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[Merged by Bors] - feat(Data/UInt, Data/Fin/Basic) declarations #90

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[Merged by Bors] - feat(Data/UInt, Data/Fin/Basic) declarations #90

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bd19997
feat(Data/UInt, Data/Fin/Basic) declarations
ammkrn Nov 10, 2021
adaf634
refactor(Data/Fin/Basic, Data/UInt)
ammkrn Nov 17, 2021
0307ddd
refactor(Data/Fin/Basic, Data/UInt)
ammkrn Nov 18, 2021
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121 changes: 95 additions & 26 deletions Mathlib/Data/Fin/Basic.lean
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Original file line number Diff line number Diff line change
@@ -1,5 +1,6 @@
import Mathlib.Data.Nat.Basic
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs

section Fin

Expand All @@ -8,21 +9,23 @@ variable {n : Nat}
instance : Zero (Fin n.succ) where
zero := Fin.ofNat 0

@[simp]
lemma Fin.val_zero : (0 : Fin n.succ).val = (0 : Nat) :=
@[simp] lemma Fin.val_zero : (0 : Fin n.succ).val = (0 : Nat) :=
show (Fin.ofNat 0).val = 0 by simp [Fin.ofNat]

instance : One (Fin n.succ) where
one := ⟨1 % n.succ, Nat.mod_lt 1 (Nat.zero_lt_succ n)⟩

@[simp]
lemma Fin.of_nat_zero : (Fin.ofNat 0 : Fin n.succ) = (0 : Fin n.succ) := by
@[simp] lemma Fin.val_one: (1 : Fin n.succ).val = (1 % n.succ : Nat) :=
show (Fin.ofNat 1).val = 1 % n.succ by simp [Fin.ofNat]

@[simp] lemma Fin.of_nat_zero : (Fin.ofNat 0 : Fin n.succ) = (0 : Fin n.succ) := by
apply Fin.eq_of_val_eq; simp only [Fin.ofNat, Nat.zero_mod, Fin.val_zero]

@[simp]
lemma Fin.val_one : (1 : Fin n.succ).val = (1 % n.succ : Nat) := by
have h0 : ∀ x, (OfNat.ofNat x : Fin n.succ) = Fin.ofNat x := by simp [OfNat.ofNat]
simp only [h0, Fin.ofNat]
instance : Neg (Fin n) where
neg a := ⟨(n - a) % n, Nat.mod_lt _ (lt_of_le_of_lt (Nat.zero_le _) a.isLt)⟩

lemma Fin.val_eq_of_lt : a < n.succ → (@Fin.ofNat n a).val = a := Nat.mod_eq_of_lt


/-- If you actually have an element of `Fin n`, then the `n` is always positive -/
lemma Fin.positive_size : ∀ (x : Fin n), 0 < n
Expand All @@ -34,6 +37,9 @@ lemma Fin.positive_size : ∀ (x : Fin n), 0 < n
lemma Fin.modn_def : ∀ (a : Fin n) (m : Nat), a % m = Fin.mk ((a.val % m) % n) (Nat.mod_lt (a.val % m) (Fin.positive_size a))
| ⟨a, pa⟩, m => by simp only [HMod.hMod, Fin.modn]

lemma Fin.mod_def : ∀ (a m : Fin n), a % m = Fin.mk ((a.val % m.val) % n) (Nat.mod_lt (a.val % m.val) (Fin.positive_size a))
| ⟨a, pa⟩, ⟨m, pm⟩ => by simp only [HMod.hMod, Mod.mod, Fin.mod]

lemma Fin.add_def : ∀ (a b : Fin n), a + b = (Fin.mk ((a.val + b.val) % n) (Nat.mod_lt _ (Fin.positive_size a)))
| ⟨a, pa⟩, ⟨b, pb⟩ => by simp only [HAdd.hAdd, Add.add, Fin.add]

Expand All @@ -48,34 +54,74 @@ lemma Fin.sub_def : ∀ (a b : Fin n), a - b = (Fin.mk ((a + (n - b)) % n) (Nat.
simp only [Fin.modn_def, Nat.mod_mod]
exact Nat.mod_eq_of_lt a.isLt

@[simp] lemma Fin.mod_eq_val (a : Fin n) : a.val % n = a.val := by
simp only [Fin.modn_def, Nat.mod_mod]
exact Nat.mod_eq_of_lt a.isLt

theorem Fin.mod_eq_of_lt {a b : Fin n} (h : a < b) : a % b = a := by
apply Fin.eq_of_val_eq
simp only [Fin.mod_def]
rw [Nat.mod_eq_of_lt h, Nat.mod_eq_of_lt a.isLt]

theorem Fin.mod_lt : ∀ (i : Fin n.succ) {m : Fin n.succ}, (0 : Fin n.succ) < m → (i % m) < m
| ⟨a, aLt⟩, ⟨m, mLt⟩, hp => by
have _ : (0 : Nat) < m := Fin.val_zero ▸ hp
have _ : a % m < m := Nat.mod_lt _ ‹0 < m›
simp only [Fin.mod_def, LT.lt]
rw [(Nat.mod_eq_of_lt (Nat.lt_trans ‹a % m < m› mLt) : a % m % n.succ = a % m)]
exact Nat.mod_lt _ ‹0 < m›

lemma Fin.add_comm (a b : Fin n) : a + b = b + a := by
apply Fin.eq_of_val_eq
simp only [Fin.add_def, Nat.add_comm]
apply Fin.eq_of_val_eq
simp only [Fin.add_def, Nat.add_comm]

@[simp] lemma Fin.add_zero (a : Fin n.succ) : a + 0 = a := by
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apply Fin.eq_of_val_eq
simp only [Fin.add_def, Fin.val_zero, Nat.add_zero]
exact Nat.mod_eq_of_lt a.isLt
apply Fin.eq_of_val_eq
simp only [Fin.add_def, Fin.val_zero, Nat.add_zero]
exact Nat.mod_eq_of_lt a.isLt

@[simp] lemma Fin.zero_add (a : Fin n.succ) : 0 + a = a := by
rw [Fin.add_comm]
exact Fin.add_zero a
@[simp] lemma Fin.zero_add (a : Fin n.succ) : 0 + a = a := (Fin.add_comm _ _) ▸ Fin.add_zero a
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lemma Fin.mul_comm (a b : Fin n) : a * b = b * a := by
apply Fin.eq_of_val_eq
simp only [Fin.mul_def, Nat.mul_comm]

@[simp] lemma Fin.zero_mul (a : Fin n.succ) : 0 * a = 0 := by
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apply Fin.eq_of_val_eq
simp only [Fin.mul_def, Fin.val_zero, Nat.zero_mul]
simp [Nat.zero_mod]
apply Fin.eq_of_val_eq
simp only [Fin.mul_def, Fin.val_zero, Nat.zero_mul]
simp [Nat.zero_mod]

lemma Fin.mul_comm (a b : Fin n) : a * b = b * a := by
@[simp] lemma Fin.mul_zero (a : Fin n.succ) : a * 0 = 0 := by
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rw [Fin.mul_comm]
exact Fin.zero_mul a

@[simp] lemma Fin.one_mul (a : Fin n.succ) : 1 * a = a := by
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apply Fin.eq_of_val_eq
simp only [Fin.mul_def, Nat.mul_comm]
simp only [Fin.mul_def]
cases n with
| zero => simp only [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ a.isLt)]
| succ n =>
have _ : (1 : Fin n.succ.succ).val = 1 := Nat.mod_eq_of_lt (Nat.succ_le_succ (Nat.succ_le_succ (Nat.zero_le n)))
rw [‹(1 : Fin n.succ.succ).val = 1›, Nat.one_mul, Nat.mod_eq_of_lt a.isLt]
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@[simp] lemma Fin.mul_one (a : Fin n.succ) : a * 1 = a := by
rw [Fin.mul_comm, Fin.one_mul]

lemma Fin.add_assoc (a b c : Fin n) : (a + b) + c = a + (b + c) := by
apply Fin.eq_of_val_eq
simp only [Fin.mul_def, Fin.add_def, Nat.mod_add_mod, Nat.add_mod_mod, Nat.add_assoc]
simp only [Fin.add_def, Nat.mod_add_mod, Nat.add_mod_mod, Nat.add_assoc]

instance : Neg (Fin n) where
neg a := ⟨(n - a) % n, Nat.mod_lt _ (lt_of_le_of_lt (Nat.zero_le _) a.isLt)⟩
lemma Fin.mul_assoc (a b c : Fin n) : (a * b) * c = a * (b * c) := by
apply Fin.eq_of_val_eq
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The mathlib proof in Lean 3 seems to be much shorter: https://github.com/leanprover-community/mathlib/blob/3578403f0c5cfbef907d1632ee91cee4b3cc6596/src/data/zmod/basic.lean#L47-L49

simp only [Fin.mul_def]
generalize lhs : ((a.val * b.val) % n * c.val) % n = l
have hl : (a.val * b.val % n) * (c.val % n) % n = (a.val * b.val * c.val) % n := (Nat.mul_mod (a.val * b.val) c.val n).symm
rw [<- Nat.mod_eq_of_lt c.isLt, hl] at lhs
generalize rhs : a.val * (b.val * c.val % n) % n = r
have hr : a.val % n * ((b.val * c.val) % n) % n = a.val * (b.val * c.val) % n := (Nat.mul_mod a.val (b.val * c.val) n).symm
rw [<- Nat.mod_eq_of_lt a.isLt, hr] at rhs
simp only [<- Nat.mul_assoc a.val b.val c.val] at rhs
rw [<- rhs, <- lhs]

@[simp] lemma Fin.add_left_neg : ∀ (a : Fin n.succ), -a + a = 0
| ⟨a, isLt⟩ => by
Expand All @@ -86,7 +132,7 @@ instance : Neg (Fin n) where
| succ a =>
have h1 : (n.succ - a.succ) % n.succ = (n.succ - a.succ) :=
Nat.mod_eq_of_lt (Nat.sub_lt (Nat.succ_pos _) (Nat.succ_pos _))
have h_aux : (Nat.succ n - Nat.succ a + Nat.succ a) = Nat.succ n := by
have h_aux : (n.succ - a.succ + a.succ) = n.succ := by
rw [Nat.add_comm]
exact Nat.add_sub_of_le (Nat.le_of_lt isLt)
rw [h1, h_aux, Nat.mod_self]
Expand All @@ -104,7 +150,7 @@ lemma Fin.nsmuls_eq (x : Nat) : ∀ (a : Fin n.succ), Fin.nsmul x a = Fin.ofNat
apply Fin.eq_of_val_eq
simp only [Fin.nsmul, Fin.ofNat, Fin.mul_def]
have hh : a % n.succ = a := Nat.mod_eq_of_lt isLt
generalize hq : x * a % Nat.succ n = q
generalize hq : x * a % n.succ = q
rw [<- hh, <- Nat.mul_mod, hq]

lemma Fin.nsmul_succ' (x : Nat) (a : Fin n.succ) : Fin.nsmul x.succ a = a + (Fin.nsmul x a) := by
Expand Down Expand Up @@ -134,6 +180,9 @@ instance : AddMonoid (Fin n.succ) where
nsmul_zero' := Fin.zero_mul
nsmul_succ' := Fin.nsmul_succ'

instance : AddCommMonoid (Fin n.succ) where
add_comm := Fin.add_comm

instance : SubNegMonoid (Fin n.succ) where
sub_eq_add_neg := Fin.sub_eq_add_neg
gsmul := Fin.gsmul
Expand All @@ -144,6 +193,19 @@ instance : SubNegMonoid (Fin n.succ) where
instance : AddGroup (Fin n.succ) where
add_left_neg := Fin.add_left_neg

instance : Monoid (Fin n.succ) where
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mul_one := Fin.mul_one
one_mul := Fin.one_mul
mul_assoc := Fin.mul_assoc
npow_zero' := fun _ => rfl
npow_succ' := fun _ _ => rfl

def Fin.addOverflows? (a b : Fin n) : Bool := n <= a.val + b.val

def Fin.mulOverflows? (a b : Fin n) : Bool := n <= a.val * b.val

def Fin.subUnderflows? (a b : Fin n) : Bool := a.val < b.val

def Fin.overflowingAdd (a b : Fin n) : (Bool × Fin n) := (n <= a.val + b.val, a + b)

def Fin.overflowingMul (a b : Fin n) : (Bool × Fin n) := (n <= a.val * b.val, a * b)
Expand Down Expand Up @@ -187,4 +249,11 @@ lemma Fin.checked_sub_spec (a b : Fin n) : (Fin.checkedSub a b).isSome = true <-
case neg => exact Nat.le_of_not_lt hx
case mpr => simp only [decide_eq_false (Nat.not_lt_of_le h : ¬a.val < b.val)]

instance : CommMonoid (Fin n.succ) where
mul_comm := Fin.mul_comm

instance : MonoidWithZero (Fin n.succ) where
zero_mul := Fin.zero_mul
mul_zero := Fin.mul_zero

end Fin