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|
signature divDefProto2Theory =
sig
type thm = Thm.thm
(* Definitions *)
val even_odd_body_def : thm
val fix_def : thm
val fix_fuel_P_def : thm
val fix_fuel_def : thm
val is_valid_fp_body_def : thm
val list_t_TY_DEF : thm
val list_t_case_def : thm
val list_t_size_def : thm
val nth_body_def : thm
(* Theorems *)
val datatype_list_t : thm
val even_def : thm
val even_odd_body_is_valid : thm
val even_odd_body_is_valid_aux : thm
val fix_fixed_diverges : thm
val fix_fixed_eq : thm
val fix_fixed_terminates : thm
val fix_fuel_P_least : thm
val fix_fuel_compute : thm
val fix_fuel_eq_fix : thm
val fix_fuel_mono : thm
val fix_fuel_mono_aux : thm
val fix_fuel_mono_least : thm
val fix_fuel_not_diverge_eq_fix : thm
val fix_fuel_not_diverge_eq_fix_aux : thm
val fix_not_diverge_implies_fix_fuel : thm
val fix_not_diverge_implies_fix_fuel_aux : thm
val is_valid_fp_body_compute : 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_body_is_valid : thm
val nth_body_is_valid_aux : thm
val nth_def : thm
val odd_def : thm
val divDefProto2_grammars : type_grammar.grammar * term_grammar.grammar
(*
[primitives] Parent theory of "divDefProto2"
[even_odd_body_def] Definition
⊢ ∀f x.
even_odd_body f x =
case x of
INL 0 => Return (INR (INR T))
| INL i =>
(case f (INR (INL (i − 1))) of
Return (INL v) => Fail Failure
| Return (INR (INL v2)) => Fail Failure
| Return (INR (INR b)) => Return (INR (INR b))
| Fail e => Fail e
| Diverge => Diverge)
| INR (INL 0) => Return (INR (INR F))
| INR (INL i) =>
(case f (INL (i − 1)) of
Return (INL v) => Fail Failure
| Return (INR (INL v2)) => Fail Failure
| Return (INR (INR b)) => Return (INR (INR b))
| Fail e => Fail e
| Diverge => Diverge)
| INR (INR v5) => Fail Failure
[fix_def] Definition
⊢ ∀f x.
fix f x =
if ∃n. fix_fuel_P f x n then
fix_fuel ($LEAST (fix_fuel_P f x)) f x
else Diverge
[fix_fuel_P_def] Definition
⊢ ∀f x n. fix_fuel_P f x n ⇔ ¬is_diverge (fix_fuel n f x)
[fix_fuel_def] Definition
⊢ (∀f x. fix_fuel 0 f x = Diverge) ∧
∀n f x. fix_fuel (SUC n) f x = f (fix_fuel n f) x
[is_valid_fp_body_def] Definition
⊢ (∀f. is_valid_fp_body 0 f ⇔ F) ∧
∀n f.
is_valid_fp_body (SUC n) f ⇔
∀x. (∀g h. f g x = f h x) ∨
∃h y.
is_valid_fp_body n h ∧ ∀g. f g x = do z <- g y; h g z od
[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_body_def] Definition
⊢ ∀f x.
nth_body f x =
case x of
INL x =>
(let
(ls,i) = x
in
case ls of
ListCons x tl =>
if u32_to_int i = 0 then Return (INR x)
else
do
i0 <- u32_sub i (int_to_u32 1);
r <- f (INL (tl,i0));
case r of
INL v => Fail Failure
| INR i1 => Return (INR i1)
od
| ListNil => Fail Failure)
| INR v2 => Fail Failure
[datatype_list_t] Theorem
⊢ DATATYPE (list_t ListCons ListNil)
[even_def] Theorem
⊢ ∀i. even i = if i = 0 then Return T else odd (i − 1)
[even_odd_body_is_valid] Theorem
⊢ is_valid_fp_body (SUC (SUC 0)) even_odd_body
[even_odd_body_is_valid_aux] Theorem
⊢ is_valid_fp_body (SUC (SUC n)) even_odd_body
[fix_fixed_diverges] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀x. ¬(∃n. fix_fuel_P f x n) ⇒ fix f x = f (fix f) x
[fix_fixed_eq] Theorem
⊢ ∀N f. is_valid_fp_body N f ⇒ ∀x. fix f x = f (fix f) x
[fix_fixed_terminates] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀x n. fix_fuel_P f x n ⇒ fix f x = f (fix f) x
[fix_fuel_P_least] Theorem
⊢ ∀f n x.
fix_fuel n f x ≠ Diverge ⇒
fix_fuel_P f x ($LEAST (fix_fuel_P f x))
[fix_fuel_compute] Theorem
⊢ (∀f x. fix_fuel 0 f x = Diverge) ∧
(∀n f x.
fix_fuel (NUMERAL (BIT1 n)) f x =
f (fix_fuel (NUMERAL (BIT1 n) − 1) f) x) ∧
∀n f x.
fix_fuel (NUMERAL (BIT2 n)) f x =
f (fix_fuel (NUMERAL (BIT1 n)) f) x
[fix_fuel_eq_fix] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀n x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix f x
[fix_fuel_mono] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀n x.
fix_fuel_P f x n ⇒ ∀m. n ≤ m ⇒ fix_fuel n f x = fix_fuel m f x
[fix_fuel_mono_aux] Theorem
⊢ ∀n N M g f.
is_valid_fp_body M f ⇒
is_valid_fp_body N g ⇒
∀x. ¬is_diverge (g (fix_fuel n f) x) ⇒
∀m. n ≤ m ⇒ g (fix_fuel n f) x = g (fix_fuel m f) x
[fix_fuel_mono_least] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀n x.
fix_fuel_P f x n ⇒
fix_fuel n f x = fix_fuel ($LEAST (fix_fuel_P f x)) f x
[fix_fuel_not_diverge_eq_fix] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀n x.
f (fix_fuel n f) x ≠ Diverge ⇒ f (fix f) x = f (fix_fuel n f) x
[fix_fuel_not_diverge_eq_fix_aux] Theorem
⊢ ∀N M g f.
is_valid_fp_body M f ⇒
is_valid_fp_body N g ⇒
∀n x.
g (fix_fuel n f) x ≠ Diverge ⇒ g (fix f) x = g (fix_fuel n f) x
[fix_not_diverge_implies_fix_fuel] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀x. f (fix f) x ≠ Diverge ⇒ ∃n. f (fix f) x = f (fix_fuel n f) x
[fix_not_diverge_implies_fix_fuel_aux] Theorem
⊢ ∀N M g f.
is_valid_fp_body M f ⇒
is_valid_fp_body N g ⇒
∀x. g (fix f) x ≠ Diverge ⇒
∃n. g (fix f) x = g (fix_fuel n f) x ∧
∀m. n ≤ m ⇒ g (fix_fuel m f) x = g (fix_fuel n f) x
[is_valid_fp_body_compute] Theorem
⊢ (∀f. is_valid_fp_body 0 f ⇔ F) ∧
(∀n f.
is_valid_fp_body (NUMERAL (BIT1 n)) f ⇔
∀x. (∀g h. f g x = f h x) ∨
∃h y.
is_valid_fp_body (NUMERAL (BIT1 n) − 1) h ∧
∀g. f g x = do z <- g y; h g z od) ∧
∀n f.
is_valid_fp_body (NUMERAL (BIT2 n)) f ⇔
∀x. (∀g h. f g x = f h x) ∨
∃h y.
is_valid_fp_body (NUMERAL (BIT1 n)) h ∧
∀g. f g x = do z <- g y; h g z od
[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_body_is_valid] Theorem
⊢ is_valid_fp_body (SUC (SUC 0)) nth_body
[nth_body_is_valid_aux] Theorem
⊢ is_valid_fp_body (SUC (SUC n)) nth_body
[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)
*)
end
|
