Solutions_S01_Implication_and_the_Universal_Quantifier.lean`

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 9 curated AI-written annotations and 7 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned Mathematics in Lean documentation, Pinned Mathematics in Lean textbook

22 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

theorem C03S01.my_lemma4 Solution: |x*y| < epsilon by a calc chain line 6 source
L4DEL explanation
Bound |x*y| by |x|*|y|, then by |x|*epsilon, then by 1*epsilon, which is epsilon.
1. intro x y ε epos ele1 xlt ylt moves the quantified variables and hypotheses into the context.
2. First calc step |x * y| = |x| * |y| is the library lemma abs_mul.
3. Next |x| * |y| <= |x| * ε uses mul_le_mul, with linarith and abs_nonneg closing the side goals.
4. Then |x| * ε < 1 * ε uses mul_lt_mul_of_pos_right and linarith, using |x| < ε <= 1 and 0 < ε.
5. The last step 1 * ε = ε is one_mul.
Sources and attribution: Pinned Lean source (dd6d752fedb1)
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, LE.le, abs, HMul.hMul
Proof uses: Trans.trans, Eq, abs_mul, mul_le_mul, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.sub_congr, HAdd.hAdd, HPow.hPow, Nat, Nat.rawCast, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, Eq.symm and 40 more
79 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:41 (same statement text; different proof or body). Qualify references as C03S01.my_lemma4@path.

calc: a chain of equalities and inequalities, each step justified by its own proof, which Lean composes into one claim.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem my_lemma4 :
    ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := by
…
```
def C03S01.FnUb line 15 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: LE.le
1 instances, coercions and auxiliary constants left out.

Also declared: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:50 (same declaration text).
def C03S01.FnLb line 18 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: LE.le
1 instances, coercions and auxiliary constants left out.

Also declared: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:53 (same declaration text).

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
example C03S01.(example at line 24) line 24 source
example C03S01.(example at line 30) line 30 source
example C03S01.(example at line 36) line 36 source
example C03S01.(example at line 50) line 50 source
example C03S01.(example at line 55) line 55 source
example C03S01.(example at line 58) line 58 source
example C03S01.(example at line 64) line 64 source
def C03S01.FnEven line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Eq, Neg.neg
1 instances, coercions and auxiliary constants left out.

Also declared: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:112 (same declaration text).
def C03S01.FnOdd line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Eq, Neg.neg
1 instances, coercions and auxiliary constants left out.

Also declared: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:115 (same declaration text).

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
example C03S01.(example at line 73) line 73 source
example C03S01.(example at line 80) line 80 source
example C03S01.(example at line 85) line 85 source
example C03S01.(example at line 96) line 96 source
theorem C03S01.Subset.trans line 102 source
no axioms what this means
Statement uses: Set, HasSubset.Subset
Proof uses: Membership.mem
3 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:146 (same statement text; different proof or body). Qualify references as C03S01.Subset.trans@path.
def C03S01.SetUb line 111 source
no axioms what this means
Statement uses: PartialOrder, Set
Proof uses: Membership.mem, LE.le
3 instances, coercions and auxiliary constants left out.

Also declared: MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean:155 (same declaration text).

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
example C03S01.(example at line 114) line 114 source
example C03S01.(example at line 118) line 118 source
example C03S01.(example at line 127) line 127 source
example C03S01.(example at line 134) line 134 source