doc: Fix (and expand) docstrings for bmod/bdiv (#6713)
The current text is missing a negative sign on the bottom of the interval that `Int.bmod` can return. While I'm here, I added illustrative example outputs to match docs for tdiv/ediv/fdiv/etc.
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@ -257,7 +257,7 @@ theorem ofNat_fdiv : ∀ m n : Nat, ↑(m / n) = fdiv ↑m ↑n
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# `bmod` ("balanced" mod)
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Balanced mod (and balanced div) are a division and modulus pair such
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that `b * (Int.bdiv a b) + Int.bmod a b = a` and `b/2 ≤ Int.bmod a b <
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that `b * (Int.bdiv a b) + Int.bmod a b = a` and `-b/2 ≤ Int.bmod a b <
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b/2` for all `a : Int` and `b > 0`.
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This is used in Omega as well as signed bitvectors.
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@ -266,10 +266,26 @@ This is used in Omega as well as signed bitvectors.
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/--
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Balanced modulus. This version of Integer modulus uses the
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balanced rounding convention, which guarantees that
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`m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
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`-m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
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to `x` modulo `m`.
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If `m = 0`, then `bmod x m = x`.
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Examples:
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```
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#eval (7 : Int).bdiv 0 -- 0
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#eval (0 : Int).bdiv 7 -- 0
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#eval (12 : Int).bdiv 6 -- 2
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#eval (12 : Int).bdiv 7 -- 2
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#eval (12 : Int).bdiv 8 -- 2
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#eval (12 : Int).bdiv 9 -- 1
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#eval (-12 : Int).bdiv 6 -- -2
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#eval (-12 : Int).bdiv 7 -- -2
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#eval (-12 : Int).bdiv 8 -- -1
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#eval (-12 : Int).bdiv 9 -- -1
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```
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-/
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def bmod (x : Int) (m : Nat) : Int :=
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let r := x % m
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@ -281,6 +297,22 @@ def bmod (x : Int) (m : Nat) : Int :=
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/--
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Balanced division. This returns the unique integer so that
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`b * (Int.bdiv a b) + Int.bmod a b = a`.
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Examples:
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```
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#eval (7 : Int).bmod 0 -- 7
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#eval (0 : Int).bmod 7 -- 0
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#eval (12 : Int).bmod 6 -- 0
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#eval (12 : Int).bmod 7 -- -2
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#eval (12 : Int).bmod 8 -- -4
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#eval (12 : Int).bmod 9 -- 3
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#eval (-12 : Int).bmod 6 -- 0
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#eval (-12 : Int).bmod 7 -- 2
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#eval (-12 : Int).bmod 8 -- -4
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#eval (-12 : Int).bmod 9 -- -3
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```
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-/
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def bdiv (x : Int) (m : Nat) : Int :=
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if m = 0 then
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