isahipster: 8866715a911da841acd55c3a0691087d68b2c3fe

     1: theory regexp_RecPlus
     2: imports Main
     3:         "$HIPSTER_HOME/IsaHipster"
     4: begin
     5: 
     6: datatype 'a list = Nil2 | Cons2 "'a" "'a list"
     7: 
     8: datatype A = X | Y
     9: 
    10: datatype R
    11:   = Nil2 | Eps | Atom "A" | Plus "R" "R" | Seq "R" "R" | Star "R"
    12: 
    13: fun seq :: "R => R => R" where
    14: "seq x y =
    15:    case x of
    16:      | other =>
    17:          case y of
    18:            | other =>
    19:                case x of
    20:                  | other =>
    21:                      case y of
    22:                        | other => Seq x y
    23:                        | Eps => x
    24:                      end
    25:                  | Eps => y
    26:                end
    27:            | Nil2 => y
    28:          end
    29:      | Nil2 => x
    30:    end"
    31: 
    32: fun plus :: "R => R => R" where
    33: "plus x y =
    34:    case x of
    35:      | other =>
    36:          case y of
    37:            | other => Plus x y
    38:            | Nil2 => x
    39:          end
    40:      | Nil2 => y
    41:    end"
    42: 
    43: fun or2 :: "bool => bool => bool" where
    44: "or2 True y = True"
    45: | "or2 False y = y"
    46: 
    47: fun eqA :: "A => A => bool" where
    48: "eqA (X) y = False"
    49: | "eqA (Y) (X) = False"
    50: | "eqA (Y) (Y) = True"
    51: 
    52: fun and2 :: "bool => bool => bool" where
    53: "and2 True y = y"
    54: | "and2 False y = False"
    55: 
    56: fun eps :: "R => bool" where
    57: "eps x =
    58:    case x of
    59:      | other => False
    60:      | Eps => True
    61:      | Plus p q => or2 (eps p) (eps q)
    62:      | Seq p2 q2 => and2 (eps p2) (eps q2)
    63:      | Star y => True
    64:    end"
    65: 
    66: fun epsR :: "R => R" where
    67: "epsR x = (if eps x then Eps else Nil2)"
    68: 
    69: fun step :: "R => A => R" where
    70: "step x y =
    71:    case x of
    72:      | other => Nil2
    73:      | Atom a => (if eqA a y then Eps else Nil2)
    74:      | Plus p q => plus (step p y) (step q y)
    75:      | Seq p2 q2 =>
    76:          plus (seq (step p2 y) q2) (seq (epsR p2) (step q2 y))
    77:      | Star p3 => seq (step p3 y) x
    78:    end"
    79: 
    80: fun recognise :: "R => A list => bool" where
    81: "recognise x (Nil2) = eps x"
    82: | "recognise x (Cons2 z xs) = recognise (step x z) xs"
    83: 
    84: (*hipster seq plus or2 eqA and2 eps epsR step recognise *)
    85: 
    86: theorem x0 :
    87:   "!! (p :: R) (q :: R) (s :: A list) .
    88:      (recognise (Plus p q) s) = (or2 (recognise p s) (recognise q s))"
    89:   by (tactic {* Subgoal.FOCUS_PARAMS (K (Tactic_Data.hard_tac @{context})) @{context} 1 *})
    90: 
    91: end