Structure patriciaTheory

signature patriciaTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val ADD_LIST_def : thm
    val BRANCH_primitive_def : thm
    val DEPTH_def : thm
    val EVERY_LEAF_def : thm
    val EXISTS_LEAF_def : thm
    val FIND_def : thm
    val INSERT_PTREE_def : thm
    val IN_PTREE_def : thm
    val IS_EMPTY_primitive_def : thm
    val IS_PTREE_def : thm
    val JOIN_def : thm
    val KEYS_def : thm
    val NUMSET_OF_PTREE_def : thm
    val PTREE_OF_NUMSET_def : thm
    val REMOVE_def : thm
    val SIZE_def : thm
    val TRANSFORM_def : thm
    val TRAVERSE_AUX_def : thm
    val TRAVERSE_def : thm
    val UNION_PTREE_def : thm
    val ptree_TY_DEF : thm
    val ptree_case_def : thm
    val ptree_size_def : thm
  
  (*  Theorems  *)
    val ADD_ADD : thm
    val ADD_ADD_SYM : thm
    val ADD_INSERT : thm
    val ADD_IS_PTREE : thm
    val ADD_LIST_IS_PTREE : thm
    val ADD_LIST_TO_EMPTY_IS_PTREE : thm
    val ADD_TRANSFORM : thm
    val ADD_def : thm
    val ADD_ind : thm
    val ALL_DISTINCT_TRAVERSE : thm
    val BRANCH : thm
    val BRANCHING_BIT : thm
    val BRANCHING_BIT_SYM : thm
    val BRANCHING_BIT_ZERO : thm
    val BRANCHING_BIT_def : thm
    val BRANCHING_BIT_ind : thm
    val BRANCH_def : thm
    val BRANCH_ind : thm
    val CARD_LIST_TO_SET : thm
    val CARD_NUMSET_OF_PTREE : thm
    val DELETE_UNION : thm
    val EMPTY_IS_PTREE : thm
    val EVERY_LEAF_ADD : thm
    val EVERY_LEAF_BRANCH : thm
    val EVERY_LEAF_PEEK : thm
    val EVERY_LEAF_REMOVE : thm
    val EVERY_LEAF_TRANSFORM : thm
    val FILTER_ALL : thm
    val FILTER_NONE : thm
    val FINITE_NUMSET_OF_PTREE : thm
    val INSERT_PTREE_IS_PTREE : thm
    val IN_NUMSET_OF_PTREE : thm
    val IN_PTREE_EMPTY : thm
    val IN_PTREE_INSERT_PTREE : thm
    val IN_PTREE_OF_NUMSET : thm
    val IN_PTREE_OF_NUMSET_EMPTY : thm
    val IN_PTREE_REMOVE : thm
    val IN_PTREE_UNION_PTREE : thm
    val IS_EMPTY_def : thm
    val IS_EMPTY_ind : thm
    val IS_PTREE_BRANCH : thm
    val IS_PTREE_PEEK : thm
    val KEYS_PEEK : thm
    val MEM_ALL_DISTINCT_IMP_PERM : thm
    val MEM_TRAVERSE : thm
    val MEM_TRAVERSE_INSERT_PTREE : thm
    val MEM_TRAVERSE_PEEK : thm
    val MONO_EVERY_LEAF : thm
    val NOT_ADD_EMPTY : thm
    val NOT_KEY_LEFT_AND_RIGHT : thm
    val NUMSET_OF_PTREE_EMPTY : thm
    val NUMSET_OF_PTREE_PTREE_OF_NUMSET : thm
    val NUMSET_OF_PTREE_PTREE_OF_NUMSET_EMPTY : thm
    val PEEK_ADD : thm
    val PEEK_INSERT_PTREE : thm
    val PEEK_NONE : thm
    val PEEK_REMOVE : thm
    val PEEK_TRANSFORM : thm
    val PEEK_def : thm
    val PEEK_ind : thm
    val PERM_ADD : thm
    val PERM_DELETE_PTREE : thm
    val PERM_INSERT_PTREE : thm
    val PERM_NOT_ADD : thm
    val PERM_NOT_REMOVE : thm
    val PERM_REMOVE : thm
    val PTREE_EQ : thm
    val PTREE_EXTENSION : thm
    val PTREE_OF_NUMSET_DELETE : thm
    val PTREE_OF_NUMSET_EMPTY : thm
    val PTREE_OF_NUMSET_INSERT : thm
    val PTREE_OF_NUMSET_INSERT_EMPTY : thm
    val PTREE_OF_NUMSET_IS_PTREE : thm
    val PTREE_OF_NUMSET_IS_PTREE_EMPTY : thm
    val PTREE_OF_NUMSET_NUMSET_OF_PTREE : thm
    val PTREE_OF_NUMSET_UNION : thm
    val PTREE_TRAVERSE_EQ : thm
    val QSORT_MEM_EQ : thm
    val REMOVE_ADD : thm
    val REMOVE_ADD_EQ : thm
    val REMOVE_IS_PTREE : thm
    val REMOVE_REMOVE : thm
    val REMOVE_TRANSFORM : thm
    val SIZE : thm
    val SIZE_ADD : thm
    val SIZE_PTREE_OF_NUMSET : thm
    val SIZE_PTREE_OF_NUMSET_EMPTY : thm
    val SIZE_REMOVE : thm
    val TRANSFORM_BRANCH : thm
    val TRANSFORM_EMPTY : thm
    val TRANSFORM_IS_PTREE : thm
    val TRAVERSE_AUX : thm
    val TRAVERSE_TRANSFORM : thm
    val UNION_PTREE_ASSOC : thm
    val UNION_PTREE_COMM : thm
    val UNION_PTREE_COMM_EMPTY : thm
    val UNION_PTREE_EMPTY : thm
    val UNION_PTREE_IS_PTREE : thm
    val datatype_ptree : thm
    val ptree_11 : thm
    val ptree_Axiom : thm
    val ptree_case_cong : thm
    val ptree_case_eq : thm
    val ptree_distinct : thm
    val ptree_induction : thm
    val ptree_nchotomy : thm
  
  val patricia_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [sorting] Parent theory of "patricia"
   
   [words] Parent theory of "patricia"
   
   [ADD_LIST_def]  Definition
      
      ⊢ $|++ = FOLDL $|+
   
   [BRANCH_primitive_def]  Definition
      
      ⊢ BRANCH =
        WFREC (@R. WF R)
          (λBRANCH a.
               case a of
                 (p,m,<{}>,t) => I t
               | (p,m,Leaf v18 v19,<{}>) => I (Leaf v18 v19)
               | (p,m,Leaf v18 v19,Leaf v30 v31) =>
                 I (Branch p m (Leaf v18 v19) (Leaf v30 v31))
               | (p,m,Leaf v18 v19,Branch v32 v33 v34 v35) =>
                 I (Branch p m (Leaf v18 v19) (Branch v32 v33 v34 v35))
               | (p,m,Branch v20 v21 v22 v23,<{}>) =>
                 I (Branch v20 v21 v22 v23)
               | (p,m,Branch v20 v21 v22 v23,Leaf v42 v43) =>
                 I (Branch p m (Branch v20 v21 v22 v23) (Leaf v42 v43))
               | (p,m,Branch v20 v21 v22 v23,Branch v44 v45 v46 v47) =>
                 I
                   (Branch p m (Branch v20 v21 v22 v23)
                      (Branch v44 v45 v46 v47)))
   
   [DEPTH_def]  Definition
      
      ⊢ (DEPTH <{}> = 0) ∧ (∀j d. DEPTH (Leaf j d) = 1) ∧
        ∀p m l r. DEPTH (Branch p m l r) = 1 + MAX (DEPTH l) (DEPTH r)
   
   [EVERY_LEAF_def]  Definition
      
      ⊢ (∀P. EVERY_LEAF P <{}> ⇔ T) ∧
        (∀P j d. EVERY_LEAF P (Leaf j d) ⇔ P j d) ∧
        ∀P p m l r.
          EVERY_LEAF P (Branch p m l r) ⇔ EVERY_LEAF P l ∧ EVERY_LEAF P r
   
   [EXISTS_LEAF_def]  Definition
      
      ⊢ (∀P. EXISTS_LEAF P <{}> ⇔ F) ∧
        (∀P j d. EXISTS_LEAF P (Leaf j d) ⇔ P j d) ∧
        ∀P p m l r.
          EXISTS_LEAF P (Branch p m l r) ⇔
          EXISTS_LEAF P l ∨ EXISTS_LEAF P r
   
   [FIND_def]  Definition
      
      ⊢ ∀t k. FIND t k = THE (t ' k)
   
   [INSERT_PTREE_def]  Definition
      
      ⊢ ∀n t. n INSERT_PTREE t = t |+ (n,())
   
   [IN_PTREE_def]  Definition
      
      ⊢ ∀n t. n IN_PTREE t ⇔ IS_SOME (t ' n)
   
   [IS_EMPTY_primitive_def]  Definition
      
      ⊢ IS_EMPTY =
        WFREC (@R. WF R)
          (λIS_EMPTY a.
               case a of
                 <{}> => I T
               | Leaf v6 v7 => I F
               | Branch v8 v9 v10 v11 => I F)
   
   [IS_PTREE_def]  Definition
      
      ⊢ (IS_PTREE <{}> ⇔ T) ∧ (∀k d. IS_PTREE (Leaf k d) ⇔ T) ∧
        ∀p m l r.
          IS_PTREE (Branch p m l r) ⇔
          p < 2 ** m ∧ IS_PTREE l ∧ IS_PTREE r ∧ l ≠ <{}> ∧ r ≠ <{}> ∧
          EVERY_LEAF (λk d. MOD_2EXP_EQ m k p ∧ BIT m k) l ∧
          EVERY_LEAF (λk d. MOD_2EXP_EQ m k p ∧ ¬BIT m k) r
   
   [JOIN_def]  Definition
      
      ⊢ ∀p0 t0 p1 t1.
          JOIN (p0,t0,p1,t1) =
          (let
             m = BRANCHING_BIT p0 p1
           in
             if BIT m p0 then Branch (MOD_2EXP m p0) m t0 t1
             else Branch (MOD_2EXP m p0) m t1 t0)
   
   [KEYS_def]  Definition
      
      ⊢ ∀t. KEYS t = QSORT $< (TRAVERSE t)
   
   [NUMSET_OF_PTREE_def]  Definition
      
      ⊢ ∀t. NUMSET_OF_PTREE t = LIST_TO_SET (TRAVERSE t)
   
   [PTREE_OF_NUMSET_def]  Definition
      
      ⊢ ∀t s. t |++ s = FOLDL (flip $INSERT_PTREE) t (SET_TO_LIST s)
   
   [REMOVE_def]  Definition
      
      ⊢ (∀k. <{}> \\ k = <{}>) ∧
        (∀j d k. Leaf j d \\ k = if j = k then <{}> else Leaf j d) ∧
        ∀p m l r k.
          Branch p m l r \\ k =
          if MOD_2EXP_EQ m k p then
            if BIT m k then BRANCH (p,m,l \\ k,r)
            else BRANCH (p,m,l,r \\ k)
          else Branch p m l r
   
   [SIZE_def]  Definition
      
      ⊢ ∀t. SIZE t = LENGTH (TRAVERSE t)
   
   [TRANSFORM_def]  Definition
      
      ⊢ (∀f. TRANSFORM f <{}> = <{}>) ∧
        (∀f j d. TRANSFORM f (Leaf j d) = Leaf j (f d)) ∧
        ∀f p m l r.
          TRANSFORM f (Branch p m l r) =
          Branch p m (TRANSFORM f l) (TRANSFORM f r)
   
   [TRAVERSE_AUX_def]  Definition
      
      ⊢ (∀a. TRAVERSE_AUX <{}> a = a) ∧
        (∀k d a. TRAVERSE_AUX (Leaf k d) a = k::a) ∧
        ∀p m l r a.
          TRAVERSE_AUX (Branch p m l r) a =
          TRAVERSE_AUX l (TRAVERSE_AUX r a)
   
   [TRAVERSE_def]  Definition
      
      ⊢ (TRAVERSE <{}> = []) ∧ (∀j d. TRAVERSE (Leaf j d) = [j]) ∧
        ∀p m l r. TRAVERSE (Branch p m l r) = TRAVERSE l ⧺ TRAVERSE r
   
   [UNION_PTREE_def]  Definition
      
      ⊢ ∀t1 t2. t1 UNION_PTREE t2 = t1 |++ NUMSET_OF_PTREE t2
   
   [ptree_TY_DEF]  Definition
      
      ⊢ ∃rep.
          TYPE_DEFINITION
            (λa0'.
                 ∀ $var$('ptree').
                   (∀a0'.
                      (a0' =
                       ind_type$CONSTR 0 (ARB,ARB,ARB)
                         (λn. ind_type$BOTTOM)) ∨
                      (∃a0 a1.
                         a0' =
                         (λa0 a1.
                              ind_type$CONSTR (SUC 0) (a0,a1,ARB)
                                (λn. ind_type$BOTTOM)) a0 a1) ∨
                      (∃a0 a1 a2 a3.
                         (a0' =
                          (λa0 a1 a2 a3.
                               ind_type$CONSTR (SUC (SUC 0)) (a0,ARB,a1)
                                 (ind_type$FCONS a2
                                    (ind_type$FCONS a3
                                       (λn. ind_type$BOTTOM)))) a0 a1 a2 a3) ∧
                         $var$('ptree') a2 ∧ $var$('ptree') a3) ⇒
                      $var$('ptree') a0') ⇒
                   $var$('ptree') a0') rep
   
   [ptree_case_def]  Definition
      
      ⊢ (∀v f f1. ptree_CASE <{}> v f f1 = v) ∧
        (∀a0 a1 v f f1. ptree_CASE (Leaf a0 a1) v f f1 = f a0 a1) ∧
        ∀a0 a1 a2 a3 v f f1.
          ptree_CASE (Branch a0 a1 a2 a3) v f f1 = f1 a0 a1 a2 a3
   
   [ptree_size_def]  Definition
      
      ⊢ (∀f. ptree_size f <{}> = 0) ∧
        (∀f a0 a1. ptree_size f (Leaf a0 a1) = 1 + (a0 + f a1)) ∧
        ∀f a0 a1 a2 a3.
          ptree_size f (Branch a0 a1 a2 a3) =
          1 + (a0 + (a1 + (ptree_size f a2 + ptree_size f a3)))
   
   [ADD_ADD]  Theorem
      
      ⊢ ∀t k d e. t |+ (k,d) |+ (k,e) = t |+ (k,e)
   
   [ADD_ADD_SYM]  Theorem
      
      ⊢ ∀t k j d e.
          IS_PTREE t ∧ k ≠ j ⇒ (t |+ (k,d) |+ (j,e) = t |+ (j,e) |+ (k,d))
   
   [ADD_INSERT]  Theorem
      
      ⊢ ∀v t n. t |+ (n,v) = n INSERT_PTREE t
   
   [ADD_IS_PTREE]  Theorem
      
      ⊢ ∀t x. IS_PTREE t ⇒ IS_PTREE (t |+ x)
   
   [ADD_LIST_IS_PTREE]  Theorem
      
      ⊢ ∀t l. IS_PTREE t ⇒ IS_PTREE (t |++ l)
   
   [ADD_LIST_TO_EMPTY_IS_PTREE]  Theorem
      
      ⊢ ∀l. IS_PTREE (<{}> |++ l)
   
   [ADD_TRANSFORM]  Theorem
      
      ⊢ ∀f t k d. TRANSFORM f (t |+ (k,d)) = TRANSFORM f t |+ (k,f d)
   
   [ADD_def]  Theorem
      
      ⊢ (∀k e. <{}> |+ (k,e) = Leaf k e) ∧
        (∀k j e d.
           Leaf j d |+ (k,e) =
           if j = k then Leaf k e else JOIN (k,Leaf k e,j,Leaf j d)) ∧
        ∀r p m l k e.
          Branch p m l r |+ (k,e) =
          if MOD_2EXP_EQ m k p then
            if BIT m k then Branch p m (l |+ (k,e)) r
            else Branch p m l (r |+ (k,e))
          else JOIN (k,Leaf k e,p,Branch p m l r)
   
   [ADD_ind]  Theorem
      
      ⊢ ∀P. (∀k e. P <{}> (k,e)) ∧ (∀j d k e. P (Leaf j d) (k,e)) ∧
            (∀p m l r k e.
               (MOD_2EXP_EQ m k p ∧ ¬BIT m k ⇒ P r (k,e)) ∧
               (MOD_2EXP_EQ m k p ∧ BIT m k ⇒ P l (k,e)) ⇒
               P (Branch p m l r) (k,e)) ⇒
            ∀v v1 v2. P v (v1,v2)
   
   [ALL_DISTINCT_TRAVERSE]  Theorem
      
      ⊢ ∀t. IS_PTREE t ⇒ ALL_DISTINCT (TRAVERSE t)
   
   [BRANCH]  Theorem
      
      ⊢ ∀p m l r.
          BRANCH (p,m,l,r) =
          if l = <{}> then r else if r = <{}> then l else Branch p m l r
   
   [BRANCHING_BIT]  Theorem
      
      ⊢ ∀a b.
          a ≠ b ⇒ (BIT (BRANCHING_BIT a b) a ⇎ BIT (BRANCHING_BIT a b) b)
   
   [BRANCHING_BIT_SYM]  Theorem
      
      ⊢ ∀a b. BRANCHING_BIT a b = BRANCHING_BIT b a
   
   [BRANCHING_BIT_ZERO]  Theorem
      
      ⊢ ∀a b. (BRANCHING_BIT a b = 0) ⇔ (ODD a ⇔ EVEN b) ∨ (a = b)
   
   [BRANCHING_BIT_def]  Theorem
      
      ⊢ ∀p1 p0.
          BRANCHING_BIT p0 p1 =
          if (ODD p0 ⇔ EVEN p1) ∨ (p0 = p1) then 0
          else SUC (BRANCHING_BIT (DIV2 p0) (DIV2 p1))
   
   [BRANCHING_BIT_ind]  Theorem
      
      ⊢ ∀P. (∀p0 p1.
               (¬((ODD p0 ⇔ EVEN p1) ∨ (p0 = p1)) ⇒ P (DIV2 p0) (DIV2 p1)) ⇒
               P p0 p1) ⇒
            ∀v v1. P v v1
   
   [BRANCH_def]  Theorem
      
      ⊢ (BRANCH (p,m,<{}>,t) = t) ∧
        (BRANCH (p,m,Leaf v6 v7,<{}>) = Leaf v6 v7) ∧
        (BRANCH (p,m,Branch v8 v9 v10 v11,<{}>) = Branch v8 v9 v10 v11) ∧
        (BRANCH (p,m,Leaf v12 v13,Leaf v24 v25) =
         Branch p m (Leaf v12 v13) (Leaf v24 v25)) ∧
        (BRANCH (p,m,Leaf v12 v13,Branch v26 v27 v28 v29) =
         Branch p m (Leaf v12 v13) (Branch v26 v27 v28 v29)) ∧
        (BRANCH (p,m,Branch v14 v15 v16 v17,Leaf v36 v37) =
         Branch p m (Branch v14 v15 v16 v17) (Leaf v36 v37)) ∧
        (BRANCH (p,m,Branch v14 v15 v16 v17,Branch v38 v39 v40 v41) =
         Branch p m (Branch v14 v15 v16 v17) (Branch v38 v39 v40 v41))
   
   [BRANCH_ind]  Theorem
      
      ⊢ ∀P. (∀p m t. P (p,m,<{}>,t)) ∧
            (∀p m v6 v7. P (p,m,Leaf v6 v7,<{}>)) ∧
            (∀p m v8 v9 v10 v11. P (p,m,Branch v8 v9 v10 v11,<{}>)) ∧
            (∀p m v12 v13 v24 v25. P (p,m,Leaf v12 v13,Leaf v24 v25)) ∧
            (∀p m v12 v13 v26 v27 v28 v29.
               P (p,m,Leaf v12 v13,Branch v26 v27 v28 v29)) ∧
            (∀p m v14 v15 v16 v17 v36 v37.
               P (p,m,Branch v14 v15 v16 v17,Leaf v36 v37)) ∧
            (∀p m v14 v15 v16 v17 v38 v39 v40 v41.
               P (p,m,Branch v14 v15 v16 v17,Branch v38 v39 v40 v41)) ⇒
            ∀v v1 v2 v3. P (v,v1,v2,v3)
   
   [CARD_LIST_TO_SET]  Theorem
      
      ⊢ ∀l. ALL_DISTINCT l ⇒ (CARD (LIST_TO_SET l) = LENGTH l)
   
   [CARD_NUMSET_OF_PTREE]  Theorem
      
      ⊢ ∀t. IS_PTREE t ⇒ (CARD (NUMSET_OF_PTREE t) = SIZE t)
   
   [DELETE_UNION]  Theorem
      
      ⊢ ∀x s1 s2. s1 ∪ s2 DELETE x = s1 DELETE x ∪ (s2 DELETE x)
   
   [EMPTY_IS_PTREE]  Theorem
      
      ⊢ IS_PTREE <{}>
   
   [EVERY_LEAF_ADD]  Theorem
      
      ⊢ ∀P t k d. P k d ∧ EVERY_LEAF P t ⇒ EVERY_LEAF P (t |+ (k,d))
   
   [EVERY_LEAF_BRANCH]  Theorem
      
      ⊢ ∀P p m l r.
          EVERY_LEAF P (BRANCH (p,m,l,r)) ⇔ EVERY_LEAF P l ∧ EVERY_LEAF P r
   
   [EVERY_LEAF_PEEK]  Theorem
      
      ⊢ ∀P t k. EVERY_LEAF P t ∧ IS_SOME (t ' k) ⇒ P k (THE (t ' k))
   
   [EVERY_LEAF_REMOVE]  Theorem
      
      ⊢ ∀P t k. EVERY_LEAF P t ⇒ EVERY_LEAF P (t \\ k)
   
   [EVERY_LEAF_TRANSFORM]  Theorem
      
      ⊢ ∀P Q f t.
          (∀k d. P k d ⇒ Q k (f d)) ∧ EVERY_LEAF P t ⇒
          EVERY_LEAF Q (TRANSFORM f t)
   
   [FILTER_ALL]  Theorem
      
      ⊢ ∀P l. (∀n. n < LENGTH l ⇒ ¬P (EL n l)) ⇔ (FILTER P l = [])
   
   [FILTER_NONE]  Theorem
      
      ⊢ ∀P l. (∀n. n < LENGTH l ⇒ P (EL n l)) ⇒ (FILTER P l = l)
   
   [FINITE_NUMSET_OF_PTREE]  Theorem
      
      ⊢ ∀t. FINITE (NUMSET_OF_PTREE t)
   
   [INSERT_PTREE_IS_PTREE]  Theorem
      
      ⊢ ∀t x. IS_PTREE t ⇒ IS_PTREE (x INSERT_PTREE t)
   
   [IN_NUMSET_OF_PTREE]  Theorem
      
      ⊢ ∀t n. IS_PTREE t ⇒ (n ∈ NUMSET_OF_PTREE t ⇔ n IN_PTREE t)
   
   [IN_PTREE_EMPTY]  Theorem
      
      ⊢ ∀n. ¬(n IN_PTREE <{}>)
   
   [IN_PTREE_INSERT_PTREE]  Theorem
      
      ⊢ ∀t m n.
          IS_PTREE t ⇒
          (n IN_PTREE m INSERT_PTREE t ⇔ (m = n) ∨ n IN_PTREE t)
   
   [IN_PTREE_OF_NUMSET]  Theorem
      
      ⊢ ∀t s n.
          IS_PTREE t ∧ FINITE s ⇒
          (n IN_PTREE t |++ s ⇔ n IN_PTREE t ∨ n ∈ s)
   
   [IN_PTREE_OF_NUMSET_EMPTY]  Theorem
      
      ⊢ ∀s n. FINITE s ⇒ (n ∈ s ⇔ n IN_PTREE <{}> |++ s)
   
   [IN_PTREE_REMOVE]  Theorem
      
      ⊢ ∀t m n. IS_PTREE t ⇒ (n IN_PTREE t \\ m ⇔ n ≠ m ∧ n IN_PTREE t)
   
   [IN_PTREE_UNION_PTREE]  Theorem
      
      ⊢ ∀t1 t2 n.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒
          (n IN_PTREE t1 UNION_PTREE t2 ⇔ n IN_PTREE t1 ∨ n IN_PTREE t2)
   
   [IS_EMPTY_def]  Theorem
      
      ⊢ (IS_EMPTY <{}> ⇔ T) ∧ (IS_EMPTY (Leaf v v1) ⇔ F) ∧
        (IS_EMPTY (Branch v2 v3 v4 v5) ⇔ F)
   
   [IS_EMPTY_ind]  Theorem
      
      ⊢ ∀P. P <{}> ∧ (∀v v1. P (Leaf v v1)) ∧
            (∀v2 v3 v4 v5. P (Branch v2 v3 v4 v5)) ⇒
            ∀v. P v
   
   [IS_PTREE_BRANCH]  Theorem
      
      ⊢ ∀p m l r.
          p < 2 ** m ∧ ¬((l = <{}>) ∧ (r = <{}>)) ∧
          EVERY_LEAF (λk d. MOD_2EXP_EQ m k p ∧ BIT m k) l ∧
          EVERY_LEAF (λk d. MOD_2EXP_EQ m k p ∧ ¬BIT m k) r ∧ IS_PTREE l ∧
          IS_PTREE r ⇒
          IS_PTREE (BRANCH (p,m,l,r))
   
   [IS_PTREE_PEEK]  Theorem
      
      ⊢ (∀k. ¬IS_SOME (<{}> ' k)) ∧
        (∀k j b. IS_SOME (Leaf j b ' k) ⇔ (j = k)) ∧
        ∀p m l r.
          IS_PTREE (Branch p m l r) ⇒
          (∃k. BIT m k ∧ IS_SOME (l ' k)) ∧
          (∃k. ¬BIT m k ∧ IS_SOME (r ' k)) ∧
          ∀k n.
            ¬MOD_2EXP_EQ m k p ∨ n < m ∧ (BIT n p ⇎ BIT n k) ⇒
            ¬IS_SOME (l ' k) ∧ ¬IS_SOME (r ' k)
   
   [KEYS_PEEK]  Theorem
      
      ⊢ ∀t1 t2.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒
          ((KEYS t1 = KEYS t2) ⇔ (TRAVERSE t1 = TRAVERSE t2))
   
   [MEM_ALL_DISTINCT_IMP_PERM]  Theorem
      
      ⊢ ∀l1 l2.
          ALL_DISTINCT l1 ∧ ALL_DISTINCT l2 ∧ (∀x. MEM x l1 ⇔ MEM x l2) ⇒
          PERM l1 l2
   
   [MEM_TRAVERSE]  Theorem
      
      ⊢ ∀t k. IS_PTREE t ⇒ (MEM k (TRAVERSE t) ⇔ k ∈ NUMSET_OF_PTREE t)
   
   [MEM_TRAVERSE_INSERT_PTREE]  Theorem
      
      ⊢ ∀t x h.
          IS_PTREE t ⇒
          (MEM x (TRAVERSE (h INSERT_PTREE t)) ⇔
           (x = h) ∨ x ≠ h ∧ MEM x (TRAVERSE t))
   
   [MEM_TRAVERSE_PEEK]  Theorem
      
      ⊢ ∀t k. IS_PTREE t ⇒ (MEM k (TRAVERSE t) ⇔ IS_SOME (t ' k))
   
   [MONO_EVERY_LEAF]  Theorem
      
      ⊢ ∀P Q t. (∀k d. P k d ⇒ Q k d) ∧ EVERY_LEAF P t ⇒ EVERY_LEAF Q t
   
   [NOT_ADD_EMPTY]  Theorem
      
      ⊢ ∀t k d. t |+ (k,d) ≠ <{}>
   
   [NOT_KEY_LEFT_AND_RIGHT]  Theorem
      
      ⊢ ∀p m l r k j.
          IS_PTREE (Branch p m l r) ∧ IS_SOME (l ' k) ∧ IS_SOME (r ' j) ⇒
          k ≠ j
   
   [NUMSET_OF_PTREE_EMPTY]  Theorem
      
      ⊢ NUMSET_OF_PTREE <{}> = ∅
   
   [NUMSET_OF_PTREE_PTREE_OF_NUMSET]  Theorem
      
      ⊢ ∀t s.
          IS_PTREE t ∧ FINITE s ⇒
          (NUMSET_OF_PTREE (t |++ s) = NUMSET_OF_PTREE t ∪ s)
   
   [NUMSET_OF_PTREE_PTREE_OF_NUMSET_EMPTY]  Theorem
      
      ⊢ ∀s. FINITE s ⇒ (NUMSET_OF_PTREE (<{}> |++ s) = s)
   
   [PEEK_ADD]  Theorem
      
      ⊢ ∀t k d j.
          IS_PTREE t ⇒ ((t |+ (k,d)) ' j = if k = j then SOME d else t ' j)
   
   [PEEK_INSERT_PTREE]  Theorem
      
      ⊢ ∀t k j.
          IS_PTREE t ⇒
          ((k INSERT_PTREE t) ' j = if k = j then SOME () else t ' j)
   
   [PEEK_NONE]  Theorem
      
      ⊢ ∀P t k. (∀d. ¬P k d) ∧ EVERY_LEAF P t ⇒ (t ' k = NONE)
   
   [PEEK_REMOVE]  Theorem
      
      ⊢ ∀t k j. IS_PTREE t ⇒ ((t \\ k) ' j = if k = j then NONE else t ' j)
   
   [PEEK_TRANSFORM]  Theorem
      
      ⊢ ∀f t k.
          TRANSFORM f t ' k =
          case t ' k of NONE => NONE | SOME x => SOME (f x)
   
   [PEEK_def]  Theorem
      
      ⊢ (∀k. <{}> ' k = NONE) ∧
        (∀k j d. Leaf j d ' k = if k = j then SOME d else NONE) ∧
        ∀r p m l k. Branch p m l r ' k = (if BIT m k then l else r) ' k
   
   [PEEK_ind]  Theorem
      
      ⊢ ∀P. (∀k. P <{}> k) ∧ (∀j d k. P (Leaf j d) k) ∧
            (∀p m l r k.
               P (if BIT m k then l else r) k ⇒ P (Branch p m l r) k) ⇒
            ∀v v1. P v v1
   
   [PERM_ADD]  Theorem
      
      ⊢ ∀t k d.
          IS_PTREE t ∧ ¬MEM k (TRAVERSE t) ⇒
          PERM (TRAVERSE (t |+ (k,d))) (k::TRAVERSE t)
   
   [PERM_DELETE_PTREE]  Theorem
      
      ⊢ ∀t k.
          IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒
          PERM (TRAVERSE (t \\ k)) (FILTER (λx. x ≠ k) (TRAVERSE t))
   
   [PERM_INSERT_PTREE]  Theorem
      
      ⊢ ∀t s.
          FINITE s ⇒
          IS_PTREE t ⇒
          PERM (TRAVERSE (FOLDL (flip $INSERT_PTREE) t (SET_TO_LIST s)))
            (SET_TO_LIST (NUMSET_OF_PTREE t ∪ s))
   
   [PERM_NOT_ADD]  Theorem
      
      ⊢ ∀t k d.
          IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒
          (TRAVERSE (t |+ (k,d)) = TRAVERSE t)
   
   [PERM_NOT_REMOVE]  Theorem
      
      ⊢ ∀t k.
          IS_PTREE t ∧ ¬MEM k (TRAVERSE t) ⇒
          (TRAVERSE (t \\ k) = TRAVERSE t)
   
   [PERM_REMOVE]  Theorem
      
      ⊢ ∀t k.
          IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒
          PERM (TRAVERSE (t \\ k)) (FILTER (λx. x ≠ k) (TRAVERSE t))
   
   [PTREE_EQ]  Theorem
      
      ⊢ ∀t1 t2.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒ ((∀k. t1 ' k = t2 ' k) ⇔ (t1 = t2))
   
   [PTREE_EXTENSION]  Theorem
      
      ⊢ ∀t1 t2.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒
          ((t1 = t2) ⇔ ∀x. x IN_PTREE t1 ⇔ x IN_PTREE t2)
   
   [PTREE_OF_NUMSET_DELETE]  Theorem
      
      ⊢ ∀s x. FINITE s ⇒ (<{}> |++ (s DELETE x) = (<{}> |++ s) \\ x)
   
   [PTREE_OF_NUMSET_EMPTY]  Theorem
      
      ⊢ ∀t. t |++ ∅ = t
   
   [PTREE_OF_NUMSET_INSERT]  Theorem
      
      ⊢ ∀t s x.
          IS_PTREE t ∧ FINITE s ⇒
          (t |++ (x INSERT s) = x INSERT_PTREE t |++ s)
   
   [PTREE_OF_NUMSET_INSERT_EMPTY]  Theorem
      
      ⊢ ∀s x.
          FINITE s ⇒ (<{}> |++ (x INSERT s) = x INSERT_PTREE <{}> |++ s)
   
   [PTREE_OF_NUMSET_IS_PTREE]  Theorem
      
      ⊢ ∀t s. IS_PTREE t ⇒ IS_PTREE (t |++ s)
   
   [PTREE_OF_NUMSET_IS_PTREE_EMPTY]  Theorem
      
      ⊢ ∀s. IS_PTREE (<{}> |++ s)
   
   [PTREE_OF_NUMSET_NUMSET_OF_PTREE]  Theorem
      
      ⊢ ∀t s.
          IS_PTREE t ∧ FINITE s ⇒
          (<{}> |++ (NUMSET_OF_PTREE t ∪ s) = t |++ s)
   
   [PTREE_OF_NUMSET_UNION]  Theorem
      
      ⊢ ∀t s1 s2.
          IS_PTREE t ∧ FINITE s1 ∧ FINITE s2 ⇒
          (t |++ (s1 ∪ s2) = t |++ s1 |++ s2)
   
   [PTREE_TRAVERSE_EQ]  Theorem
      
      ⊢ ∀t1 t2.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒
          ((∀k. MEM k (TRAVERSE t1) ⇔ MEM k (TRAVERSE t2)) ⇔
           (TRAVERSE t1 = TRAVERSE t2))
   
   [QSORT_MEM_EQ]  Theorem
      
      ⊢ ∀l2 l1 R. (QSORT R l1 = QSORT R l2) ⇒ ∀x. MEM x l1 ⇔ MEM x l2
   
   [REMOVE_ADD]  Theorem
      
      ⊢ ∀t k d j.
          IS_PTREE t ⇒
          (t |+ (k,d) \\ j = if k = j then t \\ j else t \\ j |+ (k,d))
   
   [REMOVE_ADD_EQ]  Theorem
      
      ⊢ ∀t k d. t |+ (k,d) \\ k = t \\ k
   
   [REMOVE_IS_PTREE]  Theorem
      
      ⊢ ∀t k. IS_PTREE t ⇒ IS_PTREE (t \\ k)
   
   [REMOVE_REMOVE]  Theorem
      
      ⊢ ∀t k. IS_PTREE t ⇒ (t \\ k \\ k = t \\ k)
   
   [REMOVE_TRANSFORM]  Theorem
      
      ⊢ ∀f t k. TRANSFORM f (t \\ k) = TRANSFORM f t \\ k
   
   [SIZE]  Theorem
      
      ⊢ (SIZE <{}> = 0) ∧ (∀k d. SIZE (Leaf k d) = 1) ∧
        ∀p m l r. SIZE (Branch p m l r) = SIZE l + SIZE r
   
   [SIZE_ADD]  Theorem
      
      ⊢ ∀t k d.
          IS_PTREE t ⇒
          (SIZE (t |+ (k,d)) =
           if MEM k (TRAVERSE t) then SIZE t else SIZE t + 1)
   
   [SIZE_PTREE_OF_NUMSET]  Theorem
      
      ⊢ ∀t s.
          FINITE s ⇒
          IS_PTREE t ∧ ALL_DISTINCT (TRAVERSE t ⧺ SET_TO_LIST s) ⇒
          (SIZE (t |++ s) = SIZE t + CARD s)
   
   [SIZE_PTREE_OF_NUMSET_EMPTY]  Theorem
      
      ⊢ ∀s. FINITE s ⇒ (SIZE (<{}> |++ s) = CARD s)
   
   [SIZE_REMOVE]  Theorem
      
      ⊢ ∀t k.
          IS_PTREE t ⇒
          (SIZE (t \\ k) =
           if MEM k (TRAVERSE t) then SIZE t − 1 else SIZE t)
   
   [TRANSFORM_BRANCH]  Theorem
      
      ⊢ ∀f p m l r.
          TRANSFORM f (BRANCH (p,m,l,r)) =
          BRANCH (p,m,TRANSFORM f l,TRANSFORM f r)
   
   [TRANSFORM_EMPTY]  Theorem
      
      ⊢ ∀f t. (TRANSFORM f t = <{}>) ⇔ (t = <{}>)
   
   [TRANSFORM_IS_PTREE]  Theorem
      
      ⊢ ∀f t. IS_PTREE t ⇒ IS_PTREE (TRANSFORM f t)
   
   [TRAVERSE_AUX]  Theorem
      
      ⊢ ∀t. TRAVERSE t = TRAVERSE_AUX t []
   
   [TRAVERSE_TRANSFORM]  Theorem
      
      ⊢ ∀f t. TRAVERSE (TRANSFORM f t) = TRAVERSE t
   
   [UNION_PTREE_ASSOC]  Theorem
      
      ⊢ ∀t1 t2 t3.
          IS_PTREE t1 ∧ IS_PTREE t2 ∧ IS_PTREE t3 ⇒
          (t1 UNION_PTREE (t2 UNION_PTREE t3) =
           t1 UNION_PTREE t2 UNION_PTREE t3)
   
   [UNION_PTREE_COMM]  Theorem
      
      ⊢ ∀t1 t2.
          IS_PTREE t1 ∧ IS_PTREE t2 ⇒
          (t1 UNION_PTREE t2 = t2 UNION_PTREE t1)
   
   [UNION_PTREE_COMM_EMPTY]  Theorem
      
      ⊢ ∀t. IS_PTREE t ⇒ (<{}> UNION_PTREE t = t UNION_PTREE <{}>)
   
   [UNION_PTREE_EMPTY]  Theorem
      
      ⊢ (∀t. t UNION_PTREE <{}> = t) ∧
        ∀t. IS_PTREE t ⇒ (<{}> UNION_PTREE t = t)
   
   [UNION_PTREE_IS_PTREE]  Theorem
      
      ⊢ ∀t1 t2. IS_PTREE t1 ∧ IS_PTREE t2 ⇒ IS_PTREE (t1 UNION_PTREE t2)
   
   [datatype_ptree]  Theorem
      
      ⊢ DATATYPE (ptree <{}> Leaf Branch)
   
   [ptree_11]  Theorem
      
      ⊢ (∀a0 a1 a0' a1'.
           (Leaf a0 a1 = Leaf a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')) ∧
        ∀a0 a1 a2 a3 a0' a1' a2' a3'.
          (Branch a0 a1 a2 a3 = Branch a0' a1' a2' a3') ⇔
          (a0 = a0') ∧ (a1 = a1') ∧ (a2 = a2') ∧ (a3 = a3')
   
   [ptree_Axiom]  Theorem
      
      ⊢ ∀f0 f1 f2. ∃fn.
          (fn <{}> = f0) ∧ (∀a0 a1. fn (Leaf a0 a1) = f1 a0 a1) ∧
          ∀a0 a1 a2 a3.
            fn (Branch a0 a1 a2 a3) = f2 a0 a1 a2 a3 (fn a2) (fn a3)
   
   [ptree_case_cong]  Theorem
      
      ⊢ ∀M M' v f f1.
          (M = M') ∧ ((M' = <{}>) ⇒ (v = v')) ∧
          (∀a0 a1. (M' = Leaf a0 a1) ⇒ (f a0 a1 = f' a0 a1)) ∧
          (∀a0 a1 a2 a3.
             (M' = Branch a0 a1 a2 a3) ⇒ (f1 a0 a1 a2 a3 = f1' a0 a1 a2 a3)) ⇒
          (ptree_CASE M v f f1 = ptree_CASE M' v' f' f1')
   
   [ptree_case_eq]  Theorem
      
      ⊢ (ptree_CASE x v f f1 = v') ⇔
        (x = <{}>) ∧ (v = v') ∨ (∃n a. (x = Leaf n a) ∧ (f n a = v')) ∨
        ∃n0 n p p0. (x = Branch n0 n p p0) ∧ (f1 n0 n p p0 = v')
   
   [ptree_distinct]  Theorem
      
      ⊢ (∀a1 a0. <{}> ≠ Leaf a0 a1) ∧
        (∀a3 a2 a1 a0. <{}> ≠ Branch a0 a1 a2 a3) ∧
        ∀a3 a2 a1' a1 a0' a0. Leaf a0 a1 ≠ Branch a0' a1' a2 a3
   
   [ptree_induction]  Theorem
      
      ⊢ ∀P. P <{}> ∧ (∀n a. P (Leaf n a)) ∧
            (∀p p0. P p ∧ P p0 ⇒ ∀n n0. P (Branch n0 n p p0)) ⇒
            ∀p. P p
   
   [ptree_nchotomy]  Theorem
      
      ⊢ ∀pp.
          (pp = <{}>) ∨ (∃n a. pp = Leaf n a) ∨
          ∃n0 n p p0. pp = Branch n0 n p p0
   
   
*)
end

HOL 4, Kananaskis-14