include "basic_1/getl/defs.ma".
-theorem drop1_skip_bind:
+lemma drop1_skip_bind:
\forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
(lift1 hds u)) (CHead e (Bind b) u)))))))
\def
- \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(let TMP_7 \def
-(\lambda (p: PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (let
-TMP_1 \def (Ss p) in (let TMP_2 \def (Bind b) in (let TMP_3 \def (lift1 p u)
-in (let TMP_4 \def (CHead c TMP_2 TMP_3) in (let TMP_5 \def (Bind b) in (let
-TMP_6 \def (CHead e TMP_5 u) in (drop1 TMP_1 TMP_4 TMP_6))))))))))) in (let
-TMP_16 \def (\lambda (c: C).(\lambda (u: T).(\lambda (H: (drop1 PNil c
-e)).(let H_y \def (drop1_gen_pnil c e H) in (let TMP_12 \def (\lambda (c0:
-C).(let TMP_8 \def (Bind b) in (let TMP_9 \def (CHead c0 TMP_8 u) in (let
-TMP_10 \def (Bind b) in (let TMP_11 \def (CHead e TMP_10 u) in (drop1 PNil
-TMP_9 TMP_11)))))) in (let TMP_13 \def (Bind b) in (let TMP_14 \def (CHead e
-TMP_13 u) in (let TMP_15 \def (drop1_nil TMP_14) in (eq_ind_r C e TMP_12
-TMP_15 c H_y))))))))) in (let TMP_44 \def (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (u:
-T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e
-(Bind b) u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons
-n n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) in (let H1 \def
-H_x in (let TMP_17 \def (\lambda (c2: C).(drop n n0 c c2)) in (let TMP_18
-\def (\lambda (c2: C).(drop1 p c2 e)) in (let TMP_19 \def (S n0) in (let
-TMP_20 \def (Ss p) in (let TMP_21 \def (PCons n TMP_19 TMP_20) in (let TMP_22
-\def (Bind b) in (let TMP_23 \def (lift1 p u) in (let TMP_24 \def (lift n n0
-TMP_23) in (let TMP_25 \def (CHead c TMP_22 TMP_24) in (let TMP_26 \def (Bind
-b) in (let TMP_27 \def (CHead e TMP_26 u) in (let TMP_28 \def (drop1 TMP_21
-TMP_25 TMP_27) in (let TMP_43 \def (\lambda (x: C).(\lambda (H2: (drop n n0 c
-x)).(\lambda (H3: (drop1 p x e)).(let TMP_29 \def (Bind b) in (let TMP_30
-\def (lift1 p u) in (let TMP_31 \def (lift n n0 TMP_30) in (let TMP_32 \def
-(CHead c TMP_29 TMP_31) in (let TMP_33 \def (Bind b) in (let TMP_34 \def
-(lift1 p u) in (let TMP_35 \def (CHead x TMP_33 TMP_34) in (let TMP_36 \def
-(S n0) in (let TMP_37 \def (lift1 p u) in (let TMP_38 \def (drop_skip_bind n
-n0 c x H2 b TMP_37) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead e
-TMP_39 u) in (let TMP_41 \def (Ss p) in (let TMP_42 \def (H x u H3) in
-(drop1_cons TMP_32 TMP_35 n TMP_36 TMP_38 TMP_40 TMP_41
-TMP_42)))))))))))))))))) in (ex2_ind C TMP_17 TMP_18 TMP_28 TMP_43
-H1))))))))))))))))))))))) in (PList_ind TMP_7 TMP_16 TMP_44 hds)))))).
+ \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p:
+PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p)
+(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def
+(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead
+c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c
+H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead
+c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def
+(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda
+(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S
+n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))
+(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x
+e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b)
+(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e
+(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
-theorem drop1_cons_tail:
+lemma drop1_cons_tail:
\forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
(drop1 (PConsTail hds h d) c1 c3))))))))
\def
\lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(let TMP_2 \def
-(\lambda (p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (let TMP_1 \def
-(PConsTail p h d) in (drop1 TMP_1 c1 c3))))) in (let TMP_7 \def (\lambda (c1:
-C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def (drop1_gen_pnil c1 c2 H0)
-in (let TMP_4 \def (\lambda (c: C).(let TMP_3 \def (PCons h d PNil) in (drop1
-TMP_3 c c3))) in (let TMP_5 \def (drop1_nil c3) in (let TMP_6 \def
-(drop1_cons c2 c3 h d H c3 PNil TMP_5) in (eq_ind_r C c2 TMP_4 TMP_6 c1
-H_y))))))) in (let TMP_16 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda
-(p: PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to (drop1
-(PConsTail p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n
-n0 p) c1 c2)).(let H_x \def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def
-H_x in (let TMP_8 \def (\lambda (c4: C).(drop n n0 c1 c4)) in (let TMP_9 \def
-(\lambda (c4: C).(drop1 p c4 c2)) in (let TMP_10 \def (PConsTail p h d) in
-(let TMP_11 \def (PCons n n0 TMP_10) in (let TMP_12 \def (drop1 TMP_11 c1 c3)
-in (let TMP_15 \def (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda
-(H4: (drop1 p x c2)).(let TMP_13 \def (PConsTail p h d) in (let TMP_14 \def
-(H0 x H4) in (drop1_cons c1 x n n0 H3 c3 TMP_13 TMP_14)))))) in (ex2_ind C
-TMP_8 TMP_9 TMP_12 TMP_15 H2))))))))))))))) in (PList_ind TMP_2 TMP_7 TMP_16
-hds))))))))).
+nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda
+(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
+c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def
+(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d
+PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0:
+((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
+c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x
+\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C
+(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1
+(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n
+n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3
+(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))).
theorem drop1_trans:
\forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0)
\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1
(papp is1 is2) c1 c2)))))))
\def
- \lambda (is1: PList).(let TMP_2 \def (\lambda (p: PList).(\forall (c1:
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2:
-C).((drop1 is2 c0 c2) \to (let TMP_1 \def (papp p is2) in (drop1 TMP_1 c1
-c2))))))))) in (let TMP_4 \def (\lambda (c1: C).(\lambda (c0: C).(\lambda (H:
-(drop1 PNil c1 c0)).(\lambda (is2: PList).(\lambda (c2: C).(\lambda (H0:
-(drop1 is2 c0 c2)).(let H_y \def (drop1_gen_pnil c1 c0 H) in (let TMP_3 \def
-(\lambda (c: C).(drop1 is2 c c2)) in (let H1 \def (eq_ind_r C c0 TMP_3 H0 c1
-H_y) in H1))))))))) in (let TMP_13 \def (\lambda (n: nat).(\lambda (n0:
+C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1:
+C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2:
+PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def
+(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c:
+C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0:
nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0:
C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0
c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0:
C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda
(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0
-p n n0 H0) in (let H2 \def H_x in (let TMP_5 \def (\lambda (c3: C).(drop n n0
-c1 c3)) in (let TMP_6 \def (\lambda (c3: C).(drop1 p c3 c0)) in (let TMP_7
-\def (papp p is2) in (let TMP_8 \def (PCons n n0 TMP_7) in (let TMP_9 \def
-(drop1 TMP_8 c1 c2) in (let TMP_12 \def (\lambda (x: C).(\lambda (H3: (drop n
-n0 c1 x)).(\lambda (H4: (drop1 p x c0)).(let TMP_10 \def (papp p is2) in (let
-TMP_11 \def (H x c0 H4 is2 c2 H1) in (drop1_cons c1 x n n0 H3 c2 TMP_10
-TMP_11)))))) in (ex2_ind C TMP_5 TMP_6 TMP_9 TMP_12 H2))))))))))))))))))) in
-(PList_ind TMP_2 TMP_4 TMP_13 is1)))).
+p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1
+c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1
+c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x
+c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1)))))
+H2))))))))))))) is1).
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