Merge branch 'YK-cont-alt-new' into YK-cont-alternating

archive/wiedijk_100_theorems/ascending_descending_sequences.lean

@@ -24,7 +24,6 @@ sequences, increasing, decreasing, Ramsey, Erdos-Szekeres, Erdős–Szekeres, Er
 variables {α : Type*} [linear_order α] {β : Type*}
 
 open function finset
-open_locale classical
 
 namespace theorems_100
 

src/analysis/normed_space/continuous_linear_map.lean

@@ -192,27 +192,6 @@ lemma to_span_singleton_homothety (x : E) (c : 𝕜) :
   ‖linear_map.to_span_singleton 𝕜 E x c‖ = ‖x‖ * ‖c‖ :=
 by {rw mul_comm, exact norm_smul _ _}
 
-/-- Given an element `x` of a normed space `E` over a field `𝕜`, the natural continuous
-    linear map from `𝕜` to `E` by taking multiples of `x`.-/
-def to_span_singleton (x : E) : 𝕜 →L[𝕜] E :=
-of_homothety (linear_map.to_span_singleton 𝕜 E x) ‖x‖ (to_span_singleton_homothety 𝕜 x)
-
-lemma to_span_singleton_apply (x : E) (r : 𝕜) : to_span_singleton 𝕜 x r = r • x :=
-by simp [to_span_singleton, of_homothety, linear_map.to_span_singleton]
-
-lemma to_span_singleton_add (x y : E) :
-  to_span_singleton 𝕜 (x + y) = to_span_singleton 𝕜 x + to_span_singleton 𝕜 y :=
-by { ext1, simp [to_span_singleton_apply], }
-
-lemma to_span_singleton_smul' (𝕜') [normed_field 𝕜'] [normed_space 𝕜' E]
-  [smul_comm_class 𝕜 𝕜' E] (c : 𝕜') (x : E) :
-  to_span_singleton 𝕜 (c • x) = c • to_span_singleton 𝕜 x :=
-by { ext1, rw [to_span_singleton_apply, smul_apply, to_span_singleton_apply, smul_comm], }
-
-lemma to_span_singleton_smul (c : 𝕜) (x : E) :
-  to_span_singleton 𝕜 (c • x) = c • to_span_singleton 𝕜 x :=
-to_span_singleton_smul' 𝕜 𝕜 c x
-
 end continuous_linear_map
 
 section

src/data/finset/basic.lean

@@ -570,6 +570,10 @@ lemma forall_mem_cons (h : a ∉ s) (p : α → Prop) :
   (∀ x, x ∈ cons a s h → p x) ↔ p a ∧ ∀ x, x ∈ s → p x :=
 by simp only [mem_cons, or_imp_distrib, forall_and_distrib, forall_eq]
 
+/-- Useful in proofs by induction. -/
+lemma forall_of_forall_cons {p : α → Prop} {h : a ∉ s} (H : ∀ x, x ∈ cons a s h → p x) (x)
+  (h : x ∈ s) : p x := H _ $ mem_cons.2 $ or.inr h
+
 @[simp] lemma mk_cons {s : multiset α} (h : (a ::ₘ s).nodup) :
   (⟨a ::ₘ s, h⟩ : finset α) = cons a ⟨s, (nodup_cons.1 h).2⟩ (nodup_cons.1 h).1 := rfl
 
@@ -2084,6 +2088,10 @@ lemma forall_mem_insert [decidable_eq α] (a : α) (s : finset α) (p : α → P
   (∀ x, x ∈ insert a s → p x) ↔ p a ∧ ∀ x, x ∈ s → p x :=
 by simp only [mem_insert, or_imp_distrib, forall_and_distrib, forall_eq]
 
+/-- Useful in proofs by induction. -/
+lemma forall_of_forall_insert [decidable_eq α] {p : α → Prop} {a : α} {s : finset α}
+  (H : ∀ x, x ∈ insert a s → p x) (x) (h : x ∈ s) : p x := H _ $ mem_insert_of_mem h
+
 end finset
 
 /-- Equivalence between the set of natural numbers which are `≥ k` and `ℕ`, given by `n → n - k`. -/

src/data/finset/n_ary.lean

@@ -358,6 +358,37 @@ lemma image₂_right_identity {f : γ → β → γ} {b : β} (h : ∀ a, f a b
   image₂ f s {b} = s :=
 by rw [image₂_singleton_right, funext h, image_id']
 
+/-- If each partial application of `f` is injective, and images of `s` under those partial
+applications are disjoint (but not necessarily distinct!), then the size of `t` divides the size of
+`finset.image₂ f s t`. -/
+lemma card_dvd_card_image₂_right (hf : ∀ a ∈ s, injective (f a))
+  (hs : ((λ a, t.image $ f a) '' s).pairwise_disjoint id) :
+  t.card ∣ (image₂ f s t).card :=
+begin
+  classical,
+  induction s using finset.induction with a s ha ih,
+  { simp },
+  specialize ih (forall_of_forall_insert hf)
+    (hs.subset $ set.image_subset _ $ coe_subset.2 $ subset_insert _ _),
+  rw image₂_insert_left,
+  by_cases h : disjoint (image (f a) t) (image₂ f s t),
+  { rw card_union_eq h,
+    exact (card_image_of_injective _ $ hf _ $ mem_insert_self _ _).symm.dvd.add ih },
+  simp_rw [←bUnion_image_left, disjoint_bUnion_right, not_forall] at h,
+  obtain ⟨b, hb, h⟩ := h,
+  rwa union_eq_right_iff_subset.2,
+  exact (hs.eq (set.mem_image_of_mem _ $ mem_insert_self _ _)
+    (set.mem_image_of_mem _ $ mem_insert_of_mem hb) h).trans_subset (image_subset_image₂_right hb),
+end
+
+/-- If each partial application of `f` is injective, and images of `t` under those partial
+applications are disjoint (but not necessarily distinct!), then the size of `s` divides the size of
+`finset.image₂ f s t`. -/
+lemma card_dvd_card_image₂_left (hf : ∀ b ∈ t, injective (λ a, f a b))
+  (ht : ((λ b, s.image $ λ a, f a b) '' t).pairwise_disjoint id) :
+  s.card ∣ (image₂ f s t).card :=
+by { rw ←image₂_swap, exact card_dvd_card_image₂_right hf ht }
+
 variables [decidable_eq α] [decidable_eq β]
 
 lemma image₂_inter_union_subset_union :

src/data/finset/pointwise.lean

@@ -1078,6 +1078,24 @@ coe_injective $ by { push_cast, exact set.smul_univ hs }
 @[simp, to_additive] lemma card_smul_finset (a : α) (s : finset β) : (a • s).card = s.card :=
 card_image_of_injective _ $ mul_action.injective _
 
+/-- If the left cosets of `t` by elements of `s` are disjoint (but not necessarily distinct!), then
+the size of `t` divides the size of `s * t`. -/
+@[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
+distinct!), then the size of `t` divides the size of `s + t`."]
+lemma card_dvd_card_smul_right {s : finset α} :
+  ((• t) '' (s : set α)).pairwise_disjoint id → t.card ∣ (s • t).card :=
+card_dvd_card_image₂_right (λ _ _, mul_action.injective _)
+
+variables [decidable_eq α]
+
+/-- If the right cosets of `s` by elements of `t` are disjoint (but not necessarily distinct!), then
+the size of `s` divides the size of `s * t`. -/
+@[to_additive "If the right cosets of `s` by elements of `t` are disjoint (but not necessarily
+distinct!), then the size of `s` divides the size of `s + t`."]
+lemma card_dvd_card_mul_left {s t : finset α} :
+  ((λ b, s.image $ λ a, a * b) '' (t : set α)).pairwise_disjoint id → s.card ∣ (s * t).card :=
+card_dvd_card_image₂_left (λ _ _, mul_left_injective _)
+
 end group
 
 section group_with_zero

src/order/directed.lean

@@ -131,6 +131,10 @@ lemma directed_on_of_inf_mem [semilattice_inf α] {S : set α}
   (H : ∀ ⦃i j⦄, i ∈ S → j ∈ S → i ⊓ j ∈ S) : directed_on (≥) S :=
 λ a ha b hb, ⟨a ⊓ b, H ha hb, inf_le_left, inf_le_right⟩
 
+lemma is_total.directed [is_total α r] (f : ι → α) :
+  directed r f :=
+λ i j, or.cases_on (total_of r (f i) (f j)) (λ h, ⟨j, h, refl _⟩) (λ h, ⟨i, refl _, h⟩)
+
 /-- `is_directed α r` states that for any elements `a`, `b` there exists an element `c` such that
 `r a c` and `r b c`. -/
 class is_directed (α : Type*) (r : α → α → Prop) : Prop :=
@@ -150,7 +154,7 @@ lemma directed_on_univ_iff : directed_on r set.univ ↔ is_directed α r :=
 
 @[priority 100]  -- see Note [lower instance priority]
 instance is_total.to_is_directed [is_total α r] : is_directed α r :=
-⟨λ a b, or.cases_on (total_of r a b) (λ h, ⟨b, h, refl _⟩) (λ h, ⟨a, refl _, h⟩)⟩
+by rw ← directed_id_iff; exact is_total.directed _
 
 lemma is_directed_mono [is_directed α r] (h : ∀ ⦃a b⦄, r a b → s a b) : is_directed α s :=
 ⟨λ a b, let ⟨c, ha, hb⟩ := is_directed.directed a b in ⟨c, h ha, h hb⟩⟩

src/order/monotone/basic.lean

@@ -102,6 +102,21 @@ def strict_anti_on (f : α → β) (s : set α) : Prop :=
 
 end monotone_def
 
+section decidable
+
+variables [preorder α] [preorder β] {f : α → β} {s : set α}
+
+instance [i : decidable (∀ a b, a ≤ b → f a ≤ f b)] : decidable (monotone f) := i
+instance [i : decidable (∀ a b, a ≤ b → f b ≤ f a)] : decidable (antitone f) := i
+instance [i : decidable (∀ a b ∈ s, a ≤ b → f a ≤ f b)] : decidable (monotone_on f s) := i
+instance [i : decidable (∀ a b ∈ s, a ≤ b → f b ≤ f a)] : decidable (antitone_on f s) := i
+instance [i : decidable (∀ a b, a < b → f a < f b)] : decidable (strict_mono f) := i
+instance [i : decidable (∀ a b, a < b → f b < f a)] : decidable (strict_anti f) := i
+instance [i : decidable (∀ a b ∈ s, a < b → f a < f b)] : decidable (strict_mono_on f s) := i
+instance [i : decidable (∀ a b ∈ s, a < b → f b < f a)] : decidable (strict_anti_on f s) := i
+
+end decidable
+
 /-! ### Monotonicity on the dual order
 
 Strictly, many of the `*_on.dual` lemmas in this section should use `of_dual ⁻¹' s` instead of `s`,

src/topology/algebra/module/basic.lean

@@ -1003,6 +1003,37 @@ lemma smul_right_comp [has_continuous_mul R₁] {x : M₂} {c : R₁} :
     smul_right (1 : R₁ →L[R₁] R₁) (c • x) :=
 by { ext, simp [mul_smul] }
 
+section to_span_singleton
+variables (R₁)
+variables [has_continuous_smul R₁ M₁]
+
+/-- Given an element `x` of a topological space `M` over a semiring `R`, the natural continuous
+linear map from `R` to `M` by taking multiples of `x`.-/
+def to_span_singleton (x : M₁) : R₁ →L[R₁] M₁ :=
+{ to_linear_map := linear_map.to_span_singleton R₁ M₁ x,
+  cont := continuous_id.smul continuous_const }
+
+lemma to_span_singleton_apply (x : M₁) (r : R₁) : to_span_singleton R₁ x r = r • x :=
+rfl
+
+lemma to_span_singleton_add [has_continuous_add M₁] (x y : M₁) :
+  to_span_singleton R₁ (x + y) = to_span_singleton R₁ x + to_span_singleton R₁ y :=
+by { ext1, simp [to_span_singleton_apply], }
+
+lemma to_span_singleton_smul' {α} [monoid α] [distrib_mul_action α M₁]
+  [has_continuous_const_smul α M₁]
+  [smul_comm_class R₁ α M₁] (c : α) (x : M₁) :
+  to_span_singleton R₁ (c • x) = c • to_span_singleton R₁ x :=
+by { ext1, rw [to_span_singleton_apply, smul_apply, to_span_singleton_apply, smul_comm], }
+
+/-- A special case of `to_span_singleton_smul'` for when `R` is commutative. -/
+lemma to_span_singleton_smul (R) {M₁} [comm_semiring R] [add_comm_monoid M₁] [module R M₁]
+  [topological_space R] [topological_space M₁] [has_continuous_smul R M₁] (c : R) (x : M₁) :
+  to_span_singleton R (c • x) = c • to_span_singleton R x :=
+to_span_singleton_smul' R c x
+
+end to_span_singleton
+
 end semiring
 
 section pi