Commit c92654ed authored by Robbert Krebbers's avatar Robbert Krebbers

Define `wf_guard` in terms of Coq stdlib's `Acc_intro_generator`.

parent b9ef84b4
Pipeline #5257 passed with stages
in 10 minutes and 17 seconds
......@@ -194,33 +194,8 @@ End subrel.
(** * Theorems on well founded relations *)
Notation wf := well_founded.
Section wf.
Context `{R : relation A}.
Lemma Acc_def x : Acc R x ( y : A, R y x Acc R y).
Proof. split. by destruct 1. by constructor. Qed.
(** A trick by Thomas Braibant to compute with well-founded recursions:
it lazily adds [2^n] [Acc_intro] constructors in front of a well foundedness
proof, so that the actual proof is never reached in practise. *)
Fixpoint wf_guard (n : nat) (wfR : wf R) : wf R :=
match n with
| 0 => wfR
| S n => λ x, Acc_intro x (λ y _, wf_guard n (wf_guard n wfR) y)
end.
Lemma wf_projected `(R2 : relation B) (f : A B) :
( x y, R x y R2 (f x) (f y))
wf R2 wf R.
Proof.
intros Hf Hwf.
cut ( y, Acc R2 y x, y = f x Acc R x).
{ intros aux x. apply (aux (f x)); auto. }
induction 1 as [y _ IH]. intros x ?. subst.
constructor. intros. apply (IH (f y)); auto.
Qed.
End wf.
Definition wf_guard `{R : relation A} (n : nat) (wfR : wf R) : wf R :=
Acc_intro_generator n wfR.
(* Generally we do not want [wf_guard] to be expanded (neither by tactics,
nor by conversion tests in the kernel), but in some cases we do need it for
......@@ -228,6 +203,17 @@ computation (that is, we cannot make it opaque). We use the [Strategy]
command to make its expanding behavior less eager. *)
Strategy 100 [wf_guard].
Lemma wf_projected `{R1 : relation A} `(R2 : relation B) (f : A B) :
( x y, R1 x y R2 (f x) (f y))
wf R2 wf R1.
Proof.
intros Hf Hwf.
cut ( y, Acc R2 y x, y = f x Acc R1 x).
{ intros aux x. apply (aux (f x)); auto. }
induction 1 as [y _ IH]. intros x ?. subst.
constructor. intros. apply (IH (f y)); auto.
Qed.
Lemma Fix_F_proper `{R : relation A} (B : A Type) (E : x, relation (B x))
(F : x, ( y, R y x B y) B x)
(HF : (x : A) (f g : y, R y x B y),
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment