matita-basics-lists-list.agda

open import Agda.Primitive
open import matita-basics-relations
open import matita-basics-logic
open import matita-basics-types
open import matita-basics-bool
open import matita-arithmetics-nat

data list (l1-v : Level) (A-v : Set l1-v) : Set l1-v where
  nil' : list l1-v A-v
  cons' : (X---v : A-v) -> (X--1-v : list l1-v A-v) -> list l1-v A-v

nil : (l1-v : Level) -> (A-v : Set l1-v) -> list l1-v A-v
nil _ _ = nil'

cons : (l4-v : Level) -> (A-v : Set l4-v) -> (X---v : A-v) -> (X--1-v : list l4-v A-v) -> list l4-v A-v
cons _ _ = cons'

match-list : (l8-v : Level) -> (X-A-v : Set l8-v) -> (return-sort-v : Level) -> (return-type-v : (z-v : list l8-v X-A-v) -> Set return-sort-v) -> (case-nil-v : return-type-v (nil l8-v X-A-v)) -> (case-cons-v : (X---v : X-A-v) -> (X--1-v : list l8-v X-A-v) -> return-type-v (cons l8-v X-A-v X---v X--1-v)) -> (z-v : list l8-v X-A-v) -> return-type-v z-v
match-list _ _ _ _ casenil casecons nil' = casenil
match-list _ _ _ _ casenil casecons (cons' x1 x2) = casecons x1 x2

list-ind : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-716-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-718-v : X-A-v) -> (x-717-v : list l13-v X-A-v) -> (X-x-720-v : Q--v x-717-v) -> Q--v (cons l13-v X-A-v x-718-v x-717-v)) -> (x-716-v : list l13-v X-A-v) -> Q--v x-716-v
list-ind _ _ _ _ casenil casecons nil' = casenil
list-ind A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type5 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type5 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type5 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type4 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type4 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type4 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type3 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type3 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type3 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type2 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type2 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type2 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type1 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type1 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type1 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)

list-rect-Type0 : (l13-v l10-v : Level) -> (X-A-v : Set l13-v) -> (Q--v : (X-x-721-v : list l13-v X-A-v) -> Set l10-v) -> (X-H-nil-v : Q--v (nil l13-v X-A-v)) -> (X-H-cons-v : (x-723-v : X-A-v) -> (x-722-v : list l13-v X-A-v) -> (X-x-725-v : Q--v x-722-v) -> Q--v (cons l13-v X-A-v x-723-v x-722-v)) -> (x-721-v : list l13-v X-A-v) -> Q--v x-721-v
list-rect-Type0 _ _ _ _ casenil casecons nil' = casenil
list-rect-Type0 A1 A2 A3 A4 casenil casecons (cons' x1 x2) = casecons x1 x2 (list-ind A1 A2 A3 A4 casenil casecons x2)



list-inv-ind : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1302 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1303 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-718 : x1) -> (x-717 : list l37 x1) -> (X-x-720 : (X-z1303 : eq l37 (list l37 x1) Hterm x-717) -> P x-717) -> (X-z1303 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-718 x-717)) -> P (cons l37 x1 x-718 x-717)) -> P Hterm
list-inv-ind = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1302 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1303 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-718 : x1) -> (x-717 : list l37 x1) -> (X-x-720 : (X-z1303 : eq l37 (list l37 x1) Hterm x-717) -> P x-717) -> (X-z1303 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-718 x-717)) -> P (cons l37 x1 x-718 x-717)) -> list-ind l37 (l37 ⊔ l32) x1 (λ (X-x-716 : list l37 x1) -> (X-z1303 : eq l37 (list l37 x1) Hterm X-x-716) -> P X-x-716) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-inv-rect-Type4 : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1308 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1309 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-723 : x1) -> (x-722 : list l37 x1) -> (X-x-725 : (X-z1309 : eq l37 (list l37 x1) Hterm x-722) -> P x-722) -> (X-z1309 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-723 x-722)) -> P (cons l37 x1 x-723 x-722)) -> P Hterm
list-inv-rect-Type4 = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1308 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1309 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-723 : x1) -> (x-722 : list l37 x1) -> (X-x-725 : (X-z1309 : eq l37 (list l37 x1) Hterm x-722) -> P x-722) -> (X-z1309 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-723 x-722)) -> P (cons l37 x1 x-723 x-722)) -> list-rect-Type4 l37 (l37 ⊔ l32) x1 (λ (X-x-721 : list l37 x1) -> (X-z1309 : eq l37 (list l37 x1) Hterm X-x-721) -> P X-x-721) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-inv-rect-Type3 : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1314 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1315 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-733 : x1) -> (x-732 : list l37 x1) -> (X-x-735 : (X-z1315 : eq l37 (list l37 x1) Hterm x-732) -> P x-732) -> (X-z1315 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-733 x-732)) -> P (cons l37 x1 x-733 x-732)) -> P Hterm
list-inv-rect-Type3 = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1314 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1315 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-733 : x1) -> (x-732 : list l37 x1) -> (X-x-735 : (X-z1315 : eq l37 (list l37 x1) Hterm x-732) -> P x-732) -> (X-z1315 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-733 x-732)) -> P (cons l37 x1 x-733 x-732)) -> list-rect-Type3 l37 (l37 ⊔ l32) x1 (λ (X-x-731 : list l37 x1) -> (X-z1315 : eq l37 (list l37 x1) Hterm X-x-731) -> P X-x-731) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-inv-rect-Type2 : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1320 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1321 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-738 : x1) -> (x-737 : list l37 x1) -> (X-x-740 : (X-z1321 : eq l37 (list l37 x1) Hterm x-737) -> P x-737) -> (X-z1321 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-738 x-737)) -> P (cons l37 x1 x-738 x-737)) -> P Hterm
list-inv-rect-Type2 = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1320 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1321 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-738 : x1) -> (x-737 : list l37 x1) -> (X-x-740 : (X-z1321 : eq l37 (list l37 x1) Hterm x-737) -> P x-737) -> (X-z1321 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-738 x-737)) -> P (cons l37 x1 x-738 x-737)) -> list-rect-Type2 l37 (l37 ⊔ l32) x1 (λ (X-x-736 : list l37 x1) -> (X-z1321 : eq l37 (list l37 x1) Hterm X-x-736) -> P X-x-736) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-inv-rect-Type1 : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1326 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1327 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-743 : x1) -> (x-742 : list l37 x1) -> (X-x-745 : (X-z1327 : eq l37 (list l37 x1) Hterm x-742) -> P x-742) -> (X-z1327 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-743 x-742)) -> P (cons l37 x1 x-743 x-742)) -> P Hterm
list-inv-rect-Type1 = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1326 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1327 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-743 : x1) -> (x-742 : list l37 x1) -> (X-x-745 : (X-z1327 : eq l37 (list l37 x1) Hterm x-742) -> P x-742) -> (X-z1327 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-743 x-742)) -> P (cons l37 x1 x-743 x-742)) -> list-rect-Type1 l37 (l37 ⊔ l32) x1 (λ (X-x-741 : list l37 x1) -> (X-z1327 : eq l37 (list l37 x1) Hterm X-x-741) -> P X-x-741) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-inv-rect-Type0 : (l37 l32 : Level) -> (x1 : Set l37) -> (Hterm : list l37 x1) -> (P : (X-z1332 : list l37 x1) -> Set l32) -> (X-H1 : (X-z1333 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> (X-H2 : (x-748 : x1) -> (x-747 : list l37 x1) -> (X-x-750 : (X-z1333 : eq l37 (list l37 x1) Hterm x-747) -> P x-747) -> (X-z1333 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-748 x-747)) -> P (cons l37 x1 x-748 x-747)) -> P Hterm
list-inv-rect-Type0 = λ (l37 l32 : Level) -> λ (x1 : Set l37) -> λ (Hterm : list l37 x1) -> λ (P : (X-z1332 : list l37 x1) -> Set l32) -> λ (H1 : (X-z1333 : eq l37 (list l37 x1) Hterm (nil l37 x1)) -> P (nil l37 x1)) -> λ (H2 : (x-748 : x1) -> (x-747 : list l37 x1) -> (X-x-750 : (X-z1333 : eq l37 (list l37 x1) Hterm x-747) -> P x-747) -> (X-z1333 : eq l37 (list l37 x1) Hterm (cons l37 x1 x-748 x-747)) -> P (cons l37 x1 x-748 x-747)) -> list-rect-Type0 l37 (l37 ⊔ l32) x1 (λ (X-x-746 : list l37 x1) -> (X-z1333 : eq l37 (list l37 x1) Hterm X-x-746) -> P X-x-746) H1 H2 Hterm (refl l37 (list l37 x1) Hterm)

list-discr : (l2l222 l293 : Level) -> (a1 : Set ((lsuc lzero) ⊔ (lsuc l2l222))) -> (x : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (y : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (X-e : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) x y) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) (match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> (X-z47 : P) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set (l2l222 ⊔ l293)) -> P) y) (λ (t0 : a1) -> λ (t1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set l293) -> (X-z48 : (e0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0) u0) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) (R1 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 (λ (x0 : a1) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 x0) -> list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) t1 u0 e0) u1) -> P) -> P) y) x
list-discr = λ (l2l222 l293 : Level) -> λ (a1 : Set ((lsuc lzero) ⊔ (lsuc l2l222))) -> λ (x : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> λ (y : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> λ (Deq : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) x y) -> eq-rect-Type2 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293))) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) x (λ (x-13 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> λ (X-x-14 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) x x-13) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) (match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> (X-z47 : P) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set (l2l222 ⊔ l293)) -> P) x-13) (λ (t0 : a1) -> λ (t1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set l293) -> (X-z48 : (e0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0) u0) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) (R1 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 (λ (x0 : a1) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 x0) -> list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) t1 u0 e0) u1) -> P) -> P) x-13) x) (match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293))) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) (match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> (X-z47 : P) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set (l2l222 ⊔ l293)) -> P) X--) (λ (t0 : a1) -> λ (t1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> match-list ((lsuc lzero) ⊔ (lsuc l2l222)) a1 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l2l222)) ⊔ (lsuc (lsuc l293)))) (λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> Set ((lsuc lzero) ⊔ ((lsuc l2l222) ⊔ (lsuc l293)))) ((P : Set (l2l222 ⊔ l293)) -> P) (λ (u0 : a1) -> λ (u1 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> (P : Set l293) -> (X-z48 : (e0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0) u0) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) (R1 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 (λ (x0 : a1) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 t0 x0) -> list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) t1 u0 e0) u1) -> P) -> P) X--) X--) (λ (P : Set (lzero ⊔ (l2l222 ⊔ l293))) -> λ (DH : P) -> DH) (λ (a0 : a1) -> λ (a10 : list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) -> λ (P : Set l293) -> λ (DH : (e0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0) a0) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) (R1 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0 (λ (x0 : a1) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0 x0) -> list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) a10 a0 e0) a10) -> P) -> DH (refl ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0)) (refl ((lsuc lzero) ⊔ (lsuc l2l222)) (list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) (R1 ((lsuc lzero) ⊔ (lsuc l2l222)) ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0 (λ (x0 : a1) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0 x0) -> list ((lsuc lzero) ⊔ (lsuc l2l222)) a1) a10 a0 (refl ((lsuc lzero) ⊔ (lsuc l2l222)) a1 (R0 ((lsuc lzero) ⊔ (lsuc l2l222)) a1 a0))))) x) y Deq

is-nil : (l12 l13 : Level) -> (A : Set l12) -> (X-- : list l12 A) -> Set l13
is-nil = λ (l12 l13 : Level) -> λ (A : Set l12) -> λ (l : list l12 A) -> match-list l12 A ((lsuc lzero) ⊔ (lsuc l13)) (λ (X-- : list l12 A) -> Set l13) (True l13) (λ (hd : A) -> λ (tl : list l12 A) -> False l13) l

nil-cons : (l27 : Level) -> (A : Set l27) -> (l : list l27 A) -> (a : A) -> Not l27 (eq l27 (list l27 A) (cons l27 A a l) (nil l27 A))
nil-cons = λ (l27 : Level) -> λ (A : Set l27) -> λ (l : list l27 A) -> λ (a : A) -> nmk l27 (eq l27 (list l27 A) (cons l27 A a l) (nil l27 A)) (λ (Heq : eq l27 (list l27 A) (cons l27 A a l) (nil l27 A)) -> eq-ind-r l27 l27 (list l27 A) (nil l27 A) (λ (x : list l27 A) -> λ (X-- : eq l27 (list l27 A) x (nil l27 A)) -> is-nil l27 l27 A x) (I l27) (cons l27 A a l) Heq)


append : (l5-v : Level) -> (H-v : Set l5-v) -> (X---v : list l5-v H-v) -> (X--1-v : list l5-v H-v) -> list l5-v H-v
append l12-v H-v nil' l2 = l2
append l12-v H-v (cons' x' l') l2 = cons l12-v H-v x' (append l12-v H-v l' l2)


hd : (l10 : Level) -> (A : Set l10) -> (X-l : list l10 A) -> (X-d : A) -> A
hd = λ (l10 : Level) -> λ (A : Set l10) -> λ (l : list l10 A) -> λ (d : A) -> match-list l10 A l10 (λ (X-- : list l10 A) -> A) d (λ (a : A) -> λ (X-- : list l10 A) -> a) l

tail : (l11 : Level) -> (A : Set l11) -> (X-l : list l11 A) -> list l11 A
tail = λ (l11 : Level) -> λ (A : Set l11) -> λ (l : list l11 A) -> match-list l11 A l11 (λ (X-- : list l11 A) -> list l11 A) (nil l11 A) (λ (hd-v : A) -> λ (tl : list l11 A) -> tl) l

option-hd : (l12 : Level) -> (A : Set l12) -> (X-l : list l12 A) -> option l12 A
option-hd = λ (l12 : Level) -> λ (A : Set l12) -> λ (l : list l12 A) -> match-list l12 A l12 (λ (X-- : list l12 A) -> option l12 A) (None l12 A) (λ (a : A) -> λ (X-- : list l12 A) -> Some l12 A a) l

append-nil : (l35 : Level) -> (A : Set l35) -> (l : list l35 A) -> eq l35 (list l35 A) (append l35 A l (nil l35 A)) l
append-nil = λ (l35 : Level) -> λ (A : Set l35) -> λ (l : list l35 A) -> list-ind l35 l35 A (λ (X-x-716 : list l35 A) -> eq l35 (list l35 A) (append l35 A X-x-716 (nil l35 A)) X-x-716) (refl l35 (list l35 A) (nil l35 A)) (λ (x-718 : A) -> λ (x-717 : list l35 A) -> λ (X-x-720 : eq l35 (list l35 A) (append l35 A x-717 (nil l35 A)) x-717) -> rewrite-r l35 l35 (list l35 A) x-717 (λ (X-- : list l35 A) -> eq l35 (list l35 A) (cons l35 A x-718 X--) (cons l35 A x-718 x-717)) (refl l35 (list l35 A) (cons l35 A x-718 x-717)) (append l35 A x-717 (nil l35 A)) X-x-720) l

associative-append : (l53 : Level) -> (A : Set l53) -> associative l53 (list l53 A) (append l53 A)
associative-append = λ (l53 : Level) -> λ (A : Set l53) -> λ (l1 : list l53 A) -> λ (l2 : list l53 A) -> λ (l3 : list l53 A) -> list-ind l53 l53 A (λ (X-x-716 : list l53 A) -> eq l53 (list l53 A) (append l53 A (append l53 A X-x-716 l2) l3) (append l53 A X-x-716 (append l53 A l2 l3))) (refl l53 (list l53 A) (append l53 A l2 l3)) (λ (x-718 : A) -> λ (x-717 : list l53 A) -> λ (X-x-720 : eq l53 (list l53 A) (append l53 A (append l53 A x-717 l2) l3) (append l53 A x-717 (append l53 A l2 l3))) -> rewrite-r l53 l53 (list l53 A) (append l53 A x-717 (append l53 A l2 l3)) (λ (X-- : list l53 A) -> eq l53 (list l53 A) (cons l53 A x-718 X--) (cons l53 A x-718 (append l53 A x-717 (append l53 A l2 l3)))) (refl l53 (list l53 A) (cons l53 A x-718 (append l53 A x-717 (append l53 A l2 l3)))) (append l53 A (append l53 A x-717 l2) l3) X-x-720) l1

append-cons : (l36 : Level) -> (A : Set l36) -> (a : A) -> (l : list l36 A) -> (l1 : list l36 A) -> eq l36 (list l36 A) (append l36 A l (cons l36 A a l1)) (append l36 A (append l36 A l (cons l36 A a (nil l36 A))) l1)
append-cons = λ (l36 : Level) -> λ (A : Set l36) -> λ (a : A) -> λ (l : list l36 A) -> λ (l1 : list l36 A) -> eq-ind-r l36 l36 (list l36 A) (append l36 A l (append l36 A (cons l36 A a (nil l36 A)) l1)) (λ (x : list l36 A) -> λ (X-- : eq l36 (list l36 A) x (append l36 A l (append l36 A (cons l36 A a (nil l36 A)) l1))) -> eq l36 (list l36 A) (append l36 A l (cons l36 A a l1)) x) (refl l36 (list l36 A) (append l36 A l (cons l36 A a l1))) (append l36 A (append l36 A l (cons l36 A a (nil l36 A))) l1) (associative-append l36 A l (cons l36 A a (nil l36 A)) l1)

nil-append-elim : (l73 l61 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l73))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> (P : (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> (X--1 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> Set (lzero ⊔ (l73 ⊔ l61))) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l73)) (list ((lsuc lzero) ⊔ (lsuc l73)) A) (append ((lsuc lzero) ⊔ (lsuc l73)) A l1 l2) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> (X--1 : P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> P l1 l2
nil-append-elim = λ (l73 l61 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l73))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> λ (P : (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> (X--1 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> Set (lzero ⊔ (l73 ⊔ l61))) -> match-list ((lsuc lzero) ⊔ (lsuc l73)) A ((lsuc lzero) ⊔ ((lsuc l73) ⊔ l61)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l73)) (list ((lsuc lzero) ⊔ (lsuc l73)) A) (append ((lsuc lzero) ⊔ (lsuc l73)) A X-- l2) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> (X--2 : P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> P X-- l2) (λ (auto : eq ((lsuc lzero) ⊔ (lsuc l73)) (list ((lsuc lzero) ⊔ (lsuc l73)) A) l2 (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> λ (auto' : P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> eq-coerc (l73 ⊔ l61) (P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) (P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) l2) auto' (rewrite-l ((lsuc lzero) ⊔ (lsuc l73)) ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (list ((lsuc lzero) ⊔ (lsuc l73)) A) l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> eq ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (Set (l73 ⊔ l61)) (P X-- (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) (P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) l2)) (rewrite-l ((lsuc lzero) ⊔ (lsuc l73)) ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (list ((lsuc lzero) ⊔ (lsuc l73)) A) l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> eq ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (Set (l73 ⊔ l61)) (P l2 X--) (P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) l2)) (rewrite-l ((lsuc lzero) ⊔ (lsuc l73)) ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (list ((lsuc lzero) ⊔ (lsuc l73)) A) l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> eq ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (Set (l73 ⊔ l61)) (P l2 l2) (P X-- l2)) (refl ((lsuc lzero) ⊔ ((lsuc l73) ⊔ (lsuc l61))) (Set (l73 ⊔ l61)) (P l2 l2)) (nil ((lsuc lzero) ⊔ (lsuc l73)) A) auto) (nil ((lsuc lzero) ⊔ (lsuc l73)) A) auto) (nil ((lsuc lzero) ⊔ (lsuc l73)) A) auto)) (λ (a : A) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l73)) A) -> λ (heq : eq ((lsuc lzero) ⊔ (lsuc l73)) (list ((lsuc lzero) ⊔ (lsuc l73)) A) (cons ((lsuc lzero) ⊔ (lsuc l73)) A a (append ((lsuc lzero) ⊔ (lsuc l73)) A l3 l2)) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> list-discr l73 l61 A (cons ((lsuc lzero) ⊔ (lsuc l73)) A a (append ((lsuc lzero) ⊔ (lsuc l73)) A l3 l2)) (nil ((lsuc lzero) ⊔ (lsuc l73)) A) heq ((X-- : P (nil ((lsuc lzero) ⊔ (lsuc l73)) A) (nil ((lsuc lzero) ⊔ (lsuc l73)) A)) -> P (cons ((lsuc lzero) ⊔ (lsuc l73)) A a l3) l2)) l1

nil-to-nil : (l82 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l82))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) -> And ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) l1 (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) l2 (nil ((lsuc lzero) ⊔ (lsuc l82)) A))
nil-to-nil = λ (l82 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l82))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> λ (isnil : eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) -> nil-append-elim l82 ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> And ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) X-- (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) X-0 (nil ((lsuc lzero) ⊔ (lsuc l82)) A))) (rewrite-l ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) X--) (refl ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2)) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) isnil) (conj ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (rewrite-l ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) X-- (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (rewrite-l ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) X--) (refl ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2)) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) isnil) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) isnil) (rewrite-l ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) X-- (nil ((lsuc lzero) ⊔ (lsuc l82)) A)) (rewrite-l ((lsuc lzero) ⊔ (lsuc l82)) ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l82)) A) -> eq ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2) X--) (refl ((lsuc lzero) ⊔ (lsuc l82)) (list ((lsuc lzero) ⊔ (lsuc l82)) A) (append ((lsuc lzero) ⊔ (lsuc l82)) A l1 l2)) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) isnil) (nil ((lsuc lzero) ⊔ (lsuc l82)) A) isnil))

cons-injective-l : (l94 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l94))) -> (a1 : A) -> (a2 : A) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l94)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l94)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l94)) A a1 a2
cons-injective-l = λ (l94 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l94))) -> λ (a1 : A) -> λ (a2 : A) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l94)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l94)) A) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> list-discr l94 ((lsuc lzero) ⊔ (lsuc l94)) A (cons ((lsuc lzero) ⊔ (lsuc l94)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2) Heq (eq ((lsuc lzero) ⊔ (lsuc l94)) A a1 a2) (λ (e0 : eq ((lsuc lzero) ⊔ (lsuc l94)) A (R0 ((lsuc lzero) ⊔ (lsuc l94)) A a1) a2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l94)) ((lsuc lzero) ⊔ (lsuc l94)) A a2 (λ (x : A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l94)) A x a2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A x l1) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (R1 ((lsuc lzero) ⊔ (lsuc l94)) ((lsuc lzero) ⊔ (lsuc l94)) A x (λ (x0 : A) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l94)) A x x0) -> list ((lsuc lzero) ⊔ (lsuc l94)) A) l1 a2 X--) l2) -> eq ((lsuc lzero) ⊔ (lsuc l94)) A x a2) (λ (Heq0 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l1) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> λ (e00 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (R1 ((lsuc lzero) ⊔ (lsuc l94)) ((lsuc lzero) ⊔ (lsuc l94)) A a2 (λ (x0 : A) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l94)) A a2 x0) -> list ((lsuc lzero) ⊔ (lsuc l94)) A) l1 a2 (refl ((lsuc lzero) ⊔ (lsuc l94)) A a2)) l2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l94)) ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) l2 (λ (x : list ((lsuc lzero) ⊔ (lsuc l94)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) x l2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 x) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l94)) A a2 a2) (λ (Heq1 : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> streicherK ((lsuc lzero) ⊔ (lsuc l94)) ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2) (λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l94)) (list ((lsuc lzero) ⊔ (lsuc l94)) A) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2) (cons ((lsuc lzero) ⊔ (lsuc l94)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l94)) A a2 a2) (refl ((lsuc lzero) ⊔ (lsuc l94)) A a2) Heq1) l1 e00 Heq0) a1 e0 Heq)

cons-injective-r : (l99 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l99))) -> (a1 : A) -> (a2 : A) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l99)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l99)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l1 l2
cons-injective-r = λ (l99 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l99))) -> λ (a1 : A) -> λ (a2 : A) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l99)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l99)) A) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> list-discr l99 ((lsuc lzero) ⊔ (lsuc l99)) A (cons ((lsuc lzero) ⊔ (lsuc l99)) A a1 l1) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2) Heq (eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l1 l2) (λ (e0 : eq ((lsuc lzero) ⊔ (lsuc l99)) A (R0 ((lsuc lzero) ⊔ (lsuc l99)) A a1) a2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l99)) ((lsuc lzero) ⊔ (lsuc l99)) A a2 (λ (x : A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l99)) A x a2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A x l1) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (R1 ((lsuc lzero) ⊔ (lsuc l99)) ((lsuc lzero) ⊔ (lsuc l99)) A x (λ (x0 : A) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l99)) A x x0) -> list ((lsuc lzero) ⊔ (lsuc l99)) A) l1 a2 X--) l2) -> eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l1 l2) (λ (Heq0 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l1) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> λ (e00 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (R1 ((lsuc lzero) ⊔ (lsuc l99)) ((lsuc lzero) ⊔ (lsuc l99)) A a2 (λ (x0 : A) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l99)) A a2 x0) -> list ((lsuc lzero) ⊔ (lsuc l99)) A) l1 a2 (refl ((lsuc lzero) ⊔ (lsuc l99)) A a2)) l2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l99)) ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l2 (λ (x : list ((lsuc lzero) ⊔ (lsuc l99)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) x l2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 x) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) x l2) (λ (Heq1 : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> streicherK ((lsuc lzero) ⊔ (lsuc l99)) ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2) (λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2) (cons ((lsuc lzero) ⊔ (lsuc l99)) A a2 l2)) -> eq ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l2 l2) (refl ((lsuc lzero) ⊔ (lsuc l99)) (list ((lsuc lzero) ⊔ (lsuc l99)) A) l2) Heq1) l1 e00 Heq0) a1 e0 Heq)

option-cons : (l11 : Level) -> (sig : Set l11) -> (X-c : option l11 sig) -> (X-l : list l11 sig) -> list l11 sig
option-cons = λ (l11 : Level) -> λ (sig : Set l11) -> λ (c : option l11 sig) -> λ (l : list l11 sig) -> match-option l11 sig l11 (λ (X-- : option l11 sig) -> list l11 sig) l (λ (c0 : sig) -> cons l11 sig c0 l) c

opt-cons-tail-expand : (l20 : Level) -> (A : Set l20) -> (l : list l20 A) -> eq l20 (list l20 A) l (option-cons l20 A (option-hd l20 A l) (tail l20 A l))
opt-cons-tail-expand = λ (l20 : Level) -> λ (A : Set l20) -> λ (X-clearme : list l20 A) -> match-list l20 A l20 (λ (X-- : list l20 A) -> eq l20 (list l20 A) X-- (option-cons l20 A (option-hd l20 A X--) (tail l20 A X--))) (refl l20 (list l20 A) (nil l20 A)) (λ (auto : A) -> λ (auto' : list l20 A) -> refl l20 (list l20 A) (cons l20 A auto auto')) X-clearme

compare-append : (l516 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l516))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A l1 l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4)) -> ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A l1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))
compare-append = λ (l516 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l516))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> list-ind ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) A (λ (X-x-716 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A X-x-716 l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4)) -> ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) X-x-716 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A X-x-716 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))) (λ (l2 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4)) -> ex-intro ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4)))) l3 (or-intror ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l3)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l3)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4))) (conj ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l3)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4)) (refl ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3) Heq))) (λ (a1 : A) -> λ (tl1 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (Hind : (l2 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (X-- : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l4)) -> ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl1 (append ((lsuc lzero) ⊔ (lsuc l516)) A l3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> match-list ((lsuc lzero) ⊔ (lsuc l516)) A ((lsuc lzero) ⊔ (lsuc l516)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A X-- l4)) -> ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (append ((lsuc lzero) ⊔ (lsuc l516)) A X-- l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) X-- (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))) (λ (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l4)) -> ex-intro ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4)))) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (or-introl ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1))) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1))) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l4))) (conj ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (append ((lsuc lzero) ⊔ (lsuc l516)) A (nil ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1))) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l2)) (refl ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1)) (sym-eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l2) l4 Heq)))) (λ (a3 : A) -> λ (tl3 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l2)) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l4))) -> match-ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl1 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4)))) ((lsuc lzero) ⊔ (lsuc l516)) (λ (X-- : ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl1 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))) -> ex ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a3 tl3) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a3 tl3) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A a1 tl1) l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))))) (λ (l : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> λ (X-clearme : Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl1 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4)))) -> match-Or ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl1 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l4 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l2))) (And ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) tl3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4))) ((lsuc lzero) ⊔ (lsuc 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lzero) ⊔ (lsuc l516)) A a3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l)) (append ((lsuc lzero) ⊔ (lsuc l516)) A (cons ((lsuc lzero) ⊔ (lsuc l516)) A x tl1) l)) (refl ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) (cons ((lsuc lzero) ⊔ (lsuc l516)) A a3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l))) a1 (cons-injective-l l516 A a1 a3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l4) Heq)) tl3 Heq1) (rewrite-l ((lsuc lzero) ⊔ (lsuc l516)) ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l516)) A) -> eq ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2 X--) (refl ((lsuc lzero) ⊔ (lsuc l516)) (list ((lsuc lzero) ⊔ (lsuc l516)) A) l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A l l4) Heq2)))) X-clearme0) X-clearme) (Hind l2 tl3 l4 (cons-injective-r l516 A a1 a3 (append ((lsuc lzero) ⊔ (lsuc l516)) A tl1 l2) (append ((lsuc lzero) ⊔ (lsuc l516)) A tl3 l4) Heq))) l3) l1

map : (l5-v l4-v : Level) -> (A-v : Set l5-v) -> (B-v : Set l4-v) -> (X-f-v : (X---v : A-v) -> B-v) -> (X-l-v : list l5-v A-v) -> list l4-v B-v
map l5-v l0-v A-v B-v X-f-v nil' = nil l0-v B-v
map l5-v l0-v A-v B-v X-f-v (cons' x-v tl-v) = cons l0-v B-v (X-f-v x-v) (map l5-v l0-v A-v B-v X-f-v tl-v)


map-append : (l78 l77 : Level) -> (A : Set l78) -> (B-v : Set l77) -> (f : (X-- : A) -> B-v) -> (l1 : list l78 A) -> (l2 : list l78 A) -> eq l77 (list l77 B-v) (append l77 B-v (map l78 l77 A B-v f l1) (map l78 l77 A B-v f l2)) (map l78 l77 A B-v f (append l78 A l1 l2))
map-append = λ (l78 l77 : Level) -> λ (A : Set l78) -> λ (B-v : Set l77) -> λ (f : (X-- : A) -> B-v) -> λ (l1 : list l78 A) -> list-ind l78 (l78 ⊔ l77) A (λ (X-x-716 : list l78 A) -> (l2 : list l78 A) -> eq l77 (list l77 B-v) (append l77 B-v (map l78 l77 A B-v f X-x-716) (map l78 l77 A B-v f l2)) (map l78 l77 A B-v f (append l78 A X-x-716 l2))) (λ (l2 : list l78 A) -> refl l77 (list l77 B-v) (append l77 B-v (map l78 l77 A B-v f (nil l78 A)) (map l78 l77 A B-v f l2))) (λ (h : A) -> λ (t : list l78 A) -> λ (IH : (l2 : list l78 A) -> eq l77 (list l77 B-v) (append l77 B-v (map l78 l77 A B-v f t) (map l78 l77 A B-v f l2)) (map l78 l77 A B-v f (append l78 A t l2))) -> λ (l2 : list l78 A) -> rewrite-r l77 l77 (list l77 B-v) (map l78 l77 A B-v f (append l78 A t l2)) (λ (X-- : list l77 B-v) -> eq l77 (list l77 B-v) (cons l77 B-v (f h) X--) (cons l77 B-v (f h) (map l78 l77 A B-v f (append l78 A t l2)))) (refl l77 (list l77 B-v) (cons l77 B-v (f h) (map l78 l77 A B-v f (append l78 A t l2)))) (append l77 B-v (map l78 l77 A B-v f t) (map l78 l77 A B-v f l2)) (IH l2)) l1


foldr : (l6-v l5-v : Level) -> (A-v : Set l6-v) -> (B-v : Set l5-v) -> (X-f-v : (X---v : A-v) -> (X--1-v : B-v) -> B-v) -> (X-b-v : B-v) -> (X-l-v : list l6-v A-v) -> B-v
foldr l4-v l0-v A-v B-v X-f-v X-b-v nil' = X-b-v
foldr l4-v l0-v A-v B-v X-f-v X-b-v (cons' a-v l) = X-f-v a-v (foldr l4-v l0-v A-v B-v X-f-v X-b-v l)



filter' : (l13 : Level) -> (T : Set l13) -> (X-p : (X-- : T) -> bool) -> (X-l : list l13 T) -> list l13 T
filter' = λ (l13 : Level) -> λ (T : Set l13) -> λ (p : (X-- : T) -> bool) -> foldr l13 l13 T (list l13 T) (λ (x : T) -> λ (l0 : list l13 T) -> match-bool l13 (λ (X-- : bool) -> list l13 T) (cons l13 T x l0) l0 (p x)) (nil l13 T)

compose' : (l19 l18 l17 : Level) -> (A : Set l19) -> (B-v : Set l18) -> (C : Set l17) -> (X-f : (X-- : A) -> (X--1 : B-v) -> C) -> (X-l1 : list l19 A) -> (X-l2 : list l18 B-v) -> list l17 C
compose' = λ (l19 l18 l17 : Level) -> λ (A : Set l19) -> λ (B-v : Set l18) -> λ (C : Set l17) -> λ (f : (X-- : A) -> (X--1 : B-v) -> C) -> λ (l1 : list l19 A) -> λ (l2 : list l18 B-v) -> foldr l19 l17 A (list l17 C) (λ (i : A) -> λ (acc : list l17 C) -> append l17 C (map l18 l17 B-v C (f i) l2) acc) (nil l17 C) l1

filter-true : (l276 : Level) -> (A : Set l276) -> (l : list l276 A) -> (a : A) -> (p : (X-- : A) -> bool) -> (X-- : eq lzero bool (p a) true) -> eq l276 (list l276 A) (filter' l276 A p (cons l276 A a l)) (cons l276 A a (filter' l276 A p l))
filter-true = λ (l276 : Level) -> λ (A : Set l276) -> λ (l : list l276 A) -> λ (a : A) -> λ (p : (X-- : A) -> bool) -> λ (pa : eq lzero bool (p a) true) -> list-ind l276 l276 A (λ (X-x-716 : list l276 A) -> eq l276 (list l276 A) (filter' l276 A p (cons l276 A a X-x-716)) (cons l276 A a (filter' l276 A p X-x-716))) (eq-ind-r lzero l276 bool true (λ (x : bool) -> λ (X-- : eq lzero bool x true) -> eq l276 (list l276 A) (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A a (nil l276 A)) (nil l276 A) x) (cons l276 A a (nil l276 A))) (refl l276 (list l276 A) (cons l276 A a (nil l276 A))) (p a) pa) (eq-ind-r lzero l276 bool true (λ (x : bool) -> λ (X-- : eq lzero bool x true) -> (x-718 : A) -> (x-717 : list l276 A) -> (X-x-720 : eq l276 (list l276 A) (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A a (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717)) (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717) x) (cons l276 A a (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717))) -> eq l276 (list l276 A) (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A a (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x-718 (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717)) (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717) (p x-718))) (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x-718 (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717)) (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717) (p x-718)) x) (cons l276 A a (match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x-718 (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717)) (foldr l276 l276 A (list l276 A) (λ (x0 : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-0 : bool) -> list l276 A) (cons l276 A x0 l0) l0 (p x0)) (nil l276 A) x-717) (p x-718)))) (λ (x-718 : A) -> λ (x-717 : list l276 A) -> λ (X-x-720 : eq l276 (list l276 A) (cons l276 A a (foldr l276 l276 A (list l276 A) (λ (x : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-- : bool) -> list l276 A) (cons l276 A x l0) l0 (p x)) (nil l276 A) x-717)) (cons l276 A a (foldr l276 l276 A (list l276 A) (λ (x : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-- : bool) -> list l276 A) (cons l276 A x l0) l0 (p x)) (nil l276 A) x-717))) -> refl l276 (list l276 A) (cons l276 A a (match-bool l276 (λ (X-- : bool) -> list l276 A) (cons l276 A x-718 (foldr l276 l276 A (list l276 A) (λ (x : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-- : bool) -> list l276 A) (cons l276 A x l0) l0 (p x)) (nil l276 A) x-717)) (foldr l276 l276 A (list l276 A) (λ (x : A) -> λ (l0 : list l276 A) -> match-bool l276 (λ (X-- : bool) -> list l276 A) (cons l276 A x l0) l0 (p x)) (nil l276 A) x-717) (p x-718)))) (p a) pa) l

filter-false : (l267 : Level) -> (A : Set l267) -> (l : list l267 A) -> (a : A) -> (p : (X-- : A) -> bool) -> (X-- : eq lzero bool (p a) false) -> eq l267 (list l267 A) (filter' l267 A p (cons l267 A a l)) (filter' l267 A p l)
filter-false = λ (l267 : Level) -> λ (A : Set l267) -> λ (l : list l267 A) -> λ (a : A) -> λ (p : (X-- : A) -> bool) -> λ (pa : eq lzero bool (p a) false) -> list-ind l267 l267 A (λ (X-x-716 : list l267 A) -> eq l267 (list l267 A) (filter' l267 A p (cons l267 A a X-x-716)) (filter' l267 A p X-x-716)) (eq-ind-r lzero l267 bool false (λ (x : bool) -> λ (X-- : eq lzero bool x false) -> eq l267 (list l267 A) (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A a (nil l267 A)) (nil l267 A) x) (nil l267 A)) (refl l267 (list l267 A) (nil l267 A)) (p a) pa) (eq-ind-r lzero l267 bool false (λ (x : bool) -> λ (X-- : eq lzero bool x false) -> (x-718 : A) -> (x-717 : list l267 A) -> (X-x-720 : eq l267 (list l267 A) (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A a (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717)) (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717) x) (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717)) -> eq l267 (list l267 A) (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A a (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x-718 (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717)) (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717) (p x-718))) (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x-718 (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717)) (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717) (p x-718)) x) (match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x-718 (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717)) (foldr l267 l267 A (list l267 A) (λ (x0 : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-0 : bool) -> list l267 A) (cons l267 A x0 l0) l0 (p x0)) (nil l267 A) x-717) (p x-718))) (λ (x-718 : A) -> λ (x-717 : list l267 A) -> λ (X-x-720 : eq l267 (list l267 A) (foldr l267 l267 A (list l267 A) (λ (x : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-- : bool) -> list l267 A) (cons l267 A x l0) l0 (p x)) (nil l267 A) x-717) (foldr l267 l267 A (list l267 A) (λ (x : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-- : bool) -> list l267 A) (cons l267 A x l0) l0 (p x)) (nil l267 A) x-717)) -> refl l267 (list l267 A) (match-bool l267 (λ (X-- : bool) -> list l267 A) (cons l267 A x-718 (foldr l267 l267 A (list l267 A) (λ (x : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-- : bool) -> list l267 A) (cons l267 A x l0) l0 (p x)) (nil l267 A) x-717)) (foldr l267 l267 A (list l267 A) (λ (x : A) -> λ (l0 : list l267 A) -> match-bool l267 (λ (X-- : bool) -> list l267 A) (cons l267 A x l0) l0 (p x)) (nil l267 A) x-717) (p x-718))) (p a) pa) l

eq-map : (l64 l63 : Level) -> (A : Set l64) -> (B-v : Set l63) -> (f : (X-- : A) -> B-v) -> (g : (X-- : A) -> B-v) -> (l : list l64 A) -> (X-- : (x : A) -> eq l63 B-v (f x) (g x)) -> eq l63 (list l63 B-v) (map l64 l63 A B-v f l) (map l64 l63 A B-v g l)
eq-map = λ (l64 l63 : Level) -> λ (A : Set l64) -> λ (B-v : Set l63) -> λ (f : (X-- : A) -> B-v) -> λ (g : (X-- : A) -> B-v) -> λ (l : list l64 A) -> λ (eqfg : (x : A) -> eq l63 B-v (f x) (g x)) -> list-ind l64 l63 A (λ (X-x-716 : list l64 A) -> eq l63 (list l63 B-v) (map l64 l63 A B-v f X-x-716) (map l64 l63 A B-v g X-x-716)) (refl l63 (list l63 B-v) (nil l63 B-v)) (λ (x-718 : A) -> λ (x-717 : list l64 A) -> λ (X-x-720 : eq l63 (list l63 B-v) (map l64 l63 A B-v f x-717) (map l64 l63 A B-v g x-717)) -> rewrite-l l63 l63 B-v (f x-718) (λ (X-- : B-v) -> eq l63 (list l63 B-v) (cons l63 B-v (f x-718) (map l64 l63 A B-v f x-717)) (cons l63 B-v X-- (map l64 l63 A B-v g x-717))) (rewrite-l l63 l63 (list l63 B-v) (map l64 l63 A B-v f x-717) (λ (X-- : list l63 B-v) -> eq l63 (list l63 B-v) (cons l63 B-v (f x-718) (map l64 l63 A B-v f x-717)) (cons l63 B-v (f x-718) X--)) (refl l63 (list l63 B-v) (cons l63 B-v (f x-718) (map l64 l63 A B-v f x-717))) (map l64 l63 A B-v g x-717) X-x-720) (g x-718) (eqfg x-718)) l

rev-append : (l5-v : Level) -> (H-v : Set l5-v) -> (X---v : list l5-v H-v) -> (X--1-v : list l5-v H-v) -> list l5-v H-v
rev-append l12-v H-v nil' l2-v = l2-v
rev-append l12-v H-v (cons' a-v tl-v) l2-v = rev-append l12-v H-v tl-v (cons l12-v H-v a-v l2-v)



reverse : (l4 : Level) -> (S-v : Set l4) -> (X-l : list l4 S-v) -> list l4 S-v
reverse = λ (l4 : Level) -> λ (S-v : Set l4) -> λ (l : list l4 S-v) -> rev-append l4 S-v l (nil l4 S-v)

reverse-single : (l6 : Level) -> (S-v : Set l6) -> (a : S-v) -> eq l6 (list l6 S-v) (reverse l6 S-v (cons l6 S-v a (nil l6 S-v))) (cons l6 S-v a (nil l6 S-v))
reverse-single = λ (l6 : Level) -> λ (S-v : Set l6) -> λ (a : S-v) -> refl l6 (list l6 S-v) (reverse l6 S-v (cons l6 S-v a (nil l6 S-v)))

rev-append-def : (l160 : Level) -> (S-v : Set l160) -> (l1 : list l160 S-v) -> (l2 : list l160 S-v) -> eq l160 (list l160 S-v) (rev-append l160 S-v l1 l2) (append l160 S-v (reverse l160 S-v l1) l2)
rev-append-def = λ (l160 : Level) -> λ (S-v : Set l160) -> λ (l1 : list l160 S-v) -> list-ind l160 l160 S-v (λ (X-x-716 : list l160 S-v) -> (l2 : list l160 S-v) -> eq l160 (list l160 S-v) (rev-append l160 S-v X-x-716 l2) (append l160 S-v (reverse l160 S-v X-x-716) l2)) (λ (l2 : list l160 S-v) -> refl l160 (list l160 S-v) l2) (λ (x-718 : S-v) -> λ (x-717 : list l160 S-v) -> λ (X-x-720 : (l2 : list l160 S-v) -> eq l160 (list l160 S-v) (rev-append l160 S-v x-717 l2) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) l2)) -> λ (l2 : list l160 S-v) -> rewrite-r l160 l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) X-- (append l160 S-v (rev-append l160 S-v x-717 (cons l160 S-v x-718 (nil l160 S-v))) l2)) (rewrite-r l160 l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 (nil l160 S-v))) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) (append l160 S-v X-- l2)) (rewrite-r l160 l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (append l160 S-v (cons l160 S-v x-718 (nil l160 S-v)) l2)) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) X--) (rewrite-l l160 l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) X--) (refl l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2))) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (append l160 S-v (cons l160 S-v x-718 (nil l160 S-v)) l2)) (rewrite-l l160 l160 (list l160 S-v) (append l160 S-v (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 (nil l160 S-v))) l2) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 l2)) X--) (append-cons l160 S-v x-718 (rev-append l160 S-v x-717 (nil l160 S-v)) l2) (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (append l160 S-v (cons l160 S-v x-718 (nil l160 S-v)) l2)) (associative-append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 (nil l160 S-v)) l2))) (append l160 S-v (append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 (nil l160 S-v))) l2) (associative-append l160 S-v (rev-append l160 S-v x-717 (nil l160 S-v)) (cons l160 S-v x-718 (nil l160 S-v)) l2)) (rev-append l160 S-v x-717 (cons l160 S-v x-718 (nil l160 S-v))) (X-x-720 (cons l160 S-v x-718 (nil l160 S-v)))) (rev-append l160 S-v x-717 (cons l160 S-v x-718 l2)) (X-x-720 (cons l160 S-v x-718 l2))) l1

reverse-cons : (l30 : Level) -> (S-v : Set l30) -> (a : S-v) -> (l : list l30 S-v) -> eq l30 (list l30 S-v) (reverse l30 S-v (cons l30 S-v a l)) (append l30 S-v (reverse l30 S-v l) (cons l30 S-v a (nil l30 S-v)))
reverse-cons = λ (l30 : Level) -> λ (S-v : Set l30) -> λ (a : S-v) -> λ (l : list l30 S-v) -> rewrite-r l30 l30 (list l30 S-v) (append l30 S-v (reverse l30 S-v l) (cons l30 S-v a (nil l30 S-v))) (λ (X-- : list l30 S-v) -> eq l30 (list l30 S-v) X-- (append l30 S-v (reverse l30 S-v l) (cons l30 S-v a (nil l30 S-v)))) (refl l30 (list l30 S-v) (append l30 S-v (reverse l30 S-v l) (cons l30 S-v a (nil l30 S-v)))) (rev-append l30 S-v l (cons l30 S-v a (nil l30 S-v))) (rev-append-def l30 S-v l (cons l30 S-v a (nil l30 S-v)))

reverse-append : (l235 : Level) -> (S-v : Set l235) -> (l1 : list l235 S-v) -> (l2 : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v (append l235 S-v l1 l2)) (append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v l1))
reverse-append = λ (l235 : Level) -> λ (S-v : Set l235) -> λ (l1 : list l235 S-v) -> list-ind l235 l235 S-v (λ (X-x-716 : list l235 S-v) -> (l2 : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v (append l235 S-v X-x-716 l2)) (append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v X-x-716))) (λ (l2 : list l235 S-v) -> rewrite-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v l2) (nil l235 S-v)) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) X-- (append l235 S-v (rev-append l235 S-v l2 (nil l235 S-v)) (nil l235 S-v))) (rewrite-r l235 l235 (list l235 S-v) (reverse l235 S-v l2) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) X-- (append l235 S-v (rev-append l235 S-v l2 (nil l235 S-v)) (nil l235 S-v))) (rewrite-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v l2) (nil l235 S-v)) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v l2) (append l235 S-v X-- (nil l235 S-v))) (rewrite-r l235 l235 (list l235 S-v) (reverse l235 S-v l2) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v l2) (append l235 S-v X-- (nil l235 S-v))) (rewrite-r l235 l235 (list l235 S-v) (reverse l235 S-v l2) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v l2) X--) (refl l235 (list l235 S-v) (reverse l235 S-v l2)) (append l235 S-v (reverse l235 S-v l2) (nil l235 S-v)) (append-nil l235 S-v (reverse l235 S-v l2))) (append l235 S-v (reverse l235 S-v l2) (nil l235 S-v)) (append-nil l235 S-v (reverse l235 S-v l2))) (rev-append l235 S-v l2 (nil l235 S-v)) (rev-append-def l235 S-v l2 (nil l235 S-v))) (append l235 S-v (reverse l235 S-v l2) (nil l235 S-v)) (append-nil l235 S-v (reverse l235 S-v l2))) (rev-append l235 S-v l2 (nil l235 S-v)) (rev-append-def l235 S-v l2 (nil l235 S-v))) (λ (a : S-v) -> λ (tl : list l235 S-v) -> λ (Hind : (l2 : list l235 S-v) -> eq l235 (list l235 S-v) (reverse l235 S-v (append l235 S-v tl l2)) (append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v tl))) -> λ (l2 : list l235 S-v) -> eq-ind-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v (append l235 S-v tl l2)) (cons l235 S-v a (nil l235 S-v))) (λ (x : list l235 S-v) -> λ (X-- : eq l235 (list l235 S-v) x (append l235 S-v (reverse l235 S-v (append l235 S-v tl l2)) (cons l235 S-v a (nil l235 S-v)))) -> eq l235 (list l235 S-v) x (append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v (cons l235 S-v a tl)))) (eq-ind-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v))) (λ (x : list l235 S-v) -> λ (X-- : eq l235 (list l235 S-v) x (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v)))) -> eq l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v (append l235 S-v tl l2)) (cons l235 S-v a (nil l235 S-v))) (append l235 S-v (reverse l235 S-v l2) x)) (rewrite-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v l2) (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v)))) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) X-- (append l235 S-v (reverse l235 S-v l2) (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v))))) (refl l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v l2) (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v))))) (append l235 S-v (reverse l235 S-v (append l235 S-v tl l2)) (cons l235 S-v a (nil l235 S-v))) (rewrite-r l235 l235 (list l235 S-v) (append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v tl)) (λ (X-- : list l235 S-v) -> eq l235 (list l235 S-v) (append l235 S-v X-- (cons l235 S-v a (nil l235 S-v))) (append l235 S-v (reverse l235 S-v l2) (append l235 S-v (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v))))) (associative-append l235 S-v (reverse l235 S-v l2) (reverse l235 S-v tl) (cons l235 S-v a (nil l235 S-v))) (reverse l235 S-v (append l235 S-v tl l2)) (Hind l2))) (reverse l235 S-v (cons l235 S-v a tl)) (reverse-cons l235 S-v a tl)) (reverse l235 S-v (cons l235 S-v a (append l235 S-v tl l2))) (reverse-cons l235 S-v a (append l235 S-v tl l2))) l1

reverse-reverse : (l160 : Level) -> (S-v : Set l160) -> (l : list l160 S-v) -> eq l160 (list l160 S-v) (reverse l160 S-v (reverse l160 S-v l)) l
reverse-reverse = λ (l160 : Level) -> λ (S-v : Set l160) -> λ (l : list l160 S-v) -> list-ind l160 l160 S-v (λ (X-x-716 : list l160 S-v) -> eq l160 (list l160 S-v) (reverse l160 S-v (reverse l160 S-v X-x-716)) X-x-716) (refl l160 (list l160 S-v) (reverse l160 S-v (reverse l160 S-v (nil l160 S-v)))) (λ (a : S-v) -> λ (tl : list l160 S-v) -> λ (Hind : eq l160 (list l160 S-v) (reverse l160 S-v (reverse l160 S-v tl)) tl) -> eq-ind-r l160 l160 (list l160 S-v) (append l160 S-v (reverse l160 S-v tl) (cons l160 S-v a (nil l160 S-v))) (λ (x : list l160 S-v) -> λ (X-- : eq l160 (list l160 S-v) x (append l160 S-v (reverse l160 S-v tl) (cons l160 S-v a (nil l160 S-v)))) -> eq l160 (list l160 S-v) (reverse l160 S-v x) (cons l160 S-v a tl)) (eq-ind-r l160 l160 (list l160 S-v) (append l160 S-v (reverse l160 S-v (cons l160 S-v a (nil l160 S-v))) (reverse l160 S-v (reverse l160 S-v tl))) (λ (x : list l160 S-v) -> λ (X-- : eq l160 (list l160 S-v) x (append l160 S-v (reverse l160 S-v (cons l160 S-v a (nil l160 S-v))) (reverse l160 S-v (reverse l160 S-v tl)))) -> eq l160 (list l160 S-v) x (cons l160 S-v a tl)) (rewrite-r l160 l160 (list l160 S-v) (append l160 S-v (reverse l160 S-v tl) (nil l160 S-v)) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (cons l160 S-v a (rev-append l160 S-v X-- (nil l160 S-v))) (cons l160 S-v a tl)) (rewrite-r l160 l160 (list l160 S-v) (reverse l160 S-v tl) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (cons l160 S-v a (rev-append l160 S-v X-- (nil l160 S-v))) (cons l160 S-v a tl)) (rewrite-r l160 l160 (list l160 S-v) (append l160 S-v (reverse l160 S-v (reverse l160 S-v tl)) (nil l160 S-v)) (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (cons l160 S-v a X--) (cons l160 S-v a tl)) (rewrite-r l160 l160 (list l160 S-v) tl (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (cons l160 S-v a (append l160 S-v X-- (nil l160 S-v))) (cons l160 S-v a tl)) (rewrite-r l160 l160 (list l160 S-v) tl (λ (X-- : list l160 S-v) -> eq l160 (list l160 S-v) (cons l160 S-v a X--) (cons l160 S-v a tl)) (refl l160 (list l160 S-v) (cons l160 S-v a tl)) (append l160 S-v tl (nil l160 S-v)) (append-nil l160 S-v tl)) (reverse l160 S-v (reverse l160 S-v tl)) Hind) (rev-append l160 S-v (reverse l160 S-v tl) (nil l160 S-v)) (rev-append-def l160 S-v (reverse l160 S-v tl) (nil l160 S-v))) (append l160 S-v (reverse l160 S-v tl) (nil l160 S-v)) (append-nil l160 S-v (reverse l160 S-v tl))) (rev-append l160 S-v tl (nil l160 S-v)) (rev-append-def l160 S-v tl (nil l160 S-v))) (reverse l160 S-v (append l160 S-v (reverse l160 S-v tl) (cons l160 S-v a (nil l160 S-v)))) (reverse-append l160 S-v (reverse l160 S-v tl) (cons l160 S-v a (nil l160 S-v)))) (reverse l160 S-v (cons l160 S-v a tl)) (reverse-cons l160 S-v a tl)) l

list-elim-left : (l59 l56 : Level) -> (S-v : Set l59) -> (P : (X-- : list l59 S-v) -> Set l56) -> (X-- : P (nil l59 S-v)) -> (X--1 : (a : S-v) -> (tl : list l59 S-v) -> (X--1 : P tl) -> P (append l59 S-v tl (cons l59 S-v a (nil l59 S-v)))) -> (l : list l59 S-v) -> P l
list-elim-left = λ (l59 l56 : Level) -> λ (S-v : Set l59) -> λ (P : (X-- : list l59 S-v) -> Set l56) -> λ (Pnil : P (nil l59 S-v)) -> λ (Pstep : (a : S-v) -> (tl : list l59 S-v) -> (X-- : P tl) -> P (append l59 S-v tl (cons l59 S-v a (nil l59 S-v)))) -> λ (l : list l59 S-v) -> eq-ind l59 l56 (list l59 S-v) (reverse l59 S-v (reverse l59 S-v l)) (λ (x-1 : list l59 S-v) -> λ (X-x-2 : eq l59 (list l59 S-v) (reverse l59 S-v (reverse l59 S-v l)) x-1) -> P x-1) (list-ind l59 l56 S-v (λ (X-x-716 : list l59 S-v) -> P (reverse l59 S-v X-x-716)) Pnil (λ (a : S-v) -> λ (tl : list l59 S-v) -> λ (H : P (reverse l59 S-v tl)) -> eq-ind-r l59 l56 (list l59 S-v) (append l59 S-v (reverse l59 S-v tl) (cons l59 S-v a (nil l59 S-v))) (λ (x : list l59 S-v) -> λ (X-- : eq l59 (list l59 S-v) x (append l59 S-v (reverse l59 S-v tl) (cons l59 S-v a (nil l59 S-v)))) -> P x) (Pstep a (reverse l59 S-v tl) H) (reverse l59 S-v (cons l59 S-v a tl)) (reverse-cons l59 S-v a tl)) (reverse l59 S-v l)) l (reverse-reverse l59 S-v l)



length : (l2-v : Level) -> (H-v : Set l2-v) -> (X---v : list l2-v H-v) -> nat
length l3-v H-v nil' = O
length l3-v H-v (cons' a-v tl-v) = S (length l3-v H-v tl-v)

length-tail : (l26 : Level) -> (A : Set l26) -> (l : list l26 A) -> eq lzero nat (length l26 A (tail l26 A l)) (pred (length l26 A l))
length-tail = λ (l26 : Level) -> λ (A : Set l26) -> λ (l : list l26 A) -> list-ind l26 lzero A (λ (X-x-716 : list l26 A) -> eq lzero nat (length l26 A (tail l26 A X-x-716)) (pred (length l26 A X-x-716))) (refl lzero nat (length l26 A (tail l26 A (nil l26 A)))) (λ (x-718 : A) -> λ (x-717 : list l26 A) -> λ (X-x-720 : eq lzero nat (length l26 A (tail l26 A x-717)) (pred (length l26 A x-717))) -> refl lzero nat (length l26 A (tail l26 A (cons l26 A x-718 x-717)))) l

length-tail1 : (l21 : Level) -> (A : Set l21) -> (l : list l21 A) -> (X-- : lt O (length l21 A l)) -> lt (length l21 A (tail l21 A l)) (length l21 A l)
length-tail1 = λ (l21 : Level) -> λ (A : Set l21) -> λ (X-clearme : list l21 A) -> match-list l21 A lzero (λ (X-- : list l21 A) -> (X--1 : lt O (length l21 A X--)) -> lt (length l21 A (tail l21 A X--)) (length l21 A X--)) (λ (auto : le (S O) O) -> auto) (λ (auto : A) -> λ (auto' : list l21 A) -> λ (auto'' : le (S O) (S (length l21 A auto'))) -> le-n (S (length l21 A auto'))) X-clearme

length-append : (l48 : Level) -> (A : Set l48) -> (l1 : list l48 A) -> (l2 : list l48 A) -> eq lzero nat (length l48 A (append l48 A l1 l2)) (plus (length l48 A l1) (length l48 A l2))
length-append = λ (l48 : Level) -> λ (A : Set l48) -> λ (l1 : list l48 A) -> list-ind l48 l48 A (λ (X-x-716 : list l48 A) -> (l2 : list l48 A) -> eq lzero nat (length l48 A (append l48 A X-x-716 l2)) (plus (length l48 A X-x-716) (length l48 A l2))) (λ (l2 : list l48 A) -> refl lzero nat (length l48 A (append l48 A (nil l48 A) l2))) (λ (X-x-718 : A) -> λ (x-717 : list l48 A) -> λ (X-x-720 : (l2 : list l48 A) -> eq lzero nat (length l48 A (append l48 A x-717 l2)) (plus (length l48 A x-717) (length l48 A l2))) -> λ (l2 : list l48 A) -> rewrite-l lzero lzero nat (length l48 A (append l48 A x-717 l2)) (λ (X-- : nat) -> eq lzero nat (S (length l48 A (append l48 A x-717 l2))) (S X--)) (refl lzero nat (S (length l48 A (append l48 A x-717 l2)))) (plus (length l48 A x-717) (length l48 A l2)) (X-x-720 l2)) l1

length-map : (l39 l38 : Level) -> (A : Set l39) -> (B-v : Set l38) -> (l : list l39 A) -> (f : (X-- : A) -> B-v) -> eq lzero nat (length l38 B-v (map l39 l38 A B-v f l)) (length l39 A l)
length-map = λ (l39 l38 : Level) -> λ (A : Set l39) -> λ (B-v : Set l38) -> λ (l : list l39 A) -> λ (f : (X-- : A) -> B-v) -> list-ind l39 lzero A (λ (X-x-716 : list l39 A) -> eq lzero nat (length l38 B-v (map l39 l38 A B-v f X-x-716)) (length l39 A X-x-716)) (refl lzero nat (length l38 B-v (map l39 l38 A B-v f (nil l39 A)))) (λ (a : A) -> λ (tl : list l39 A) -> λ (Hind : eq lzero nat (length l38 B-v (map l39 l38 A B-v f tl)) (length l39 A tl)) -> rewrite-r lzero lzero nat (length l39 A tl) (λ (X-- : nat) -> eq lzero nat (S X--) (S (length l39 A tl))) (refl lzero nat (S (length l39 A tl))) (length l38 B-v (map l39 l38 A B-v f tl)) Hind) l

length-reverse : (l127 : Level) -> (A : Set l127) -> (l : list l127 A) -> eq lzero nat (length l127 A (reverse l127 A l)) (length l127 A l)
length-reverse = λ (l127 : Level) -> λ (A : Set l127) -> λ (l : list l127 A) -> list-ind l127 lzero A (λ (X-x-716 : list l127 A) -> eq lzero nat (length l127 A (reverse l127 A X-x-716)) (length l127 A X-x-716)) (refl lzero nat (length l127 A (reverse l127 A (nil l127 A)))) (λ (a : A) -> λ (l0 : list l127 A) -> λ (IH : eq lzero nat (length l127 A (reverse l127 A l0)) (length l127 A l0)) -> eq-ind-r l127 lzero (list l127 A) (append l127 A (reverse l127 A l0) (cons l127 A a (nil l127 A))) (λ (x : list l127 A) -> λ (X-- : eq l127 (list l127 A) x (append l127 A (reverse l127 A l0) (cons l127 A a (nil l127 A)))) -> eq lzero nat (length l127 A x) (length l127 A (cons l127 A a l0))) (eq-ind-r lzero lzero nat (plus (length l127 A (reverse l127 A l0)) (length l127 A (cons l127 A a (nil l127 A)))) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus (length l127 A (reverse l127 A l0)) (length l127 A (cons l127 A a (nil l127 A))))) -> eq lzero nat x (length l127 A (cons l127 A a l0))) (rewrite-r l127 lzero (list l127 A) (append l127 A (reverse l127 A l0) (nil l127 A)) (λ (X-- : list l127 A) -> eq lzero nat (plus (length l127 A X--) (S O)) (S (length l127 A l0))) (rewrite-r l127 lzero (list l127 A) (reverse l127 A l0) (λ (X-- : list l127 A) -> eq lzero nat (plus (length l127 A X--) (S O)) (S (length l127 A l0))) (rewrite-r lzero lzero nat (length l127 A l0) (λ (X-- : nat) -> eq lzero nat (plus X-- (S O)) (S (length l127 A l0))) (rewrite-l lzero lzero nat (S (length l127 A l0)) (λ (X-- : nat) -> eq lzero nat X-- (S (length l127 A l0))) (refl lzero nat (S (length l127 A l0))) (plus (length l127 A l0) (S O)) (rewrite-r lzero lzero nat (plus (length l127 A l0) O) (λ (X-- : nat) -> eq lzero nat (S X--) (plus (length l127 A l0) (S O))) (plus-n-Sm (length l127 A l0) O) (length l127 A l0) (plus-n-O (length l127 A l0)))) (length l127 A (reverse l127 A l0)) IH) (append l127 A (reverse l127 A l0) (nil l127 A)) (append-nil l127 A (reverse l127 A l0))) (rev-append l127 A l0 (nil l127 A)) (rev-append-def l127 A l0 (nil l127 A))) (length l127 A (append l127 A (reverse l127 A l0) (cons l127 A a (nil l127 A)))) (length-append l127 A (reverse l127 A l0) (cons l127 A a (nil l127 A)))) (reverse l127 A (cons l127 A a l0)) (reverse-cons l127 A a l0)) l

lenght-to-nil : (l35 : Level) -> (A : Set l35) -> (l : list l35 A) -> (X-- : eq lzero nat (length l35 A l) O) -> eq l35 (list l35 A) l (nil l35 A)
lenght-to-nil = λ (l35 : Level) -> λ (A : Set l35) -> λ (X-clearme : list l35 A) -> match-list l35 A l35 (λ (X-- : list l35 A) -> (X--1 : eq lzero nat (length l35 A X--) O) -> eq l35 (list l35 A) X-- (nil l35 A)) (λ (auto : eq lzero nat (length l35 A (nil l35 A)) O) -> refl l35 (list l35 A) (nil l35 A)) (λ (a : A) -> λ (tl : list l35 A) -> λ (H : eq lzero nat (S (length l35 A tl)) O) -> nat-discr l35 (S (length l35 A tl)) O H (eq l35 (list l35 A) (cons l35 A a tl) (nil l35 A))) X-clearme

lists-length-split : (l1018 : Level) -> (A : Set l1018) -> (l1 : list l1018 A) -> (l2 : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A l1)) (eq l1018 (list l1018 A) l2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A l2)) (eq l1018 (list l1018 A) l1 (append l1018 A la lb)))))
lists-length-split = λ (l1018 : Level) -> λ (A : Set l1018) -> λ (l1 : list l1018 A) -> list-ind l1018 l1018 A (λ (X-x-716 : list l1018 A) -> (l2 : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A X-x-716)) (eq l1018 (list l1018 A) l2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A l2)) (eq l1018 (list l1018 A) X-x-716 (append l1018 A la lb)))))) (λ (l2 : list l1018 A) -> ex-intro l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) l2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A l2)) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A la lb))))) (nil l1018 A) (ex-intro l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) l2 (append l1018 A (nil l1018 A) lb))) (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A l2)) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A (nil l1018 A) lb)))) l2 (or-introl l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) l2 (append l1018 A (nil l1018 A) l2))) (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A l2)) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A (nil l1018 A) l2))) (conj lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) l2 (append l1018 A (nil l1018 A) l2)) (refl lzero nat (length l1018 A (nil l1018 A))) (refl l1018 (list l1018 A) l2))))) (λ (hd1 : A) -> λ (tl1 : list l1018 A) -> λ (IH : (l2 : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A tl1)) (eq l1018 (list l1018 A) l2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A l2)) (eq l1018 (list l1018 A) tl1 (append l1018 A la lb)))))) -> λ (X-clearme : list l1018 A) -> match-list l1018 A l1018 (λ (X-- : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) X-- (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A X--)) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (ex-intro l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb))))) (nil l1018 A) (ex-intro l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A (nil l1018 A) lb))) (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (nil l1018 A) lb)))) (cons l1018 A hd1 tl1) (or-intror l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (nil l1018 A) (append l1018 A (nil l1018 A) (cons l1018 A hd1 tl1)))) (And lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (nil l1018 A) (cons l1018 A hd1 tl1)))) (conj lzero l1018 (eq lzero nat (length l1018 A (nil l1018 A)) (length l1018 A (nil l1018 A))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (nil l1018 A) (cons l1018 A hd1 tl1))) (refl lzero nat (length l1018 A (nil l1018 A))) (refl l1018 (list l1018 A) (cons l1018 A hd1 tl1)))))) (λ (hd2 : A) -> λ (tl2 : list l1018 A) -> match-ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A la lb))))) l1018 (λ (X-- : ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A la lb)))))) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (λ (x : list l1018 A) -> λ (X-clearme0 : ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x lb))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x lb))))) -> match-ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x lb))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x lb)))) l1018 (λ (X-- : ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x lb))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x lb))))) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (λ (y : list l1018 A) -> λ (X-clearme1 : Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y)))) -> match-Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y))) l1018 (λ (X-- : Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y))) (And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y)))) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (λ (X-clearme2 : And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y))) -> match-And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y)) l1018 (λ (X-- : And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl1)) (eq l1018 (list l1018 A) tl2 (append l1018 A x y))) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (λ (IH1 : eq lzero nat (length l1018 A x) (length l1018 A tl1)) -> λ (IH2 : eq l1018 (list l1018 A) tl2 (append l1018 A x y)) -> ex-intro l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb))))) (cons l1018 A hd2 x) (ex-intro l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd2 x)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A (cons l1018 A hd2 x) lb))) (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd2 x)) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (cons l1018 A hd2 x) lb)))) y (or-introl l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd2 x)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A (cons l1018 A hd2 x) y))) (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd2 x)) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (cons l1018 A hd2 x) y))) (conj lzero l1018 (eq lzero nat (S (length l1018 A x)) (S (length l1018 A tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (cons l1018 A hd2 (append l1018 A x y))) (rewrite-r lzero lzero nat (length l1018 A tl1) (λ (X-- : nat) -> eq lzero nat (S X--) (S (length l1018 A tl1))) (refl lzero nat (S (length l1018 A tl1))) (length l1018 A x) IH1) (rewrite-l l1018 l1018 (list l1018 A) tl2 (λ (X-- : list l1018 A) -> eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (cons l1018 A hd2 X--)) (refl l1018 (list l1018 A) (cons l1018 A hd2 tl2)) (append l1018 A x y) IH2))))) X-clearme2) (λ (X-clearme2 : And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y))) -> match-And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y)) l1018 (λ (X-- : And lzero l1018 (eq lzero nat (length l1018 A x) (length l1018 A tl2)) (eq l1018 (list l1018 A) tl1 (append l1018 A x y))) -> ex l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb)))))) (λ (IH1 : eq lzero nat (length l1018 A x) (length l1018 A tl2)) -> λ (IH2 : eq l1018 (list l1018 A) tl1 (append l1018 A x y)) -> ex-intro l1018 l1018 (list l1018 A) (λ (la : list l1018 A) -> ex l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A la lb))) (And lzero l1018 (eq lzero nat (length l1018 A la) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A la lb))))) (cons l1018 A hd1 x) (ex-intro l1018 l1018 (list l1018 A) (λ (lb : list l1018 A) -> Or l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd1 x)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A (cons l1018 A hd1 x) lb))) (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd1 x)) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (cons l1018 A hd1 x) lb)))) y (or-intror l1018 l1018 (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd1 x)) (length l1018 A (cons l1018 A hd1 tl1))) (eq l1018 (list l1018 A) (cons l1018 A hd2 tl2) (append l1018 A (cons l1018 A hd1 x) y))) (And lzero l1018 (eq lzero nat (length l1018 A (cons l1018 A hd1 x)) (length l1018 A (cons l1018 A hd2 tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (append l1018 A (cons l1018 A hd1 x) y))) (conj lzero l1018 (eq lzero nat (S (length l1018 A x)) (S (length l1018 A tl2))) (eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (cons l1018 A hd1 (append l1018 A x y))) (rewrite-r lzero lzero nat (length l1018 A tl2) (λ (X-- : nat) -> eq lzero nat (S X--) (S (length l1018 A tl2))) (refl lzero nat (S (length l1018 A tl2))) (length l1018 A x) IH1) (rewrite-l l1018 l1018 (list l1018 A) tl1 (λ (X-- : list l1018 A) -> eq l1018 (list l1018 A) (cons l1018 A hd1 tl1) (cons l1018 A hd1 X--)) (refl l1018 (list l1018 A) (cons l1018 A hd1 tl1)) (append l1018 A x y) IH2))))) X-clearme2) X-clearme1) X-clearme0) (IH tl2)) X-clearme) l1

list-ind2 : (l143 l142 l133 : Level) -> (T1 : Set l143) -> (T2 : Set l142) -> (l1 : list l143 T1) -> (l2 : list l142 T2) -> (P : (X-- : list l143 T1) -> (X--1 : list l142 T2) -> Set l133) -> (X-- : eq lzero nat (length l143 T1 l1) (length l142 T2 l2)) -> (X--1 : P (nil l143 T1) (nil l142 T2)) -> (X--2 : (tl1 : list l143 T1) -> (tl2 : list l142 T2) -> (hd1 : T1) -> (hd2 : T2) -> (X--2 : P tl1 tl2) -> P (cons l143 T1 hd1 tl1) (cons l142 T2 hd2 tl2)) -> P l1 l2
list-ind2 = λ (l143 l142 l133 : Level) -> λ (T1 : Set l143) -> λ (T2 : Set l142) -> λ (l1 : list l143 T1) -> λ (l2 : list l142 T2) -> λ (P : (X-- : list l143 T1) -> (X--1 : list l142 T2) -> Set l133) -> λ (Hl : eq lzero nat (length l143 T1 l1) (length l142 T2 l2)) -> λ (Pnil : P (nil l143 T1) (nil l142 T2)) -> λ (Pcons : (tl1 : list l143 T1) -> (tl2 : list l142 T2) -> (hd1 : T1) -> (hd2 : T2) -> (X-- : P tl1 tl2) -> P (cons l143 T1 hd1 tl1) (cons l142 T2 hd2 tl2)) -> list-ind l143 (l142 ⊔ l133) T1 (λ (X-x-716 : list l143 T1) -> (l : list l142 T2) -> (X-- : eq lzero nat (length l143 T1 X-x-716) (length l142 T2 l)) -> P X-x-716 l) (λ (l20 : list l142 T2) -> match-list l142 T2 l133 (λ (X-- : list l142 T2) -> (X--1 : eq lzero nat (length l143 T1 (nil l143 T1)) (length l142 T2 X--)) -> P (nil l143 T1) X--) (λ (auto : eq lzero nat (length l143 T1 (nil l143 T1)) (length l142 T2 (nil l142 T2))) -> Pnil) (λ (t2 : T2) -> λ (tl2 : list l142 T2) -> λ (H : eq lzero nat O (S (length l142 T2 tl2))) -> nat-discr l133 O (S (length l142 T2 tl2)) H (P (nil l143 T1) (cons l142 T2 t2 tl2))) l20) (λ (t1 : T1) -> λ (tl1 : list l143 T1) -> λ (IH : (l : list l142 T2) -> (X-- : eq lzero nat (length l143 T1 tl1) (length l142 T2 l)) -> P tl1 l) -> λ (l20 : list l142 T2) -> match-list l142 T2 l133 (λ (X-- : list l142 T2) -> (X--1 : eq lzero nat (length l143 T1 (cons l143 T1 t1 tl1)) (length l142 T2 X--)) -> P (cons l143 T1 t1 tl1) X--) (λ (H : eq lzero nat (S (length l143 T1 tl1)) O) -> nat-discr l133 (S (length l143 T1 tl1)) O H (P (cons l143 T1 t1 tl1) (nil l142 T2))) (λ (t2 : T2) -> λ (tl2 : list l142 T2) -> λ (H : eq lzero nat (length l143 T1 (cons l143 T1 t1 tl1)) (length l142 T2 (cons l142 T2 t2 tl2))) -> Pcons tl1 tl2 t1 t2 (IH tl2 (nat-discr lzero (S (length l143 T1 tl1)) (S (length l142 T2 tl2)) H (eq lzero nat (length l143 T1 tl1) (length l142 T2 tl2)) (λ (e0 : eq lzero nat (R0 lzero nat (length l143 T1 tl1)) (length l142 T2 tl2)) -> rewrite-l lzero lzero nat (length l143 T1 tl1) (λ (X-- : nat) -> eq lzero nat (length l143 T1 tl1) X--) (refl lzero nat (length l143 T1 tl1)) (length l142 T2 tl2) e0)))) l20) l1 l2 Hl

list-cases2 : (l196 l195 l190 : Level) -> (T1 : Set l196) -> (T2 : Set l195) -> (l1 : list l196 T1) -> (l2 : list l195 T2) -> (P : Set l190) -> (X-- : eq lzero nat (length l196 T1 l1) (length l195 T2 l2)) -> (X--1 : (X--1 : eq l196 (list l196 T1) l1 (nil l196 T1)) -> (X--2 : eq l195 (list l195 T2) l2 (nil l195 T2)) -> P) -> (X--2 : (hd1 : T1) -> (hd2 : T2) -> (tl1 : list l196 T1) -> (tl2 : list l195 T2) -> (X--2 : eq l196 (list l196 T1) l1 (cons l196 T1 hd1 tl1)) -> (X--3 : eq l195 (list l195 T2) l2 (cons l195 T2 hd2 tl2)) -> P) -> P
list-cases2 = λ (l196 l195 l190 : Level) -> λ (T1 : Set l196) -> λ (T2 : Set l195) -> λ (l1 : list l196 T1) -> λ (l2 : list l195 T2) -> λ (P : Set l190) -> λ (Hl : eq lzero nat (length l196 T1 l1) (length l195 T2 l2)) -> list-ind2 l196 l195 ((l196 ⊔ l195) ⊔ l190) T1 T2 l1 l2 (λ (X-- : list l196 T1) -> λ (X-0 : list l195 T2) -> (X--1 : (X--1 : eq l196 (list l196 T1) X-- (nil l196 T1)) -> (X--2 : eq l195 (list l195 T2) X-0 (nil l195 T2)) -> P) -> (X--2 : (hd1 : T1) -> (hd2 : T2) -> (tl1 : list l196 T1) -> (tl2 : list l195 T2) -> (X--2 : eq l196 (list l196 T1) X-- (cons l196 T1 hd1 tl1)) -> (X--3 : eq l195 (list l195 T2) X-0 (cons l195 T2 hd2 tl2)) -> P) -> P) Hl (λ (Pnil : (X-- : eq l196 (list l196 T1) (nil l196 T1) (nil l196 T1)) -> (X--1 : eq l195 (list l195 T2) (nil l195 T2) (nil l195 T2)) -> P) -> λ (Pcons : (hd1 : T1) -> (hd2 : T2) -> (tl1 : list l196 T1) -> (tl2 : list l195 T2) -> (X-- : eq l196 (list l196 T1) (nil l196 T1) (cons l196 T1 hd1 tl1)) -> (X--1 : eq l195 (list l195 T2) (nil l195 T2) (cons l195 T2 hd2 tl2)) -> P) -> Pnil (refl l196 (list l196 T1) (nil l196 T1)) (refl l195 (list l195 T2) (nil l195 T2))) (λ (tl1 : list l196 T1) -> λ (tl2 : list l195 T2) -> λ (hd1 : T1) -> λ (hd2 : T2) -> λ (IH1 : (X-- : (X-- : eq l196 (list l196 T1) tl1 (nil l196 T1)) -> (X--1 : eq l195 (list l195 T2) tl2 (nil l195 T2)) -> P) -> (X--1 : (hd10 : T1) -> (hd20 : T2) -> (tl10 : list l196 T1) -> (tl20 : list l195 T2) -> (X--1 : eq l196 (list l196 T1) tl1 (cons l196 T1 hd10 tl10)) -> (X--2 : eq l195 (list l195 T2) tl2 (cons l195 T2 hd20 tl20)) -> P) -> P) -> λ (IH2 : (X-- : eq l196 (list l196 T1) (cons l196 T1 hd1 tl1) (nil l196 T1)) -> (X--1 : eq l195 (list l195 T2) (cons l195 T2 hd2 tl2) (nil l195 T2)) -> P) -> λ (Hp : (hd10 : T1) -> (hd20 : T2) -> (tl10 : list l196 T1) -> (tl20 : list l195 T2) -> (X-- : eq l196 (list l196 T1) (cons l196 T1 hd1 tl1) (cons l196 T1 hd10 tl10)) -> (X--1 : eq l195 (list l195 T2) (cons l195 T2 hd2 tl2) (cons l195 T2 hd20 tl20)) -> P) -> Hp hd1 hd2 tl1 tl2 (refl l196 (list l196 T1) (cons l196 T1 hd1 tl1)) (refl l195 (list l195 T2) (cons l195 T2 hd2 tl2)))

append-l1-injective : (l125 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l125))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l125)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l125)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l125)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l125)) A) -> (X-- : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l125)) A l1) (length ((lsuc lzero) ⊔ (lsuc l125)) A l2)) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) A) (append ((lsuc lzero) ⊔ (lsuc l125)) A l1 l3) (append ((lsuc lzero) ⊔ (lsuc l125)) A l2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) A) l1 l2
append-l1-injective = λ (l125 : Level) -> λ (a : Set ((lsuc lzero) ⊔ (lsuc l125))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (Hlen : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l125)) a l1) (length ((lsuc lzero) ⊔ (lsuc l125)) a l2)) -> list-ind2 ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) a a l1 l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a X-- l3) (append ((lsuc lzero) ⊔ (lsuc l125)) a X-0 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) X-- X-0) Hlen (λ (auto : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a (nil ((lsuc lzero) ⊔ (lsuc l125)) a) l3) (append ((lsuc lzero) ⊔ (lsuc l125)) a (nil ((lsuc lzero) ⊔ (lsuc l125)) a) l4)) -> refl ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (nil ((lsuc lzero) ⊔ (lsuc l125)) a)) (λ (tl1 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (tl2 : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> λ (hd1 : a) -> λ (hd2 : a) -> λ (IH : (X-- : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) tl1 tl2) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd1 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4))) -> list-discr l125 ((lsuc lzero) ⊔ (lsuc l125)) a (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd1 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4)) Heq (eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd1 tl1) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 tl2)) (λ (e0 : eq ((lsuc lzero) ⊔ (lsuc l125)) a (R0 ((lsuc lzero) ⊔ (lsuc l125)) a hd1) hd2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (λ (x : a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l125)) a x hd2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a x (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4))) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (R1 ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) a x (λ (x0 : a) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l125)) a x x0) -> list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3) hd2 X--) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a x tl1) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 tl2)) (λ (Heq0 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4))) -> λ (e00 : eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (R1 ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) a hd2 (λ (x0 : a) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l125)) a hd2 x0) -> list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3) hd2 (refl ((lsuc lzero) ⊔ (lsuc l125)) a hd2)) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4)) -> eq-f ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (cons ((lsuc lzero) ⊔ (lsuc l125)) a hd2) tl1 tl2 (IH (rewrite-l ((lsuc lzero) ⊔ (lsuc l125)) ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l125)) a) -> eq ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3) X--) (refl ((lsuc lzero) ⊔ (lsuc l125)) (list ((lsuc lzero) ⊔ (lsuc l125)) a) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl1 l3)) (append ((lsuc lzero) ⊔ (lsuc l125)) a tl2 l4) e00))) hd1 e0 Heq))

append-l2-injective : (l118 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l118))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (X-- : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) A l1) (length ((lsuc lzero) ⊔ (lsuc l118)) A l2)) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) (append ((lsuc lzero) ⊔ (lsuc l118)) A l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) A l2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) l3 l4
append-l2-injective = λ (l118 : Level) -> λ (a : Set ((lsuc lzero) ⊔ (lsuc l118))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (Hlen : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l1) (length ((lsuc lzero) ⊔ (lsuc l118)) a l2)) -> list-ind2 ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) a a l1 l2 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (X-0 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a X-- l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a X-0 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) Hlen (λ (auto : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) -> rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 X--) (refl ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3) l4 auto) (λ (tl1 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (tl2 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (hd1 : a) -> λ (hd2 : a) -> λ (IH : (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd1 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4))) -> list-discr l118 ((lsuc lzero) ⊔ (lsuc l118)) a (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd1 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4)) Heq (eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) (λ (e0 : eq ((lsuc lzero) ⊔ (lsuc l118)) a (R0 ((lsuc lzero) ⊔ (lsuc l118)) a hd1) hd2) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (λ (x : a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) a x hd2) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (cons ((lsuc lzero) ⊔ (lsuc l118)) a x (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4))) -> (X-e1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (R1 ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) a x (λ (x0 : a) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l118)) a x x0) -> list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3) hd2 X--) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) (λ (Heq0 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3)) (cons ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4))) -> λ (e00 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (R1 ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) a hd2 (λ (x0 : a) -> λ (p0 : eq ((lsuc lzero) ⊔ (lsuc l118)) a hd2 x0) -> list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3) hd2 (refl ((lsuc lzero) ⊔ (lsuc l118)) a hd2)) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4)) -> IH (rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3) X--) (refl ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl1 l3)) (append ((lsuc lzero) ⊔ (lsuc l118)) a tl2 l4) e00)) hd1 e0 Heq))

append-l1-injective-r : (l118 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l118))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (X-- : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) A l3) (length ((lsuc lzero) ⊔ (lsuc l118)) A l4)) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) (append ((lsuc lzero) ⊔ (lsuc l118)) A l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) A l2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) l1 l2
append-l1-injective-r = λ (l118 : Level) -> λ (a : Set ((lsuc lzero) ⊔ (lsuc l118))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (Hlen : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (length ((lsuc lzero) ⊔ (lsuc l118)) a l4)) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) (λ (x : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1))) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4))) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1 l2) (eq-ind-r ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2)) (λ (x : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) x) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1 l2) (λ (Heq1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) -> rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1 X--) (refl ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1) l2 (rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l1 X--) (rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) X-- (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) (eq-f ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2) (append-l2-injective l118 a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2) (rewrite-r lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (λ (X-- : nat) -> eq lzero nat X-- (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4))) (rewrite-r lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l4) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) X--) (rewrite-l lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) X--) (refl lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3)) (length ((lsuc lzero) ⊔ (lsuc l118)) a l4) Hlen) (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4)) (length-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4)) (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) (length-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) Heq1)) l1 (reverse-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) l2 (reverse-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) (reverse-append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3)) (reverse-append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (eq-f ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4) Heq)

append-l2-injective-r : (l118 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l118))) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) A) -> (X-- : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) A l3) (length ((lsuc lzero) ⊔ (lsuc l118)) A l4)) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) (append ((lsuc lzero) ⊔ (lsuc l118)) A l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) A l2 l4)) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) A) l3 l4
append-l2-injective-r = λ (l118 : Level) -> λ (a : Set ((lsuc lzero) ⊔ (lsuc l118))) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l3 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (l4 : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (Hlen : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (length ((lsuc lzero) ⊔ (lsuc l118)) a l4)) -> λ (Heq : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) -> eq-ind-r ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) (λ (x : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1))) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4))) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) (eq-ind-r ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2)) (λ (x : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) x (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) x) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 l4) (λ (Heq1 : eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1)) (append ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2))) -> rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 X--) (refl ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3) l4 (rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) l3 X--) (rewrite-l ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l118)) a) -> eq ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) X-- (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4))) (eq-f ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (append-l1-injective l118 a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l1) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l2) (rewrite-r lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (λ (X-- : nat) -> eq lzero nat X-- (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4))) (rewrite-r lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l4) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) X--) (rewrite-l lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3) X--) (refl lzero nat (length ((lsuc lzero) ⊔ (lsuc l118)) a l3)) (length ((lsuc lzero) ⊔ (lsuc l118)) a l4) Hlen) (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4)) (length-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4)) (length ((lsuc lzero) ⊔ (lsuc l118)) a (reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) (length-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) Heq1)) l3 (reverse-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l3)) l4 (reverse-reverse ((lsuc lzero) ⊔ (lsuc l118)) a l4))) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) (reverse-append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4)) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3)) (reverse-append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (eq-f ((lsuc lzero) ⊔ (lsuc l118)) ((lsuc lzero) ⊔ (lsuc l118)) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (list ((lsuc lzero) ⊔ (lsuc l118)) a) (reverse ((lsuc lzero) ⊔ (lsuc l118)) a) (append ((lsuc lzero) ⊔ (lsuc l118)) a l1 l3) (append ((lsuc lzero) ⊔ (lsuc l118)) a l2 l4) Heq)

length-rev-append : (l68 : Level) -> (A : Set l68) -> (l : list l68 A) -> (acc : list l68 A) -> eq lzero nat (length l68 A (rev-append l68 A l acc)) (plus (length l68 A l) (length l68 A acc))
length-rev-append = λ (l68 : Level) -> λ (A : Set l68) -> λ (l : list l68 A) -> list-ind l68 l68 A (λ (X-x-716 : list l68 A) -> (acc : list l68 A) -> eq lzero nat (length l68 A (rev-append l68 A X-x-716 acc)) (plus (length l68 A X-x-716) (length l68 A acc))) (λ (acc : list l68 A) -> refl lzero nat (length l68 A (rev-append l68 A (nil l68 A) acc))) (λ (a : A) -> λ (tl : list l68 A) -> λ (Hind : (acc : list l68 A) -> eq lzero nat (length l68 A (rev-append l68 A tl acc)) (plus (length l68 A tl) (length l68 A acc))) -> λ (acc : list l68 A) -> eq-ind-r lzero lzero nat (plus (length l68 A tl) (length l68 A (cons l68 A a acc))) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus (length l68 A tl) (length l68 A (cons l68 A a acc)))) -> eq lzero nat x (S (plus (length l68 A tl) (length l68 A acc)))) (rewrite-l lzero lzero nat (S (plus (length l68 A tl) (length l68 A acc))) (λ (X-- : nat) -> eq lzero nat X-- (S (plus (length l68 A tl) (length l68 A acc)))) (refl lzero nat (S (plus (length l68 A tl) (length l68 A acc)))) (plus (length l68 A tl) (S (length l68 A acc))) (plus-n-Sm (length l68 A tl) (length l68 A acc))) (length l68 A (rev-append l68 A tl (cons l68 A a acc))) (Hind (cons l68 A a acc))) l


mem : (l0-v : Level) -> (H-v : Set l0-v) -> (X---v : H-v) -> (X--1-v : list l0-v H-v) -> Set l0-v
mem ll0-v H-v X---v nil' = False ll0-v
mem ll0-v H-v X---v (cons' hd-v tl-v) = Or ll0-v ll0-v (eq ll0-v H-v X---v hd-v) (mem ll0-v H-v X---v tl-v)



mem-append : (l117 : Level) -> (A : Set l117) -> (a : A) -> (l1 : list l117 A) -> (l2 : list l117 A) -> (X-- : mem l117 A a (append l117 A l1 l2)) -> Or l117 l117 (mem l117 A a l1) (mem l117 A a l2)
mem-append = λ (l117 : Level) -> λ (A : Set l117) -> λ (a : A) -> λ (l1 : list l117 A) -> list-ind l117 l117 A (λ (X-x-716 : list l117 A) -> (l2 : list l117 A) -> (X-- : mem l117 A a (append l117 A X-x-716 l2)) -> Or l117 l117 (mem l117 A a X-x-716) (mem l117 A a l2)) (λ (l2 : list l117 A) -> λ (mema : mem l117 A a (append l117 A (nil l117 A) l2)) -> or-intror l117 l117 (mem l117 A a (nil l117 A)) (mem l117 A a l2) mema) (λ (b : A) -> λ (tl : list l117 A) -> λ (Hind : (l2 : list l117 A) -> (X-- : mem l117 A a (append l117 A tl l2)) -> Or l117 l117 (mem l117 A a tl) (mem l117 A a l2)) -> λ (l2 : list l117 A) -> λ (X-clearme : mem l117 A a (append l117 A (cons l117 A b tl) l2)) -> match-Or l117 l117 (eq l117 A a b) (mem l117 A a (append l117 A tl l2)) l117 (λ (X-- : Or l117 l117 (eq l117 A a b) (mem l117 A a (append l117 A tl l2))) -> Or l117 l117 (mem l117 A a (cons l117 A b tl)) (mem l117 A a l2)) (λ (eqab : eq l117 A a b) -> or-introl l117 l117 (mem l117 A a (cons l117 A b tl)) (mem l117 A a l2) (or-introl l117 l117 (eq l117 A a b) (mem l117 A a tl) eqab)) (λ (Hmema : mem l117 A a (append l117 A tl l2)) -> match-Or l117 l117 (mem l117 A a tl) (mem l117 A a l2) l117 (λ (X-- : Or l117 l117 (mem l117 A a tl) (mem l117 A a l2)) -> Or l117 l117 (mem l117 A a (cons l117 A b tl)) (mem l117 A a l2)) (λ (Hmema0 : mem l117 A a tl) -> or-introl l117 l117 (mem l117 A a (cons l117 A b tl)) (mem l117 A a l2) (or-intror l117 l117 (eq l117 A a b) (mem l117 A a tl) Hmema0)) (λ (Hmema0 : mem l117 A a l2) -> or-intror l117 l117 (mem l117 A a (cons l117 A b tl)) (mem l117 A a l2) Hmema0) (Hind l2 Hmema)) X-clearme) l1

mem-append-l1 : (l61 : Level) -> (A : Set l61) -> (a : A) -> (l1 : list l61 A) -> (l2 : list l61 A) -> (X-- : mem l61 A a l1) -> mem l61 A a (append l61 A l1 l2)
mem-append-l1 = λ (l61 : Level) -> λ (A : Set l61) -> λ (a : A) -> λ (l1 : list l61 A) -> λ (l2 : list l61 A) -> list-ind l61 l61 A (λ (X-x-716 : list l61 A) -> (X-- : mem l61 A a X-x-716) -> mem l61 A a (append l61 A X-x-716 l2)) (False-ind l61 l61 (λ (X-x-66 : False l61) -> mem l61 A a (append l61 A (nil l61 A) l2))) (λ (b : A) -> λ (tl : list l61 A) -> λ (Hind : (X-- : mem l61 A a tl) -> mem l61 A a (append l61 A tl l2)) -> λ (X-clearme : mem l61 A a (cons l61 A b tl)) -> match-Or l61 l61 (eq l61 A a b) (mem l61 A a tl) l61 (λ (X-- : Or l61 l61 (eq l61 A a b) (mem l61 A a tl)) -> mem l61 A a (append l61 A (cons l61 A b tl) l2)) (λ (eqab : eq l61 A a b) -> or-introl l61 l61 (eq l61 A a b) (mem l61 A a (append l61 A tl l2)) eqab) (λ (Hmema : mem l61 A a tl) -> or-intror l61 l61 (eq l61 A a b) (mem l61 A a (append l61 A tl l2)) (Hind Hmema)) X-clearme) l1

mem-append-l2 : (l33 : Level) -> (A : Set l33) -> (a : A) -> (l1 : list l33 A) -> (l2 : list l33 A) -> (X-- : mem l33 A a l2) -> mem l33 A a (append l33 A l1 l2)
mem-append-l2 = λ (l33 : Level) -> λ (A : Set l33) -> λ (a : A) -> λ (l1 : list l33 A) -> λ (l2 : list l33 A) -> list-ind l33 l33 A (λ (X-x-716 : list l33 A) -> (X-- : mem l33 A a l2) -> mem l33 A a (append l33 A X-x-716 l2)) (λ (auto : mem l33 A a l2) -> auto) (λ (b : A) -> λ (tl : list l33 A) -> λ (Hind : (X-- : mem l33 A a l2) -> mem l33 A a (append l33 A tl l2)) -> λ (Hmema : mem l33 A a l2) -> or-intror l33 l33 (eq l33 A a b) (mem l33 A a (append l33 A tl l2)) (Hind Hmema)) l1

mem-single : (l31 : Level) -> (A : Set l31) -> (a : A) -> (b : A) -> (X-- : mem l31 A a (cons l31 A b (nil l31 A))) -> eq l31 A a b
mem-single = λ (l31 : Level) -> λ (A : Set l31) -> λ (a : A) -> λ (b : A) -> λ (X-clearme : mem l31 A a (cons l31 A b (nil l31 A))) -> match-Or l31 l31 (eq l31 A a b) (mem l31 A a (nil l31 A)) l31 (λ (X-- : Or l31 l31 (eq l31 A a b) (mem l31 A a (nil l31 A))) -> eq l31 A a b) (λ (auto : eq l31 A a b) -> rewrite-l l31 l31 A a (λ (X-- : A) -> eq l31 A a X--) (refl l31 A a) b auto) (False-ind l31 l31 (λ (X-x-66 : False l31) -> eq l31 A a b)) X-clearme

mem-map : (l209 l202 : Level) -> (A : Set l209) -> (B-v : Set l202) -> (f : (X-- : A) -> B-v) -> (l : list l209 A) -> (b : B-v) -> (X-- : mem l202 B-v b (map l209 l202 A B-v f l)) -> ex l209 (l209 ⊔ l202) A (λ (a : A) -> And l209 l202 (mem l209 A a l) (eq l202 B-v (f a) b))
mem-map = λ (l209 l202 : Level) -> λ (A : Set l209) -> λ (B-v : Set l202) -> λ (f : (X-- : A) -> B-v) -> λ (l : list l209 A) -> list-ind l209 (l209 ⊔ l202) A (λ (X-x-716 : list l209 A) -> (b : B-v) -> (X-- : mem l202 B-v b (map l209 l202 A B-v f X-x-716)) -> ex l209 (l209 ⊔ l202) A (λ (a : A) -> And l209 l202 (mem l209 A a X-x-716) (eq l202 B-v (f a) b))) (λ (b : B-v) -> False-ind l202 (l209 ⊔ l202) (λ (X-x-66 : False l202) -> ex l209 (l209 ⊔ l202) A (λ (a : A) -> And l209 l202 (False l209) (eq l202 B-v (f a) b)))) (λ (a : A) -> λ (tl : list l209 A) -> λ (Hind : (b : B-v) -> (X-- : mem l202 B-v b (map l209 l202 A B-v f tl)) -> ex l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b))) -> λ (b : B-v) -> λ (X-clearme : Or l202 l202 (eq l202 B-v b (f a)) (mem l202 B-v b (map l209 l202 A B-v f tl))) -> match-Or l202 l202 (eq l202 B-v b (f a)) (mem l202 B-v b (map l209 l202 A B-v f tl)) (l209 ⊔ l202) (λ (X-- : Or l202 l202 (eq l202 B-v b (f a)) (mem l202 B-v b (map l209 l202 A B-v f tl))) -> ex l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (Or l209 l209 (eq l209 A a0 a) (mem l209 A a0 tl)) (eq l202 B-v (f a0) b))) (λ (eqb-v : eq l202 B-v b (f a)) -> ex-intro l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (Or l209 l209 (eq l209 A a0 a) (mem l209 A a0 tl)) (eq l202 B-v (f a0) b)) a (conj l209 l202 (Or l209 l209 (eq l209 A a a) (mem l209 A a tl)) (eq l202 B-v (f a) b) (or-introl l209 l209 (eq l209 A a a) (mem l209 A a tl) (refl l209 A a)) (rewrite-l l202 l202 B-v b (λ (X-- : B-v) -> eq l202 B-v X-- b) (refl l202 B-v b) (f a) eqb-v))) (λ (memb : mem l202 B-v b (map l209 l202 A B-v f tl)) -> match-ex l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b)) (l209 ⊔ l202) (λ (X-- : ex l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b))) -> ex l209 (l209 ⊔ l202) A (λ (a0 : A) -> And l209 l202 (Or l209 l209 (eq l209 A a0 a) (mem l209 A a0 tl)) (eq l202 B-v (f a0) b))) (λ (a0 : A) -> λ (X-clearme0 : And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b)) -> match-And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b) (l209 ⊔ l202) (λ (X-- : And l209 l202 (mem l209 A a0 tl) (eq l202 B-v (f a0) b)) -> ex l209 (l209 ⊔ l202) A (λ (a00 : A) -> And l209 l202 (Or l209 l209 (eq l209 A a00 a) (mem l209 A a00 tl)) (eq l202 B-v (f a00) b))) (λ (mema : mem l209 A a0 tl) -> λ (eqb-v : eq l202 B-v (f a0) b) -> ex-intro l209 (l209 ⊔ l202) A (λ (a00 : A) -> And l209 l202 (Or l209 l209 (eq l209 A a00 a) (mem l209 A a00 tl)) (eq l202 B-v (f a00) b)) a0 (conj l209 l202 (Or l209 l209 (eq l209 A a0 a) (mem l209 A a0 tl)) (eq l202 B-v (f a0) b) (or-intror l209 l209 (eq l209 A a0 a) (mem l209 A a0 tl) mema) (rewrite-r l202 l202 B-v b (λ (X-- : B-v) -> eq l202 B-v X-- b) (refl l202 B-v b) (f a0) eqb-v))) X-clearme0) (Hind b memb)) X-clearme) l

mem-map-forward : (l75 l74 : Level) -> (A : Set l75) -> (B-v : Set l74) -> (f : (X-- : A) -> B-v) -> (a : A) -> (l : list l75 A) -> (X-- : mem l75 A a l) -> mem l74 B-v (f a) (map l75 l74 A B-v f l)
mem-map-forward = λ (l75 l74 : Level) -> λ (A : Set l75) -> λ (B-v : Set l74) -> λ (f : (X-- : A) -> B-v) -> λ (a : A) -> λ (l : list l75 A) -> list-ind l75 (l75 ⊔ l74) A (λ (X-x-716 : list l75 A) -> (X-- : mem l75 A a X-x-716) -> mem l74 B-v (f a) (map l75 l74 A B-v f X-x-716)) (False-ind l75 l74 (λ (X-x-66 : False l75) -> False l74)) (λ (b : A) -> λ (tl : list l75 A) -> λ (Hind : (X-- : mem l75 A a tl) -> mem l74 B-v (f a) (map l75 l74 A B-v f tl)) -> λ (X-clearme : mem l75 A a (cons l75 A b tl)) -> match-Or l75 l75 (eq l75 A a b) (mem l75 A a tl) l74 (λ (X-- : Or l75 l75 (eq l75 A a b) (mem l75 A a tl)) -> mem l74 B-v (f a) (map l75 l74 A B-v f (cons l75 A b tl))) (λ (eqab : eq l75 A a b) -> eq-ind l75 l74 A a (λ (x-1 : A) -> λ (X-x-2 : eq l75 A a x-1) -> mem l74 B-v (f a) (map l75 l74 A B-v f (cons l75 A x-1 tl))) (or-introl l74 l74 (eq l74 B-v (f a) (f a)) (mem l74 B-v (f a) (map l75 l74 A B-v f tl)) (refl l74 B-v (f a))) b eqab) (λ (memtl : mem l75 A a tl) -> or-intror l74 l74 (eq l74 B-v (f a) (f b)) (mem l74 B-v (f a) (map l75 l74 A B-v f tl)) (Hind memtl)) X-clearme) l

mem-filter : (l182 : Level) -> (S-v : Set l182) -> (f : (X-- : S-v) -> bool) -> (a : S-v) -> (l : list l182 S-v) -> (X-- : mem l182 S-v a (filter' l182 S-v f l)) -> mem l182 S-v a l
mem-filter = λ (l182 : Level) -> λ (S-v : Set l182) -> λ (f : (X-- : S-v) -> bool) -> λ (a : S-v) -> λ (l : list l182 S-v) -> list-ind l182 l182 S-v (λ (X-x-716 : list l182 S-v) -> (X-- : mem l182 S-v a (filter' l182 S-v f X-x-716)) -> mem l182 S-v a X-x-716) (λ (auto : False l182) -> auto) (λ (b : S-v) -> λ (tl : list l182 S-v) -> λ (Hind : (X-- : mem l182 S-v a (filter' l182 S-v f tl)) -> mem l182 S-v a tl) -> match-bool l182 (λ (X-- : bool) -> (X--1 : mem l182 S-v a (match-bool l182 (λ (X-0 : bool) -> list l182 S-v) (cons l182 S-v b (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-0 : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl)) (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-0 : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl) X--)) -> Or l182 l182 (eq l182 S-v a b) (mem l182 S-v a tl)) (λ (X-clearme : Or l182 l182 (eq l182 S-v a b) (mem l182 S-v a (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-- : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl))) -> match-Or l182 l182 (eq l182 S-v a b) (mem l182 S-v a (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-- : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl)) l182 (λ (X-- : Or l182 l182 (eq l182 S-v a b) (mem l182 S-v a (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-- : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl))) -> Or l182 l182 (eq l182 S-v a b) (mem l182 S-v a tl)) (λ (eqab : eq l182 S-v a b) -> or-introl l182 l182 (eq l182 S-v a b) (mem l182 S-v a tl) eqab) (λ (H : mem l182 S-v a (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-- : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl)) -> or-intror l182 l182 (eq l182 S-v a b) (mem l182 S-v a tl) (Hind H)) X-clearme) (λ (H : mem l182 S-v a (foldr l182 l182 S-v (list l182 S-v) (λ (x : S-v) -> λ (l0 : list l182 S-v) -> match-bool l182 (λ (X-- : bool) -> list l182 S-v) (cons l182 S-v x l0) l0 (f x)) (nil l182 S-v) tl)) -> or-intror l182 l182 (eq l182 S-v a b) (mem l182 S-v a tl) (Hind H)) (f b)) l

mem-filter-true : (l240 : Level) -> (S-v : Set l240) -> (f : (X-- : S-v) -> bool) -> (a : S-v) -> (l : list l240 S-v) -> (X-- : mem l240 S-v a (filter' l240 S-v f l)) -> eq lzero bool (f a) true
mem-filter-true = λ (l240 : Level) -> λ (S-v : Set l240) -> λ (f : (X-- : S-v) -> bool) -> λ (a : S-v) -> λ (l : list l240 S-v) -> list-ind l240 l240 S-v (λ (X-x-716 : list l240 S-v) -> (X-- : mem l240 S-v a (filter' l240 S-v f X-x-716)) -> eq lzero bool (f a) true) (False-ind l240 lzero (λ (X-x-66 : False l240) -> eq lzero bool (f a) true)) (λ (b : S-v) -> λ (tl : list l240 S-v) -> λ (Hind : (X-- : mem l240 S-v a (filter' l240 S-v f tl)) -> eq lzero bool (f a) true) -> match-Or lzero lzero (eq lzero bool (f b) true) (eq lzero bool (f b) false) l240 (λ (X-- : Or lzero lzero (eq lzero bool (f b) true) (eq lzero bool (f b) false)) -> (X--1 : mem l240 S-v a (filter' l240 S-v f (cons l240 S-v b tl))) -> eq lzero bool (f a) true) (λ (H : eq lzero bool (f b) true) -> eq-ind-r lzero l240 bool true (λ (x : bool) -> λ (X-- : eq lzero bool x true) -> (X--1 : mem l240 S-v a (match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v b (foldr l240 l240 S-v (list l240 S-v) (λ (x0 : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v x0 l0) l0 (f x0)) (nil l240 S-v) tl)) (foldr l240 l240 S-v (list l240 S-v) (λ (x0 : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v x0 l0) l0 (f x0)) (nil l240 S-v) tl) x)) -> eq lzero bool (f a) true) (λ (X-clearme : Or l240 l240 (eq l240 S-v a b) (mem l240 S-v a (foldr l240 l240 S-v (list l240 S-v) (λ (x : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-- : bool) -> list l240 S-v) (cons l240 S-v x l0) l0 (f x)) (nil l240 S-v) tl))) -> match-Or l240 l240 (eq l240 S-v a b) (mem l240 S-v a (foldr l240 l240 S-v (list l240 S-v) (λ (x : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-- : bool) -> list l240 S-v) (cons l240 S-v x l0) l0 (f x)) (nil l240 S-v) tl)) lzero (λ (X-- : Or l240 l240 (eq l240 S-v a b) (mem l240 S-v a (foldr l240 l240 S-v (list l240 S-v) (λ (x : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-- : bool) -> list l240 S-v) (cons l240 S-v x l0) l0 (f x)) (nil l240 S-v) tl))) -> eq lzero bool (f a) true) (λ (eqab : eq l240 S-v a b) -> rewrite-r lzero lzero bool true (λ (X-- : bool) -> eq lzero bool X-- true) (refl lzero bool true) (f a) (rewrite-r l240 lzero S-v b (λ (X-- : S-v) -> eq lzero bool (f X--) true) H a eqab)) Hind X-clearme) (f b) H) (λ (H : eq lzero bool (f b) false) -> eq-ind-r lzero l240 bool false (λ (x : bool) -> λ (X-- : eq lzero bool x false) -> (X--1 : mem l240 S-v a (match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v b (foldr l240 l240 S-v (list l240 S-v) (λ (x0 : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v x0 l0) l0 (f x0)) (nil l240 S-v) tl)) (foldr l240 l240 S-v (list l240 S-v) (λ (x0 : S-v) -> λ (l0 : list l240 S-v) -> match-bool l240 (λ (X-0 : bool) -> list l240 S-v) (cons l240 S-v x0 l0) l0 (f x0)) (nil l240 S-v) tl) x)) -> eq lzero bool (f a) true) Hind (f b) H) (true-or-false (f b))) l

mem-filter-l : (l135 : Level) -> (S-v : Set l135) -> (f : (X-- : S-v) -> bool) -> (x : S-v) -> (l : list l135 S-v) -> (X-- : eq lzero bool (f x) true) -> (X--1 : mem l135 S-v x l) -> mem l135 S-v x (filter' l135 S-v f l)
mem-filter-l = λ (l135 : Level) -> λ (S-v : Set l135) -> λ (f : (X-- : S-v) -> bool) -> λ (x : S-v) -> λ (l : list l135 S-v) -> λ (fx : eq lzero bool (f x) true) -> list-ind l135 l135 S-v (λ (X-x-716 : list l135 S-v) -> (X-- : mem l135 S-v x X-x-716) -> mem l135 S-v x (filter' l135 S-v f X-x-716)) (False-ind l135 l135 (λ (X-x-66 : False l135) -> mem l135 S-v x (filter' l135 S-v f (nil l135 S-v)))) (λ (b : S-v) -> λ (tl : list l135 S-v) -> λ (Hind : (X-- : mem l135 S-v x tl) -> mem l135 S-v x (filter' l135 S-v f tl)) -> λ (X-clearme : mem l135 S-v x (cons l135 S-v b tl)) -> match-Or l135 l135 (eq l135 S-v x b) (mem l135 S-v x tl) l135 (λ (X-- : Or l135 l135 (eq l135 S-v x b) (mem l135 S-v x tl)) -> mem l135 S-v x (filter' l135 S-v f (cons l135 S-v b tl))) (λ (eqxb : eq l135 S-v x b) -> eq-ind l135 l135 S-v x (λ (x-1 : S-v) -> λ (X-x-2 : eq l135 S-v x x-1) -> mem l135 S-v x (filter' l135 S-v f (cons l135 S-v x-1 tl))) (eq-ind-r l135 l135 (list l135 S-v) (cons l135 S-v x (filter' l135 S-v f tl)) (λ (x0 : list l135 S-v) -> λ (X-- : eq l135 (list l135 S-v) x0 (cons l135 S-v x (filter' l135 S-v f tl))) -> mem l135 S-v x x0) (or-introl l135 l135 (eq l135 S-v x x) (mem l135 S-v x (filter' l135 S-v f tl)) (refl l135 S-v x)) (filter' l135 S-v f (cons l135 S-v x tl)) (filter-true l135 S-v tl x f fx)) b eqxb) (λ (Htl : mem l135 S-v x tl) -> match-Or lzero lzero (eq lzero bool (f b) true) (eq lzero bool (f b) false) l135 (λ (X-- : Or lzero lzero (eq lzero bool (f b) true) (eq lzero bool (f b) false)) -> mem l135 S-v x (filter' l135 S-v f (cons l135 S-v b tl))) (λ (fb : eq lzero bool (f b) true) -> eq-ind-r l135 l135 (list l135 S-v) (cons l135 S-v b (filter' l135 S-v f tl)) (λ (x0 : list l135 S-v) -> λ (X-- : eq l135 (list l135 S-v) x0 (cons l135 S-v b (filter' l135 S-v f tl))) -> mem l135 S-v x x0) (or-intror l135 l135 (eq l135 S-v x b) (mem l135 S-v x (filter' l135 S-v f tl)) (Hind Htl)) (filter' l135 S-v f (cons l135 S-v b tl)) (filter-true l135 S-v tl b f fb)) (λ (fb : eq lzero bool (f b) false) -> eq-ind-r l135 l135 (list l135 S-v) (filter' l135 S-v f tl) (λ (x0 : list l135 S-v) -> λ (X-- : eq l135 (list l135 S-v) x0 (filter' l135 S-v f tl)) -> mem l135 S-v x x0) (Hind Htl) (filter' l135 S-v f (cons l135 S-v b tl)) (filter-false l135 S-v tl b f fb)) (true-or-false (f b))) X-clearme) l

filter-case : (l385 : Level) -> (A : Set l385) -> (p : (X-- : A) -> bool) -> (l : list l385 A) -> (x : A) -> (X-- : mem l385 A x l) -> Or l385 l385 (mem l385 A x (filter' l385 A p l)) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) l))
filter-case = λ (l385 : Level) -> λ (A : Set l385) -> λ (p : (X-- : A) -> bool) -> λ (l : list l385 A) -> list-ind l385 l385 A (λ (X-x-716 : list l385 A) -> (x : A) -> (X-- : mem l385 A x X-x-716) -> Or l385 l385 (mem l385 A x (filter' l385 A p X-x-716)) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) X-x-716))) (λ (x : A) -> False-ind l385 l385 (λ (X-x-66 : False l385) -> Or l385 l385 (mem l385 A x (filter' l385 A p (nil l385 A))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (nil l385 A))))) (λ (a : A) -> λ (tl : list l385 A) -> λ (Hind : (x : A) -> (X-- : mem l385 A x tl) -> Or l385 l385 (mem l385 A x (filter' l385 A p tl)) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl))) -> λ (x : A) -> λ (X-clearme : mem l385 A x (cons l385 A a tl)) -> match-Or l385 l385 (eq l385 A x a) (mem l385 A x tl) l385 (λ (X-- : Or l385 l385 (eq l385 A x a) (mem l385 A x tl)) -> Or l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)))) (λ (eqxa : eq l385 A x a) -> eq-ind-r l385 l385 A a (λ (x0 : A) -> λ (X-- : eq l385 A x0 a) -> Or l385 l385 (mem l385 A x0 (filter' l385 A p (cons l385 A a tl))) (mem l385 A x0 (filter' l385 A (λ (x00 : A) -> notb (p x00)) (cons l385 A a tl)))) (match-Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false) l385 (λ (X-- : Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false)) -> Or l385 l385 (mem l385 A a (filter' l385 A p (cons l385 A a tl))) (mem l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)))) (λ (Hcase : eq lzero bool (p a) true) -> or-introl l385 l385 (mem l385 A a (filter' l385 A p (cons l385 A a tl))) (mem l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl))) (eq-ind-r l385 l385 (list l385 A) (cons l385 A a (filter' l385 A p tl)) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (cons l385 A a (filter' l385 A p tl))) -> mem l385 A a x0) (or-introl l385 l385 (eq l385 A a a) (mem l385 A a (filter' l385 A p tl)) (refl l385 A a)) (filter' l385 A p (cons l385 A a tl)) (filter-true l385 A tl a p Hcase))) (λ (Hcase : eq lzero bool (p a) false) -> or-intror l385 l385 (mem l385 A a (filter' l385 A p (cons l385 A a tl))) (mem l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl))) (eq-ind-r l385 l385 (list l385 A) (cons l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (cons l385 A a (filter' l385 A (λ (x00 : A) -> notb (p x00)) tl))) -> mem l385 A a x0) (or-introl l385 l385 (eq l385 A a a) (mem l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) (refl l385 A a)) (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)) (filter-true l385 A tl a (λ (x0 : A) -> notb (p x0)) (eq-ind-r lzero lzero bool false (λ (x0 : bool) -> λ (X-- : eq lzero bool x0 false) -> eq lzero bool (notb x0) true) (refl lzero bool (notb false)) (p a) Hcase)))) (true-or-false (p a))) x eqxa) (λ (memx : mem l385 A x tl) -> match-Or l385 l385 (mem l385 A x (filter' l385 A p tl)) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) l385 (λ (X-- : Or l385 l385 (mem l385 A x (filter' l385 A p tl)) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl))) -> Or l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)))) (λ (memx0 : mem l385 A x (filter' l385 A p tl)) -> or-introl l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl))) (match-Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false) l385 (λ (X-- : Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false)) -> mem l385 A x (filter' l385 A p (cons l385 A a tl))) (λ (Hpa : eq lzero bool (p a) true) -> eq-ind-r l385 l385 (list l385 A) (cons l385 A a (filter' l385 A p tl)) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (cons l385 A a (filter' l385 A p tl))) -> mem l385 A x x0) (or-intror l385 l385 (eq l385 A x a) (mem l385 A x (filter' l385 A p tl)) memx0) (filter' l385 A p (cons l385 A a tl)) (filter-true l385 A tl a p Hpa)) (λ (Hpa : eq lzero bool (p a) false) -> eq-ind-r l385 l385 (list l385 A) (filter' l385 A p tl) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (filter' l385 A p tl)) -> mem l385 A x x0) memx0 (filter' l385 A p (cons l385 A a tl)) (filter-false l385 A tl a p Hpa)) (true-or-false (p a)))) (λ (memx0 : mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) -> match-Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false) l385 (λ (X-- : Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false)) -> Or l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)))) (λ (Hcase : eq lzero bool (p a) true) -> or-intror l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl))) (eq-ind-r l385 l385 (list l385 A) (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (filter' l385 A (λ (x00 : A) -> notb (p x00)) tl)) -> mem l385 A x x0) memx0 (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)) (filter-false l385 A tl a (λ (x0 : A) -> notb (p x0)) (eq-ind-r lzero lzero bool true (λ (x0 : bool) -> λ (X-- : eq lzero bool x0 true) -> eq lzero bool (notb x0) false) (refl lzero bool (notb true)) (p a) Hcase)))) (λ (Hcase : eq lzero bool (p a) false) -> or-intror l385 l385 (mem l385 A x (filter' l385 A p (cons l385 A a tl))) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl))) (eq-ind-r l385 l385 (list l385 A) (cons l385 A a (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) (λ (x0 : list l385 A) -> λ (X-- : eq l385 (list l385 A) x0 (cons l385 A a (filter' l385 A (λ (x00 : A) -> notb (p x00)) tl))) -> mem l385 A x x0) (or-intror l385 l385 (eq l385 A x a) (mem l385 A x (filter' l385 A (λ (x0 : A) -> notb (p x0)) tl)) memx0) (filter' l385 A (λ (x0 : A) -> notb (p x0)) (cons l385 A a tl)) (filter-true l385 A tl a (λ (x0 : A) -> notb (p x0)) (eq-ind-r lzero lzero bool false (λ (x0 : bool) -> λ (X-- : eq lzero bool x0 false) -> eq lzero bool (notb x0) true) (refl lzero bool (notb false)) (p a) Hcase)))) (true-or-false (p a))) (Hind x memx)) X-clearme) l

filter-length2 : (l210 : Level) -> (A : Set l210) -> (p : (X-- : A) -> bool) -> (l : list l210 A) -> eq lzero nat (plus (length l210 A (filter' l210 A p l)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) l))) (length l210 A l)
filter-length2 = λ (l210 : Level) -> λ (A : Set l210) -> λ (p : (X-- : A) -> bool) -> λ (l : list l210 A) -> list-ind l210 lzero A (λ (X-x-716 : list l210 A) -> eq lzero nat (plus (length l210 A (filter' l210 A p X-x-716)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) X-x-716))) (length l210 A X-x-716)) (refl lzero nat (plus (length l210 A (filter' l210 A p (nil l210 A))) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) (nil l210 A))))) (λ (a : A) -> λ (tl : list l210 A) -> λ (Hind : eq lzero nat (plus (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl))) (length l210 A tl)) -> match-Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false) lzero (λ (X-- : Or lzero lzero (eq lzero bool (p a) true) (eq lzero bool (p a) false)) -> eq lzero nat (plus (length l210 A (filter' l210 A p (cons l210 A a tl))) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) (cons l210 A a tl)))) (length l210 A (cons l210 A a tl))) (λ (Hcase : eq lzero bool (p a) true) -> eq-ind-r l210 lzero (list l210 A) (cons l210 A a (filter' l210 A p tl)) (λ (x : list l210 A) -> λ (X-- : eq l210 (list l210 A) x (cons l210 A a (filter' l210 A p tl))) -> eq lzero nat (plus (length l210 A x) (length l210 A (filter' l210 A (λ (x0 : A) -> notb (p x0)) (cons l210 A a tl)))) (length l210 A (cons l210 A a tl))) (eq-ind-r l210 lzero (list l210 A) (filter' l210 A (λ (x : A) -> notb (p x)) tl) (λ (x : list l210 A) -> λ (X-- : eq l210 (list l210 A) x (filter' l210 A (λ (x0 : A) -> notb (p x0)) tl)) -> eq lzero nat (plus (length l210 A (cons l210 A a (filter' l210 A p tl))) (length l210 A x)) (length l210 A (cons l210 A a tl))) (eq-f lzero lzero nat nat S (plus (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl))) (length l210 A tl) Hind) (filter' l210 A (λ (x : A) -> notb (p x)) (cons l210 A a tl)) (filter-false l210 A tl a (λ (x : A) -> notb (p x)) (eq-ind-r lzero lzero bool true (λ (x : bool) -> λ (X-- : eq lzero bool x true) -> eq lzero bool (notb x) false) (refl lzero bool (notb true)) (p a) Hcase))) (filter' l210 A p (cons l210 A a tl)) (filter-true l210 A tl a p Hcase)) (λ (Hcase : eq lzero bool (p a) false) -> eq-ind-r l210 lzero (list l210 A) (filter' l210 A p tl) (λ (x : list l210 A) -> λ (X-- : eq l210 (list l210 A) x (filter' l210 A p tl)) -> eq lzero nat (plus (length l210 A x) (length l210 A (filter' l210 A (λ (x0 : A) -> notb (p x0)) (cons l210 A a tl)))) (length l210 A (cons l210 A a tl))) (eq-ind-r l210 lzero (list l210 A) (cons l210 A a (filter' l210 A (λ (x : A) -> notb (p x)) tl)) (λ (x : list l210 A) -> λ (X-- : eq l210 (list l210 A) x (cons l210 A a (filter' l210 A (λ (x0 : A) -> notb (p x0)) tl))) -> eq lzero nat (plus (length l210 A (filter' l210 A p tl)) (length l210 A x)) (length l210 A (cons l210 A a tl))) (eq-ind lzero lzero nat (S (plus (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl)))) (λ (x-1 : nat) -> λ (X-x-2 : eq lzero nat (S (plus (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl)))) x-1) -> eq lzero nat x-1 (length l210 A (cons l210 A a tl))) (eq-f lzero lzero nat nat S (plus (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl))) (length l210 A tl) Hind) (plus (length l210 A (filter' l210 A p tl)) (S (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl)))) (plus-n-Sm (length l210 A (filter' l210 A p tl)) (length l210 A (filter' l210 A (λ (x : A) -> notb (p x)) tl)))) (filter' l210 A (λ (x : A) -> notb (p x)) (cons l210 A a tl)) (filter-true l210 A tl a (λ (x : A) -> notb (p x)) (eq-ind-r lzero lzero bool false (λ (x : bool) -> λ (X-- : eq lzero bool x false) -> eq lzero bool (notb x) true) (refl lzero bool (notb false)) (p a) Hcase))) (filter' l210 A p (cons l210 A a tl)) (filter-false l210 A tl a p Hcase)) (true-or-false (p a))) l


unique : (l2-v : Level) -> (H-v : Set l2-v) -> (X---v : list l2-v H-v) -> Set l2-v
unique ll0-v H-v nil' = True ll0-v
unique ll0-v H-v (cons' a-v tl-v) = And ll0-v ll0-v (Not ll0-v (mem ll0-v H-v a-v tl-v)) (unique ll0-v H-v tl-v)


unique-filter : (l113 : Level) -> (S-v : Set l113) -> (l : list l113 S-v) -> (f : (X-- : S-v) -> bool) -> (X-- : unique l113 S-v l) -> unique l113 S-v (filter' l113 S-v f l)
unique-filter = λ (l113 : Level) -> λ (S-v : Set l113) -> λ (l : list l113 S-v) -> λ (f : (X-- : S-v) -> bool) -> list-ind l113 l113 S-v (λ (X-x-716 : list l113 S-v) -> (X-- : unique l113 S-v X-x-716) -> unique l113 S-v (filter' l113 S-v f X-x-716)) (λ (auto : unique l113 S-v (nil l113 S-v)) -> auto) (λ (a : S-v) -> λ (tl : list l113 S-v) -> λ (Hind : (X-- : unique l113 S-v tl) -> unique l113 S-v (filter' l113 S-v f tl)) -> λ (X-clearme : unique l113 S-v (cons l113 S-v a tl)) -> match-And l113 l113 (Not l113 (mem l113 S-v a tl)) (unique l113 S-v tl) l113 (λ (X-- : And l113 l113 (Not l113 (mem l113 S-v a tl)) (unique l113 S-v tl)) -> unique l113 S-v (filter' l113 S-v f (cons l113 S-v a tl))) (λ (memba : Not l113 (mem l113 S-v a tl)) -> λ (uniquetl : unique l113 S-v tl) -> match-Or lzero lzero (eq lzero bool (f a) true) (eq lzero bool (f a) false) l113 (λ (X-- : Or lzero lzero (eq lzero bool (f a) true) (eq lzero bool (f a) false)) -> unique l113 S-v (filter' l113 S-v f (cons l113 S-v a tl))) (λ (Hfa : eq lzero bool (f a) true) -> eq-ind-r l113 l113 (list l113 S-v) (cons l113 S-v a (filter' l113 S-v f tl)) (λ (x : list l113 S-v) -> λ (X-- : eq l113 (list l113 S-v) x (cons l113 S-v a (filter' l113 S-v f tl))) -> unique l113 S-v x) (conj l113 l113 (Not l113 (mem l113 S-v a (filter' l113 S-v f tl))) (unique l113 S-v (filter' l113 S-v f tl)) (not-to-not l113 (mem l113 S-v a (filter' l113 S-v f tl)) (mem l113 S-v a tl) (mem-filter l113 S-v f a tl) memba) (Hind uniquetl)) (filter' l113 S-v f (cons l113 S-v a tl)) (filter-true l113 S-v tl a f Hfa)) (λ (Hfa : eq lzero bool (f a) false) -> eq-ind-r l113 l113 (list l113 S-v) (filter' l113 S-v f tl) (λ (x : list l113 S-v) -> λ (X-- : eq l113 (list l113 S-v) x (filter' l113 S-v f tl)) -> unique l113 S-v x) (Hind uniquetl) (filter' l113 S-v f (cons l113 S-v a tl)) (filter-false l113 S-v tl a f (rewrite-r lzero lzero bool false (λ (X-- : bool) -> eq lzero bool X-- false) (refl lzero bool false) (f a) Hfa))) (true-or-false (f a))) X-clearme) l

filter-eqb : (m : nat) -> (l : list lzero nat) -> (X-- : unique lzero nat l) -> Or lzero lzero (And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat)))
filter-eqb = λ (m : nat) -> λ (l : list lzero nat) -> list-ind lzero lzero nat (λ (X-x-716 : list lzero nat) -> (X-- : unique lzero nat X-x-716) -> Or lzero lzero (And lzero lzero (mem lzero nat m X-x-716) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) X-x-716) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m X-x-716)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) X-x-716) (nil lzero nat)))) (λ (X-- : unique lzero nat (nil lzero nat)) -> or-intror lzero lzero (And lzero lzero (mem lzero nat m (nil lzero nat)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (nil lzero nat)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (nil lzero nat))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (nil lzero nat)) (nil lzero nat))) (conj lzero lzero (Not lzero (mem lzero nat m (nil lzero nat))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (nil lzero nat)) (nil lzero nat)) (nmk lzero (mem lzero nat m (nil lzero nat)) (False-ind lzero lzero (λ (X-x-66 : False lzero) -> False lzero))) (refl lzero (list lzero nat) (filter' lzero nat (eqb m) (nil lzero nat))))) (λ (a : nat) -> λ (tl : list lzero nat) -> λ (Hind : (X-- : unique lzero nat tl) -> Or lzero lzero (And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat)))) -> λ (X-clearme : unique lzero nat (cons lzero nat a tl)) -> match-And lzero lzero (Not lzero (mem lzero nat a tl)) (unique lzero nat tl) lzero (λ (X-- : And lzero lzero (Not lzero (mem lzero nat a tl)) (unique lzero nat tl)) -> Or lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)))) (λ (Hmema : Not lzero (mem lzero nat a tl)) -> λ (Hunique : unique lzero nat tl) -> match-Or lzero lzero (And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat))) lzero (λ (X-- : Or lzero lzero (And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat)))) -> Or lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)))) (λ (X-clearme0 : And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat)))) -> match-And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat))) lzero (λ (X-- : And lzero lzero (mem lzero nat m tl) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat)))) -> Or lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)))) (λ (Hmemm : mem lzero nat m tl) -> λ (Hind0 : eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (cons lzero nat m (nil lzero nat))) -> or-introl lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat))) (conj lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat))) (or-intror lzero lzero (eq lzero nat m a) (mem lzero nat m tl) Hmemm) (eq-ind-r lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (x : list lzero nat) -> λ (X-- : eq lzero (list lzero nat) x (filter' lzero nat (eqb m) tl)) -> eq lzero (list lzero nat) x (cons lzero nat m (nil lzero nat))) (rewrite-l lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (X-- : list lzero nat) -> eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) X--) (refl lzero (list lzero nat) (filter' lzero nat (eqb m) tl)) (cons lzero nat m (nil lzero nat)) Hind0) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (filter-false lzero nat tl a (eqb m) (not-eq-to-eqb-false m a (nmk lzero (eq lzero nat m a) (λ (eqma : eq lzero nat m a) -> absurd lzero (mem lzero nat m tl) Hmemm (eq-coerc lzero (Not lzero (mem lzero nat a tl)) (Not lzero (mem lzero nat m tl)) Hmema (rewrite-l lzero (lsuc lzero) nat m (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (Not lzero (mem lzero nat X-- tl)) (Not lzero (mem lzero nat m tl))) (refl (lsuc lzero) (Set (lzero)) (Not lzero (mem lzero nat m tl))) a eqma))))))))) X-clearme0) (λ (X-clearme0 : And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat))) -> match-And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat)) lzero (λ (X-- : And lzero lzero (Not lzero (mem lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat))) -> Or lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)))) (λ (Hmemm : Not lzero (mem lzero nat m tl)) -> λ (Hind0 : eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (nil lzero nat)) -> match-Or lzero lzero (eq lzero nat m a) (Not lzero (eq lzero nat m a)) lzero (λ (X-- : Or lzero lzero (eq lzero nat m a) (Not lzero (eq lzero nat m a))) -> Or lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)))) (λ (eqma : eq lzero nat m a) -> or-introl lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat))) (eq-ind lzero lzero nat m (λ (x-1 : nat) -> λ (X-x-2 : eq lzero nat m x-1) -> And lzero lzero (mem lzero nat m (cons lzero nat x-1 tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat x-1 tl)) (cons lzero nat m (nil lzero nat)))) (conj lzero lzero (mem lzero nat m (cons lzero nat m tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat m tl)) (cons lzero nat m (nil lzero nat))) (or-introl lzero lzero (eq lzero nat m m) (mem lzero nat m tl) (refl lzero nat m)) (eq-ind-r lzero lzero (list lzero nat) (cons lzero nat m (filter' lzero nat (eqb m) tl)) (λ (x : list lzero nat) -> λ (X-- : eq lzero (list lzero nat) x (cons lzero nat m (filter' lzero nat (eqb m) tl))) -> eq lzero (list lzero nat) x (cons lzero nat m (nil lzero nat))) (eq-ind-r lzero lzero (list lzero nat) (nil lzero nat) (λ (x : list lzero nat) -> λ (X-- : eq lzero (list lzero nat) x (nil lzero nat)) -> eq lzero (list lzero nat) (cons lzero nat m x) (cons lzero nat m (nil lzero nat))) (rewrite-l lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (X-- : list lzero nat) -> eq lzero (list lzero nat) (cons lzero nat m X--) (cons lzero nat m (nil lzero nat))) (rewrite-l lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (X-- : list lzero nat) -> eq lzero (list lzero nat) (cons lzero nat m (filter' lzero nat (eqb m) tl)) (cons lzero nat m X--)) (refl lzero (list lzero nat) (cons lzero nat m (filter' lzero nat (eqb m) tl))) (nil lzero nat) Hind0) (nil lzero nat) Hind0) (filter' lzero nat (eqb m) tl) Hind0) (filter' lzero nat (eqb m) (cons lzero nat m tl)) (filter-true lzero nat tl m (eqb m) (eq-to-eqb-true m m (refl lzero nat m))))) a eqma)) (λ (eqma : Not lzero (eq lzero nat m a)) -> or-intror lzero lzero (And lzero lzero (mem lzero nat m (cons lzero nat a tl)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat))) (conj lzero lzero (Not lzero (mem lzero nat m (cons lzero nat a tl))) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (nil lzero nat)) (not-to-not lzero (mem lzero nat m (cons lzero nat a tl)) (mem lzero nat m tl) (λ (X-clearme1 : mem lzero nat m (cons lzero nat a tl)) -> match-Or lzero lzero (eq lzero nat m a) (mem lzero nat m tl) lzero (λ (X-- : Or lzero lzero (eq lzero nat m a) (mem lzero nat m tl)) -> mem lzero nat m tl) (λ (H : eq lzero nat m a) -> False-ind lzero lzero (λ (X-x-66 : False lzero) -> mem lzero nat m tl) (absurd lzero (eq lzero nat m a) H eqma)) (λ (auto : mem lzero nat m tl) -> auto) X-clearme1) Hmemm) (eq-ind-r lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (x : list lzero nat) -> λ (X-- : eq lzero (list lzero nat) x (filter' lzero nat (eqb m) tl)) -> eq lzero (list lzero nat) x (nil lzero nat)) (rewrite-l lzero lzero (list lzero nat) (filter' lzero nat (eqb m) tl) (λ (X-- : list lzero nat) -> eq lzero (list lzero nat) (filter' lzero nat (eqb m) tl) X--) (refl lzero (list lzero nat) (filter' lzero nat (eqb m) tl)) (nil lzero nat) Hind0) (filter' lzero nat (eqb m) (cons lzero nat a tl)) (filter-false lzero nat tl a (eqb m) (not-eq-to-eqb-false m a eqma))))) (decidable-eq-nat m a)) X-clearme0) (Hind Hunique)) X-clearme) l

length-filter-eqb : (m : nat) -> (l : list lzero nat) -> (X-- : unique lzero nat l) -> le (length lzero nat (filter' lzero nat (eqb m) l)) (S O)
length-filter-eqb = λ (m : nat) -> λ (l : list lzero nat) -> λ (Huni : unique lzero nat l) -> match-Or lzero lzero (And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat))) lzero (λ (X-- : Or lzero lzero (And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat)))) (And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat)))) -> le (length lzero nat (filter' lzero nat (eqb m) l)) (S O)) (λ (X-clearme : And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat)))) -> match-And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat))) lzero (λ (X-- : And lzero lzero (mem lzero nat m l) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat)))) -> le (length lzero nat (filter' lzero nat (eqb m) l)) (S O)) (λ (X-- : mem lzero nat m l) -> λ (H : eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (cons lzero nat m (nil lzero nat))) -> eq-ind-r lzero lzero (list lzero nat) (cons lzero nat m (nil lzero nat)) (λ (x : list lzero nat) -> λ (X-0 : eq lzero (list lzero nat) x (cons lzero nat m (nil lzero nat))) -> le (length lzero nat x) (S O)) (le-n (length lzero nat (cons lzero nat m (nil lzero nat)))) (filter' lzero nat (eqb m) l) H) X-clearme) (λ (X-clearme : And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat))) -> match-And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat)) lzero (λ (X-- : And lzero lzero (Not lzero (mem lzero nat m l)) (eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat))) -> le (length lzero nat (filter' lzero nat (eqb m) l)) (S O)) (λ (X-- : Not lzero (mem lzero nat m l)) -> λ (H : eq lzero (list lzero nat) (filter' lzero nat (eqb m) l) (nil lzero nat)) -> eq-ind-r lzero lzero (list lzero nat) (nil lzero nat) (λ (x : list lzero nat) -> λ (X-0 : eq lzero (list lzero nat) x (nil lzero nat)) -> le (length lzero nat x) (S O)) (le-n-Sn (length lzero nat (nil lzero nat))) (filter' lzero nat (eqb m) l) H) X-clearme) (filter-eqb m l Huni)


split-rev : (l9-v : Level) -> (H-v : Set l9-v) -> (X---v : list l9-v H-v) -> (X--1-v : list l9-v H-v) -> (X--2-v : nat) -> Prod l9-v l9-v (list l9-v H-v) (list l9-v H-v)
split-rev l16-v H-v X X--1-v O = mk-Prod l16-v l16-v (list l16-v H-v) (list l16-v H-v) X--1-v X
split-rev l16-v H-v nil' X--1-v (S m) = mk-Prod l16-v l16-v (list l16-v H-v) (list l16-v H-v) X--1-v (nil l16-v H-v)
split-rev l16-v H-v (cons' a-v tl-v) X--1-v (S m) = split-rev l16-v H-v tl-v (cons l16-v H-v a-v X--1-v) m


split : (l28 : Level) -> (A : Set l28) -> (X-l : list l28 A) -> (X-n : nat) -> Prod l28 l28 (list l28 A) (list l28 A)
split = λ (l28 : Level) -> λ (A : Set l28) -> λ (l : list l28 A) -> λ (n : nat) -> match-Prod l28 l28 (list l28 A) (list l28 A) l28 (λ (X-- : Prod l28 l28 (list l28 A) (list l28 A)) -> Prod l28 l28 (list l28 A) (list l28 A)) (λ (l1 : list l28 A) -> λ (l2 : list l28 A) -> mk-Prod l28 l28 (list l28 A) (list l28 A) (reverse l28 A l1) l2) (split-rev l28 A l (nil l28 A) n)

split-rev-len : (l128 : Level) -> (A : Set l128) -> (n : nat) -> (l : list l128 A) -> (acc : list l128 A) -> (X-- : le n (length l128 A l)) -> eq lzero nat (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A l acc n))) (plus n (length l128 A acc))
split-rev-len = λ (l128 : Level) -> λ (A : Set l128) -> λ (n : nat) -> nat-ind l128 (λ (X-x-365 : nat) -> (l : list l128 A) -> (acc : list l128 A) -> (X-- : le X-x-365 (length l128 A l)) -> eq lzero nat (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A l acc X-x-365))) (plus X-x-365 (length l128 A acc))) (λ (l : list l128 A) -> λ (acc : list l128 A) -> λ (auto : le O (length l128 A l)) -> refl lzero nat (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A l acc O)))) (λ (m : nat) -> λ (Hind : (l : list l128 A) -> (acc : list l128 A) -> (X-- : le m (length l128 A l)) -> eq lzero nat (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A l acc m))) (plus m (length l128 A acc))) -> λ (X-clearme : list l128 A) -> match-list l128 A l128 (λ (X-- : list l128 A) -> (acc : list l128 A) -> (X--1 : le (S m) (length l128 A X--)) -> eq lzero nat (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A X-- acc (S m)))) (plus (S m) (length l128 A acc))) (λ (acc : list l128 A) -> λ (Hfalse : le (S m) O) -> False-ind lzero lzero (λ (X-x-66 : False lzero) -> eq lzero nat (length l128 A acc) (S (plus m (length l128 A acc)))) (absurd lzero (le (S m) O) Hfalse (not-le-Sn-O m))) (λ (a : A) -> λ (tl : list l128 A) -> λ (acc : list l128 A) -> λ (Hlen : le (S m) (length l128 A (cons l128 A a tl))) -> eq-ind-r lzero lzero nat (plus m (length l128 A (cons l128 A a acc))) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus m (length l128 A (cons l128 A a acc)))) -> eq lzero nat x (S (plus m (length l128 A acc)))) (rewrite-r lzero lzero nat (plus m (S (length l128 A acc))) (λ (X-- : nat) -> eq lzero nat (plus m (S (length l128 A acc))) X--) (refl lzero nat (plus m (S (length l128 A acc)))) (S (plus m (length l128 A acc))) (plus-n-Sm m (length l128 A acc))) (length l128 A (fst l128 l128 (list l128 A) (list l128 A) (split-rev l128 A tl (cons l128 A a acc) m))) (Hind tl (cons l128 A a acc) (le-S-S-to-le m (length l128 A tl) Hlen))) X-clearme) n

split-len : (l151 : Level) -> (A : Set l151) -> (n : nat) -> (l : list l151 A) -> (X-- : le n (length l151 A l)) -> eq lzero nat (length l151 A (fst l151 l151 (list l151 A) (list l151 A) (split l151 A l n))) n
split-len = λ (l151 : Level) -> λ (A : Set l151) -> λ (n : nat) -> λ (l : list l151 A) -> λ (Hlen : le n (length l151 A l)) -> eq-ind-r l151 lzero (Prod l151 l151 (list l151 A) (list l151 A)) (mk-Prod l151 l151 (list l151 A) (list l151 A) (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)) (snd l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n))) (λ (x : Prod l151 l151 (list l151 A) (list l151 A)) -> λ (X-- : eq l151 (Prod l151 l151 (list l151 A) (list l151 A)) x (mk-Prod l151 l151 (list l151 A) (list l151 A) (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)) (snd l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)))) -> eq lzero nat (length l151 A (fst l151 l151 (list l151 A) (list l151 A) (match-Prod l151 l151 (list l151 A) (list l151 A) l151 (λ (X-0 : Prod l151 l151 (list l151 A) (list l151 A)) -> Prod l151 l151 (list l151 A) (list l151 A)) (λ (l1 : list l151 A) -> λ (l2 : list l151 A) -> mk-Prod l151 l151 (list l151 A) (list l151 A) (rev-append l151 A l1 (nil l151 A)) l2) x))) n) (eq-ind-r lzero lzero nat (length l151 A (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n))) (λ (x : nat) -> λ (X-- : eq lzero nat x (length l151 A (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)))) -> eq lzero nat x n) (eq-ind-r lzero lzero nat (plus n (length l151 A (nil l151 A))) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus n (length l151 A (nil l151 A)))) -> eq lzero nat x n) (rewrite-l lzero lzero nat n (λ (X-- : nat) -> eq lzero nat X-- n) (refl lzero nat n) (plus n O) (plus-n-O n)) (length l151 A (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n))) (split-rev-len l151 A n l (nil l151 A) Hlen)) (length l151 A (reverse l151 A (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)))) (length-reverse l151 A (fst l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n)))) (split-rev l151 A l (nil l151 A) n) (eq-pair-fst-snd l151 l151 (list l151 A) (list l151 A) (split-rev l151 A l (nil l151 A) n))

split-rev-eq : (l236 : Level) -> (A : Set l236) -> (n : nat) -> (l : list l236 A) -> (acc : list l236 A) -> (X-- : le n (length l236 A l)) -> eq l236 (list l236 A) (append l236 A (reverse l236 A acc) l) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc n))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc n)))
split-rev-eq = λ (l236 : Level) -> λ (A : Set l236) -> λ (n : nat) -> nat-ind l236 (λ (X-x-365 : nat) -> (l : list l236 A) -> (acc : list l236 A) -> (X-- : le X-x-365 (length l236 A l)) -> eq l236 (list l236 A) (append l236 A (reverse l236 A acc) l) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc X-x-365))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc X-x-365)))) (λ (l : list l236 A) -> λ (acc : list l236 A) -> λ (auto : le O (length l236 A l)) -> refl l236 (list l236 A) (append l236 A (reverse l236 A acc) l)) (λ (m : nat) -> λ (Hind : (l : list l236 A) -> (acc : list l236 A) -> (X-- : le m (length l236 A l)) -> eq l236 (list l236 A) (append l236 A (reverse l236 A acc) l) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc m))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A l acc m)))) -> λ (X-clearme : list l236 A) -> match-list l236 A l236 (λ (X-- : list l236 A) -> (acc : list l236 A) -> (X--1 : le (S m) (length l236 A X--)) -> eq l236 (list l236 A) (append l236 A (reverse l236 A acc) X--) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A X-- acc (S m)))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A X-- acc (S m))))) (λ (acc : list l236 A) -> λ (False-ind-v : le (S m) O) -> refl l236 (list l236 A) (append l236 A (reverse l236 A acc) (nil l236 A))) (λ (a : A) -> λ (tl : list l236 A) -> λ (acc : list l236 A) -> λ (Hlen : le (S m) (length l236 A (cons l236 A a tl))) -> eq-ind-r l236 l236 (list l236 A) (append l236 A (append l236 A (reverse l236 A acc) (cons l236 A a (nil l236 A))) tl) (λ (x : list l236 A) -> λ (X-- : eq l236 (list l236 A) x (append l236 A (append l236 A (reverse l236 A acc) (cons l236 A a (nil l236 A))) tl)) -> eq l236 (list l236 A) x (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m)))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m))))) (eq-ind l236 l236 (list l236 A) (reverse l236 A (cons l236 A a (nil l236 A))) (λ (x-1 : list l236 A) -> λ (X-x-2 : eq l236 (list l236 A) (reverse l236 A (cons l236 A a (nil l236 A))) x-1) -> eq l236 (list l236 A) (append l236 A (append l236 A (reverse l236 A acc) x-1) tl) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m)))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m))))) (eq-ind l236 l236 (list l236 A) (reverse l236 A (append l236 A (cons l236 A a (nil l236 A)) acc)) (λ (x-1 : list l236 A) -> λ (X-x-2 : eq l236 (list l236 A) (reverse l236 A (append l236 A (cons l236 A a (nil l236 A)) acc)) x-1) -> eq l236 (list l236 A) (append l236 A x-1 tl) (append l236 A (reverse l236 A (fst l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m)))) (snd l236 l236 (list l236 A) (list l236 A) (split-rev l236 A (cons l236 A a tl) acc (S m))))) (Hind tl (append l236 A (cons l236 A a (nil l236 A)) acc) (le-S-S-to-le m (length l236 A tl) Hlen)) (append l236 A (reverse l236 A acc) (reverse l236 A (cons l236 A a (nil l236 A)))) (reverse-append l236 A (cons l236 A a (nil l236 A)) acc)) (cons l236 A a (nil l236 A)) (reverse-single l236 A a)) (append l236 A (reverse l236 A acc) (cons l236 A a tl)) (append-cons l236 A a (reverse l236 A acc) tl)) X-clearme) n

split-eq : (l203 : Level) -> (A : Set l203) -> (n : nat) -> (l : list l203 A) -> (X-- : le n (length l203 A l)) -> eq l203 (list l203 A) l (append l203 A (fst l203 l203 (list l203 A) (list l203 A) (split l203 A l n)) (snd l203 l203 (list l203 A) (list l203 A) (split l203 A l n)))
split-eq = λ (l203 : Level) -> λ (A : Set l203) -> λ (n : nat) -> λ (l : list l203 A) -> λ (Hlen : le n (length l203 A l)) -> eq-ind-r l203 l203 (list l203 A) (append l203 A (reverse l203 A (fst l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n))) (snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n))) (λ (x : list l203 A) -> λ (X-- : eq l203 (list l203 A) x (append l203 A (reverse l203 A (fst l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n))) (snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)))) -> eq l203 (list l203 A) x (append l203 A (fst l203 l203 (list l203 A) (list l203 A) (split l203 A l n)) (snd l203 l203 (list l203 A) (list l203 A) (split l203 A l n)))) (eq-ind-r l203 l203 (Prod l203 l203 (list l203 A) (list l203 A)) (mk-Prod l203 l203 (list l203 A) (list l203 A) (fst l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)) (snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n))) (λ (x : Prod l203 l203 (list l203 A) (list l203 A)) -> λ (X-- : eq l203 (Prod l203 l203 (list l203 A) (list l203 A)) x (mk-Prod l203 l203 (list l203 A) (list l203 A) (fst l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)) (snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)))) -> eq l203 (list l203 A) (append l203 A (rev-append l203 A (fst l203 l203 (list l203 A) (list l203 A) x) (nil l203 A)) (snd l203 l203 (list l203 A) (list l203 A) x)) (append l203 A (fst l203 l203 (list l203 A) (list l203 A) (match-Prod l203 l203 (list l203 A) (list l203 A) l203 (λ (X-0 : Prod l203 l203 (list l203 A) (list l203 A)) -> Prod l203 l203 (list l203 A) (list l203 A)) (λ (l1 : list l203 A) -> λ (l2 : list l203 A) -> mk-Prod l203 l203 (list l203 A) (list l203 A) (rev-append l203 A l1 (nil l203 A)) l2) x)) (snd l203 l203 (list l203 A) (list l203 A) (match-Prod l203 l203 (list l203 A) (list l203 A) l203 (λ (X-0 : Prod l203 l203 (list l203 A) (list l203 A)) -> Prod l203 l203 (list l203 A) (list l203 A)) (λ (l1 : list l203 A) -> λ (l2 : list l203 A) -> mk-Prod l203 l203 (list l203 A) (list l203 A) (rev-append l203 A l1 (nil l203 A)) l2) x)))) (refl l203 (list l203 A) (append l203 A (rev-append l203 A (fst l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)) (nil l203 A)) (snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n)))) (split-rev l203 A l (nil l203 A) n) (eq-pair-fst-snd l203 l203 (list l203 A) (list l203 A) (split-rev l203 A l (nil l203 A) n))) (append l203 A (reverse l203 A (nil l203 A)) l) (split-rev-eq l203 A n l (nil l203 A) Hlen)

split-exists : (l94 : Level) -> (A : Set l94) -> (n : nat) -> (l : list l94 A) -> (X-- : le n (length l94 A l)) -> ex l94 l94 (list l94 A) (λ (l1 : list l94 A) -> ex l94 l94 (list l94 A) (λ (l2 : list l94 A) -> And l94 lzero (eq l94 (list l94 A) l (append l94 A l1 l2)) (eq lzero nat (length l94 A l1) n)))
split-exists = λ (l94 : Level) -> λ (A : Set l94) -> λ (n : nat) -> λ (l : list l94 A) -> λ (Hlen : le n (length l94 A l)) -> ex-intro l94 l94 (list l94 A) (λ (l1 : list l94 A) -> ex l94 l94 (list l94 A) (λ (l2 : list l94 A) -> And l94 lzero (eq l94 (list l94 A) l (append l94 A l1 l2)) (eq lzero nat (length l94 A l1) n))) (fst l94 l94 (list l94 A) (list l94 A) (split l94 A l n)) (ex-intro l94 l94 (list l94 A) (λ (l2 : list l94 A) -> And l94 lzero (eq l94 (list l94 A) l (append l94 A (fst l94 l94 (list l94 A) (list l94 A) (split l94 A l n)) l2)) (eq lzero nat (length l94 A (fst l94 l94 (list l94 A) (list l94 A) (split l94 A l n))) n)) (snd l94 l94 (list l94 A) (list l94 A) (split l94 A l n)) (conj l94 lzero (eq l94 (list l94 A) l (append l94 A (fst l94 l94 (list l94 A) (list l94 A) (split l94 A l n)) (snd l94 l94 (list l94 A) (list l94 A) (split l94 A l n)))) (eq lzero nat (length l94 A (fst l94 l94 (list l94 A) (list l94 A) (split l94 A l n))) n) (split-eq l94 A n l Hlen) (split-len l94 A n l Hlen)))

flatten : (l6 : Level) -> (A : Set l6) -> (X-l : list l6 (list l6 A)) -> list l6 A
flatten = λ (l6 : Level) -> λ (A : Set l6) -> foldr l6 l6 (list l6 A) (list l6 A) (append l6 A) (nil l6 A)

let-clause-1227 : (l157 l143 : Level) -> (A : Set l157) -> (n : nat) -> (l : list l157 (list l157 A)) -> (l1 : list l157 A) -> (l2 : list l157 A) -> (a : list l157 A) -> (posn : le (S O) n) -> (Hlen : (x : list l157 A) -> (X-- : False l143) -> eq lzero nat (length l157 A x) n) -> (Ha : eq lzero nat (length l157 A a) n) -> (Hnil : eq l157 (list l157 A) (nil l157 A) (append l157 A l1 (append l157 A a l2))) -> eq lzero nat O n
let-clause-1227 = λ (l157 l143 : Level) -> λ (A : Set l157) -> λ (n : nat) -> λ (l : list l157 (list l157 A)) -> λ (l1 : list l157 A) -> λ (l2 : list l157 A) -> λ (a : list l157 A) -> λ (posn : le (S O) n) -> λ (Hlen : (x : list l157 A) -> (X-- : False l143) -> eq lzero nat (length l157 A x) n) -> λ (Ha : eq lzero nat (length l157 A a) n) -> λ (Hnil : eq l157 (list l157 A) (nil l157 A) (append l157 A l1 (append l157 A a l2))) -> rewrite-l lzero lzero nat (length l157 A a) (λ (X-- : nat) -> eq lzero nat X-- n) Ha O (sym-eq lzero nat O (length l157 A a) (le-n-O-to-eq (length l157 A a) (transitive-le (length l157 A a) (length l157 A (nil l157 A)) O (eq-ind-r l157 lzero (list l157 A) (append l157 A l1 (append l157 A a l2)) (λ (x : list l157 A) -> λ (X-- : eq l157 (list l157 A) x (append l157 A l1 (append l157 A a l2))) -> le (length l157 A a) (length l157 A x)) (eq-ind-r lzero lzero nat (plus (length l157 A l1) (length l157 A (append l157 A a l2))) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus (length l157 A l1) (length l157 A (append l157 A a l2)))) -> le (length l157 A a) x) (eq-ind-r lzero lzero nat (plus (length l157 A a) (length l157 A l2)) (λ (x : nat) -> λ (X-- : eq lzero nat x (plus (length l157 A a) (length l157 A l2))) -> le (length l157 A a) (plus (length l157 A l1) x)) (eq-coerc lzero (le n (plus n (length l157 A (append l157 A l1 l2)))) (le (length l157 A a) (plus (length l157 A l1) (plus (length l157 A a) (length l157 A l2)))) (le-plus-n-r (length l157 A (append l157 A l1 l2)) n) (rewrite-r lzero (lsuc lzero) nat n (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le n (plus n (length l157 A (append l157 A l1 l2)))) (le X-- (plus (length l157 A l1) (plus (length l157 A a) (length l157 A l2))))) (rewrite-r lzero (lsuc lzero) nat n (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le n (plus n (length l157 A (append l157 A l1 l2)))) (le n (plus (length l157 A l1) (plus X-- (length l157 A l2))))) (rewrite-r lzero (lsuc lzero) nat (plus n (plus (length l157 A l1) (length l157 A l2))) (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le n (plus n (length l157 A (append l157 A l1 l2)))) (le n X--)) (rewrite-l lzero (lsuc lzero) nat (length l157 A (append l157 A l1 l2)) (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le n (plus n (length l157 A (append l157 A l1 l2)))) (le n (plus n X--))) (refl (lsuc lzero) (Set (lzero)) (le n (plus n (length l157 A (append l157 A l1 l2))))) (plus (length l157 A l1) (length l157 A l2)) (length-append l157 A l1 l2)) (plus (length l157 A l1) (plus n (length l157 A l2))) (rewrite-l lzero lzero nat (plus (plus n (length l157 A l1)) (length l157 A l2)) (λ (X-- : nat) -> eq lzero nat (plus (length l157 A l1) (plus n (length l157 A l2))) X--) (assoc-plus1 (length l157 A l2) n (length l157 A l1)) (plus n (plus (length l157 A l1) (length l157 A l2))) (associative-plus n (length l157 A l1) (length l157 A l2)))) (length l157 A a) Ha) (length l157 A a) Ha)) (length l157 A (append l157 A a l2)) (length-append l157 A a l2)) (length l157 A (append l157 A l1 (append l157 A a l2))) (length-append l157 A l1 (append l157 A a l2))) (nil l157 A) Hnil) (le-n (length l157 A (nil l157 A))))))

let-clause-1222 : (l153 : Level) -> (A : Set l153) -> (n : nat) -> (l : list l153 (list l153 A)) -> (hd-v : list l153 A) -> (tl : list l153 (list l153 A)) -> (Hind : (l1 : list l153 A) -> (l2 : list l153 A) -> (a : list l153 A) -> (X-- : lt O n) -> (X--1 : (x : list l153 A) -> (X--1 : mem l153 (list l153 A) x tl) -> eq lzero nat (length l153 A x) n) -> (X--2 : eq lzero nat (length l153 A a) n) -> (X--3 : eq l153 (list l153 A) (flatten l153 A tl) (append l153 A l1 (append l153 A a l2))) -> (X--4 : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length l153 A l1) (times n q))) -> mem l153 (list l153 A) a tl) -> (l1 : list l153 A) -> (l2 : list l153 A) -> (a : list l153 A) -> (posn : lt O n) -> (Hlen : (x : list l153 A) -> (X-- : mem l153 (list l153 A) x (cons l153 (list l153 A) hd-v tl)) -> eq lzero nat (length l153 A x) n) -> (Ha : eq lzero nat (length l153 A a) n) -> (Hflat : eq l153 (list l153 A) (append l153 A hd-v (foldr l153 l153 (list l153 A) (list l153 A) (append l153 A) (nil l153 A) tl)) (append l153 A l1 (append l153 A a l2))) -> (X-clearme : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length l153 A l1) (times n q))) -> (q : nat) -> (q1 : nat) -> (Hl1 : eq lzero nat (length l153 A l1) (times n (S q1))) -> (l11 : list l153 A) -> (X-clearme0 : ex l153 l153 (list l153 A) (λ (l20 : list l153 A) -> And l153 lzero (eq l153 (list l153 A) l1 (append l153 A l11 l20)) (eq lzero nat (length l153 A l11) n))) -> (l12 : list l153 A) -> (X-clearme1 : And l153 lzero (eq l153 (list l153 A) l1 (append l153 A l11 l12)) (eq lzero nat (length l153 A l11) n)) -> (Heql1 : eq l153 (list l153 A) l1 (append l153 A l11 l12)) -> (Hlenl11 : eq lzero nat (length l153 A l11) n) -> eq lzero nat (length l153 A l1) (plus n (times n q1))
let-clause-1222 = λ (l153 : Level) -> λ (A : Set l153) -> λ (n : nat) -> λ (l : list l153 (list l153 A)) -> λ (hd-v : list l153 A) -> λ (tl : list l153 (list l153 A)) -> λ (Hind : (l1 : list l153 A) -> (l2 : list l153 A) -> (a : list l153 A) -> (X-- : lt O n) -> (X--1 : (x : list l153 A) -> (X--1 : mem l153 (list l153 A) x tl) -> eq lzero nat (length l153 A x) n) -> (X--2 : eq lzero nat (length l153 A a) n) -> (X--3 : eq l153 (list l153 A) (flatten l153 A tl) (append l153 A l1 (append l153 A a l2))) -> (X--4 : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length l153 A l1) (times n q))) -> mem l153 (list l153 A) a tl) -> λ (l1 : list l153 A) -> λ (l2 : list l153 A) -> λ (a : list l153 A) -> λ (posn : lt O n) -> λ (Hlen : (x : list l153 A) -> (X-- : mem l153 (list l153 A) x (cons l153 (list l153 A) hd-v tl)) -> eq lzero nat (length l153 A x) n) -> λ (Ha : eq lzero nat (length l153 A a) n) -> λ (Hflat : eq l153 (list l153 A) (append l153 A hd-v (foldr l153 l153 (list l153 A) (list l153 A) (append l153 A) (nil l153 A) tl)) (append l153 A l1 (append l153 A a l2))) -> λ (X-clearme : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length l153 A l1) (times n q))) -> λ (q : nat) -> λ (q1 : nat) -> λ (Hl1 : eq lzero nat (length l153 A l1) (times n (S q1))) -> λ (l11 : list l153 A) -> λ (X-clearme0 : ex l153 l153 (list l153 A) (λ (l20 : list l153 A) -> And l153 lzero (eq l153 (list l153 A) l1 (append l153 A l11 l20)) (eq lzero nat (length l153 A l11) n))) -> λ (l12 : list l153 A) -> λ (X-clearme1 : And l153 lzero (eq l153 (list l153 A) l1 (append l153 A l11 l12)) (eq lzero nat (length l153 A l11) n)) -> λ (Heql1 : eq l153 (list l153 A) l1 (append l153 A l11 l12)) -> λ (Hlenl11 : eq lzero nat (length l153 A l11) n) -> rewrite-r lzero lzero nat (times n (S q1)) (λ (X-- : nat) -> eq lzero nat (length l153 A l1) X--) Hl1 (plus n (times n q1)) (times-n-Sm n q1)

flatten-to-mem : (l263 : Level) -> (A : Set ((lsuc lzero) ⊔ (lsuc l263))) -> (n : nat) -> (l : list ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A)) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (a : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X-- : lt O n) -> (X--1 : (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X--1 : mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x l) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x) n) -> (X--2 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A a) n) -> (X--3 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (flatten ((lsuc lzero) ⊔ (lsuc l263)) A l) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> (X--4 : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a l
flatten-to-mem = λ (l263 : Level) -> λ (A : Set ((lsuc lzero) ⊔ (lsuc l263))) -> λ (n : nat) -> λ (l : list ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A)) -> list-ind ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (X-x-716 : list ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A)) -> (l1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (a : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X-- : lt O n) -> (X--1 : (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X--1 : mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x X-x-716) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x) n) -> (X--2 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A a) n) -> (X--3 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (flatten ((lsuc lzero) ⊔ (lsuc l263)) A X-x-716) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> (X--4 : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a X-x-716) (λ (l1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (a : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (posn : le (S O) n) -> λ (Hlen : (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X-- : False ((lsuc lzero) ⊔ (lsuc l263))) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x) n) -> λ (Ha : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A a) n) -> λ (Hnil : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> False-ind lzero ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-x-66 : False lzero) -> (X-- : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> False ((lsuc lzero) ⊔ (lsuc l263))) (absurd lzero (le (S n) O) (eq-coerc lzero (le (S O) n) (le (S n) O) posn (rewrite-r lzero (lsuc lzero) nat n (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le (S X--) n) (le (S n) O)) (rewrite-r lzero (lsuc lzero) nat n (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le (S n) n) (le (S n) X--)) (refl (lsuc lzero) (Set (lzero)) (le (S n) n)) O (let-clause-1227 ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) A n l l1 l2 a posn Hlen Ha Hnil)) O (let-clause-1227 ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) A n l l1 l2 a posn Hlen Ha Hnil))) (not-le-Sn-O n))) (λ (hd-v : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (tl : list ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A)) -> λ (Hind : (l1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (l2 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (a : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X-- : lt O n) -> (X--1 : (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X--1 : mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x tl) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x) n) -> (X--2 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A a) n) -> (X--3 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (flatten ((lsuc lzero) ⊔ (lsuc l263)) A tl) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> (X--4 : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a tl) -> λ (l1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (l2 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (a : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (posn : lt O n) -> λ (Hlen : (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> (X-- : mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x) n) -> λ (Ha : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A a) n) -> λ (Hflat : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A hd-v (foldr ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) tl)) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> λ (X-clearme : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> match-ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q)) ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-- : ex lzero lzero nat (λ (q : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n q))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (λ (q : nat) -> match-nat ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-- : nat) -> (X--1 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n X--)) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (eq-ind lzero ((lsuc lzero) ⊔ (lsuc l263)) nat O (λ (x-1 : nat) -> λ (X-x-2 : eq lzero nat O x-1) -> (X-- : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) x-1) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (λ (Hl1 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) O) -> or-introl ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a hd-v) (mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a tl) (rewrite-r ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v (λ (X-- : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) X-- hd-v) (refl ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v) a (eq-ind-r ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (nil ((lsuc lzero) ⊔ (lsuc l263)) A)) -> (X--1 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A hd-v (foldr ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) tl)) (append ((lsuc lzero) ⊔ (lsuc l263)) A x (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a hd-v) (λ (Hflat0 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A hd-v (foldr ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) tl)) (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)) -> sym-eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v a (append-l1-injective l263 A hd-v a (foldr ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) tl) l2 (eq-ind-r lzero lzero nat n (λ (x : nat) -> λ (X-- : eq lzero nat x n) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A hd-v) x) (eq-ind-r lzero lzero nat n (λ (x : nat) -> λ (X-- : eq lzero nat x n) -> eq lzero nat x n) (refl lzero nat n) (length ((lsuc lzero) ⊔ (lsuc l263)) A hd-v) (Hlen hd-v (or-introl ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v hd-v) (mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl) (refl ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v)))) (length ((lsuc lzero) ⊔ (lsuc l263)) A a) Ha) Hflat0)) l1 (lenght-to-nil ((lsuc lzero) ⊔ (lsuc l263)) A l1 Hl1) Hflat))) (times n O) (times-n-O n)) (λ (q1 : nat) -> λ (Hl1 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (times n (S q1))) -> match-ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l11 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l21 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l21)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n))) ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-- : ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l10 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l20 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l10 l20)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l10) n)))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (λ (l11 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-clearme0 : ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l20 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l20)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n))) -> match-ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l20 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l20)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n)) ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-- : ex ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (λ (l20 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l20)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n))) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (λ (l12 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-clearme1 : And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n)) -> match-And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n) ((lsuc lzero) ⊔ (lsuc l263)) (λ (X-- : And ((lsuc lzero) ⊔ (lsuc l263)) lzero (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) (eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n)) -> mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a (cons ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl)) (λ (Heql1 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) -> λ (Hlenl11 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) n) -> or-intror ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a hd-v) (mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) a tl) (Hind l12 l2 a posn (λ (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (memx : mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x tl) -> Hlen x (or-intror ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x hd-v) (mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x tl) memx)) Ha (append-l2-injective l263 A hd-v l11 (flatten ((lsuc lzero) ⊔ (lsuc l263)) A tl) (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)) (eq-ind-r lzero lzero nat n (λ (x : nat) -> λ (X-- : eq lzero nat x n) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A hd-v) x) (Hlen hd-v (or-introl ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v hd-v) (mem ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v tl) (refl ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) hd-v))) (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) Hlenl11) (eq-ind-r ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)) (λ (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (append ((lsuc lzero) ⊔ (lsuc l263)) A l1 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) -> eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)))) (eq-ind-r ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12) (λ (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) -> eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A x (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)) (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)))) (eq-ind-r ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) (λ (x : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-- : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)))) -> eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) x (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)))) (refl ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 (append ((lsuc lzero) ⊔ (lsuc l263)) A l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)))) (append ((lsuc lzero) ⊔ (lsuc l263)) A (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12) (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2)) (associative-append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12 (append ((lsuc lzero) ⊔ (lsuc l263)) A a l2))) l1 Heql1) (append ((lsuc lzero) ⊔ (lsuc l263)) A hd-v (foldr ((lsuc lzero) ⊔ (lsuc l263)) ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (list ((lsuc lzero) ⊔ (lsuc l263)) A) (append ((lsuc lzero) ⊔ (lsuc l263)) A) (nil ((lsuc lzero) ⊔ (lsuc l263)) A) tl)) Hflat)) (ex-intro lzero lzero nat (λ (q0 : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l12) (times n q0)) q1 (injective-plus-r n (length ((lsuc lzero) ⊔ (lsuc l263)) A l12) (times n q1) (eq-ind lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) (λ (x-1 : nat) -> λ (X-x-2 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) x-1) -> eq lzero nat (plus x-1 (length ((lsuc lzero) ⊔ (lsuc l263)) A l12)) (plus n (times n q1))) (eq-ind lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) (λ (x-1 : nat) -> λ (X-x-2 : eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) x-1) -> eq lzero nat x-1 (plus n (times n q1))) (eq-ind ((lsuc lzero) ⊔ (lsuc l263)) lzero (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 (λ (x-1 : list ((lsuc lzero) ⊔ (lsuc l263)) A) -> λ (X-x-2 : eq ((lsuc lzero) ⊔ (lsuc l263)) (list ((lsuc lzero) ⊔ (lsuc l263)) A) l1 x-1) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A x-1) (plus n (times n q1))) (eq-ind-r lzero lzero nat (times n (S q1)) (λ (x : nat) -> λ (X-- : eq lzero nat x (times n (S q1))) -> eq lzero nat x (plus n (times n q1))) (rewrite-l lzero lzero nat (plus n (times n q1)) (λ (X-- : nat) -> eq lzero nat X-- (plus n (times n q1))) (rewrite-l lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (λ (X-- : nat) -> eq lzero nat X-- (plus n (times n q1))) (rewrite-l lzero lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) X--) (refl lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1)) (plus n (times n q1)) (let-clause-1222 ((lsuc lzero) ⊔ (lsuc l263)) A n l hd-v tl Hind l1 l2 a posn Hlen Ha Hflat X-clearme q q1 Hl1 l11 X-clearme0 l12 X-clearme1 Heql1 Hlenl11)) (plus n (times n q1)) (let-clause-1222 ((lsuc lzero) ⊔ (lsuc l263)) A n l hd-v tl Hind l1 l2 a posn Hlen Ha Hflat X-clearme q q1 Hl1 l11 X-clearme0 l12 X-clearme1 Heql1 Hlenl11)) (times n (S q1)) (times-n-Sm n q1)) (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) Hl1) (append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12) Heql1) (plus (length ((lsuc lzero) ⊔ (lsuc l263)) A l11) (length ((lsuc lzero) ⊔ (lsuc l263)) A l12)) (length-append ((lsuc lzero) ⊔ (lsuc l263)) A l11 l12)) n Hlenl11))))) X-clearme1) X-clearme0) (split-exists ((lsuc lzero) ⊔ (lsuc l263)) A n l1 (eq-coerc lzero (le (minus (plus n (times n q1)) (plus O (times n q1))) (plus n (times n q1))) (le n (length ((lsuc lzero) ⊔ (lsuc l263)) A l1)) (minus-le (plus n (times n q1)) (plus O (times n q1))) (rewrite-r lzero (lsuc lzero) nat (minus n O) (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le X-- (plus n (times n q1))) (le n (length ((lsuc lzero) ⊔ (lsuc l263)) A l1))) (rewrite-l lzero (lsuc lzero) nat n (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le X-- (plus n (times n q1))) (le n (length ((lsuc lzero) ⊔ (lsuc l263)) A l1))) (rewrite-l lzero (lsuc lzero) nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) (λ (X-- : nat) -> eq (lsuc lzero) (Set (lzero)) (le n X--) (le n (length ((lsuc lzero) ⊔ (lsuc l263)) A l1))) (refl (lsuc lzero) (Set (lzero)) (le n (length ((lsuc lzero) ⊔ (lsuc l263)) A l1))) (plus n (times n q1)) (rewrite-r lzero lzero nat (times n (S q1)) (λ (X-- : nat) -> eq lzero nat (length ((lsuc lzero) ⊔ (lsuc l263)) A l1) X--) Hl1 (plus n (times n q1)) (times-n-Sm n q1))) (minus n O) (minus-n-O n)) (minus (plus n (times n q1)) (plus O (times n q1))) (minus-plus-plus-l n O (times n q1)))))) q) X-clearme) l


nth : (l3-v : Level) -> (X---v : nat) -> (H-v : Set l3-v) -> (X--1-v : list l3-v H-v) -> (X--2-v : H-v) -> H-v
nth l10-v O A-v l-v d-v = hd l10-v A-v l-v d-v
nth l10-v (S m-v) A-v l-v d-v = nth l10-v m-v A-v (tail l10-v A-v l-v) d-v


nth-nil : (l20 : Level) -> (A : Set l20) -> (a : A) -> (i : nat) -> eq l20 A (nth l20 i A (nil l20 A) a) a
nth-nil = λ (l20 : Level) -> λ (A : Set l20) -> λ (a : A) -> λ (i : nat) -> nat-ind l20 (λ (X-x-365 : nat) -> eq l20 A (nth l20 X-x-365 A (nil l20 A) a) a) (refl l20 A a) (λ (x-366 : nat) -> λ (X-x-368 : eq l20 A (nth l20 x-366 A (nil l20 A) a) a) -> rewrite-r l20 l20 A a (λ (X-- : A) -> eq l20 A X-- a) (refl l20 A a) (nth l20 x-366 A (nil l20 A) a) X-x-368) i


nth-opt : (l3-v : Level) -> (A-v : Set l3-v) -> (X-n-v : nat) -> (X-l-v : list l3-v A-v) -> option l3-v A-v
nth-opt l4-v A-v X-n-v nil' = None l4-v A-v
nth-opt l4-v A-v X-n-v (cons' h-v t-v) = match-nat l4-v (λ (X---v : nat) -> option l4-v A-v) (Some l4-v A-v h-v) (λ (m-v : nat) -> nth-opt l4-v A-v m-v t-v) X-n-v

All : (l4-v l3-v : Level) -> (A-v : Set l4-v) -> (X-P-v : (X---v : A-v) -> Set l3-v) -> (X-l-v : list l4-v A-v) -> Set (lzero ⊔ (l4-v ⊔ l3-v))
All l5-v l10-v A-v X-P-v nil' = True (l10-v ⊔ l5-v)
All l5-v l10-v A-v X-P-v (cons' h-v t-v) = And l10-v (l10-v ⊔ l5-v) (X-P-v h-v) (All l5-v l10-v A-v X-P-v t-v)


All-mp : (l61 l59 l57 : Level) -> (A : Set l61) -> (P : (X-- : A) -> Set l59) -> (Q : (X-- : A) -> Set l57) -> (X-- : (a : A) -> (X-- : P a) -> Q a) -> (l : list l61 A) -> (X--1 : All l61 l59 A P l) -> All l61 l57 A Q l
All-mp = λ (l61 l59 l57 : Level) -> λ (A : Set l61) -> λ (P : (X-- : A) -> Set l59) -> λ (Q : (X-- : A) -> Set l57) -> λ (H : (a : A) -> (X-- : P a) -> Q a) -> λ (l : list l61 A) -> list-ind l61 ((l61 ⊔ l59) ⊔ l57) A (λ (X-x-716 : list l61 A) -> (X-- : All l61 l59 A P X-x-716) -> All l61 l57 A Q X-x-716) (λ (auto : True (l61 ⊔ l59)) -> I (l61 ⊔ l57)) (λ (h : A) -> λ (t : list l61 A) -> λ (IH : (X-- : All l61 l59 A P t) -> All l61 l57 A Q t) -> λ (X-clearme : And l59 (l61 ⊔ l59) (P h) (All l61 l59 A P t)) -> match-And l59 (l61 ⊔ l59) (P h) (All l61 l59 A P t) (l61 ⊔ l57) (λ (X-- : And l59 (l61 ⊔ l59) (P h) (All l61 l59 A P t)) -> And l57 (l61 ⊔ l57) (Q h) (All l61 l57 A Q t)) (λ (auto : P h) -> λ (auto' : All l61 l59 A P t) -> conj l57 (l61 ⊔ l57) (Q h) (All l61 l57 A Q t) (H h auto) (IH auto')) X-clearme) l

All-nth : (l242 : Level) -> (A : Set (lzero)) -> (P : (X-- : A) -> Set l242) -> (n : nat) -> (l : list lzero A) -> (X-- : All lzero l242 A P l) -> (a : A) -> (X--1 : eq lzero (option lzero A) (nth-opt lzero A n l) (Some lzero A a)) -> P a
All-nth = λ (l242 : Level) -> λ (A : Set (lzero)) -> λ (P : (X-- : A) -> Set l242) -> λ (n : nat) -> nat-ind l242 (λ (X-x-365 : nat) -> (l : list lzero A) -> (X-- : All lzero l242 A P l) -> (a : A) -> (X--1 : eq lzero (option lzero A) (nth-opt lzero A X-x-365 l) (Some lzero A a)) -> P a) (λ (X-clearme : list lzero A) -> match-list lzero A l242 (λ (X-- : list lzero A) -> (X--1 : All lzero l242 A P X--) -> (a : A) -> (X--2 : eq lzero (option lzero A) (nth-opt lzero A O X--) (Some lzero A a)) -> P a) (λ (X-- : All lzero l242 A P (nil lzero A)) -> λ (a : A) -> λ (E : eq lzero (option lzero A) (nth-opt lzero A O (nil lzero A)) (Some lzero A a)) -> option-discr l242 A (None lzero A) (Some lzero A a) E (P a)) (λ (hd-v : A) -> λ (tl : list lzero A) -> λ (X-clearme0 : All lzero l242 A P (cons lzero A hd-v tl)) -> match-And l242 l242 (P hd-v) (All lzero l242 A P tl) l242 (λ (X-- : And l242 l242 (P hd-v) (All lzero l242 A P tl)) -> (a : A) -> (X--1 : eq lzero (option lzero A) (nth-opt lzero A O (cons lzero A hd-v tl)) (Some lzero A a)) -> P a) (λ (H : P hd-v) -> λ (X-- : All lzero l242 A P tl) -> λ (a : A) -> λ (E : eq lzero (option lzero A) (nth-opt lzero A O (cons lzero A hd-v tl)) (Some lzero A a)) -> option-discr l242 A (Some lzero A hd-v) (Some lzero A a) E (P a) (λ (e0 : eq lzero A (R0 lzero A hd-v) a) -> eq-ind-r lzero l242 A a (λ (x : A) -> λ (X-0 : eq lzero A x a) -> (X--1 : P x) -> (X--2 : eq lzero (option lzero A) (Some lzero A x) (Some lzero A a)) -> P a) (λ (H0 : P a) -> λ (E0 : eq lzero (option lzero A) (Some lzero A a) (Some lzero A a)) -> streicherK lzero l242 (option lzero A) (Some lzero A a) (λ (X--1 : eq lzero (option lzero A) (Some lzero A a) (Some lzero A a)) -> P a) H0 E0) hd-v e0 H E)) X-clearme0) X-clearme) (λ (m : nat) -> λ (IH : (l : list lzero A) -> (X-- : All lzero l242 A P l) -> (a : A) -> (X--1 : eq lzero (option lzero A) (nth-opt lzero A m l) (Some lzero A a)) -> P a) -> λ (X-clearme : list lzero A) -> match-list lzero A l242 (λ (X-- : list lzero A) -> (X--1 : All lzero l242 A P X--) -> (a : A) -> (X--2 : eq lzero (option lzero A) (nth-opt lzero A (S m) X--) (Some lzero A a)) -> P a) (λ (X-- : All lzero l242 A P (nil lzero A)) -> λ (a : A) -> λ (E : eq lzero (option lzero A) (nth-opt lzero A (S m) (nil lzero A)) (Some lzero A a)) -> option-discr l242 A (None lzero A) (Some lzero A a) E (P a)) (λ (hd-v : A) -> λ (tl : list lzero A) -> λ (X-clearme0 : All lzero l242 A P (cons lzero A hd-v tl)) -> match-And l242 l242 (P hd-v) (All lzero l242 A P tl) l242 (λ (X-- : And l242 l242 (P hd-v) (All lzero l242 A P tl)) -> (a : A) -> (X--1 : eq lzero (option lzero A) (nth-opt lzero A (S m) (cons lzero A hd-v tl)) (Some lzero A a)) -> P a) (λ (X-- : P hd-v) -> IH tl) X-clearme0) X-clearme) n

All-append : (l93 l91 : Level) -> (A : Set l93) -> (P : (X-- : A) -> Set l91) -> (l1 : list l93 A) -> (l2 : list l93 A) -> (X-- : All l93 l91 A P l1) -> (X--1 : All l93 l91 A P l2) -> All l93 l91 A P (append l93 A l1 l2)
All-append = λ (l93 l91 : Level) -> λ (A : Set l93) -> λ (P : (X-- : A) -> Set l91) -> λ (l1 : list l93 A) -> list-ind l93 (l93 ⊔ l91) A (λ (X-x-716 : list l93 A) -> (l2 : list l93 A) -> (X-- : All l93 l91 A P X-x-716) -> (X--1 : All l93 l91 A P l2) -> All l93 l91 A P (append l93 A X-x-716 l2)) (λ (l2 : list l93 A) -> λ (auto : All l93 l91 A P (nil l93 A)) -> λ (auto' : All l93 l91 A P l2) -> auto') (λ (a : A) -> λ (l10 : list l93 A) -> λ (IHl1 : (l2 : list l93 A) -> (X-- : All l93 l91 A P l10) -> (X--1 : All l93 l91 A P l2) -> All l93 l91 A P (append l93 A l10 l2)) -> λ (l2 : list l93 A) -> λ (X-clearme : All l93 l91 A P (cons l93 A a l10)) -> match-And l91 (l93 ⊔ l91) (P a) (All l93 l91 A P l10) (l93 ⊔ l91) (λ (X-- : And l91 (l93 ⊔ l91) (P a) (All l93 l91 A P l10)) -> (X--1 : All l93 l91 A P l2) -> All l93 l91 A P (append l93 A (cons l93 A a l10) l2)) (λ (auto : P a) -> λ (auto' : All l93 l91 A P l10) -> λ (auto'' : All l93 l91 A P l2) -> conj l91 (l93 ⊔ l91) (P a) (All l93 l91 A P (append l93 A l10 l2)) auto (IHl1 l2 auto' auto'')) X-clearme) l1

All-inv-append : (l128 l126 : Level) -> (A : Set l128) -> (P : (X-- : A) -> Set l126) -> (l1 : list l128 A) -> (l2 : list l128 A) -> (X-- : All l128 l126 A P (append l128 A l1 l2)) -> And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P l1) (All l128 l126 A P l2)
All-inv-append = λ (l128 l126 : Level) -> λ (A : Set l128) -> λ (P : (X-- : A) -> Set l126) -> λ (l1 : list l128 A) -> list-ind l128 (l128 ⊔ l126) A (λ (X-x-716 : list l128 A) -> (l2 : list l128 A) -> (X-- : All l128 l126 A P (append l128 A X-x-716 l2)) -> And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P X-x-716) (All l128 l126 A P l2)) (λ (l2 : list l128 A) -> λ (auto : All l128 l126 A P (append l128 A (nil l128 A) l2)) -> conj (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P (nil l128 A)) (All l128 l126 A P l2) (I (l128 ⊔ l126)) auto) (λ (a : A) -> λ (l10 : list l128 A) -> λ (IHl1 : (l2 : list l128 A) -> (X-- : All l128 l126 A P (append l128 A l10 l2)) -> And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P l10) (All l128 l126 A P l2)) -> λ (l2 : list l128 A) -> λ (X-clearme : All l128 l126 A P (append l128 A (cons l128 A a l10) l2)) -> match-And l126 (l128 ⊔ l126) (P a) (All l128 l126 A P (append l128 A l10 l2)) (l128 ⊔ l126) (λ (X-- : And l126 (l128 ⊔ l126) (P a) (All l128 l126 A P (append l128 A l10 l2))) -> And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P (cons l128 A a l10)) (All l128 l126 A P l2)) (λ (Ha : P a) -> λ (Hl12 : All l128 l126 A P (append l128 A l10 l2)) -> And-ind (l128 ⊔ l126) (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P l10) (All l128 l126 A P l2) (λ (X-x-118 : And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P l10) (All l128 l126 A P l2)) -> And (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P (cons l128 A a l10)) (All l128 l126 A P l2)) (λ (X-x-120 : All l128 l126 A P l10) -> λ (X-x-119 : All l128 l126 A P l2) -> conj (l128 ⊔ l126) (l128 ⊔ l126) (All l128 l126 A P (cons l128 A a l10)) (All l128 l126 A P l2) (conj l126 (l128 ⊔ l126) (P a) (All l128 l126 A P l10) Ha X-x-120) X-x-119) (IHl1 l2 Hl12)) X-clearme) l1

Allr : (l4-v l3-v : Level) -> (A-v : Set l4-v) -> (X-R-v : relation l4-v l3-v A-v) -> (X-l-v : list l4-v A-v) -> Set (lzero ⊔ (l4-v ⊔ l3-v))
Allr l5-v l15-v A-v X-R-v nil' = True (l15-v ⊔ l5-v)
Allr l5-v l15-v A-v X-R-v (cons' a1-v l0-v) = match-list l5-v A-v ((lsuc lzero) ⊔ ((lsuc l15-v) ⊔ (lsuc l5-v))) (λ (X---v : list l5-v A-v) -> Set (l15-v ⊔ l5-v)) (True (l15-v ⊔ l5-v)) (λ (a2-v : A-v) -> λ (X---v : list l5-v A-v) -> And l15-v (l15-v ⊔ l5-v) (X-R-v a1-v a2-v) (Allr l5-v l15-v A-v X-R-v l0-v)) l0-v


Allr-fwd-append-sn : (l157 l154 : Level) -> (A : Set l157) -> (R : relation l157 l154 A) -> (l1 : list l157 A) -> (l2 : list l157 A) -> (X-- : Allr l157 l154 A R (append l157 A l1 l2)) -> Allr l157 l154 A R l1
Allr-fwd-append-sn = λ (l157 l154 : Level) -> λ (A : Set l157) -> λ (R : relation l157 l154 A) -> λ (l1 : list l157 A) -> list-ind l157 (l157 ⊔ l154) A (λ (X-x-716 : list l157 A) -> (l2 : list l157 A) -> (X-- : Allr l157 l154 A R (append l157 A X-x-716 l2)) -> Allr l157 l154 A R X-x-716) (λ (l2 : list l157 A) -> λ (auto : Allr l157 l154 A R (append l157 A (nil l157 A) l2)) -> I (l157 ⊔ l154)) (λ (a1 : A) -> λ (X-clearme : list l157 A) -> match-list l157 A (l157 ⊔ l154) (λ (X-- : list l157 A) -> (X-x-720 : (l2 : list l157 A) -> (X--1 : Allr l157 l154 A R (append l157 A X-- l2)) -> Allr l157 l154 A R X--) -> (l2 : list l157 A) -> (X--1 : Allr l157 l154 A R (append l157 A (cons l157 A a1 X--) l2)) -> Allr l157 l154 A R (cons l157 A a1 X--)) (λ (X-x-720 : (l2 : list l157 A) -> (X-- : Allr l157 l154 A R (append l157 A (nil l157 A) l2)) -> Allr l157 l154 A R (nil l157 A)) -> λ (l2 : list l157 A) -> λ (auto : Allr l157 l154 A R (append l157 A (cons l157 A a1 (nil l157 A)) l2)) -> I (l157 ⊔ l154)) (λ (a2 : A) -> λ (l10 : list l157 A) -> λ (IHl1 : (l2 : list l157 A) -> (X-- : Allr l157 l154 A R (append l157 A (cons l157 A a2 l10) l2)) -> Allr l157 l154 A R (cons l157 A a2 l10)) -> λ (l2 : list l157 A) -> λ (X-clearme0 : Allr l157 l154 A R (append l157 A (cons l157 A a1 (cons l157 A a2 l10)) l2)) -> match-And l154 (l157 ⊔ l154) (R a1 a2) (Allr l157 l154 A R (append l157 A (cons l157 A a2 l10) l2)) (l157 ⊔ l154) (λ (X-- : And l154 (l157 ⊔ l154) (R a1 a2) (Allr l157 l154 A R (append l157 A (cons l157 A a2 l10) l2))) -> Allr l157 l154 A R (cons l157 A a1 (cons l157 A a2 l10))) (λ (auto : R a1 a2) -> λ (auto' : Allr l157 l154 A R (append l157 A (cons l157 A a2 l10) l2)) -> conj l154 (l157 ⊔ l154) (R a1 a2) (Allr l157 l154 A R (cons l157 A a2 l10)) auto (IHl1 l2 auto')) X-clearme0) X-clearme) l1

Allr-fwd-cons : (l54 l51 : Level) -> (A : Set l54) -> (R : relation l54 l51 A) -> (a : A) -> (l : list l54 A) -> (X-- : Allr l54 l51 A R (cons l54 A a l)) -> Allr l54 l51 A R l
Allr-fwd-cons = λ (l54 l51 : Level) -> λ (A : Set l54) -> λ (R : relation l54 l51 A) -> λ (a : A) -> λ (X-clearme : list l54 A) -> match-list l54 A (l54 ⊔ l51) (λ (X-- : list l54 A) -> (X--1 : Allr l54 l51 A R (cons l54 A a X--)) -> Allr l54 l51 A R X--) (λ (auto : Allr l54 l51 A R (cons l54 A a (nil l54 A))) -> auto) (λ (a0 : A) -> λ (l : list l54 A) -> λ (X-clearme0 : Allr l54 l51 A R (cons l54 A a (cons l54 A a0 l))) -> match-And l51 (l54 ⊔ l51) (R a a0) (Allr l54 l51 A R (cons l54 A a0 l)) (l54 ⊔ l51) (λ (X-- : And l51 (l54 ⊔ l51) (R a a0) (Allr l54 l51 A R (cons l54 A a0 l))) -> Allr l54 l51 A R (cons l54 A a0 l)) (λ (auto : R a a0) -> λ (auto' : Allr l54 l51 A R (cons l54 A a0 l)) -> auto') X-clearme0) X-clearme

Allr-fwd-append-dx : (l54 l51 : Level) -> (A : Set l54) -> (R : relation l54 l51 A) -> (l1 : list l54 A) -> (l2 : list l54 A) -> (X-- : Allr l54 l51 A R (append l54 A l1 l2)) -> Allr l54 l51 A R l2
Allr-fwd-append-dx = λ (l54 l51 : Level) -> λ (A : Set l54) -> λ (R : relation l54 l51 A) -> λ (l1 : list l54 A) -> list-ind l54 (l54 ⊔ l51) A (λ (X-x-716 : list l54 A) -> (l2 : list l54 A) -> (X-- : Allr l54 l51 A R (append l54 A X-x-716 l2)) -> Allr l54 l51 A R l2) (λ (l2 : list l54 A) -> λ (auto : Allr l54 l51 A R (append l54 A (nil l54 A) l2)) -> auto) (λ (a1 : A) -> λ (l10 : list l54 A) -> λ (IHl1 : (l2 : list l54 A) -> (X-- : Allr l54 l51 A R (append l54 A l10 l2)) -> Allr l54 l51 A R l2) -> λ (l2 : list l54 A) -> λ (H : Allr l54 l51 A R (append l54 A (cons l54 A a1 l10) l2)) -> IHl1 l2 (Allr-fwd-cons l54 l51 A R a1 (append l54 A l10 l2) H)) l1



Exists : (l4-v l3-v : Level) -> (A-v : Set l4-v) -> (X-P-v : (X---v : A-v) -> Set l3-v) -> (X-l-v : list l4-v A-v) -> Set (lzero ⊔ (l4-v ⊔ l3-v))
Exists l5-v l10-v A-v X-P-v nil' = False (l10-v ⊔ l5-v)
Exists l5-v l10-v A-v X-P-v (cons' h-v t-v) = Or l10-v (l10-v ⊔ l5-v) (X-P-v h-v) (Exists l5-v l10-v A-v X-P-v t-v)

Exists-append : (l141 l139 : Level) -> (A : Set l141) -> (P : (X-- : A) -> Set l139) -> (l1 : list l141 A) -> (l2 : list l141 A) -> (X-- : Exists l141 l139 A P (append l141 A l1 l2)) -> Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P l1) (Exists l141 l139 A P l2)
Exists-append = λ (l141 l139 : Level) -> λ (A : Set l141) -> λ (P : (X-- : A) -> Set l139) -> λ (l1 : list l141 A) -> list-ind l141 (l141 ⊔ l139) A (λ (X-x-716 : list l141 A) -> (l2 : list l141 A) -> (X-- : Exists l141 l139 A P (append l141 A X-x-716 l2)) -> Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P X-x-716) (Exists l141 l139 A P l2)) (λ (l2 : list l141 A) -> λ (auto : Exists l141 l139 A P l2) -> or-intror (l141 ⊔ l139) (l141 ⊔ l139) (False (l141 ⊔ l139)) (Exists l141 l139 A P l2) auto) (λ (h : A) -> λ (t : list l141 A) -> λ (IH : (l2 : list l141 A) -> (X-- : Exists l141 l139 A P (append l141 A t l2)) -> Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P t) (Exists l141 l139 A P l2)) -> λ (l2 : list l141 A) -> λ (X-clearme : Exists l141 l139 A P (append l141 A (cons l141 A h t) l2)) -> match-Or l139 (l141 ⊔ l139) (P h) (Exists l141 l139 A P (append l141 A t l2)) (l141 ⊔ l139) (λ (X-- : Or l139 (l141 ⊔ l139) (P h) (Exists l141 l139 A P (append l141 A t l2))) -> Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P (cons l141 A h t)) (Exists l141 l139 A P l2)) (λ (H : P h) -> or-introl (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P (cons l141 A h t)) (Exists l141 l139 A P l2) (or-introl l139 (l141 ⊔ l139) (P h) (Exists l141 l139 A P t) H)) (λ (H : Exists l141 l139 A P (append l141 A t l2)) -> match-Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P t) (Exists l141 l139 A P l2) (l141 ⊔ l139) (λ (X-- : Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P t) (Exists l141 l139 A P l2)) -> Or (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P (cons l141 A h t)) (Exists l141 l139 A P l2)) (λ (auto : Exists l141 l139 A P t) -> or-introl (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P (cons l141 A h t)) (Exists l141 l139 A P l2) (or-intror l139 (l141 ⊔ l139) (P h) (Exists l141 l139 A P t) auto)) (λ (auto : Exists l141 l139 A P l2) -> or-intror (l141 ⊔ l139) (l141 ⊔ l139) (Exists l141 l139 A P (cons l141 A h t)) (Exists l141 l139 A P l2) auto) (IH l2 H)) X-clearme) l1

Exists-append-l : (l74 l72 : Level) -> (A : Set l74) -> (P : (X-- : A) -> Set l72) -> (l1 : list l74 A) -> (l2 : list l74 A) -> (X-- : Exists l74 l72 A P l1) -> Exists l74 l72 A P (append l74 A l1 l2)
Exists-append-l = λ (l74 l72 : Level) -> λ (A : Set l74) -> λ (P : (X-- : A) -> Set l72) -> λ (l1 : list l74 A) -> λ (l2 : list l74 A) -> list-ind l74 (l74 ⊔ l72) A (λ (X-x-716 : list l74 A) -> (X-- : Exists l74 l72 A P X-x-716) -> Exists l74 l72 A P (append l74 A X-x-716 l2)) (λ (X-clearme : Exists l74 l72 A P (nil l74 A)) -> match-False (l74 ⊔ l72) (l74 ⊔ l72) (λ (X-- : False (l74 ⊔ l72)) -> Exists l74 l72 A P (append l74 A (nil l74 A) l2)) X-clearme) (λ (h : A) -> λ (t : list l74 A) -> λ (IH : (X-- : Exists l74 l72 A P t) -> Exists l74 l72 A P (append l74 A t l2)) -> λ (X-clearme : Exists l74 l72 A P (cons l74 A h t)) -> match-Or l72 (l74 ⊔ l72) (P h) (Exists l74 l72 A P t) (l74 ⊔ l72) (λ (X-- : Or l72 (l74 ⊔ l72) (P h) (Exists l74 l72 A P t)) -> Exists l74 l72 A P (append l74 A (cons l74 A h t) l2)) (λ (H : P h) -> or-introl l72 (l74 ⊔ l72) (P h) (Exists l74 l72 A P (append l74 A t l2)) H) (λ (H : Exists l74 l72 A P t) -> or-intror l72 (l74 ⊔ l72) (P h) (Exists l74 l72 A P (append l74 A t l2)) (IH H)) X-clearme) l1

Exists-append-r : (l41 l39 : Level) -> (A : Set l41) -> (P : (X-- : A) -> Set l39) -> (l1 : list l41 A) -> (l2 : list l41 A) -> (X-- : Exists l41 l39 A P l2) -> Exists l41 l39 A P (append l41 A l1 l2)
Exists-append-r = λ (l41 l39 : Level) -> λ (A : Set l41) -> λ (P : (X-- : A) -> Set l39) -> λ (l1 : list l41 A) -> λ (l2 : list l41 A) -> list-ind l41 (l41 ⊔ l39) A (λ (X-x-716 : list l41 A) -> (X-- : Exists l41 l39 A P l2) -> Exists l41 l39 A P (append l41 A X-x-716 l2)) (λ (H : Exists l41 l39 A P l2) -> H) (λ (h : A) -> λ (t : list l41 A) -> λ (IH : (X-- : Exists l41 l39 A P l2) -> Exists l41 l39 A P (append l41 A t l2)) -> λ (H : Exists l41 l39 A P l2) -> or-intror l39 (l41 ⊔ l39) (P h) (Exists l41 l39 A P (append l41 A t l2)) (IH H)) l1

Exists-add : (l84 l82 : Level) -> (A : Set l84) -> (P : (X-- : A) -> Set l82) -> (l1 : list l84 A) -> (x : A) -> (l2 : list l84 A) -> (X-- : Exists l84 l82 A P (append l84 A l1 l2)) -> Exists l84 l82 A P (append l84 A l1 (cons l84 A x l2))
Exists-add = λ (l84 l82 : Level) -> λ (A : Set l84) -> λ (P : (X-- : A) -> Set l82) -> λ (l1 : list l84 A) -> λ (x : A) -> λ (l2 : list l84 A) -> list-ind l84 (l84 ⊔ l82) A (λ (X-x-716 : list l84 A) -> (X-- : Exists l84 l82 A P (append l84 A X-x-716 l2)) -> Exists l84 l82 A P (append l84 A X-x-716 (cons l84 A x l2))) (λ (H : Exists l84 l82 A P l2) -> or-intror l82 (l84 ⊔ l82) (P x) (Exists l84 l82 A P l2) H) (λ (h : A) -> λ (t : list l84 A) -> λ (IH : (X-- : Exists l84 l82 A P (append l84 A t l2)) -> Exists l84 l82 A P (append l84 A t (cons l84 A x l2))) -> λ (X-clearme : Or l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t l2))) -> match-Or l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t l2)) (l84 ⊔ l82) (λ (X-- : Or l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t l2))) -> Or l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t (cons l84 A x l2)))) (λ (H : P h) -> or-introl l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t (cons l84 A x l2))) H) (λ (H : Exists l84 l82 A P (append l84 A t l2)) -> or-intror l82 (l84 ⊔ l82) (P h) (Exists l84 l82 A P (append l84 A t (cons l84 A x l2))) (IH H)) X-clearme) l1

Exists-mid : (l34 l32 : Level) -> (A : Set l34) -> (P : (X-- : A) -> Set l32) -> (l1 : list l34 A) -> (x : A) -> (l2 : list l34 A) -> (X-- : P x) -> Exists l34 l32 A P (append l34 A l1 (cons l34 A x l2))
Exists-mid = λ (l34 l32 : Level) -> λ (A : Set l34) -> λ (P : (X-- : A) -> Set l32) -> λ (l1 : list l34 A) -> λ (x : A) -> λ (l2 : list l34 A) -> λ (H : P x) -> list-ind l34 (l34 ⊔ l32) A (λ (X-x-716 : list l34 A) -> Exists l34 l32 A P (append l34 A X-x-716 (cons l34 A x l2))) (or-introl l32 (l34 ⊔ l32) (P x) (Exists l34 l32 A P l2) H) (λ (h : A) -> λ (t : list l34 A) -> λ (IH : Exists l34 l32 A P (append l34 A t (cons l34 A x l2))) -> or-intror l32 (l34 ⊔ l32) (P h) (Exists l34 l32 A P (append l34 A t (cons l34 A x l2))) IH) l1

Exists-map : (l107 l103 : Level) -> (A : Set (lzero ⊔ (l107 ⊔ l103))) -> (B-v : Set l107) -> (P : (X-- : A) -> Set (lzero ⊔ (l107 ⊔ l103))) -> (Q : (X-- : B-v) -> Set l103) -> (f : (X-- : A) -> B-v) -> (l : list (l107 ⊔ l103) A) -> (X-- : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P l) -> (X--1 : (a : A) -> (X--1 : P a) -> Q (f a)) -> Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f l)
Exists-map = λ (l107 l103 : Level) -> λ (A : Set (lzero ⊔ (l107 ⊔ l103))) -> λ (B-v : Set l107) -> λ (P : (X-- : A) -> Set (lzero ⊔ (l107 ⊔ l103))) -> λ (Q : (X-- : B-v) -> Set l103) -> λ (f : (X-- : A) -> B-v) -> λ (l : list (l107 ⊔ l103) A) -> list-ind (l107 ⊔ l103) (l107 ⊔ l103) A (λ (X-x-716 : list (l107 ⊔ l103) A) -> (X-- : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P X-x-716) -> (X--1 : (a : A) -> (X--1 : P a) -> Q (f a)) -> Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f X-x-716)) (λ (auto : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P (nil (l107 ⊔ l103) A)) -> λ (auto' : (a : A) -> (X-- : P a) -> Q (f a)) -> auto) (λ (h : A) -> λ (t : list (l107 ⊔ l103) A) -> λ (IH : (X-- : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P t) -> (X--1 : (a : A) -> (X--1 : P a) -> Q (f a)) -> Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f t)) -> λ (X-clearme : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P (cons (l107 ⊔ l103) A h t)) -> match-Or (l107 ⊔ l103) (l107 ⊔ l103) (P h) (Exists (l107 ⊔ l103) (l107 ⊔ l103) A P t) (l107 ⊔ l103) (λ (X-- : Or (l107 ⊔ l103) (l107 ⊔ l103) (P h) (Exists (l107 ⊔ l103) (l107 ⊔ l103) A P t)) -> (X--1 : (a : A) -> (X--1 : P a) -> Q (f a)) -> Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f (cons (l107 ⊔ l103) A h t))) (λ (H : P h) -> λ (F : (a : A) -> (X-- : P a) -> Q (f a)) -> or-introl l103 (l107 ⊔ l103) (Q (f h)) (Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f t)) (F h H)) (λ (H : Exists (l107 ⊔ l103) (l107 ⊔ l103) A P t) -> λ (F : (a : A) -> (X-- : P a) -> Q (f a)) -> or-intror l103 (l107 ⊔ l103) (Q (f h)) (Exists l107 l103 B-v Q (map (l107 ⊔ l103) l107 A B-v f t)) (IH H F)) X-clearme) l

Exists-All : (l155 l149 l147 : Level) -> (A : Set l155) -> (P : (X-- : A) -> Set l149) -> (Q : (X-- : A) -> Set l147) -> (l : list l155 A) -> (X-- : Exists l155 l149 A P l) -> (X--1 : All l155 l147 A Q l) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))
Exists-All = λ (l155 l149 l147 : Level) -> λ (A : Set l155) -> λ (P : (X-- : A) -> Set l149) -> λ (Q : (X-- : A) -> Set l147) -> λ (l : list l155 A) -> list-ind l155 ((l155 ⊔ l149) ⊔ l147) A (λ (X-x-716 : list l155 A) -> (X-- : Exists l155 l149 A P X-x-716) -> (X--1 : All l155 l147 A Q X-x-716) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) (λ (X-clearme : Exists l155 l149 A P (nil l155 A)) -> match-False (l155 ⊔ l149) ((l155 ⊔ l149) ⊔ l147) (λ (X-- : False (l155 ⊔ l149)) -> (X--1 : All l155 l147 A Q (nil l155 A)) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) X-clearme) (λ (hd-v : A) -> λ (tl : list l155 A) -> λ (IH : (X-- : Exists l155 l149 A P tl) -> (X--1 : All l155 l147 A Q tl) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) -> λ (X-clearme : Exists l155 l149 A P (cons l155 A hd-v tl)) -> match-Or l149 (l155 ⊔ l149) (P hd-v) (Exists l155 l149 A P tl) ((l155 ⊔ l149) ⊔ l147) (λ (X-- : Or l149 (l155 ⊔ l149) (P hd-v) (Exists l155 l149 A P tl)) -> (X--1 : All l155 l147 A Q (cons l155 A hd-v tl)) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) (λ (H1 : P hd-v) -> λ (X-clearme0 : All l155 l147 A Q (cons l155 A hd-v tl)) -> match-And l147 (l155 ⊔ l147) (Q hd-v) (All l155 l147 A Q tl) ((l155 ⊔ l149) ⊔ l147) (λ (X-- : And l147 (l155 ⊔ l147) (Q hd-v) (All l155 l147 A Q tl)) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) (λ (H2 : Q hd-v) -> λ (X-- : All l155 l147 A Q tl) -> ex-intro l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x)) hd-v (conj l149 l147 (P hd-v) (Q hd-v) H1 H2)) X-clearme0) (λ (H1 : Exists l155 l149 A P tl) -> λ (X-clearme0 : All l155 l147 A Q (cons l155 A hd-v tl)) -> match-And l147 (l155 ⊔ l147) (Q hd-v) (All l155 l147 A Q tl) ((l155 ⊔ l149) ⊔ l147) (λ (X-- : And l147 (l155 ⊔ l147) (Q hd-v) (All l155 l147 A Q tl)) -> ex l155 (l149 ⊔ l147) A (λ (x : A) -> And l149 l147 (P x) (Q x))) (λ (X-- : Q hd-v) -> λ (H2 : All l155 l147 A Q tl) -> IH H1 H2) X-clearme0) X-clearme) l


fold : (l6-v l10-v : Level) -> (A-v : Set l6-v) -> (B-v : Set l10-v) -> (X-op-v : (X---v : B-v) -> (X--1-v : B-v) -> B-v) -> (X-b-v : B-v) -> (X-p-v : (X---v : A-v) -> bool) -> (X-f-v : (X---v : A-v) -> B-v) -> (X-l-v : list l6-v A-v) -> B-v
fold l4-v l0-v A-v B-v X-op-v X-b-v X-p-v X-f-v nil' = X-b-v
fold l4-v l0-v A-v B-v X-op-v X-b-v X-p-v X-f-v (cons' a-v l-v) = match-bool l0-v (λ (X---v : bool) -> B-v) (X-op-v (X-f-v a-v) (fold l4-v l0-v A-v B-v X-op-v X-b-v X-p-v X-f-v l-v)) (fold l4-v l0-v A-v B-v X-op-v X-b-v X-p-v X-f-v l-v) (X-p-v a-v)


fold-true : (l47 l44 : Level) -> (A : Set l47) -> (B-v : Set l44) -> (a : A) -> (l : list l47 A) -> (p : (X-- : A) -> bool) -> (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> (nil-v : B-v) -> (f : (X-- : A) -> B-v) -> (X-- : eq lzero bool (p a) true) -> eq l44 B-v (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l47 A a l)) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l))
fold-true = λ (l47 l44 : Level) -> λ (A : Set l47) -> λ (B-v : Set l44) -> λ (a : A) -> λ (l : list l47 A) -> λ (p : (X-- : A) -> bool) -> λ (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> λ (nil-v : B-v) -> λ (f : (X-- : A) -> B-v) -> λ (pa : eq lzero bool (p a) true) -> eq-ind-r lzero l44 bool true (λ (x : bool) -> λ (X-- : eq lzero bool x true) -> eq l44 B-v (match-bool l44 (λ (X-0 : bool) -> B-v) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l) x) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l))) (refl l44 B-v (match-bool l44 (λ (X-- : bool) -> B-v) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l) true)) (p a) pa

fold-false : (l47 l44 : Level) -> (A : Set l47) -> (B-v : Set l44) -> (a : A) -> (l : list l47 A) -> (p : (X-- : A) -> bool) -> (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> (nil-v : B-v) -> (f : (X-- : A) -> B-v) -> (X-- : eq lzero bool (p a) false) -> eq l44 B-v (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l47 A a l)) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)
fold-false = λ (l47 l44 : Level) -> λ (A : Set l47) -> λ (B-v : Set l44) -> λ (a : A) -> λ (l : list l47 A) -> λ (p : (X-- : A) -> bool) -> λ (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> λ (nil-v : B-v) -> λ (f : (X-- : A) -> B-v) -> λ (pa : eq lzero bool (p a) false) -> eq-ind-r lzero l44 bool false (λ (x : bool) -> λ (X-- : eq lzero bool x false) -> eq l44 B-v (match-bool l44 (λ (X-0 : bool) -> B-v) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l) x) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)) (refl l44 B-v (match-bool l44 (λ (X-- : bool) -> B-v) (op-v (f a) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l)) (fold l47 l44 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l) false)) (p a) pa

fold-filter : (l270 l267 : Level) -> (A : Set l270) -> (B-v : Set l267) -> (X-a : A) -> (l : list l270 A) -> (p : (X-- : A) -> bool) -> (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> (nil-v : B-v) -> (f : (X-- : A) -> B-v) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) l) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p l))
fold-filter = λ (l270 l267 : Level) -> λ (A : Set l270) -> λ (B-v : Set l267) -> λ (a : A) -> λ (l : list l270 A) -> λ (p : (X-- : A) -> bool) -> λ (op-v : (X-- : B-v) -> (X--1 : B-v) -> B-v) -> λ (nil-v : B-v) -> λ (f : (X-- : A) -> B-v) -> list-ind l270 l267 A (λ (X-x-716 : list l270 A) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) X-x-716) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p X-x-716))) (refl l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (nil l270 A))) (λ (a0 : A) -> λ (tl : list l270 A) -> λ (Hind : eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl))) -> match-Or lzero lzero (eq lzero bool (p a0) true) (eq lzero bool (p a0) false) l267 (λ (X-- : Or lzero lzero (eq lzero bool (p a0) true) (eq lzero bool (p a0) false)) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l270 A a0 tl)) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p (cons l270 A a0 tl)))) (λ (pa : eq lzero bool (p a0) true) -> eq-ind-r l270 l267 (list l270 A) (cons l270 A a0 (filter' l270 A p tl)) (λ (x : list l270 A) -> λ (X-- : eq l270 (list l270 A) x (cons l270 A a0 (filter' l270 A p tl))) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l270 A a0 tl)) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) x)) (eq-ind-r l267 l267 B-v (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl)) (λ (x : B-v) -> λ (X-- : eq l267 B-v x (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl))) -> eq l267 B-v x (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (cons l270 A a0 (filter' l270 A p tl)))) (eq-ind-r l267 l267 B-v (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl))) (λ (x : B-v) -> λ (X-- : eq l267 B-v x (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl)))) -> eq l267 B-v (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl)) x) (rewrite-l l267 l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl) (λ (X-- : B-v) -> eq l267 B-v (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl)) (op-v (f a0) X--)) (refl l267 B-v (op-v (f a0) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl))) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl)) Hind) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (cons l270 A a0 (filter' l270 A p tl))) (fold-true l270 l267 A B-v a0 (filter' l270 A p tl) (λ (X-- : A) -> true) op-v nil-v f (refl lzero bool true))) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l270 A a0 tl)) (fold-true l270 l267 A B-v a0 tl p op-v nil-v f (rewrite-r lzero lzero bool true (λ (X-- : bool) -> eq lzero bool X-- true) (refl lzero bool true) (p a0) pa))) (filter' l270 A p (cons l270 A a0 tl)) (filter-true l270 A tl a0 p (rewrite-r lzero lzero bool true (λ (X-- : bool) -> eq lzero bool X-- true) (refl lzero bool true) (p a0) pa))) (λ (pa : eq lzero bool (p a0) false) -> eq-ind-r l270 l267 (list l270 A) (filter' l270 A p tl) (λ (x : list l270 A) -> λ (X-- : eq l270 (list l270 A) x (filter' l270 A p tl)) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l270 A a0 tl)) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) x)) (eq-ind-r l267 l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl) (λ (x : B-v) -> λ (X-- : eq l267 B-v x (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl)) -> eq l267 B-v x (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl))) (rewrite-l l267 l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl) (λ (X-- : B-v) -> eq l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl) X--) (refl l267 B-v (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) tl)) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (filter' l270 A p tl)) Hind) (fold l270 l267 A B-v op-v nil-v (λ (i : A) -> p i) (λ (i : A) -> f i) (cons l270 A a0 tl)) (fold-false l270 l267 A B-v a0 tl p op-v nil-v f (rewrite-r lzero lzero bool false (λ (X-- : bool) -> eq lzero bool X-- false) (refl lzero bool false) (p a0) pa))) (filter' l270 A p (cons l270 A a0 tl)) (filter-false l270 A tl a0 p (rewrite-r lzero lzero bool false (λ (X-- : bool) -> eq lzero bool X-- false) (refl lzero bool false) (p a0) pa))) (true-or-false (p a0))) l

data Aop' (l11-v : Level) (A-v : Set l11-v) (nil-v : A-v) : Set l11-v where
  mk-Aop'' : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l11-v A-v (op-v nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l11-v A-v (op-v a-v nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l11-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Aop' l11-v A-v nil-v

mk-Aop' : (l11-v : Level) -> (A-v : Set l11-v) -> (nil-v : A-v) -> (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l11-v A-v (op-v nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l11-v A-v (op-v a-v nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l11-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Aop' l11-v A-v nil-v
mk-Aop' _ _ _ = mk-Aop''

match-Aop' : (l18-v : Level) -> (A-v : Set l18-v) -> (X-nil-v : A-v) -> (return-sort-v : Level) -> (return-type-v : (z-v : Aop' l18-v A-v X-nil-v) -> Set return-sort-v) -> (case-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l18-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l18-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l18-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> return-type-v (mk-Aop' l18-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (z-v : Aop' l18-v A-v X-nil-v) -> return-type-v z-v
match-Aop' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-ind' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-781-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-781-v : Aop' l21-v A-v X-nil-v) -> Q--v x-781-v
Aop-ind' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type5' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type5' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type4' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type4' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type3' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type3' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type2' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type2' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type1' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type1' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

Aop-rect-Type0' : (l21-v l4-v : Level) -> (A-v : Set l21-v) -> (X-nil-v : A-v) -> (Q--v : (X-x-783-v : Aop' l21-v A-v X-nil-v) -> Set l4-v) -> (X-H-mk-Aop-v : (op-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> (X-nill-v : (a-v : A-v) -> eq l21-v A-v (op-v X-nil-v a-v) a-v) -> (X-nilr-v : (a-v : A-v) -> eq l21-v A-v (op-v a-v X-nil-v) a-v) -> (X-assoc-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l21-v A-v (op-v a-v (op-v b-v c-v)) (op-v (op-v a-v b-v) c-v)) -> Q--v (mk-Aop' l21-v A-v X-nil-v op-v X-nill-v X-nilr-v X-assoc-v)) -> (x-783-v : Aop' l21-v A-v X-nil-v) -> Q--v x-783-v
Aop-rect-Type0' _ _ _ _ _ casemk (mk-Aop'' x1 x2 x3 x4) = casemk x1 x2 x3 x4

op' : (l4-v : Level) -> (A-v : Set l4-v) -> (nil-v : A-v) -> (X-xxx-v : Aop' l4-v A-v nil-v) -> (X-x-807-v : A-v) -> (X-x-808-v : A-v) -> A-v
op' l9-v A-v nil-v X-xxx-v = match-Aop' l9-v A-v nil-v l9-v (λ (xxx0-v : Aop' l9-v A-v nil-v) -> (X-x-807-v : A-v) -> (X-x-808-v : A-v) -> A-v) (λ (yyy-v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> λ (X---v : (a-v : A-v) -> eq l9-v A-v (yyy-v nil-v a-v) a-v) -> λ (X-0-v : (a-v : A-v) -> eq l9-v A-v (yyy-v a-v nil-v) a-v) -> λ (X-1-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l9-v A-v (yyy-v a-v (yyy-v b-v c-v)) (yyy-v (yyy-v a-v b-v) c-v)) -> yyy-v) X-xxx-v

nill' : (l2-v : Level) -> (A-v : Set l2-v) -> (nil-v : A-v) -> (xxx-v : Aop' l2-v A-v nil-v) -> (a-v : A-v) -> eq l2-v A-v (op' l2-v A-v nil-v xxx-v nil-v a-v) a-v
nill' l8-v A-v nil-v xxx-v = match-Aop' l8-v A-v nil-v l8-v (λ (xxx0-v : Aop' l8-v A-v nil-v) -> (a-v : A-v) -> eq l8-v A-v (op' l8-v A-v nil-v xxx0-v nil-v a-v) a-v) (λ (X---v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> λ (yyy-v : (a-v : A-v) -> eq l8-v A-v (X---v nil-v a-v) a-v) -> λ (X-0-v : (a-v : A-v) -> eq l8-v A-v (X---v a-v nil-v) a-v) -> λ (X-1-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l8-v A-v (X---v a-v (X---v b-v c-v)) (X---v (X---v a-v b-v) c-v)) -> yyy-v) xxx-v

nilr' : (l2-v : Level) -> (A-v : Set l2-v) -> (nil-v : A-v) -> (xxx-v : Aop' l2-v A-v nil-v) -> (a-v : A-v) -> eq l2-v A-v (op' l2-v A-v nil-v xxx-v a-v nil-v) a-v
nilr' l8-v A-v nil-v xxx-v = match-Aop' l8-v A-v nil-v l8-v (λ (xxx0-v : Aop' l8-v A-v nil-v) -> (a-v : A-v) -> eq l8-v A-v (op' l8-v A-v nil-v xxx0-v a-v nil-v) a-v) (λ (X---v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> λ (X-0-v : (a-v : A-v) -> eq l8-v A-v (X---v nil-v a-v) a-v) -> λ (yyy-v : (a-v : A-v) -> eq l8-v A-v (X---v a-v nil-v) a-v) -> λ (X-1-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l8-v A-v (X---v a-v (X---v b-v c-v)) (X---v (X---v a-v b-v) c-v)) -> yyy-v) xxx-v

assoc' : (l5-v : Level) -> (A-v : Set l5-v) -> (nil-v : A-v) -> (xxx-v : Aop' l5-v A-v nil-v) -> (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l5-v A-v (op' l5-v A-v nil-v xxx-v a-v (op' l5-v A-v nil-v xxx-v b-v c-v)) (op' l5-v A-v nil-v xxx-v (op' l5-v A-v nil-v xxx-v a-v b-v) c-v)
assoc' l17-v A-v nil-v xxx-v = match-Aop' l17-v A-v nil-v l17-v (λ (xxx0-v : Aop' l17-v A-v nil-v) -> (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l17-v A-v (op' l17-v A-v nil-v xxx0-v a-v (op' l17-v A-v nil-v xxx0-v b-v c-v)) (op' l17-v A-v nil-v xxx0-v (op' l17-v A-v nil-v xxx0-v a-v b-v) c-v)) (λ (X---v : (X---v : A-v) -> (X--1-v : A-v) -> A-v) -> λ (X-0-v : (a-v : A-v) -> eq l17-v A-v (X---v nil-v a-v) a-v) -> λ (X-1-v : (a-v : A-v) -> eq l17-v A-v (X---v a-v nil-v) a-v) -> λ (yyy-v : (a-v : A-v) -> (b-v : A-v) -> (c-v : A-v) -> eq l17-v A-v (X---v a-v (X---v b-v c-v)) (X---v (X---v a-v b-v) c-v)) -> yyy-v) xxx-v


Aop-inv-ind' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1368 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1369 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-ind' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1368 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1369 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-ind' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-781 : Aop' l39 x1 x2) -> (X-z1369 : eq l39 (Aop' l39 x1 x2) Hterm X-x-781) -> P X-x-781) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-inv-rect-Type4' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1374 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1375 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-rect-Type4' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1374 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1375 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-rect-Type4' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-783 : Aop' l39 x1 x2) -> (X-z1375 : eq l39 (Aop' l39 x1 x2) Hterm X-x-783) -> P X-x-783) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-inv-rect-Type3' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1380 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1381 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-rect-Type3' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1380 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1381 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-rect-Type3' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-787 : Aop' l39 x1 x2) -> (X-z1381 : eq l39 (Aop' l39 x1 x2) Hterm X-x-787) -> P X-x-787) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-inv-rect-Type2' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1386 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1387 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-rect-Type2' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1386 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1387 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-rect-Type2' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-789 : Aop' l39 x1 x2) -> (X-z1387 : eq l39 (Aop' l39 x1 x2) Hterm X-x-789) -> P X-x-789) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-inv-rect-Type1' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1392 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1393 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-rect-Type1' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1392 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1393 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-rect-Type1' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-791 : Aop' l39 x1 x2) -> (X-z1393 : eq l39 (Aop' l39 x1 x2) Hterm X-x-791) -> P X-x-791) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-inv-rect-Type0' : (l39 l33 : Level) -> (x1 : Set l39) -> (x2 : x1) -> (Hterm : Aop' l39 x1 x2) -> (P : (X-z1398 : Aop' l39 x1 x2) -> Set l33) -> (X-H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1399 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P Hterm
Aop-inv-rect-Type0' = λ (l39 l33 : Level) -> λ (x1 : Set l39) -> λ (x2 : x1) -> λ (Hterm : Aop' l39 x1 x2) -> λ (P : (X-z1398 : Aop' l39 x1 x2) -> Set l33) -> λ (H1 : (op-v : (X-- : x1) -> (X--1 : x1) -> x1) -> (X-nill : (a : x1) -> eq l39 x1 (op-v x2 a) a) -> (X-nilr : (a : x1) -> eq l39 x1 (op-v a x2) a) -> (X-assoc : (a : x1) -> (b : x1) -> (c : x1) -> eq l39 x1 (op-v a (op-v b c)) (op-v (op-v a b) c)) -> (X-z1399 : eq l39 (Aop' l39 x1 x2) Hterm (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> P (mk-Aop' l39 x1 x2 op-v X-nill X-nilr X-assoc)) -> Aop-rect-Type0' l39 (l39 ⊔ l33) x1 x2 (λ (X-x-793 : Aop' l39 x1 x2) -> (X-z1399 : eq l39 (Aop' l39 x1 x2) Hterm X-x-793) -> P X-x-793) H1 Hterm (refl l39 (Aop' l39 x1 x2) Hterm)

Aop-discr' : (l3178 l4236 : Level) -> (a1 : Set l3178) -> (a2 : a1) -> (x : Aop' l3178 a1 a2) -> (y : Aop' l3178 a1 a2) -> (X-e : eq l3178 (Aop' l3178 a1 a2) x y) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-- : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (t0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (t1 : (a : a1) -> eq l3178 a1 (t0 a2 a) a) -> λ (t2 : (a : a1) -> eq l3178 a1 (t0 a a2) a) -> λ (t3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (t0 a (t0 b c)) (t0 (t0 a b) c)) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-- : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (u0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (u1 : (a : a1) -> eq l3178 a1 (u0 a2 a) a) -> λ (u2 : (a : a1) -> eq l3178 a1 (u0 a a2) a) -> λ (u3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) -> (P : Set l4236) -> (X-z51 : (e0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0) u0) -> (e1 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 u0 e0) u1) -> (e2 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 u0 e0 u1 e1) u2) -> (X-e3 : eq l3178 ((a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) (R3 l3178 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (p1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> λ (x2 : (a : a1) -> eq l3178 a1 (x0 a a2) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x01) -> (a : a1) -> eq l3178 a1 (x01 a2 a) a) t1 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p01 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x01) -> λ (x11 : (a : a1) -> eq l3178 a1 (x01 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x01 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x01 p01) x11) -> (a : a1) -> eq l3178 a1 (x01 a a2) a) t2 x0 p0 x1 p1) x2) -> (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (x0 a (x0 b c)) (x0 (x0 a b) c)) t3 u0 e0 u1 e1 u2 e2) u3) -> P) -> P) y) x
Aop-discr' = λ (l3178 l4236 : Level) -> λ (a1 : Set l3178) -> λ (a2 : a1) -> λ (x : Aop' l3178 a1 a2) -> λ (y : Aop' l3178 a1 a2) -> λ (Deq : eq l3178 (Aop' l3178 a1 a2) x y) -> eq-rect-Type2 l3178 ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178)) (Aop' l3178 a1 a2) x (λ (x-13 : Aop' l3178 a1 a2) -> λ (X-x-14 : eq l3178 (Aop' l3178 a1 a2) x x-13) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-- : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (t0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (t1 : (a : a1) -> eq l3178 a1 (t0 a2 a) a) -> λ (t2 : (a : a1) -> eq l3178 a1 (t0 a a2) a) -> λ (t3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (t0 a (t0 b c)) (t0 (t0 a b) c)) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-- : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (u0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (u1 : (a : a1) -> eq l3178 a1 (u0 a2 a) a) -> λ (u2 : (a : a1) -> eq l3178 a1 (u0 a a2) a) -> λ (u3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) -> (P : Set l4236) -> (X-z51 : (e0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0) u0) -> (e1 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 u0 e0) u1) -> (e2 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 u0 e0 u1 e1) u2) -> (X-e3 : eq l3178 ((a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) (R3 l3178 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (p1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> λ (x2 : (a : a1) -> eq l3178 a1 (x0 a a2) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x01) -> (a : a1) -> eq l3178 a1 (x01 a2 a) a) t1 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p01 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x01) -> λ (x11 : (a : a1) -> eq l3178 a1 (x01 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x01 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x01 p01) x11) -> (a : a1) -> eq l3178 a1 (x01 a a2) a) t2 x0 p0 x1 p1) x2) -> (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (x0 a (x0 b c)) (x0 (x0 a b) c)) t3 u0 e0 u1 e1 u2 e2) u3) -> P) -> P) x-13) x) (match-Aop' l3178 a1 a2 ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178)) (λ (X-- : Aop' l3178 a1 a2) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-0 : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (t0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (t1 : (a : a1) -> eq l3178 a1 (t0 a2 a) a) -> λ (t2 : (a : a1) -> eq l3178 a1 (t0 a a2) a) -> λ (t3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (t0 a (t0 b c)) (t0 (t0 a b) c)) -> match-Aop' l3178 a1 a2 ((lsuc (lsuc lzero)) ⊔ ((lsuc (lsuc l4236)) ⊔ (lsuc l3178))) (λ (X-0 : Aop' l3178 a1 a2) -> Set ((lsuc lzero) ⊔ ((lsuc l4236) ⊔ l3178))) (λ (u0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (u1 : (a : a1) -> eq l3178 a1 (u0 a2 a) a) -> λ (u2 : (a : a1) -> eq l3178 a1 (u0 a a2) a) -> λ (u3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) -> (P : Set l4236) -> (X-z51 : (e0 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) (R0 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0) u0) -> (e1 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a2 a) a) (R1 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x-19 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 u0 e0) u1) -> (e2 : eq l3178 ((a : a1) -> eq l3178 a1 (u0 a a2) a) (R2 l3178 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X--1 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (p0 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X--1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x-19 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 u0 e0 u1 e1) u2) -> (X-e3 : eq l3178 ((a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (u0 a (u0 b c)) (u0 (u0 a b) c)) (R3 l3178 l3178 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X--1 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) t1 (λ (x0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (p0 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X--1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x-19 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) t2 (λ (x0 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (p0 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (p1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x-19 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x0 p0) x1) -> λ (x2 : (a : a1) -> eq l3178 a1 (x0 a a2) a) -> λ (X--1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a a2) a) (R2 l3178 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x01 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X--1 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x01) -> (a : a1) -> eq l3178 a1 (x01 a2 a) a) t1 (λ (x01 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (p01 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x01) -> λ (x11 : (a : a1) -> eq l3178 a1 (x01 a2 a) a) -> λ (X--1 : eq l3178 ((a : a1) -> eq l3178 a1 (x01 a2 a) a) (R1 l3178 l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 (λ (x-19 : (X--1 : a1) -> (X--2 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X--1 : a1) -> (X--2 : a1) -> a1) t0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) t1 x01 p01) x11) -> (a : a1) -> eq l3178 a1 (x01 a a2) a) t2 x0 p0 x1 p1) x2) -> (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (x0 a (x0 b c)) (x0 (x0 a b) c)) t3 u0 e0 u1 e1 u2 e2) u3) -> P) -> P) X--) X--) (λ (a0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (a10 : (a : a1) -> eq l3178 a1 (a0 a2 a) a) -> λ (a20 : (a : a1) -> eq l3178 a1 (a0 a a2) a) -> λ (a3 : (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (a0 a (a0 b c)) (a0 (a0 a b) c)) -> λ (P : Set l4236) -> λ (DH : (e0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0) a0) -> (e1 : eq l3178 ((a : a1) -> eq l3178 a1 (a0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 a0 e0) a10) -> (e2 : eq l3178 ((a : a1) -> eq l3178 a1 (a0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) a10 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) a20 a0 e0 a10 e1) a20) -> (X-e3 : eq l3178 ((a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (a0 a (a0 b c)) (a0 (a0 a b) c)) (R3 l3178 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) a10 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) a20 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (p1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> λ (x2 : (a : a1) -> eq l3178 a1 (x0 a a2) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x01) -> (a : a1) -> eq l3178 a1 (x01 a2 a) a) a10 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p01 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x01) -> λ (x11 : (a : a1) -> eq l3178 a1 (x01 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x01 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x01 p01) x11) -> (a : a1) -> eq l3178 a1 (x01 a a2) a) a20 x0 p0 x1 p1) x2) -> (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (x0 a (x0 b c)) (x0 (x0 a b) c)) a3 a0 e0 a10 e1 a20 e2) a3) -> P) -> DH (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)) (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)))) (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) a10 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) a20 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)) a10 (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)))))) (refl l3178 ((a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (a0 a (a0 b c)) (a0 (a0 a b) c)) (R3 l3178 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) a10 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) a20 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (p1 : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> λ (x2 : (a : a1) -> eq l3178 a1 (x0 a a2) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x01) -> (a : a1) -> eq l3178 a1 (x01 a2 a) a) a10 (λ (x01 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p01 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x01) -> λ (x11 : (a : a1) -> eq l3178 a1 (x01 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x01 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x01 p01) x11) -> (a : a1) -> eq l3178 a1 (x01 a a2) a) a20 x0 p0 x1 p1) x2) -> (a : a1) -> (b : a1) -> (c : a1) -> eq l3178 a1 (x0 a (x0 b c)) (x0 (x0 a b) c)) a3 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)) a10 (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)))) a20 (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a a2) a) (R2 l3178 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-- : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> (a : a1) -> eq l3178 a1 (x0 a2 a) a) a10 (λ (x0 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (p0 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x0) -> λ (x1 : (a : a1) -> eq l3178 a1 (x0 a2 a) a) -> λ (X-- : eq l3178 ((a : a1) -> eq l3178 a1 (x0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 x0 p0) x1) -> (a : a1) -> eq l3178 a1 (x0 a a2) a) a20 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0)) a10 (refl l3178 ((a : a1) -> eq l3178 a1 (a0 a2 a) a) (R1 l3178 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 (λ (x-19 : (X-- : a1) -> (X--1 : a1) -> a1) -> λ (X-x-20 : eq l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0 x-19) -> (a : a1) -> eq l3178 a1 (x-19 a2 a) a) a10 a0 (refl l3178 ((X-- : a1) -> (X--1 : a1) -> a1) (R0 l3178 ((X-- : a1) -> (X--1 : a1) -> a1) a0))))))))) x) y Deq

dpi1--o--op' : (l12 l8 : Level) -> (x0 : Set l12) -> (x1 : x0) -> (x2 : (X-- : Aop' l12 x0 x1) -> Set l8) -> (x3 : DPair l12 l8 (Aop' l12 x0 x1) x2) -> (X-x-807 : x0) -> (X-x-808 : x0) -> x0
dpi1--o--op' = λ (l12 l8 : Level) -> λ (x0 : Set l12) -> λ (x1 : x0) -> λ (x2 : (X-- : Aop' l12 x0 x1) -> Set l8) -> λ (x3 : DPair l12 l8 (Aop' l12 x0 x1) x2) -> op' l12 x0 x1 (dpi1 l12 l8 (Aop' l12 x0 x1) x2 x3)

fold-sum : (l203 l199 : Level) -> (A : Set l203) -> (B-v : Set l199) -> (I-v : list l203 A) -> (J : list l203 A) -> (nil-v : B-v) -> (op-v : Aop' l199 B-v nil-v) -> (f : (X-- : A) -> B-v) -> eq l199 B-v (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) I-v) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (append l203 A I-v J))
fold-sum = λ (l203 l199 : Level) -> λ (A : Set l203) -> λ (B-v : Set l199) -> λ (I-v : list l203 A) -> λ (J : list l203 A) -> λ (nil-v : B-v) -> λ (op-v : Aop' l199 B-v nil-v) -> λ (f : (X-- : A) -> B-v) -> list-ind l203 l199 A (λ (X-x-716 : list l203 A) -> eq l199 B-v (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) X-x-716) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (append l203 A X-x-716 J))) (eq-ind-r l199 l199 B-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J) (λ (x : B-v) -> λ (X-- : eq l199 B-v x (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) -> eq l199 B-v x (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (refl l199 B-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (op' l199 B-v nil-v op-v nil-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (nill' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J))) (λ (a : A) -> λ (tl : list l203 A) -> λ (Hind : eq l199 B-v (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (append l203 A tl J))) -> eq-ind l199 l199 B-v (op' l199 B-v nil-v op-v (f a) (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J))) (λ (x-1 : B-v) -> λ (X-x-2 : eq l199 B-v (op' l199 B-v nil-v op-v (f a) (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J))) x-1) -> eq l199 B-v x-1 (op' l199 B-v nil-v op-v (f a) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (append l203 A tl J)))) (rewrite-l l199 l199 B-v (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (λ (X-- : B-v) -> eq l199 B-v (op' l199 B-v nil-v op-v (f a) (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J))) (op' l199 B-v nil-v op-v (f a) X--)) (refl l199 B-v (op' l199 B-v nil-v op-v (f a) (op' l199 B-v nil-v op-v (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)))) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) (append l203 A tl J)) Hind) (op' l199 B-v nil-v op-v (op' l199 B-v nil-v op-v (f a) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl)) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J)) (assoc' l199 B-v nil-v op-v (f a) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) tl) (fold l203 l199 A B-v (op' l199 B-v nil-v op-v) nil-v (λ (i : A) -> true) (λ (i : A) -> f i) J))) I-v


lhd : (l4-v : Level) -> (H-v : Set l4-v) -> (X---v : list l4-v H-v) -> (X--1-v : nat) -> list l4-v H-v
lhd l7-v H-v X---v O = nil l7-v H-v
lhd l7-v H-v X---v (S n0-v) = match-list l7-v H-v l7-v (λ (X--2-v : list l7-v H-v) -> list l7-v H-v) (nil l7-v H-v) (λ (a-v : H-v) -> λ (l0-v : list l7-v H-v) -> cons l7-v H-v a-v (lhd l7-v H-v l0-v n0-v)) X---v

ltl : (l4-v : Level) -> (H-v : Set l4-v) -> (X---v : list l4-v H-v) -> (X--1-v : nat) -> list l4-v H-v
ltl l0-v H-v X---v O = X---v
ltl l0-v H-v X---v (S n0-v) = ltl l0-v H-v (tail l0-v H-v X---v) n0-v



lhd-nil : (l23 : Level) -> (A : Set l23) -> (n : nat) -> eq l23 (list l23 A) (lhd l23 A (nil l23 A) n) (nil l23 A)
lhd-nil = λ (l23 : Level) -> λ (A : Set l23) -> λ (n : nat) -> nat-ind l23 (λ (X-x-365 : nat) -> eq l23 (list l23 A) (lhd l23 A (nil l23 A) X-x-365) (nil l23 A)) (refl l23 (list l23 A) (lhd l23 A (nil l23 A) O)) (λ (x-366 : nat) -> λ (X-x-368 : eq l23 (list l23 A) (lhd l23 A (nil l23 A) x-366) (nil l23 A)) -> refl l23 (list l23 A) (lhd l23 A (nil l23 A) (S x-366))) n

ltl-nil : (l32 : Level) -> (A : Set l32) -> (n : nat) -> eq l32 (list l32 A) (ltl l32 A (nil l32 A) n) (nil l32 A)
ltl-nil = λ (l32 : Level) -> λ (A : Set l32) -> λ (n : nat) -> nat-ind l32 (λ (X-x-365 : nat) -> eq l32 (list l32 A) (ltl l32 A (nil l32 A) X-x-365) (nil l32 A)) (refl l32 (list l32 A) (nil l32 A)) (λ (x-366 : nat) -> λ (X-x-368 : eq l32 (list l32 A) (ltl l32 A (nil l32 A) x-366) (nil l32 A)) -> rewrite-r l32 l32 (list l32 A) (nil l32 A) (λ (X-- : list l32 A) -> eq l32 (list l32 A) X-- (nil l32 A)) (refl l32 (list l32 A) (nil l32 A)) (ltl l32 A (nil l32 A) x-366) X-x-368) n

lhd-cons-ltl : (l98 : Level) -> (A : Set l98) -> (n : nat) -> (l : list l98 A) -> eq l98 (list l98 A) (append l98 A (lhd l98 A l n) (ltl l98 A l n)) l
lhd-cons-ltl = λ (l98 : Level) -> λ (A : Set l98) -> λ (n : nat) -> nat-ind l98 (λ (X-x-365 : nat) -> (l : list l98 A) -> eq l98 (list l98 A) (append l98 A (lhd l98 A l X-x-365) (ltl l98 A l X-x-365)) l) (λ (l : list l98 A) -> refl l98 (list l98 A) (append l98 A (lhd l98 A l O) (ltl l98 A l O))) (λ (n0 : nat) -> λ (IHn : (l : list l98 A) -> eq l98 (list l98 A) (append l98 A (lhd l98 A l n0) (ltl l98 A l n0)) l) -> λ (l : list l98 A) -> list-ind l98 l98 A (λ (X-x-716 : list l98 A) -> eq l98 (list l98 A) (append l98 A (lhd l98 A X-x-716 (S n0)) (ltl l98 A X-x-716 (S n0))) X-x-716) (rewrite-r l98 l98 (list l98 A) (nil l98 A) (λ (X-- : list l98 A) -> eq l98 (list l98 A) X-- (nil l98 A)) (refl l98 (list l98 A) (nil l98 A)) (ltl l98 A (nil l98 A) n0) (ltl-nil l98 A n0)) (λ (x-718 : A) -> λ (x-717 : list l98 A) -> λ (X-x-720 : eq l98 (list l98 A) (append l98 A (match-list l98 A l98 (λ (X-- : list l98 A) -> list l98 A) (nil l98 A) (λ (a : A) -> λ (l0 : list l98 A) -> cons l98 A a (lhd l98 A l0 n0)) x-717) (ltl l98 A (match-list l98 A l98 (λ (X-- : list l98 A) -> list l98 A) (nil l98 A) (λ (hd-v : A) -> λ (tl : list l98 A) -> tl) x-717) n0)) x-717) -> rewrite-r l98 l98 (list l98 A) x-717 (λ (X-- : list l98 A) -> eq l98 (list l98 A) (cons l98 A x-718 X--) (cons l98 A x-718 x-717)) (refl l98 (list l98 A) (cons l98 A x-718 x-717)) (append l98 A (lhd l98 A x-717 n0) (ltl l98 A x-717 n0)) (IHn x-717)) l) n

length-ltl : (l84 : Level) -> (A : Set l84) -> (n : nat) -> (l : list l84 A) -> eq lzero nat (length l84 A (ltl l84 A l n)) (minus (length l84 A l) n)
length-ltl = λ (l84 : Level) -> λ (A : Set l84) -> λ (n : nat) -> nat-ind l84 (λ (X-x-365 : nat) -> (l : list l84 A) -> eq lzero nat (length l84 A (ltl l84 A l X-x-365)) (minus (length l84 A l) X-x-365)) (λ (l : list l84 A) -> minus-n-O (length l84 A (ltl l84 A l O))) (λ (n0 : nat) -> λ (IHn : (l : list l84 A) -> eq lzero nat (length l84 A (ltl l84 A l n0)) (minus (length l84 A l) n0)) -> λ (X-clearme : list l84 A) -> match-list l84 A lzero (λ (X-- : list l84 A) -> eq lzero nat (length l84 A (ltl l84 A X-- (S n0))) (minus (length l84 A X--) (S n0))) (sym-eq lzero nat O (length l84 A (ltl l84 A (nil l84 A) n0)) (eq-coerc lzero (eq lzero nat O (length l84 A (nil l84 A))) (eq lzero nat O (length l84 A (ltl l84 A (nil l84 A) n0))) (le-n-O-to-eq (length l84 A (nil l84 A)) (le-n (length l84 A (nil l84 A)))) (rewrite-r l84 (lsuc lzero) (list l84 A) (nil l84 A) (λ (X-- : list l84 A) -> eq (lsuc lzero) (Set (lzero)) (eq lzero nat O (length l84 A (nil l84 A))) (eq lzero nat O (length l84 A X--))) (refl (lsuc lzero) (Set (lzero)) (eq lzero nat O (length l84 A (nil l84 A)))) (ltl l84 A (nil l84 A) n0) (ltl-nil l84 A n0)))) (λ (auto : A) -> λ (auto' : list l84 A) -> rewrite-r lzero lzero nat (minus (length l84 A auto') n0) (λ (X-- : nat) -> eq lzero nat X-- (minus (length l84 A auto') n0)) (refl lzero nat (minus (length l84 A auto') n0)) (length l84 A (ltl l84 A auto' n0)) (IHn auto')) X-clearme) n


find : (l6-v l5-v : Level) -> (A-v : Set l6-v) -> (B-v : Set l5-v) -> (X-f-v : (X---v : A-v) -> option l5-v B-v) -> (X-l-v : list l6-v A-v) -> option l5-v B-v
find l5-v l10-v A-v B-v X-f-v nil' = None l10-v B-v
find l5-v l10-v A-v B-v X-f-v (cons' h-v t-v) = match-option l10-v B-v l10-v (λ (X---v : option l10-v B-v) -> option l10-v B-v) (find l5-v l10-v A-v B-v X-f-v t-v) (λ (b-v : B-v) -> Some l10-v B-v b-v) (X-f-v h-v)


position-of-aux : (l6-v : Level) -> (A-v : Set l6-v) -> (X-found-v : (X---v : A-v) -> bool) -> (X-l-v : list l6-v A-v) -> (X-acc-v : nat) -> option lzero nat
position-of-aux l4-v A-v X-found-v nil' acc-v = None lzero nat
position-of-aux l4-v A-v X-found-v (cons' h-v t-v) acc-v = match-bool lzero (λ (X---v : bool) -> option lzero nat) (Some lzero nat acc-v) (position-of-aux l4-v A-v X-found-v t-v (S acc-v)) (X-found-v h-v)


position-of : (l5 : Level) -> (A : Set l5) -> (X-- : (X-- : A) -> bool) -> (X--1 : list l5 A) -> option lzero nat
position-of = λ (l5 : Level) -> λ (A : Set l5) -> λ (found : (X-- : A) -> bool) -> λ (l : list l5 A) -> position-of-aux l5 A found l O


make-list : (l3-v : Level) -> (A-v : Set l3-v) -> (X-a-v : A-v) -> (X-n-v : nat) -> list l3-v A-v
make-list l0-v A-v X-a-v O = nil l0-v A-v
make-list l0-v A-v X-a-v (S m-v) = cons l0-v A-v X-a-v (make-list l0-v A-v X-a-v m-v)