feat: grind annotations for List.Pairwise/Nodup (#8482)
This PR adds preliminary `@[grind]` annotations for `List.Pairwise` and `List.Nodup`.
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Kim Morrison
GitHub
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1 changed files with 25 additions and 25 deletions
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@ -24,7 +24,7 @@ open Nat
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/-! ### Pairwise -/
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theorem Pairwise.sublist : l₁ <+ l₂ → l₂.Pairwise R → l₁.Pairwise R
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@[grind →] theorem Pairwise.sublist : l₁ <+ l₂ → l₂.Pairwise R → l₁.Pairwise R
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| .slnil, h => h
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| .cons _ s, .cons _ h₂ => h₂.sublist s
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| .cons₂ _ s, .cons h₁ h₂ => (h₂.sublist s).cons fun _ h => h₁ _ (s.subset h)
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@ -37,11 +37,11 @@ theorem Pairwise.imp {α R S} (H : ∀ {a b}, R a b → S a b) :
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theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
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(pairwise_cons.1 p).1 _
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theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
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@[grind →] theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
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(pairwise_cons.1 p).2
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set_option linter.unusedVariables false in
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theorem Pairwise.tail : ∀ {l : List α} (h : Pairwise R l), Pairwise R l.tail
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@[grind] theorem Pairwise.tail : ∀ {l : List α} (h : Pairwise R l), Pairwise R l.tail
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| [], h => h
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| _ :: _, h => h.of_cons
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@ -101,11 +101,11 @@ theorem Pairwise.forall_of_forall_of_flip (h₁ : ∀ x ∈ l, R x x) (h₂ : Pa
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· exact h₃.1 _ hx
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· exact ih (fun x hx => h₁ _ <| mem_cons_of_mem _ hx) h₂.2 h₃.2 hx hy
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theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by simp
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@[grind] theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by simp
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theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by simp
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@[grind =] theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by simp
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theorem pairwise_map {l : List α} :
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@[grind =] theorem pairwise_map {l : List α} :
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(l.map f).Pairwise R ↔ l.Pairwise fun a b => R (f a) (f b) := by
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induction l
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· simp
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@ -115,11 +115,11 @@ theorem Pairwise.of_map {S : β → β → Prop} (f : α → β) (H : ∀ a b :
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(p : Pairwise S (map f l)) : Pairwise R l :=
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(pairwise_map.1 p).imp (H _ _)
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theorem Pairwise.map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
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@[grind] theorem Pairwise.map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
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(p : Pairwise R l) : Pairwise S (map f l) :=
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pairwise_map.2 <| p.imp (H _ _)
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theorem pairwise_filterMap {f : β → Option α} {l : List β} :
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@[grind =] theorem pairwise_filterMap {f : β → Option α} {l : List β} :
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Pairwise R (filterMap f l) ↔ Pairwise (fun a a' : β => ∀ b, f a = some b → ∀ b', f a' = some b' → R b b') l := by
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let _S (a a' : β) := ∀ b, f a = some b → ∀ b', f a' = some b' → R b b'
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induction l with
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@ -134,20 +134,20 @@ theorem pairwise_filterMap {f : β → Option α} {l : List β} :
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simpa [IH, e] using fun _ =>
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⟨fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab⟩
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theorem Pairwise.filterMap {S : β → β → Prop} (f : α → Option β)
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@[grind] theorem Pairwise.filterMap {S : β → β → Prop} (f : α → Option β)
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(H : ∀ a a' : α, R a a' → ∀ b, f a = some b → ∀ b', f a' = some b' → S b b') {l : List α} (p : Pairwise R l) :
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Pairwise S (filterMap f l) :=
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pairwise_filterMap.2 <| p.imp (H _ _)
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theorem pairwise_filter {p : α → Prop} [DecidablePred p] {l : List α} :
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@[grind =] theorem pairwise_filter {p : α → Bool} {l : List α} :
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Pairwise R (filter p l) ↔ Pairwise (fun x y => p x → p y → R x y) l := by
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rw [← filterMap_eq_filter, pairwise_filterMap]
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simp
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theorem Pairwise.filter (p : α → Bool) : Pairwise R l → Pairwise R (filter p l) :=
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@[grind] theorem Pairwise.filter (p : α → Bool) : Pairwise R l → Pairwise R (filter p l) :=
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Pairwise.sublist filter_sublist
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theorem pairwise_append {l₁ l₂ : List α} :
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@[grind =] theorem pairwise_append {l₁ l₂ : List α} :
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(l₁ ++ l₂).Pairwise R ↔ l₁.Pairwise R ∧ l₂.Pairwise R ∧ ∀ a ∈ l₁, ∀ b ∈ l₂, R a b := by
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induction l₁ <;> simp [*, or_imp, forall_and, and_assoc, and_left_comm]
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@ -157,13 +157,13 @@ theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y →
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(x : α) (xm : x ∈ l₂) (y : α) (ym : y ∈ l₁) : R x y := s (H y ym x xm)
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simp only [pairwise_append, and_left_comm]; rw [Iff.intro (this l₁ l₂) (this l₂ l₁)]
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theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
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@[grind =] theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
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Pairwise R (l₁ ++ a :: l₂) ↔ Pairwise R (a :: (l₁ ++ l₂)) := by
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show Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
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rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
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simp only [mem_append, or_comm]
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theorem pairwise_flatten {L : List (List α)} :
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@[grind =] theorem pairwise_flatten {L : List (List α)} :
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Pairwise R (flatten L) ↔
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(∀ l ∈ L, Pairwise R l) ∧ Pairwise (fun l₁ l₂ => ∀ x ∈ l₁, ∀ y ∈ l₂, R x y) L := by
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induction L with
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@ -174,16 +174,16 @@ theorem pairwise_flatten {L : List (List α)} :
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rw [and_comm, and_congr_left_iff]
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intros; exact ⟨fun h l' b c d e => h c d e l' b, fun h c d e l' b => h l' b c d e⟩
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theorem pairwise_flatMap {R : β → β → Prop} {l : List α} {f : α → List β} :
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@[grind =] theorem pairwise_flatMap {R : β → β → Prop} {l : List α} {f : α → List β} :
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List.Pairwise R (l.flatMap f) ↔
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(∀ a ∈ l, Pairwise R (f a)) ∧ Pairwise (fun a₁ a₂ => ∀ x ∈ f a₁, ∀ y ∈ f a₂, R x y) l := by
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simp [List.flatMap, pairwise_flatten, pairwise_map]
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theorem pairwise_reverse {l : List α} :
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@[grind =] theorem pairwise_reverse {l : List α} :
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l.reverse.Pairwise R ↔ l.Pairwise (fun a b => R b a) := by
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induction l <;> simp [*, pairwise_append, and_comm]
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@[simp] theorem pairwise_replicate {n : Nat} {a : α} :
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@[simp, grind =] theorem pairwise_replicate {n : Nat} {a : α} :
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(replicate n a).Pairwise R ↔ n ≤ 1 ∨ R a a := by
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induction n with
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| zero => simp
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@ -205,10 +205,10 @@ theorem pairwise_reverse {l : List α} :
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simp
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· exact ⟨fun _ => h, Or.inr h⟩
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theorem Pairwise.drop {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.drop i) :=
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@[grind] theorem Pairwise.drop {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.drop i) :=
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h.sublist (drop_sublist _ _)
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theorem Pairwise.take {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.take i) :=
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@[grind] theorem Pairwise.take {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.take i) :=
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h.sublist (take_sublist _ _)
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theorem pairwise_iff_forall_sublist : l.Pairwise R ↔ (∀ {a b}, [a,b] <+ l → R a b) := by
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@ -247,7 +247,7 @@ theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h :
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intro a b hab
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apply h <;> (apply hab.subset; simp)
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theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
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@[grind =] theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
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Pairwise R (l.pmap f h) ↔
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Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by
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induction l with
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@ -259,7 +259,7 @@ theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h
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rintro H _ b hb rfl
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exact H b hb _ _
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theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
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@[grind] theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
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(h : ∀ x ∈ l, p x) {S : β → β → Prop}
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(hS : ∀ ⦃x⦄ (hx : p x) ⦃y⦄ (hy : p y), R x y → S (f x hx) (f y hy)) :
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Pairwise S (l.pmap f h) := by
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@ -268,15 +268,15 @@ theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f :
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/-! ### Nodup -/
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@[simp]
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@[simp, grind]
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theorem nodup_nil : @Nodup α [] :=
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Pairwise.nil
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@[simp]
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@[simp, grind =]
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theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by
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simp only [Nodup, pairwise_cons, forall_mem_ne]
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theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
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@[grind →] theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
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Pairwise.sublist
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theorem Sublist.nodup : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
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@ -303,7 +303,7 @@ theorem getElem?_inj {xs : List α}
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rw [mem_iff_getElem?]
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exact ⟨_, h₂⟩; exact ⟨_ , h₂.symm⟩
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@[simp] theorem nodup_replicate {n : Nat} {a : α} :
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@[simp, grind =] theorem nodup_replicate {n : Nat} {a : α} :
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(replicate n a).Nodup ↔ n ≤ 1 := by simp [Nodup]
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end List
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