branch for universe

[helm.git] / matita / contribs / LAMBDA-TYPES / LambdaDelta-1 / r / props.ma

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma
new file mode 100644 (file)
index 0000000..a768ab9
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "LambdaDelta-1/r/defs.ma".
+
+include "LambdaDelta-1/s/defs.ma".
+
+theorem 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).
+
+theorem 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).
+
+theorem 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).
+
+theorem 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)))).
+
+theorem 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).
+
+theorem 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).
+
+theorem 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))).
+
+theorem 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)))).
+