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signature paperTheory =
sig
type thm = Thm.thm
(* Definitions *)
val call_choose_fwd_def : thm
val choose_back_def : thm
val choose_fwd_def : thm
val list_nth_mut_back_def : thm
val list_nth_mut_fwd_def : thm
val list_t_TY_DEF : thm
val list_t_case_def : thm
val list_t_size_def : thm
val ref_incr_fwd_back_def : thm
val sum_fwd_def : thm
val test_choose_fwd_def : thm
val test_incr_fwd_def : thm
val test_nth_fwd_def : thm
(* Theorems *)
val datatype_list_t : 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 paper_grammars : type_grammar.grammar * term_grammar.grammar
(*
[divDef] Parent theory of "paper"
[call_choose_fwd_def] Definition
⊢ ∀p. call_choose_fwd p =
(let
(px,py) = p
in
do
pz <- choose_fwd T px py;
pz0 <- u32_add pz (int_to_u32 1);
(px0,_) <- choose_back T px py pz0;
Return px0
od)
[choose_back_def] Definition
⊢ ∀b x y ret.
choose_back b x y ret =
if b then Return (ret,y) else Return (x,ret)
[choose_fwd_def] Definition
⊢ ∀b x y. choose_fwd b x y = if b then Return x else Return y
[list_nth_mut_back_def] Definition
⊢ ∀l i ret.
list_nth_mut_back l i ret =
case l of
ListCons x tl =>
if i = int_to_u32 0 then Return (ListCons ret tl)
else
do
i0 <- u32_sub i (int_to_u32 1);
tl0 <- list_nth_mut_back tl i0 ret;
Return (ListCons x tl0)
od
| ListNil => Fail Failure
[list_nth_mut_fwd_def] Definition
⊢ ∀l i.
list_nth_mut_fwd l i =
case l of
ListCons x tl =>
if i = int_to_u32 0 then Return x
else
do
i0 <- u32_sub i (int_to_u32 1);
list_nth_mut_fwd tl i0
od
| ListNil => Fail Failure
[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
[ref_incr_fwd_back_def] Definition
⊢ ∀x. ref_incr_fwd_back x = i32_add x (int_to_i32 1)
[sum_fwd_def] Definition
⊢ ∀l. sum_fwd l =
case l of
ListCons x tl => do i <- sum_fwd tl; i32_add x i od
| ListNil => Return (int_to_i32 0)
[test_choose_fwd_def] Definition
⊢ test_choose_fwd =
do
z <- choose_fwd T (int_to_i32 0) (int_to_i32 0);
z0 <- i32_add z (int_to_i32 1);
if z0 ≠ int_to_i32 1 then Fail Failure
else
do
(x,y) <- choose_back T (int_to_i32 0) (int_to_i32 0) z0;
if x ≠ int_to_i32 1 then Fail Failure
else if y ≠ int_to_i32 0 then Fail Failure
else Return ()
od
od
[test_incr_fwd_def] Definition
⊢ test_incr_fwd =
do
x <- ref_incr_fwd_back (int_to_i32 0);
if x ≠ int_to_i32 1 then Fail Failure else Return ()
od
[test_nth_fwd_def] Definition
⊢ test_nth_fwd =
(let
l = ListNil;
l0 = ListCons (int_to_i32 3) l;
l1 = ListCons (int_to_i32 2) l0
in
do
x <-
list_nth_mut_fwd (ListCons (int_to_i32 1) l1) (int_to_u32 2);
x0 <- i32_add x (int_to_i32 1);
l2 <-
list_nth_mut_back (ListCons (int_to_i32 1) l1)
(int_to_u32 2) x0;
i <- sum_fwd l2;
if i ≠ int_to_i32 7 then Fail Failure else Return ()
od)
[datatype_list_t] Theorem
⊢ DATATYPE (list_t ListCons ListNil)
[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
*)
end
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