# Structure quotient_listTheory

Source File Identifier index Theory binding index

```signature quotient_listTheory =
sig
type thm = Thm.thm

(*  Theorems  *)
val ALL_EL_PRS : thm
val ALL_EL_RSP : thm
val APPEND_PRS : thm
val APPEND_RSP : thm
val CONS_PRS : thm
val CONS_RSP : thm
val FILTER_PRS : thm
val FILTER_RSP : thm
val FLAT_PRS : thm
val FLAT_RSP : thm
val FOLDL_PRS : thm
val FOLDL_RSP : thm
val FOLDR_PRS : thm
val FOLDR_RSP : thm
val LENGTH_PRS : thm
val LENGTH_RSP : thm
val LIST_EQUIV : thm
val LIST_MAP_I : thm
val LIST_QUOTIENT : thm
val LIST_REL_EQ : thm
val LIST_REL_REFL : thm
val LIST_REL_REL : thm
val MAP_PRS : thm
val MAP_RSP : thm
val NIL_PRS : thm
val NIL_RSP : thm
val NULL_PRS : thm
val NULL_RSP : thm
val REVERSE_PRS : thm
val REVERSE_RSP : thm
val SOME_EL_PRS : thm
val SOME_EL_RSP : thm

val quotient_list_grammars : type_grammar.grammar * term_grammar.grammar
(*
[quotient] Parent theory of "quotient_list"

[ALL_EL_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l P. EVERY P l ⇔ EVERY ((abs --> I) P) (MAP rep l)

[ALL_EL_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2 P1 P2.
(R ===> \$<=>) P1 P2 ∧ LIST_REL R l1 l2 ⇒
(EVERY P1 l1 ⇔ EVERY P2 l2)

[APPEND_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l m. l ⧺ m = MAP abs (MAP rep l ⧺ MAP rep m)

[APPEND_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2 m1 m2.
LIST_REL R l1 l2 ∧ LIST_REL R m1 m2 ⇒
LIST_REL R (l1 ⧺ m1) (l2 ⧺ m2)

[CONS_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀t h. h::t = MAP abs (rep h::MAP rep t)

[CONS_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀t1 t2 h1 h2.
R h1 h2 ∧ LIST_REL R t1 t2 ⇒ LIST_REL R (h1::t1) (h2::t2)

[FILTER_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀P l. FILTER P l = MAP abs (FILTER ((abs --> I) P) (MAP rep l))

[FILTER_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀P1 P2 l1 l2.
(R ===> \$<=>) P1 P2 ∧ LIST_REL R l1 l2 ⇒
LIST_REL R (FILTER P1 l1) (FILTER P2 l2)

[FLAT_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l. FLAT l = MAP abs (FLAT (MAP (MAP rep) l))

[FLAT_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2.
LIST_REL (LIST_REL R) l1 l2 ⇒
LIST_REL R (FLAT l1) (FLAT l2)

[FOLDL_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l f e.
FOLDL f e l =
abs1
(FOLDL ((abs1 --> abs2 --> rep1) f) (rep1 e)
(MAP rep2 l))

[FOLDL_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l1 l2 f1 f2 e1 e2.
(R1 ===> R2 ===> R1) f1 f2 ∧ R1 e1 e2 ∧
LIST_REL R2 l1 l2 ⇒
R1 (FOLDL f1 e1 l1) (FOLDL f2 e2 l2)

[FOLDR_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l f e.
FOLDR f e l =
abs2
(FOLDR ((abs1 --> abs2 --> rep2) f) (rep2 e)
(MAP rep1 l))

[FOLDR_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l1 l2 f1 f2 e1 e2.
(R1 ===> R2 ===> R2) f1 f2 ∧ R2 e1 e2 ∧
LIST_REL R1 l1 l2 ⇒
R2 (FOLDR f1 e1 l1) (FOLDR f2 e2 l2)

[LENGTH_PRS]  Theorem

⊢ ∀R abs rep. QUOTIENT R abs rep ⇒ ∀l. LENGTH l = LENGTH (MAP rep l)

[LENGTH_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2. LIST_REL R l1 l2 ⇒ (LENGTH l1 = LENGTH l2)

[LIST_EQUIV]  Theorem

⊢ ∀R. EQUIV R ⇒ EQUIV (LIST_REL R)

[LIST_MAP_I]  Theorem

⊢ MAP I = I

[LIST_QUOTIENT]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒ QUOTIENT (LIST_REL R) (MAP abs) (MAP rep)

[LIST_REL_EQ]  Theorem

⊢ LIST_REL \$= = \$=

[LIST_REL_REFL]  Theorem

⊢ ∀R. (∀x y. R x y ⇔ (R x = R y)) ⇒ ∀x. LIST_REL R x x

[LIST_REL_REL]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀r s.
LIST_REL R r s ⇔
LIST_REL R r r ∧ LIST_REL R s s ∧ (MAP abs r = MAP abs s)

[MAP_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l f.
MAP f l =
MAP abs2 (MAP ((abs1 --> rep2) f) (MAP rep1 l))

[MAP_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀l1 l2 f1 f2.
(R1 ===> R2) f1 f2 ∧ LIST_REL R1 l1 l2 ⇒
LIST_REL R2 (MAP f1 l1) (MAP f2 l2)

[NIL_PRS]  Theorem

⊢ ∀R abs rep. QUOTIENT R abs rep ⇒ ([] = MAP abs [])

[NIL_RSP]  Theorem

⊢ ∀R abs rep. QUOTIENT R abs rep ⇒ LIST_REL R [] []

[NULL_PRS]  Theorem

⊢ ∀R abs rep. QUOTIENT R abs rep ⇒ ∀l. NULL l ⇔ NULL (MAP rep l)

[NULL_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2. LIST_REL R l1 l2 ⇒ (NULL l1 ⇔ NULL l2)

[REVERSE_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l. REVERSE l = MAP abs (REVERSE (MAP rep l))

[REVERSE_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2. LIST_REL R l1 l2 ⇒ LIST_REL R (REVERSE l1) (REVERSE l2)

[SOME_EL_PRS]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l P. EXISTS P l ⇔ EXISTS ((abs --> I) P) (MAP rep l)

[SOME_EL_RSP]  Theorem

⊢ ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2 P1 P2.
(R ===> \$<=>) P1 P2 ∧ LIST_REL R l1 l2 ⇒
(EXISTS P1 l1 ⇔ EXISTS P2 l2)

*)
end

```

Source File Identifier index Theory binding index