+(\lambda (a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_:
+((\forall (i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4:
+A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i:
+nat).(\forall (b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1
+b2)) (\lambda (b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2:
+A).(\lambda (H4: (aprem i (AHead a3 a5) b2)).(let TMP_12 \def (\lambda (n:
+nat).((aprem n (AHead a3 a5) b2) \to (let TMP_9 \def (\lambda (b1: A).(leq g
+b1 b2)) in (let TMP_11 \def (\lambda (b1: A).(let TMP_10 \def (AHead a0 a4)
+in (aprem n TMP_10 b1))) in (ex2 A TMP_9 TMP_11))))) in (let TMP_22 \def
+(\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_O a3
+a5 b2 H5) in (let TMP_16 \def (\lambda (a: A).(let TMP_13 \def (\lambda (b1:
+A).(leq g b1 a)) in (let TMP_15 \def (\lambda (b1: A).(let TMP_14 \def (AHead
+a0 a4) in (aprem O TMP_14 b1))) in (ex2 A TMP_13 TMP_15)))) in (let TMP_17
+\def (\lambda (b1: A).(leq g b1 a3)) in (let TMP_19 \def (\lambda (b1:
+A).(let TMP_18 \def (AHead a0 a4) in (aprem O TMP_18 b1))) in (let TMP_20
+\def (aprem_zero a0 a4) in (let TMP_21 \def (ex_intro2 A TMP_17 TMP_19 a0 H0
+TMP_20) in (eq_ind_r A a3 TMP_16 TMP_21 b2 H_y)))))))) in (let TMP_36 \def
+(\lambda (i0: nat).(\lambda (_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 (AHead a0 a4)
+b1)))))).(\lambda (H5: (aprem (S i0) (AHead a3 a5) b2)).(let H_y \def
+(aprem_gen_head_S a3 a5 b2 i0 H5) in (let H_x \def (H3 i0 b2 H_y) in (let H6
+\def H_x in (let TMP_23 \def (\lambda (b1: A).(leq g b1 b2)) in (let TMP_24
+\def (\lambda (b1: A).(aprem i0 a4 b1)) in (let TMP_25 \def (\lambda (b1:
+A).(leq g b1 b2)) in (let TMP_28 \def (\lambda (b1: A).(let TMP_26 \def (S
+i0) in (let TMP_27 \def (AHead a0 a4) in (aprem TMP_26 TMP_27 b1)))) in (let
+TMP_29 \def (ex2 A TMP_25 TMP_28) in (let TMP_35 \def (\lambda (x:
+A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(let TMP_30
+\def (\lambda (b1: A).(leq g b1 b2)) in (let TMP_33 \def (\lambda (b1:
+A).(let TMP_31 \def (S i0) in (let TMP_32 \def (AHead a0 a4) in (aprem TMP_31
+TMP_32 b1)))) in (let TMP_34 \def (aprem_succ a4 x i0 H8 a0) in (ex_intro2 A
+TMP_30 TMP_33 x H7 TMP_34))))))) in (ex2_ind A TMP_23 TMP_24 TMP_29 TMP_35
+H6))))))))))))) in (nat_ind TMP_12 TMP_22 TMP_36 i H4))))))))))))))) in
+(leq_ind g TMP_3 TMP_8 TMP_37 a1 a2 H))))))).
theorem aprem_asucc:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i
a1 a2) \to (aprem i (asucc g a1) a2)))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
-(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
-(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
-A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
-(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
-a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2
-H))))).
-(* COMMENTS
-Initial nodes: 101
-END *)
+(H: (aprem i a1 a2)).(let TMP_2 \def (\lambda (n: nat).(\lambda (a:
+A).(\lambda (a0: A).(let TMP_1 \def (asucc g a) in (aprem n TMP_1 a0))))) in
+(let TMP_4 \def (\lambda (a0: A).(\lambda (a3: A).(let TMP_3 \def (asucc g
+a3) in (aprem_zero a0 TMP_3)))) in (let TMP_6 \def (\lambda (a0: A).(\lambda
+(a: A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem
+i0 (asucc g a0) a)).(\lambda (a3: A).(let TMP_5 \def (asucc g a0) in
+(aprem_succ TMP_5 a i0 H1 a3)))))))) in (aprem_ind TMP_2 TMP_4 TMP_6 i a1 a2
+H)))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/A/defs.ma".
+include "basic_1/A/defs.ma".
-include "Basic-1/G/defs.ma".
+include "basic_1/G/defs.ma".
-definition asucc:
- G \to (A \to A)
-\def
- let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n)
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h)
-\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g
-a2))]) in asucc.
+let rec asucc (g: G) (l: A) on l: A \def match l with [(ASort n0 n)
+\Rightarrow (match n0 with [O \Rightarrow (let TMP_2 \def (next g n) in
+(ASort O TMP_2)) | (S h) \Rightarrow (ASort h n)]) | (AHead a1 a2)
+\Rightarrow (let TMP_1 \def (asucc g a2) in (AHead a1 TMP_1))].
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/asucc/defs.ma".
+include "basic_1/asucc/defs.ma".
+
+include "basic_1/A/fwd.ma".
theorem asucc_gen_sort:
\forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
nat).(eq A a (ASort h0 n0)))))))))
\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind
-(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda
-(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0:
-nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0
-n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0
-with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0
-n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A
-(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1)))))))
-(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat
-nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0
-n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1))
-\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0
-n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2
-\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda
-(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2)))))))
-a)))).
-(* COMMENTS
-Initial nodes: 317
-END *)
+ \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(let
+TMP_3 \def (\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (let TMP_2
+\def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_1 \def (ASort h0 n0) in
+(eq A a0 TMP_1)))) in (ex_2 nat nat TMP_2)))) in (let TMP_13 \def (\lambda
+(n0: nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort
+n0 n1)))).(let TMP_4 \def (\lambda (e: A).e) in (let TMP_5 \def (ASort h n)
+in (let TMP_7 \def (match n0 with [O \Rightarrow (let TMP_6 \def (next g n1)
+in (ASort O TMP_6)) | (S h0) \Rightarrow (ASort h0 n1)]) in (let H0 \def
+(f_equal A A TMP_4 TMP_5 TMP_7 H) in (let TMP_10 \def (\lambda (h0:
+nat).(\lambda (n2: nat).(let TMP_8 \def (ASort n0 n1) in (let TMP_9 \def
+(ASort h0 n2) in (eq A TMP_8 TMP_9))))) in (let TMP_11 \def (ASort n0 n1) in
+(let TMP_12 \def (refl_equal A TMP_11) in (ex_2_intro nat nat TMP_10 n0 n1
+TMP_12))))))))))) in (let TMP_22 \def (\lambda (a0: A).(\lambda (_: (((eq A
+(ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
+nat).(eq A a0 (ASort h0 n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A
+(ASort h n) (asucc g a1)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
+nat).(eq A a1 (ASort h0 n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g
+(AHead a0 a1)))).(let TMP_14 \def (ASort h n) in (let TMP_15 \def (\lambda
+(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) in (let TMP_16 \def (AHead a0 a1) in (let TMP_17 \def
+(asucc g TMP_16) in (let H2 \def (eq_ind A TMP_14 TMP_15 I TMP_17 H1) in (let
+TMP_20 \def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_18 \def (AHead a0
+a1) in (let TMP_19 \def (ASort h0 n0) in (eq A TMP_18 TMP_19))))) in (let
+TMP_21 \def (ex_2 nat nat TMP_20) in (False_ind TMP_21 H2))))))))))))) in
+(A_ind TMP_3 TMP_13 TMP_22 a))))))).
theorem asucc_gen_head:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A
(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1
a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind
-(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
-A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3))))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc
-g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g
-(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead
-a1 a2) (asucc g (ASort O n0)))).(let H1 \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 O (next g n0))
-H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0)))
-(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda
-(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0:
-A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
-a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1)
-n0)))).(let H1 \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 n1 n0) H0) in (False_ind (ex2 A (\lambda (a0:
-A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
-a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2)
-(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda
-(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A
-(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1
-a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead
-a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
-(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in
-((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
-A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4]))
-(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let
-H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4))
-\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A
-a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A
-(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A
-a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead
-a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5)))
-(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7
-\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to
-(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4
-(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda
-(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda
-(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead
-a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3
-(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4))))
-H2))))))) a)))).
-(* COMMENTS
-Initial nodes: 957
-END *)
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(let TMP_5
+\def (\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (let TMP_2 \def
+(\lambda (a3: A).(let TMP_1 \def (AHead a1 a3) in (eq A a0 TMP_1))) in (let
+TMP_4 \def (\lambda (a3: A).(let TMP_3 \def (asucc g a3) in (eq A a2 TMP_3)))
+in (ex2 A TMP_2 TMP_4))))) in (let TMP_34 \def (\lambda (n: nat).(\lambda
+(n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc g (ASort n n0)))).(let
+TMP_11 \def (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g (ASort n1 n0)))
+\to (let TMP_8 \def (\lambda (a0: A).(let TMP_6 \def (ASort n1 n0) in (let
+TMP_7 \def (AHead a1 a0) in (eq A TMP_6 TMP_7)))) in (let TMP_10 \def
+(\lambda (a0: A).(let TMP_9 \def (asucc g a0) in (eq A a2 TMP_9))) in (ex2 A
+TMP_8 TMP_10))))) in (let TMP_22 \def (\lambda (H0: (eq A (AHead a1 a2)
+(asucc g (ASort O n0)))).(let TMP_12 \def (AHead a1 a2) in (let TMP_13 \def
+(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) in (let TMP_14 \def (next g n0) in (let TMP_15 \def
+(ASort O TMP_14) in (let H1 \def (eq_ind A TMP_12 TMP_13 I TMP_15 H0) in (let
+TMP_18 \def (\lambda (a0: A).(let TMP_16 \def (ASort O n0) in (let TMP_17
+\def (AHead a1 a0) in (eq A TMP_16 TMP_17)))) in (let TMP_20 \def (\lambda
+(a0: A).(let TMP_19 \def (asucc g a0) in (eq A a2 TMP_19))) in (let TMP_21
+\def (ex2 A TMP_18 TMP_20) in (False_ind TMP_21 H1)))))))))) in (let TMP_33
+\def (\lambda (n1: nat).(\lambda (_: (((eq A (AHead a1 a2) (asucc g (ASort n1
+n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 a0)))
+(\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H0: (eq A (AHead a1
+a2) (asucc g (ASort (S n1) n0)))).(let TMP_23 \def (AHead a1 a2) in (let
+TMP_24 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) in (let TMP_25 \def (ASort n1 n0) in (let H1
+\def (eq_ind A TMP_23 TMP_24 I TMP_25 H0) in (let TMP_29 \def (\lambda (a0:
+A).(let TMP_26 \def (S n1) in (let TMP_27 \def (ASort TMP_26 n0) in (let
+TMP_28 \def (AHead a1 a0) in (eq A TMP_27 TMP_28))))) in (let TMP_31 \def
+(\lambda (a0: A).(let TMP_30 \def (asucc g a0) in (eq A a2 TMP_30))) in (let
+TMP_32 \def (ex2 A TMP_29 TMP_31) in (False_ind TMP_32 H1))))))))))) in
+(nat_ind TMP_11 TMP_22 TMP_33 n H))))))) in (let TMP_86 \def (\lambda (a0:
+A).(\lambda (H: (((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
+A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g
+a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A (AHead a1 a2) (asucc g a3))
+\to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 a4))) (\lambda (a4: A).(eq A
+a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead a1 a2) (asucc g (AHead a0
+a3)))).(let TMP_35 \def (\lambda (e: A).(match e with [(ASort _ _)
+\Rightarrow a1 | (AHead a4 _) \Rightarrow a4])) in (let TMP_36 \def (AHead a1
+a2) in (let TMP_37 \def (asucc g a3) in (let TMP_38 \def (AHead a0 TMP_37) in
+(let H2 \def (f_equal A A TMP_35 TMP_36 TMP_38 H1) in (let TMP_39 \def
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a2 | (AHead _ a4)
+\Rightarrow a4])) in (let TMP_40 \def (AHead a1 a2) in (let TMP_41 \def
+(asucc g a3) in (let TMP_42 \def (AHead a0 TMP_41) in (let H3 \def (f_equal A
+A TMP_39 TMP_40 TMP_42 H1) in (let TMP_85 \def (\lambda (H4: (eq A a1
+a0)).(let TMP_47 \def (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) \to
+(let TMP_44 \def (\lambda (a5: A).(let TMP_43 \def (AHead a1 a5) in (eq A a4
+TMP_43))) in (let TMP_46 \def (\lambda (a5: A).(let TMP_45 \def (asucc g a5)
+in (eq A a2 TMP_45))) in (ex2 A TMP_44 TMP_46))))) in (let H5 \def (eq_ind_r
+A a0 TMP_47 H a1 H4) in (let TMP_53 \def (\lambda (a4: A).(let TMP_50 \def
+(\lambda (a5: A).(let TMP_48 \def (AHead a4 a3) in (let TMP_49 \def (AHead a1
+a5) in (eq A TMP_48 TMP_49)))) in (let TMP_52 \def (\lambda (a5: A).(let
+TMP_51 \def (asucc g a5) in (eq A a2 TMP_51))) in (ex2 A TMP_50 TMP_52)))) in
+(let TMP_58 \def (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a3)) \to (let
+TMP_55 \def (\lambda (a5: A).(let TMP_54 \def (AHead a1 a5) in (eq A a3
+TMP_54))) in (let TMP_57 \def (\lambda (a5: A).(let TMP_56 \def (asucc g a5)
+in (eq A a4 TMP_56))) in (ex2 A TMP_55 TMP_57))))) in (let TMP_59 \def (asucc
+g a3) in (let H6 \def (eq_ind A a2 TMP_58 H0 TMP_59 H3) in (let TMP_64 \def
+(\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to (let TMP_61 \def
+(\lambda (a5: A).(let TMP_60 \def (AHead a1 a5) in (eq A a1 TMP_60))) in (let
+TMP_63 \def (\lambda (a5: A).(let TMP_62 \def (asucc g a5) in (eq A a4
+TMP_62))) in (ex2 A TMP_61 TMP_63))))) in (let TMP_65 \def (asucc g a3) in
+(let H7 \def (eq_ind A a2 TMP_64 H5 TMP_65 H3) in (let TMP_66 \def (asucc g
+a3) in (let TMP_72 \def (\lambda (a4: A).(let TMP_69 \def (\lambda (a5:
+A).(let TMP_67 \def (AHead a1 a3) in (let TMP_68 \def (AHead a1 a5) in (eq A
+TMP_67 TMP_68)))) in (let TMP_71 \def (\lambda (a5: A).(let TMP_70 \def
+(asucc g a5) in (eq A a4 TMP_70))) in (ex2 A TMP_69 TMP_71)))) in (let TMP_75
+\def (\lambda (a4: A).(let TMP_73 \def (AHead a1 a3) in (let TMP_74 \def
+(AHead a1 a4) in (eq A TMP_73 TMP_74)))) in (let TMP_78 \def (\lambda (a4:
+A).(let TMP_76 \def (asucc g a3) in (let TMP_77 \def (asucc g a4) in (eq A
+TMP_76 TMP_77)))) in (let TMP_79 \def (AHead a1 a3) in (let TMP_80 \def
+(refl_equal A TMP_79) in (let TMP_81 \def (asucc g a3) in (let TMP_82 \def
+(refl_equal A TMP_81) in (let TMP_83 \def (ex_intro2 A TMP_75 TMP_78 a3
+TMP_80 TMP_82) in (let TMP_84 \def (eq_ind_r A TMP_66 TMP_72 TMP_83 a2 H3) in
+(eq_ind A a1 TMP_53 TMP_84 a0 H4))))))))))))))))))))) in (TMP_85
+H2))))))))))))))))) in (A_ind TMP_5 TMP_34 TMP_86 a))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/A/defs.ma".
+include "basic_1/A/defs.ma".
-include "Basic-1/G/defs.ma".
+include "basic_1/G/defs.ma".
definition gz:
G
--- /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 "basic_1/ex0/defs.ma".
+
+let rec leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: nat).(\forall
+(h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus h1 n2) (plus
+h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall (a1:
+A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leqz a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead
+a2 a4))))))))))) (a: A) (a0: A) (l: leqz a a0) on l: P a a0 \def match l with
+[(leqz_sort h1 h2 n1 n2 e) \Rightarrow (f h1 h2 n1 n2 e) | (leqz_head a1 a2
+l0 a3 a4 l1) \Rightarrow (let TMP_1 \def ((leqz_ind P f f0) a1 a2 l0) in (let
+TMP_2 \def ((leqz_ind P f f0) a3 a4 l1) in (f0 a1 a2 l0 TMP_1 a3 a4 l1
+TMP_2)))].
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/ex0/defs.ma".
+include "basic_1/ex0/fwd.ma".
-include "Basic-1/leq/defs.ma".
+include "basic_1/leq/fwd.ma".
-include "Basic-1/aplus/props.ma".
+include "basic_1/aplus/props.ma".
theorem aplus_gz_le:
\forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A
(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
\def
- \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0:
-nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h)
-n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y
-\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n)
-(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0:
-nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A
-(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h:
-nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A
-(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O
-\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n:
-nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n))
-k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O
-(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0
-n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n)))
-(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0:
-nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S
-(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0))
-(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz
-(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n:
-nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
+ \lambda (k: nat).(let TMP_6 \def (\lambda (n: nat).(\forall (h:
+nat).(\forall (n0: nat).((le h n) \to (let TMP_1 \def (ASort h n0) in (let
+TMP_2 \def (aplus gz TMP_1 n) in (let TMP_3 \def (minus n h) in (let TMP_4
+\def (plus TMP_3 n0) in (let TMP_5 \def (ASort O TMP_4) in (eq A TMP_2
+TMP_5)))))))))) in (let TMP_12 \def (\lambda (h: nat).(\lambda (n:
+nat).(\lambda (H: (le h O)).(let H_y \def (le_n_O_eq h H) in (let TMP_9 \def
+(\lambda (n0: nat).(let TMP_7 \def (ASort n0 n) in (let TMP_8 \def (ASort O
+n) in (eq A TMP_7 TMP_8)))) in (let TMP_10 \def (ASort O n) in (let TMP_11
+\def (refl_equal A TMP_10) in (eq_ind nat O TMP_9 TMP_11 h H_y)))))))) in
+(let TMP_103 \def (\lambda (k0: nat).(\lambda (IH: ((\forall (h:
+nat).(\forall (n: nat).((le h k0) \to (eq A (aplus gz (ASort h n) k0) (ASort
+O (plus (minus k0 h) n)))))))).(\lambda (h: nat).(let TMP_19 \def (\lambda
+(n: nat).(\forall (n0: nat).((le n (S k0)) \to (let TMP_13 \def (ASort n n0)
+in (let TMP_14 \def (aplus gz TMP_13 k0) in (let TMP_15 \def (asucc gz
+TMP_14) in (let TMP_16 \def (match n with [O \Rightarrow (S k0) | (S l)
+\Rightarrow (minus k0 l)]) in (let TMP_17 \def (plus TMP_16 n0) in (let
+TMP_18 \def (ASort O TMP_17) in (eq A TMP_15 TMP_18)))))))))) in (let TMP_72
+\def (\lambda (n: nat).(\lambda (_: (le O (S k0))).(let TMP_20 \def (ASort O
+n) in (let TMP_21 \def (asucc gz TMP_20) in (let TMP_22 \def (aplus gz TMP_21
+k0) in (let TMP_26 \def (\lambda (a: A).(let TMP_23 \def (plus k0 n) in (let
+TMP_24 \def (S TMP_23) in (let TMP_25 \def (ASort O TMP_24) in (eq A a
+TMP_25))))) in (let TMP_27 \def (minus k0 O) in (let TMP_28 \def (S n) in
+(let TMP_29 \def (plus TMP_27 TMP_28) in (let TMP_30 \def (ASort O TMP_29) in
+(let TMP_34 \def (\lambda (a: A).(let TMP_31 \def (plus k0 n) in (let TMP_32
+\def (S TMP_31) in (let TMP_33 \def (ASort O TMP_32) in (eq A a TMP_33)))))
+in (let TMP_41 \def (\lambda (n0: nat).(let TMP_35 \def (S n) in (let TMP_36
+\def (plus n0 TMP_35) in (let TMP_37 \def (ASort O TMP_36) in (let TMP_38
+\def (plus k0 n) in (let TMP_39 \def (S TMP_38) in (let TMP_40 \def (ASort O
+TMP_39) in (eq A TMP_37 TMP_40)))))))) in (let TMP_42 \def (plus k0 n) in
+(let TMP_43 \def (S TMP_42) in (let TMP_48 \def (\lambda (n0: nat).(let
+TMP_44 \def (ASort O n0) in (let TMP_45 \def (plus k0 n) in (let TMP_46 \def
+(S TMP_45) in (let TMP_47 \def (ASort O TMP_46) in (eq A TMP_44 TMP_47))))))
+in (let TMP_49 \def (plus k0 n) in (let TMP_50 \def (S TMP_49) in (let TMP_51
+\def (ASort O TMP_50) in (let TMP_52 \def (refl_equal A TMP_51) in (let
+TMP_53 \def (S n) in (let TMP_54 \def (plus k0 TMP_53) in (let TMP_55 \def
+(plus_n_Sm k0 n) in (let TMP_56 \def (eq_ind nat TMP_43 TMP_48 TMP_52 TMP_54
+TMP_55) in (let TMP_57 \def (minus k0 O) in (let TMP_58 \def (minus_n_O k0)
+in (let TMP_59 \def (eq_ind nat k0 TMP_41 TMP_56 TMP_57 TMP_58) in (let
+TMP_60 \def (S n) in (let TMP_61 \def (ASort O TMP_60) in (let TMP_62 \def
+(aplus gz TMP_61 k0) in (let TMP_63 \def (S n) in (let TMP_64 \def (le_O_n
+k0) in (let TMP_65 \def (IH O TMP_63 TMP_64) in (let TMP_66 \def (eq_ind_r A
+TMP_30 TMP_34 TMP_59 TMP_62 TMP_65) in (let TMP_67 \def (ASort O n) in (let
+TMP_68 \def (aplus gz TMP_67 k0) in (let TMP_69 \def (asucc gz TMP_68) in
+(let TMP_70 \def (ASort O n) in (let TMP_71 \def (aplus_asucc gz k0 TMP_70)
+in (eq_ind A TMP_22 TMP_26 TMP_66 TMP_69
+TMP_71))))))))))))))))))))))))))))))))))))))) in (let TMP_102 \def (\lambda
+(n: nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
-(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz
-(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0)
-k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
-A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0)
-k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S
-n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
-(* COMMENTS
-Initial nodes: 683
-END *)
+(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (let TMP_73 \def
+(ASort n n0) in (let TMP_74 \def (aplus gz TMP_73 k0) in (let TMP_79 \def
+(\lambda (a: A).(let TMP_75 \def (S n) in (let TMP_76 \def (ASort TMP_75 n0)
+in (let TMP_77 \def (aplus gz TMP_76 k0) in (let TMP_78 \def (asucc gz
+TMP_77) in (eq A TMP_78 a)))))) in (let TMP_80 \def (S n) in (let TMP_81 \def
+(ASort TMP_80 n0) in (let TMP_82 \def (asucc gz TMP_81) in (let TMP_83 \def
+(aplus gz TMP_82 k0) in (let TMP_86 \def (\lambda (a: A).(let TMP_84 \def
+(ASort n n0) in (let TMP_85 \def (aplus gz TMP_84 k0) in (eq A a TMP_85))))
+in (let TMP_87 \def (ASort n n0) in (let TMP_88 \def (aplus gz TMP_87 k0) in
+(let TMP_89 \def (refl_equal A TMP_88) in (let TMP_90 \def (S n) in (let
+TMP_91 \def (ASort TMP_90 n0) in (let TMP_92 \def (aplus gz TMP_91 k0) in
+(let TMP_93 \def (asucc gz TMP_92) in (let TMP_94 \def (S n) in (let TMP_95
+\def (ASort TMP_94 n0) in (let TMP_96 \def (aplus_asucc gz k0 TMP_95) in (let
+TMP_97 \def (eq_ind A TMP_83 TMP_86 TMP_89 TMP_93 TMP_96) in (let TMP_98 \def
+(minus k0 n) in (let TMP_99 \def (plus TMP_98 n0) in (let TMP_100 \def (ASort
+O TMP_99) in (let TMP_101 \def (IH n n0 H_y) in (eq_ind A TMP_74 TMP_79
+TMP_97 TMP_100 TMP_101))))))))))))))))))))))))))))) in (nat_ind TMP_19 TMP_72
+TMP_102 h))))))) in (nat_ind TMP_6 TMP_12 TMP_103 k)))).
theorem aplus_gz_ge:
\forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A
(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
\def
- \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h:
-nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0)
-n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0:
-nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O)
-(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0
-h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda
-(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz
-(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le
-(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0:
-nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
-(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0
-x)).(let H2 \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 x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O
-n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0)
-n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
-n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in
-(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus
-gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n))
-k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus
-gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc
-gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
-(* COMMENTS
-Initial nodes: 524
-END *)
+ \lambda (n: nat).(\lambda (k: nat).(let TMP_5 \def (\lambda (n0:
+nat).(\forall (h: nat).((le n0 h) \to (let TMP_1 \def (ASort h n) in (let
+TMP_2 \def (aplus gz TMP_1 n0) in (let TMP_3 \def (minus h n0) in (let TMP_4
+\def (ASort TMP_3 n) in (eq A TMP_2 TMP_4)))))))) in (let TMP_13 \def
+(\lambda (h: nat).(\lambda (_: (le O h)).(let TMP_8 \def (\lambda (n0:
+nat).(let TMP_6 \def (ASort h n) in (let TMP_7 \def (ASort n0 n) in (eq A
+TMP_6 TMP_7)))) in (let TMP_9 \def (ASort h n) in (let TMP_10 \def
+(refl_equal A TMP_9) in (let TMP_11 \def (minus h O) in (let TMP_12 \def
+(minus_n_O h) in (eq_ind nat h TMP_8 TMP_10 TMP_11 TMP_12)))))))) in (let
+TMP_68 \def (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 h)
+\to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda (h:
+nat).(let TMP_20 \def (\lambda (n0: nat).((le (S k0) n0) \to (let TMP_14 \def
+(ASort n0 n) in (let TMP_15 \def (aplus gz TMP_14 k0) in (let TMP_16 \def
+(asucc gz TMP_15) in (let TMP_17 \def (S k0) in (let TMP_18 \def (minus n0
+TMP_17) in (let TMP_19 \def (ASort TMP_18 n) in (eq A TMP_16 TMP_19)))))))))
+in (let TMP_38 \def (\lambda (H: (le (S k0) 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 k0 n0)) in (let TMP_24 \def (ASort O n) in (let TMP_25
+\def (aplus gz TMP_24 k0) in (let TMP_26 \def (asucc gz TMP_25) in (let
+TMP_27 \def (ASort O n) in (let TMP_28 \def (eq A TMP_26 TMP_27) in (let
+TMP_36 \def (\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_:
+(le k0 x)).(let TMP_29 \def (\lambda (ee: nat).(match ee with [O \Rightarrow
+True | (S _) \Rightarrow False])) in (let TMP_30 \def (S x) in (let H2 \def
+(eq_ind nat O TMP_29 I TMP_30 H0) in (let TMP_31 \def (ASort O n) in (let
+TMP_32 \def (aplus gz TMP_31 k0) in (let TMP_33 \def (asucc gz TMP_32) in
+(let TMP_34 \def (ASort O n) in (let TMP_35 \def (eq A TMP_33 TMP_34) in
+(False_ind TMP_35 H2)))))))))))) in (let TMP_37 \def (le_gen_S k0 O H) in
+(ex2_ind nat TMP_22 TMP_23 TMP_28 TMP_36 TMP_37))))))))))) in (let TMP_67
+\def (\lambda (n0: nat).(\lambda (_: (((le (S k0) n0) \to (eq A (asucc gz
+(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n))))).(\lambda (H0: (le
+(S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in (let TMP_39 \def (ASort n0
+n) in (let TMP_40 \def (aplus gz TMP_39 k0) in (let TMP_45 \def (\lambda (a:
+A).(let TMP_41 \def (S n0) in (let TMP_42 \def (ASort TMP_41 n) in (let
+TMP_43 \def (aplus gz TMP_42 k0) in (let TMP_44 \def (asucc gz TMP_43) in (eq
+A TMP_44 a)))))) in (let TMP_46 \def (S n0) in (let TMP_47 \def (ASort TMP_46
+n) in (let TMP_48 \def (asucc gz TMP_47) in (let TMP_49 \def (aplus gz TMP_48
+k0) in (let TMP_52 \def (\lambda (a: A).(let TMP_50 \def (ASort n0 n) in (let
+TMP_51 \def (aplus gz TMP_50 k0) in (eq A a TMP_51)))) in (let TMP_53 \def
+(ASort n0 n) in (let TMP_54 \def (aplus gz TMP_53 k0) in (let TMP_55 \def
+(refl_equal A TMP_54) in (let TMP_56 \def (S n0) in (let TMP_57 \def (ASort
+TMP_56 n) in (let TMP_58 \def (aplus gz TMP_57 k0) in (let TMP_59 \def (asucc
+gz TMP_58) in (let TMP_60 \def (S n0) in (let TMP_61 \def (ASort TMP_60 n) in
+(let TMP_62 \def (aplus_asucc gz k0 TMP_61) in (let TMP_63 \def (eq_ind A
+TMP_49 TMP_52 TMP_55 TMP_59 TMP_62) in (let TMP_64 \def (minus n0 k0) in (let
+TMP_65 \def (ASort TMP_64 n) in (let TMP_66 \def (IH n0 H_y) in (eq_ind A
+TMP_40 TMP_45 TMP_63 TMP_65 TMP_66))))))))))))))))))))))))))) in (nat_ind
+TMP_20 TMP_38 TMP_67 h))))))) in (nat_ind TMP_5 TMP_13 TMP_68 k))))).
theorem next_plus_gz:
\forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
\def
- \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat
-(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0:
-nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat
-S (next_plus gz n n0) (plus n0 n) H))) h)).
-(* COMMENTS
-Initial nodes: 77
-END *)
+ \lambda (n: nat).(\lambda (h: nat).(let TMP_3 \def (\lambda (n0: nat).(let
+TMP_1 \def (next_plus gz n n0) in (let TMP_2 \def (plus n0 n) in (eq nat
+TMP_1 TMP_2)))) in (let TMP_4 \def (refl_equal nat n) in (let TMP_7 \def
+(\lambda (n0: nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0
+n))).(let TMP_5 \def (next_plus gz n n0) in (let TMP_6 \def (plus n0 n) in
+(f_equal nat nat S TMP_5 TMP_6 H))))) in (nat_ind TMP_3 TMP_4 TMP_7 h))))).
theorem leqz_leq:
\forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz
-(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
-(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k
-h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2
-(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def
-(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort
-h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1
-(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k)
-(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2)
-(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \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 k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow
-(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2
-k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in
-(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n:
-nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n:
-nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal
-nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1))
-(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2
-k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a
-(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1
-(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort
-h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus
-(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat
-(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2)
-n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind
-A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return
-(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])
-| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in
-(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1
-k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k
-h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A
-a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
-(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2)
-k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort
-(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in
-(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k)
-n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A
-(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k)))
-(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k)))
-n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort
-n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow
-False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1
-n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A
-(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2)
-k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4
-\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O
-(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le
-k h2 n2 H2)) in (let H5 \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 plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
-plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus
-(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in
-(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3:
-A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4:
-A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4
-a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-(* COMMENTS
-Initial nodes: 1375
-END *)
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(let TMP_1
+\def (\lambda (a: A).(\lambda (a0: A).(leqz a a0))) in (let TMP_225 \def
+(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus gz (ASort h1 n1) k) (aplus
+gz (ASort h2 n2) k))).(let TMP_2 \def (ASort h1 n1) in (let TMP_3 \def (ASort
+h2 n2) in (let TMP_4 \def (leqz TMP_2 TMP_3) in (let TMP_136 \def (\lambda
+(H1: (lt k h1)).(let TMP_5 \def (ASort h1 n1) in (let TMP_6 \def (ASort h2
+n2) in (let TMP_7 \def (leqz TMP_5 TMP_6) in (let TMP_86 \def (\lambda (H2:
+(lt k h2)).(let TMP_8 \def (ASort h1 n1) in (let TMP_9 \def (aplus gz TMP_8
+k) in (let TMP_12 \def (\lambda (a: A).(let TMP_10 \def (ASort h2 n2) in (let
+TMP_11 \def (aplus gz TMP_10 k) in (eq A a TMP_11)))) in (let TMP_13 \def
+(minus h1 k) in (let TMP_14 \def (ASort TMP_13 n1) in (let TMP_15 \def (S k)
+in (let TMP_16 \def (S h1) in (let TMP_17 \def (S k) in (let TMP_18 \def (S
+TMP_17) in (let TMP_19 \def (S h1) in (let TMP_20 \def (S k) in (let TMP_21
+\def (le_n_S TMP_20 h1 H1) in (let TMP_22 \def (le_S TMP_18 TMP_19 TMP_21) in
+(let TMP_23 \def (le_S_n TMP_15 TMP_16 TMP_22) in (let TMP_24 \def (le_S_n k
+h1 TMP_23) in (let TMP_25 \def (aplus_gz_ge n1 k h1 TMP_24) in (let H3 \def
+(eq_ind A TMP_9 TMP_12 H0 TMP_14 TMP_25) in (let TMP_26 \def (ASort h2 n2) in
+(let TMP_27 \def (aplus gz TMP_26 k) in (let TMP_30 \def (\lambda (a: A).(let
+TMP_28 \def (minus h1 k) in (let TMP_29 \def (ASort TMP_28 n1) in (eq A
+TMP_29 a)))) in (let TMP_31 \def (minus h2 k) in (let TMP_32 \def (ASort
+TMP_31 n2) in (let TMP_33 \def (S k) in (let TMP_34 \def (S h2) in (let
+TMP_35 \def (S k) in (let TMP_36 \def (S TMP_35) in (let TMP_37 \def (S h2)
+in (let TMP_38 \def (S k) in (let TMP_39 \def (le_n_S TMP_38 h2 H2) in (let
+TMP_40 \def (le_S TMP_36 TMP_37 TMP_39) in (let TMP_41 \def (le_S_n TMP_33
+TMP_34 TMP_40) in (let TMP_42 \def (le_S_n k h2 TMP_41) in (let TMP_43 \def
+(aplus_gz_ge n2 k h2 TMP_42) in (let H4 \def (eq_ind A TMP_27 TMP_30 H3
+TMP_32 TMP_43) in (let TMP_44 \def (\lambda (e: A).(match e with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow (minus h1 k)])) in (let TMP_45 \def
+(minus h1 k) in (let TMP_46 \def (ASort TMP_45 n1) in (let TMP_47 \def (minus
+h2 k) in (let TMP_48 \def (ASort TMP_47 n2) in (let H5 \def (f_equal A nat
+TMP_44 TMP_46 TMP_48 H4) in (let TMP_49 \def (\lambda (e: A).(match e with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) in (let TMP_50
+\def (minus h1 k) in (let TMP_51 \def (ASort TMP_50 n1) in (let TMP_52 \def
+(minus h2 k) in (let TMP_53 \def (ASort TMP_52 n2) in (let H6 \def (f_equal A
+nat TMP_49 TMP_51 TMP_53 H4) in (let TMP_85 \def (\lambda (H7: (eq nat (minus
+h1 k) (minus h2 k))).(let TMP_56 \def (\lambda (n: nat).(let TMP_54 \def
+(ASort h1 n1) in (let TMP_55 \def (ASort h2 n) in (leqz TMP_54 TMP_55)))) in
+(let TMP_59 \def (\lambda (n: nat).(let TMP_57 \def (ASort h1 n1) in (let
+TMP_58 \def (ASort n n1) in (leqz TMP_57 TMP_58)))) in (let TMP_60 \def (plus
+h1 n1) in (let TMP_61 \def (refl_equal nat TMP_60) in (let TMP_62 \def
+(leqz_sort h1 h1 n1 n1 TMP_61) in (let TMP_63 \def (S k) in (let TMP_64 \def
+(S h1) in (let TMP_65 \def (S k) in (let TMP_66 \def (S TMP_65) in (let
+TMP_67 \def (S h1) in (let TMP_68 \def (S k) in (let TMP_69 \def (le_n_S
+TMP_68 h1 H1) in (let TMP_70 \def (le_S TMP_66 TMP_67 TMP_69) in (let TMP_71
+\def (le_S_n TMP_63 TMP_64 TMP_70) in (let TMP_72 \def (le_S_n k h1 TMP_71)
+in (let TMP_73 \def (S k) in (let TMP_74 \def (S h2) in (let TMP_75 \def (S
+k) in (let TMP_76 \def (S TMP_75) in (let TMP_77 \def (S h2) in (let TMP_78
+\def (S k) in (let TMP_79 \def (le_n_S TMP_78 h2 H2) in (let TMP_80 \def
+(le_S TMP_76 TMP_77 TMP_79) in (let TMP_81 \def (le_S_n TMP_73 TMP_74 TMP_80)
+in (let TMP_82 \def (le_S_n k h2 TMP_81) in (let TMP_83 \def (minus_minus k
+h1 h2 TMP_72 TMP_82 H7) in (let TMP_84 \def (eq_ind nat h1 TMP_59 TMP_62 h2
+TMP_83) in (eq_ind nat n1 TMP_56 TMP_84 n2 H6))))))))))))))))))))))))))))) in
+(TMP_85 H5))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_135
+\def (\lambda (H2: (le h2 k)).(let TMP_87 \def (ASort h1 n1) in (let TMP_88
+\def (aplus gz TMP_87 k) in (let TMP_91 \def (\lambda (a: A).(let TMP_89 \def
+(ASort h2 n2) in (let TMP_90 \def (aplus gz TMP_89 k) in (eq A a TMP_90))))
+in (let TMP_92 \def (minus h1 k) in (let TMP_93 \def (ASort TMP_92 n1) in
+(let TMP_94 \def (S k) in (let TMP_95 \def (S h1) in (let TMP_96 \def (S k)
+in (let TMP_97 \def (S TMP_96) in (let TMP_98 \def (S h1) in (let TMP_99 \def
+(S k) in (let TMP_100 \def (le_n_S TMP_99 h1 H1) in (let TMP_101 \def (le_S
+TMP_97 TMP_98 TMP_100) in (let TMP_102 \def (le_S_n TMP_94 TMP_95 TMP_101) in
+(let TMP_103 \def (le_S_n k h1 TMP_102) in (let TMP_104 \def (aplus_gz_ge n1
+k h1 TMP_103) in (let H3 \def (eq_ind A TMP_88 TMP_91 H0 TMP_93 TMP_104) in
+(let TMP_105 \def (ASort h2 n2) in (let TMP_106 \def (aplus gz TMP_105 k) in
+(let TMP_109 \def (\lambda (a: A).(let TMP_107 \def (minus h1 k) in (let
+TMP_108 \def (ASort TMP_107 n1) in (eq A TMP_108 a)))) in (let TMP_110 \def
+(minus k h2) in (let TMP_111 \def (plus TMP_110 n2) in (let TMP_112 \def
+(ASort O TMP_111) in (let TMP_113 \def (aplus_gz_le k h2 n2 H2) in (let H4
+\def (eq_ind A TMP_106 TMP_109 H3 TMP_112 TMP_113) in (let TMP_114 \def
+(minus h1 k) in (let TMP_119 \def (\lambda (n: nat).(let TMP_115 \def (ASort
+n n1) in (let TMP_116 \def (minus k h2) in (let TMP_117 \def (plus TMP_116
+n2) in (let TMP_118 \def (ASort O TMP_117) in (eq A TMP_115 TMP_118)))))) in
+(let TMP_120 \def (S k) in (let TMP_121 \def (minus h1 TMP_120) in (let
+TMP_122 \def (S TMP_121) in (let TMP_123 \def (minus_x_Sy h1 k H1) in (let H5
+\def (eq_ind nat TMP_114 TMP_119 H4 TMP_122 TMP_123) in (let TMP_124 \def (S
+k) in (let TMP_125 \def (minus h1 TMP_124) in (let TMP_126 \def (S TMP_125)
+in (let TMP_127 \def (ASort TMP_126 n1) in (let TMP_128 \def (\lambda (ee:
+A).(match ee with [(ASort n _) \Rightarrow (match n with [O \Rightarrow False
+| (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow False])) in (let TMP_129
+\def (minus k h2) in (let TMP_130 \def (plus TMP_129 n2) in (let TMP_131 \def
+(ASort O TMP_130) in (let H6 \def (eq_ind A TMP_127 TMP_128 I TMP_131 H5) in
+(let TMP_132 \def (ASort h1 n1) in (let TMP_133 \def (ASort h2 n2) in (let
+TMP_134 \def (leqz TMP_132 TMP_133) in (False_ind TMP_134
+H6)))))))))))))))))))))))))))))))))))))))))))))) in (lt_le_e k h2 TMP_7
+TMP_86 TMP_135))))))) in (let TMP_224 \def (\lambda (H1: (le h1 k)).(let
+TMP_137 \def (ASort h1 n1) in (let TMP_138 \def (ASort h2 n2) in (let TMP_139
+\def (leqz TMP_137 TMP_138) in (let TMP_194 \def (\lambda (H2: (lt k
+h2)).(let TMP_140 \def (ASort h1 n1) in (let TMP_141 \def (aplus gz TMP_140
+k) in (let TMP_144 \def (\lambda (a: A).(let TMP_142 \def (ASort h2 n2) in
+(let TMP_143 \def (aplus gz TMP_142 k) in (eq A a TMP_143)))) in (let TMP_145
+\def (minus k h1) in (let TMP_146 \def (plus TMP_145 n1) in (let TMP_147 \def
+(ASort O TMP_146) in (let TMP_148 \def (aplus_gz_le k h1 n1 H1) in (let H3
+\def (eq_ind A TMP_141 TMP_144 H0 TMP_147 TMP_148) in (let TMP_149 \def
+(ASort h2 n2) in (let TMP_150 \def (aplus gz TMP_149 k) in (let TMP_154 \def
+(\lambda (a: A).(let TMP_151 \def (minus k h1) in (let TMP_152 \def (plus
+TMP_151 n1) in (let TMP_153 \def (ASort O TMP_152) in (eq A TMP_153 a))))) in
+(let TMP_155 \def (minus h2 k) in (let TMP_156 \def (ASort TMP_155 n2) in
+(let TMP_157 \def (S k) in (let TMP_158 \def (S h2) in (let TMP_159 \def (S
+k) in (let TMP_160 \def (S TMP_159) in (let TMP_161 \def (S h2) in (let
+TMP_162 \def (S k) in (let TMP_163 \def (le_n_S TMP_162 h2 H2) in (let
+TMP_164 \def (le_S TMP_160 TMP_161 TMP_163) in (let TMP_165 \def (le_S_n
+TMP_157 TMP_158 TMP_164) in (let TMP_166 \def (le_S_n k h2 TMP_165) in (let
+TMP_167 \def (aplus_gz_ge n2 k h2 TMP_166) in (let H4 \def (eq_ind A TMP_150
+TMP_154 H3 TMP_156 TMP_167) in (let TMP_168 \def (minus k h1) in (let TMP_169
+\def (plus TMP_168 n1) in (let TMP_170 \def (ASort O TMP_169) in (let TMP_171
+\def (minus h2 k) in (let TMP_172 \def (ASort TMP_171 n2) in (let H5 \def
+(sym_eq A TMP_170 TMP_172 H4) in (let TMP_173 \def (minus h2 k) in (let
+TMP_178 \def (\lambda (n: nat).(let TMP_174 \def (ASort n n2) in (let TMP_175
+\def (minus k h1) in (let TMP_176 \def (plus TMP_175 n1) in (let TMP_177 \def
+(ASort O TMP_176) in (eq A TMP_174 TMP_177)))))) in (let TMP_179 \def (S k)
+in (let TMP_180 \def (minus h2 TMP_179) in (let TMP_181 \def (S TMP_180) in
+(let TMP_182 \def (minus_x_Sy h2 k H2) in (let H6 \def (eq_ind nat TMP_173
+TMP_178 H5 TMP_181 TMP_182) in (let TMP_183 \def (S k) in (let TMP_184 \def
+(minus h2 TMP_183) in (let TMP_185 \def (S TMP_184) in (let TMP_186 \def
+(ASort TMP_185 n2) in (let TMP_187 \def (\lambda (ee: A).(match ee with
+[(ASort n _) \Rightarrow (match n with [O \Rightarrow False | (S _)
+\Rightarrow True]) | (AHead _ _) \Rightarrow False])) in (let TMP_188 \def
+(minus k h1) in (let TMP_189 \def (plus TMP_188 n1) in (let TMP_190 \def
+(ASort O TMP_189) in (let H7 \def (eq_ind A TMP_186 TMP_187 I TMP_190 H6) in
+(let TMP_191 \def (ASort h1 n1) in (let TMP_192 \def (ASort h2 n2) in (let
+TMP_193 \def (leqz TMP_191 TMP_192) in (False_ind TMP_193
+H7)))))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_223 \def
+(\lambda (H2: (le h2 k)).(let TMP_195 \def (ASort h1 n1) in (let TMP_196 \def
+(aplus gz TMP_195 k) in (let TMP_199 \def (\lambda (a: A).(let TMP_197 \def
+(ASort h2 n2) in (let TMP_198 \def (aplus gz TMP_197 k) in (eq A a
+TMP_198)))) in (let TMP_200 \def (minus k h1) in (let TMP_201 \def (plus
+TMP_200 n1) in (let TMP_202 \def (ASort O TMP_201) in (let TMP_203 \def
+(aplus_gz_le k h1 n1 H1) in (let H3 \def (eq_ind A TMP_196 TMP_199 H0 TMP_202
+TMP_203) in (let TMP_204 \def (ASort h2 n2) in (let TMP_205 \def (aplus gz
+TMP_204 k) in (let TMP_209 \def (\lambda (a: A).(let TMP_206 \def (minus k
+h1) in (let TMP_207 \def (plus TMP_206 n1) in (let TMP_208 \def (ASort O
+TMP_207) in (eq A TMP_208 a))))) in (let TMP_210 \def (minus k h2) in (let
+TMP_211 \def (plus TMP_210 n2) in (let TMP_212 \def (ASort O TMP_211) in (let
+TMP_213 \def (aplus_gz_le k h2 n2 H2) in (let H4 \def (eq_ind A TMP_205
+TMP_209 H3 TMP_212 TMP_213) in (let TMP_216 \def (\lambda (e: A).(match e
+with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow (let TMP_215 \def
+(minus k h1) in (plus TMP_215 n1))])) in (let TMP_217 \def (minus k h1) in
+(let TMP_218 \def (plus TMP_217 n1) in (let TMP_219 \def (ASort O TMP_218) in
+(let TMP_220 \def (minus k h2) in (let TMP_221 \def (plus TMP_220 n2) in (let
+TMP_222 \def (ASort O TMP_221) in (let H5 \def (f_equal A nat TMP_216 TMP_219
+TMP_222 H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in (leqz_sort
+h1 h2 n1 n2 H_y))))))))))))))))))))))))))) in (lt_le_e k h2 TMP_139 TMP_194
+TMP_223))))))) in (lt_le_e k h1 TMP_4 TMP_136 TMP_224)))))))))))) in (let
+TMP_226 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leq gz a0
+a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda
+(_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5
+H3))))))))) in (leq_ind gz TMP_1 TMP_225 TMP_226 a1 a2 H)))))).
theorem leq_leqz:
\forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind
-(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus
-h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A
-(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1)))
-(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A
-(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2)))
-(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus
-(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus
-h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2))
-(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n:
-nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus
-h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq
-A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz
-n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1
-h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n:
-nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus
-h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal
-nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2
-n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2)
-(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2)
-(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2)
-(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus
-(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2))
-(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1
-h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0:
-A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0
-a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda
-(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-(* COMMENTS
-Initial nodes: 717
-END *)
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(let TMP_1 \def
+(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) in (let TMP_113 \def
+(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (H0: (eq nat (plus h1 n2) (plus h2 n1))).(let TMP_2 \def (plus
+h1 h2) in (let TMP_3 \def (plus h1 h2) in (let TMP_4 \def (minus h1 TMP_3) in
+(let TMP_5 \def (plus h1 h2) in (let TMP_6 \def (minus TMP_5 h1) in (let
+TMP_7 \def (next_plus gz n1 TMP_6) in (let TMP_8 \def (ASort TMP_4 TMP_7) in
+(let TMP_12 \def (\lambda (a: A).(let TMP_9 \def (ASort h2 n2) in (let TMP_10
+\def (plus h1 h2) in (let TMP_11 \def (aplus gz TMP_9 TMP_10) in (eq A a
+TMP_11))))) in (let TMP_13 \def (plus h1 h2) in (let TMP_14 \def (minus h2
+TMP_13) in (let TMP_15 \def (plus h1 h2) in (let TMP_16 \def (minus TMP_15
+h2) in (let TMP_17 \def (next_plus gz n2 TMP_16) in (let TMP_18 \def (ASort
+TMP_14 TMP_17) in (let TMP_25 \def (\lambda (a: A).(let TMP_19 \def (plus h1
+h2) in (let TMP_20 \def (minus h1 TMP_19) in (let TMP_21 \def (plus h1 h2) in
+(let TMP_22 \def (minus TMP_21 h1) in (let TMP_23 \def (next_plus gz n1
+TMP_22) in (let TMP_24 \def (ASort TMP_20 TMP_23) in (eq A TMP_24 a))))))))
+in (let TMP_36 \def (\lambda (n: nat).(let TMP_26 \def (plus h1 h2) in (let
+TMP_27 \def (minus h1 TMP_26) in (let TMP_28 \def (next_plus gz n1 n) in (let
+TMP_29 \def (ASort TMP_27 TMP_28) in (let TMP_30 \def (plus h1 h2) in (let
+TMP_31 \def (minus h2 TMP_30) in (let TMP_32 \def (plus h1 h2) in (let TMP_33
+\def (minus TMP_32 h2) in (let TMP_34 \def (next_plus gz n2 TMP_33) in (let
+TMP_35 \def (ASort TMP_31 TMP_34) in (eq A TMP_29 TMP_35)))))))))))) in (let
+TMP_45 \def (\lambda (n: nat).(let TMP_37 \def (plus h1 h2) in (let TMP_38
+\def (minus h1 TMP_37) in (let TMP_39 \def (next_plus gz n1 h2) in (let
+TMP_40 \def (ASort TMP_38 TMP_39) in (let TMP_41 \def (plus h1 h2) in (let
+TMP_42 \def (minus h2 TMP_41) in (let TMP_43 \def (next_plus gz n2 n) in (let
+TMP_44 \def (ASort TMP_42 TMP_43) in (eq A TMP_40 TMP_44)))))))))) in (let
+TMP_52 \def (\lambda (n: nat).(let TMP_46 \def (next_plus gz n1 h2) in (let
+TMP_47 \def (ASort n TMP_46) in (let TMP_48 \def (plus h1 h2) in (let TMP_49
+\def (minus h2 TMP_48) in (let TMP_50 \def (next_plus gz n2 h1) in (let
+TMP_51 \def (ASort TMP_49 TMP_50) in (eq A TMP_47 TMP_51)))))))) in (let
+TMP_57 \def (\lambda (n: nat).(let TMP_53 \def (next_plus gz n1 h2) in (let
+TMP_54 \def (ASort O TMP_53) in (let TMP_55 \def (next_plus gz n2 h1) in (let
+TMP_56 \def (ASort n TMP_55) in (eq A TMP_54 TMP_56)))))) in (let TMP_58 \def
+(plus h2 n1) in (let TMP_62 \def (\lambda (n: nat).(let TMP_59 \def (ASort O
+n) in (let TMP_60 \def (next_plus gz n2 h1) in (let TMP_61 \def (ASort O
+TMP_60) in (eq A TMP_59 TMP_61))))) in (let TMP_63 \def (plus h1 n2) in (let
+TMP_67 \def (\lambda (n: nat).(let TMP_64 \def (plus h2 n1) in (let TMP_65
+\def (ASort O TMP_64) in (let TMP_66 \def (ASort O n) in (eq A TMP_65
+TMP_66))))) in (let TMP_68 \def (ASort O) in (let TMP_69 \def (plus h2 n1) in
+(let TMP_70 \def (plus h1 n2) in (let TMP_71 \def (plus h1 n2) in (let TMP_72
+\def (plus h2 n1) in (let TMP_73 \def (sym_eq nat TMP_71 TMP_72 H0) in (let
+TMP_74 \def (f_equal nat A TMP_68 TMP_69 TMP_70 TMP_73) in (let TMP_75 \def
+(next_plus gz n2 h1) in (let TMP_76 \def (next_plus_gz n2 h1) in (let TMP_77
+\def (eq_ind_r nat TMP_63 TMP_67 TMP_74 TMP_75 TMP_76) in (let TMP_78 \def
+(next_plus gz n1 h2) in (let TMP_79 \def (next_plus_gz n1 h2) in (let TMP_80
+\def (eq_ind_r nat TMP_58 TMP_62 TMP_77 TMP_78 TMP_79) in (let TMP_81 \def
+(plus h1 h2) in (let TMP_82 \def (minus h2 TMP_81) in (let TMP_83 \def (plus
+h1 h2) in (let TMP_84 \def (le_plus_r h1 h2) in (let TMP_85 \def (O_minus h2
+TMP_83 TMP_84) in (let TMP_86 \def (eq_ind_r nat O TMP_57 TMP_80 TMP_82
+TMP_85) in (let TMP_87 \def (plus h1 h2) in (let TMP_88 \def (minus h1
+TMP_87) in (let TMP_89 \def (plus h1 h2) in (let TMP_90 \def (le_plus_l h1
+h2) in (let TMP_91 \def (O_minus h1 TMP_89 TMP_90) in (let TMP_92 \def
+(eq_ind_r nat O TMP_52 TMP_86 TMP_88 TMP_91) in (let TMP_93 \def (plus h1 h2)
+in (let TMP_94 \def (minus TMP_93 h2) in (let TMP_95 \def (minus_plus_r h1
+h2) in (let TMP_96 \def (eq_ind_r nat h1 TMP_45 TMP_92 TMP_94 TMP_95) in (let
+TMP_97 \def (plus h1 h2) in (let TMP_98 \def (minus TMP_97 h1) in (let TMP_99
+\def (minus_plus h1 h2) in (let TMP_100 \def (eq_ind_r nat h2 TMP_36 TMP_96
+TMP_98 TMP_99) in (let TMP_101 \def (ASort h2 n2) in (let TMP_102 \def (plus
+h1 h2) in (let TMP_103 \def (aplus gz TMP_101 TMP_102) in (let TMP_104 \def
+(plus h1 h2) in (let TMP_105 \def (aplus_asort_simpl gz TMP_104 h2 n2) in
+(let TMP_106 \def (eq_ind_r A TMP_18 TMP_25 TMP_100 TMP_103 TMP_105) in (let
+TMP_107 \def (ASort h1 n1) in (let TMP_108 \def (plus h1 h2) in (let TMP_109
+\def (aplus gz TMP_107 TMP_108) in (let TMP_110 \def (plus h1 h2) in (let
+TMP_111 \def (aplus_asort_simpl gz TMP_110 h1 n1) in (let TMP_112 \def
+(eq_ind_r A TMP_8 TMP_12 TMP_106 TMP_109 TMP_111) in (leq_sort gz h1 h2 n1 n2
+TMP_2
+TMP_112)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))) in (let TMP_114 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leqz
+a0 a3)).(\lambda (H1: (leq gz a0 a3)).(\lambda (a4: A).(\lambda (a5:
+A).(\lambda (_: (leqz a4 a5)).(\lambda (H3: (leq gz a4 a5)).(leq_head gz a0
+a3 H1 a4 a5 H3))))))))) in (leqz_ind TMP_1 TMP_113 TMP_114 a1 a2 H)))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/leq/props.ma".
+include "basic_1/leq/props.ma".
theorem asucc_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
(asucc g a1) (asucc g a2)))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
-a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
-nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
-(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
-(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
-(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
-A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
-nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
-(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
-h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
-(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
-A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
-(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
-A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
-(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
-(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
-H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
-(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
+a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def (asucc
+g a) in (let TMP_2 \def (asucc g a0) in (leq g TMP_1 TMP_2))))) in (let
+TMP_186 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(let TMP_8 \def (\lambda (n: nat).((eq
+A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (let TMP_5 \def
+(match n with [O \Rightarrow (let TMP_4 \def (next g n1) in (ASort O TMP_4))
+| (S h) \Rightarrow (ASort h n1)]) in (let TMP_7 \def (match h2 with [O
+\Rightarrow (let TMP_6 \def (next g n2) in (ASort O TMP_6)) | (S h)
+\Rightarrow (ASort h n2)]) in (leq g TMP_5 TMP_7))))) in (let TMP_97 \def
+(\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(let
+TMP_13 \def (\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort
+n n2) k)) \to (let TMP_9 \def (next g n1) in (let TMP_10 \def (ASort O TMP_9)
+in (let TMP_12 \def (match n with [O \Rightarrow (let TMP_11 \def (next g n2)
+in (ASort O TMP_11)) | (S h) \Rightarrow (ASort h n2)]) in (leq g TMP_10
+TMP_12)))))) in (let TMP_54 \def (\lambda (H2: (eq A (aplus g (ASort O n1) k)
+(aplus g (ASort O n2) k))).(let TMP_14 \def (next g n1) in (let TMP_15 \def
+(next g n2) in (let TMP_16 \def (ASort O n1) in (let TMP_17 \def (S k) in
+(let TMP_18 \def (aplus g TMP_16 TMP_17) in (let TMP_22 \def (\lambda (a:
+A).(let TMP_19 \def (next g n2) in (let TMP_20 \def (ASort O TMP_19) in (let
+TMP_21 \def (aplus g TMP_20 k) in (eq A a TMP_21))))) in (let TMP_23 \def
+(ASort O n2) in (let TMP_24 \def (S k) in (let TMP_25 \def (aplus g TMP_23
+TMP_24) in (let TMP_29 \def (\lambda (a: A).(let TMP_26 \def (ASort O n1) in
+(let TMP_27 \def (S k) in (let TMP_28 \def (aplus g TMP_26 TMP_27) in (eq A
+TMP_28 a))))) in (let TMP_30 \def (ASort O n2) in (let TMP_31 \def (aplus g
+TMP_30 k) in (let TMP_36 \def (\lambda (a: A).(let TMP_32 \def (asucc g a) in
+(let TMP_33 \def (ASort O n2) in (let TMP_34 \def (aplus g TMP_33 k) in (let
+TMP_35 \def (asucc g TMP_34) in (eq A TMP_32 TMP_35)))))) in (let TMP_37 \def
+(ASort O n2) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def
+(asucc g TMP_38) in (let TMP_40 \def (refl_equal A TMP_39) in (let TMP_41
+\def (ASort O n1) in (let TMP_42 \def (aplus g TMP_41 k) in (let TMP_43 \def
+(eq_ind_r A TMP_31 TMP_36 TMP_40 TMP_42 H2) in (let TMP_44 \def (next g n2)
+in (let TMP_45 \def (ASort O TMP_44) in (let TMP_46 \def (aplus g TMP_45 k)
+in (let TMP_47 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_48 \def (eq_ind
+A TMP_25 TMP_29 TMP_43 TMP_46 TMP_47) in (let TMP_49 \def (next g n1) in (let
+TMP_50 \def (ASort O TMP_49) in (let TMP_51 \def (aplus g TMP_50 k) in (let
+TMP_52 \def (aplus_sort_O_S_simpl g n1 k) in (let TMP_53 \def (eq_ind A
+TMP_18 TMP_22 TMP_48 TMP_51 TMP_52) in (leq_sort g O O TMP_14 TMP_15 k
+TMP_53)))))))))))))))))))))))))))))))) in (let TMP_96 \def (\lambda (h3:
nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
-(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
-n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
-(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
-A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
-n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
-k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
-O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
-(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
-(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
-(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
-n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
-O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
-(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
-(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
-((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
-(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
-(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
-\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
-\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
-(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
+(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(let TMP_55 \def (next g n1)
+in (let TMP_56 \def (ASort O n1) in (let TMP_57 \def (S k) in (let TMP_58
+\def (aplus g TMP_56 TMP_57) in (let TMP_61 \def (\lambda (a: A).(let TMP_59
+\def (ASort h3 n2) in (let TMP_60 \def (aplus g TMP_59 k) in (eq A a
+TMP_60)))) in (let TMP_62 \def (S h3) in (let TMP_63 \def (ASort TMP_62 n2)
+in (let TMP_64 \def (S k) in (let TMP_65 \def (aplus g TMP_63 TMP_64) in (let
+TMP_69 \def (\lambda (a: A).(let TMP_66 \def (ASort O n1) in (let TMP_67 \def
+(S k) in (let TMP_68 \def (aplus g TMP_66 TMP_67) in (eq A TMP_68 a))))) in
+(let TMP_70 \def (S h3) in (let TMP_71 \def (ASort TMP_70 n2) in (let TMP_72
+\def (aplus g TMP_71 k) in (let TMP_78 \def (\lambda (a: A).(let TMP_73 \def
+(asucc g a) in (let TMP_74 \def (S h3) in (let TMP_75 \def (ASort TMP_74 n2)
+in (let TMP_76 \def (aplus g TMP_75 k) in (let TMP_77 \def (asucc g TMP_76)
+in (eq A TMP_73 TMP_77))))))) in (let TMP_79 \def (S h3) in (let TMP_80 \def
+(ASort TMP_79 n2) in (let TMP_81 \def (aplus g TMP_80 k) in (let TMP_82 \def
+(asucc g TMP_81) in (let TMP_83 \def (refl_equal A TMP_82) in (let TMP_84
+\def (ASort O n1) in (let TMP_85 \def (aplus g TMP_84 k) in (let TMP_86 \def
+(eq_ind_r A TMP_72 TMP_78 TMP_83 TMP_85 H2) in (let TMP_87 \def (ASort h3 n2)
+in (let TMP_88 \def (aplus g TMP_87 k) in (let TMP_89 \def
+(aplus_sort_S_S_simpl g n2 h3 k) in (let TMP_90 \def (eq_ind A TMP_65 TMP_69
+TMP_86 TMP_88 TMP_89) in (let TMP_91 \def (next g n1) in (let TMP_92 \def
+(ASort O TMP_91) in (let TMP_93 \def (aplus g TMP_92 k) in (let TMP_94 \def
+(aplus_sort_O_S_simpl g n1 k) in (let TMP_95 \def (eq_ind A TMP_58 TMP_61
+TMP_90 TMP_93 TMP_94) in (leq_sort g O h3 TMP_55 n2 k
+TMP_95))))))))))))))))))))))))))))))))))) in (nat_ind TMP_13 TMP_54 TMP_96 h2
+H1))))) in (let TMP_185 \def (\lambda (h3: nat).(\lambda (IHh1: (((eq A
+(aplus g (ASort h3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match h3
+with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)])
+(match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow
+(ASort h n2)]))))).(\lambda (H1: (eq A (aplus g (ASort (S h3) n1) k) (aplus g
+(ASort h2 n2) k))).(let TMP_101 \def (\lambda (n: nat).((eq A (aplus g (ASort
+(S h3) n1) k) (aplus g (ASort n n2) k)) \to ((((eq A (aplus g (ASort h3 n1)
+k) (aplus g (ASort n n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort
+O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match n with [O
+\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to
+(let TMP_98 \def (ASort h3 n1) in (let TMP_100 \def (match n with [O
+\Rightarrow (let TMP_99 \def (next g n2) in (ASort O TMP_99)) | (S h)
+\Rightarrow (ASort h n2)]) in (leq g TMP_98 TMP_100)))))) in (let TMP_141
+\def (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
-\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
-(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
-(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
-(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
-(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
-n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
-(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
-(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
-(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
-h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
-\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
+\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(let TMP_102 \def (next
+g n2) in (let TMP_103 \def (ASort O n2) in (let TMP_104 \def (S k) in (let
+TMP_105 \def (aplus g TMP_103 TMP_104) in (let TMP_108 \def (\lambda (a:
+A).(let TMP_106 \def (ASort h3 n1) in (let TMP_107 \def (aplus g TMP_106 k)
+in (eq A TMP_107 a)))) in (let TMP_109 \def (S h3) in (let TMP_110 \def
+(ASort TMP_109 n1) in (let TMP_111 \def (S k) in (let TMP_112 \def (aplus g
+TMP_110 TMP_111) in (let TMP_116 \def (\lambda (a: A).(let TMP_113 \def
+(ASort O n2) in (let TMP_114 \def (S k) in (let TMP_115 \def (aplus g TMP_113
+TMP_114) in (eq A a TMP_115))))) in (let TMP_117 \def (ASort O n2) in (let
+TMP_118 \def (aplus g TMP_117 k) in (let TMP_123 \def (\lambda (a: A).(let
+TMP_119 \def (asucc g a) in (let TMP_120 \def (ASort O n2) in (let TMP_121
+\def (aplus g TMP_120 k) in (let TMP_122 \def (asucc g TMP_121) in (eq A
+TMP_119 TMP_122)))))) in (let TMP_124 \def (ASort O n2) in (let TMP_125 \def
+(aplus g TMP_124 k) in (let TMP_126 \def (asucc g TMP_125) in (let TMP_127
+\def (refl_equal A TMP_126) in (let TMP_128 \def (S h3) in (let TMP_129 \def
+(ASort TMP_128 n1) in (let TMP_130 \def (aplus g TMP_129 k) in (let TMP_131
+\def (eq_ind_r A TMP_118 TMP_123 TMP_127 TMP_130 H2) in (let TMP_132 \def
+(ASort h3 n1) in (let TMP_133 \def (aplus g TMP_132 k) in (let TMP_134 \def
+(aplus_sort_S_S_simpl g n1 h3 k) in (let TMP_135 \def (eq_ind A TMP_112
+TMP_116 TMP_131 TMP_133 TMP_134) in (let TMP_136 \def (next g n2) in (let
+TMP_137 \def (ASort O TMP_136) in (let TMP_138 \def (aplus g TMP_137 k) in
+(let TMP_139 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_140 \def (eq_ind
+A TMP_105 TMP_108 TMP_135 TMP_138 TMP_139) in (leq_sort g h3 O n1 TMP_102 k
+TMP_140))))))))))))))))))))))))))))))))) in (let TMP_184 \def (\lambda (h4:
+nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort h4
+n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) \to
+(leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
-g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
-n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
-(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
-(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
-(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
-(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
-h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
-(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
-(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
-(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
-(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
-(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
-a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-(* COMMENTS
-Initial nodes: 1907
-END *)
+g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(let TMP_142 \def
+(S h3) in (let TMP_143 \def (ASort TMP_142 n1) in (let TMP_144 \def (S k) in
+(let TMP_145 \def (aplus g TMP_143 TMP_144) in (let TMP_148 \def (\lambda (a:
+A).(let TMP_146 \def (ASort h4 n2) in (let TMP_147 \def (aplus g TMP_146 k)
+in (eq A a TMP_147)))) in (let TMP_149 \def (S h4) in (let TMP_150 \def
+(ASort TMP_149 n2) in (let TMP_151 \def (S k) in (let TMP_152 \def (aplus g
+TMP_150 TMP_151) in (let TMP_157 \def (\lambda (a: A).(let TMP_153 \def (S
+h3) in (let TMP_154 \def (ASort TMP_153 n1) in (let TMP_155 \def (S k) in
+(let TMP_156 \def (aplus g TMP_154 TMP_155) in (eq A TMP_156 a)))))) in (let
+TMP_158 \def (S h4) in (let TMP_159 \def (ASort TMP_158 n2) in (let TMP_160
+\def (aplus g TMP_159 k) in (let TMP_166 \def (\lambda (a: A).(let TMP_161
+\def (asucc g a) in (let TMP_162 \def (S h4) in (let TMP_163 \def (ASort
+TMP_162 n2) in (let TMP_164 \def (aplus g TMP_163 k) in (let TMP_165 \def
+(asucc g TMP_164) in (eq A TMP_161 TMP_165))))))) in (let TMP_167 \def (S h4)
+in (let TMP_168 \def (ASort TMP_167 n2) in (let TMP_169 \def (aplus g TMP_168
+k) in (let TMP_170 \def (asucc g TMP_169) in (let TMP_171 \def (refl_equal A
+TMP_170) in (let TMP_172 \def (S h3) in (let TMP_173 \def (ASort TMP_172 n1)
+in (let TMP_174 \def (aplus g TMP_173 k) in (let TMP_175 \def (eq_ind_r A
+TMP_160 TMP_166 TMP_171 TMP_174 H2) in (let TMP_176 \def (ASort h4 n2) in
+(let TMP_177 \def (aplus g TMP_176 k) in (let TMP_178 \def
+(aplus_sort_S_S_simpl g n2 h4 k) in (let TMP_179 \def (eq_ind A TMP_152
+TMP_157 TMP_175 TMP_177 TMP_178) in (let TMP_180 \def (ASort h3 n1) in (let
+TMP_181 \def (aplus g TMP_180 k) in (let TMP_182 \def (aplus_sort_S_S_simpl g
+n1 h3 k) in (let TMP_183 \def (eq_ind A TMP_145 TMP_148 TMP_179 TMP_181
+TMP_182) in (leq_sort g h3 h4 n1 n2 k
+TMP_183)))))))))))))))))))))))))))))))))))) in (nat_ind TMP_101 TMP_141
+TMP_184 h2 H1 IHh1))))))) in (nat_ind TMP_8 TMP_97 TMP_185 h1 H0)))))))))) in
+(let TMP_189 \def (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3
+a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5:
+A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g
+a5) (asucc g a6))).(let TMP_187 \def (asucc g a5) in (let TMP_188 \def (asucc
+g a6) in (leq_head g a3 a4 H0 TMP_187 TMP_188 H3))))))))))) in (leq_ind g
+TMP_3 TMP_186 TMP_189 a1 a2 H))))))).
theorem asucc_inj:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
g a2)) \to (leq g a1 a2))))
\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
-(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
-(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
-(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
-n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
-(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
-n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
-(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
-(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1
-g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind
-nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
-h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
-(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3
-O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g
-(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in
-(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a:
-A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S
-x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
-a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
-(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
-nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2)))
-\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g
-(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O
-(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g
-(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
-n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
-x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
-(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
-n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8
-n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
-(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0)
-(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10
-(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in
-(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0))
-(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g
-(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq
-g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda
-(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to
-((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort
-n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))
-(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
-n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
-\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3
-n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
-(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
+ \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall
+(a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) in (let TMP_315
+\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(let TMP_3 \def
+(\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (let TMP_2
+\def (ASort n n0) in (leq g TMP_2 a)))) in (let TMP_260 \def (\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g
+(ASort n1 n2)))).(let TMP_6 \def (\lambda (n3: nat).((leq g (asucc g (ASort
+n3 n0)) (asucc g (ASort n1 n2))) \to (let TMP_4 \def (ASort n3 n0) in (let
+TMP_5 \def (ASort n1 n2) in (leq g TMP_4 TMP_5))))) in (let TMP_133 \def
+(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).(let
+TMP_9 \def (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort
+n3 n2))) \to (let TMP_7 \def (ASort O n0) in (let TMP_8 \def (ASort n3 n2) in
+(leq g TMP_7 TMP_8))))) in (let TMP_73 \def (\lambda (H1: (leq g (asucc g
+(ASort O n0)) (asucc g (ASort O n2)))).(let TMP_10 \def (next g n0) in (let
+TMP_11 \def (next g n2) in (let TMP_12 \def (ASort O TMP_11) in (let H_x \def
+(leq_gen_sort1 g O TMP_10 TMP_12 H1) in (let H2 \def H_x in (let TMP_18 \def
+(\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_13 \def
+(next g n0) in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (aplus g
+TMP_14 k) in (let TMP_16 \def (ASort h2 n3) in (let TMP_17 \def (aplus g
+TMP_16 k) in (eq A TMP_15 TMP_17))))))))) in (let TMP_22 \def (\lambda (n3:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_19 \def (next g n2) in
+(let TMP_20 \def (ASort O TMP_19) in (let TMP_21 \def (ASort h2 n3) in (eq A
+TMP_20 TMP_21))))))) in (let TMP_23 \def (ASort O n0) in (let TMP_24 \def
+(ASort O n2) in (let TMP_25 \def (leq g TMP_23 TMP_24) in (let TMP_72 \def
+(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A
+(aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4:
+(eq A (ASort O (next g n2)) (ASort x1 x0))).(let TMP_26 \def (\lambda (e:
+A).(match e with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O]))
+in (let TMP_27 \def (next g n2) in (let TMP_28 \def (ASort O TMP_27) in (let
+TMP_29 \def (ASort x1 x0) in (let H5 \def (f_equal A nat TMP_26 TMP_28 TMP_29
+H4) in (let TMP_31 \def (\lambda (e: A).(match e with [(ASort _ n3)
+\Rightarrow n3 | (AHead _ _) \Rightarrow (let TMP_30 \def (match g with
+[(mk_G next _) \Rightarrow next]) in (TMP_30 n2))])) in (let TMP_32 \def
+(next g n2) in (let TMP_33 \def (ASort O TMP_32) in (let TMP_34 \def (ASort
+x1 x0) in (let H6 \def (f_equal A nat TMP_31 TMP_33 TMP_34 H4) in (let TMP_71
+\def (\lambda (H7: (eq nat O x1)).(let TMP_40 \def (\lambda (n3: nat).(let
+TMP_35 \def (next g n0) in (let TMP_36 \def (ASort O TMP_35) in (let TMP_37
+\def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort n3 x0) in (let TMP_39
+\def (aplus g TMP_38 x2) in (eq A TMP_37 TMP_39))))))) in (let H8 \def
+(eq_ind_r nat x1 TMP_40 H3 O H7) in (let TMP_46 \def (\lambda (n3: nat).(let
+TMP_41 \def (next g n0) in (let TMP_42 \def (ASort O TMP_41) in (let TMP_43
+\def (aplus g TMP_42 x2) in (let TMP_44 \def (ASort O n3) in (let TMP_45 \def
+(aplus g TMP_44 x2) in (eq A TMP_43 TMP_45))))))) in (let TMP_47 \def (next g
+n2) in (let H9 \def (eq_ind_r nat x0 TMP_46 H8 TMP_47 H6) in (let TMP_48 \def
+(next g n0) in (let TMP_49 \def (ASort O TMP_48) in (let TMP_50 \def (aplus g
+TMP_49 x2) in (let TMP_54 \def (\lambda (a: A).(let TMP_51 \def (next g n2)
+in (let TMP_52 \def (ASort O TMP_51) in (let TMP_53 \def (aplus g TMP_52 x2)
+in (eq A a TMP_53))))) in (let TMP_55 \def (ASort O n0) in (let TMP_56 \def
+(S x2) in (let TMP_57 \def (aplus g TMP_55 TMP_56) in (let TMP_58 \def
+(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_50 TMP_54 H9
+TMP_57 TMP_58) in (let TMP_59 \def (next g n2) in (let TMP_60 \def (ASort O
+TMP_59) in (let TMP_61 \def (aplus g TMP_60 x2) in (let TMP_65 \def (\lambda
+(a: A).(let TMP_62 \def (ASort O n0) in (let TMP_63 \def (S x2) in (let
+TMP_64 \def (aplus g TMP_62 TMP_63) in (eq A TMP_64 a))))) in (let TMP_66
+\def (ASort O n2) in (let TMP_67 \def (S x2) in (let TMP_68 \def (aplus g
+TMP_66 TMP_67) in (let TMP_69 \def (aplus_sort_O_S_simpl g n2 x2) in (let H11
+\def (eq_ind_r A TMP_61 TMP_65 H10 TMP_68 TMP_69) in (let TMP_70 \def (S x2)
+in (leq_sort g O O n0 n2 TMP_70 H11)))))))))))))))))))))))))) in (TMP_71
+H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_18 TMP_22 TMP_25 TMP_72
+H2))))))))))))) in (let TMP_132 \def (\lambda (n3: nat).(\lambda (_: (((leq g
+(asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0)
+(ASort n3 n2))))).(\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort
+(S n3) n2)))).(let TMP_74 \def (next g n0) in (let TMP_75 \def (ASort n3 n2)
+in (let H_x \def (leq_gen_sort1 g O TMP_74 TMP_75 H1) in (let H2 \def H_x in
+(let TMP_81 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(let
+TMP_76 \def (next g n0) in (let TMP_77 \def (ASort O TMP_76) in (let TMP_78
+\def (aplus g TMP_77 k) in (let TMP_79 \def (ASort h2 n4) in (let TMP_80 \def
+(aplus g TMP_79 k) in (eq A TMP_78 TMP_80))))))))) in (let TMP_84 \def
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_82 \def
+(ASort n3 n2) in (let TMP_83 \def (ASort h2 n4) in (eq A TMP_82 TMP_83))))))
+in (let TMP_85 \def (ASort O n0) in (let TMP_86 \def (S n3) in (let TMP_87
+\def (ASort TMP_86 n2) in (let TMP_88 \def (leq g TMP_85 TMP_87) in (let
+TMP_131 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
+nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
+x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 x0))).(let TMP_89
+\def (\lambda (e: A).(match e with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
+\Rightarrow n3])) in (let TMP_90 \def (ASort n3 n2) in (let TMP_91 \def
+(ASort x1 x0) in (let H5 \def (f_equal A nat TMP_89 TMP_90 TMP_91 H4) in (let
+TMP_92 \def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 |
+(AHead _ _) \Rightarrow n2])) in (let TMP_93 \def (ASort n3 n2) in (let
+TMP_94 \def (ASort x1 x0) in (let H6 \def (f_equal A nat TMP_92 TMP_93 TMP_94
+H4) in (let TMP_130 \def (\lambda (H7: (eq nat n3 x1)).(let TMP_100 \def
+(\lambda (n4: nat).(let TMP_95 \def (next g n0) in (let TMP_96 \def (ASort O
+TMP_95) in (let TMP_97 \def (aplus g TMP_96 x2) in (let TMP_98 \def (ASort n4
+x0) in (let TMP_99 \def (aplus g TMP_98 x2) in (eq A TMP_97 TMP_99))))))) in
+(let H8 \def (eq_ind_r nat x1 TMP_100 H3 n3 H7) in (let TMP_106 \def (\lambda
+(n4: nat).(let TMP_101 \def (next g n0) in (let TMP_102 \def (ASort O
+TMP_101) in (let TMP_103 \def (aplus g TMP_102 x2) in (let TMP_104 \def
+(ASort n3 n4) in (let TMP_105 \def (aplus g TMP_104 x2) in (eq A TMP_103
+TMP_105))))))) in (let H9 \def (eq_ind_r nat x0 TMP_106 H8 n2 H6) in (let
+TMP_107 \def (next g n0) in (let TMP_108 \def (ASort O TMP_107) in (let
+TMP_109 \def (aplus g TMP_108 x2) in (let TMP_112 \def (\lambda (a: A).(let
+TMP_110 \def (ASort n3 n2) in (let TMP_111 \def (aplus g TMP_110 x2) in (eq A
+a TMP_111)))) in (let TMP_113 \def (ASort O n0) in (let TMP_114 \def (S x2)
+in (let TMP_115 \def (aplus g TMP_113 TMP_114) in (let TMP_116 \def
+(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_109 TMP_112
+H9 TMP_115 TMP_116) in (let TMP_117 \def (ASort n3 n2) in (let TMP_118 \def
+(aplus g TMP_117 x2) in (let TMP_122 \def (\lambda (a: A).(let TMP_119 \def
+(ASort O n0) in (let TMP_120 \def (S x2) in (let TMP_121 \def (aplus g
+TMP_119 TMP_120) in (eq A TMP_121 a))))) in (let TMP_123 \def (S n3) in (let
+TMP_124 \def (ASort TMP_123 n2) in (let TMP_125 \def (S x2) in (let TMP_126
+\def (aplus g TMP_124 TMP_125) in (let TMP_127 \def (aplus_sort_S_S_simpl g
+n2 n3 x2) in (let H11 \def (eq_ind_r A TMP_118 TMP_122 H10 TMP_126 TMP_127)
+in (let TMP_128 \def (S n3) in (let TMP_129 \def (S x2) in (leq_sort g O
+TMP_128 n0 n2 TMP_129 H11)))))))))))))))))))))))))) in (TMP_130
+H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_81 TMP_84 TMP_88 TMP_131
+H2))))))))))))))) in (nat_ind TMP_9 TMP_73 TMP_132 n1 H0))))) in (let TMP_259
+\def (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0))
+(asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda
+(H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(let
+TMP_137 \def (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
+(ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4
+n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (let TMP_134 \def (S n3)
+in (let TMP_135 \def (ASort TMP_134 n0) in (let TMP_136 \def (ASort n4 n2) in
+(leq g TMP_135 TMP_136))))))) in (let TMP_200 \def (\lambda (H1: (leq g
+(asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(\lambda (_: (((leq g
+(asucc g (ASort n3 n0)) (asucc g (ASort O n2))) \to (leq g (ASort n3 n0)
+(ASort O n2))))).(let TMP_138 \def (next g n2) in (let TMP_139 \def (ASort O
+TMP_138) in (let H_x \def (leq_gen_sort1 g n3 n0 TMP_139 H1) in (let H2 \def
+H_x in (let TMP_144 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(let TMP_140 \def (ASort n3 n0) in (let TMP_141 \def (aplus g TMP_140 k)
+in (let TMP_142 \def (ASort h2 n4) in (let TMP_143 \def (aplus g TMP_142 k)
+in (eq A TMP_141 TMP_143)))))))) in (let TMP_148 \def (\lambda (n4:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_145 \def (next g n2) in
+(let TMP_146 \def (ASort O TMP_145) in (let TMP_147 \def (ASort h2 n4) in (eq
+A TMP_146 TMP_147))))))) in (let TMP_149 \def (S n3) in (let TMP_150 \def
+(ASort TMP_149 n0) in (let TMP_151 \def (ASort O n2) in (let TMP_152 \def
+(leq g TMP_150 TMP_151) in (let TMP_199 \def (\lambda (x0: nat).(\lambda (x1:
nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4:
-nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7)
-in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3
-n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def
-(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
-(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
-x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
-(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4:
-nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
-n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
-g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
-n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
-n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
-n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
-(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
-nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
-(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
-n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
-(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
-n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
-\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
-n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1
-x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1
-(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0)
-x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
-(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
-\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
-(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2))
-a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2))
-in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1
-H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n
-n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda
-(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
-a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
-a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
+x0))).(let TMP_153 \def (\lambda (e: A).(match e with [(ASort n4 _)
+\Rightarrow n4 | (AHead _ _) \Rightarrow O])) in (let TMP_154 \def (next g
+n2) in (let TMP_155 \def (ASort O TMP_154) in (let TMP_156 \def (ASort x1 x0)
+in (let H5 \def (f_equal A nat TMP_153 TMP_155 TMP_156 H4) in (let TMP_158
+\def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
+\Rightarrow (let TMP_157 \def (match g with [(mk_G next _) \Rightarrow next])
+in (TMP_157 n2))])) in (let TMP_159 \def (next g n2) in (let TMP_160 \def
+(ASort O TMP_159) in (let TMP_161 \def (ASort x1 x0) in (let H6 \def (f_equal
+A nat TMP_158 TMP_160 TMP_161 H4) in (let TMP_198 \def (\lambda (H7: (eq nat
+O x1)).(let TMP_166 \def (\lambda (n4: nat).(let TMP_162 \def (ASort n3 n0)
+in (let TMP_163 \def (aplus g TMP_162 x2) in (let TMP_164 \def (ASort n4 x0)
+in (let TMP_165 \def (aplus g TMP_164 x2) in (eq A TMP_163 TMP_165)))))) in
+(let H8 \def (eq_ind_r nat x1 TMP_166 H3 O H7) in (let TMP_171 \def (\lambda
+(n4: nat).(let TMP_167 \def (ASort n3 n0) in (let TMP_168 \def (aplus g
+TMP_167 x2) in (let TMP_169 \def (ASort O n4) in (let TMP_170 \def (aplus g
+TMP_169 x2) in (eq A TMP_168 TMP_170)))))) in (let TMP_172 \def (next g n2)
+in (let H9 \def (eq_ind_r nat x0 TMP_171 H8 TMP_172 H6) in (let TMP_173 \def
+(ASort n3 n0) in (let TMP_174 \def (aplus g TMP_173 x2) in (let TMP_178 \def
+(\lambda (a: A).(let TMP_175 \def (next g n2) in (let TMP_176 \def (ASort O
+TMP_175) in (let TMP_177 \def (aplus g TMP_176 x2) in (eq A a TMP_177))))) in
+(let TMP_179 \def (S n3) in (let TMP_180 \def (ASort TMP_179 n0) in (let
+TMP_181 \def (S x2) in (let TMP_182 \def (aplus g TMP_180 TMP_181) in (let
+TMP_183 \def (aplus_sort_S_S_simpl g n0 n3 x2) in (let H10 \def (eq_ind_r A
+TMP_174 TMP_178 H9 TMP_182 TMP_183) in (let TMP_184 \def (next g n2) in (let
+TMP_185 \def (ASort O TMP_184) in (let TMP_186 \def (aplus g TMP_185 x2) in
+(let TMP_191 \def (\lambda (a: A).(let TMP_187 \def (S n3) in (let TMP_188
+\def (ASort TMP_187 n0) in (let TMP_189 \def (S x2) in (let TMP_190 \def
+(aplus g TMP_188 TMP_189) in (eq A TMP_190 a)))))) in (let TMP_192 \def
+(ASort O n2) in (let TMP_193 \def (S x2) in (let TMP_194 \def (aplus g
+TMP_192 TMP_193) in (let TMP_195 \def (aplus_sort_O_S_simpl g n2 x2) in (let
+H11 \def (eq_ind_r A TMP_186 TMP_191 H10 TMP_194 TMP_195) in (let TMP_196
+\def (S n3) in (let TMP_197 \def (S x2) in (leq_sort g TMP_196 O n0 n2
+TMP_197 H11))))))))))))))))))))))))))) in (TMP_198 H5))))))))))))))))) in
+(ex2_3_ind nat nat nat TMP_144 TMP_148 TMP_152 TMP_199 H2)))))))))))))) in
+(let TMP_258 \def (\lambda (n4: nat).(\lambda (_: (((leq g (asucc g (ASort (S
+n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0))
+(asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (leq g
+(ASort (S n3) n0) (ASort n4 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S
+n3) n0)) (asucc g (ASort (S n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort
+n3 n0)) (asucc g (ASort (S n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4)
+n2))))).(let TMP_201 \def (ASort n4 n2) in (let H_x \def (leq_gen_sort1 g n3
+n0 TMP_201 H1) in (let H2 \def H_x in (let TMP_206 \def (\lambda (n5:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_202 \def (ASort n3 n0) in
+(let TMP_203 \def (aplus g TMP_202 k) in (let TMP_204 \def (ASort h2 n5) in
+(let TMP_205 \def (aplus g TMP_204 k) in (eq A TMP_203 TMP_205)))))))) in
+(let TMP_209 \def (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (_:
+nat).(let TMP_207 \def (ASort n4 n2) in (let TMP_208 \def (ASort h2 n5) in
+(eq A TMP_207 TMP_208)))))) in (let TMP_210 \def (S n3) in (let TMP_211 \def
+(ASort TMP_210 n0) in (let TMP_212 \def (S n4) in (let TMP_213 \def (ASort
+TMP_212 n2) in (let TMP_214 \def (leq g TMP_211 TMP_213) in (let TMP_257 \def
+(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A
+(aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A
+(ASort n4 n2) (ASort x1 x0))).(let TMP_215 \def (\lambda (e: A).(match e with
+[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) in (let TMP_216
+\def (ASort n4 n2) in (let TMP_217 \def (ASort x1 x0) in (let H5 \def
+(f_equal A nat TMP_215 TMP_216 TMP_217 H4) in (let TMP_218 \def (\lambda (e:
+A).(match e with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2]))
+in (let TMP_219 \def (ASort n4 n2) in (let TMP_220 \def (ASort x1 x0) in (let
+H6 \def (f_equal A nat TMP_218 TMP_219 TMP_220 H4) in (let TMP_256 \def
+(\lambda (H7: (eq nat n4 x1)).(let TMP_225 \def (\lambda (n5: nat).(let
+TMP_221 \def (ASort n3 n0) in (let TMP_222 \def (aplus g TMP_221 x2) in (let
+TMP_223 \def (ASort n5 x0) in (let TMP_224 \def (aplus g TMP_223 x2) in (eq A
+TMP_222 TMP_224)))))) in (let H8 \def (eq_ind_r nat x1 TMP_225 H3 n4 H7) in
+(let TMP_230 \def (\lambda (n5: nat).(let TMP_226 \def (ASort n3 n0) in (let
+TMP_227 \def (aplus g TMP_226 x2) in (let TMP_228 \def (ASort n4 n5) in (let
+TMP_229 \def (aplus g TMP_228 x2) in (eq A TMP_227 TMP_229)))))) in (let H9
+\def (eq_ind_r nat x0 TMP_230 H8 n2 H6) in (let TMP_231 \def (ASort n3 n0) in
+(let TMP_232 \def (aplus g TMP_231 x2) in (let TMP_235 \def (\lambda (a:
+A).(let TMP_233 \def (ASort n4 n2) in (let TMP_234 \def (aplus g TMP_233 x2)
+in (eq A a TMP_234)))) in (let TMP_236 \def (S n3) in (let TMP_237 \def
+(ASort TMP_236 n0) in (let TMP_238 \def (S x2) in (let TMP_239 \def (aplus g
+TMP_237 TMP_238) in (let TMP_240 \def (aplus_sort_S_S_simpl g n0 n3 x2) in
+(let H10 \def (eq_ind_r A TMP_232 TMP_235 H9 TMP_239 TMP_240) in (let TMP_241
+\def (ASort n4 n2) in (let TMP_242 \def (aplus g TMP_241 x2) in (let TMP_247
+\def (\lambda (a: A).(let TMP_243 \def (S n3) in (let TMP_244 \def (ASort
+TMP_243 n0) in (let TMP_245 \def (S x2) in (let TMP_246 \def (aplus g TMP_244
+TMP_245) in (eq A TMP_246 a)))))) in (let TMP_248 \def (S n4) in (let TMP_249
+\def (ASort TMP_248 n2) in (let TMP_250 \def (S x2) in (let TMP_251 \def
+(aplus g TMP_249 TMP_250) in (let TMP_252 \def (aplus_sort_S_S_simpl g n2 n4
+x2) in (let H11 \def (eq_ind_r A TMP_242 TMP_247 H10 TMP_251 TMP_252) in (let
+TMP_253 \def (S n3) in (let TMP_254 \def (S n4) in (let TMP_255 \def (S x2)
+in (leq_sort g TMP_253 TMP_254 n0 n2 TMP_255 H11)))))))))))))))))))))))))))
+in (TMP_256 H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_206 TMP_209
+TMP_214 TMP_257 H2)))))))))))))))) in (nat_ind TMP_137 TMP_200 TMP_258 n1 H0
+IHn))))))) in (nat_ind TMP_6 TMP_133 TMP_259 n H))))))) in (let TMP_314 \def
+(\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to
+(leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g
+(ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq
+g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).(let TMP_263 \def (\lambda
+(n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1
+n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g
+(ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a
+a0))) \to (let TMP_261 \def (ASort n1 n0) in (let TMP_262 \def (AHead a a0)
+in (leq g TMP_261 TMP_262))))))) in (let TMP_288 \def (\lambda (_: (((leq g
+(asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda
+(_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0)
+a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a
+a0)))).(let TMP_264 \def (next g n0) in (let TMP_265 \def (asucc g a0) in
+(let TMP_266 \def (AHead a TMP_265) in (let H_x \def (leq_gen_sort1 g O
+TMP_264 TMP_266 H4) in (let H5 \def H_x in (let TMP_272 \def (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_267 \def (next g n0) in
+(let TMP_268 \def (ASort O TMP_267) in (let TMP_269 \def (aplus g TMP_268 k)
+in (let TMP_270 \def (ASort h2 n2) in (let TMP_271 \def (aplus g TMP_270 k)
+in (eq A TMP_269 TMP_271))))))))) in (let TMP_276 \def (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_273 \def (asucc g a0) in
+(let TMP_274 \def (AHead a TMP_273) in (let TMP_275 \def (ASort h2 n2) in (eq
+A TMP_274 TMP_275))))))) in (let TMP_277 \def (ASort O n0) in (let TMP_278
+\def (AHead a a0) in (let TMP_279 \def (leq g TMP_277 TMP_278) in (let
+TMP_287 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
+nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1
+x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let
+TMP_280 \def (asucc g a0) in (let TMP_281 \def (AHead a TMP_280) in (let
+TMP_282 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) in (let TMP_283 \def (ASort x1 x0) in (let H8
+\def (eq_ind A TMP_281 TMP_282 I TMP_283 H7) in (let TMP_284 \def (ASort O
+n0) in (let TMP_285 \def (AHead a a0) in (let TMP_286 \def (leq g TMP_284
+TMP_285) in (False_ind TMP_286 H8)))))))))))))) in (ex2_3_ind nat nat nat
+TMP_272 TMP_276 TMP_279 TMP_287 H5))))))))))))))) in (let TMP_313 \def
+(\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g
a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
-n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
-(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
-n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
-g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
-(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
-g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
-n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
-(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
-(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
-(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
-x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
-(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
-(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
-(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
-\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
-a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
-(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
-n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
-(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
-a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
-a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
-(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
-(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
-(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
-(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
-(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
-g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
-x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
-H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
-A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
-(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
-(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
-(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
-(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
-(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
-(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
-in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
-(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
-(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
-(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
-x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
-g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
-(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
-nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
-\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
-(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
-(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
-(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
-g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
-(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
-a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
-n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
-(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
-(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
-(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
-(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
-(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
-A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
-(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
-a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
-a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8
-\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3
-(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
-((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n)
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h)
-\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0
-a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g
-a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def
-(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9)
-in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in
-(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2))))))
-a1)).
-(* COMMENTS
-Initial nodes: 4697
-END *)
+n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a
+a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to
+(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
+n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g
+(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let TMP_289 \def (asucc
+g a0) in (let TMP_290 \def (AHead a TMP_289) in (let H_x \def (leq_gen_sort1
+g n1 n0 TMP_290 H4) in (let H5 \def H_x in (let TMP_295 \def (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_291 \def (ASort n1 n0) in
+(let TMP_292 \def (aplus g TMP_291 k) in (let TMP_293 \def (ASort h2 n2) in
+(let TMP_294 \def (aplus g TMP_293 k) in (eq A TMP_292 TMP_294)))))))) in
+(let TMP_299 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_:
+nat).(let TMP_296 \def (asucc g a0) in (let TMP_297 \def (AHead a TMP_296) in
+(let TMP_298 \def (ASort h2 n2) in (eq A TMP_297 TMP_298))))))) in (let
+TMP_300 \def (S n1) in (let TMP_301 \def (ASort TMP_300 n0) in (let TMP_302
+\def (AHead a a0) in (let TMP_303 \def (leq g TMP_301 TMP_302) in (let
+TMP_312 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
+nat).(\lambda (_: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0)
+x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let TMP_304
+\def (asucc g a0) in (let TMP_305 \def (AHead a TMP_304) in (let TMP_306 \def
+(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) in (let TMP_307 \def (ASort x1 x0) in (let H8 \def
+(eq_ind A TMP_305 TMP_306 I TMP_307 H7) in (let TMP_308 \def (S n1) in (let
+TMP_309 \def (ASort TMP_308 n0) in (let TMP_310 \def (AHead a a0) in (let
+TMP_311 \def (leq g TMP_309 TMP_310) in (False_ind TMP_311 H8)))))))))))))))
+in (ex2_3_ind nat nat nat TMP_295 TMP_299 TMP_303 TMP_312 H5)))))))))))))))))
+in (nat_ind TMP_263 TMP_288 TMP_313 n H H0 H1))))))))) in (A_ind TMP_3
+TMP_260 TMP_314 a2))))))) in (let TMP_396 \def (\lambda (a: A).(\lambda (_:
+((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
+a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
+(asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(let TMP_317 \def
+(\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (let
+TMP_316 \def (AHead a a0) in (leq g TMP_316 a3)))) in (let TMP_364 \def
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
+a0)) (asucc g (ASort n n0)))).(let TMP_320 \def (\lambda (n1: nat).((leq g
+(asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (let TMP_318 \def (AHead
+a a0) in (let TMP_319 \def (ASort n1 n0) in (leq g TMP_318 TMP_319))))) in
+(let TMP_342 \def (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort
+O n0)))).(let TMP_321 \def (asucc g a0) in (let TMP_322 \def (next g n0) in
+(let TMP_323 \def (ASort O TMP_322) in (let H_x \def (leq_gen_head1 g a
+TMP_321 TMP_323 H2) in (let H3 \def H_x in (let TMP_324 \def (\lambda (a3:
+A).(\lambda (_: A).(leq g a a3))) in (let TMP_326 \def (\lambda (_:
+A).(\lambda (a4: A).(let TMP_325 \def (asucc g a0) in (leq g TMP_325 a4))))
+in (let TMP_330 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_327 \def
+(next g n0) in (let TMP_328 \def (ASort O TMP_327) in (let TMP_329 \def
+(AHead a3 a4) in (eq A TMP_328 TMP_329)))))) in (let TMP_331 \def (AHead a
+a0) in (let TMP_332 \def (ASort O n0) in (let TMP_333 \def (leq g TMP_331
+TMP_332) in (let TMP_341 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_:
+(leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A
+(ASort O (next g n0)) (AHead x0 x1))).(let TMP_334 \def (next g n0) in (let
+TMP_335 \def (ASort O TMP_334) in (let TMP_336 \def (\lambda (ee: A).(match
+ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) in
+(let TMP_337 \def (AHead x0 x1) in (let H7 \def (eq_ind A TMP_335 TMP_336 I
+TMP_337 H6) in (let TMP_338 \def (AHead a a0) in (let TMP_339 \def (ASort O
+n0) in (let TMP_340 \def (leq g TMP_338 TMP_339) in (False_ind TMP_340
+H7)))))))))))))) in (ex3_2_ind A A TMP_324 TMP_326 TMP_330 TMP_333 TMP_341
+H3)))))))))))))) in (let TMP_363 \def (\lambda (n1: nat).(\lambda (_: (((leq
+g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0)
+(ASort n1 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort
+(S n1) n0)))).(let TMP_343 \def (asucc g a0) in (let TMP_344 \def (ASort n1
+n0) in (let H_x \def (leq_gen_head1 g a TMP_343 TMP_344 H2) in (let H3 \def
+H_x in (let TMP_345 \def (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) in
+(let TMP_347 \def (\lambda (_: A).(\lambda (a4: A).(let TMP_346 \def (asucc g
+a0) in (leq g TMP_346 a4)))) in (let TMP_350 \def (\lambda (a3: A).(\lambda
+(a4: A).(let TMP_348 \def (ASort n1 n0) in (let TMP_349 \def (AHead a3 a4) in
+(eq A TMP_348 TMP_349))))) in (let TMP_351 \def (AHead a a0) in (let TMP_352
+\def (S n1) in (let TMP_353 \def (ASort TMP_352 n0) in (let TMP_354 \def (leq
+g TMP_351 TMP_353) in (let TMP_362 \def (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda
+(H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let TMP_355 \def (ASort n1 n0) in
+(let TMP_356 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow
+True | (AHead _ _) \Rightarrow False])) in (let TMP_357 \def (AHead x0 x1) in
+(let H7 \def (eq_ind A TMP_355 TMP_356 I TMP_357 H6) in (let TMP_358 \def
+(AHead a a0) in (let TMP_359 \def (S n1) in (let TMP_360 \def (ASort TMP_359
+n0) in (let TMP_361 \def (leq g TMP_358 TMP_360) in (False_ind TMP_361
+H7)))))))))))))) in (ex3_2_ind A A TMP_345 TMP_347 TMP_350 TMP_354 TMP_362
+H3)))))))))))))))) in (nat_ind TMP_320 TMP_342 TMP_363 n H1))))))) in (let
+TMP_395 \def (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0))
+(asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_:
+(((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
+a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
+a4)))).(let TMP_365 \def (asucc g a0) in (let TMP_366 \def (asucc g a4) in
+(let TMP_367 \def (AHead a3 TMP_366) in (let H_x \def (leq_gen_head1 g a
+TMP_365 TMP_367 H3) in (let H4 \def H_x in (let TMP_368 \def (\lambda (a5:
+A).(\lambda (_: A).(leq g a a5))) in (let TMP_370 \def (\lambda (_:
+A).(\lambda (a6: A).(let TMP_369 \def (asucc g a0) in (leq g TMP_369 a6))))
+in (let TMP_374 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_371 \def
+(asucc g a4) in (let TMP_372 \def (AHead a3 TMP_371) in (let TMP_373 \def
+(AHead a5 a6) in (eq A TMP_372 TMP_373)))))) in (let TMP_375 \def (AHead a
+a0) in (let TMP_376 \def (AHead a3 a4) in (let TMP_377 \def (leq g TMP_375
+TMP_376) in (let TMP_394 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5:
+(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A
+(AHead a3 (asucc g a4)) (AHead x0 x1))).(let TMP_378 \def (\lambda (e:
+A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5]))
+in (let TMP_379 \def (asucc g a4) in (let TMP_380 \def (AHead a3 TMP_379) in
+(let TMP_381 \def (AHead x0 x1) in (let H8 \def (f_equal A A TMP_378 TMP_380
+TMP_381 H7) in (let TMP_384 \def (\lambda (e: A).(match e with [(ASort _ _)
+\Rightarrow (asucc g a4) | (AHead _ a5) \Rightarrow a5])) in (let TMP_385
+\def (asucc g a4) in (let TMP_386 \def (AHead a3 TMP_385) in (let TMP_387
+\def (AHead x0 x1) in (let H9 \def (f_equal A A TMP_384 TMP_386 TMP_387 H7)
+in (let TMP_393 \def (\lambda (H10: (eq A a3 x0)).(let TMP_389 \def (\lambda
+(a5: A).(let TMP_388 \def (asucc g a0) in (leq g TMP_388 a5))) in (let
+TMP_390 \def (asucc g a4) in (let H11 \def (eq_ind_r A x1 TMP_389 H6 TMP_390
+H9) in (let TMP_391 \def (\lambda (a5: A).(leq g a a5)) in (let H12 \def
+(eq_ind_r A x0 TMP_391 H5 a3 H10) in (let TMP_392 \def (H0 a4 H11) in
+(leq_head g a a3 H12 a0 a4 TMP_392)))))))) in (TMP_393 H8))))))))))))))))) in
+(ex3_2_ind A A TMP_368 TMP_370 TMP_374 TMP_377 TMP_394 H4))))))))))))))))))
+in (A_ind TMP_317 TMP_364 TMP_395 a2))))))))) in (A_ind TMP_1 TMP_315 TMP_396
+a1))))).
theorem leq_asucc:
\forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
a0)))))
\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
-A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
-A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
-(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
-(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
-(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
-(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
-(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
-g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
-(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
-a)).
-(* COMMENTS
-Initial nodes: 221
-END *)
+ \lambda (g: G).(\lambda (a: A).(let TMP_3 \def (\lambda (a0: A).(let TMP_2
+\def (\lambda (a1: A).(let TMP_1 \def (asucc g a1) in (leq g a0 TMP_1))) in
+(ex A TMP_2))) in (let TMP_11 \def (\lambda (n: nat).(\lambda (n0: nat).(let
+TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (ASort n n0) in (let TMP_5 \def
+(asucc g a0) in (leq g TMP_4 TMP_5)))) in (let TMP_7 \def (S n) in (let TMP_8
+\def (ASort TMP_7 n0) in (let TMP_9 \def (ASort n n0) in (let TMP_10 \def
+(leq_refl g TMP_9) in (ex_intro A TMP_6 TMP_8 TMP_10)))))))) in (let TMP_26
+\def (\lambda (a0: A).(\lambda (_: (ex A (\lambda (a1: A).(leq g a0 (asucc g
+a1))))).(\lambda (a1: A).(\lambda (H0: (ex A (\lambda (a2: A).(leq g a1
+(asucc g a2))))).(let H1 \def H0 in (let TMP_13 \def (\lambda (a2: A).(let
+TMP_12 \def (asucc g a2) in (leq g a1 TMP_12))) in (let TMP_16 \def (\lambda
+(a2: A).(let TMP_14 \def (AHead a0 a1) in (let TMP_15 \def (asucc g a2) in
+(leq g TMP_14 TMP_15)))) in (let TMP_17 \def (ex A TMP_16) in (let TMP_25
+\def (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc g x))).(let TMP_20 \def
+(\lambda (a2: A).(let TMP_18 \def (AHead a0 a1) in (let TMP_19 \def (asucc g
+a2) in (leq g TMP_18 TMP_19)))) in (let TMP_21 \def (AHead a0 x) in (let
+TMP_22 \def (leq_refl g a0) in (let TMP_23 \def (asucc g x) in (let TMP_24
+\def (leq_head g a0 a0 TMP_22 a1 TMP_23 H2) in (ex_intro A TMP_20 TMP_21
+TMP_24)))))))) in (ex_ind A TMP_13 TMP_17 TMP_25 H1)))))))))) in (A_ind TMP_3
+TMP_11 TMP_26 a))))).
theorem leq_ahead_asucc_false:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
(asucc g a1)) \to (\forall (P: Prop).P))))
\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
-(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
-\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
-nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
-(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
-(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1
-g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
-(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+ \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall
+(a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) in
+(let TMP_34 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2:
+A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P:
+Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2)
+(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
+(ASort h n0)])) \to P)) in (let TMP_18 \def (\lambda (H0: (leq g (AHead
+(ASort O n0) a2) (ASort O (next g n0)))).(let TMP_3 \def (ASort O n0) in (let
+TMP_4 \def (next g n0) in (let TMP_5 \def (ASort O TMP_4) in (let H_x \def
+(leq_gen_head1 g TMP_3 a2 TMP_5 H0) in (let H1 \def H_x in (let TMP_7 \def
+(\lambda (a3: A).(\lambda (_: A).(let TMP_6 \def (ASort O n0) in (leq g TMP_6
+a3)))) in (let TMP_8 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in
+(let TMP_12 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_9 \def (next g
+n0) in (let TMP_10 \def (ASort O TMP_9) in (let TMP_11 \def (AHead a3 a4) in
+(eq A TMP_10 TMP_11)))))) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1:
A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
-(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
-(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
+(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let TMP_13 \def (next g n0)
+in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (\lambda (ee:
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) in (let TMP_16 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_14
+TMP_15 I TMP_16 H4) in (False_ind P H5))))))))))) in (ex3_2_ind A A TMP_7
+TMP_8 TMP_12 P TMP_17 H1))))))))))) in (let TMP_33 \def (\lambda (n1:
nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
-H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1
-\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S
-n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_:
-(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
-(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
-a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
-(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
-Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
-(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g
-(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
-(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
-x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g
-a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc
-(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0
-with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) |
-(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead
-_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
-H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
-(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6)))))))
-H2)))))))))) a1)).
-(* COMMENTS
-Initial nodes: 927
-END *)
+TMP_19 \def (S n1) in (let TMP_20 \def (ASort TMP_19 n0) in (let TMP_21 \def
+(ASort n1 n0) in (let H_x \def (leq_gen_head1 g TMP_20 a2 TMP_21 H0) in (let
+H1 \def H_x in (let TMP_24 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_22
+\def (S n1) in (let TMP_23 \def (ASort TMP_22 n0) in (leq g TMP_23 a3))))) in
+(let TMP_25 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in (let
+TMP_28 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_26 \def (ASort n1 n0)
+in (let TMP_27 \def (AHead a3 a4) in (eq A TMP_26 TMP_27))))) in (let TMP_32
+\def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0)
+x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0
+x1))).(let TMP_29 \def (ASort n1 n0) in (let TMP_30 \def (\lambda (ee:
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) in (let TMP_31 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_29
+TMP_30 I TMP_31 H4) in (False_ind P H5)))))))))) in (ex3_2_ind A A TMP_24
+TMP_25 TMP_28 P TMP_32 H1))))))))))))) in (nat_ind TMP_2 TMP_18 TMP_33 n
+H))))))))) in (let TMP_61 \def (\lambda (a: A).(\lambda (_: ((\forall (a2:
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda
+(a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0))
+\to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead
+(AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let TMP_35 \def
+(AHead a a0) in (let TMP_36 \def (asucc g a0) in (let TMP_37 \def (AHead a
+TMP_36) in (let H_x \def (leq_gen_head1 g TMP_35 a2 TMP_37 H1) in (let H2
+\def H_x in (let TMP_39 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_38
+\def (AHead a a0) in (leq g TMP_38 a3)))) in (let TMP_40 \def (\lambda (_:
+A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_44 \def (\lambda (a3:
+A).(\lambda (a4: A).(let TMP_41 \def (asucc g a0) in (let TMP_42 \def (AHead
+a TMP_41) in (let TMP_43 \def (AHead a3 a4) in (eq A TMP_42 TMP_43)))))) in
+(let TMP_60 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g
+(AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq A (AHead a
+(asucc g a0)) (AHead x0 x1))).(let TMP_45 \def (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_46
+\def (asucc g a0) in (let TMP_47 \def (AHead a TMP_46) in (let TMP_48 \def
+(AHead x0 x1) in (let H6 \def (f_equal A A TMP_45 TMP_47 TMP_48 H5) in (let
+TMP_51 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g
+a0) | (AHead _ a3) \Rightarrow a3])) in (let TMP_52 \def (asucc g a0) in (let
+TMP_53 \def (AHead a TMP_52) in (let TMP_54 \def (AHead x0 x1) in (let H7
+\def (f_equal A A TMP_51 TMP_53 TMP_54 H5) in (let TMP_59 \def (\lambda (H8:
+(eq A a x0)).(let TMP_55 \def (\lambda (a3: A).(leq g a2 a3)) in (let TMP_56
+\def (asucc g a0) in (let H9 \def (eq_ind_r A x1 TMP_55 H4 TMP_56 H7) in (let
+TMP_58 \def (\lambda (a3: A).(let TMP_57 \def (AHead a a0) in (leq g TMP_57
+a3))) in (let H10 \def (eq_ind_r A x0 TMP_58 H3 a H8) in (leq_ahead_false_1 g
+a a0 H10 P))))))) in (TMP_59 H6))))))))))))))))) in (ex3_2_ind A A TMP_39
+TMP_40 TMP_44 P TMP_60 H2))))))))))))))))) in (A_ind TMP_1 TMP_34 TMP_61
+a1))))).
theorem leq_asucc_false:
\forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
Prop).P)))
\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
-a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
-(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
-\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
-(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
-n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
-(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O
-(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
-(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
-(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow
-n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
-with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r
-nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g
-(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7
-n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
-(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O
-n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S
-x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1:
-nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda
-(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next
-g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda
-(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g
-n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
+ \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).((leq g
+(asucc g a0) a0) \to (\forall (P: Prop).P))) in (let TMP_103 \def (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n
+n0))).(\lambda (P: Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (match n1
+with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])
+(ASort n1 n0)) \to P)) in (let TMP_50 \def (\lambda (H0: (leq g (ASort O
+(next g n0)) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def
+(ASort O n0) in (let H_x \def (leq_gen_sort1 g O TMP_3 TMP_4 H0) in (let H1
+\def H_x in (let TMP_10 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(k: nat).(let TMP_5 \def (next g n0) in (let TMP_6 \def (ASort O TMP_5) in
+(let TMP_7 \def (aplus g TMP_6 k) in (let TMP_8 \def (ASort h2 n2) in (let
+TMP_9 \def (aplus g TMP_8 k) in (eq A TMP_7 TMP_9))))))))) in (let TMP_13
+\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_11 \def
+(ASort O n0) in (let TMP_12 \def (ASort h2 n2) in (eq A TMP_11 TMP_12))))))
+in (let TMP_49 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
+nat).(\lambda (H2: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
+x1 x0) x2))).(\lambda (H3: (eq A (ASort O n0) (ASort x1 x0))).(let TMP_14
+\def (\lambda (e: A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _)
+\Rightarrow O])) in (let TMP_15 \def (ASort O n0) in (let TMP_16 \def (ASort
+x1 x0) in (let H4 \def (f_equal A nat TMP_14 TMP_15 TMP_16 H3) in (let TMP_17
+\def (\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _)
+\Rightarrow n0])) in (let TMP_18 \def (ASort O n0) in (let TMP_19 \def (ASort
+x1 x0) in (let H5 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H3) in (let TMP_48
+\def (\lambda (H6: (eq nat O x1)).(let TMP_25 \def (\lambda (n1: nat).(let
+TMP_20 \def (next g n0) in (let TMP_21 \def (ASort O TMP_20) in (let TMP_22
+\def (aplus g TMP_21 x2) in (let TMP_23 \def (ASort n1 x0) in (let TMP_24
+\def (aplus g TMP_23 x2) in (eq A TMP_22 TMP_24))))))) in (let H7 \def
+(eq_ind_r nat x1 TMP_25 H2 O H6) in (let TMP_31 \def (\lambda (n1: nat).(let
+TMP_26 \def (next g n0) in (let TMP_27 \def (ASort O TMP_26) in (let TMP_28
+\def (aplus g TMP_27 x2) in (let TMP_29 \def (ASort O n1) in (let TMP_30 \def
+(aplus g TMP_29 x2) in (eq A TMP_28 TMP_30))))))) in (let H8 \def (eq_ind_r
+nat x0 TMP_31 H7 n0 H5) in (let TMP_32 \def (next g n0) in (let TMP_33 \def
+(ASort O TMP_32) in (let TMP_34 \def (aplus g TMP_33 x2) in (let TMP_37 \def
+(\lambda (a0: A).(let TMP_35 \def (ASort O n0) in (let TMP_36 \def (aplus g
+TMP_35 x2) in (eq A a0 TMP_36)))) in (let TMP_38 \def (ASort O n0) in (let
+TMP_39 \def (S x2) in (let TMP_40 \def (aplus g TMP_38 TMP_39) in (let TMP_41
+\def (aplus_sort_O_S_simpl g n0 x2) in (let H9 \def (eq_ind_r A TMP_34 TMP_37
+H8 TMP_40 TMP_41) in (let TMP_42 \def (S x2) in (let TMP_43 \def (ASort O n0)
+in (let H_y \def (aplus_inj g TMP_42 x2 TMP_43 H9) in (let TMP_44 \def
+(\lambda (n1: nat).(le n1 x2)) in (let TMP_45 \def (le_n x2) in (let TMP_46
+\def (S x2) in (let TMP_47 \def (eq_ind_r nat x2 TMP_44 TMP_45 TMP_46 H_y) in
+(le_Sx_x x2 TMP_47 P)))))))))))))))))))))) in (TMP_48 H4))))))))))))))) in
+(ex2_3_ind nat nat nat TMP_10 TMP_13 P TMP_49 H1))))))))) in (let TMP_102
+\def (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to
+P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let TMP_51 \def
+(S n1) in (let TMP_52 \def (ASort TMP_51 n0) in (let H_x \def (leq_gen_sort1
+g n1 n0 TMP_52 H0) in (let H1 \def H_x in (let TMP_57 \def (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_53 \def (ASort n1 n0) in
+(let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2 n2) in (let
+TMP_56 \def (aplus g TMP_55 k) in (eq A TMP_54 TMP_56)))))))) in (let TMP_61
+\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_58 \def
+(S n1) in (let TMP_59 \def (ASort TMP_58 n0) in (let TMP_60 \def (ASort h2
+n2) in (eq A TMP_59 TMP_60))))))) in (let TMP_101 \def (\lambda (x0:
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
-n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e
-in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _
-_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1)
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def
-(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g
-(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda
-(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2)))
-H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda
-(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1)
-n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g
-(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2:
-nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H)))))
-(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
-Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
-(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
-a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
-(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
-x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
-(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
-\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1)
-H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2:
-A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0
-(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6)))))))
-H2))))))))) a)).
-(* COMMENTS
-Initial nodes: 1327
-END *)
+n1) n0) (ASort x1 x0))).(let TMP_62 \def (\lambda (e: A).(match e with
+[(ASort n2 _) \Rightarrow n2 | (AHead _ _) \Rightarrow (S n1)])) in (let
+TMP_63 \def (S n1) in (let TMP_64 \def (ASort TMP_63 n0) in (let TMP_65 \def
+(ASort x1 x0) in (let H4 \def (f_equal A nat TMP_62 TMP_64 TMP_65 H3) in (let
+TMP_66 \def (\lambda (e: A).(match e with [(ASort _ n2) \Rightarrow n2 |
+(AHead _ _) \Rightarrow n0])) in (let TMP_67 \def (S n1) in (let TMP_68 \def
+(ASort TMP_67 n0) in (let TMP_69 \def (ASort x1 x0) in (let H5 \def (f_equal
+A nat TMP_66 TMP_68 TMP_69 H3) in (let TMP_100 \def (\lambda (H6: (eq nat (S
+n1) x1)).(let TMP_74 \def (\lambda (n2: nat).(let TMP_70 \def (ASort n1 n0)
+in (let TMP_71 \def (aplus g TMP_70 x2) in (let TMP_72 \def (ASort n2 x0) in
+(let TMP_73 \def (aplus g TMP_72 x2) in (eq A TMP_71 TMP_73)))))) in (let
+TMP_75 \def (S n1) in (let H7 \def (eq_ind_r nat x1 TMP_74 H2 TMP_75 H6) in
+(let TMP_81 \def (\lambda (n2: nat).(let TMP_76 \def (ASort n1 n0) in (let
+TMP_77 \def (aplus g TMP_76 x2) in (let TMP_78 \def (S n1) in (let TMP_79
+\def (ASort TMP_78 n2) in (let TMP_80 \def (aplus g TMP_79 x2) in (eq A
+TMP_77 TMP_80))))))) in (let H8 \def (eq_ind_r nat x0 TMP_81 H7 n0 H5) in
+(let TMP_82 \def (ASort n1 n0) in (let TMP_83 \def (aplus g TMP_82 x2) in
+(let TMP_87 \def (\lambda (a0: A).(let TMP_84 \def (S n1) in (let TMP_85 \def
+(ASort TMP_84 n0) in (let TMP_86 \def (aplus g TMP_85 x2) in (eq A a0
+TMP_86))))) in (let TMP_88 \def (S n1) in (let TMP_89 \def (ASort TMP_88 n0)
+in (let TMP_90 \def (S x2) in (let TMP_91 \def (aplus g TMP_89 TMP_90) in
+(let TMP_92 \def (aplus_sort_S_S_simpl g n0 n1 x2) in (let H9 \def (eq_ind_r
+A TMP_83 TMP_87 H8 TMP_91 TMP_92) in (let TMP_93 \def (S x2) in (let TMP_94
+\def (S n1) in (let TMP_95 \def (ASort TMP_94 n0) in (let H_y \def (aplus_inj
+g TMP_93 x2 TMP_95 H9) in (let TMP_96 \def (\lambda (n2: nat).(le n2 x2)) in
+(let TMP_97 \def (le_n x2) in (let TMP_98 \def (S x2) in (let TMP_99 \def
+(eq_ind_r nat x2 TMP_96 TMP_97 TMP_98 H_y) in (le_Sx_x x2 TMP_99
+P)))))))))))))))))))))))) in (TMP_100 H4))))))))))))))))) in (ex2_3_ind nat
+nat nat TMP_57 TMP_61 P TMP_101 H1))))))))))) in (nat_ind TMP_2 TMP_50
+TMP_102 n H)))))))) in (let TMP_123 \def (\lambda (a0: A).(\lambda (_: (((leq
+g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda
+(H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1:
+(leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let
+TMP_104 \def (asucc g a1) in (let TMP_105 \def (AHead a0 a1) in (let H_x \def
+(leq_gen_head1 g a0 TMP_104 TMP_105 H1) in (let H2 \def H_x in (let TMP_106
+\def (\lambda (a3: A).(\lambda (_: A).(leq g a0 a3))) in (let TMP_108 \def
+(\lambda (_: A).(\lambda (a4: A).(let TMP_107 \def (asucc g a1) in (leq g
+TMP_107 a4)))) in (let TMP_111 \def (\lambda (a3: A).(\lambda (a4: A).(let
+TMP_109 \def (AHead a0 a1) in (let TMP_110 \def (AHead a3 a4) in (eq A
+TMP_109 TMP_110))))) in (let TMP_122 \def (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (leq g a0 x0)).(\lambda (H4: (leq g (asucc g a1)
+x1)).(\lambda (H5: (eq A (AHead a0 a1) (AHead x0 x1))).(let TMP_112 \def
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
+\Rightarrow a2])) in (let TMP_113 \def (AHead a0 a1) in (let TMP_114 \def
+(AHead x0 x1) in (let H6 \def (f_equal A A TMP_112 TMP_113 TMP_114 H5) in
+(let TMP_115 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a1 |
+(AHead _ a2) \Rightarrow a2])) in (let TMP_116 \def (AHead a0 a1) in (let
+TMP_117 \def (AHead x0 x1) in (let H7 \def (f_equal A A TMP_115 TMP_116
+TMP_117 H5) in (let TMP_121 \def (\lambda (H8: (eq A a0 x0)).(let TMP_119
+\def (\lambda (a2: A).(let TMP_118 \def (asucc g a1) in (leq g TMP_118 a2)))
+in (let H9 \def (eq_ind_r A x1 TMP_119 H4 a1 H7) in (let TMP_120 \def
+(\lambda (a2: A).(leq g a0 a2)) in (let H10 \def (eq_ind_r A x0 TMP_120 H3 a0
+H8) in (H0 H9 P)))))) in (TMP_121 H6))))))))))))))) in (ex3_2_ind A A TMP_106
+TMP_108 TMP_111 P TMP_122 H2))))))))))))))) in (A_ind TMP_1 TMP_103 TMP_123
+a))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/aplus/defs.ma".
+include "basic_1/aplus/defs.ma".
inductive leq (g: G): A \to (A \to Prop) \def
| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/leq/defs.ma".
+include "basic_1/leq/defs.ma".
+
+let rec leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1:
+nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k:
+nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (P
+(ASort h1 n1) (ASort h2 n2))))))))) (f0: (\forall (a1: A).(\forall (a2:
+A).((leq g a1 a2) \to ((P a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leq
+g a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead a2 a4))))))))))) (a: A)
+(a0: A) (l: leq g a a0) on l: P a a0 \def match l with [(leq_sort h1 h2 n1 n2
+k e) \Rightarrow (f h1 h2 n1 n2 k e) | (leq_head a1 a2 l0 a3 a4 l1)
+\Rightarrow (let TMP_1 \def ((leq_ind g P f f0) a1 a2 l0) in (let TMP_2 \def
+((leq_ind g P f f0) a3 a4 l1) in (f0 a1 a2 l0 TMP_1 a3 a4 l1 TMP_2)))].
theorem leq_gen_sort1:
\forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
(ASort h2 n2))))))))))
\def
\lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort
-h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A
-a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g
-(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat
-nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a
-k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(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 h0])) (ASort h0 n0) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0
-h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort
-n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0
-H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let TMP_1 \def (ASort h1 n1) in
+(let TMP_2 \def (\lambda (a: A).(leq g a a2)) in (let TMP_9 \def (\lambda (a:
+A).(let TMP_6 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(let TMP_3 \def (aplus g a k) in (let TMP_4 \def (ASort h2 n2) in (let
+TMP_5 \def (aplus g TMP_4 k) in (eq A TMP_3 TMP_5))))))) in (let TMP_8 \def
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_7 \def
+(ASort h2 n2) in (eq A a2 TMP_7))))) in (ex2_3 nat nat nat TMP_6 TMP_8)))) in
+(let TMP_78 \def (\lambda (y: A).(\lambda (H0: (leq g y a2)).(let TMP_16 \def
+(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (let TMP_13 \def
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_10 \def
+(aplus g a k) in (let TMP_11 \def (ASort h2 n2) in (let TMP_12 \def (aplus g
+TMP_11 k) in (eq A TMP_10 TMP_12))))))) in (let TMP_15 \def (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_14 \def (ASort h2 n2) in
+(eq A a0 TMP_14))))) in (ex2_3 nat nat nat TMP_13 TMP_15)))))) in (let TMP_64
+\def (\lambda (h0: nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g
+(ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let
+TMP_17 \def (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead
+_ _) \Rightarrow h0])) in (let TMP_18 \def (ASort h0 n0) in (let TMP_19 \def
+(ASort h1 n1) in (let H3 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H2) in (let
+TMP_20 \def (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead
+_ _) \Rightarrow n0])) in (let TMP_21 \def (ASort h0 n0) in (let TMP_22 \def
+(ASort h1 n1) in (let H4 \def (f_equal A nat TMP_20 TMP_21 TMP_22 H2) in (let
+TMP_63 \def (\lambda (H5: (eq nat h0 h1)).(let TMP_27 \def (\lambda (n:
+nat).(let TMP_23 \def (ASort h0 n) in (let TMP_24 \def (aplus g TMP_23 k) in
+(let TMP_25 \def (ASort h2 n2) in (let TMP_26 \def (aplus g TMP_25 k) in (eq
+A TMP_24 TMP_26)))))) in (let H6 \def (eq_ind nat n0 TMP_27 H1 n1 H4) in (let
+TMP_36 \def (\lambda (n: nat).(let TMP_32 \def (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(let TMP_28 \def (ASort h0 n) in (let TMP_29
+\def (aplus g TMP_28 k0) in (let TMP_30 \def (ASort h3 n3) in (let TMP_31
+\def (aplus g TMP_30 k0) in (eq A TMP_29 TMP_31)))))))) in (let TMP_35 \def
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_33 \def
+(ASort h2 n2) in (let TMP_34 \def (ASort h3 n3) in (eq A TMP_33 TMP_34))))))
+in (ex2_3 nat nat nat TMP_32 TMP_35)))) in (let TMP_41 \def (\lambda (n:
+nat).(let TMP_37 \def (ASort n n1) in (let TMP_38 \def (aplus g TMP_37 k) in
+(let TMP_39 \def (ASort h2 n2) in (let TMP_40 \def (aplus g TMP_39 k) in (eq
+A TMP_38 TMP_40)))))) in (let H7 \def (eq_ind nat h0 TMP_41 H6 h1 H5) in (let
+TMP_50 \def (\lambda (n: nat).(let TMP_46 \def (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(let TMP_42 \def (ASort n n1) in (let TMP_43
+\def (aplus g TMP_42 k0) in (let TMP_44 \def (ASort h3 n3) in (let TMP_45
+\def (aplus g TMP_44 k0) in (eq A TMP_43 TMP_45)))))))) in (let TMP_49 \def
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_47 \def
+(ASort h2 n2) in (let TMP_48 \def (ASort h3 n3) in (eq A TMP_47 TMP_48))))))
+in (ex2_3 nat nat nat TMP_46 TMP_49)))) in (let TMP_55 \def (\lambda (n3:
+nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_51 \def (ASort h1 n1) in
+(let TMP_52 \def (aplus g TMP_51 k0) in (let TMP_53 \def (ASort h3 n3) in
+(let TMP_54 \def (aplus g TMP_53 k0) in (eq A TMP_52 TMP_54)))))))) in (let
+TMP_58 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let
+TMP_56 \def (ASort h2 n2) in (let TMP_57 \def (ASort h3 n3) in (eq A TMP_56
+TMP_57)))))) in (let TMP_59 \def (ASort h2 n2) in (let TMP_60 \def
+(refl_equal A TMP_59) in (let TMP_61 \def (ex2_3_intro nat nat nat TMP_55
+TMP_58 n2 h2 k H7 TMP_60) in (let TMP_62 \def (eq_ind_r nat h1 TMP_50 TMP_61
+h0 H5) in (eq_ind_r nat n1 TMP_36 TMP_62 n0 H4)))))))))))))) in (TMP_63
+H3))))))))))))))))) in (let TMP_77 \def (\lambda (a1: A).(\lambda (a3:
+A).(\lambda (_: (leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to
(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0)))
-H))))).
-(* COMMENTS
-Initial nodes: 913
-END *)
+nat).(eq A (aplus g a1 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2
+n2))))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4
+a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda
+(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a4 k) (aplus g
+(ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_:
+nat).(eq A a5 (ASort h2 n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort
+h1 n1))).(let TMP_65 \def (AHead a1 a4) in (let TMP_66 \def (\lambda (ee:
+A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) in (let TMP_67 \def (ASort h1 n1) in (let H6 \def (eq_ind A TMP_65
+TMP_66 I TMP_67 H5) in (let TMP_72 \def (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(let TMP_68 \def (AHead a1 a4) in (let TMP_69 \def
+(aplus g TMP_68 k) in (let TMP_70 \def (ASort h2 n2) in (let TMP_71 \def
+(aplus g TMP_70 k) in (eq A TMP_69 TMP_71)))))))) in (let TMP_75 \def
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_73 \def
+(AHead a3 a5) in (let TMP_74 \def (ASort h2 n2) in (eq A TMP_73 TMP_74))))))
+in (let TMP_76 \def (ex2_3 nat nat nat TMP_72 TMP_75) in (False_ind TMP_76
+H6))))))))))))))))) in (leq_ind g TMP_16 TMP_64 TMP_77 y a2 H0)))))) in
+(insert_eq A TMP_1 TMP_2 TMP_9 TMP_78 H))))))))).
theorem leq_gen_head1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a3 (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 h1 n1)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\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 a1 a3))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
-a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4)
-(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 a4 | (AHead _ a6)
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0
-a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6:
-A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5)
-(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5)))))))))
-H6))))))))))) y a H0))) H))))).
-(* COMMENTS
-Initial nodes: 797
-END *)
+(H: (leq g (AHead a1 a2) a)).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def
+(\lambda (a0: A).(leq g a0 a)) in (let TMP_7 \def (\lambda (_: A).(let TMP_3
+\def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_4 \def
+(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) 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 y a)).(let TMP_12 \def (\lambda (a0: A).(\lambda (a3:
+A).((eq A a0 (AHead a1 a2)) \to (let TMP_8 \def (\lambda (a4: A).(\lambda (_:
+A).(leq g a1 a4))) in (let TMP_9 \def (\lambda (_: A).(\lambda (a5: A).(leq g
+a2 a5))) in (let TMP_11 \def (\lambda (a4: A).(\lambda (a5: A).(let TMP_10
+\def (AHead a4 a5) in (eq A a3 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 h1 n1) (AHead a1 a2))).(let TMP_13 \def (ASort h1 n1) 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 a1 a3))) in (let TMP_17 \def (\lambda (_:
+A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_20 \def (\lambda (a3:
+A).(\lambda (a4: A).(let TMP_18 \def (ASort h2 n2) 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 a0 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_:
+A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda
+(a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))))))).(\lambda (a4:
+A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a4
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a1
+a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda (a6:
+A).(\lambda (a7: A).(eq A a5 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead
+a0 a4) (AHead a1 a2))).(let TMP_23 \def (\lambda (e: A).(match e with [(ASort
+_ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) in (let TMP_24 \def
+(AHead a0 a4) 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 a4 | (AHead _ a6) \Rightarrow a6])) in (let TMP_27
+\def (AHead a0 a4) 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 a0
+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 a1 a7))) in (let TMP_30
+\def (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) in (let TMP_32 \def
+(\lambda (a7: A).(\lambda (a8: A).(let TMP_31 \def (AHead a7 a8) in (eq A a5
+TMP_31)))) in (ex3_2 A A TMP_29 TMP_30 TMP_32)))))) in (let H9 \def (eq_ind A
+a4 TMP_33 H4 a2 H7) in (let TMP_34 \def (\lambda (a6: A).(leq g a6 a5)) in
+(let H10 \def (eq_ind A a4 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 a1 a7))) in (let TMP_36 \def (\lambda (_: A).(\lambda (a8:
+A).(leq g a2 a8))) in (let TMP_38 \def (\lambda (a7: A).(\lambda (a8: A).(let
+TMP_37 \def (AHead a7 a8) in (eq A a3 TMP_37)))) in (ex3_2 A A TMP_35 TMP_36
+TMP_38)))))) in (let H11 \def (eq_ind A a0 TMP_39 H2 a1 H8) in (let TMP_40
+\def (\lambda (a6: A).(leq g a6 a3)) in (let H12 \def (eq_ind A a0 TMP_40 H1
+a1 H8) in (let TMP_41 \def (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6)))
+in (let TMP_42 \def (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) in (let
+TMP_45 \def (\lambda (a6: A).(\lambda (a7: A).(let TMP_43 \def (AHead a3 a5)
+in (let TMP_44 \def (AHead a6 a7) in (eq A TMP_43 TMP_44))))) in (let TMP_46
+\def (AHead a3 a5) in (let TMP_47 \def (refl_equal A TMP_46) in (ex3_2_intro
+A A TMP_41 TMP_42 TMP_45 a3 a5 H12 H10 TMP_47))))))))))))))) in (TMP_48
+H6))))))))))))))))))) in (leq_ind g TMP_12 TMP_22 TMP_49 y a H0)))))) in
+(insert_eq A TMP_1 TMP_2 TMP_7 TMP_50 H))))))))).
theorem leq_gen_sort2:
\forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
(ASort h2 n2))))))))))
\def
\lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq
-A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind
-g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat
-nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus
-g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(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 h2])) (ASort h2 n2) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2
-h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort
-h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1
-n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2
-H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat
+A).(\lambda (H: (leq g a2 (ASort h1 n1))).(let TMP_1 \def (ASort h1 n1) in
+(let TMP_2 \def (\lambda (a: A).(leq g a2 a)) in (let TMP_9 \def (\lambda (a:
+A).(let TMP_6 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(let TMP_3 \def (ASort h2 n2) in (let TMP_4 \def (aplus g TMP_3 k) in
+(let TMP_5 \def (aplus g a k) in (eq A TMP_4 TMP_5))))))) in (let TMP_8 \def
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_7 \def
+(ASort h2 n2) in (eq A a2 TMP_7))))) in (ex2_3 nat nat nat TMP_6 TMP_8)))) in
+(let TMP_78 \def (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(let TMP_16 \def
+(\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (let TMP_13
+\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_10 \def
+(ASort h2 n2) in (let TMP_11 \def (aplus g TMP_10 k) in (let TMP_12 \def
+(aplus g a0 k) in (eq A TMP_11 TMP_12))))))) in (let TMP_15 \def (\lambda
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_14 \def (ASort h2 n2)
+in (eq A a TMP_14))))) in (ex2_3 nat nat nat TMP_13 TMP_15)))))) in (let
+TMP_64 \def (\lambda (h0: nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda
+(n2: nat).(\lambda (k: nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k)
+(aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1
+n1))).(let TMP_17 \def (\lambda (e: A).(match e with [(ASort n _) \Rightarrow
+n | (AHead _ _) \Rightarrow h2])) in (let TMP_18 \def (ASort h2 n2) in (let
+TMP_19 \def (ASort h1 n1) in (let H3 \def (f_equal A nat TMP_17 TMP_18 TMP_19
+H2) in (let TMP_20 \def (\lambda (e: A).(match e with [(ASort _ n)
+\Rightarrow n | (AHead _ _) \Rightarrow n2])) in (let TMP_21 \def (ASort h2
+n2) in (let TMP_22 \def (ASort h1 n1) in (let H4 \def (f_equal A nat TMP_20
+TMP_21 TMP_22 H2) in (let TMP_63 \def (\lambda (H5: (eq nat h2 h1)).(let
+TMP_27 \def (\lambda (n: nat).(let TMP_23 \def (ASort h0 n0) in (let TMP_24
+\def (aplus g TMP_23 k) in (let TMP_25 \def (ASort h2 n) in (let TMP_26 \def
+(aplus g TMP_25 k) in (eq A TMP_24 TMP_26)))))) in (let H6 \def (eq_ind nat
+n2 TMP_27 H1 n1 H4) in (let TMP_36 \def (\lambda (n: nat).(let TMP_32 \def
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_28 \def
+(ASort h3 n3) in (let TMP_29 \def (aplus g TMP_28 k0) in (let TMP_30 \def
+(ASort h2 n) in (let TMP_31 \def (aplus g TMP_30 k0) in (eq A TMP_29
+TMP_31)))))))) in (let TMP_35 \def (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (_: nat).(let TMP_33 \def (ASort h0 n0) in (let TMP_34 \def
+(ASort h3 n3) in (eq A TMP_33 TMP_34)))))) in (ex2_3 nat nat nat TMP_32
+TMP_35)))) in (let TMP_41 \def (\lambda (n: nat).(let TMP_37 \def (ASort h0
+n0) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def (ASort n n1)
+in (let TMP_40 \def (aplus g TMP_39 k) in (eq A TMP_38 TMP_40)))))) in (let
+H7 \def (eq_ind nat h2 TMP_41 H6 h1 H5) in (let TMP_50 \def (\lambda (n:
+nat).(let TMP_46 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
+nat).(let TMP_42 \def (ASort h3 n3) in (let TMP_43 \def (aplus g TMP_42 k0)
+in (let TMP_44 \def (ASort n n1) in (let TMP_45 \def (aplus g TMP_44 k0) in
+(eq A TMP_43 TMP_45)))))))) in (let TMP_49 \def (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (_: nat).(let TMP_47 \def (ASort h0 n0) in (let TMP_48
+\def (ASort h3 n3) in (eq A TMP_47 TMP_48)))))) in (ex2_3 nat nat nat TMP_46
+TMP_49)))) in (let TMP_55 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(k0: nat).(let TMP_51 \def (ASort h3 n3) in (let TMP_52 \def (aplus g TMP_51
+k0) in (let TMP_53 \def (ASort h1 n1) in (let TMP_54 \def (aplus g TMP_53 k0)
+in (eq A TMP_52 TMP_54)))))))) in (let TMP_58 \def (\lambda (n3:
+nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_56 \def (ASort h0 n0) in
+(let TMP_57 \def (ASort h3 n3) in (eq A TMP_56 TMP_57)))))) in (let TMP_59
+\def (ASort h0 n0) in (let TMP_60 \def (refl_equal A TMP_59) in (let TMP_61
+\def (ex2_3_intro nat nat nat TMP_55 TMP_58 n0 h0 k H7 TMP_60) in (let TMP_62
+\def (eq_ind_r nat h1 TMP_50 TMP_61 h2 H5) in (eq_ind_r nat n1 TMP_36 TMP_62
+n2 H4)))))))))))))) in (TMP_63 H3))))))))))))))))) in (let TMP_77 \def
+(\lambda (a1: A).(\lambda (a3: A).(\lambda (_: (leq g a1 a3)).(\lambda (_:
+(((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a3
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a1
+(ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g
+a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to (ex2_3 nat nat nat
(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0)))
-H))))).
-(* COMMENTS
-Initial nodes: 913
-END *)
+h2 n2) k) (aplus g a5 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A a4 (ASort h2 n2))))))))).(\lambda (H5: (eq A (AHead a3 a5)
+(ASort h1 n1))).(let TMP_65 \def (AHead a3 a5) in (let TMP_66 \def (\lambda
+(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) in (let TMP_67 \def (ASort h1 n1) in (let H6 \def (eq_ind
+A TMP_65 TMP_66 I TMP_67 H5) in (let TMP_72 \def (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (k: nat).(let TMP_68 \def (ASort h2 n2) in (let TMP_69
+\def (aplus g TMP_68 k) in (let TMP_70 \def (AHead a3 a5) in (let TMP_71 \def
+(aplus g TMP_70 k) in (eq A TMP_69 TMP_71)))))))) in (let TMP_75 \def
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_73 \def
+(AHead a1 a4) in (let TMP_74 \def (ASort h2 n2) in (eq A TMP_73 TMP_74))))))
+in (let TMP_76 \def (ex2_3 nat nat nat TMP_72 TMP_75) in (False_ind TMP_76
+H6))))))))))))))))) in (leq_ind g TMP_16 TMP_64 TMP_77 a2 y H0)))))) in
+(insert_eq A TMP_1 TMP_2 TMP_9 TMP_78 H))))))))).
theorem leq_gen_head2:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a
A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(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)))))))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/leq/fwd.ma".
+include "basic_1/leq/fwd.ma".
-include "Basic-1/aplus/props.ma".
-
-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 H_x \def
-(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A
-(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda
-(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4)
-(AHead a5 a6)))) (leq g a2 a4) (\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 H4 \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 a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3)
-in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a:
-A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a:
-A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
-(* COMMENTS
-Initial nodes: 259
-END *)
+include "basic_1/aplus/props.ma".
theorem leq_refl:
\forall (g: G).(\forall (a: A).(leq g a a))
\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
-(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
-(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
-a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
-H0))))) a)).
-(* COMMENTS
-Initial nodes: 87
-END *)
+ \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).(leq g a0
+a0)) in (let TMP_5 \def (\lambda (n: nat).(\lambda (n0: nat).(let TMP_2 \def
+(ASort n n0) in (let TMP_3 \def (aplus g TMP_2 O) in (let TMP_4 \def
+(refl_equal A TMP_3) in (leq_sort g n n n0 n0 O TMP_4)))))) in (let TMP_6
+\def (\lambda (a0: A).(\lambda (H: (leq g a0 a0)).(\lambda (a1: A).(\lambda
+(H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 H0))))) in (A_ind TMP_1 TMP_5
+TMP_6 a))))).
theorem leq_eq:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
a2))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
-a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
-(* COMMENTS
-Initial nodes: 39
-END *)
+a2)).(let TMP_1 \def (\lambda (a: A).(leq g a1 a)) in (let TMP_2 \def
+(leq_refl g a1) in (eq_ind A a1 TMP_1 TMP_2 a2 H)))))).
theorem leq_sym:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
a2 a1))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
-nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
-(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
-A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
-H1 a6 a5 H3))))))))) a1 a2 H)))).
-(* COMMENTS
-Initial nodes: 173
-END *)
+a2)).(let TMP_1 \def (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) in (let
+TMP_7 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda
+(n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k)
+(aplus g (ASort h2 n2) k))).(let TMP_2 \def (ASort h1 n1) in (let TMP_3 \def
+(aplus g TMP_2 k) in (let TMP_4 \def (ASort h2 n2) in (let TMP_5 \def (aplus
+g TMP_4 k) in (let TMP_6 \def (sym_eq A TMP_3 TMP_5 H0) in (leq_sort g h2 h1
+n2 n1 k TMP_6)))))))))))) in (let TMP_8 \def (\lambda (a3: A).(\lambda (a4:
+A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5:
+A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6
+a5)).(leq_head g a4 a3 H1 a6 a5 H3))))))))) in (leq_ind g TMP_1 TMP_7 TMP_8
+a1 a2 H))))))).
theorem leq_trans:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
-a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
-(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2
-\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3
-(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus
-g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def
-(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1
-x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1
-n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g
-(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r
-nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n)
-(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k)
-x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus
-g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7
-H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
-(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus
-(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
-(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1
-x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g
-(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3:
+a2)).(let TMP_1 \def (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq
+g a0 a3) \to (leq g a a3))))) in (let TMP_63 \def (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
+(H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda
+(a3: A).(\lambda (H1: (leq g (ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1
+g h2 n2 a3 H1) in (let H2 \def H_x in (let TMP_6 \def (\lambda (n3:
+nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_2 \def (ASort h2 n2) in
+(let TMP_3 \def (aplus g TMP_2 k0) in (let TMP_4 \def (ASort h3 n3) in (let
+TMP_5 \def (aplus g TMP_4 k0) in (eq A TMP_3 TMP_5)))))))) in (let TMP_8 \def
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_7 \def
+(ASort h3 n3) in (eq A a3 TMP_7))))) in (let TMP_9 \def (ASort h1 n1) in (let
+TMP_10 \def (leq g TMP_9 a3) in (let TMP_62 \def (\lambda (x0: nat).(\lambda
+(x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2)
+(aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let
+TMP_11 \def (\lambda (e: A).e) in (let TMP_12 \def (ASort x1 x0) in (let H5
+\def (f_equal A A TMP_11 a3 TMP_12 H4) in (let TMP_13 \def (ASort x1 x0) in
+(let TMP_15 \def (\lambda (a: A).(let TMP_14 \def (ASort h1 n1) in (leq g
+TMP_14 a))) in (let TMP_16 \def (ASort h1 n1) in (let TMP_17 \def (ASort x1
+x0) in (let TMP_18 \def (leq g TMP_16 TMP_17) in (let TMP_41 \def (\lambda
+(H6: (lt k x2)).(let TMP_19 \def (ASort h1 n1) in (let TMP_20 \def (ASort h2
+n2) in (let TMP_21 \def (minus x2 k) in (let H_y \def (aplus_reg_r g TMP_19
+TMP_20 k k H0 TMP_21) in (let TMP_22 \def (minus x2 k) in (let TMP_23 \def
+(plus TMP_22 k) in (let TMP_28 \def (\lambda (n: nat).(let TMP_24 \def (ASort
+h1 n1) in (let TMP_25 \def (aplus g TMP_24 n) in (let TMP_26 \def (ASort h2
+n2) in (let TMP_27 \def (aplus g TMP_26 n) in (eq A TMP_25 TMP_27)))))) in
+(let TMP_29 \def (S k) in (let TMP_30 \def (le_n k) in (let TMP_31 \def (le_S
+k k TMP_30) in (let TMP_32 \def (le_trans k TMP_29 x2 TMP_31 H6) in (let
+TMP_33 \def (le_plus_minus_sym k x2 TMP_32) in (let H7 \def (eq_ind_r nat
+TMP_23 TMP_28 H_y x2 TMP_33) in (let TMP_34 \def (ASort h1 n1) in (let TMP_35
+\def (aplus g TMP_34 x2) in (let TMP_36 \def (ASort h2 n2) in (let TMP_37
+\def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort x1 x0) in (let TMP_39
+\def (aplus g TMP_38 x2) in (let TMP_40 \def (trans_eq A TMP_35 TMP_37 TMP_39
+H7 H3) in (leq_sort g h1 x1 n1 x0 x2 TMP_40)))))))))))))))))))))) in (let
+TMP_60 \def (\lambda (H6: (le x2 k)).(let TMP_42 \def (ASort h2 n2) in (let
+TMP_43 \def (ASort x1 x0) in (let TMP_44 \def (minus k x2) in (let H_y \def
+(aplus_reg_r g TMP_42 TMP_43 x2 x2 H3 TMP_44) in (let TMP_45 \def (minus k
+x2) in (let TMP_46 \def (plus TMP_45 x2) in (let TMP_51 \def (\lambda (n:
+nat).(let TMP_47 \def (ASort h2 n2) in (let TMP_48 \def (aplus g TMP_47 n) in
+(let TMP_49 \def (ASort x1 x0) in (let TMP_50 \def (aplus g TMP_49 n) in (eq
+A TMP_48 TMP_50)))))) in (let TMP_52 \def (le_plus_minus_sym x2 k H6) in (let
+H7 \def (eq_ind_r nat TMP_46 TMP_51 H_y k TMP_52) in (let TMP_53 \def (ASort
+h1 n1) in (let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2
+n2) in (let TMP_56 \def (aplus g TMP_55 k) in (let TMP_57 \def (ASort x1 x0)
+in (let TMP_58 \def (aplus g TMP_57 k) in (let TMP_59 \def (trans_eq A TMP_54
+TMP_56 TMP_58 H0 H7) in (leq_sort g h1 x1 n1 x0 k TMP_59)))))))))))))))))) in
+(let TMP_61 \def (lt_le_e k x2 TMP_18 TMP_41 TMP_60) in (eq_ind_r A TMP_13
+TMP_15 TMP_61 a3 H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_6 TMP_8
+TMP_10 TMP_62 H2)))))))))))))))) in (let TMP_79 \def (\lambda (a3:
A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5:
A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6:
A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7)
\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6)
-a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in
-(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8:
-(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0
-(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
-a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
-H5))))))))))))) a1 a2 H)))).
-(* COMMENTS
-Initial nodes: 869
-END *)
+a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in (let
+TMP_64 \def (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) in (let TMP_65
+\def (\lambda (_: A).(\lambda (a8: A).(leq g a6 a8))) in (let TMP_67 \def
+(\lambda (a7: A).(\lambda (a8: A).(let TMP_66 \def (AHead a7 a8) in (eq A a0
+TMP_66)))) in (let TMP_68 \def (AHead a3 a5) in (let TMP_69 \def (leq g
+TMP_68 a0) in (let TMP_78 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: (eq A a0
+(AHead x0 x1))).(let TMP_70 \def (\lambda (e: A).e) in (let TMP_71 \def
+(AHead x0 x1) in (let H9 \def (f_equal A A TMP_70 a0 TMP_71 H8) in (let
+TMP_72 \def (AHead x0 x1) in (let TMP_74 \def (\lambda (a: A).(let TMP_73
+\def (AHead a3 a5) in (leq g TMP_73 a))) in (let TMP_75 \def (H1 x0 H6) in
+(let TMP_76 \def (H3 x1 H7) in (let TMP_77 \def (leq_head g a3 x0 TMP_75 a5
+x1 TMP_76) in (eq_ind_r A TMP_72 TMP_74 TMP_77 a0 H9)))))))))))))) in
+(ex3_2_ind A A TMP_64 TMP_65 TMP_67 TMP_69 TMP_78 H5))))))))))))))))))) in
+(leq_ind g TMP_1 TMP_63 TMP_79 a1 a2 H))))))).
theorem leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
\to (\forall (P: Prop).P))))
\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
-n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
-(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
-(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0)
-a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4:
-A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+ \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall
+(a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) in (let TMP_34
+\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq
+g (AHead (ASort n n0) a2) (ASort n n0))).(\lambda (P: Prop).(let TMP_2 \def
+(\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) in
+(let TMP_15 \def (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O
+n0))).(let TMP_3 \def (ASort O n0) in (let TMP_4 \def (ASort O n0) in (let
+H_x \def (leq_gen_head1 g TMP_3 a2 TMP_4 H0) in (let H1 \def H_x in (let
+TMP_6 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_5 \def (ASort O n0) in
+(leq g TMP_5 a3)))) in (let TMP_7 \def (\lambda (_: A).(\lambda (a4: A).(leq
+g a2 a4))) in (let TMP_10 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_8
+\def (ASort O n0) in (let TMP_9 \def (AHead a3 a4) in (eq A TMP_8 TMP_9)))))
+in (let TMP_14 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O
-n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee:
-A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
-True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
-H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1
-n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0)
-a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2
-(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
-n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
-(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort
-(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
-(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H:
-((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
-Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
-a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
-g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
-(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3)))
-(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda
-(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
-x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) 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 _ a3) \Rightarrow a3]))
-(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
-(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def
-(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0
-H10 P))))) H6))))))) H2)))))))))) a1)).
-(* COMMENTS
-Initial nodes: 797
-END *)
+n0) (AHead x0 x1))).(let TMP_11 \def (ASort O n0) in (let TMP_12 \def
+(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def
+(eq_ind A TMP_11 TMP_12 I TMP_13 H4) in (False_ind P H5)))))))))) in
+(ex3_2_ind A A TMP_6 TMP_7 TMP_10 P TMP_14 H1)))))))))) in (let TMP_33 \def
+(\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (ASort n1
+n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1)
+n0))).(let TMP_16 \def (S n1) in (let TMP_17 \def (ASort TMP_16 n0) in (let
+TMP_18 \def (S n1) in (let TMP_19 \def (ASort TMP_18 n0) in (let H_x \def
+(leq_gen_head1 g TMP_17 a2 TMP_19 H0) in (let H1 \def H_x in (let TMP_22 \def
+(\lambda (a3: A).(\lambda (_: A).(let TMP_20 \def (S n1) in (let TMP_21 \def
+(ASort TMP_20 n0) in (leq g TMP_21 a3))))) in (let TMP_23 \def (\lambda (_:
+A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_27 \def (\lambda (a3:
+A).(\lambda (a4: A).(let TMP_24 \def (S n1) in (let TMP_25 \def (ASort TMP_24
+n0) in (let TMP_26 \def (AHead a3 a4) in (eq A TMP_25 TMP_26)))))) in (let
+TMP_32 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S
+n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort (S n1)
+n0) (AHead x0 x1))).(let TMP_28 \def (S n1) in (let TMP_29 \def (ASort TMP_28
+n0) in (let TMP_30 \def (\lambda (ee: A).(match ee with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) in (let TMP_31 \def
+(AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 TMP_30 I TMP_31 H4) in
+(False_ind P H5))))))))))) in (ex3_2_ind A A TMP_22 TMP_23 TMP_27 P TMP_32
+H1)))))))))))))) in (nat_ind TMP_2 TMP_15 TMP_33 n H))))))))) in (let TMP_54
+\def (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a)
+\to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2:
+A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2:
+A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P:
+Prop).(let TMP_35 \def (AHead a a0) in (let TMP_36 \def (AHead a a0) in (let
+H_x \def (leq_gen_head1 g TMP_35 a2 TMP_36 H1) in (let H2 \def H_x in (let
+TMP_38 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_37 \def (AHead a a0) in
+(leq g TMP_37 a3)))) in (let TMP_39 \def (\lambda (_: A).(\lambda (a4:
+A).(leq g a2 a4))) in (let TMP_42 \def (\lambda (a3: A).(\lambda (a4: A).(let
+TMP_40 \def (AHead a a0) in (let TMP_41 \def (AHead a3 a4) in (eq A TMP_40
+TMP_41))))) in (let TMP_53 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq
+A (AHead a a0) (AHead x0 x1))).(let TMP_43 \def (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_44
+\def (AHead a a0) in (let TMP_45 \def (AHead x0 x1) in (let H6 \def (f_equal
+A A TMP_43 TMP_44 TMP_45 H5) in (let TMP_46 \def (\lambda (e: A).(match e
+with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) in (let
+TMP_47 \def (AHead a a0) in (let TMP_48 \def (AHead x0 x1) in (let H7 \def
+(f_equal A A TMP_46 TMP_47 TMP_48 H5) in (let TMP_52 \def (\lambda (H8: (eq A
+a x0)).(let TMP_49 \def (\lambda (a3: A).(leq g a2 a3)) in (let H9 \def
+(eq_ind_r A x1 TMP_49 H4 a0 H7) in (let TMP_51 \def (\lambda (a3: A).(let
+TMP_50 \def (AHead a a0) in (leq g TMP_50 a3))) in (let H10 \def (eq_ind_r A
+x0 TMP_51 H3 a H8) in (H a0 H10 P)))))) in (TMP_52 H6))))))))))))))) in
+(ex3_2_ind A A TMP_38 TMP_39 TMP_42 P TMP_53 H2)))))))))))))))) in (A_ind
+TMP_1 TMP_34 TMP_54 a1))))).
theorem leq_ahead_false_2:
\forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
\to (\forall (P: Prop).P))))
\def
- \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
-A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
-n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
-(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
-a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O
-n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
+ \lambda (g: G).(\lambda (a2: A).(let TMP_1 \def (\lambda (a: A).(\forall
+(a1: A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) in (let TMP_34
+\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq
+g (AHead a1 (ASort n n0)) (ASort n n0))).(\lambda (P: Prop).(let TMP_2 \def
+(\lambda (n1: nat).((leq g (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) in
+(let TMP_15 \def (\lambda (H0: (leq g (AHead a1 (ASort O n0)) (ASort O
+n0))).(let TMP_3 \def (ASort O n0) in (let TMP_4 \def (ASort O n0) in (let
+H_x \def (leq_gen_head1 g a1 TMP_3 TMP_4 H0) in (let H1 \def H_x in (let
+TMP_5 \def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_7
+\def (\lambda (_: A).(\lambda (a4: A).(let TMP_6 \def (ASort O n0) in (leq g
+TMP_6 a4)))) in (let TMP_10 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_8
+\def (ASort O n0) in (let TMP_9 \def (AHead a3 a4) in (eq A TMP_8 TMP_9)))))
+in (let TMP_14 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0)
-(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
-H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1
-n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1)
-n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0)
-(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
-n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
-a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort
-(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
-(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_:
-((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P:
-Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead
-a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq
-g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
-(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4:
-A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0)
-x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) 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 _ a3) \Rightarrow a3]))
-(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
-(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let
-H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9
-P))))) H6))))))) H2)))))))))) a2)).
-(* COMMENTS
-Initial nodes: 797
-END *)
+(AHead x0 x1))).(let TMP_11 \def (ASort O n0) in (let TMP_12 \def (\lambda
+(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def
+(eq_ind A TMP_11 TMP_12 I TMP_13 H4) in (False_ind P H5)))))))))) in
+(ex3_2_ind A A TMP_5 TMP_7 TMP_10 P TMP_14 H1)))))))))) in (let TMP_33 \def
+(\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 n0)) (ASort n1
+n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) n0)) (ASort (S n1)
+n0))).(let TMP_16 \def (S n1) in (let TMP_17 \def (ASort TMP_16 n0) in (let
+TMP_18 \def (S n1) in (let TMP_19 \def (ASort TMP_18 n0) in (let H_x \def
+(leq_gen_head1 g a1 TMP_17 TMP_19 H0) in (let H1 \def H_x in (let TMP_20 \def
+(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_23 \def (\lambda
+(_: A).(\lambda (a4: A).(let TMP_21 \def (S n1) in (let TMP_22 \def (ASort
+TMP_21 n0) in (leq g TMP_22 a4))))) in (let TMP_27 \def (\lambda (a3:
+A).(\lambda (a4: A).(let TMP_24 \def (S n1) in (let TMP_25 \def (ASort TMP_24
+n0) in (let TMP_26 \def (AHead a3 a4) in (eq A TMP_25 TMP_26)))))) in (let
+TMP_32 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
+x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort (S
+n1) n0) (AHead x0 x1))).(let TMP_28 \def (S n1) in (let TMP_29 \def (ASort
+TMP_28 n0) in (let TMP_30 \def (\lambda (ee: A).(match ee with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) in (let TMP_31 \def
+(AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 TMP_30 I TMP_31 H4) in
+(False_ind P H5))))))))))) in (ex3_2_ind A A TMP_20 TMP_23 TMP_27 P TMP_32
+H1)))))))))))))) in (nat_ind TMP_2 TMP_15 TMP_33 n H))))))))) in (let TMP_54
+\def (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a)
+\to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
+A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
+A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
+Prop).(let TMP_35 \def (AHead a a0) in (let TMP_36 \def (AHead a a0) in (let
+H_x \def (leq_gen_head1 g a1 TMP_35 TMP_36 H1) in (let H2 \def H_x in (let
+TMP_37 \def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_39
+\def (\lambda (_: A).(\lambda (a4: A).(let TMP_38 \def (AHead a a0) in (leq g
+TMP_38 a4)))) in (let TMP_42 \def (\lambda (a3: A).(\lambda (a4: A).(let
+TMP_40 \def (AHead a a0) in (let TMP_41 \def (AHead a3 a4) in (eq A TMP_40
+TMP_41))))) in (let TMP_53 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) x1)).(\lambda (H5: (eq
+A (AHead a a0) (AHead x0 x1))).(let TMP_43 \def (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_44
+\def (AHead a a0) in (let TMP_45 \def (AHead x0 x1) in (let H6 \def (f_equal
+A A TMP_43 TMP_44 TMP_45 H5) in (let TMP_46 \def (\lambda (e: A).(match e
+with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) in (let
+TMP_47 \def (AHead a a0) in (let TMP_48 \def (AHead x0 x1) in (let H7 \def
+(f_equal A A TMP_46 TMP_47 TMP_48 H5) in (let TMP_52 \def (\lambda (H8: (eq A
+a x0)).(let TMP_50 \def (\lambda (a3: A).(let TMP_49 \def (AHead a a0) in
+(leq g TMP_49 a3))) in (let H9 \def (eq_ind_r A x1 TMP_50 H4 a0 H7) in (let
+TMP_51 \def (\lambda (a3: A).(leq g a1 a3)) in (let H10 \def (eq_ind_r A x0
+TMP_51 H3 a H8) in (H0 a H9 P)))))) in (TMP_52 H6))))))))))))))) in
+(ex3_2_ind A A TMP_37 TMP_39 TMP_42 P TMP_53 H2)))))))))))))))) in (A_ind
+TMP_1 TMP_34 TMP_54 a2))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/A/defs.ma".
+include "basic_1/A/defs.ma".
-definition lweight:
- A \to nat
-\def
- let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow
-O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in
-lweight.
+let rec lweight (a: A) on a: nat \def match a with [(ASort _ _) \Rightarrow O
+| (AHead a1 a2) \Rightarrow (let TMP_1 \def (lweight a1) in (let TMP_2 \def
+(lweight a2) in (let TMP_3 \def (plus TMP_1 TMP_2) in (S TMP_3))))].
definition llt:
A \to (A \to Prop)
\def
- \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
+ \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a1) in (let TMP_2
+\def (lweight a2) in (lt TMP_1 TMP_2)))).
--- /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 "basic_1/llt/defs.ma".
+
+theorem llt_wf__q_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
+Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
+a))))) P n))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
+n) \to (P a)))))).(\lambda (a: A).(let TMP_1 \def (lweight a) in (let TMP_2
+\def (lweight a) in (let TMP_3 \def (refl_equal nat TMP_2) in (H TMP_1 a
+TMP_3))))))).
+
+theorem llt_wf_ind:
+ \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
+a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(let TMP_1 \def
+(\lambda (a0: A).(P a0)) in (let TMP_11 \def (\lambda (n: nat).(let TMP_2
+\def (\lambda (a0: A).(P a0)) in (let TMP_3 \def (Q TMP_2) in (let TMP_10
+\def (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) \to (Q
+(\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
+(lweight a0) n0)).(let TMP_4 \def (\lambda (n1: nat).(\forall (m: nat).((lt m
+n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P a1)))))) in (let
+TMP_5 \def (lweight a0) in (let H2 \def (eq_ind_r nat n0 TMP_4 H0 TMP_5 H1)
+in (let TMP_9 \def (\lambda (a1: A).(\lambda (H3: (lt (lweight a1) (lweight
+a0))).(let TMP_6 \def (lweight a1) in (let TMP_7 \def (lweight a1) in (let
+TMP_8 \def (refl_equal nat TMP_7) in (H2 TMP_6 H3 a1 TMP_8)))))) in (H a0
+TMP_9))))))))) in (lt_wf_ind n TMP_3 TMP_10))))) in (llt_wf__q_ind TMP_1
+TMP_11 a)))))).
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/llt/defs.ma".
+include "basic_1/llt/defs.ma".
-include "Basic-1/leq/defs.ma".
+include "basic_1/leq/fwd.ma".
theorem lweight_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
(lweight a1) (lweight a2)))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
-a0)))) (\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))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
-A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
-a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
-(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
-a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
-(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
-H)))).
-(* COMMENTS
-Initial nodes: 189
-END *)
+a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def
+(lweight a) in (let TMP_2 \def (lweight a0) in (eq nat TMP_1 TMP_2))))) in
+(let TMP_4 \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))).(refl_equal nat O))))))) in (let TMP_16
+\def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leq g a0 a3)).(\lambda
+(H1: (eq nat (lweight a0) (lweight a3))).(\lambda (a4: A).(\lambda (a5:
+A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: (eq nat (lweight a4) (lweight
+a5))).(let TMP_5 \def (lweight a0) in (let TMP_6 \def (lweight a4) in (let
+TMP_7 \def (plus TMP_5 TMP_6) in (let TMP_8 \def (lweight a3) in (let TMP_9
+\def (lweight a5) in (let TMP_10 \def (plus TMP_8 TMP_9) in (let TMP_11 \def
+(lweight a0) in (let TMP_12 \def (lweight a3) in (let TMP_13 \def (lweight
+a4) in (let TMP_14 \def (lweight a5) in (let TMP_15 \def (f_equal2 nat nat
+nat plus TMP_11 TMP_12 TMP_13 TMP_14 H1 H3) in (f_equal nat nat S TMP_7
+TMP_10 TMP_15)))))))))))))))))))) in (leq_ind g TMP_3 TMP_4 TMP_16 a1 a2
+H))))))).
theorem llt_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
(a3: A).((llt a1 a3) \to (llt a2 a3))))))
\def
\lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
-\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
-(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let
+TMP_1 \def (lweight a1) in (let TMP_3 \def (\lambda (n: nat).(let TMP_2 \def
+(lweight a3) in (lt n TMP_2))) in (let TMP_4 \def (lweight a2) in (let TMP_5
+\def (lweight_repl g a1 a2 H) in (let H1 \def (eq_ind nat TMP_1 TMP_3 H0
+TMP_4 TMP_5) in H1)))))))))).
theorem llt_trans:
\forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
a3) \to (llt a1 a3)))))
\def
\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
-a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
-(lweight a1) (lweight a2) (lweight a3) H H0))))).
-(* COMMENTS
-Initial nodes: 43
-END *)
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(let TMP_1
+\def (lweight a1) in (let TMP_2 \def (lweight a2) in (let TMP_3 \def (lweight
+a3) in (lt_trans TMP_1 TMP_2 TMP_3 H H0)))))))).
theorem llt_head_sx:
\forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1)
-(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
-(* COMMENTS
-Initial nodes: 29
-END *)
+ \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a1) in (let TMP_2
+\def (lweight a1) in (let TMP_3 \def (lweight a2) in (let TMP_4 \def (plus
+TMP_2 TMP_3) in (let TMP_5 \def (lweight a1) in (let TMP_6 \def (lweight a2)
+in (let TMP_7 \def (le_plus_l TMP_5 TMP_6) in (le_n_S TMP_1 TMP_4
+TMP_7))))))))).
theorem llt_head_dx:
\forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1)
-(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
-(* COMMENTS
-Initial nodes: 29
-END *)
-
-theorem llt_wf__q_ind:
- \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
-Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
-a))))) P n))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
-n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
-a)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
-
-theorem llt_wf_ind:
- \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
-a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
-Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
-(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
-(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
-A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
-(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
-a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
-(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
-a1))))))))))))) a)))).
-(* COMMENTS
-Initial nodes: 179
-END *)
+ \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a2) in (let TMP_2
+\def (lweight a1) in (let TMP_3 \def (lweight a2) in (let TMP_4 \def (plus
+TMP_2 TMP_3) in (let TMP_5 \def (lweight a1) in (let TMP_6 \def (lweight a2)
+in (let TMP_7 \def (le_plus_r TMP_5 TMP_6) in (le_n_S TMP_1 TMP_4
+TMP_7))))))))).
projects / helm.git / commitdiff