feat: missing Fin @[simp] lemmas (#5380)
This commit is contained in:
Kim Morrison
GitHub
parent
ddd471223c
commit
e417ad8a70
3 changed files with 98 additions and 6 deletions
|
|
@ -54,7 +54,12 @@ theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
|
|||
@[simp] theorem val_ofNat' (n : Nat) [NeZero n] (a : Nat) :
|
||||
(Fin.ofNat' n a).val = a % n := rfl
|
||||
|
||||
@[simp] theorem ofNat'_val_eq_self [NeZero n](x : Fin n) : (Fin.ofNat' n x) = x := by
|
||||
@[simp] theorem ofNat'_self {n : Nat} [NeZero n] : Fin.ofNat' n n = 0 := by
|
||||
ext
|
||||
simp
|
||||
congr
|
||||
|
||||
@[simp] theorem ofNat'_val_eq_self [NeZero n] (x : Fin n) : (Fin.ofNat' n x) = x := by
|
||||
ext
|
||||
rw [val_ofNat', Nat.mod_eq_of_lt]
|
||||
exact x.2
|
||||
|
|
@ -123,7 +128,7 @@ theorem mk_le_of_le_val {b : Fin n} {a : Nat} (h : a ≤ b) :
|
|||
|
||||
@[simp] theorem mk_lt_mk {x y : Nat} {hx hy} : (⟨x, hx⟩ : Fin n) < ⟨y, hy⟩ ↔ x < y := .rfl
|
||||
|
||||
@[simp] theorem val_zero (n : Nat) : (0 : Fin (n + 1)).1 = 0 := rfl
|
||||
@[simp] theorem val_zero (n : Nat) [NeZero n] : ((0 : Fin n) : Nat) = 0 := rfl
|
||||
|
||||
@[simp] theorem mk_zero : (⟨0, Nat.succ_pos n⟩ : Fin (n + 1)) = 0 := rfl
|
||||
|
||||
|
|
@ -170,8 +175,24 @@ theorem rev_eq {n a : Nat} (i : Fin (n + 1)) (h : n = a + i) :
|
|||
@[simp] theorem rev_lt_rev {i j : Fin n} : rev i < rev j ↔ j < i := by
|
||||
rw [← Fin.not_le, ← Fin.not_le, rev_le_rev]
|
||||
|
||||
/-! ### last -/
|
||||
|
||||
@[simp] theorem val_last (n : Nat) : last n = n := rfl
|
||||
|
||||
@[simp] theorem last_zero : (Fin.last 0 : Fin 1) = 0 := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem zero_eq_last_iff {n : Nat} : (0 : Fin (n + 1)) = last n ↔ n = 0 := by
|
||||
constructor
|
||||
· intro h
|
||||
simp_all [Fin.ext_iff]
|
||||
· rintro rfl
|
||||
simp
|
||||
|
||||
@[simp] theorem last_eq_zero_iff {n : Nat} : Fin.last n = 0 ↔ n = 0 := by
|
||||
simp [eq_comm (a := Fin.last n)]
|
||||
|
||||
theorem le_last (i : Fin (n + 1)) : i ≤ last n := Nat.le_of_lt_succ i.is_lt
|
||||
|
||||
theorem last_pos : (0 : Fin (n + 2)) < last (n + 1) := Nat.succ_pos _
|
||||
|
|
@ -205,10 +226,28 @@ instance subsingleton_one : Subsingleton (Fin 1) := subsingleton_iff_le_one.2 (b
|
|||
|
||||
theorem fin_one_eq_zero (a : Fin 1) : a = 0 := Subsingleton.elim a 0
|
||||
|
||||
@[simp] theorem zero_eq_one_iff {n : Nat} [NeZero n] : (0 : Fin n) = 1 ↔ n = 1 := by
|
||||
constructor
|
||||
· intro h
|
||||
simp [Fin.ext_iff] at h
|
||||
change 0 % n = 1 % n at h
|
||||
rw [eq_comm] at h
|
||||
simpa using h
|
||||
· rintro rfl
|
||||
simp
|
||||
|
||||
@[simp] theorem one_eq_zero_iff {n : Nat} [NeZero n] : (1 : Fin n) = 0 ↔ n = 1 := by
|
||||
rw [eq_comm]
|
||||
simp
|
||||
|
||||
theorem add_def (a b : Fin n) : a + b = Fin.mk ((a + b) % n) (Nat.mod_lt _ a.size_pos) := rfl
|
||||
|
||||
theorem val_add (a b : Fin n) : (a + b).val = (a.val + b.val) % n := rfl
|
||||
|
||||
@[simp] protected theorem zero_add {n : Nat} [NeZero n] (i : Fin n) : (0 : Fin n) + i = i := by
|
||||
ext
|
||||
simp [Fin.add_def, Nat.mod_eq_of_lt i.2]
|
||||
|
||||
theorem val_add_one_of_lt {n : Nat} {i : Fin n.succ} (h : i < last _) : (i + 1).1 = i + 1 := by
|
||||
match n with
|
||||
| 0 => cases h
|
||||
|
|
@ -332,6 +371,10 @@ theorem succ_succ_ne_one (a : Fin n) : Fin.succ (Fin.succ a) ≠ 1 :=
|
|||
|
||||
@[simp] theorem cast_mk (h : n = m) (i : Nat) (hn : i < n) : cast h ⟨i, hn⟩ = ⟨i, h ▸ hn⟩ := rfl
|
||||
|
||||
@[simp] theorem cast_refl (n : Nat) (h : n = n) : cast h = id := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem cast_trans {k : Nat} (h : n = m) (h' : m = k) {i : Fin n} :
|
||||
cast h' (cast h i) = cast (Eq.trans h h') i := rfl
|
||||
|
||||
|
|
@ -440,6 +483,10 @@ theorem succ_castSucc {n : Nat} (i : Fin n) : i.castSucc.succ = castSucc i.succ
|
|||
|
||||
@[simp] theorem coe_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl
|
||||
|
||||
@[simp] theorem addNat_zero (n : Nat) (i : Fin n) : addNat i 0 = i := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem addNat_one {i : Fin n} : addNat i 1 = i.succ := rfl
|
||||
|
||||
theorem le_coe_addNat (m : Nat) (i : Fin n) : m ≤ addNat i m :=
|
||||
|
|
@ -469,7 +516,7 @@ theorem cast_addNat_left {n n' m : Nat} (i : Fin n') (h : n' + m = n + m) :
|
|||
|
||||
theorem le_coe_natAdd (m : Nat) (i : Fin n) : m ≤ natAdd m i := Nat.le_add_right ..
|
||||
|
||||
theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
|
||||
@[simp] theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
|
||||
|
||||
/-- For rewriting in the reverse direction, see `Fin.cast_natAdd_right`. -/
|
||||
theorem natAdd_cast {n n' : Nat} (m : Nat) (i : Fin n') (h : n' = n) :
|
||||
|
|
@ -507,9 +554,19 @@ theorem cast_addNat {n : Nat} (m : Nat) (i : Fin n) :
|
|||
|
||||
@[simp] theorem natAdd_last {m n : Nat} : natAdd n (last m) = last (n + m) := rfl
|
||||
|
||||
@[simp] theorem addNat_last (n : Nat) :
|
||||
addNat (last n) m = cast (by omega) (last (n + m)) := by
|
||||
ext
|
||||
simp
|
||||
|
||||
theorem natAdd_castSucc {m n : Nat} {i : Fin m} : natAdd n (castSucc i) = castSucc (natAdd n i) :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem natAdd_eq_addNat (n : Nat) (i : Fin n) : Fin.natAdd n i = i.addNat n := by
|
||||
ext
|
||||
simp
|
||||
omega
|
||||
|
||||
theorem rev_castAdd (k : Fin n) (m : Nat) : rev (castAdd m k) = addNat (rev k) m := Fin.ext <| by
|
||||
rw [val_rev, coe_castAdd, coe_addNat, val_rev, Nat.sub_add_comm (Nat.succ_le_of_lt k.is_lt)]
|
||||
|
||||
|
|
@ -575,6 +632,15 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
|
|||
@[simp] theorem subNat_mk {i : Nat} (h₁ : i < n + m) (h₂ : m ≤ i) :
|
||||
subNat m ⟨i, h₁⟩ h₂ = ⟨i - m, Nat.sub_lt_right_of_lt_add h₂ h₁⟩ := rfl
|
||||
|
||||
@[simp] theorem subNat_zero (i : Fin n) (h : 0 ≤ (i : Nat)): Fin.subNat 0 i h = i := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem subNat_one_succ (i : Fin (n + 1)) (h : 1 ≤ ↑i) : (subNat 1 i h).succ = i := by
|
||||
ext
|
||||
simp
|
||||
omega
|
||||
|
||||
@[simp] theorem pred_castSucc_succ (i : Fin n) :
|
||||
pred (castSucc i.succ) (Fin.ne_of_gt (castSucc_pos i.succ_pos)) = castSucc i := rfl
|
||||
|
||||
|
|
@ -585,7 +651,7 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
|
|||
subNat m (addNat i m) h = i := Fin.ext <| Nat.add_sub_cancel i m
|
||||
|
||||
@[simp] theorem natAdd_subNat_cast {i : Fin (n + m)} (h : n ≤ i) :
|
||||
natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]; rfl
|
||||
natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]
|
||||
|
||||
/-! ### recursion and induction principles -/
|
||||
|
||||
|
|
@ -778,6 +844,11 @@ theorem sub_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
|||
apply Fin.eq_of_val_eq
|
||||
simp [Fin.ofNat', Fin.sub_def]
|
||||
|
||||
@[simp] protected theorem sub_self [NeZero n] {x : Fin n} : x - x = 0 := by
|
||||
ext
|
||||
rw [Fin.sub_def]
|
||||
simp
|
||||
|
||||
private theorem _root_.Nat.mod_eq_sub_of_lt_two_mul {x n} (h₁ : n ≤ x) (h₂ : x < 2 * n) :
|
||||
x % n = x - n := by
|
||||
rw [Nat.mod_eq, if_pos (by omega), Nat.mod_eq_of_lt (by omega)]
|
||||
|
|
|
|||
|
|
@ -134,6 +134,19 @@ theorem mod_eq_of_lt {a b : Nat} (h : a < b) : a % b = a :=
|
|||
if_neg h'
|
||||
(mod_eq a b).symm ▸ this
|
||||
|
||||
@[simp] theorem one_mod_eq_zero_iff {n : Nat} : 1 % n = 0 ↔ n = 1 := by
|
||||
match n with
|
||||
| 0 => simp
|
||||
| 1 => simp
|
||||
| n + 2 =>
|
||||
rw [mod_eq_of_lt (by exact Nat.lt_of_sub_eq_succ rfl)]
|
||||
simp only [add_one_ne_zero, false_iff, ne_eq]
|
||||
exact ne_of_beq_eq_false rfl
|
||||
|
||||
@[simp] theorem Nat.zero_eq_one_mod_iff {n : Nat} : 0 = 1 % n ↔ n = 1 := by
|
||||
rw [eq_comm]
|
||||
simp
|
||||
|
||||
theorem mod_eq_sub_mod {a b : Nat} (h : a ≥ b) : a % b = (a - b) % b :=
|
||||
match eq_zero_or_pos b with
|
||||
| Or.inl h₁ => h₁.symm ▸ (Nat.sub_zero a).symm ▸ rfl
|
||||
|
|
@ -157,6 +170,13 @@ theorem mod_lt (x : Nat) {y : Nat} : y > 0 → x % y < y := by
|
|||
rw [mod_eq_sub_mod h₁]
|
||||
exact h₂ h₃
|
||||
|
||||
@[simp] protected theorem sub_mod_add_mod_cancel (a b : Nat) [NeZero a] : a - b % a + b % a = a := by
|
||||
rw [Nat.sub_add_cancel]
|
||||
cases a with
|
||||
| zero => simp_all
|
||||
| succ a =>
|
||||
exact Nat.le_of_lt (mod_lt b (zero_lt_succ a))
|
||||
|
||||
theorem mod_le (x y : Nat) : x % y ≤ x := by
|
||||
match Nat.lt_or_ge x y with
|
||||
| Or.inl h₁ => rw [mod_eq_of_lt h₁]; apply Nat.le_refl
|
||||
|
|
@ -197,7 +217,6 @@ decreasing_by apply div_rec_lemma; assumption
|
|||
theorem div_eq_sub_div (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 := by
|
||||
rw [div_eq a, if_pos]; constructor <;> assumption
|
||||
|
||||
|
||||
theorem mod_add_div (m k : Nat) : m % k + k * (m / k) = m := by
|
||||
induction m, k using mod.inductionOn with rw [div_eq, mod_eq]
|
||||
| base x y h => simp [h]
|
||||
|
|
|
|||
|
|
@ -21,13 +21,15 @@ example : Not
|
|||
(@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))) := by
|
||||
simp
|
||||
|
||||
@[simp] theorem OfNat.ofNat_ofNat : OfNat.ofNat (no_index OfNat.ofNat n) = OfNat.ofNat n := rfl
|
||||
|
||||
example : Not
|
||||
(@Eq.{1} Nat
|
||||
(@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod)
|
||||
(@OfNat.ofNat.{0} Nat 1 (@One.toOfNat1.{0} Nat (@One.ofOfNat1.{0} Nat (instOfNatNat 1))))
|
||||
(@OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
|
||||
(@OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) := by
|
||||
simp
|
||||
simp only [reduceCtorEq, not_false_eq_true]
|
||||
|
||||
def WithTop (α : Type) := Option α
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue