include "LambdaDelta-1/arity/props.ma".
-include "LambdaDelta-1/drop1/defs.ma".
+include "LambdaDelta-1/drop1/fwd.ma".
theorem arity_lift1:
\forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
-(H0: (arity g c2 t a)).(let H1 \def (match H in drop1 return (\lambda (p:
-PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
-PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a))))))))
-with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda
-(H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0:
-C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C
-c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C
-c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda
-(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda
-(H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\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 c4 c2) \to ((drop h d c0 c3) \to ((drop1
-hds0 c3 c4) \to (arity g c1 t a))))) 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 (H: ((\forall (c1:
+(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C
+c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1:
C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1
(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1
-(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H2 \def (match H0
-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 (arity g c1 (lift n n0 (lift1 p t)) a)))))))) 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 (arity g
-c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4
-hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
-p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 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 h d 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 d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d 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 h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
-p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n1 d c0 c3) \to ((drop1
-hds0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) (\lambda (H10:
-(eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq
-C c0 c1) \to ((eq C c4 c2) \to ((drop n n1 c0 c3) \to ((drop1 hds0 c3 c4) \to
-(arity g c1 (lift n n0 (lift1 p t)) a))))))) (\lambda (H11: (eq PList hds0
-p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c4 c2) \to
-((drop n n0 c0 c3) \to ((drop1 p0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p
-t)) a)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C
-c4 c2) \to ((drop n n0 c c3) \to ((drop1 p c3 c4) \to (arity g c1 (lift n n0
-(lift1 p t)) a))))) (\lambda (H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c:
-C).((drop n n0 c1 c3) \to ((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p
-t)) a)))) (\lambda (H14: (drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3
-c2)).(arity_lift g c3 (lift1 p t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq
-C c4 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds0 (sym_eq PList hds0 p H11))) d
-(sym_eq nat d n0 H10))) h (sym_eq nat h 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)))).
+(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def
+(drop1_gen_pcons c1 c2 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 c2)) (arity g c1
+(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1
+x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1)
+c1 n n0 H3)))) H2))))))))))) hds)))).
projects / helm.git / blobdiff