ok up to arity assignment

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

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
index a47a50e0771a52126b18aa89015dadd152d531ba..eeec1fc7d76e9e2c55a721b613b98d40c99d40f5 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
@@ -18,6 +18,10 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props".
 
 include "lift1/defs.ma".
 
+include "lift/props.ma".
+
+include "drop1/defs.ma".
+
 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))))
@@ -53,3 +57,59 @@ t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
 (lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) 
 (lift (S O) O t)) (lift1_xhg hds t))))) ts)).
 
+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)))))
+\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: 
+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))))) 
+(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)))) 
+(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).
+