Add pick-any, pick-min, pick-max

util/all.v

@@ -16,3 +16,4 @@ Require Export rt.util.sum.
 Require Export rt.util.minmax.
 Require Export rt.util.seqset.
 Require Export rt.util.step_function.
+Require Export rt.util.pick.
\ No newline at end of file

util/pick.v

+From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype.
+
+(* In this file, we define functions for picking numbers in an interval [0, n). *)
+
+(** Auxiliary Functions *)
+
+Definition default0 {n} (x: option 'I_n) : nat := if x is Some y then y else 0.
+
+Definition arg_pred_nat n (P: pred 'I_n) ord :=
+  [pred i | P i & [forall j: 'I_n, P j ==> ord i j]].
+
+Definition pred_min_nat n (P: pred 'I_n) := arg_pred_nat n P leq.
+Definition pred_max_nat n (P: pred 'I_n) := arg_pred_nat n P (fun x y => geq x y).
+
+(** Defining Pick functions *)
+
+(* (pick_any n P) returns some number < n that satisfies P, or 0 if it cannot be found. *)
+Definition pick_any n (P: pred 'I_n) := default0 (pick P).
+
+(* (pick_min n P) returns the smallest number < n that satisfies P, or 0 if it cannot be found. *)
+Definition pick_min n (P: pred 'I_n) := default0 (pick (pred_min_nat n P)).
+
+(* (pick_max n P) returns the largest number < n that satisfies P, or 0 if it cannot be found. *)
+Definition pick_max n (P: pred 'I_n) := default0 (pick (pred_max_nat n P)).
+
+(** Improved notation *)
+
+(* Next we provide the following notation for the variations of pick:
+     [pick-any x <= N | P], [pick-any x < N | P]
+     [pick-min x <= N | P], [pick-min x < N | P]
+     [pick-max x <= N | P], [pick-max x < N | P]. *)
+Notation "[ 'pick-any' x <= N | P ]" :=
+  (pick_any N.+1 (fun x : 'I_N.+1 => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+    
+Notation "[ 'pick-any' x < N | P ]" :=
+  (pick_any N (fun x : 'I_N => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+
+Notation "[ 'pick-min' x <= N | P ]" :=
+  (pick_min N.+1 (fun x : 'I_N.+1 => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+
+Notation "[ 'pick-min' x < N | P ]" :=
+  (pick_min N (fun x : 'I_N => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+
+Notation "[ 'pick-max' x <= N | P ]" :=
+  (pick_max N.+1 (fun x : 'I_N.+1 => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+
+Notation "[ 'pick-max' x < N | P ]" :=
+  (pick_max N (fun x : 'I_N => P%B))
+  (at level 0, x ident, only parsing) : form_scope.
+
+(** Lemmas *)
+
+(* First, we show that any property P of (pick_any n p) can be proven by showing that
+   P holds for any number < n that satisfies p. *)
+Section PickAny.
+
+  Variable n: nat.
+  Variable p: pred 'I_n.
+
+  Variable P: nat -> Prop.
+
+  Hypothesis EX: exists x:'I_n, p x.
+
+  Hypothesis HOLDS: forall x, p x -> P x.
+
+  Lemma pick_any_holds: P (pick_any n p).
+  Proof.
+    rewrite /pick_any /default0.
+    case: pickP; first by intros x PRED; apply HOLDS.
+    intros NONE; red in NONE; exfalso.
+    move: EX => [x PRED].
+    by specialize (NONE x); rewrite PRED in NONE.
+  Qed.
+
+End PickAny.
+
+(* Next, we show that any property P of (pick_min n p) can be proven by showing that
+   P holds for the smallest number < n that satisfies p. *)
+Section PickMin.
+
+  Variable n: nat.
+  Variable p: pred 'I_n.
+
+  Variable P: nat -> Prop.
+
+  Hypothesis EX: exists x:'I_n, p x.
+
+  Hypothesis MIN:
+    forall x,
+      p x ->
+      (forall y, p y -> x <= y) ->
+      P x.
+  
+  Lemma pick_min_holds: P (pick_min n p).
+  Proof.
+    rewrite /pick_min /odflt /oapp.
+    case: pickP.
+    {
+      move => x /andP [PRED /forallP ALL].
+      apply MIN; first by done.
+      by intros y Py; specialize (ALL y); move: ALL => /implyP ALL; apply ALL.
+    }
+    {
+      intros NONE; red in NONE; exfalso.
+      move: EX => [x PRED]; clear EX.
+      set argmin := arg_min x p id.
+      specialize (NONE argmin).
+      suff ARGMIN: (pred_min_nat n p) argmin by rewrite ARGMIN in NONE.
+      rewrite /argmin; case: arg_minP; first by done.
+      intros y Py MINy.
+      apply/andP; split; first by done.
+      by apply/forallP; intros y0; apply/implyP; intros Py0; apply MINy.
+    }
+  Qed.
+
+End PickMin.
+
+(* Next, we show that any property P of (pick_max n p) can be proven by showing that
+   P holds for the largest number < n that satisfies p. *)
+Section PickMax.
+
+  Variable n: nat.
+  Variable p: pred 'I_n.
+
+  Variable P: nat -> Prop.
+
+  Hypothesis EX: exists x:'I_n, p x.
+
+  Hypothesis MAX:
+    forall x,
+      p x ->
+      (forall y, p y -> x >= y) ->
+      P x.
+  
+  Lemma pick_max_holds: P (pick_max n p).
+  Proof.
+    rewrite /pick_max /odflt /oapp.
+    case: pickP.
+    {
+      move => x /andP [PRED /forallP ALL].
+      apply MAX; first by done.
+      by intros y Py; specialize (ALL y); move: ALL => /implyP ALL; apply ALL.
+    }
+    {
+      intros NONE; red in NONE; exfalso.
+      move: EX => [x PRED]; clear EX.
+      set argmax := arg_max x p id.
+      specialize (NONE argmax).
+      suff ARGMAX: (pred_max_nat n p) argmax by rewrite ARGMAX in NONE.
+      rewrite /argmax; case: arg_maxP; first by done.
+      intros y Py MAXy.
+      apply/andP; split; first by done.
+      by apply/forallP; intros y0; apply/implyP; intros Py0; apply MAXy.
+    }
+  Qed.
+
+End PickMax.
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