chore: move BitVec.udiv/umod/sdiv/smod after add/sub/mul/lt (#5623)
Divison proofs are more likely to depend on add/sub/mul proofs than the other way around. This cleans up https://github.com/leanprover/lean4/pull/5609, which added division proofs that rely on negation to already be defined.
This commit is contained in:
Tobias Grosser
GitHub
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47e0430b07
commit
7fd2aa04ae
1 changed files with 110 additions and 110 deletions
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@ -1445,104 +1445,6 @@ theorem msb_sshiftRight' {x y: BitVec w} :
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(x.sshiftRight' y).msb = x.msb := by
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simp [BitVec.sshiftRight', BitVec.msb_sshiftRight]
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/-! ### udiv -/
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theorem udiv_def {x y : BitVec n} : x / y = BitVec.ofNat n (x.toNat / y.toNat) := by
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have h : x.toNat / y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
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rw [← udiv_eq]
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simp [udiv, bv_toNat, h, Nat.mod_eq_of_lt]
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@[simp, bv_toNat]
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theorem toNat_udiv {x y : BitVec n} : (x / y).toNat = x.toNat / y.toNat := by
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rw [udiv_def]
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by_cases h : y = 0
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· simp [h]
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· rw [toNat_ofNat, Nat.mod_eq_of_lt]
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exact Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
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@[simp]
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theorem udiv_zero {x : BitVec n} : x / 0#n = 0#n := by
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simp [udiv_def]
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/-! ### umod -/
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theorem umod_def {x y : BitVec n} :
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x % y = BitVec.ofNat n (x.toNat % y.toNat) := by
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rw [← umod_eq]
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have h : x.toNat % y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.mod_le _ _) x.isLt
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simp [umod, bv_toNat, Nat.mod_eq_of_lt h]
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@[simp, bv_toNat]
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theorem toNat_umod {x y : BitVec n} :
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(x % y).toNat = x.toNat % y.toNat := rfl
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@[simp]
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theorem umod_zero {x : BitVec n} : x % 0#n = x := by
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simp [umod_def]
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/-! ### sdiv -/
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/-- Equation theorem for `sdiv` in terms of `udiv`. -/
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theorem sdiv_eq (x y : BitVec w) : x.sdiv y =
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match x.msb, y.msb with
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| false, false => udiv x y
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| false, true => - (x.udiv (- y))
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| true, false => - ((- x).udiv y)
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| true, true => (- x).udiv (- y) := by
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rw [BitVec.sdiv]
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rcases x.msb <;> rcases y.msb <;> simp
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@[bv_toNat]
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theorem toNat_sdiv {x y : BitVec w} : (x.sdiv y).toNat =
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match x.msb, y.msb with
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| false, false => (udiv x y).toNat
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| false, true => (- (x.udiv (- y))).toNat
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| true, false => (- ((- x).udiv y)).toNat
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| true, true => ((- x).udiv (- y)).toNat := by
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simp only [sdiv_eq, toNat_udiv]
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by_cases h : x.msb <;> by_cases h' : y.msb <;> simp [h, h']
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theorem sdiv_eq_and (x y : BitVec 1) : x.sdiv y = x &&& y := by
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have hx : x = 0#1 ∨ x = 1#1 := by bv_omega
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have hy : y = 0#1 ∨ y = 1#1 := by bv_omega
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rcases hx with rfl | rfl <;>
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rcases hy with rfl | rfl <;>
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rfl
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/-! ### smod -/
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/-- Equation theorem for `smod` in terms of `umod`. -/
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theorem smod_eq (x y : BitVec w) : x.smod y =
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match x.msb, y.msb with
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| false, false => x.umod y
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| false, true =>
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let u := x.umod (- y)
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(if u = 0#w then u else u + y)
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| true, false =>
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let u := umod (- x) y
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(if u = 0#w then u else y - u)
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| true, true => - ((- x).umod (- y)) := by
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rw [BitVec.smod]
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rcases x.msb <;> rcases y.msb <;> simp
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@[bv_toNat]
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theorem toNat_smod {x y : BitVec w} : (x.smod y).toNat =
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match x.msb, y.msb with
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| false, false => (x.umod y).toNat
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| false, true =>
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let u := x.umod (- y)
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(if u = 0#w then u.toNat else (u + y).toNat)
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| true, false =>
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let u := (-x).umod y
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(if u = 0#w then u.toNat else (y - u).toNat)
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| true, true => (- ((- x).umod (- y))).toNat := by
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simp only [smod_eq, toNat_umod]
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by_cases h : x.msb <;> by_cases h' : y.msb
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<;> by_cases h'' : (-x).umod y = 0#w <;> by_cases h''' : x.umod (-y) = 0#w
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<;> simp only [h, h', h'', h''']
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<;> simp only [umod, toNat_eq, toNat_ofNatLt, toNat_ofNat, Nat.zero_mod] at h'' h'''
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<;> simp [h'', h''']
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/-! ### signExtend -/
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/-- Equation theorem for `Int.sub` when both arguments are `Int.ofNat` -/
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@ -2171,18 +2073,6 @@ theorem sub_eq_xor {a b : BitVec 1} : a - b = a ^^^ b := by
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have hb : b = 0 ∨ b = 1 := eq_zero_or_eq_one _
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rcases ha with h | h <;> (rcases hb with h' | h' <;> (simp [h, h']))
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@[simp]
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theorem sdiv_zero {x : BitVec n} : x.sdiv 0#n = 0#n := by
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simp only [sdiv_eq, msb_zero]
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rcases x.msb with msb | msb <;> apply eq_of_toNat_eq <;> simp
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@[simp]
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theorem smod_zero {x : BitVec n} : x.smod 0#n = x := by
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simp only [smod_eq, msb_zero]
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rcases x.msb with msb | msb <;> apply eq_of_toNat_eq
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· simp
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· by_cases h : x = 0#n <;> simp [h]
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theorem not_neg (x : BitVec w) : ~~~(-x) = x + -1#w := by
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rcases w with _ | w
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· apply Subsingleton.elim
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@ -2341,6 +2231,116 @@ theorem not_lt_iff_le {x y : BitVec w} : (¬ x < y) ↔ y ≤ x := by
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constructor <;>
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(intro h; simp only [lt_def, Nat.not_lt, le_def] at h ⊢; omega)
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/-! ### udiv -/
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theorem udiv_def {x y : BitVec n} : x / y = BitVec.ofNat n (x.toNat / y.toNat) := by
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have h : x.toNat / y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
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rw [← udiv_eq]
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simp [udiv, bv_toNat, h, Nat.mod_eq_of_lt]
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@[simp, bv_toNat]
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theorem toNat_udiv {x y : BitVec n} : (x / y).toNat = x.toNat / y.toNat := by
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rw [udiv_def]
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by_cases h : y = 0
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· simp [h]
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· rw [toNat_ofNat, Nat.mod_eq_of_lt]
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exact Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
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@[simp]
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theorem udiv_zero {x : BitVec n} : x / 0#n = 0#n := by
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simp [udiv_def]
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/-! ### umod -/
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theorem umod_def {x y : BitVec n} :
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x % y = BitVec.ofNat n (x.toNat % y.toNat) := by
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rw [← umod_eq]
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have h : x.toNat % y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.mod_le _ _) x.isLt
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simp [umod, bv_toNat, Nat.mod_eq_of_lt h]
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@[simp, bv_toNat]
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theorem toNat_umod {x y : BitVec n} :
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(x % y).toNat = x.toNat % y.toNat := rfl
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@[simp]
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theorem umod_zero {x : BitVec n} : x % 0#n = x := by
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simp [umod_def]
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/-! ### sdiv -/
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/-- Equation theorem for `sdiv` in terms of `udiv`. -/
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theorem sdiv_eq (x y : BitVec w) : x.sdiv y =
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match x.msb, y.msb with
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| false, false => udiv x y
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| false, true => - (x.udiv (- y))
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| true, false => - ((- x).udiv y)
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| true, true => (- x).udiv (- y) := by
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rw [BitVec.sdiv]
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rcases x.msb <;> rcases y.msb <;> simp
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@[bv_toNat]
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theorem toNat_sdiv {x y : BitVec w} : (x.sdiv y).toNat =
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match x.msb, y.msb with
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| false, false => (udiv x y).toNat
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| false, true => (- (x.udiv (- y))).toNat
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| true, false => (- ((- x).udiv y)).toNat
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| true, true => ((- x).udiv (- y)).toNat := by
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simp only [sdiv_eq, toNat_udiv]
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by_cases h : x.msb <;> by_cases h' : y.msb <;> simp [h, h']
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theorem sdiv_eq_and (x y : BitVec 1) : x.sdiv y = x &&& y := by
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have hx : x = 0#1 ∨ x = 1#1 := by bv_omega
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have hy : y = 0#1 ∨ y = 1#1 := by bv_omega
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rcases hx with rfl | rfl <;>
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rcases hy with rfl | rfl <;>
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rfl
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@[simp]
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theorem sdiv_zero {x : BitVec n} : x.sdiv 0#n = 0#n := by
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simp only [sdiv_eq, msb_zero]
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rcases x.msb with msb | msb <;> apply eq_of_toNat_eq <;> simp
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/-! ### smod -/
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/-- Equation theorem for `smod` in terms of `umod`. -/
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theorem smod_eq (x y : BitVec w) : x.smod y =
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match x.msb, y.msb with
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| false, false => x.umod y
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| false, true =>
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let u := x.umod (- y)
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(if u = 0#w then u else u + y)
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| true, false =>
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let u := umod (- x) y
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(if u = 0#w then u else y - u)
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| true, true => - ((- x).umod (- y)) := by
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rw [BitVec.smod]
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rcases x.msb <;> rcases y.msb <;> simp
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@[bv_toNat]
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theorem toNat_smod {x y : BitVec w} : (x.smod y).toNat =
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match x.msb, y.msb with
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| false, false => (x.umod y).toNat
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| false, true =>
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let u := x.umod (- y)
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(if u = 0#w then u.toNat else (u + y).toNat)
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| true, false =>
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let u := (-x).umod y
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(if u = 0#w then u.toNat else (y - u).toNat)
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| true, true => (- ((- x).umod (- y))).toNat := by
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simp only [smod_eq, toNat_umod]
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by_cases h : x.msb <;> by_cases h' : y.msb
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<;> by_cases h'' : (-x).umod y = 0#w <;> by_cases h''' : x.umod (-y) = 0#w
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<;> simp only [h, h', h'', h''']
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<;> simp only [umod, toNat_eq, toNat_ofNatLt, toNat_ofNat, Nat.zero_mod] at h'' h'''
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<;> simp [h'', h''']
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@[simp]
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theorem smod_zero {x : BitVec n} : x.smod 0#n = x := by
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simp only [smod_eq, msb_zero]
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rcases x.msb with msb | msb <;> apply eq_of_toNat_eq
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· simp
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· by_cases h : x = 0#n <;> simp [h]
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/-! ### ofBoolList -/
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@[simp] theorem getMsbD_ofBoolListBE : (ofBoolListBE bs).getMsbD i = bs.getD i false := by
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