Structure quotient_listTheory

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

HOL 4, Kananaskis-14