Structure EncodeTheory


Source File Identifier index Theory binding index

signature EncodeTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val biprefix_def : thm
    val collision_free_def : thm
    val encode_blist_def : thm
    val encode_bnum_def : thm
    val encode_bool_def : thm
    val encode_list_def : thm
    val encode_num_primitive_def : thm
    val encode_option_def : thm
    val encode_prod_def : thm
    val encode_sum_def : thm
    val encode_unit_def : thm
    val lift_blist_def : thm
    val lift_option_def : thm
    val lift_prod_def : thm
    val lift_sum_def : thm
    val tree_TY_DEF : thm
    val tree_case_def : thm
    val tree_size_def : thm
    val wf_encoder_def : thm
    val wf_pred_bnum_def : thm
    val wf_pred_def : thm
  
  (*  Theorems  *)
    val biprefix_append : thm
    val biprefix_appends : thm
    val biprefix_cons : thm
    val biprefix_refl : thm
    val biprefix_sym : thm
    val datatype_tree : thm
    val encode_blist_compute : thm
    val encode_bnum_compute : thm
    val encode_bnum_inj : thm
    val encode_bnum_length : thm
    val encode_list_cong : thm
    val encode_num_def : thm
    val encode_num_ind : thm
    val encode_prod_alt : thm
    val encode_tree_def : thm
    val lift_blist_suc : thm
    val lift_tree_def : thm
    val tree_11 : thm
    val tree_Axiom : thm
    val tree_case_cong : thm
    val tree_case_eq : thm
    val tree_ind : thm
    val tree_induction : thm
    val tree_nchotomy : thm
    val tree_size_eq : thm
    val wf_encode_blist : thm
    val wf_encode_bnum : thm
    val wf_encode_bnum_collision_free : thm
    val wf_encode_bool : thm
    val wf_encode_list : thm
    val wf_encode_num : thm
    val wf_encode_option : thm
    val wf_encode_prod : thm
    val wf_encode_sum : thm
    val wf_encode_tree : thm
    val wf_encode_unit : thm
    val wf_encoder_alt : thm
    val wf_encoder_eq : thm
    val wf_encoder_total : thm
    val wf_pred_bnum : thm
    val wf_pred_bnum_total : thm
  
  val Encode_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [indexedLists] Parent theory of "Encode"
   
   [patternMatches] Parent theory of "Encode"
   
   [biprefix_def]  Definition
      
      ⊢ ∀a b. biprefix a b ⇔ b ≼ a ∨ a ≼ b
   
   [collision_free_def]  Definition
      
      ⊢ ∀m p.
          collision_free m p ⇔
          ∀x y. p x ∧ p y ∧ x MOD 2 ** m = y MOD 2 ** m ⇒ x = y
   
   [encode_blist_def]  Definition
      
      ⊢ (∀e l. encode_blist 0 e l = []) ∧
        ∀m e l.
          encode_blist (SUC m) e l = e (HD l) ⧺ encode_blist m e (TL l)
   
   [encode_bnum_def]  Definition
      
      ⊢ (∀n. encode_bnum 0 n = []) ∧
        ∀m n. encode_bnum (SUC m) n = ¬EVEN n::encode_bnum m (n DIV 2)
   
   [encode_bool_def]  Definition
      
      ⊢ ∀x. encode_bool x = [x]
   
   [encode_list_def]  Definition
      
      ⊢ (∀xb. encode_list xb [] = [F]) ∧
        ∀xb x xs. encode_list xb (x::xs) = T::(xb x ⧺ encode_list xb xs)
   
   [encode_num_primitive_def]  Definition
      
      ⊢ encode_num =
        WFREC
          (@R. WF R ∧ (∀n. n ≠ 0 ∧ EVEN n ⇒ R ((n − 2) DIV 2) n) ∧
               ∀n. n ≠ 0 ∧ ¬EVEN n ⇒ R ((n − 1) DIV 2) n)
          (λencode_num a.
               I
                 (if a = 0 then [T; T]
                  else if EVEN a then F::encode_num ((a − 2) DIV 2)
                  else T::F::encode_num ((a − 1) DIV 2)))
   
   [encode_option_def]  Definition
      
      ⊢ (∀xb. encode_option xb NONE = [F]) ∧
        ∀xb x. encode_option xb (SOME x) = T::xb x
   
   [encode_prod_def]  Definition
      
      ⊢ ∀xb yb x y. encode_prod xb yb (x,y) = xb x ⧺ yb y
   
   [encode_sum_def]  Definition
      
      ⊢ (∀xb yb x. encode_sum xb yb (INL x) = T::xb x) ∧
        ∀xb yb y. encode_sum xb yb (INR y) = F::yb y
   
   [encode_unit_def]  Definition
      
      ⊢ ∀v0. encode_unit v0 = []
   
   [lift_blist_def]  Definition
      
      ⊢ ∀m p x. lift_blist m p x ⇔ EVERY p x ∧ LENGTH x = m
   
   [lift_option_def]  Definition
      
      ⊢ ∀p x. lift_option p x ⇔ case x of NONE => T | SOME y => p y
   
   [lift_prod_def]  Definition
      
      ⊢ ∀p1 p2 x. lift_prod p1 p2 x ⇔ p1 (FST x) ∧ p2 (SND x)
   
   [lift_sum_def]  Definition
      
      ⊢ ∀p1 p2 x.
          lift_sum p1 p2 x ⇔ case x of INL x1 => p1 x1 | INR x2 => p2 x2
   
   [tree_TY_DEF]  Definition
      
      ⊢ ∃rep.
          TYPE_DEFINITION
            (λa0'.
                 ∀ $var$('tree') $var$('@temp@ind_typeEncode0list').
                   (∀a0'.
                      (∃a0 a1.
                         a0' =
                         (λa0 a1.
                              ind_type$CONSTR 0 a0
                                (ind_type$FCONS a1 (λn. ind_type$BOTTOM)))
                           a0 a1 ∧ $var$('@temp@ind_typeEncode0list') a1) ⇒
                      $var$('tree') a0') ∧
                   (∀a1'.
                      a1' =
                      ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ∨
                      (∃a0 a1.
                         a1' =
                         (λa0 a1.
                              ind_type$CONSTR (SUC (SUC 0)) ARB
                                (ind_type$FCONS a0
                                   (ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
                           a0 a1 ∧ $var$('tree') a0 ∧
                         $var$('@temp@ind_typeEncode0list') a1) ⇒
                      $var$('@temp@ind_typeEncode0list') a1') ⇒
                   $var$('tree') a0') rep
   
   [tree_case_def]  Definition
      
      ⊢ ∀a0 a1 f. tree_CASE (Node a0 a1) f = f a0 a1
   
   [tree_size_def]  Definition
      
      ⊢ (∀f a0 a1. tree_size f (Node a0 a1) = 1 + (f a0 + tree1_size f a1)) ∧
        (∀f. tree1_size f [] = 0) ∧
        ∀f a0 a1.
          tree1_size f (a0::a1) = 1 + (tree_size f a0 + tree1_size f a1)
   
   [wf_encoder_def]  Definition
      
      ⊢ ∀p e. wf_encoder p e ⇔ ∀x y. p x ∧ p y ∧ e y ≼ e x ⇒ x = y
   
   [wf_pred_bnum_def]  Definition
      
      ⊢ ∀m p. wf_pred_bnum m p ⇔ wf_pred p ∧ ∀x. p x ⇒ x < 2 ** m
   
   [wf_pred_def]  Definition
      
      ⊢ ∀p. wf_pred p ⇔ ∃x. p x
   
   [biprefix_append]  Theorem
      
      ⊢ ∀a b c d. biprefix (a ⧺ b) (c ⧺ d) ⇒ biprefix a c
   
   [biprefix_appends]  Theorem
      
      ⊢ ∀a b c. biprefix (a ⧺ b) (a ⧺ c) ⇔ biprefix b c
   
   [biprefix_cons]  Theorem
      
      ⊢ ∀a b c d. biprefix (a::b) (c::d) ⇔ a = c ∧ biprefix b d
   
   [biprefix_refl]  Theorem
      
      ⊢ ∀x. biprefix x x
   
   [biprefix_sym]  Theorem
      
      ⊢ ∀x y. biprefix x y ⇒ biprefix y x
   
   [datatype_tree]  Theorem
      
      ⊢ DATATYPE (tree Node)
   
   [encode_blist_compute]  Theorem
      
      ⊢ (∀e l. encode_blist 0 e l = []) ∧
        (∀m e l.
           encode_blist (NUMERAL (BIT1 m)) e l =
           e (HD l) ⧺ encode_blist (NUMERAL (BIT1 m) − 1) e (TL l)) ∧
        ∀m e l.
          encode_blist (NUMERAL (BIT2 m)) e l =
          e (HD l) ⧺ encode_blist (NUMERAL (BIT1 m)) e (TL l)
   
   [encode_bnum_compute]  Theorem
      
      ⊢ (∀n. encode_bnum 0 n = []) ∧
        (∀m n.
           encode_bnum (NUMERAL (BIT1 m)) n =
           ¬EVEN n::encode_bnum (NUMERAL (BIT1 m) − 1) (n DIV 2)) ∧
        ∀m n.
          encode_bnum (NUMERAL (BIT2 m)) n =
          ¬EVEN n::encode_bnum (NUMERAL (BIT1 m)) (n DIV 2)
   
   [encode_bnum_inj]  Theorem
      
      ⊢ ∀m x y.
          x < 2 ** m ∧ y < 2 ** m ∧ encode_bnum m x = encode_bnum m y ⇒
          x = y
   
   [encode_bnum_length]  Theorem
      
      ⊢ ∀m n. LENGTH (encode_bnum m n) = m
   
   [encode_list_cong]  Theorem
      
      ⊢ ∀l1 l2 f1 f2.
          l1 = l2 ∧ (∀x. MEM x l2 ⇒ f1 x = f2 x) ⇒
          encode_list f1 l1 = encode_list f2 l2
   
   [encode_num_def]  Theorem
      
      ⊢ encode_num n =
        if n = 0 then [T; T]
        else if EVEN n then F::encode_num ((n − 2) DIV 2)
        else T::F::encode_num ((n − 1) DIV 2)
   
   [encode_num_ind]  Theorem
      
      ⊢ ∀P. (∀n. (n ≠ 0 ∧ EVEN n ⇒ P ((n − 2) DIV 2)) ∧
                 (n ≠ 0 ∧ ¬EVEN n ⇒ P ((n − 1) DIV 2)) ⇒
                 P n) ⇒
            ∀v. P v
   
   [encode_prod_alt]  Theorem
      
      ⊢ ∀xb yb p. encode_prod xb yb p = xb (FST p) ⧺ yb (SND p)
   
   [encode_tree_def]  Theorem
      
      ⊢ encode_tree e (Node a ts) = e a ⧺ encode_list (encode_tree e) ts
   
   [lift_blist_suc]  Theorem
      
      ⊢ ∀n p h t. lift_blist (SUC n) p (h::t) ⇔ p h ∧ lift_blist n p t
   
   [lift_tree_def]  Theorem
      
      ⊢ lift_tree p (Node a ts) ⇔ p a ∧ EVERY (lift_tree p) ts
   
   [tree_11]  Theorem
      
      ⊢ ∀a0 a1 a0' a1'. Node a0 a1 = Node a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
   
   [tree_Axiom]  Theorem
      
      ⊢ ∀f0 f1 f2. ∃fn0 fn1.
          (∀a0 a1. fn0 (Node a0 a1) = f0 a0 a1 (fn1 a1)) ∧ fn1 [] = f1 ∧
          ∀a0 a1. fn1 (a0::a1) = f2 a0 a1 (fn0 a0) (fn1 a1)
   
   [tree_case_cong]  Theorem
      
      ⊢ ∀M M' f.
          M = M' ∧ (∀a0 a1. M' = Node a0 a1 ⇒ f a0 a1 = f' a0 a1) ⇒
          tree_CASE M f = tree_CASE M' f'
   
   [tree_case_eq]  Theorem
      
      ⊢ tree_CASE x f = v ⇔ ∃a l. x = Node a l ∧ f a l = v
   
   [tree_ind]  Theorem
      
      ⊢ ∀p. (∀a ts. (∀t. MEM t ts ⇒ p t) ⇒ p (Node a ts)) ⇒ ∀t. p t
   
   [tree_induction]  Theorem
      
      ⊢ ∀P0 P1.
          (∀l. P1 l ⇒ ∀a. P0 (Node a l)) ∧ P1 [] ∧
          (∀t l. P0 t ∧ P1 l ⇒ P1 (t::l)) ⇒
          (∀t. P0 t) ∧ ∀l. P1 l
   
   [tree_nchotomy]  Theorem
      
      ⊢ ∀tt. ∃a l. tt = Node a l
   
   [tree_size_eq]  Theorem
      
      ⊢ tree1_size f = list_size (tree_size f)
   
   [wf_encode_blist]  Theorem
      
      ⊢ ∀m p e.
          wf_encoder p e ⇒ wf_encoder (lift_blist m p) (encode_blist m e)
   
   [wf_encode_bnum]  Theorem
      
      ⊢ ∀m p. wf_pred_bnum m p ⇒ wf_encoder p (encode_bnum m)
   
   [wf_encode_bnum_collision_free]  Theorem
      
      ⊢ ∀m p. wf_encoder p (encode_bnum m) ⇔ collision_free m p
   
   [wf_encode_bool]  Theorem
      
      ⊢ ∀p. wf_encoder p encode_bool
   
   [wf_encode_list]  Theorem
      
      ⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (EVERY p) (encode_list e)
   
   [wf_encode_num]  Theorem
      
      ⊢ ∀p. wf_encoder p encode_num
   
   [wf_encode_option]  Theorem
      
      ⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (lift_option p) (encode_option e)
   
   [wf_encode_prod]  Theorem
      
      ⊢ ∀p1 p2 e1 e2.
          wf_encoder p1 e1 ∧ wf_encoder p2 e2 ⇒
          wf_encoder (lift_prod p1 p2) (encode_prod e1 e2)
   
   [wf_encode_sum]  Theorem
      
      ⊢ ∀p1 p2 e1 e2.
          wf_encoder p1 e1 ∧ wf_encoder p2 e2 ⇒
          wf_encoder (lift_sum p1 p2) (encode_sum e1 e2)
   
   [wf_encode_tree]  Theorem
      
      ⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (lift_tree p) (encode_tree e)
   
   [wf_encode_unit]  Theorem
      
      ⊢ ∀p. wf_encoder p encode_unit
   
   [wf_encoder_alt]  Theorem
      
      ⊢ wf_encoder p e ⇔ ∀x y. p x ∧ p y ∧ biprefix (e x) (e y) ⇒ x = y
   
   [wf_encoder_eq]  Theorem
      
      ⊢ ∀p e f. wf_encoder p e ∧ (∀x. p x ⇒ e x = f x) ⇒ wf_encoder p f
   
   [wf_encoder_total]  Theorem
      
      ⊢ ∀p e. wf_encoder (K T) e ⇒ wf_encoder p e
   
   [wf_pred_bnum]  Theorem
      
      ⊢ ∀m p. wf_pred_bnum m p ⇒ collision_free m p
   
   [wf_pred_bnum_total]  Theorem
      
      ⊢ ∀m. wf_pred_bnum m (λx. x < 2 ** m)
   
   
*)
end


Source File Identifier index Theory binding index

HOL 4, Trindemossen-1