Commit a4d57101 authored by Sergey Bozhko's avatar Sergey Bozhko

Add lemmas about ex_minn

parent 80bd79d5
......@@ -15,8 +15,9 @@ From rt.util Require Import tactics.
*)
Section ArgSearch.
(* Given a function [f] that maps the naturals to elements of type [T]... *)
Context {T: Type}.
Context {T : Type}.
Variable f: nat -> T.
(* ... a predicate [P] on [T] ... *)
......@@ -28,7 +29,7 @@ Section ArgSearch.
(* ... we define the procedure [search_arg] to iterate a given search space
[a, b), while checking each element whether [f] satisfies [P] at that
point and returning the extremum as defined by [R]. *)
Fixpoint search_arg (a b: nat): option nat :=
Fixpoint search_arg (a b : nat) : option nat :=
if a < b then
match b with
| 0 => None
......@@ -202,3 +203,36 @@ Section ArgSearch.
Qed.
End ArgSearch.
Section ExMinn.
(* We show that the fact that the minimal satisfying argument [ex_minn ex] of
a predicate [pred] satisfies another predicate [P] implies the existence
of a minimal element that satisfies both [pred] and [P]. *)
Lemma prop_on_ex_minn:
forall (P : nat -> Prop) (pred : nat -> bool) (ex : exists n, pred n),
P (ex_minn ex) ->
exists n, P n /\ pred n /\ (forall n', pred n' -> n <= n').
Proof.
intros.
exists (ex_minn ex); repeat split; auto.
all: have MIN := ex_minnP ex; move: MIN => [n Pn MIN]; auto.
Qed.
(* As a corollary, we show that if there is a constant [c] such
that [P c], then the minimal satisfying argument [ex_minn ex]
of a predicate [P] is less than or equal to [c]. *)
Corollary ex_minn_le_ex:
forall (P : nat -> bool) (exP : exists n, P n) (c : nat),
P c ->
ex_minn exP <= c.
Proof.
intros ? ? ? EX.
rewrite leqNgt; apply/negP; intros GT.
pattern (ex_minn (P:=P) exP) in GT;
apply prop_on_ex_minn in GT; move: GT => [n [LT [Pn MIN]]].
specialize (MIN c EX).
by move: MIN; rewrite leqNgt; move => /negP MIN; apply: MIN.
Qed.
End ExMinn.
\ No newline at end of file
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