mirror of
https://github.com/leanprover/lean4.git
synced 2026-03-17 18:34:06 +00:00
9bd4dfb696
This PR deprecated the handful of names in core involving the component `cons₂` in favor of `cons_cons`.
509 lines
13 KiB
Lean4
509 lines
13 KiB
Lean4
set_option autoImplicit false -- For compatibility with downstream projects, so we can retest after Mathlib.
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set_option relaxedAutoImplicit false
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open List
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variable {α : Type _}
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variable {x y z : α}
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variable (l l₁ l₂ l₃ : List α)
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variable (L₁ L₂ : List (List α))
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variable {β : Type _}
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variable {f g : α → β}
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variable {γ : Type _}
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variable {f' : β → γ}
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variable (m n : Nat)
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/-! ## Preliminaries -/
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/-! ### cons -/
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/-! ### length -/
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/-! ### L[i] and L[i]? -/
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/-! ### mem -/
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/-! ### set -/
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/-! ### foldlM and foldrM -/
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/-! ### foldl and foldr -/
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/-! ### Equality -/
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/-! ### Lexicographic order -/
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/-! ## Getters -/
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#check_simp [x, y, x, y][0] ~> x
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#check_simp [x, y, x, y][1] ~> y
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#check_simp [x, y, x, y][2] ~> x
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#check_simp [x, y, x, y][3] ~> y
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#check_simp [x, y, x, y][0]? ~> some x
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#check_simp [x, y, x, y][1]? ~> some y
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#check_simp [x, y, x, y][2]? ~> some x
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#check_simp [x, y, x, y][3]? ~> some y
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/-! ### get, get!, get?, getD -/
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/-! ### getLast, getLast!, getLast?, getLastD -/
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/-! ## Head and tail -/
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/-! ### head, head!, head?, headD -/
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#check_simp l.headD x ~> l.head?.getD x
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#check_simp l.head? = none ~> l = []
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/-! ### tail!, tail?, tailD -/
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/-! ## Basic operations -/
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/-! ### map -/
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#check_simp l.map id ~> l
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#check_simp l.map (fun x => x) ~> l
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#check_simp [].map f ~> []
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#check_simp [x].map f ~> [f x]
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#check_simp map f l = map g l ~> ∀ a ∈ l, f a = g a
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variable (l : List Nat) in
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#check_simp map (· + 1) l = map (·.succ) l ~> True
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variable (l : List Nat) in
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#check_simp map (0 * ·) l ~> map (fun _ => 0) l
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variable (l : List String) in
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#check_simp map (fun s => s ++ s) ("a" :: l) ~> "aa" :: map (fun s => s ++ s) l
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#check_simp l.map f = [] ~> l = []
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variable (w : l ≠ []) in
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#check_simp head (l.map f) (by simpa) ~> f (head l (by simpa))
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variable (l : List String) in
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#check_simp head (("a" :: l).map fun s => s ++ s) (by simp) ~> "aa"
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variable (w : l ≠ []) in
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#check_simp getLast (l.map f) (by simpa) ~> f (getLast l (by simpa))
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#check_simp (l₁ ++ l₂).map f ~> l₁.map f ++ l₂.map f
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#check_simp (l.map f).map f' ~> l.map (f' ∘ f)
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#check_simp (concat l x).map f ~> map f l ++ [f x]
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variable (L : List (List α)) in
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#check_simp L.flatten.map f ~> (L.map (map f)).flatten
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#check_simp [l₁, l₂].flatten.map f ~> map f l₁ ++ map f l₂
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#check_simp l.map (Function.const α "1") ~> replicate l.length "1"
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#check_simp [x, y].map (Function.const α "1") ~> ["1", "1"]
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#check_simp l.reverse.map f ~> (l.map f).reverse
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#check_simp (l.take 3).map f ~> (l.map f).take 3
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#check_simp (l.drop 3).map f ~> (l.map f).drop 3
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#check_simp l.dropLast.map f ~> (l.map f).dropLast
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variable (p : β → Bool) in
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#check_simp (l.map f).find? p ~> (l.find? (p ∘ f)).map f
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/-! ### filter -/
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/-! ### filterMap -/
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/-! ### append -/
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variable (w : l₁ ≠ []) in
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#check_tactic head (l₁ ++ l₂) (by simp_all) ~> head l₁ w by simp_all
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#check_simp (l₁ ++ l₂).head? ~> l₁.head?.or l₂.head?
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#check_simp (l₁ ++ l₂).getLast? ~> l₂.getLast?.or l₁.getLast?
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/-! ### concat -/
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/-! ### flatten -/
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#check_simp (L₁ ++ L₂).flatten ~> L₁.flatten ++ L₂.flatten
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/-! ### bind -/
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/-! ### replicate -/
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#check_simp replicate 0 x ~> []
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#check_simp replicate 1 x ~> [x]
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#check_simp replicate 5 x ~> [x, x, x, x, x]
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-- `∈` and `contains
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#check_simp y ∈ replicate 0 x ~> False
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variable [BEq α] in
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#check_simp (replicate 0 x).contains y ~> false
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variable [BEq α] [LawfulBEq α] in
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#check_simp (replicate 0 x).contains y ~> false
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#check_simp y ∈ replicate 7 x ~> y = x
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variable [BEq α] in
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#check_simp (replicate 7 x).contains y ~> y == x
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variable [BEq α] [LawfulBEq α] in
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#check_simp (replicate 7 x).contains y ~> y == x
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-- `getElem` and `getElem?`
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variable (h : n < m) (w) in
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#check_tactic (replicate m x)[n]'w ~> x by simp
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variable (h : n < m) in
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#check_tactic (replicate m x)[n]? ~> some x by simp [h]
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#check_simp (replicate 7 x)[5] ~> x
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#check_simp (replicate 7 x)[5]? ~> some x
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variable (w : replicate n x ≠ []) in
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#check_tactic (replicate n x).head w ~> x by simp_all
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variable (w : replicate n x ≠ []) in
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#check_tactic (replicate n x).getLast w ~> x by simp_all
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-- injectivity
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#check_simp replicate 3 x = replicate 7 x ~> False
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#check_simp replicate 3 x = replicate 3 y ~> x = y
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#check_simp replicate 3 "1" = replicate 3 "1" ~> True
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#check_simp replicate n x = replicate m y ~> n = m ∧ (n = 0 ∨ x = y)
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-- append
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#check_simp replicate n x ++ replicate m x ~> replicate (n + m) x
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-- map
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#check_simp (replicate n "x").map (fun s => s ++ s) ~> replicate n "xx"
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-- filter
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#check_simp (replicate n [1]).filter (fun s => s.length = 1) ~> replicate n [1]
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#check_simp (replicate n [1]).filter (fun s => s.length = 2) ~> []
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-- filterMap
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#check_simp (replicate n [1]).filterMap (fun s => if s.length = 1 then some s else none) ~> replicate n [1]
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#check_simp (replicate n [1]).filterMap (fun s => if s.length = 2 then some s else none) ~> []
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-- join
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#check_simp (replicate n (replicate m x)).flatten ~> replicate (n * m) x
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#check_simp (replicate 1 (replicate m x)).flatten ~> replicate m x
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#check_simp (replicate n (replicate 1 x)).flatten ~> replicate n x
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#check_simp (replicate n (replicate 0 x)).flatten ~> []
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#check_simp (replicate 0 (replicate m x)).flatten ~> []
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#check_simp (replicate 0 (replicate 0 x)).flatten ~> []
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-- isEmpty
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#check_simp (replicate (n + 1) x).isEmpty ~> false
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#check_simp (replicate 0 x).isEmpty ~> true
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variable (h : ¬ n = 0) in -- It would be nice if this also worked with `h : 0 < n`
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#check_tactic (replicate n x).isEmpty ~> false by simp [h]
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-- reverse
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#check_simp (replicate n x).reverse ~> replicate n x
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-- dropLast
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#check_simp (replicate 0 x).dropLast ~> []
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#check_simp (replicate n x).dropLast ~> replicate (n-1) x
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#check_simp (replicate (n+1) x).dropLast ~> replicate n x
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-- isPrefixOf
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variable [BEq α] [LawfulBEq α] in
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#check_simp isPrefixOf [x, y, x] (replicate n x) ~> decide (3 ≤ n) && y == x
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attribute [local simp] isPrefixOf_cons_cons in
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variable [BEq α] [LawfulBEq α] in
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#check_simp isPrefixOf [x, y, x] (replicate (n+3) x) ~> y == x
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-- isSuffixOf
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variable [BEq α] [LawfulBEq α] in
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#check_simp isSuffixOf [x, y, x] (replicate n x) ~> decide (3 ≤ n) && y == x
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-- rotateLeft
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#check_simp (replicate n x).rotateLeft m ~> replicate n x
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-- rotateRight
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#check_simp (replicate n x).rotateRight m ~> replicate n x
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-- replace
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variable [BEq α] [LawfulBEq α] in
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#check_simp (replicate (n+1) x).replace x y ~> y :: replicate n x
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#check_simp (replicate n "1").replace "2" "3" ~> (replicate n "1")
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-- insert
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variable [BEq α] [LawfulBEq α] (h : 0 < n) in
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#check_tactic (replicate n x).insert x ~> replicate n x by simp [h]
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#check_simp (replicate n "1").insert "2" ~> "2" :: replicate n "1"
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-- erase
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variable [BEq α] [LawfulBEq α] in
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#check_simp (replicate (n+1) x).erase x ~> replicate n x
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#check_simp (replicate n "1").erase "2" ~> replicate n "1"
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-- find?
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#check_simp (replicate (n+1) x).find? (fun _ => true) ~> some x
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#check_simp (replicate (n+1) x).find? (fun _ => false) ~> none
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variable {p : α → Bool} (w : p x) in
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#check_tactic (replicate (n+1) x).find? p ~> some x by simp [w]
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variable {p : α → Bool} (w : ¬ p x) in
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#check_tactic (replicate (n+1) x).find? p ~> none by simp [w]
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variable (h : 0 < n) in
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#check_tactic (replicate n x).find? (fun _ => true) ~> some x by simp [h]
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variable (h : 0 < n) in
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#check_tactic (replicate n x).find? (fun _ => false) ~> none by simp [h]
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variable {p : α → Bool} (w : p x) (h : 0 < n) in
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#check_tactic (replicate n x).find? p ~> some x by simp [w, h]
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variable {p : α → Bool} (w : ¬ p x) (h : 0 < n) in
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#check_tactic (replicate n x).find? p ~> none by simp [w, h]
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-- findSome?
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#check_simp (replicate (n+1) x).findSome? (fun x => some x) ~> some x
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#check_simp (replicate (n+1) x).findSome? (fun _ => (none : Option β)) ~> none
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variable {f : α → Option β} in
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#check_tactic (replicate (n+1) x).findSome? f ~> f x by simp
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variable {f : α → Option β} (w : (f x).isNone) in
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#check_tactic (replicate (n+1) x).findSome? f ~> none by simp_all
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variable (h : 0 < n) in
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#check_tactic (replicate n x).findSome? (fun x => some x) ~> some x by simp [h]
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variable (h : 0 < n) in
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#check_tactic (replicate n x).findSome? (fun _ => (none : Option β)) ~> none by simp [h]
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variable {f : α → Option β} (w : (f x).isSome) (h : 0 < n) in
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#check_tactic (replicate n x).findSome? f ~> f x by simp [h]
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variable {f : α → Option β} (w : (f x).isNone) (h : 0 < n) in
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#check_tactic (replicate n x).findSome? f ~> none by simp_all
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-- lookup
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variable [BEq α] [LawfulBEq α] in
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#check_simp (replicate (n+1) (x, y)).lookup x ~> some y
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variable [BEq α] [LawfulBEq α] (h : 0 < n) in
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#check_tactic (replicate n (x, y)).lookup x ~> some y by simp [h]
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#check_simp (replicate n ("1", "2")).lookup "3" ~> none
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-- zip
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#check_simp (replicate n x).zip (replicate n y) ~> replicate n (x, y)
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#check_simp (replicate n x).zip (replicate m y) ~> replicate (min n m) (x, y)
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variable (h : n ≤ m) in
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#check_tactic (replicate n x).zip (replicate m y) ~> replicate n (x, y) by simp [h, Nat.min_eq_left]
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-- zipWith
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section
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variable (f : α → α → α)
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#check_simp zipWith f (replicate n x) (replicate n y) ~> replicate n (f x y)
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#check_simp zipWith f (replicate n x) (replicate m y) ~> replicate (min n m) (f x y)
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variable (h : n ≤ m) in
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#check_tactic zipWith f (replicate n x) (replicate m y) ~> replicate n (f x y) by simp [h, Nat.min_eq_left]
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-- unzip
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#check_simp unzip (replicate n (x, y)) ~> (replicate n x, replicate n y)
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-- min?
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-- Note this relies on the fact that we do not have `replicate_succ` as a `@[simp]` lemma
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#check_simp (replicate (n+1) 7).min? ~> some 7
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variable (h : 0 < n) in
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#check_tactic (replicate n 7).min? ~> some 7 by simp [h]
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-- max?
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-- Note this relies on the fact that we do not have `replicate_succ` as a `@[simp]` lemma
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#check_simp (replicate (n+1) 7).max? ~> some 7
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variable (h : 0 < n) in
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#check_tactic (replicate n 7).max? ~> some 7 by simp [h]
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end
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/-! ### reverse -/
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variable (p : α → Bool) in
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#check_simp (l.reverse.filter p) ~> (l.filter p).reverse
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variable (f : α → Option β) in
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#check_simp (l.reverse.filterMap f) ~> (l.filterMap f).reverse
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#check_simp l.reverse.head? ~> l.getLast?
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#check_simp l.reverse.getLast? ~> l.head?
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variable (h : l.reverse ≠ []) in
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#check_simp l.reverse.head h ~> l.getLast (by simp_all)
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variable (h : l.reverse ≠ []) in
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#check_simp l.reverse.getLast h ~> l.head (by simp_all)
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/-! ## List membership -/
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/-! ### elem / contains -/
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/-! ## Sublists -/
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/-! ### take and drop -/
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/-! ### takeWhile and dropWhile -/
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/-! ### partition -/
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/-! ### dropLast -/
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/-! ### isPrefixOf -/
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/-! ### isSuffixOf -/
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variable [BEq α] in
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#check_simp ([] : List α).isSuffixOf l ~> true
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/-! ### rotateLeft -/
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/-! ### rotateRight -/
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/-! ## Pairwise and Nodup -/
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/-! ### Pairwise -/
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section Pairwise
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variable (R : α → α → Prop)
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#check_simp Pairwise R [] ~> True
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#check_simp Pairwise R (x :: l) ~> (∀ (a' : α), a' ∈ l → R x a') ∧ Pairwise R l
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#check_simp Pairwise R [x, y, z] ~> (R x y ∧ R x z) ∧ R y z
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#check_simp Pairwise R (replicate n x) ~> n ≤ 1 ∨ R x x
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#check_simp Pairwise R (replicate 1 x) ~> True
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#check_simp Pairwise R (replicate (n+2) x) ~> R x x
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#check_simp Pairwise (· < ·) (replicate 2 m) ~> False
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#check_simp Pairwise (· < ·) (replicate n m) ~> n ≤ 1
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#check_simp Pairwise (· < ·) (replicate (n + 2) m) ~> False
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#check_simp Pairwise (· = ·) (replicate 2 m) ~> True
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#check_simp Pairwise (· = ·) (replicate n m) ~> True
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end Pairwise
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/-! ### Nodup -/
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#check_simp Nodup [] ~> True
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#check_simp Nodup (x :: l) ~> ¬x ∈ l ∧ l.Nodup
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#check_simp Nodup [x, y, z] ~> (¬x = y ∧ ¬x = z) ∧ ¬y = z
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#check_simp Nodup (replicate (n+2) x) ~> False
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#check_simp Nodup (replicate 2 x) ~> False
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/-! ## Manipulating elements -/
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/-! ### replace -/
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/-! ### insert -/
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/-! ### erase -/
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/-! ### find? -/
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/-! ### findSome? -/
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/-! ### lookup -/
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/-! ## Logic -/
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/-! ### any / all -/
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/-! ## Zippers -/
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/-! ### zip -/
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/-! ### zipWith -/
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/-! ### zipWithAll -/
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/-! ## Ranges and enumeration -/
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example : (List.range 1).sum = 0 := by simp
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/-! ### enumFrom -/
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/-! ### min? -/
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/-! ### max? -/
|
||
|
||
/-! ## ofFn -/
|
||
|
||
example (f : Fin 3 → Nat) : List.ofFn f = [f 0, f 1, f 2] := rfl
|
||
|
||
-- Out of place, but lets check that `Fin.foldl` is semireducible too.
|
||
-- Well, no longer the case
|
||
-- example (f : Fin 3 → Nat) : Fin.foldl 3 (fun acc i => f i :: acc) [] = [f 2, f 1, f 0] := rfl
|
||
|
||
/-! ## Monadic operations -/
|
||
|
||
#check_simp
|
||
(Id.run do
|
||
let mut s := 0
|
||
for i in [1,2,3,4] do
|
||
s := s + i
|
||
pure s) ~> 10
|
||
|
||
#check_simp
|
||
(Id.run do
|
||
let mut s := 0
|
||
for h : i in [1,2,3,4] do
|
||
s := s + i
|
||
pure s) ~> 10
|
||
|
||
variable (l : List α) (k m : Nat) in
|
||
#check_simp
|
||
(Id.run do
|
||
let mut x := m
|
||
for _ in l do
|
||
x := x + k
|
||
pure x) ~> m + k * l.length
|
||
|
||
-- as above, but for an arbitrary monad
|
||
variable (l : List α) (k m : Nat) {M} [Monad M] [LawfulMonad M] in
|
||
#check_simp
|
||
(show M _ from do
|
||
let mut x := m
|
||
for _ in l do
|
||
x := x + k
|
||
pure x) ~> pure (m + k * l.length)
|
||
|
||
/-! ### mapM -/
|
||
|
||
/-! ### forM -/
|