Structure dftTheory

signature dftTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val Rel_def : thm
  
  (*  Theorems  *)
    val DFT_ALL_DISTINCT : thm
    val DFT_CONS : thm
    val DFT_FOLD : thm
    val DFT_REACH_1 : thm
    val DFT_REACH_2 : thm
    val DFT_REACH_THM : thm
    val DFT_def : thm
    val DFT_ind : thm
    val DFT_ind0 : thm
    val def : thm
  
  val dft_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [dirGraph] Parent theory of "dft"
   
   [Rel_def]  Definition
      
      ⊢ ∀G f seen to_visit acc.
          Rel (G,f,seen,to_visit,acc) =
          (CARD (Parents G DIFF set seen),LENGTH to_visit)
   
   [DFT_ALL_DISTINCT]  Theorem
      
      ⊢ ∀G seen to_visit.
          FINITE (Parents G) ⇒ ALL_DISTINCT (DFT G CONS seen to_visit [])
   
   [DFT_CONS]  Theorem
      
      ⊢ ∀G f seen to_visit acc a b.
          FINITE (Parents G) ∧ f = CONS ∧ acc = a ⧺ b ⇒
          DFT G f seen to_visit acc = DFT G f seen to_visit a ⧺ b
   
   [DFT_FOLD]  Theorem
      
      ⊢ ∀G f seen to_visit acc.
          FINITE (Parents G) ⇒
          DFT G f seen to_visit acc =
          FOLDR f acc (DFT G CONS seen to_visit [])
   
   [DFT_REACH_1]  Theorem
      
      ⊢ ∀G f seen to_visit acc.
          FINITE (Parents G) ∧ f = CONS ⇒
          ∀x. MEM x (DFT G f seen to_visit acc) ⇒
              x ∈ REACH_LIST G to_visit ∨ MEM x acc
   
   [DFT_REACH_2]  Theorem
      
      ⊢ ∀G f seen to_visit acc x.
          FINITE (Parents G) ∧ f = CONS ∧
          x ∈ REACH_LIST (EXCLUDE G (set seen)) to_visit ∧ ¬MEM x seen ⇒
          MEM x (DFT G f seen to_visit acc)
   
   [DFT_REACH_THM]  Theorem
      
      ⊢ ∀G to_visit.
          FINITE (Parents G) ⇒
          ∀x. x ∈ REACH_LIST G to_visit ⇔ MEM x (DFT G CONS [] to_visit [])
   
   [DFT_def]  Theorem
      
      ⊢ FINITE (Parents G) ⇒
        DFT G f seen [] acc = acc ∧
        DFT G f seen (visit_now::visit_later) acc =
        if MEM visit_now seen then DFT G f seen visit_later acc
        else
          DFT G f (visit_now::seen) (G visit_now ⧺ visit_later)
            (f visit_now acc)
   
   [DFT_ind]  Theorem
      
      ⊢ ∀P. (∀G f seen visit_now visit_later acc.
               P G f seen [] acc ∧
               ((FINITE (Parents G) ∧ ¬MEM visit_now seen ⇒
                 P G f (visit_now::seen) (G visit_now ⧺ visit_later)
                   (f visit_now acc)) ∧
                (FINITE (Parents G) ∧ MEM visit_now seen ⇒
                 P G f seen visit_later acc) ⇒
                P G f seen (visit_now::visit_later) acc)) ⇒
            ∀v v1 v2 v3 v4. P v v1 v2 v3 v4
   
   [DFT_ind0]  Theorem
      
      ⊢ ∀P. (∀G f seen to_visit acc.
               (∀visit_now visit_later.
                  FINITE (Parents G) ∧ to_visit = visit_now::visit_later ∧
                  ¬MEM visit_now seen ⇒
                  P G f (visit_now::seen) (G visit_now ⧺ visit_later)
                    (f visit_now acc)) ∧
               (∀visit_now visit_later.
                  FINITE (Parents G) ∧ to_visit = visit_now::visit_later ∧
                  MEM visit_now seen ⇒
                  P G f seen visit_later acc) ⇒
               P G f seen to_visit acc) ⇒
            ∀v v1 v2 v3 v4. P v v1 v2 v3 v4
   
   [def]  Theorem
      
      ⊢ ∀to_visit seen f acc G.
          DFT G f seen to_visit acc =
          if FINITE (Parents G) then
            case to_visit of
              [] => acc
            | visit_now::visit_later =>
              if MEM visit_now seen then DFT G f seen visit_later acc
              else
                DFT G f (visit_now::seen) (G visit_now ⧺ visit_later)
                  (f visit_now acc)
          else ARB
   
   
*)
end

HOL 4, Trindemossen-1