mirror of
https://github.com/achlipala/frap.git
synced 2024-11-10 00:07:51 +00:00
FirstClassFunctions: move bruisingly long proof to end
This commit is contained in:
parent
63836dad24
commit
399e8f7228
1 changed files with 115 additions and 112 deletions
|
@ -123,118 +123,6 @@ Definition sublistSummingToK (ns : list nat) (target : nat) : option (list nat)
|
|||
|
||||
Time Compute sublistSummingToK (countingDown 20) 1.
|
||||
|
||||
Theorem allSublistsK_ok : forall {A B} (ls : list A) (failed : unit -> B) found,
|
||||
(forall sol, (exists ans, (forall failed', found sol failed' = ans)
|
||||
/\ ans <> failed tt)
|
||||
\/ (forall failed', found sol failed' = failed' tt))
|
||||
-> (exists sol ans, In sol (allSublists ls)
|
||||
/\ (forall failed', found sol failed' = ans)
|
||||
/\ allSublistsK ls failed found = ans
|
||||
/\ ans <> failed tt)
|
||||
\/ ((forall sol, In sol (allSublists ls)
|
||||
-> forall failed', found sol failed' = failed' tt)
|
||||
/\ allSublistsK ls failed found = failed tt).
|
||||
Proof.
|
||||
induct ls; simplify.
|
||||
|
||||
specialize (H []).
|
||||
first_order.
|
||||
right.
|
||||
propositional.
|
||||
subst.
|
||||
trivial.
|
||||
trivial.
|
||||
|
||||
assert (let found := (fun (sol : list A) (failed' : unit -> B) =>
|
||||
found sol (fun _ : unit => found (a :: sol) failed')) in
|
||||
(exists (sol : list A) (ans : B),
|
||||
In sol (allSublists ls) /\
|
||||
(forall failed' : unit -> B, found sol failed' = ans) /\
|
||||
allSublistsK ls failed found = ans /\ ans <> failed tt) \/
|
||||
(forall sol : list A,
|
||||
In sol (allSublists ls) -> forall failed' : unit -> B, found sol failed' = failed' tt) /\
|
||||
allSublistsK ls failed found = failed tt).
|
||||
apply IHls.
|
||||
first_order.
|
||||
generalize (H sol).
|
||||
first_order.
|
||||
specialize (H (a :: sol)).
|
||||
first_order.
|
||||
left.
|
||||
exists x; propositional.
|
||||
rewrite H0.
|
||||
trivial.
|
||||
right.
|
||||
simplify.
|
||||
rewrite H0.
|
||||
trivial.
|
||||
|
||||
clear IHls.
|
||||
simplify.
|
||||
first_order.
|
||||
|
||||
generalize (H x); first_order.
|
||||
left; exists x, x1; propositional.
|
||||
apply in_or_app; propositional.
|
||||
specialize (H1 failed).
|
||||
specialize (H4 (fun _ => found (a :: x) failed)).
|
||||
equality.
|
||||
left; exists (a :: x), x0; propositional.
|
||||
apply in_or_app; right; apply in_map_iff.
|
||||
first_order.
|
||||
specialize (H1 failed').
|
||||
rewrite H4 in H1.
|
||||
trivial.
|
||||
|
||||
right; propositional.
|
||||
apply in_app_or in H2; propositional.
|
||||
|
||||
generalize (H sol); first_order.
|
||||
apply H0 with (failed' := failed') in H3.
|
||||
rewrite H2 in H3.
|
||||
equality.
|
||||
|
||||
apply in_map_iff in H3.
|
||||
first_order.
|
||||
subst.
|
||||
generalize (H x); first_order.
|
||||
apply H0 with (failed' := failed) in H3.
|
||||
equality.
|
||||
apply H0 with (failed' := failed') in H3.
|
||||
rewrite H2 in H3; trivial.
|
||||
Qed.
|
||||
|
||||
Theorem sublistSummingToK_ok : forall ns target,
|
||||
match sublistSummingToK ns target with
|
||||
| None => forall sol, In sol (allSublists ns) -> sum sol <> target
|
||||
| Some sol => In sol (allSublists ns) /\ sum sol = target
|
||||
end.
|
||||
Proof.
|
||||
simplify.
|
||||
unfold sublistSummingToK.
|
||||
pose proof (allSublistsK_ok ns (fun _ => None)
|
||||
(fun sol failed => if sum sol ==n target then Some sol else failed tt)).
|
||||
cases H.
|
||||
|
||||
simplify.
|
||||
cases (sum sol ==n target).
|
||||
left; exists (Some sol); equality.
|
||||
propositional.
|
||||
|
||||
first_order.
|
||||
specialize (H0 (fun _ => None)).
|
||||
cases (sum x ==n target); try equality.
|
||||
subst.
|
||||
rewrite H1.
|
||||
propositional.
|
||||
|
||||
first_order.
|
||||
rewrite H0.
|
||||
simplify.
|
||||
apply H with (failed' := fun _ => None) in H1.
|
||||
cases (sum sol ==n target); equality.
|
||||
Qed.
|
||||
|
||||
|
||||
(** * The classics in continuation-passing style *)
|
||||
|
||||
|
@ -516,3 +404,118 @@ Proof.
|
|||
rewrite flattenS_flattenKD.
|
||||
apply flattenKD_ok.
|
||||
Qed.
|
||||
|
||||
|
||||
(** * Proof of our motivating example *)
|
||||
|
||||
Theorem allSublistsK_ok : forall {A B} (ls : list A) (failed : unit -> B) found,
|
||||
(forall sol, (exists ans, (forall failed', found sol failed' = ans)
|
||||
/\ ans <> failed tt)
|
||||
\/ (forall failed', found sol failed' = failed' tt))
|
||||
-> (exists sol ans, In sol (allSublists ls)
|
||||
/\ (forall failed', found sol failed' = ans)
|
||||
/\ allSublistsK ls failed found = ans
|
||||
/\ ans <> failed tt)
|
||||
\/ ((forall sol, In sol (allSublists ls)
|
||||
-> forall failed', found sol failed' = failed' tt)
|
||||
/\ allSublistsK ls failed found = failed tt).
|
||||
Proof.
|
||||
induct ls; simplify.
|
||||
|
||||
specialize (H []).
|
||||
first_order.
|
||||
right.
|
||||
propositional.
|
||||
subst.
|
||||
trivial.
|
||||
trivial.
|
||||
|
||||
assert (let found := (fun (sol : list A) (failed' : unit -> B) =>
|
||||
found sol (fun _ : unit => found (a :: sol) failed')) in
|
||||
(exists (sol : list A) (ans : B),
|
||||
In sol (allSublists ls) /\
|
||||
(forall failed' : unit -> B, found sol failed' = ans) /\
|
||||
allSublistsK ls failed found = ans /\ ans <> failed tt) \/
|
||||
(forall sol : list A,
|
||||
In sol (allSublists ls) -> forall failed' : unit -> B, found sol failed' = failed' tt) /\
|
||||
allSublistsK ls failed found = failed tt).
|
||||
apply IHls.
|
||||
first_order.
|
||||
generalize (H sol).
|
||||
first_order.
|
||||
specialize (H (a :: sol)).
|
||||
first_order.
|
||||
left.
|
||||
exists x; propositional.
|
||||
rewrite H0.
|
||||
trivial.
|
||||
right.
|
||||
simplify.
|
||||
rewrite H0.
|
||||
trivial.
|
||||
|
||||
clear IHls.
|
||||
simplify.
|
||||
first_order.
|
||||
|
||||
generalize (H x); first_order.
|
||||
left; exists x, x1; propositional.
|
||||
apply in_or_app; propositional.
|
||||
specialize (H1 failed).
|
||||
specialize (H4 (fun _ => found (a :: x) failed)).
|
||||
equality.
|
||||
left; exists (a :: x), x0; propositional.
|
||||
apply in_or_app; right; apply in_map_iff.
|
||||
first_order.
|
||||
specialize (H1 failed').
|
||||
rewrite H4 in H1.
|
||||
trivial.
|
||||
|
||||
right; propositional.
|
||||
apply in_app_or in H2; propositional.
|
||||
|
||||
generalize (H sol); first_order.
|
||||
apply H0 with (failed' := failed') in H3.
|
||||
rewrite H2 in H3.
|
||||
equality.
|
||||
|
||||
apply in_map_iff in H3.
|
||||
first_order.
|
||||
subst.
|
||||
generalize (H x); first_order.
|
||||
apply H0 with (failed' := failed) in H3.
|
||||
equality.
|
||||
apply H0 with (failed' := failed') in H3.
|
||||
rewrite H2 in H3; trivial.
|
||||
Qed.
|
||||
|
||||
Theorem sublistSummingToK_ok : forall ns target,
|
||||
match sublistSummingToK ns target with
|
||||
| None => forall sol, In sol (allSublists ns) -> sum sol <> target
|
||||
| Some sol => In sol (allSublists ns) /\ sum sol = target
|
||||
end.
|
||||
Proof.
|
||||
simplify.
|
||||
unfold sublistSummingToK.
|
||||
pose proof (allSublistsK_ok ns (fun _ => None)
|
||||
(fun sol failed => if sum sol ==n target then Some sol else failed tt)).
|
||||
cases H.
|
||||
|
||||
simplify.
|
||||
cases (sum sol ==n target).
|
||||
left; exists (Some sol); equality.
|
||||
propositional.
|
||||
|
||||
first_order.
|
||||
specialize (H0 (fun _ => None)).
|
||||
cases (sum x ==n target); try equality.
|
||||
subst.
|
||||
rewrite H1.
|
||||
propositional.
|
||||
|
||||
first_order.
|
||||
rewrite H0.
|
||||
simplify.
|
||||
apply H with (failed' := fun _ => None) in H1.
|
||||
cases (sum sol ==n target); equality.
|
||||
Qed.
|
||||
|
|
Loading…
Reference in a new issue