(* 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".
-theorem aplus_reg_r:
+include "basic_1/next_plus/props.ma".
+
+lemma aplus_reg_r:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
(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 *)
-theorem aplus_assoc:
+lemma 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
(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 *)
-theorem aplus_asucc:
+lemma 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
(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 *)
-theorem aplus_sort_O_S_simpl:
+lemma 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
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 *)
-theorem aplus_sort_S_S_simpl:
+lemma 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
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 *)
-theorem aplus_asort_O_simpl:
+lemma 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
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 *)
-theorem aplus_asort_le_simpl:
+lemma 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
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 *)
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x)
+H1) in (False_ind (eq A (asucc g (aplus g (ASort O n) h0)) (ASort (minus O (S
+h0)) n)) H3))))) (le_gen_S h0 O H0)))) (\lambda (n: nat).(\lambda (_:
+((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n n0)
+h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: (le (S
+h0) (S n))).(eq_ind A (aplus g (asucc g (ASort (S n) n0)) h0) (\lambda (a:
+A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n n0 (le_S_n h0 n H1)) (asucc
+g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g h0 (ASort (S n) n0)))))))
+k)))) h)).
-theorem aplus_asort_simpl:
+lemma 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
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)
+(le_S_n (S k) (S h) (le_S (S (S k)) (S h) (le_n_S (S k) h H)))))) (minus k k)
+(minus_n_n k)) (aplus g (ASort k n) k) (aplus_asort_le_simpl g k k n (le_n
+k))) (aplus g (ASort k n) (plus k (minus h k))) (aplus_assoc g (ASort k n) k
+(minus h k))) h (le_plus_minus k h (le_S_n k h (le_S_n (S k) (S h) (le_S (S
+(S k)) (S h) (le_n_S (S k) h H))))))) (\lambda (H: (le h k)).(eq_ind_r A
+(ASort (minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g
+n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h)
n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h)
(aplus_asort_le_simpl g h k n H))))))).
-(* COMMENTS
-Initial nodes: 587
-END *)
-theorem aplus_ahead_simpl:
+lemma 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
(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 *)
-theorem aplus_asucc_false:
+lemma 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
\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
-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 *)
+(\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _)
+\Rightarrow (next_plus g (next g n0) (minus h O))])) (ASort (minus O h)
+(next_plus g (next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def
+(eq_ind_r nat (minus h O) (\lambda (n1: nat).(eq nat (next_plus g (next g n0)
+n1) n0)) H2 h (minus_n_O h)) in (le_lt_false n0 n0 (le_n n0) (eq_ind nat
+(next_plus g (next g n0) h) (\lambda (n1: nat).(lt n0 n1)) (next_plus_lt g h
+n0) n0 H3) P))))) (\lambda (n1: nat).(\lambda (_: (((eq A (aplus g (match n1
+with [O \Rightarrow (ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0
+n0)]) h) (ASort n1 n0)) \to P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0)
+h) (ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h)
+(\lambda (a0: A).(eq A a0 (ASort (S n1) n0))) H0 (ASort (minus n1 h)
+(next_plus g n0 (minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def
+(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 |
+(AHead _ _) \Rightarrow (minus n1 h)])) (ASort (minus n1 h) (next_plus g n0
+(minus h n1))) (ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda
+(e: A).(match e with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow
+(next_plus g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h
+n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S
+n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
+(minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: A).(\lambda
+(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
+Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
+g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
+nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
+a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
+a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 (AHead a0 (aplus g
+(asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (aplus g
+(asucc g a1) h) | (AHead _ a2) \Rightarrow a2])) (AHead a0 (aplus g (asucc g
+a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
-theorem aplus_inj:
+lemma 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
(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 *)
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