+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/r/defs.ma".
-
-include "basic_1/s/defs.ma".
-
-lemma r_S:
- \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
-i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
-f) i))))) k).
-
-lemma r_plus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
-(plus (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
-(Flat f) i) j))))) k).
-
-lemma r_plus_sym:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
-(plus i (r k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
-F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
-
-lemma r_minus:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
-(minus (r k i) (S n)) (r k (minus i (S n)))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
-K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
-n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
-F).(minus_x_Sy i n H)) k)))).
-
-lemma r_dis:
- \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
-\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
-nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
-(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
-((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
-nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
-((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
-(i: nat).(refl_equal nat (S i)))))))) k).
-
-lemma s_r:
- \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
-i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
-(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
-
-lemma r_arith0:
- \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
-\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
-nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
-nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
-(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
-
-lemma r_arith1:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
-i)) (S j)) (minus (r k i) j))))
-\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
-(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
-(minus (r k i) j)) (r k (S i)) (r_S k i)))).
-
-lemma r_arith2:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to
-(le (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(\lambda (H: (le (S i) (S j))).(let H_y \def
-(le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le (S i) j)).H)))) k).
-
-lemma r_arith3:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to
-(le j (r k i)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(\lambda (H: (le (S j) (S i))).(let H_y \def
-(le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le j (S i))).H)))) k).
-
-lemma r_arith4:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k
-j)) (minus (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (minus (S i) (s k0 j)) (minus (r k0 i) j))))) (\lambda (b:
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i)
-j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
-(minus (r (Flat f) i) j))))) k).
-
-lemma r_arith5:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to
-(lt j (r k i)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H)))))
-(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
-i))).H)))) k).
-
-lemma r_arith6:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S
-j)) (minus i (s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (minus (r k0 i) (S j)) (minus i (s k0 j)))))) (\lambda (b:
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b)
-j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
-(minus i (s (Flat f) j)))))) k).
-
-lemma r_arith7:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j))
-\to (eq nat (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((eq nat (S i) (s k0 j)) \to (eq nat (r k0 i) j))))) (\lambda (_:
-B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (S i) (S
-j))).(eq_add_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (eq nat (S i) j)).H)))) k).
-
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