119 lines
3.8 KiB
Text
119 lines
3.8 KiB
Text
/-
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Copyright (c) 2017 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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prelude
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import init.data.nat init.data.fin.basic
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namespace fin
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open nat
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variable {n : nat}
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protected def succ : fin n → fin (succ n)
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| ⟨a, h⟩ := ⟨nat.succ a, succ_lt_succ h⟩
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def of_nat {n : nat} (a : nat) : fin (succ n) :=
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⟨a % succ n, nat.mod_lt _ (nat.zero_lt_succ _)⟩
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private lemma mlt {n b : nat} : ∀ {a}, n > a → b % n < n
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| 0 h := nat.mod_lt _ h
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| (a+1) h :=
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have n > 0, from lt.trans (nat.zero_lt_succ _) h,
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nat.mod_lt _ this
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protected def add : fin n → fin n → fin n
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| ⟨a, h⟩ ⟨b, _⟩ := ⟨(a + b) % n, mlt h⟩
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protected def mul : fin n → fin n → fin n
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| ⟨a, h⟩ ⟨b, _⟩ := ⟨(a * b) % n, mlt h⟩
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private lemma sublt {a b n : nat} (h : a < n) : a - b < n :=
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lt_of_le_of_lt (nat.sub_le a b) h
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protected def sub : fin n → fin n → fin n
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| ⟨a, h⟩ ⟨b, _⟩ := ⟨a - b, sublt h⟩
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private lemma modlt {a b n : nat} (h₁ : a < n) (h₂ : b < n) : a % b < n :=
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begin
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cases b with b,
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{simp [mod_zero], assumption},
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{have h : a % (succ b) < succ b,
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apply nat.mod_lt _ (nat.zero_lt_succ _),
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exact lt.trans h h₂}
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end
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protected def mod : fin n → fin n → fin n
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| ⟨a, h₁⟩ ⟨b, h₂⟩ := ⟨a % b, modlt h₁ h₂⟩
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private lemma divlt {a b n : nat} (h : a < n) : a / b < n :=
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lt_of_le_of_lt (nat.div_le_self a b) h
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protected def div : fin n → fin n → fin n
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| ⟨a, h⟩ ⟨b, _⟩ := ⟨a / b, divlt h⟩
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protected def lt : fin n → fin n → Prop
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| ⟨a, _⟩ ⟨b, _⟩ := a < b
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protected def le : fin n → fin n → Prop
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| ⟨a, _⟩ ⟨b, _⟩ := a ≤ b
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instance : has_zero (fin (succ n)) := ⟨⟨0, succ_pos n⟩⟩
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instance : has_one (fin (succ n)) := ⟨of_nat 1⟩
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instance : has_add (fin n) := ⟨fin.add⟩
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instance : has_sub (fin n) := ⟨fin.sub⟩
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instance : has_mul (fin n) := ⟨fin.mul⟩
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instance : has_mod (fin n) := ⟨fin.mod⟩
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instance : has_div (fin n) := ⟨fin.div⟩
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instance : has_lt (fin n) := ⟨fin.lt⟩
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instance : has_le (fin n) := ⟨fin.le⟩
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instance decidable_lt : ∀ (a b : fin n), decidable (a < b)
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| ⟨a, _⟩ ⟨b, _⟩ := by apply nat.decidable_lt
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instance decidable_le : ∀ (a b : fin n), decidable (a ≤ b)
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| ⟨a, _⟩ ⟨b, _⟩ := by apply nat.decidable_le
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lemma of_nat_zero : @of_nat n 0 = 0 :=
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fin.eq_of_veq $ zero_mod n.succ
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lemma add_def (a b : fin n) : (a + b).val = (a.val + b.val) % n :=
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show (fin.add a b).val = (a.val + b.val) % n, from
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by cases a; cases b; simp [fin.add]
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lemma mul_def (a b : fin n) : (a * b).val = (a.val * b.val) % n :=
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show (fin.mul a b).val = (a.val * b.val) % n, from
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by cases a; cases b; simp [fin.mul]
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lemma sub_def (a b : fin n) : (a - b).val = a.val - b.val :=
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show (fin.sub a b).val = a.val - b.val, from
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by cases a; cases b; simp [fin.sub]
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lemma mod_def (a b : fin n) : (a % b).val = a.val % b.val :=
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show (fin.mod a b).val = a.val % b.val, from
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by cases a; cases b; simp [fin.mod]
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lemma div_def (a b : fin n) : (a / b).val = a.val / b.val :=
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show (fin.div a b).val = a.val / b.val, from
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by cases a; cases b; simp [fin.div]
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lemma lt_def (a b : fin n) : (a < b) = (a.val < b.val) :=
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show (fin.lt a b) = (a.val < b.val), from
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by cases a; cases b; simp [fin.lt]
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lemma le_def (a b : fin n) : (a ≤ b) = (a.val ≤ b.val) :=
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show (fin.le a b) = (a.val ≤ b.val), from
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by cases a; cases b; simp [fin.le]
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lemma val_zero : (0 : fin (succ n)).val = 0 := rfl
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def pred {n : nat} : ∀ i : fin (succ n), i ≠ 0 → fin n
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| ⟨a, h₁⟩ h₂ := ⟨a.pred,
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begin
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have this : a ≠ 0,
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{ have aux₁ := vne_of_ne h₂,
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dsimp at aux₁, rw val_zero at aux₁, exact aux₁ },
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exact nat.pred_lt_pred this (nat.succ_ne_zero n) h₁
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end⟩
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end fin
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