Theorems / mh_ineq_5013c9

Mathlib held-out

Inequalities · easy

Statement, exactly as the model and Lean see it
theorem lg_target (R : Type u_1) [Semiring R] [PartialOrder R] [IsOrderedRing R] : Subsemiring.nonneg R = Subsemiring.nonneg R := by
Mathlib declaration
Subsemiring.nonneg.congr_simp
Held-out module
Mathlib.Algebra.Order.Ring.Cone
Banned modules (itself + downstream)
1
Reference proof premises (reachable / held-out)
0 / 0

Traces

Each configuration's full record: what was retrieved, what the model wrote, what Lean said back.

Verified4 model calls · 4 Lean checks · $0.00070

Retrieved lemmas

  1. 1. IsStrictOrderedRing.toIsOrderedRing
  2. 2. IsOrderedRing.toIsOrderedAddMonoid
  3. 3. conjugate_nonneg_of_nonneg
  4. 4. CanonicallyOrderedAdd.toIsOrderedRing
  5. 5. star_nonneg_iff
  6. 6. Nonneg.isOrderedRing
  7. 7. Set.Icc.nonneg
  8. 8. instIsDirectedOrder

Plan

### Explanation of the Proof: The theorem `lg_target` is trivial because it asserts that a subsemiring is equal to itself. This is a tautology and requires no meaningful computation or reasoning. However, the context includes typeclass assumptions such as `[Semiring R]`, `[PartialOrder R]`, and `[IsOrderedRing R]`, which are used in the background for consistency checks or for compatibility with other lemmas. Since the goal is simply to show that `Subsemiring.nonneg R = Subsemiring.nonneg R`, we can proceed by directly applying the reflexivity of equality. No additional lemmas or steps are needed, though the typeclass assumptions ensure that `Subsemiring.nonneg R` is well-defined. --- ### Lean Proof Skeleton: This proof uses `rfl` (reflexivity) to show that the subsemiring of non-negative elements is equal to itself.

Skeleton — Lean reported problems
import Algebra.Subsemiring
import Order.Lattice

open Subsemiring

theorem lg_target (R : Type u_1) [Semiring R] [PartialOrder R] [IsOrderedRing R] : Subsemiring.nonneg R = Subsemiring.nonneg R := by
  rfl

Attempts

  1. sample 0 · round 0Forbidden token2.1 s model time
    Proof
    import Algebra.Subsemiring
import Order.Lattice

open Subsemiring

theorem lg_target (R : Type u_1) [Semiring R] [PartialOrder R] [IsOrderedRing R] : Subsemiring.nonneg R = Subsemiring.nonneg R := by
  rfl
    What Lean printed (this is all the model sees on repair)
    rejected before compilation: forbidden token(s): command
  2. sample 0 · round 1Forbidden token2.2 s model time
    Proof
    import Algebra.Subsemiring
import Order.Lattice

open Subsemiring

theorem lg_target (R : Type u_1) [Semiring R] [PartialOrder R] [IsOrderedRing R] : Subsemiring.nonneg R = Subsemiring.nonneg R := by
  rfl
    What Lean printed (this is all the model sees on repair)
    rejected before compilation: forbidden token(s): command
  3. sample 0 · round 2Verified2.3 s model time
    Proof
    rfl

Verified proof

theorem lg_target (R : Type u_1) [Semiring R] [PartialOrder R] [IsOrderedRing R] : Subsemiring.nonneg R = Subsemiring.nonneg R := by
  rfl

Axioms used: none

Reference proof

Mathlib's own source for this declaration, shown for comparison. It may use lemmas the prover is not allowed to use.