new theorems added. does not comile well yet :(( problems found in

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / lift1 / props.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
index eeec1fc7d76e9e2c55a721b613b98d40c99d40f5..5b92ff26a0a52481b1908938ca282eab15568518 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
@@ -22,6 +22,23 @@ include "lift/props.ma".
 
 include "drop1/defs.ma".
 
+theorem lift1_lift1:
+ \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
+(lift1 is2 t)) (lift1 (papp is1 is2) t))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
+PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) 
+t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
+((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 
+(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(sym_eq T (lift n 
+n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 t))) (sym_eq T 
+(lift n n0 (lift1 p (lift1 is2 t))) (lift n n0 (lift1 (papp p is2) t)) 
+(sym_eq T (lift n n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 
+t))) (f_equal3 nat nat T T lift n n n0 n0 (lift1 (papp p is2) t) (lift1 p 
+(lift1 is2 t)) (refl_equal nat n) (refl_equal nat n0) (sym_eq T (lift1 p 
+(lift1 is2 t)) (lift1 (papp p is2) t) (H is2 t)))))))))))) is1).
+
 theorem lift1_xhg:
  \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
 (lift (S O) O (lift1 hds t))))
@@ -59,57 +76,60 @@ t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
 
 theorem lift1_free:
  \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
-(lift (S i) O t)) (lift (S (trans hds i)) O (ctrans hds i t)))))
+(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
 \def
  \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: 
 nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O 
-(ctrans p i t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T (lift (S 
-i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
+(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T 
+(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
 PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 
-(lift (S i) O t)) (lift (S (trans hds0 i)) O (ctrans hds0 i t))))))).(\lambda 
-(i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 i)) O (ctrans hds0 
-i t)) (\lambda (t0: T).(eq T (lift h d t0) (lift (S (match (blt (trans hds0 
-i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans 
-hds0 i) h)])) O (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h 
-(minus d (S (trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans 
-hds0 i t)])))) (xinduction bool (blt (trans hds0 i) d) (\lambda (b: bool).(eq 
-T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (match b 
-with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 
-i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i t)])))) (\lambda 
-(x_x: bool).(bool_ind (\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) 
-\to (eq T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S 
-(match b with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus 
-(trans hds0 i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S 
-(trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i 
-t)]))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat 
-(plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq 
-T (lift h n (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (trans 
-hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) 
+(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) 
+t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 
+i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift 
+(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | 
+false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 
+i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans 
+hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt 
+(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) 
+O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans 
+hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with 
+[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
+false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind 
+(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d 
+(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) 
+(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S 
+(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h 
+n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
 (eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 i)) O 
-(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) (refl_equal T 
-(lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 
-i t)))) (lift h (plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (lift 
-(S (trans hds0 i)) O (ctrans hds0 i t))) (lift_d (ctrans hds0 i t) h (S 
-(trans hds0 i)) (minus d (S (trans hds0 i))) O (le_O_n (minus d (S (trans 
-hds0 i)))))) d (le_plus_minus (S (trans hds0 i)) d (bge_le (S (trans hds0 i)) 
-d (le_bge (S (trans hds0 i)) d (lt_le_S (trans hds0 i) d (blt_lt d (trans 
-hds0 i) H0))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) 
-false)).(eq_ind_r T (lift (plus h (S (trans hds0 i))) O (ctrans hds0 i t)) 
-(\lambda (t0: T).(eq T t0 (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i 
-t)))) (eq_ind nat (S (plus h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n 
-O (ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
+(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
+(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S 
+(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) 
+(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans 
+hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S 
+(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d 
+(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: 
+(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans 
+hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S 
+(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus 
+h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) 
+t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) 
 (eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
-(ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
-(refl_equal T (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))) (plus h 
-(trans hds0 i)) (plus_comm h (trans hds0 i))) (plus h (S (trans hds0 i))) 
-(plus_n_Sm h (trans hds0 i))) (lift h d (lift (S (trans hds0 i)) O (ctrans 
-hds0 i t))) (lift_free (ctrans hds0 i t) (S (trans hds0 i)) h O d (eq_ind nat 
-(S (plus O (trans hds0 i))) (\lambda (n: nat).(le d n)) (eq_ind_r nat (plus 
-(trans hds0 i) O) (\lambda (n: nat).(le d (S n))) (le_S d (plus (trans hds0 
-i) O) (le_plus_trans d (trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus 
-O (trans hds0 i)) (plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i))) 
-(plus_n_Sm O (trans hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O 
-t)) (H i t)))))))) hds).
+(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans 
+hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 
+(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_comm h (trans hds0 i))) 
+(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S 
+(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 
+i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) 
+(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda 
+(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d 
+(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) 
+(plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans 
+hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) 
+hds).