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_tree_curried_def : thm
val encode_tree_tupled_primitive_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 lift_tree_curried_def : thm
val lift_tree_tupled_primitive_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_def_compute : thm
val encode_bnum_def_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_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
(*
[quantHeuristics] Parent theory of "Encode"
[rich_list] 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 n.
I
(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_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_tree_curried_def] Definition
|- ∀x x1. encode_tree x x1 = encode_tree_tupled (x,x1)
[encode_tree_tupled_primitive_def] Definition
|- encode_tree_tupled =
WFREC (@R. WF R ∧ ∀a e ts a'. MEM a' ts ⇒ R (e,a') (e,Node a ts))
(λencode_tree_tupled a'.
case a' of
(e,Node a ts) =>
I (e a ++ encode_list (λa. encode_tree_tupled (e,a)) ts))
[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
[lift_tree_curried_def] Definition
|- ∀x x1. lift_tree x x1 ⇔ lift_tree_tupled (x,x1)
[lift_tree_tupled_primitive_def] Definition
|- lift_tree_tupled =
WFREC (@R. WF R ∧ ∀a p ts a'. MEM a' ts ⇒ R (p,a') (p,Node a ts))
(λlift_tree_tupled a'.
case a' of
(p,Node a ts) =>
I (p a ∧ EVERY (λa. lift_tree_tupled (p,a)) ts))
[tree_TY_DEF] Definition
|- ∃rep.
TYPE_DEFINITION
(λa0'.
∀'tree' '@temp @ind_typeEncode0list' .
(∀a0'.
(∃a0 a1.
(a0' =
(λa0 a1.
ind_type$CONSTR 0 a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM))) a0
a1) ∧ '@temp @ind_typeEncode0list' a1) ⇒
'tree' a0') ∧
(∀a1'.
(a1' =
ind_type$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) ∧
'tree' a0 ∧ '@temp @ind_typeEncode0list' a1) ⇒
'@temp @ind_typeEncode0list' a1') ⇒
'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_def_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_def_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_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-11