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. *)
(* 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
by giving an injection from the natural numbers into the type. It is mostly
......@@ -25,7 +25,7 @@ Qed.
(** * Fresh elements *)
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.
Proof. set_solver. Qed.
......@@ -47,36 +47,35 @@ Section Fresh.
apply relfg.
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:
fresh_generic_fix s n = fresh_generic_body s (λ s' _ n, fresh_generic_fix s' n) n.
Lemma fresh_generic_fixpoint_unfold u s n:
fresh_generic_fix u s n = fresh_generic_body s (λ s' _ n, fresh_generic_fix u s' n) n.
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).
Qed.
Lemma fresh_generic_fixpoint_spec s n:
m, n m fresh_generic_fix s n = inject m inject m s i, n i < m inject i s.
Lemma fresh_generic_fixpoint_spec u s n:
m, n m fresh_generic_fix u s n = inject m inject m s i, n i < m inject i s.
Proof.
revert n.
induction s as [s IH] using (well_founded_ind collection_wf); intro.
setoid_rewrite fresh_generic_fixpoint_unfold; unfold fresh_generic_body.
destruct decide as [case|case].
- destruct (IH _ (subset_difference_in case) (S n)) as [m [mbound [eqfix [notin inbelow]]]].
exists m; repeat split; auto.
+ omega.
+ rewrite not_elem_of_difference, elem_of_singleton in notin.
destruct notin as [?|?%inject_injective]; auto; omega.
+ intros i ibound.
destruct (decide (i = n)) as [<-|neq]; auto.
enough (inject i s {[inject n]}) by set_solver.
apply inbelow; omega.
- exists n; repeat split; auto.
intros; omega.
destruct decide as [case|case]; eauto with omega.
destruct (IH _ (subset_difference_in case) (S n)) as [m [mbound [eqfix [notin inbelow]]]].
exists m; repeat split; auto with omega.
- rewrite not_elem_of_difference, elem_of_singleton in notin.
destruct notin as [?|?%inject_injective]; auto with omega.
- intros i ibound.
destruct (decide (i = n)) as [<-|neq]; auto.
enough (inject i s {[inject n]}) by set_solver.
apply inbelow; omega.
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.
Proof.
......@@ -84,20 +83,23 @@ Section Fresh.
- apply _.
- intros * eqXY.
unfold fresh, fresh_generic.
destruct (fresh_generic_fixpoint_spec X 0) as [mX [_ [-> [notinX belowinX]]]].
destruct (fresh_generic_fixpoint_spec Y 0) as [mY [_ [-> [notinY belowinY]]]].
destruct (fresh_generic_fixpoint_spec (1 + Nat.log2 (size X)) X 0)
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.
+ contradict notinX; rewrite eqXY; apply belowinY; omega.
+ contradict notinY; rewrite <- eqXY; apply belowinX; omega.
- intro.
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.
End Fresh.
(** Derive fresh instances. *)
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_spec: FreshSpec string C := _.
Global Instance string_fresh_spec: FreshSpec string C := fresh_generic_spec.
End StringFresh.
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