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|
signature divDefLibTestTheory =
sig
type thm = Thm.thm
(* Definitions *)
val btree_TY_DEF : thm
val btree_case_def : thm
val btree_height_def : thm
val btree_size_def : thm
val even_def : thm
val list_t_TY_DEF : thm
val list_t_case_def : thm
val list_t_size_def : thm
val node_TY_DEF : thm
val node_case_def : thm
val nth0_def : thm
val nth_def : thm
val odd_def : thm
val tree_TY_DEF : thm
val tree_case_def : thm
val tree_height_def : thm
val tree_nodes_height_def : thm
val tree_size_def : thm
(* Theorems *)
val btree_11 : thm
val btree_Axiom : thm
val btree_case_cong : thm
val btree_case_eq : thm
val btree_distinct : thm
val btree_induction : thm
val btree_nchotomy : thm
val datatype_btree : thm
val datatype_list_t : thm
val datatype_tree : thm
val list_t_11 : thm
val list_t_Axiom : thm
val list_t_case_cong : thm
val list_t_case_eq : thm
val list_t_distinct : thm
val list_t_induction : thm
val list_t_nchotomy : thm
val node_11 : thm
val node_Axiom : thm
val node_case_cong : thm
val node_case_eq : thm
val node_induction : thm
val node_nchotomy : thm
val tree_11 : thm
val tree_Axiom : thm
val tree_case_cong : thm
val tree_case_eq : thm
val tree_distinct : thm
val tree_induction : thm
val tree_nchotomy : thm
val tree_size_eq : thm
val divDefLibTest_grammars : type_grammar.grammar * term_grammar.grammar
(*
[divDef] Parent theory of "divDefLibTest"
[btree_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa0'.
∀ $var$('btree').
(∀a0'.
(∃a. a0' =
(λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM))
a) ∨
(∃a0 a1.
a0' =
(λa0 a1.
ind_type$CONSTR (SUC 0) ARB
(ind_type$FCONS a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
a0 a1 ∧ $var$('btree') a0 ∧ $var$('btree') a1) ⇒
$var$('btree') a0') ⇒
$var$('btree') a0') rep
[btree_case_def] Definition
⊢ (∀a f f1. btree_CASE (BLeaf a) f f1 = f a) ∧
∀a0 a1 f f1. btree_CASE (BNode a0 a1) f f1 = f1 a0 a1
[btree_height_def] Definition
⊢ ∀tree.
btree_height tree =
case tree of
BLeaf v => Return 1
| BNode l r =>
do
hl <- btree_height l;
hr <- btree_height r;
Return (hl + hr)
od
[btree_size_def] Definition
⊢ (∀f a. btree_size f (BLeaf a) = 1 + f a) ∧
∀f a0 a1.
btree_size f (BNode a0 a1) =
1 + (btree_size f a0 + btree_size f a1)
[even_def] Definition
⊢ ∀i. even i = if i = 0 then Return T else odd (i − 1)
[list_t_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa0'.
∀ $var$('list_t').
(∀a0'.
(∃a0 a1.
a0' =
(λa0 a1.
ind_type$CONSTR 0 a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM)))
a0 a1 ∧ $var$('list_t') a1) ∨
a0' =
ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ⇒
$var$('list_t') a0') ⇒
$var$('list_t') a0') rep
[list_t_case_def] Definition
⊢ (∀a0 a1 f v. list_t_CASE (ListCons a0 a1) f v = f a0 a1) ∧
∀f v. list_t_CASE ListNil f v = v
[list_t_size_def] Definition
⊢ (∀f a0 a1.
list_t_size f (ListCons a0 a1) = 1 + (f a0 + list_t_size f a1)) ∧
∀f. list_t_size f ListNil = 0
[node_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa1'.
∀ $var$('tree') $var$('node')
$var$('@temp @ind_typedivDefLibTest8list').
(∀a0'.
(∃a. a0' =
(λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM))
a) ∨
(∃a. a0' =
(λa.
ind_type$CONSTR (SUC 0) ARB
(ind_type$FCONS a (λn. ind_type$BOTTOM)))
a ∧ $var$('node') a) ⇒
$var$('tree') a0') ∧
(∀a1'.
(∃a. a1' =
(λa.
ind_type$CONSTR (SUC (SUC 0)) ARB
(ind_type$FCONS a (λn. ind_type$BOTTOM)))
a ∧
$var$('@temp @ind_typedivDefLibTest8list') a) ⇒
$var$('node') a1') ∧
(∀a2.
a2 =
ind_type$CONSTR (SUC (SUC (SUC 0))) ARB
(λn. ind_type$BOTTOM) ∨
(∃a0 a1.
a2 =
(λa0 a1.
ind_type$CONSTR (SUC (SUC (SUC (SUC 0)))) ARB
(ind_type$FCONS a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
a0 a1 ∧ $var$('tree') a0 ∧
$var$('@temp @ind_typedivDefLibTest8list') a1) ⇒
$var$('@temp @ind_typedivDefLibTest8list') a2) ⇒
$var$('node') a1') rep
[node_case_def] Definition
⊢ ∀a f. node_CASE (Node a) f = f a
[nth0_def] Definition
⊢ ∀ls i.
nth0 ls i =
case ls of
ListCons x tl => if i = 0 then Return x else nth0 tl (i − 1)
| ListNil => Fail Failure
[nth_def] Definition
⊢ ∀ls i.
nth ls i =
case ls of
ListCons x tl =>
if u32_to_int i = 0 then Return x
else do i0 <- u32_sub i (int_to_u32 1); nth tl i0 od
| ListNil => Fail Failure
[odd_def] Definition
⊢ ∀i. odd i = if i = 0 then Return F else even (i − 1)
[tree_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa0'.
∀ $var$('tree') $var$('node')
$var$('@temp @ind_typedivDefLibTest8list').
(∀a0'.
(∃a. a0' =
(λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM))
a) ∨
(∃a. a0' =
(λa.
ind_type$CONSTR (SUC 0) ARB
(ind_type$FCONS a (λn. ind_type$BOTTOM)))
a ∧ $var$('node') a) ⇒
$var$('tree') a0') ∧
(∀a1'.
(∃a. a1' =
(λa.
ind_type$CONSTR (SUC (SUC 0)) ARB
(ind_type$FCONS a (λn. ind_type$BOTTOM)))
a ∧
$var$('@temp @ind_typedivDefLibTest8list') a) ⇒
$var$('node') a1') ∧
(∀a2.
a2 =
ind_type$CONSTR (SUC (SUC (SUC 0))) ARB
(λn. ind_type$BOTTOM) ∨
(∃a0 a1.
a2 =
(λa0 a1.
ind_type$CONSTR (SUC (SUC (SUC (SUC 0)))) ARB
(ind_type$FCONS a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
a0 a1 ∧ $var$('tree') a0 ∧
$var$('@temp @ind_typedivDefLibTest8list') a1) ⇒
$var$('@temp @ind_typedivDefLibTest8list') a2) ⇒
$var$('tree') a0') rep
[tree_case_def] Definition
⊢ (∀a f f1. tree_CASE (TLeaf a) f f1 = f a) ∧
∀a f f1. tree_CASE (TNode a) f f1 = f1 a
[tree_height_def] Definition
⊢ ∀tree.
tree_height tree =
case tree of
TLeaf v => Return 1
| TNode (Node ls) => tree_nodes_height ls
[tree_nodes_height_def] Definition
⊢ ∀ls.
tree_nodes_height ls =
case ls of
[] => Return 0
| t::tl =>
do
h1 <- tree_height t;
h2 <- tree_nodes_height tl;
Return (h1 + h2)
od
[tree_size_def] Definition
⊢ (∀f a. tree_size f (TLeaf a) = 1 + f a) ∧
(∀f a. tree_size f (TNode a) = 1 + node_size f a) ∧
(∀f a. node_size f (Node a) = 1 + tree1_size f a) ∧
(∀f. tree1_size f [] = 0) ∧
∀f a0 a1.
tree1_size f (a0::a1) = 1 + (tree_size f a0 + tree1_size f a1)
[btree_11] Theorem
⊢ (∀a a'. BLeaf a = BLeaf a' ⇔ a = a') ∧
∀a0 a1 a0' a1'. BNode a0 a1 = BNode a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
[btree_Axiom] Theorem
⊢ ∀f0 f1. ∃fn.
(∀a. fn (BLeaf a) = f0 a) ∧
∀a0 a1. fn (BNode a0 a1) = f1 a0 a1 (fn a0) (fn a1)
[btree_case_cong] Theorem
⊢ ∀M M' f f1.
M = M' ∧ (∀a. M' = BLeaf a ⇒ f a = f' a) ∧
(∀a0 a1. M' = BNode a0 a1 ⇒ f1 a0 a1 = f1' a0 a1) ⇒
btree_CASE M f f1 = btree_CASE M' f' f1'
[btree_case_eq] Theorem
⊢ btree_CASE x f f1 = v ⇔
(∃a. x = BLeaf a ∧ f a = v) ∨ ∃b b0. x = BNode b b0 ∧ f1 b b0 = v
[btree_distinct] Theorem
⊢ ∀a1 a0 a. BLeaf a ≠ BNode a0 a1
[btree_induction] Theorem
⊢ ∀P. (∀a. P (BLeaf a)) ∧ (∀b b0. P b ∧ P b0 ⇒ P (BNode b b0)) ⇒
∀b. P b
[btree_nchotomy] Theorem
⊢ ∀bb. (∃a. bb = BLeaf a) ∨ ∃b b0. bb = BNode b b0
[datatype_btree] Theorem
⊢ DATATYPE (btree BLeaf BNode)
[datatype_list_t] Theorem
⊢ DATATYPE (list_t ListCons ListNil)
[datatype_tree] Theorem
⊢ DATATYPE (tree TLeaf TNode ∧ node Node)
[list_t_11] Theorem
⊢ ∀a0 a1 a0' a1'.
ListCons a0 a1 = ListCons a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
[list_t_Axiom] Theorem
⊢ ∀f0 f1. ∃fn.
(∀a0 a1. fn (ListCons a0 a1) = f0 a0 a1 (fn a1)) ∧
fn ListNil = f1
[list_t_case_cong] Theorem
⊢ ∀M M' f v.
M = M' ∧ (∀a0 a1. M' = ListCons a0 a1 ⇒ f a0 a1 = f' a0 a1) ∧
(M' = ListNil ⇒ v = v') ⇒
list_t_CASE M f v = list_t_CASE M' f' v'
[list_t_case_eq] Theorem
⊢ list_t_CASE x f v = v' ⇔
(∃t l. x = ListCons t l ∧ f t l = v') ∨ x = ListNil ∧ v = v'
[list_t_distinct] Theorem
⊢ ∀a1 a0. ListCons a0 a1 ≠ ListNil
[list_t_induction] Theorem
⊢ ∀P. (∀l. P l ⇒ ∀t. P (ListCons t l)) ∧ P ListNil ⇒ ∀l. P l
[list_t_nchotomy] Theorem
⊢ ∀ll. (∃t l. ll = ListCons t l) ∨ ll = ListNil
[node_11] Theorem
⊢ ∀a a'. Node a = Node a' ⇔ a = a'
[node_Axiom] Theorem
⊢ ∀f0 f1 f2 f3 f4. ∃fn0 fn1 fn2.
(∀a. fn0 (TLeaf a) = f0 a) ∧ (∀a. fn0 (TNode a) = f1 a (fn1 a)) ∧
(∀a. fn1 (Node a) = f2 a (fn2 a)) ∧ fn2 [] = f3 ∧
∀a0 a1. fn2 (a0::a1) = f4 a0 a1 (fn0 a0) (fn2 a1)
[node_case_cong] Theorem
⊢ ∀M M' f.
M = M' ∧ (∀a. M' = Node a ⇒ f a = f' a) ⇒
node_CASE M f = node_CASE M' f'
[node_case_eq] Theorem
⊢ node_CASE x f = v ⇔ ∃l. x = Node l ∧ f l = v
[node_induction] Theorem
⊢ ∀P0 P1 P2.
(∀a. P0 (TLeaf a)) ∧ (∀n. P1 n ⇒ P0 (TNode n)) ∧
(∀l. P2 l ⇒ P1 (Node l)) ∧ P2 [] ∧
(∀t l. P0 t ∧ P2 l ⇒ P2 (t::l)) ⇒
(∀t. P0 t) ∧ (∀n. P1 n) ∧ ∀l. P2 l
[node_nchotomy] Theorem
⊢ ∀nn. ∃l. nn = Node l
[tree_11] Theorem
⊢ (∀a a'. TLeaf a = TLeaf a' ⇔ a = a') ∧
∀a a'. TNode a = TNode a' ⇔ a = a'
[tree_Axiom] Theorem
⊢ ∀f0 f1 f2 f3 f4. ∃fn0 fn1 fn2.
(∀a. fn0 (TLeaf a) = f0 a) ∧ (∀a. fn0 (TNode a) = f1 a (fn1 a)) ∧
(∀a. fn1 (Node a) = f2 a (fn2 a)) ∧ fn2 [] = f3 ∧
∀a0 a1. fn2 (a0::a1) = f4 a0 a1 (fn0 a0) (fn2 a1)
[tree_case_cong] Theorem
⊢ ∀M M' f f1.
M = M' ∧ (∀a. M' = TLeaf a ⇒ f a = f' a) ∧
(∀a. M' = TNode a ⇒ f1 a = f1' a) ⇒
tree_CASE M f f1 = tree_CASE M' f' f1'
[tree_case_eq] Theorem
⊢ tree_CASE x f f1 = v ⇔
(∃a. x = TLeaf a ∧ f a = v) ∨ ∃n. x = TNode n ∧ f1 n = v
[tree_distinct] Theorem
⊢ ∀a' a. TLeaf a ≠ TNode a'
[tree_induction] Theorem
⊢ ∀P0 P1 P2.
(∀a. P0 (TLeaf a)) ∧ (∀n. P1 n ⇒ P0 (TNode n)) ∧
(∀l. P2 l ⇒ P1 (Node l)) ∧ P2 [] ∧
(∀t l. P0 t ∧ P2 l ⇒ P2 (t::l)) ⇒
(∀t. P0 t) ∧ (∀n. P1 n) ∧ ∀l. P2 l
[tree_nchotomy] Theorem
⊢ ∀tt. (∃a. tt = TLeaf a) ∨ ∃n. tt = TNode n
[tree_size_eq] Theorem
⊢ tree1_size f = list_size (tree_size f)
*)
end
|
