--- /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 "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/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).(f_equal3 nat nat
+T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal
+nat n) (refl_equal nat n0) (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))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T
+(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t:
+T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p)
+(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T
+(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S
+O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n:
+nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
+(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda
+(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift
+(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1
+p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S
+d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
+
+theorem lifts1_xhg:
+ \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts
+(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+\def
+ \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq
+TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t))))
+(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq
+TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds
+t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList
+(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1
+hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O
+(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1
+hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds
+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 (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
+(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 (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)))
+(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
+(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_sym 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_sym 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).
+
projects / helm.git / blobdiff