(* This file was automatically generated: do not edit *********************)
-include "Basic-1/lift/props.ma".
+include "basic_1/lift1/defs.ma".
-include "Basic-1/drop1/defs.ma".
+include "basic_1/lift/props.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))))
+lemma lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
\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).
-(* COMMENTS
-Initial nodes: 145
-END *)
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
-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))))
+lemma lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
+(trans hds i))))
\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).
-(* COMMENTS
-Initial nodes: 371
-END *)
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) 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))))
+lemma lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
+hds) t))))))
\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)).
-(* COMMENTS
-Initial nodes: 307
-END *)
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p)
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t)))))
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)))
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u
+t)) (H u t)))))))) hds)).
-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)))))
+lemma lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
+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).
-(* COMMENTS
-Initial nodes: 1339
-END *)
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u:
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t))
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u))
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f)
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f)
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+lemma lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
+h d) t) H))))) hds)))).
+
+lemma lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
+(THeads (Flat f) t1 t)))))) ts)))).
+
+lemma lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
+PNil t0) H)))) ts).
+
+lemma lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
+H)))) ts)))).
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