alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
alias id "lt_S_n" = "cic:/Coq/Arith/Lt/lt_S_n.con".
+alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
theorem f_equal: \forall A,B:Type. \forall f:A \to B.
\forall x,y:A. x = y \to f x = f y.
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/A/defs".
+
+include "../Base/theory.ma".
+
+inductive A: Set \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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/defs".
+
+include "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.
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props".
+
+include "aplus/defs.ma".
+
+include "next_plus/props.ma".
+
+theorem 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)))))))))
+\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)))).(sym_equal A (asucc g (aplus g a2 (plus n
+h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1
+(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus
+g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g
+g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq
+A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+
+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)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0))))
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0))))
+(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a
+(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))
+H0))))) (plus n (S n0)) (plus_n_Sm n n0)))) h2)))) h1))).
+
+theorem aplus_asucc:
+ \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
+h) (asucc g (aplus g a h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
+(aplus_assoc g a (S O) h)))).
+
+theorem aplus_sort_O_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
+n) (S k)) (aplus g (ASort O (next g n)) k))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
+k)) (aplus_asucc g k (ASort O n))))).
+
+theorem aplus_sort_S_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
+(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind
+A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
+(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g
+(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
+
+theorem aplus_asort_O_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
+n) h) (ASort O (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
+nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda
+(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall
+(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0
+n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n)
+(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat
+(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next
+g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
+(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n
+(ASort O n0)))))) h)).
+
+theorem aplus_asort_le_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
+k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(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)).
+
+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))))))).
+
+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)).
+
+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 h)
+\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in
+nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0))
+\to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g
+n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0))
+h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g
+(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus
+(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S
+i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0)
+(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
+(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n:
+nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in
+(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
+n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
+(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1)
+\Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1)
+n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A
+a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
+(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
+| (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def
+(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m
+with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1
+h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0)
+H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def
+(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n
+i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g
+n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h)
+(S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
+(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda
+(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
+Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
+g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
+nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
+a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
+a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc
+g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with
+[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g
+(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g
+a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+
+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)).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props".
+
+include "aplus/defs.ma".
+
+theorem 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)))))))))
+\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)))).(sym_equal A (asucc g (aplus g a2 (plus n
+h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1
+(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus
+g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g
+g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq
+A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+
+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)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0))))
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0))))
+(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a
+(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))
+H0))))) (plus n (S n0)) (plus_n_Sm n n0)))) h2)))) h1))).
+
+theorem aplus_asucc:
+ \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
+h) (asucc g (aplus g a h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
+(aplus_assoc g a (S O) h)))).
+
+theorem aplus_sort_O_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
+n) (S k)) (aplus g (ASort O (next g n)) k))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
+k)) (aplus_asucc g k (ASort O n))))).
+
+theorem aplus_sort_S_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
+(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(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)))))).
+
+alias id "next_plus_next" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props/next_plus_next.con".
+alias id "next_plus" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs/next_plus.con".
+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)).
+
+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)).
+
+alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
+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))))))).
+
+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)).
+
+alias id "next_plus_lt" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props/next_plus_lt.con".
+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 h)
+\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in
+nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0))
+\to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g
+n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0))
+h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g
+(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus
+(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S
+i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0)
+(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
+(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n:
+nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in
+(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
+n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
+(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1)
+\Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1)
+n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A
+a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
+(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
+| (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def
+(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m
+with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1
+h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0)
+H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def
+(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n
+i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g
+n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h)
+(S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
+(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda
+(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
+Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
+g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
+nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
+a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
+a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc
+g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with
+[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g
+(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g
+a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+
+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)).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/asucc/defs".
+
+include "A/defs.ma".
+
+include "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.
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/asucc/fwd".
+
+include "asucc/defs.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 h) \Rightarrow (ASort h 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)))).
+
+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 (a2: A).(eq A a0 (AHead a1 a2))) (\lambda
+(a0: A).(eq A a2 (asucc g a0))))))).(\lambda (a3: A).(\lambda (H0: (((eq A
+(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\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 a _) \Rightarrow a])) (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 _ a) \Rightarrow a])) (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 (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)))))) 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 (a: A).((eq A (AHead a1 a)
+(asucc g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1 a0))) (\lambda
+(a0: A).(eq A a (asucc g a0)))))) H0 (asucc g a3) H3) in (let H7 \def (eq_ind
+A a2 (\lambda (a: A).((eq A (AHead a1 a) (asucc g a1)) \to (ex2 A (\lambda
+(a0: A).(eq A a1 (AHead a1 a0))) (\lambda (a0: A).(eq A a (asucc g a0))))))
+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)))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc".
+
+include "leq/defs.ma".
+
+include "aplus/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))).((match h1 in nat return (\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)])))) with [O \Rightarrow (\lambda (H1: (eq A (aplus g (ASort O n1) k)
+(aplus g (ASort h2 n2) k))).((match h2 in nat return (\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)])))) with [O \Rightarrow (\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)))) | (S n) \Rightarrow
+(\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort (S n) n2)
+k))).(leq_sort g O n (next g n1) n2 k (eq_ind A (aplus g (ASort O n1) (S k))
+(\lambda (a: A).(eq A a (aplus g (ASort n n2) k))) (eq_ind A (aplus g (ASort
+(S n) n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a))
+(eq_ind_r A (aplus g (ASort (S n) n2) k) (\lambda (a: A).(eq A (asucc g a)
+(asucc g (aplus g (ASort (S n) n2) k)))) (refl_equal A (asucc g (aplus g
+(ASort (S n) n2) k))) (aplus g (ASort O n1) k) H2) (aplus g (ASort n n2) k)
+(aplus_sort_S_S_simpl g n2 n k)) (aplus g (ASort O (next g n1)) k)
+(aplus_sort_O_S_simpl g n1 k))))]) H1)) | (S n) \Rightarrow (\lambda (H1: (eq
+A (aplus g (ASort (S n) n1) k) (aplus g (ASort h2 n2) k))).((match h2 in nat
+return (\lambda (n0: nat).((eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort
+n0 n2) k)) \to (leq g (ASort n n1) (match n0 with [O \Rightarrow (ASort O
+(next g n2)) | (S h) \Rightarrow (ASort h n2)])))) with [O \Rightarrow
+(\lambda (H2: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort O n2)
+k))).(leq_sort g n O n1 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k))
+(\lambda (a: A).(eq A (aplus g (ASort n n1) k) a)) (eq_ind A (aplus g (ASort
+(S n) 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 n) n1) k) H2) (aplus g (ASort n n1) k)
+(aplus_sort_S_S_simpl g n1 n k)) (aplus g (ASort O (next g n2)) k)
+(aplus_sort_O_S_simpl g n2 k)))) | (S n0) \Rightarrow (\lambda (H2: (eq A
+(aplus g (ASort (S n) n1) k) (aplus g (ASort (S n0) n2) k))).(leq_sort g n n0
+n1 n2 k (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a
+(aplus g (ASort n0 n2) k))) (eq_ind A (aplus g (ASort (S n0) n2) (S k))
+(\lambda (a: A).(eq A (aplus g (ASort (S n) n1) (S k)) a)) (eq_ind_r A (aplus
+g (ASort (S n0) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g
+(ASort (S n0) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n0) n2)
+k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n0 n2) k)
+(aplus_sort_S_S_simpl g n2 n0 k)) (aplus g (ASort n n1) k)
+(aplus_sort_S_S_simpl g n1 n k))))]) 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)))).
+
+axiom asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
+g a2)) \to (leq g a1 a2))))
+.
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs".
+
+include "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
+(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
+| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2
+a4))))))).
+
include "tau0/props.ma".
+include "A/defs.ma".
+
+include "asucc/defs.ma".
+
+include "asucc/fwd.ma".
+
+include "aplus/defs.ma".
+
+include "aplus/props.ma".
+
+include "leq/defs.ma".
+
+include "leq/asucc.ma".
+
(**************************************************************************)
(* Problematic objects for disambiguation/typechecking ********************)
-(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem iso_trans:
+ \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
+t3) \to (iso t1 t3)))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
+(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq
+T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_:
+nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T
+(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
+(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T
+(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in
+(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) |
+(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1)
+(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T
+(THead k v1 t1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2)
+t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2))
+(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
+T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
+(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
+i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
+\Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T
+(TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef
+_) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in
+(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) |
+(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef
+i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
+i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind
+T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0
+(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2)
+t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in
+(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))])
+in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
+K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
+(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
+t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
+t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
+(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T
+(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2)
+t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef
+i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0)
+in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1)))
+| (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0)
+(THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
+\Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda
+(k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to
+(iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2
+(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead
+k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
+T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T
+(THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k
+v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0
+(sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1
+(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
(**************************************************************************)
(* Problematic objects for disambiguation/typechecking ********************)
-(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
+(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
+(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i
+v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to
+(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
+(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C
+c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))
+(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2:
+C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i
+c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
+H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3:
+(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
+(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
+(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
+hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2)
+(refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda
+(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2
+c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i:
+nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl
+(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq
+C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
+h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow
+(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c
+c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2)
+in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3
+H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq
+PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0
+c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
+(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
+d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
+in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
+n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9
+\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
+(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
+(PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
+nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4
+c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2:
+C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match
+(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))))
+(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0
+hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1
+hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d)
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
+i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0
+hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1)
+\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda
+(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d
+c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt
+(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow
+(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d)
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4
+c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c)
+\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift
+h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans
+hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
+hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex
+C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
+false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x:
+bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
+(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt
+(trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16
+\def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
+(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2
+(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind
+b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d
+(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d)
+(S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H
+\def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1
+(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda
+(e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2:
+C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0
+i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d
+(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda
+(H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1
+v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0
+i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda
+(x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0
+i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x
+(Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus
+(trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d
+(trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq
+C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10)))
+h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
+PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1)))))))))))))))
+hds).
+
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+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)))).((match n in nat return (\lambda (n3: nat).((leq g
+(asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0)
+(ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O
+n0)) (asucc g (ASort n1 n2)))).((match n1 in nat return (\lambda (n3:
+nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g
+(ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc
+g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq return
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O
+(next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0)
+(ASort O n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda
+(H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort
+h2 n3) (ASort O (next g n2)))).((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 n1])) (ASort h1 n1) (ASort O (next g n0)) H1) 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 h1]))
+(ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n:
+nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort O (next g n2)))
+\to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g
+(ASort O n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind
+nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2)))
+\to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g
+(ASort O n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O
+(next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \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 n3)
+(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
+(next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n
+n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H9: (eq nat n3
+(next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to (leq g (ASort O n0)
+(ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus
+g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next
+g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10
+(aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def
+(eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g
+(ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl
+g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2)
+H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1
+(sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
+n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4
+\def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
+O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0)
+(ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g
+n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) \Rightarrow (\lambda
+(H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def
+(match H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ?
+a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq
+g (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n1 n3 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g
+n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 n2))).((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 n1])) (ASort h1 n1)
+(ASort O (next g n0)) H1) 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 h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in
+(eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2
+n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3)
+k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H5: (eq nat n1
+(next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3)
+(ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k))
+\to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: (eq A (ASort h2
+n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 \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 n3) (ASort n3 n2)
+H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to ((eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0)
+(ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat n2 (\lambda
+(n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n) k))
+\to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def (eq_ind_r A
+(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3
+n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in
+(let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A
+(aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k))
+(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11))))
+n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1
+(next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2
+H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
+n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def
+(eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
+n3 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort
+(S n3) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0)))
+(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0:
+(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 in
+nat return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
+(ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+O n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O
+(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort
+h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3
+n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((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 n1])) (ASort h1 n1)
+(ASort n3 n0) H1) 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 h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to
+((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort
+(S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A
+(aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3)
+n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g
+n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \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 n3)
+(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
+(next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k))
+\to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next
+g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3
+n0) k) (aplus g (ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))
+(\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2))
+k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A
+a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
+k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in
+(leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9)))
+h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1
+n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda
+(H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4)
+(ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind
+((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3
+a4) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2
+(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4)
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+(S n4) n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda
+(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0
+(ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with
+[(leq_sort h1 h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3)
+(ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((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 n3])) (ASort h1 n3)
+(ASort n3 n0) H1) 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 h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A
+(aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g
+(ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0)
+(ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let
+H7 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
+A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4]))
+(ASort h2 n4) (ASort n4 n2) H6) in ((let H8 \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 n4) (ASort n4 n2) H6) in (eq_ind
+nat n4 (\lambda (n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k)
+(aplus g (ASort n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))
+(\lambda (H9: (eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus
+g (ASort n3 n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0)
+(ASort (S n4) n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g
+(ASort n4 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda
+(a: A).(eq A a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S
+k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a))
+H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in
+(leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2
+(sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3
+H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
+(eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort
+n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4)
+(ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3
+n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) 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)))).((match n in nat return (\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
+(leq g (ASort n1 n0) (AHead a a0)))))) with [O \Rightarrow (\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 H5 \def (match H4 in leq return (\lambda (a1:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort O (next g
+n0))) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
+a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n1) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2)
+(AHead a (asucc g a0)))).((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 n1])) (ASort h1 n1) (ASort O (next g n0)) H3) 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 h1])) (ASort h1 n1)
+(ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1
+(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
+(aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0)))))) (\lambda (H7: (eq nat n1 (next g n0))).(eq_ind nat (next g
+n0) (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
+(aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g
+a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus
+g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0))) H9))) n1 (sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O
+H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4:
+(eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4)
+(AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in
+(False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2)
+\to ((leq g a3 a4) \to (leq g (ASort O n0) (AHead a a0))))) H6)) H5 H2
+H3)))]) in (H5 (refl_equal A (ASort O (next g n0))) (refl_equal A (AHead a
+(asucc g a0)))))))) | (S n1) \Rightarrow (\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 H5 \def (match H4 in leq return (\lambda (a1:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort n1 n0)) \to
+((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a
+a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a
+(asucc g a0)))).((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 n1])) (ASort h1 n1) (ASort n1 n0) H3) 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 h1])) (ASort h1 n1) (ASort n1 n0)
+H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2
+n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
+(H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2)
+(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: (eq A
+(ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2)
+(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
+H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))
+\to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7)))
+h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3)
+\Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5:
+(eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead
+a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq
+g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
+H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
+(asucc g a0))))))))]) 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)))).((match n in nat return (\lambda (n1:
+nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g
+(AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\lambda (H2: (leq g (asucc
+g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def (match H2 in leq return
+(\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1
+(AHead a (asucc g a0))) \to ((eq A a2 (ASort O (next g n0))) \to (leq g
+(AHead a a0) (ASort O n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
+(\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq
+A (ASort h2 n2) (ASort O (next g n0)))).((let H5 \def (eq_ind A (ASort h1 n1)
+(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
+H3) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A
+(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0)
+(ASort O n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow
+(\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq
+A (AHead a2 a4) (ASort O (next g n0)))).((let H6 \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 a1 a3) (AHead a (asucc g a0)) H4) in
+((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead
+a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3
+(asucc g a0)) \to ((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5
+a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda
+(H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A
+(AHead a2 a4) (ASort O (next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to
+(leq g (AHead a a0) (ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4)
+(ASort O (next g n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H9) in
+(False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0)
+(ASort O n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a
+H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0)))
+(refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2:
+(leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def
+(match H2 in leq return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ?
+a1 a2)).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort n1 n0)) \to
+(leq g (AHead a a0) (ASort (S n1) n0))))))) with [(leq_sort h1 h2 n1 n2 k H2)
+\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g
+a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort n1 n0))).((let H5 \def
+(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2)
+(ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) | (leq_head a1
+a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc
+g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 n0))).((let H6 \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 a1 a3)
+(AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
+(eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4)
+(ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0)
+(ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A
+(asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g
+a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))
+(\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A
+(AHead a2 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g
+(AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8)))
+a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a
+(asucc g a0))) (refl_equal A (ASort n1 n0)))))]) 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 H4 \def (match H3 in leq
+return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A
+a1 (AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g
+(AHead a a0) (AHead a3 a4))))))) with [(leq_sort h1 h2 n1 n2 k H4)
+\Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g
+a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2)
+(AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head
+a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a
+(asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g
+a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow
+a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \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 a5) (AHead a (asucc g
+a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A
+(AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6)
+\to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g
+a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3
+(asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0)
+(AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g
+a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow
+a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \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 a4 a6) (AHead a3 (asucc
+g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to
+((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3
+a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4)
+(\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead
+a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g
+(asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6
+(sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq
+A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4
+(refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g
+a4)))))))))) a2)))))) a1)).
+
(* iso_trans (in problems-1)
* drop1_getl_trans (in problems-2)
+ * asucc_inj (in problems-3)
*)
(* Problem 2: assertion failure raised by type checker on this object *)
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+(* Problem 1: disambiguation errors with these objects *)
+
+(* iso_trans (in problems-1)
+ * drop1_getl_trans (in problems-2)
+ *)
+
+(* Problem 2: assertion failure raised by type checker on this object *)
+
+inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def
+| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
+| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c
+t t2) \to (tau1 g c t1 t2)))).
projects / helm.git / commitdiff