refactor(*): use 'lemma' iff statement is private

algebra/lattice/basic.lean

@@ -20,7 +20,7 @@ section
   variable {α : Type u}
 
   -- TODO: this seems crazy, but it also seems to work reasonably well
-  @[ematch] lemma le_antisymm' [weak_order α] : ∀ {a b : α}, (: a ≤ b :) → b ≤ a → a = b :=
+  @[ematch] theorem le_antisymm' [weak_order α] : ∀ {a b : α}, (: a ≤ b :) → b ≤ a → a = b :=
   weak_order.le_antisymm
 end
 
@@ -47,17 +47,17 @@ class order_top (α : Type u) extends has_top α, weak_order α :=
 section order_top
 variables {α : Type u} [order_top α] {a : α}
 
-@[simp] lemma le_top : a ≤ ⊤ :=
+@[simp] theorem le_top : a ≤ ⊤ :=
 order_top.le_top a
 
-lemma top_unique (h : ⊤ ≤ a) : a = ⊤ :=
+theorem top_unique (h : ⊤ ≤ a) : a = ⊤ :=
 le_antisymm le_top h
 
 -- TODO: delete in favor of the next?
-lemma eq_top_iff : a = ⊤ ↔ ⊤ ≤ a :=
+theorem eq_top_iff : a = ⊤ ↔ ⊤ ≤ a :=
 ⟨assume eq, eq.symm ▸ le_refl ⊤, top_unique⟩
 
-@[simp] lemma top_le_iff : ⊤ ≤ a ↔ a = ⊤ :=
+@[simp] theorem top_le_iff : ⊤ ≤ a ↔ a = ⊤ :=
 ⟨top_unique, λ h, h.symm ▸ le_refl ⊤⟩
 
 end order_top
@@ -68,19 +68,19 @@ class order_bot (α : Type u) extends has_bot α, weak_order α :=
 section order_bot
 variables {α : Type u} [order_bot α] {a : α}
 
-@[simp] lemma bot_le : ⊥ ≤ a := order_bot.bot_le a
+@[simp] theorem bot_le : ⊥ ≤ a := order_bot.bot_le a
 
-lemma bot_unique (h : a ≤ ⊥) : a = ⊥ :=
+theorem bot_unique (h : a ≤ ⊥) : a = ⊥ :=
 le_antisymm h bot_le
 
 -- TODO: delete?
-lemma eq_bot_iff : a = ⊥ ↔ a ≤ ⊥ :=
+theorem eq_bot_iff : a = ⊥ ↔ a ≤ ⊥ :=
 ⟨assume eq, eq.symm ▸ le_refl ⊥, bot_unique⟩
 
-@[simp] lemma le_bot_iff : a ≤ ⊥ ↔ a = ⊥ :=
+@[simp] theorem le_bot_iff : a ≤ ⊥ ↔ a = ⊥ :=
 ⟨bot_unique, assume h, h.symm ▸ le_refl ⊥⟩
 
-lemma neq_bot_of_le_neq_bot {a b : α} (hb : b ≠ ⊥) (hab : b ≤ a) : a ≠ ⊥ :=
+theorem neq_bot_of_le_neq_bot {a b : α} (hb : b ≠ ⊥) (hab : b ≤ a) : a ≠ ⊥ :=
 assume ha, hb $ bot_unique $ ha ▸ hab
 
 end order_bot
@@ -93,55 +93,55 @@ class semilattice_sup (α : Type u) extends has_sup α, weak_order α :=
 section semilattice_sup
 variables {α : Type u} [semilattice_sup α] {a b c d : α}
 
-lemma le_sup_left : a ≤ a ⊔ b :=
+theorem le_sup_left : a ≤ a ⊔ b :=
 semilattice_sup.le_sup_left a b
 
-@[ematch] lemma le_sup_left' : a ≤ (: a ⊔ b :) :=
+@[ematch] theorem le_sup_left' : a ≤ (: a ⊔ b :) :=
 semilattice_sup.le_sup_left a b
 
-lemma le_sup_right : b ≤ a ⊔ b :=
+theorem le_sup_right : b ≤ a ⊔ b :=
 semilattice_sup.le_sup_right a b
 
-@[ematch] lemma le_sup_right' : b ≤ (: a ⊔ b :) :=
+@[ematch] theorem le_sup_right' : b ≤ (: a ⊔ b :) :=
 semilattice_sup.le_sup_right a b
 
-lemma le_sup_left_of_le (h : c ≤ a) : c ≤ a ⊔ b :=
+theorem le_sup_left_of_le (h : c ≤ a) : c ≤ a ⊔ b :=
 by finish
 
-lemma le_sup_right_of_le (h : c ≤ b) : c ≤ a ⊔ b :=
+theorem le_sup_right_of_le (h : c ≤ b) : c ≤ a ⊔ b :=
 by finish
 
-lemma sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
+theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
 semilattice_sup.sup_le a b c
 
-@[simp] lemma sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
+@[simp] theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
 ⟨assume h : a ⊔ b ≤ c, ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
   assume ⟨h₁, h₂⟩, sup_le h₁ h₂⟩
 
 -- TODO: if we just write le_antisymm, Lean doesn't know which ≤ we want to use
 -- Can we do anything about that?
-lemma sup_of_le_left (h : b ≤ a) : a ⊔ b = a :=
+theorem sup_of_le_left (h : b ≤ a) : a ⊔ b = a :=
 by apply le_antisymm; finish
 
-lemma sup_of_le_right (h : a ≤ b) : a ⊔ b = b :=
+theorem sup_of_le_right (h : a ≤ b) : a ⊔ b = b :=
 by apply le_antisymm; finish
 
-lemma sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
+theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
 by finish
 
-lemma le_of_sup_eq (h : a ⊔ b = b) : a ≤ b :=
+theorem le_of_sup_eq (h : a ⊔ b = b) : a ≤ b :=
 by finish
 
-@[simp] lemma sup_idem : a ⊔ a = a :=
+@[simp] theorem sup_idem : a ⊔ a = a :=
 by apply le_antisymm; finish
 
-lemma sup_comm : a ⊔ b = b ⊔ a :=
+theorem sup_comm : a ⊔ b = b ⊔ a :=
 by apply le_antisymm; finish
 
 instance semilattice_sup_to_is_commutative [semilattice_sup α] : is_commutative α (⊔) :=
 ⟨@sup_comm _ _⟩
 
-lemma sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
+theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
 by apply le_antisymm; finish
 
 instance semilattice_sup_to_is_associative [semilattice_sup α] : is_associative α (⊔) :=
@@ -157,53 +157,53 @@ class semilattice_inf (α : Type u) extends has_inf α, weak_order α :=
 section semilattice_inf
 variables {α : Type u} [semilattice_inf α] {a b c d : α}
 
-lemma inf_le_left : a ⊓ b ≤ a :=
+theorem inf_le_left : a ⊓ b ≤ a :=
 semilattice_inf.inf_le_left a b
 
-@[ematch] lemma inf_le_left' : (: a ⊓ b :) ≤ a :=
+@[ematch] theorem inf_le_left' : (: a ⊓ b :) ≤ a :=
 semilattice_inf.inf_le_left a b
 
-lemma inf_le_right : a ⊓ b ≤ b :=
+theorem inf_le_right : a ⊓ b ≤ b :=
 semilattice_inf.inf_le_right a b
 
-@[ematch] lemma inf_le_right' : (: a ⊓ b :) ≤ b :=
+@[ematch] theorem inf_le_right' : (: a ⊓ b :) ≤ b :=
 semilattice_inf.inf_le_right a b
 
-lemma le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
+theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
 semilattice_inf.le_inf a b c
 
-lemma inf_le_left_of_le (h : a ≤ c) : a ⊓ b ≤ c :=
+theorem inf_le_left_of_le (h : a ≤ c) : a ⊓ b ≤ c :=
 le_trans inf_le_left h
 
-lemma inf_le_right_of_le (h : b ≤ c) : a ⊓ b ≤ c :=
+theorem inf_le_right_of_le (h : b ≤ c) : a ⊓ b ≤ c :=
 le_trans inf_le_right h
 
-@[simp] lemma le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
+@[simp] theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
 ⟨assume h : a ≤ b ⊓ c, ⟨le_trans h inf_le_left, le_trans h inf_le_right⟩,
   assume ⟨h₁, h₂⟩, le_inf h₁ h₂⟩
 
-lemma inf_of_le_left (h : a ≤ b) : a ⊓ b = a :=
+theorem inf_of_le_left (h : a ≤ b) : a ⊓ b = a :=
 by apply le_antisymm; finish
 
-lemma inf_of_le_right (h : b ≤ a) : a ⊓ b = b :=
+theorem inf_of_le_right (h : b ≤ a) : a ⊓ b = b :=
 by apply le_antisymm; finish
 
-lemma inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
+theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
 by finish
 
-lemma le_of_inf_eq (h : a ⊓ b = a) : a ≤ b :=
+theorem le_of_inf_eq (h : a ⊓ b = a) : a ≤ b :=
 by finish
 
-@[simp] lemma inf_idem : a ⊓ a = a :=
+@[simp] theorem inf_idem : a ⊓ a = a :=
 by apply le_antisymm; finish
 
-lemma inf_comm : a ⊓ b = b ⊓ a :=
+theorem inf_comm : a ⊓ b = b ⊓ a :=
 by apply le_antisymm; finish
 
 instance semilattice_inf_to_is_commutative [semilattice_inf α] : is_commutative α (⊓) :=
 ⟨@inf_comm _ _⟩
 
-lemma inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
+theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
 by apply le_antisymm; finish
 
 instance semilattice_inf_to_is_associative [semilattice_inf α] : is_associative α (⊓) :=
@@ -216,10 +216,10 @@ class semilattice_sup_top (α : Type u) extends order_top α, semilattice_sup α
 section semilattice_sup_top
 variables {α : Type u} [semilattice_sup_top α] {a : α}
 
-@[simp] lemma top_sup_eq : ⊤ ⊔ a = ⊤ :=
+@[simp] theorem top_sup_eq : ⊤ ⊔ a = ⊤ :=
 sup_of_le_left le_top
 
-@[simp] lemma sup_top_eq : a ⊔ ⊤ = ⊤ :=
+@[simp] theorem sup_top_eq : a ⊔ ⊤ = ⊤ :=
 sup_of_le_right le_top
 
 end semilattice_sup_top
@@ -229,13 +229,13 @@ class semilattice_sup_bot (α : Type u) extends order_bot α, semilattice_sup α
 section semilattice_sup_bot
 variables {α : Type u} [semilattice_sup_bot α] {a b : α}
 
-@[simp] lemma bot_sup_eq : ⊥ ⊔ a = a :=
+@[simp] theorem bot_sup_eq : ⊥ ⊔ a = a :=
 sup_of_le_right bot_le
 
-@[simp] lemma sup_bot_eq : a ⊔ ⊥ = a :=
+@[simp] theorem sup_bot_eq : a ⊔ ⊥ = a :=
 sup_of_le_left bot_le
 
-@[simp] lemma sup_eq_bot_iff : a ⊔ b = ⊥ ↔ (a = ⊥ ∧ b = ⊥) :=
+@[simp] theorem sup_eq_bot_iff : a ⊔ b = ⊥ ↔ (a = ⊥ ∧ b = ⊥) :=
 by rw [eq_bot_iff, sup_le_iff]; simp
 
 end semilattice_sup_bot
@@ -245,13 +245,13 @@ class semilattice_inf_top (α : Type u) extends order_top α, semilattice_inf α
 section semilattice_inf_top
 variables {α : Type u} [semilattice_inf_top α] {a b : α}
 
-@[simp] lemma top_inf_eq : ⊤ ⊓ a = a :=
+@[simp] theorem top_inf_eq : ⊤ ⊓ a = a :=
 inf_of_le_right le_top
 
-@[simp] lemma inf_top_eq : a ⊓ ⊤ = a :=
+@[simp] theorem inf_top_eq : a ⊓ ⊤ = a :=
 inf_of_le_left le_top
 
-@[simp] lemma inf_eq_top_iff : a ⊓ b = ⊤ ↔ (a = ⊤ ∧ b = ⊤) :=
+@[simp] theorem inf_eq_top_iff : a ⊓ b = ⊤ ↔ (a = ⊤ ∧ b = ⊤) :=
 by rw [eq_top_iff, le_inf_iff]; simp
 
 end semilattice_inf_top
@@ -261,10 +261,10 @@ class semilattice_inf_bot (α : Type u) extends order_bot α, semilattice_inf α
 section semilattice_inf_bot
 variables {α : Type u} [semilattice_inf_bot α] {a : α}
 
-@[simp] lemma bot_inf_eq : ⊥ ⊓ a = ⊥ :=
+@[simp] theorem bot_inf_eq : ⊥ ⊓ a = ⊥ :=
 inf_of_le_left bot_le
 
-@[simp] lemma inf_bot_eq : a ⊓ ⊥ = ⊥ :=
+@[simp] theorem inf_bot_eq : a ⊓ ⊥ = ⊥ :=
 inf_of_le_right bot_le
 
 end semilattice_inf_bot
@@ -278,16 +278,16 @@ variables {α : Type u} [lattice α] {a b c d : α}
 
 /- Distributivity laws -/
 /- TODO: better names? -/
-lemma sup_inf_le : a ⊔ (b ⊓ c) ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
+theorem sup_inf_le : a ⊔ (b ⊓ c) ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
 by finish
 
-lemma le_inf_sup : (a ⊓ b) ⊔ (a ⊓ c) ≤ a ⊓ (b ⊔ c) :=
+theorem le_inf_sup : (a ⊓ b) ⊔ (a ⊓ c) ≤ a ⊓ (b ⊔ c) :=
 by finish
 
-lemma inf_sup_self : a ⊓ (a ⊔ b) = a :=
+theorem inf_sup_self : a ⊓ (a ⊔ b) = a :=
 le_antisymm (by finish) (by finish)
 
-lemma sup_inf_self : a ⊔ (a ⊓ b) = a :=
+theorem sup_inf_self : a ⊔ (a ⊓ b) = a :=
 le_antisymm (by finish) (by finish)
 
 end lattice