Theory patricia

⊢ $|++ = FOLDL $|+

⊢ 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 <{}> = 0) ∧ (∀j d. DEPTH (Leaf j d) = 1) ∧
  ∀p m l r. DEPTH (Branch p m l r) = 1 + MAX (DEPTH l) (DEPTH r)

⊢ (∀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

⊢ (∀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

⊢ ∀t k. FIND t k = THE (t ' k)

⊢ ∀n t. n INSERT_PTREE t = t |+ (n,())

⊢ ∀n t. n IN_PTREE t ⇔ IS_SOME (t ' n)

⊢ 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 <{}> ⇔ 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

⊢ ∀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)

⊢ ∀t. KEYS t = QSORT $< (TRAVERSE t)

⊢ ∀t. NUMSET_OF_PTREE t = LIST_TO_SET (TRAVERSE t)

⊢ ∀t s. t |++ s = FOLDL (flip $INSERT_PTREE) t (SET_TO_LIST s)

⊢ (∀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

⊢ ∀t. SIZE t = LENGTH (TRAVERSE t)

⊢ (∀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)

⊢ (∀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 <{}> = []) ∧ (∀j d. TRAVERSE (Leaf j d) = [j]) ∧
  ∀p m l r. TRAVERSE (Branch p m l r) = TRAVERSE l ⧺ TRAVERSE r

⊢ ∀t1 t2. t1 UNION_PTREE t2 = t1 |++ NUMSET_OF_PTREE t2

⊢ ∃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

⊢ (∀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

⊢ (∀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)))

⊢ ∀t k d e. t |+ (k,d) |+ (k,e) = t |+ (k,e)

⊢ ∀t k j d e.
    IS_PTREE t ∧ k ≠ j ⇒ (t |+ (k,d) |+ (j,e) = t |+ (j,e) |+ (k,d))

⊢ ∀v t n. t |+ (n,v) = n INSERT_PTREE t

⊢ ∀t x. IS_PTREE t ⇒ IS_PTREE (t |+ x)

⊢ ∀t l. IS_PTREE t ⇒ IS_PTREE (t |++ l)

⊢ ∀l. IS_PTREE (<{}> |++ l)

⊢ ∀f t k d. TRANSFORM f (t |+ (k,d)) = TRANSFORM f t |+ (k,f d)

⊢ (∀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)

⊢ ∀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)

⊢ ∀t. IS_PTREE t ⇒ ALL_DISTINCT (TRAVERSE t)

⊢ ∀p m l r.
    BRANCH (p,m,l,r) =
    if l = <{}> then r else if r = <{}> then l else Branch p m l r

⊢ ∀a b. a ≠ b ⇒ (BIT (BRANCHING_BIT a b) a ⇎ BIT (BRANCHING_BIT a b) b)

⊢ ∀a b. BRANCHING_BIT a b = BRANCHING_BIT b a

⊢ ∀a b. (BRANCHING_BIT a b = 0) ⇔ (ODD a ⇔ EVEN b) ∨ (a = b)

⊢ ∀p1 p0.
    BRANCHING_BIT p0 p1 =
    if (ODD p0 ⇔ EVEN p1) ∨ (p0 = p1) then 0
    else SUC (BRANCHING_BIT (DIV2 p0) (DIV2 p1))

⊢ ∀P. (∀p0 p1.
         (¬((ODD p0 ⇔ EVEN p1) ∨ (p0 = p1)) ⇒ P (DIV2 p0) (DIV2 p1)) ⇒
         P p0 p1) ⇒
      ∀v v1. P v v1

⊢ (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))

⊢ ∀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)

⊢ ∀l. ALL_DISTINCT l ⇒ (CARD (LIST_TO_SET l) = LENGTH l)

⊢ ∀t. IS_PTREE t ⇒ (CARD (NUMSET_OF_PTREE t) = SIZE t)

⊢ ∀x s1 s2. s1 ∪ s2 DELETE x = s1 DELETE x ∪ (s2 DELETE x)

⊢ IS_PTREE <{}>

⊢ ∀P t k d. P k d ∧ EVERY_LEAF P t ⇒ EVERY_LEAF P (t |+ (k,d))

⊢ ∀P p m l r.
    EVERY_LEAF P (BRANCH (p,m,l,r)) ⇔ EVERY_LEAF P l ∧ EVERY_LEAF P r

⊢ ∀P t k. EVERY_LEAF P t ∧ IS_SOME (t ' k) ⇒ P k (THE (t ' k))

⊢ ∀P t k. EVERY_LEAF P t ⇒ EVERY_LEAF P (t \\ k)

⊢ ∀P Q f t.
    (∀k d. P k d ⇒ Q k (f d)) ∧ EVERY_LEAF P t ⇒
    EVERY_LEAF Q (TRANSFORM f t)

⊢ ∀P l. (∀n. n < LENGTH l ⇒ ¬P l❲n❳) ⇔ (FILTER P l = [])

⊢ ∀P l. (∀n. n < LENGTH l ⇒ P l❲n❳) ⇒ (FILTER P l = l)

⊢ ∀t. FINITE (NUMSET_OF_PTREE t)

⊢ ∀t x. IS_PTREE t ⇒ IS_PTREE (x INSERT_PTREE t)

⊢ ∀t n. IS_PTREE t ⇒ (n ∈ NUMSET_OF_PTREE t ⇔ n IN_PTREE t)

⊢ ∀n. ¬(n IN_PTREE <{}>)

⊢ ∀t m n.
    IS_PTREE t ⇒ (n IN_PTREE m INSERT_PTREE t ⇔ (m = n) ∨ n IN_PTREE t)

⊢ ∀t s n.
    IS_PTREE t ∧ FINITE s ⇒ (n IN_PTREE t |++ s ⇔ n IN_PTREE t ∨ n ∈ s)

⊢ ∀s n. FINITE s ⇒ (n ∈ s ⇔ n IN_PTREE <{}> |++ s)

⊢ ∀t m n. IS_PTREE t ⇒ (n IN_PTREE t \\ m ⇔ n ≠ m ∧ n IN_PTREE t)

⊢ ∀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 <{}> ⇔ T) ∧ (IS_EMPTY (Leaf v v1) ⇔ F) ∧
  (IS_EMPTY (Branch v2 v3 v4 v5) ⇔ F)

⊢ ∀P. P <{}> ∧ (∀v v1. P (Leaf v v1)) ∧
      (∀v2 v3 v4 v5. P (Branch v2 v3 v4 v5)) ⇒
      ∀v. P v

⊢ ∀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))

⊢ (∀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)

⊢ ∀t1 t2.
    IS_PTREE t1 ∧ IS_PTREE t2 ⇒
    ((KEYS t1 = KEYS t2) ⇔ (TRAVERSE t1 = TRAVERSE t2))

⊢ ∀l1 l2.
    ALL_DISTINCT l1 ∧ ALL_DISTINCT l2 ∧ (∀x. MEM x l1 ⇔ MEM x l2) ⇒
    PERM l1 l2

⊢ ∀t k. IS_PTREE t ⇒ (MEM k (TRAVERSE t) ⇔ k ∈ NUMSET_OF_PTREE t)

⊢ ∀t x h.
    IS_PTREE t ⇒
    (MEM x (TRAVERSE (h INSERT_PTREE t)) ⇔
     (x = h) ∨ x ≠ h ∧ MEM x (TRAVERSE t))

⊢ ∀t k. IS_PTREE t ⇒ (MEM k (TRAVERSE t) ⇔ IS_SOME (t ' k))

⊢ ∀P Q t. (∀k d. P k d ⇒ Q k d) ∧ EVERY_LEAF P t ⇒ EVERY_LEAF Q t

⊢ ∀t k d. t |+ (k,d) ≠ <{}>

⊢ ∀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 <{}> = ∅

⊢ ∀t s.
    IS_PTREE t ∧ FINITE s ⇒
    (NUMSET_OF_PTREE (t |++ s) = NUMSET_OF_PTREE t ∪ s)

⊢ ∀s. FINITE s ⇒ (NUMSET_OF_PTREE (<{}> |++ s) = s)

⊢ ∀t k d j.
    IS_PTREE t ⇒ ((t |+ (k,d)) ' j = if k = j then SOME d else t ' j)

⊢ ∀t k j.
    IS_PTREE t ⇒
    ((k INSERT_PTREE t) ' j = if k = j then SOME () else t ' j)

⊢ ∀P t k. (∀d. ¬P k d) ∧ EVERY_LEAF P t ⇒ (t ' k = NONE)

⊢ ∀t k j. IS_PTREE t ⇒ ((t \\ k) ' j = if k = j then NONE else t ' j)

⊢ ∀f t k.
    TRANSFORM f t ' k = case t ' k of NONE => NONE | SOME x => SOME (f x)

⊢ (∀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

⊢ ∀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

⊢ ∀t k d.
    IS_PTREE t ∧ ¬MEM k (TRAVERSE t) ⇒
    PERM (TRAVERSE (t |+ (k,d))) (k::TRAVERSE t)

⊢ ∀t k.
    IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒
    PERM (TRAVERSE (t \\ k)) (FILTER (λx. x ≠ k) (TRAVERSE t))

⊢ ∀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))

⊢ ∀t k d.
    IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒ (TRAVERSE (t |+ (k,d)) = TRAVERSE t)

⊢ ∀t k. IS_PTREE t ∧ ¬MEM k (TRAVERSE t) ⇒ (TRAVERSE (t \\ k) = TRAVERSE t)

⊢ ∀t k.
    IS_PTREE t ∧ MEM k (TRAVERSE t) ⇒
    PERM (TRAVERSE (t \\ k)) (FILTER (λx. x ≠ k) (TRAVERSE t))

⊢ ∀t1 t2. IS_PTREE t1 ∧ IS_PTREE t2 ⇒ ((∀k. t1 ' k = t2 ' k) ⇔ (t1 = t2))

⊢ ∀t1 t2.
    IS_PTREE t1 ∧ IS_PTREE t2 ⇒
    ((t1 = t2) ⇔ ∀x. x IN_PTREE t1 ⇔ x IN_PTREE t2)

⊢ ∀s x. FINITE s ⇒ (<{}> |++ (s DELETE x) = (<{}> |++ s) \\ x)

⊢ ∀t. t |++ ∅ = t

⊢ ∀t s x.
    IS_PTREE t ∧ FINITE s ⇒ (t |++ (x INSERT s) = x INSERT_PTREE t |++ s)

⊢ ∀s x. FINITE s ⇒ (<{}> |++ (x INSERT s) = x INSERT_PTREE <{}> |++ s)

⊢ ∀t s. IS_PTREE t ⇒ IS_PTREE (t |++ s)

⊢ T

⊢ ∀t s.
    IS_PTREE t ∧ FINITE s ⇒ (<{}> |++ (NUMSET_OF_PTREE t ∪ s) = t |++ s)

⊢ ∀t s1 s2.
    IS_PTREE t ∧ FINITE s1 ∧ FINITE s2 ⇒
    (t |++ (s1 ∪ s2) = t |++ s1 |++ s2)

⊢ ∀t1 t2.
    IS_PTREE t1 ∧ IS_PTREE t2 ⇒
    ((∀k. MEM k (TRAVERSE t1) ⇔ MEM k (TRAVERSE t2)) ⇔
     (TRAVERSE t1 = TRAVERSE t2))

⊢ ∀l2 l1 R. (QSORT R l1 = QSORT R l2) ⇒ ∀x. MEM x l1 ⇔ MEM x l2

⊢ ∀t k d j.
    IS_PTREE t ⇒
    (t |+ (k,d) \\ j = if k = j then t \\ j else t \\ j |+ (k,d))

⊢ ∀t k d. t |+ (k,d) \\ k = t \\ k

⊢ ∀t k. IS_PTREE t ⇒ IS_PTREE (t \\ k)

⊢ ∀t k. IS_PTREE t ⇒ (t \\ k \\ k = t \\ k)

⊢ ∀f t k. TRANSFORM f (t \\ k) = TRANSFORM f t \\ k

⊢ (SIZE <{}> = 0) ∧ (∀k d. SIZE (Leaf k d) = 1) ∧
  ∀p m l r. SIZE (Branch p m l r) = SIZE l + SIZE r

⊢ ∀t k d.
    IS_PTREE t ⇒
    (SIZE (t |+ (k,d)) = if MEM k (TRAVERSE t) then SIZE t else SIZE t + 1)

⊢ ∀t s.
    FINITE s ⇒
    IS_PTREE t ∧ ALL_DISTINCT (TRAVERSE t ⧺ SET_TO_LIST s) ⇒
    (SIZE (t |++ s) = SIZE t + CARD s)

⊢ ∀s. FINITE s ⇒ (SIZE (<{}> |++ s) = CARD s)

⊢ ∀t k.
    IS_PTREE t ⇒
    (SIZE (t \\ k) = if MEM k (TRAVERSE t) then SIZE t − 1 else SIZE t)

⊢ ∀f p m l r.
    TRANSFORM f (BRANCH (p,m,l,r)) =
    BRANCH (p,m,TRANSFORM f l,TRANSFORM f r)

⊢ ∀f t. (TRANSFORM f t = <{}>) ⇔ (t = <{}>)

⊢ ∀f t. IS_PTREE t ⇒ IS_PTREE (TRANSFORM f t)

⊢ ∀t. TRAVERSE t = TRAVERSE_AUX t []

⊢ ∀f t. TRAVERSE (TRANSFORM f t) = TRAVERSE t

⊢ ∀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)

⊢ ∀t1 t2.
    IS_PTREE t1 ∧ IS_PTREE t2 ⇒ (t1 UNION_PTREE t2 = t2 UNION_PTREE t1)

⊢ ∀t. IS_PTREE t ⇒ (<{}> UNION_PTREE t = t UNION_PTREE <{}>)

⊢ (∀t. t UNION_PTREE <{}> = t) ∧ ∀t. IS_PTREE t ⇒ (<{}> UNION_PTREE t = t)

⊢ ∀t1 t2. IS_PTREE t1 ∧ IS_PTREE t2 ⇒ IS_PTREE (t1 UNION_PTREE t2)

⊢ DATATYPE (ptree <{}> Leaf Branch)

⊢ (∀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')

⊢ ∀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)

⊢ ∀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 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')

⊢ (∀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

⊢ ∀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

⊢ ∀pp.
    (pp = <{}>) ∨ (∃n a. pp = Leaf n a) ∨ ∃n0 n p p0. pp = Branch n0 n p p0