Commit a62a60ae authored by Johannes Kloos's avatar Johannes Kloos

Compile fixes and addition of wf_guard.

parent 34e58198
(* Copyright (c) 2012-2017, Coq-std++ developers. *) (* Copyright (c) 2012-2017, Coq-std++ developers. *)
(* This file is distributed under the terms of the BSD license. *) (* This file is distributed under the terms of the BSD license. *)
From stdpp Require Import pretty fin_collections. From stdpp Require Import pretty fin_collections relations prelude.
(** The class [Infinite] axiomatizes types with infinitely many elements (** The class [Infinite] axiomatizes types with infinitely many elements
by giving an injection from the natural numbers into the type. It is mostly by giving an injection from the natural numbers into the type. It is mostly
...@@ -25,7 +25,7 @@ Qed. ...@@ -25,7 +25,7 @@ Qed.
(** * Fresh elements *) (** * Fresh elements *)
Section Fresh. Section Fresh.
Context `{Infinite A, (x: A) (s: C), Decision (x s)}. Context `{FinCollection A C} `{Infinite A, !RelDecision (@elem_of A C _)}.
Lemma subset_difference_in {x: A} {s: C} (inx: x s): s {[ x ]} s. Lemma subset_difference_in {x: A} {s: C} (inx: x s): s {[ x ]} s.
Proof. set_solver. Qed. Proof. set_solver. Qed.
...@@ -47,36 +47,35 @@ Section Fresh. ...@@ -47,36 +47,35 @@ Section Fresh.
apply relfg. apply relfg.
Qed. Qed.
Definition fresh_generic_fix := Fix collection_wf (const (nat A)) fresh_generic_body. Definition fresh_generic_fix u :=
Fix (wf_guard u collection_wf) (const (nat A)) fresh_generic_body.
Lemma fresh_generic_fixpoint_unfold s n: Lemma fresh_generic_fixpoint_unfold u s n:
fresh_generic_fix s n = fresh_generic_body s (λ s' _ n, fresh_generic_fix s' n) n. fresh_generic_fix u s n = fresh_generic_body s (λ s' _ n, fresh_generic_fix u s' n) n.
Proof. Proof.
apply (Fix_unfold_rel collection_wf (const (nat A)) (const (pointwise_relation nat (=))) apply (Fix_unfold_rel (wf_guard u collection_wf)
(const (nat A)) (const (pointwise_relation nat (=)))
fresh_generic_body fresh_generic_body_proper s n). fresh_generic_body fresh_generic_body_proper s n).
Qed. Qed.
Lemma fresh_generic_fixpoint_spec s n: Lemma fresh_generic_fixpoint_spec u s n:
m, n m fresh_generic_fix s n = inject m inject m s i, n i < m inject i s. m, n m fresh_generic_fix u s n = inject m inject m s i, n i < m inject i s.
Proof. Proof.
revert n. revert n.
induction s as [s IH] using (well_founded_ind collection_wf); intro. induction s as [s IH] using (well_founded_ind collection_wf); intro.
setoid_rewrite fresh_generic_fixpoint_unfold; unfold fresh_generic_body. setoid_rewrite fresh_generic_fixpoint_unfold; unfold fresh_generic_body.
destruct decide as [case|case]. destruct decide as [case|case]; eauto with omega.
- destruct (IH _ (subset_difference_in case) (S n)) as [m [mbound [eqfix [notin inbelow]]]]. destruct (IH _ (subset_difference_in case) (S n)) as [m [mbound [eqfix [notin inbelow]]]].
exists m; repeat split; auto. exists m; repeat split; auto with omega.
+ omega. - rewrite not_elem_of_difference, elem_of_singleton in notin.
+ rewrite not_elem_of_difference, elem_of_singleton in notin. destruct notin as [?|?%inject_injective]; auto with omega.
destruct notin as [?|?%inject_injective]; auto; omega. - intros i ibound.
+ intros i ibound. destruct (decide (i = n)) as [<-|neq]; auto.
destruct (decide (i = n)) as [<-|neq]; auto. enough (inject i s {[inject n]}) by set_solver.
enough (inject i s {[inject n]}) by set_solver. apply inbelow; omega.
apply inbelow; omega.
- exists n; repeat split; auto.
intros; omega.
Qed. Qed.
Instance fresh_generic: Fresh A C | 20 := λ s, fresh_generic_fix s 0. Instance fresh_generic: Fresh A C | 20 := λ s, fresh_generic_fix (1 + Nat.log2 (size s)) s 0.
Instance fresh_generic_spec: FreshSpec A C. Instance fresh_generic_spec: FreshSpec A C.
Proof. Proof.
...@@ -84,20 +83,23 @@ Section Fresh. ...@@ -84,20 +83,23 @@ Section Fresh.
- apply _. - apply _.
- intros * eqXY. - intros * eqXY.
unfold fresh, fresh_generic. unfold fresh, fresh_generic.
destruct (fresh_generic_fixpoint_spec X 0) as [mX [_ [-> [notinX belowinX]]]]. destruct (fresh_generic_fixpoint_spec (1 + Nat.log2 (size X)) X 0)
destruct (fresh_generic_fixpoint_spec Y 0) as [mY [_ [-> [notinY belowinY]]]]. as [mX [_ [-> [notinX belowinX]]]].
destruct (fresh_generic_fixpoint_spec (1 + Nat.log2 (size Y)) Y 0)
as [mY [_ [-> [notinY belowinY]]]].
destruct (Nat.lt_trichotomy mX mY) as [case|[->|case]]; auto. destruct (Nat.lt_trichotomy mX mY) as [case|[->|case]]; auto.
+ contradict notinX; rewrite eqXY; apply belowinY; omega. + contradict notinX; rewrite eqXY; apply belowinY; omega.
+ contradict notinY; rewrite <- eqXY; apply belowinX; omega. + contradict notinY; rewrite <- eqXY; apply belowinX; omega.
- intro. - intro.
unfold fresh, fresh_generic. unfold fresh, fresh_generic.
destruct (fresh_generic_fixpoint_spec X 0) as [m [_ [-> [notinX belowinX]]]]; auto. destruct (fresh_generic_fixpoint_spec (1 + Nat.log2 (size X)) X 0)
as [m [_ [-> [notinX belowinX]]]]; auto.
Qed. Qed.
End Fresh. End Fresh.
(** Derive fresh instances. *) (** Derive fresh instances. *)
Section StringFresh. Section StringFresh.
Context `{FinCollection string C, (x: string) (s: C), Decision (x s)}. Context `{FinCollection string C, !RelDecision elem_of}.
Global Instance string_fresh: Fresh string C := fresh_generic. Global Instance string_fresh: Fresh string C := fresh_generic.
Global Instance string_fresh_spec: FreshSpec string C := _. Global Instance string_fresh_spec: FreshSpec string C := fresh_generic_spec.
End StringFresh. End StringFresh.
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