[Merged by Bors] - feat: Extending convex functions

Mathlib/Algebra/Order/Module.lean

@@ -5,7 +5,7 @@
 -/
 import Mathlib.Algebra.Order.SMul
 
-#align_import algebra.order.module from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
+#align_import algebra.order.module from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
 
 /-!
 # Ordered module
@@ -217,6 +217,19 @@
 
 end OrderedRing
 
+section LinearOrderedRing
+variable [LinearOrderedRing k] [LinearOrderedAddCommGroup M] [Module k M] [OrderedSMul k M] {a : k}
+
+theorem smul_max_of_nonpos (ha : a ≤ 0) (b₁ b₂ : M) : a • max b₁ b₂ = min (a • b₁) (a • b₂) :=
+  (antitone_smul_left ha : Antitone (_ : M → M)).map_max
+#align smul_max_of_nonpos smul_max_of_nonpos
+
+theorem smul_min_of_nonpos (ha : a ≤ 0) (b₁ b₂ : M) : a • min b₁ b₂ = max (a • b₁) (a • b₂) :=
+  (antitone_smul_left ha : Antitone (_ : M → M)).map_min
+#align smul_min_of_nonpos smul_min_of_nonpos
+
+end LinearOrderedRing
+
 section LinearOrderedField
 
 variable [LinearOrderedField k] [OrderedAddCommGroup M] [Module k M] [OrderedSMul k M] {s : Set M}

Mathlib/Algebra/Order/Monoid/Lemmas.lean

@@ -9,7 +9,7 @@
 import Mathlib.Order.MinMax
 import Mathlib.Tactic.Contrapose
 
-#align_import algebra.order.monoid.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
+#align_import algebra.order.monoid.lemmas from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
 
 /-!
 # Ordered monoids
@@ -315,6 +315,27 @@
 #align min_le_max_of_add_le_add min_le_max_of_add_le_add
 #align min_le_max_of_mul_le_mul min_le_max_of_mul_le_mul
 
+end LinearOrder
+
+section LinearOrder
+variable [LinearOrder α] [CovariantClass α α HMul.hMul LE.le]
+  [CovariantClass α α (swap HMul.hMul) LE.le] {a b c d : α}
+
+@[to_additive max_add_add_le_max_add_max]
+theorem max_mul_mul_le_max_mul_max' : max (a * b) (c * d) ≤ max a c * max b d :=
+  max_le (mul_le_mul' (le_max_left _ _) <| le_max_left _ _) <|
+    mul_le_mul' (le_max_right _ _) <| le_max_right _ _
+#align max_mul_mul_le_max_mul_max' max_mul_mul_le_max_mul_max'
+#align max_add_add_le_max_add_max max_add_add_le_max_add_max
+
+--TODO: Also missing `min_mul_min_le_min_mul_mul`
+@[to_additive min_add_min_le_min_add_add]
+theorem min_mul_min_le_min_mul_mul' : min a c * min b d ≤ min (a * b) (c * d) :=
+  le_min (mul_le_mul' (min_le_left _ _) <| min_le_left _ _) <|
+    mul_le_mul' (min_le_right _ _) <| min_le_right _ _
+#align min_mul_min_le_min_mul_mul' min_mul_min_le_min_mul_mul'
+#align min_add_min_le_min_add_add min_add_min_le_min_add_add
+
 end LinearOrder
 end Mul
 

Mathlib/Algebra/Order/SMul.lean

@@ -11,7 +11,7 @@
 import Mathlib.Tactic.GCongr.Core
 import Mathlib.Tactic.Positivity
 
-#align_import algebra.order.smul from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
+#align_import algebra.order.smul from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
 
 /-!
 # Ordered scalar product
@@ -186,12 +186,25 @@
     · cases (Int.negSucc_not_pos _).1 hn
 #align int.ordered_smul Int.orderedSMul
 
+section LinearOrderedSemiring
+variable [LinearOrderedSemiring R] [LinearOrderedAddCommMonoid M] [SMulWithZero R M]
+  [OrderedSMul R M] {a : R}
+
 -- TODO: `LinearOrderedField M → OrderedSMul ℚ M`
-instance LinearOrderedSemiring.toOrderedSMul {R : Type*} [LinearOrderedSemiring R] :
-    OrderedSMul R R :=
+instance LinearOrderedSemiring.toOrderedSMul : OrderedSMul R R :=
   OrderedSMul.mk'' fun _ => strictMono_mul_left_of_pos
 #align linear_ordered_semiring.to_ordered_smul LinearOrderedSemiring.toOrderedSMul
 
+theorem smul_max (ha : 0 ≤ a) (b₁ b₂ : M) : a • max b₁ b₂ = max (a • b₁) (a • b₂) :=
+  (monotone_smul_left ha : Monotone (_ : M → M)).map_max
+#align smul_max smul_max
+
+theorem smul_min (ha : 0 ≤ a) (b₁ b₂ : M) : a • min b₁ b₂ = min (a • b₁) (a • b₂) :=
+  (monotone_smul_left ha : Monotone (_ : M → M)).map_min
+#align smul_min smul_min
+
+end LinearOrderedSemiring
+
 section LinearOrderedSemifield
 
 variable [LinearOrderedSemifield 𝕜] [OrderedAddCommMonoid M] [OrderedAddCommMonoid N]

Mathlib/Data/Set/Intervals/Basic.lean

@@ -6,7 +6,7 @@
 import Mathlib.Order.MinMax
 import Mathlib.Data.Set.Prod
 
-#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"4367b192b58a665b6f18773f73eb492eb4df7990"
+#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
 
 /-!
 # Intervals

Mathlib/Data/Set/Intervals/Image.lean

@@ -64,6 +64,7 @@ variable [Preorder α] [Preorder β]
 theorem Monotone.image_Icc_subset (h : Monotone f) :
     f '' Icc a b ⊆ Icc (f a) (f b) :=
   h.mapsTo_Icc.image_subset
+#align monotone.image_Icc_subset Monotone.image_Icc_subset
 
 theorem StrictMono.image_Ioo_subset (h : StrictMono f) :
     f '' Ioo a b ⊆ Ioo (f a) (f b) :=

Mathlib/Order/Lattice.lean

@@ -9,7 +9,7 @@
 import Mathlib.Order.ULift
 import Mathlib.Tactic.GCongr.Core
 
-#align_import order.lattice from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
+#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
 
 /-!
 # (Semi-)lattices
@@ -1135,6 +1135,7 @@
 end Monotone
 
 namespace MonotoneOn
+variable {f : α → β} {s : Set α} {x y : α}
 
 /-- Pointwise supremum of two monotone functions is a monotone function. -/
 protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
@@ -1160,6 +1161,32 @@
   hf.inf hg
 #align monotone_on.min MonotoneOn.min
 
+theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
+    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
+  inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
+#align monotone_on.of_map_inf MonotoneOn.of_map_inf
+
+theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
+    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
+  (@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
+#align monotone_on.of_map_sup MonotoneOn.of_map_sup
+
+variable [LinearOrder α]
+
+theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
+    f (x ⊔ y) = f x ⊔ f y := by
+  cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
+    first
+    | assumption
+    | simp only [*, sup_of_le_left, sup_of_le_right]
+
+#align monotone_on.map_sup MonotoneOn.map_sup
+
+theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
+    f (x ⊓ y) = f x ⊓ f y :=
+  hf.dual.map_sup hx hy
+#align monotone_on.map_inf MonotoneOn.map_inf
+
 end MonotoneOn
 
 namespace Antitone
@@ -1215,6 +1242,7 @@
 end Antitone
 
 namespace AntitoneOn
+variable {f : α → β} {s : Set α} {x y : α}
 
 /-- Pointwise supremum of two antitone functions is an antitone function. -/
 protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
@@ -1240,6 +1268,31 @@
   hf.inf hg
 #align antitone_on.min AntitoneOn.min
 
+theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
+    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
+  sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
+#align antitone_on.of_map_inf AntitoneOn.of_map_inf
+
+theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
+    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
+  (@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
+#align antitone_on.of_map_sup AntitoneOn.of_map_sup
+
+variable [LinearOrder α]
+
+theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
+    f (x ⊔ y) = f x ⊓ f y := by
+  cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
+    first
+    | assumption
+    | simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
+#align antitone_on.map_sup AntitoneOn.map_sup
+
+theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
+    f (x ⊓ y) = f x ⊔ f y :=
+  hf.dual.map_sup hx hy
+#align antitone_on.map_inf AntitoneOn.map_inf
+
 end AntitoneOn
 
 /-!