Structure EncodeTheory
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 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
[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
HOL 4, Kananaskis-13