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signature divDefTheory =
sig
type thm = Thm.thm
(* Definitions *)
val fix_def : thm
val fix_exec_def : thm
val fix_fuel_P_def : thm
val fix_fuel_def : thm
val fix_nexec_def : thm
val is_valid_fp_body_def : thm
(* Theorems *)
val case_result_switch_eq : thm
val fix_exec_fixed_eq : 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_nexec_eq_fix : 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 divDef_grammars : type_grammar.grammar * term_grammar.grammar
(*
[primitives] Parent theory of "divDef"
[fix_def] Definition
⊢ ∀f x.
fix f x =
if ∃n. fix_fuel_P f x n then
fix_fuel ($LEAST (fix_fuel_P f x)) f x
else Diverge
[fix_exec_def] Definition
⊢ fix_exec = fix_nexec 1000000
[fix_fuel_P_def] Definition
⊢ ∀f x n. fix_fuel_P f x n ⇔ ¬is_diverge (fix_fuel n f x)
[fix_fuel_def] Definition
⊢ (∀f x. fix_fuel 0 f x = Diverge) ∧
∀n f x. fix_fuel (SUC n) f x = f (fix_fuel n f) x
[fix_nexec_def] Definition
⊢ ∀n f x.
fix_nexec n f x =
if fix_fuel_P f x n then fix_fuel n f x else fix f x
[is_valid_fp_body_def] Definition
⊢ (∀f. is_valid_fp_body 0 f ⇔ F) ∧
∀n f.
is_valid_fp_body (SUC n) f ⇔
∀x. (∀g h. f g x = f h x) ∨
∃h y.
is_valid_fp_body n h ∧ ∀g. f g x = do z <- g y; h g z od
[case_result_switch_eq] Theorem
⊢ (case
case x of
Return y => f y
| Fail e => Fail e
| Diverge => Diverge
of
Return y => g y
| Fail e => Fail e
| Diverge => Diverge) =
case x of
Return y =>
(case f y of
Return y => g y
| Fail e => Fail e
| Diverge => Diverge)
| Fail e => Fail e
| Diverge => Diverge
[fix_exec_fixed_eq] Theorem
⊢ ∀N f. is_valid_fp_body N f ⇒ ∀x. fix_exec f x = f (fix_exec f) x
[fix_fixed_diverges] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀x. ¬(∃n. fix_fuel_P f x n) ⇒ fix f x = f (fix f) x
[fix_fixed_eq] Theorem
⊢ ∀N f. is_valid_fp_body N f ⇒ ∀x. fix f x = f (fix f) x
[fix_fixed_terminates] Theorem
⊢ ∀N f.
is_valid_fp_body N f ⇒
∀x n. fix_fuel_P f x n ⇒ fix f x = f (fix f) x
[fix_fuel_P_least] Theorem
⊢ ∀f n x.
fix_fuel n f x ≠ Diverge ⇒
fix_fuel_P f x ($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_nexec_eq_fix] Theorem
⊢ ∀N f n. is_valid_fp_body N f ⇒ fix_nexec n f = fix f
[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
*)
end
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