from MathComp-Analysis' unstable.v

CHANGELOG_UNRELEASED.md

@@ -9,6 +9,9 @@
 
 - Added a theory of finite permutations in `finperm.v`
 
+- in `finmap.v`:
+  + lemma `fset_nat_maximum`
+
 ### Changed
 
 ### Renamed

finmap.v

@@ -147,10 +147,10 @@
 Reserved Notation "A `=P` B" (at level 70, no associativity).
 
 Reserved Notation "f @`[ key ] A" (at level 24, key at level 0).
 Reserved Notation "f @2`[ key ] ( A , B )"
   (at level 24, format "f  @2`[ key ]  ( A ,  B )").
 Reserved Notation "f @` A" (at level 24).
 Reserved Notation "f @2` ( A , B )" (at level 24, format "f  @2`  ( A ,  B )").
 
 (* unary *)
 Reserved Notation "[ 'fset' E | x : T 'in' A ]" (at level 0, E, x at level 99).
@@ -168,9 +168,9 @@
 Reserved Notation "[ 'fset' x 'in' A | P & Q ]" (at level 0, x at level 99).
 
 (* binary *)
 Reserved Notation "[ 'fset' E | x : T 'in' A , y : T' 'in' B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99).
 Reserved Notation "[ 'fset' E | x 'in' A , y 'in' B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99).
 
 (* keyed parse only *)
@@ -183,11 +183,11 @@
   (at level 0, E, x at level 99).
 Reserved Notation "[ 'fset[' key ] E | x : A & P ]"
   (at level 0, E, x at level 99).
 Reserved Notation "[ 'fset[' key ] E | x : T 'in' A , y : T' 'in' B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99).
 Reserved Notation "[ 'fset[' key ] E | x 'in' A , y 'in' B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99).
 Reserved Notation "[ 'fset[' key ] E | x : A , y : B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99).
 Reserved Notation "[ 'fset[' key ] E | x : A , y 'in' B ]"
   (at level 0, E, x, y at level 99).
@@ -233,7 +233,7 @@
 Reserved Notation "[ 'f' 'set' x 'in' A | P & Q ]"
   (at level 0, x at level 99, format "[ 'f' 'set'  x  'in'  A  |  P  &  Q ]").
 (* binary printing only *)
 Reserved Notation "[ 'f' 'set' E | x 'in' A , y 'in' B ]"
   (at level 0, E, x at level 99, A at level 200, y at level 99,
    format "[ '[hv' 'f' 'set'  E '/ '  |  x  'in'  A , '/' y  'in'  B ] ']'").
 
@@ -357,7 +357,7 @@
 Defined.
 Arguments PredType [T pT] toP.
 
 Local Notation predOfType T := (pred_of_simpl (@pred_of_argType T)).
 
 Definition oextract (T : Type) (o : option T) : o -> T :=
   if o is Some t return o -> T then fun=> t else False_rect T \o notF.
@@ -420,7 +420,7 @@
 
 #[global] Hint Resolve SortKeys.perm SortKeys.uniq SortKeys.E : core.
 
 Notation sort_keys      := SortKeys.f.
 Notation sort_keys_perm := SortKeys.perm.
 Notation sort_keys_uniq := SortKeys.uniq.
 Notation sort_keysE     := SortKeys.E.
@@ -484,7 +484,7 @@
 
 End Def.
 
 Arguments finSet K%type_scope.
 
 #[warning="-deprecated-reference"]
 Identity Coercion type_of_finset : finset_of >-> finSet.
@@ -1784,6 +1784,15 @@
 by split=> [] [a aP]; [exists (val a); apply: valP|exists [` aP]].
 Qed.
 
+Lemma fset_nat_maximum A (f : K -> nat) : A != fset0 ->
+  (exists i, i \in A /\ forall j, j \in A -> f j <= f i)%N.
+Proof.
+move=> /fset0Pn[x Ax].
+have [/= y _ /(_ _ isT) mf] := @arg_maxnP _ [` Ax]%fset xpredT (f \o val) isT.
+exists (val y); split; first exact: valP.
+by move=> z Az; have := mf [` Az]%fset.
+Qed.
+
 Lemma cardfs_gt0 A : (0 < #|` A|)%N = (A != fset0).
 Proof. by rewrite lt0n cardfs_eq0. Qed.
 
@@ -2976,7 +2985,7 @@
 
 End DefMap.
 
 Arguments finMap K%type_scope V%type_scope.
 
 Notation "{ 'fmap' K -> T }" := (finMap K T) : type_scope.