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|
signature divDefNoFixLibTestTheory =
sig
type thm = Thm.thm
(* Definitions *)
val even___E_def : thm
val even___P_def : thm
val even___fuel0_UNION_extract0_def : thm
val even___fuel0_UNION_extract1_def : thm
val even___fuel0_UNION_primitive_def : thm
val even___fuel_UNION_extract0_def : thm
val even___fuel_UNION_extract1_def : thm
val even___fuel_UNION_primitive_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 nth_mut_fwd___E_def : thm
val nth_mut_fwd___P_def : thm
val nth_mut_fwd_def : thm
val odd___E_def : thm
val odd___P_def : thm
val odd_def : thm
(* Theorems *)
val datatype_list_t : thm
val even___fuel0_def : thm
val even___fuel0_ind : thm
val even___fuel_def : thm
val even___fuel_ind : 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 nth_mut_fwd___fuel0_def : thm
val nth_mut_fwd___fuel0_ind : thm
val nth_mut_fwd___fuel_def : thm
val nth_mut_fwd___fuel_ind : thm
val divDefNoFixLibTest_grammars : type_grammar.grammar * term_grammar.grammar
(*
[primitives] Parent theory of "divDefNoFixLibTest"
[even___E_def] Definition
⊢ ∀even odd i.
even___E even odd i =
if i = 0 then do b <- Return T; Return b od
else do b <- odd (i − 1); Return b od
[even___P_def] Definition
⊢ ∀i $var$($n).
even___P i $var$($n) ⇔ ¬is_diverge (even___fuel0 $var$($n) i)
[even___fuel0_UNION_extract0_def] Definition
⊢ ∀x x0. even___fuel0 x x0 = even___fuel0_UNION (INL (x,x0))
[even___fuel0_UNION_extract1_def] Definition
⊢ ∀x x0. odd___fuel0 x x0 = even___fuel0_UNION (INR (x,x0))
[even___fuel0_UNION_primitive_def] Definition
⊢ even___fuel0_UNION =
WFREC
(@R. WF R ∧
(∀i $var$($n) $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒
R (INR ($var$($m),i − 1)) (INL ($var$($n),i))) ∧
∀i $var$($n) $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒
R (INL ($var$($m),i − 1)) (INR ($var$($n),i)))
(λeven___fuel0_UNION a.
case a of
INL ($var$($n),i) =>
I
(case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then do b <- Return T; Return b od
else
do
b <- even___fuel0_UNION (INR ($var$($m),i − 1));
Return b
od)
| INR ($var$($n'),i') =>
I
(case $var$($n') of
0 => Diverge
| SUC $var$($m) =>
if i' = 0 then do b <- Return F; Return b od
else
do
b <- even___fuel0_UNION (INL ($var$($m),i' − 1));
Return b
od))
[even___fuel_UNION_extract0_def] Definition
⊢ ∀x x0. even___fuel x x0 = even___fuel_UNION (INL (x,x0))
[even___fuel_UNION_extract1_def] Definition
⊢ ∀x x0. odd___fuel x x0 = even___fuel_UNION (INR (x,x0))
[even___fuel_UNION_primitive_def] Definition
⊢ even___fuel_UNION =
WFREC
(@R. WF R ∧
(∀i $var$($n) $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒
R (INR ($var$($m),i − 1)) (INL ($var$($n),i))) ∧
∀i $var$($n) $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒
R (INL ($var$($m),i − 1)) (INR ($var$($n),i)))
(λeven___fuel_UNION a.
case a of
INL ($var$($n),i) =>
I
(case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then Return T
else even___fuel_UNION (INR ($var$($m),i − 1)))
| INR ($var$($n'),i') =>
I
(case $var$($n') of
0 => Diverge
| SUC $var$($m) =>
if i' = 0 then Return F
else even___fuel_UNION (INL ($var$($m),i' − 1))))
[even_def] Definition
⊢ ∀i. even i =
if ∃ $var$($n). even___P i $var$($n) then
even___fuel0 ($LEAST (even___P i)) i
else Diverge
[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
[nth_mut_fwd___E_def] Definition
⊢ ∀nth_mut_fwd ls i.
nth_mut_fwd___E nth_mut_fwd 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);
x <- nth_mut_fwd tl i0;
Return x
od
| ListNil => Fail Failure
[nth_mut_fwd___P_def] Definition
⊢ ∀ls i $var$($n).
nth_mut_fwd___P ls i $var$($n) ⇔
¬is_diverge (nth_mut_fwd___fuel0 $var$($n) ls i)
[nth_mut_fwd_def] Definition
⊢ ∀ls i.
nth_mut_fwd ls i =
if ∃ $var$($n). nth_mut_fwd___P ls i $var$($n) then
nth_mut_fwd___fuel0 ($LEAST (nth_mut_fwd___P ls i)) ls i
else Diverge
[odd___E_def] Definition
⊢ ∀even odd i.
odd___E even odd i =
if i = 0 then do b <- Return F; Return b od
else do b <- even (i − 1); Return b od
[odd___P_def] Definition
⊢ ∀i $var$($n).
odd___P i $var$($n) ⇔ ¬is_diverge (odd___fuel0 $var$($n) i)
[odd_def] Definition
⊢ ∀i. odd i =
if ∃ $var$($n). odd___P i $var$($n) then
odd___fuel0 ($LEAST (odd___P i)) i
else Diverge
[datatype_list_t] Theorem
⊢ DATATYPE (list_t ListCons ListNil)
[even___fuel0_def] Theorem
⊢ (∀i $var$($n).
even___fuel0 $var$($n) i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then do b <- Return T; Return b od
else do b <- odd___fuel0 $var$($m) (i − 1); Return b od) ∧
∀i $var$($n).
odd___fuel0 $var$($n) i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then do b <- Return F; Return b od
else do b <- even___fuel0 $var$($m) (i − 1); Return b od
[even___fuel0_ind] Theorem
⊢ ∀P0 P1.
(∀ $var$($n) i.
(∀ $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P1 $var$($m) (i − 1)) ⇒
P0 $var$($n) i) ∧
(∀ $var$($n) i.
(∀ $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P0 $var$($m) (i − 1)) ⇒
P1 $var$($n) i) ⇒
(∀v0 v1. P0 v0 v1) ∧ ∀v0 v1. P1 v0 v1
[even___fuel_def] Theorem
⊢ (∀i $var$($n).
even___fuel $var$($n) i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then Return T else odd___fuel $var$($m) (i − 1)) ∧
∀i $var$($n).
odd___fuel $var$($n) i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
if i = 0 then Return F else even___fuel $var$($m) (i − 1)
[even___fuel_ind] Theorem
⊢ ∀P0 P1.
(∀ $var$($n) i.
(∀ $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P1 $var$($m) (i − 1)) ⇒
P0 $var$($n) i) ∧
(∀ $var$($n) i.
(∀ $var$($m).
$var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P0 $var$($m) (i − 1)) ⇒
P1 $var$($n) i) ⇒
(∀v0 v1. P0 v0 v1) ∧ ∀v0 v1. P1 v0 v1
[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
[nth_mut_fwd___fuel0_def] Theorem
⊢ ∀ls i $var$($n).
nth_mut_fwd___fuel0 $var$($n) ls i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
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);
x <- nth_mut_fwd___fuel0 $var$($m) tl i0;
Return x
od
| ListNil => Fail Failure
[nth_mut_fwd___fuel0_ind] Theorem
⊢ ∀P. (∀ $var$($n) ls i.
(∀ $var$($m) x tl i0.
$var$($n) = SUC $var$($m) ∧ ls = ListCons x tl ∧
u32_to_int i ≠ 0 ⇒
P $var$($m) tl i0) ⇒
P $var$($n) ls i) ⇒
∀v v1 v2. P v v1 v2
[nth_mut_fwd___fuel_def] Theorem
⊢ ∀ls i $var$($n).
nth_mut_fwd___fuel $var$($n) ls i =
case $var$($n) of
0 => Diverge
| SUC $var$($m) =>
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_mut_fwd___fuel $var$($m) tl i0
od
| ListNil => Fail Failure
[nth_mut_fwd___fuel_ind] Theorem
⊢ ∀P. (∀ $var$($n) ls i.
(∀ $var$($m) x tl i0.
$var$($n) = SUC $var$($m) ∧ ls = ListCons x tl ∧
u32_to_int i ≠ 0 ⇒
P $var$($m) tl i0) ⇒
P $var$($n) ls i) ⇒
∀v v1 v2. P v v1 v2
*)
end
|
