- LIST_MAP_I
-
|- MAP I = I
- LIST_REL_EQ
-
|- LIST_REL $= = $=
- LIST_REL_REFL
-
|- ∀R. (∀x y. R x y ⇔ (R x = R y)) ⇒ ∀x. LIST_REL R x x
- LIST_EQUIV
-
|- ∀R. EQUIV R ⇒ EQUIV (LIST_REL R)
- LIST_REL_REL
-
|- ∀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)
- LIST_QUOTIENT
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ QUOTIENT (LIST_REL R) (MAP abs) (MAP rep)
- CONS_PRS
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ ∀t h. h::t = MAP abs (rep h::MAP rep t)
- CONS_RSP
-
|- ∀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)
- NIL_PRS
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ ([] = MAP abs [])
- NIL_RSP
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ LIST_REL R [] []
- MAP_PRS
-
|- ∀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
-
|- ∀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)
- LENGTH_PRS
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ ∀l. LENGTH l = LENGTH (MAP rep l)
- LENGTH_RSP
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l1 l2. LIST_REL R l1 l2 ⇒ (LENGTH l1 = LENGTH l2)
- APPEND_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l m. l ++ m = MAP abs (MAP rep l ++ MAP rep m)
- APPEND_RSP
-
|- ∀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)
- FLAT_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l. FLAT l = MAP abs (FLAT (MAP (MAP rep) l))
- FLAT_RSP
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2. LIST_REL (LIST_REL R) l1 l2 ⇒ LIST_REL R (FLAT l1) (FLAT l2)
- REVERSE_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l. REVERSE l = MAP abs (REVERSE (MAP rep l))
- REVERSE_RSP
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l1 l2. LIST_REL R l1 l2 ⇒ LIST_REL R (REVERSE l1) (REVERSE l2)
- FILTER_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒
∀P l. FILTER P l = MAP abs (FILTER ((abs --> I) P) (MAP rep l))
- FILTER_RSP
-
|- ∀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)
- NULL_PRS
-
|- ∀R abs rep. QUOTIENT R abs rep ⇒ ∀l. NULL l ⇔ NULL (MAP rep l)
- NULL_RSP
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l1 l2. LIST_REL R l1 l2 ⇒ (NULL l1 ⇔ NULL l2)
- ALL_EL_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒ ∀l P. EVERY P l ⇔ EVERY ((abs --> I) P) (MAP rep l)
- ALL_EL_RSP
-
|- ∀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)
- SOME_EL_PRS
-
|- ∀R abs rep.
QUOTIENT R abs rep ⇒
∀l P. EXISTS P l ⇔ EXISTS ((abs --> I) P) (MAP rep l)
- SOME_EL_RSP
-
|- ∀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)
- FOLDL_PRS
-
|- ∀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
-
|- ∀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
-
|- ∀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
-
|- ∀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)