summaryrefslogtreecommitdiff
path: root/backends/hol4/divDefLibTestTheory.sig
diff options
context:
space:
mode:
Diffstat (limited to '')
-rw-r--r--backends/hol4/divDefLibTestTheory.sig434
1 files changed, 434 insertions, 0 deletions
diff --git a/backends/hol4/divDefLibTestTheory.sig b/backends/hol4/divDefLibTestTheory.sig
new file mode 100644
index 00000000..8da736ab
--- /dev/null
+++ b/backends/hol4/divDefLibTestTheory.sig
@@ -0,0 +1,434 @@
+signature divDefLibTestTheory =
+sig
+ type thm = Thm.thm
+
+ (* Definitions *)
+ val btree_TY_DEF : thm
+ val btree_case_def : thm
+ val btree_size_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 tree_TY_DEF : thm
+ val tree_case_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_height_def : thm
+ val btree_induction : thm
+ val btree_nchotomy : thm
+ val datatype_btree : thm
+ val datatype_list_t : thm
+ val datatype_tree : thm
+ val even_def : 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 nth_def : thm
+ val odd_def : 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_height_def : thm
+ val tree_induction : thm
+ val tree_nchotomy : thm
+ val tree_nodes_height_def : 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_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)
+
+ [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
+
+ [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_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_height_def] Theorem
+
+ ⊢ ∀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_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)
+
+ [even_def] Theorem
+
+ ⊢ ∀i. even i = if i = 0 then Return T else odd (i − 1)
+
+ [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
+
+ [nth_def] Theorem
+
+ ⊢ ∀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] Theorem
+
+ ⊢ ∀i. odd i = if i = 0 then Return F else even (i − 1)
+
+ [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_height_def] Theorem
+
+ ⊢ ∀tree.
+ tree_height tree =
+ case tree of
+ TLeaf v => Return 1
+ | TNode (Node ls) => tree_nodes_height ls
+
+ [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_nodes_height_def] Theorem
+
+ ⊢ ∀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_eq] Theorem
+
+ ⊢ tree1_size f = list_size (tree_size f)
+
+
+*)
+end
generated by cgit v1.2.3 (git 2.25.1) at 2024-10-19 16:30:49 +0000