X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fr%2Fprops.ma;h=e67f789c51f0b3d618dc0b7a06594ebef598f9bb;hb=6d1bb99e7f355d826c07285ba46b6b13a4abaefc;hp=0815aaf5e8f042a0c78610639874b09323c36dc9;hpb=538c5526a6b3c3af44f92c9cc67d82f28995da96;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
index 0815aaf5e..e67f789c5 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
@@ -14,104 +14,187 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/r/defs.ma".
+include "basic_1/r/defs.ma".
 
-include "Basic-1/s/defs.ma".
+include "basic_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).
-(* COMMENTS
-Initial nodes: 65
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1 
+\def (S i) in (let TMP_2 \def (r k0 TMP_1) in (let TMP_3 \def (r k0 i) in 
+(let TMP_4 \def (S TMP_3) in (eq nat TMP_2 TMP_4))))))) in (let TMP_9 \def 
+(\lambda (b: B).(\lambda (i: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def 
+(r TMP_6 i) in (let TMP_8 \def (S TMP_7) in (refl_equal nat TMP_8)))))) in 
+(let TMP_13 \def (\lambda (f: F).(\lambda (i: nat).(let TMP_10 \def (Flat f) 
+in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (S TMP_11) in 
+(refl_equal nat TMP_12)))))) in (K_ind TMP_5 TMP_9 TMP_13 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).
-(* COMMENTS
-Initial nodes: 79
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let 
+TMP_3 \def (r k0 i) in (let TMP_4 \def (plus TMP_3 j) in (eq nat TMP_2 
+TMP_4)))))))) in (let TMP_9 \def (\lambda (b: B).(\lambda (i: nat).(\lambda 
+(j: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def (r TMP_6 i) in (let 
+TMP_8 \def (plus TMP_7 j) in (refl_equal nat TMP_8))))))) in (let TMP_13 \def 
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_10 \def (Flat f) 
+in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (plus TMP_11 j) in 
+(refl_equal nat TMP_12))))))) in (K_ind TMP_5 TMP_9 TMP_13 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).
-(* COMMENTS
-Initial nodes: 63
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let 
+TMP_3 \def (r k0 j) in (let TMP_4 \def (plus i TMP_3) in (eq nat TMP_2 
+TMP_4)))))))) in (let TMP_7 \def (\lambda (_: B).(\lambda (i: nat).(\lambda 
+(j: nat).(let TMP_6 \def (plus i j) in (refl_equal nat TMP_6))))) in (let 
+TMP_8 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i 
+j)))) in (K_ind TMP_5 TMP_7 TMP_8 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)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
+K).(let TMP_7 \def (\lambda (k0: K).(let TMP_1 \def (r k0 i) in (let TMP_2 
+\def (S n) in (let TMP_3 \def (minus TMP_1 TMP_2) in (let TMP_4 \def (S n) in 
+(let TMP_5 \def (minus i TMP_4) in (let TMP_6 \def (r k0 TMP_5) in (eq nat 
+TMP_3 TMP_6)))))))) in (let TMP_10 \def (\lambda (_: B).(let TMP_8 \def (S n) 
+in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat TMP_9)))) in (let 
+TMP_11 \def (\lambda (_: F).(minus_x_Sy i n H)) in (K_ind TMP_7 TMP_10 TMP_11 
+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).
-(* COMMENTS
-Initial nodes: 151
-END *)
+ \lambda (k: K).(let TMP_1 \def (\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)))) in (let TMP_3 \def 
+(\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))).(let TMP_2 \def (\lambda (i: nat).(refl_equal 
+nat i)) in (H TMP_2)))))) in (let TMP_6 \def (\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))).(let TMP_5 \def (\lambda (i: nat).(let TMP_4 \def (S i) in (refl_equal 
+nat TMP_4))) in (H0 TMP_5)))))) in (K_ind TMP_1 TMP_3 TMP_6 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).
-(* COMMENTS
-Initial nodes: 51
-END *)
+ \lambda (k: K).(let TMP_4 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1 
+\def (r k0 i) in (let TMP_2 \def (s k0 TMP_1) in (let TMP_3 \def (S i) in (eq 
+nat TMP_2 TMP_3)))))) in (let TMP_6 \def (\lambda (_: B).(\lambda (i: 
+nat).(let TMP_5 \def (S i) in (refl_equal nat TMP_5)))) in (let TMP_8 \def 
+(\lambda (_: F).(\lambda (i: nat).(let TMP_7 \def (S i) in (refl_equal nat 
+TMP_7)))) in (K_ind TMP_4 TMP_6 TMP_8 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))).
-(* COMMENTS
-Initial nodes: 105
-END *)
+ \lambda (k: K).(\lambda (i: nat).(let TMP_1 \def (r k i) in (let TMP_2 \def 
+(S TMP_1) in (let TMP_6 \def (\lambda (n: nat).(let TMP_3 \def (S O) in (let 
+TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in (eq nat TMP_4 
+TMP_5))))) in (let TMP_7 \def (r k i) in (let TMP_9 \def (\lambda (n: 
+nat).(let TMP_8 \def (r k i) in (eq nat n TMP_8))) in (let TMP_10 \def (r k 
+i) in (let TMP_11 \def (refl_equal nat TMP_10) in (let TMP_12 \def (r k i) in 
+(let TMP_13 \def (S TMP_12) in (let TMP_14 \def (S O) in (let TMP_15 \def 
+(minus TMP_13 TMP_14) in (let TMP_16 \def (r k i) in (let TMP_17 \def 
+(minus_Sx_SO TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_9 TMP_11 
+TMP_15 TMP_17) in (let TMP_19 \def (S i) in (let TMP_20 \def (r k TMP_19) in 
+(let TMP_21 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_6 TMP_18 TMP_20 
+TMP_21))))))))))))))))))).
 
 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)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(let TMP_1 \def (r k i) 
+in (let TMP_2 \def (S TMP_1) in (let TMP_7 \def (\lambda (n: nat).(let TMP_3 
+\def (S j) in (let TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in 
+(let TMP_6 \def (minus TMP_5 j) in (eq nat TMP_4 TMP_6)))))) in (let TMP_8 
+\def (r k i) in (let TMP_9 \def (minus TMP_8 j) in (let TMP_10 \def 
+(refl_equal nat TMP_9) in (let TMP_11 \def (S i) in (let TMP_12 \def (r k 
+TMP_11) in (let TMP_13 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_7 TMP_10 
+TMP_12 TMP_13)))))))))))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).((le (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (le TMP_1 
+j)))))) in (let TMP_3 \def (\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))))) 
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (le (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).((le (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (le j 
+TMP_1)))))) in (let TMP_3 \def (\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))))) 
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (le j (S i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).(let TMP_1 \def (S i) in (let TMP_2 \def (s k0 j) in (let TMP_3 \def 
+(minus TMP_1 TMP_2) in (let TMP_4 \def (r k0 i) in (let TMP_5 \def (minus 
+TMP_4 j) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b: 
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8 
+\def (r TMP_7 i) in (let TMP_9 \def (minus TMP_8 j) in (refl_equal nat 
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda 
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (r TMP_11 i) in (let 
+TMP_13 \def (minus TMP_12 j) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6 
+TMP_10 TMP_14 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).((lt (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (lt j 
+TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) in (let TMP_4 \def 
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S 
+i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).(let TMP_1 \def (r k0 i) in (let TMP_2 \def (S j) in (let TMP_3 \def 
+(minus TMP_1 TMP_2) in (let TMP_4 \def (s k0 j) in (let TMP_5 \def (minus i 
+TMP_4) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b: 
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8 
+\def (s TMP_7 j) in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat 
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda 
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (s TMP_11 j) in (let 
+TMP_13 \def (minus i TMP_12) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6 
+TMP_10 TMP_14 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall 
+(j: nat).((eq nat (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (eq nat 
+TMP_1 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda 
+(j: nat).(\lambda (H: (eq nat (S i) (S j))).(eq_add_S i j H))))) in (let 
+TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: 
+(eq nat (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).