include "drop/props.ma".
+include "getl/defs.ma".
+
theorem 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)
[(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1:
(eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C
c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda
-(H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u)
-(CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3)))
-c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1)
-\Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq
+(H3: (eq C c e)).(eq_ind C e (\lambda (c1: C).(drop1 PNil (CHead c1 (Bind b)
+u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e
+H3))) c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds0 H1)
+\Rightarrow (\lambda (H2: (eq PList (PCons h d hds0) PNil)).(\lambda (H3: (eq
C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d
-hds) (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).Prop)
-with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2)
-in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1
-hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))
-H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c)
-(refl_equal C e)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+hds0) (\lambda (e0: PList).(match e0 in PList return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True]))
+I PNil H2) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2)
+\to ((drop1 hds0 c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind
+b) u)))))) H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal
+C c) (refl_equal C e)))))) (\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)
(_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow
False])) I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to
(drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u)))
-(CHead e (Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2)
-\Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda
-(H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList
-PList (\lambda (e0: PList).(match e0 in PList return (\lambda (_:
-PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p]))
-(PCons h d hds) (PCons n n0 p) H3) in ((let H7 \def (f_equal PList nat
+(CHead e (Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds0 H2)
+\Rightarrow (\lambda (H3: (eq PList (PCons h d hds0) (PCons n n0
+p))).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def
+(f_equal PList PList (\lambda (e0: PList).(match e0 in PList return (\lambda
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
+p0])) (PCons h d hds0) (PCons n n0 p) H3) in ((let H7 \def (f_equal PList nat
(\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with
-[PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n
-n0 p) H3) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0
-in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n
-_ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in (eq_ind nat n
-(\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to
-((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons
-n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b)
-u))))))))) (\lambda (H9: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1:
-nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1
-c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind
-b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))) (\lambda (H10: (eq
-PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3
-e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (drop1 (PCons n (S n0)
-(Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))
-(\lambda (H11: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to
-((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (drop1 (PCons n (S n0) (Ss p))
-(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))) (\lambda
-(H12: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1
-p c2 c0) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0
-(lift1 p u))) (CHead e (Bind b) u))))) (\lambda (H13: (drop n n0 c
-c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons (CHead c (Bind b) (lift n n0
-(lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n (S n0) (drop_skip_bind n n0 c
-c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss p) (H c2 u H14)))) c3 (sym_eq
-C c3 e H12))) c1 (sym_eq C c1 c H11))) hds (sym_eq PList hds p H10))) d
-(sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) H7)) H6)) H4 H5 H1 H2))))]) in
-(H1 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C
-e)))))))))) hds))).
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
+(PCons n n0 p) H3) in ((let H8 \def (f_equal PList nat (\lambda (e0:
+PList).(match e0 in PList return (\lambda (_: PList).nat) with [PNil
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
+p) H3) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
+p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds0
+c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1
+p u))) (CHead e (Bind b) u))))))))) (\lambda (H9: (eq nat d n0)).(eq_ind nat
+n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e)
+\to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3) \to (drop1 (PCons n (S n0) (Ss
+p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))
+(\lambda (H10: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: PList).((eq C
+c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to
+(drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u)))
+(CHead e (Bind b) u))))))) (\lambda (H11: (eq C c1 c)).(eq_ind C c (\lambda
+(c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (drop1
+(PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e
+(Bind b) u)))))) (\lambda (H12: (eq C c3 e)).(eq_ind C e (\lambda (c0:
+C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (drop1 (PCons n (S n0) (Ss p))
+(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))) (\lambda
+(H13: (drop n n0 c c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons (CHead c
+(Bind b) (lift n n0 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n (S n0)
+(drop_skip_bind n n0 c c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss p) (H
+c2 u H14)))) c3 (sym_eq C c3 e H12))) c1 (sym_eq C c1 c H11))) hds0 (sym_eq
+PList hds0 p H10))) d (sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) H7))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList (PCons n n0 p)) (refl_equal C
+c) (refl_equal C e)))))))))) hds))).
theorem drop1_cons_tail:
\forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
\to (drop1 (PCons h d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2
(\lambda (c0: C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3
PNil (drop1_nil c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) |
-(drop1_cons c0 c4 h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList
-(PCons h0 d0 hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5
-c2)).((let H6 \def (eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match
-e in PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow False |
-(PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to
-((eq C c5 c2) \to ((drop h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1
-(PCons h d PNil) c1 c3))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
-PNil) (refl_equal C c1) (refl_equal C c2))))) (\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 H2 \def (match H1 in drop1 return (\lambda (p0:
-PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq
-PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons
-n n0 (PConsTail p h d)) c1 c3)))))))) with [(drop1_nil c) \Rightarrow
-(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c
-c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e:
+(drop1_cons c0 c4 h0 d0 H1 c5 hds0 H2) \Rightarrow (\lambda (H3: (eq PList
+(PCons h0 d0 hds0) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5
+c2)).((let H6 \def (eq_ind PList (PCons h0 d0 hds0) (\lambda (e:
PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in
-(False_ind ((eq C c c1) \to ((eq C c c2) \to (drop1 (PCons n n0 (PConsTail p
-h d)) c1 c3))) H5)) H3 H4)))) | (drop1_cons c0 c4 h0 d0 H2 c5 hds H3)
-\Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds) (PCons n n0
-p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c5 c2)).((let H7 \def
-(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p]))
-(PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
-[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds)
-(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
-\Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds
-p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1
-hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda
-(H10: (eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to
-((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5)
-\to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq
-PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C
-c5 c2) \to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0
-(PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1
-(\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to
-(drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5
-c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to
-(drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1
-c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail
-p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds
-(sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n
-H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
+(False_ind ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop h0 d0 c0 c4) \to
+((drop1 hds0 c4 c5) \to (drop1 (PCons h d PNil) c1 c3))))) H6)) H4 H5 H1
+H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C
+c2))))) (\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 H2
+\def (match H1 in drop1 return (\lambda (p0: PList).(\lambda (c: C).(\lambda
+(c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq
+C c c1) \to ((eq C c0 c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1
+c3)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil
+(PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let
+H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList return
+(\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq
+C c c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))) H5)) H3 H4)))) |
+(drop1_cons c0 c4 h0 d0 H2 c5 hds0 H3) \Rightarrow (\lambda (H4: (eq PList
+(PCons h0 d0 hds0) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6:
+(eq C c5 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e
+in PList return (\lambda (_: PList).PList) with [PNil \Rightarrow hds0 |
+(PCons _ _ p0) \Rightarrow p0])) (PCons h0 d0 hds0) (PCons n n0 p) H4) in
+((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e in PList return
+(\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _ n1 _)
+\Rightarrow n1])) (PCons h0 d0 hds0) (PCons n n0 p) H4) in ((let H9 \def
+(f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda (_:
+PList).nat) with [PNil \Rightarrow h0 | (PCons n1 _ _) \Rightarrow n1]))
+(PCons h0 d0 hds0) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1:
+nat).((eq nat d0 n0) \to ((eq PList hds0 p) \to ((eq C c0 c1) \to ((eq C c5
+c2) \to ((drop n1 d0 c0 c4) \to ((drop1 hds0 c4 c5) \to (drop1 (PCons n n0
+(PConsTail p h d)) c1 c3)))))))) (\lambda (H10: (eq nat d0 n0)).(eq_ind nat
+n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c0 c1) \to ((eq C c5 c2)
+\to ((drop n n1 c0 c4) \to ((drop1 hds0 c4 c5) \to (drop1 (PCons n n0
+(PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind
+PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n0
+c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1
+c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c5
+c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to (drop1 (PCons n n0
+(PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5 c2)).(eq_ind C c2
+(\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to (drop1 (PCons n n0
+(PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1 c4)).(\lambda (H15:
+(drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail p h d) (H0 c4
+H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds0 (sym_eq
+PList hds0 p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n H9)))
+H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
(refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
projects / helm.git / blobdiff