Lean 4 fork for HoTT-compatible kernel extensions (Path types, transport, HITs). Maintained against upstream leanprover/lean4.
b875627198
The theorem
```lean
namespace Int
theorem ediv_nonneg_of_nonpos_of_nonpos {a b : Int} (Ha : a ≤ 0) (Hb : b ≤ 0) : 0 ≤ a / b := by
match a, b with
| ofNat a, b =>
match Int.le_antisymm Ha (ofNat_zero_le a) with
| h1 =>
rw [h1, zero_ediv,]
exact Int.le_refl 0
| a, ofNat b =>
match Int.le_antisymm Hb (ofNat_zero_le b) with
| h1 =>
rw [h1, Int.ediv_zero]
exact Int.le_refl 0
| negSucc a, negSucc b =>
rw [Int.div_def, ediv]
have le_succ {a: Int} : a ≤ a+1 := (le_add_one (Int.le_refl a))
have h2: 0 ≤ ((↑b:Int) + 1) := Int.le_trans (ofNat_zero_le b) le_succ
have h3: (0:Int) ≤ ↑a / (↑b + 1) := (ediv_nonneg (ofNat_zero_le a) h2)
exact Int.le_trans h3 le_succ
```
is nontrivial to prove from existing theorems and would be nice to add
as standard theorem in DivModLemmas.
See the zullip conversation
[here](https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Adding.20theorem.20theorem.20ediv_nonneg'.20for.20negative.20a.20and.20b)
---------
Co-authored-by: Kim Morrison <kim@tqft.net>
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