`Carleson/Calculations.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 981 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 project documentation, Pinned Carleson blueprint

29 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.ProofData

Imported by 6 files

lemma sixteen_times_le_cube line 20 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HMul.hMul, HPow.hPow
Proof uses: Trans.trans, Eq, rfl, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.atom_pf, Eq.mpr, id, congrArg, Eq.symm, Mathlib.Tactic.Ring.Common.add_mul, Mathlib.Tactic.Ring.Common.mul_add, Mathlib.Tactic.Ring.Common.mul_pf_right, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_mul and 16 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma four_times_sq_le_cube line 26 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HMul.hMul, HPow.hPow
Proof uses: Trans.trans, Eq, rfl, mul_le_mul', le_refl, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.pow_add, Mathlib.Tactic.Ring.Common.single_pow and 15 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma add_le_pow_two line 31 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, PartialOrder, IsOrderedRing, Nat, LE.le, HAdd.hAdd, OfNat.ofNat, HPow.hPow
Proof uses: le_imp_le_of_le_of_le, add_le_add, pow_le_pow_right₀, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Bool.true, le_refl, Eq.mpr, HMul.hMul, congrArg, Eq.symm, mul_two, pow_succ
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma add_le_pow_two₃ line 36 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, PartialOrder, IsOrderedRing, Nat, LE.le, HAdd.hAdd, OfNat.ofNat, HPow.hPow
Proof uses: Trans.trans, Eq, rfl, add_le_add, add_le_pow_two, le_rfl, le_refl, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring and 44 more
98 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma add_le_pow_two_add_cube line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Semiring, PartialOrder, IsOrderedRing, LE.le, OfNat.ofNat, HAdd.hAdd, HPow.hPow
Proof uses: add_le_pow_two, one_le_pow₀, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HMul.hMul, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos and 59 more
107 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma calculation_1 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, Real, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD, Neg.neg, HAdd.hAdd
Proof uses: LT.lt, realD_pos, Trans.trans, rfl, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.atom_pf, Eq.mpr, id, congrArg, Eq.symm, Mathlib.Tactic.Ring.Common.add_mul, Mathlib.Tactic.Ring.Common.mul_add, Mathlib.Tactic.Ring.Common.mul_pf_right, Mathlib.Meta.NormNum.IsNat.to_raw_eq and 44 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma calculation_2 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, Real, HMul.hMul, Inv.inv, OfNat.ofNat, HPow.hPow, defaultD, Neg.neg, HAdd.hAdd
Proof uses: LT.lt, realD_pos, Trans.trans, rfl, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, NNRat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.inv_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.inv_single, Mathlib.Meta.NormNum.IsNNRat.to_raw_eq, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.IsNat.of_raw, Mathlib.Tactic.Ring.Common.atom_pf, Eq.mpr and 40 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma calculation_10 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Real, OfNat.ofNat, defaultD, HMul.hMul, HAdd.hAdd, HPow.hPow, Neg.neg, Inv.inv
Proof uses: Trans.trans, LE.le, Eq, rfl, id, mul_le_mul, add_le_add, le_refl, mul_le_mul_of_nonneg_left, Real.rpow_le_rpow_of_nonpos, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Bool.false, LT.lt.le, Mathlib.Meta.NormNum.isInt_le_true, Int.negOfNat and 33 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma calculation_3 line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LT.lt, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow, defaultD, Neg.neg, Inv.inv
Proof uses: Eq.mpr, id, Eq, congrArg, Eq.symm, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.instAtLeastTwo, right_distrib, add_lt_add_left, LE.le, distrib_three_right, Trans.trans, HSub.hSub, rfl, mul_le_mul_of_nonneg_left, zpow_le_zpow_right₀, le_of_not_gt and 77 more
154 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma calculation_4 line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LT.lt, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD, HAdd.hAdd, Inv.inv, Eq
Proof uses: Trans.trans, LE.le, Neg.neg, rfl, add_le_add, le_of_lt, le_refl, Eq.mpr, id, congrArg, Eq.symm, calculation_1, Eq.refl, calculation_2, calculation_3
53 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma calculation_logD_64 line 133 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LT.lt, Real.logb, defaultD, OfNat.ofNat
Proof uses: Iff.mpr, HPow.hPow, Real.logb_lt_iff_lt_rpow, lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, HMul.hMul, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr and 51 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma calculation_5 line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, HMul.hMul, HPow.hPow, OfNat.ofNat, Neg.neg, 𝕔, HSub.hSub
Proof uses: Iff.mp, mul_le_mul_iff_right₀, LT.lt, Real.rpow_pos_of_pos, Mathlib.Meta.Positivity.pos_of_isNat, inferInstance, PartialOrder, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, Eq.mpr, id, Eq, congrArg, Eq.symm, mul_assoc, neg_mul, Inv.inv and 58 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma calculation_6 line 151 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, Real, HPow.hPow, defaultD, HAdd.hAdd, OfNat.ofNat, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, zpow_add₀, Not.intro, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, LT.lt, HSub.hSub, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.symm and 28 more
74 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma calculation_7 line 157 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, Real, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD, HAdd.hAdd, defaultA, 𝕔
Proof uses: Eq.mpr, id, congrArg, Eq.symm, mul_assoc, mul_comm, HMul, Eq.rec, Eq.refl, Eq.trans, pow_mul, defaultD.eq_1, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl and 19 more
75 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma calculation_8 line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, HMul.hMul, HPow.hPow, OfNat.ofNat, 𝕔, Neg.neg
Proof uses: Eq.mpr, id, Eq, congrArg, neg_mul, Inv.inv, Real.rpow_neg, le_of_lt, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, mul_comm, Iff.mpr, le_mul_inv_iff₀, Real.rpow_pos_of_pos, inferInstance, PartialOrder
69 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma calculation_9 line 172 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LE.le, OfNat.ofNat, HPow.hPow, HMul.hMul, Neg.neg, HSub.hSub, 𝕔, False
Proof uses: LT.lt, Real.rpow_lt_rpow_of_neg, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Nat.cast_one, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.instAtLeastTwo, Eq.mpr, id, Eq, Eq.trans, congrFun', congrArg, neg_sub, sub_mul, HAdd.hAdd, congr and 54 more
149 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma calculation_11 line 185 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LT.lt, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD
Proof uses: Eq.mpr, id, Eq, congrArg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.isNat_add, right_distrib, add_lt_add_right, mul_lt_mul'', lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, HSub.hSub, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr and 49 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma calculation_12 line 194 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD, HAdd.hAdd, 𝕔
Proof uses: id, Trans.trans, rfl, congrArg, Real.rpow_mul, le_of_lt, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, Eq.mpr, congr, congrFun', Eq.trans, Eq.mp, Real.rpow_natCast, Mathlib.Tactic.Ring.of_eq, Nat.rawCast and 67 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma calculation_13 line 229 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Real, HPow.hPow, OfNat.ofNat, HAdd.hAdd, HMul.hMul, 𝕔, defaultA
Proof uses: Eq.mpr, id, congrArg, congrFun', Nat.cast_pow, Real.rpow_mul, le_of_lt, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, Eq.trans, congr, Eq.symm, Eq.mp, Real.rpow_natCast, Mathlib.Tactic.Ring.of_eq, Nat.rawCast and 28 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma calculation_14 line 235 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LE.le, HPow.hPow, OfNat.ofNat, HSub.hSub, HDiv.hDiv, HMul.hMul, defaultZ, HAdd.hAdd, 𝕔
Proof uses: Real.rpow_le_rpow_of_exponent_le, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.neg_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Mathlib.Tactic.Ring.Common.neg_add and 71 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma calculation_15 line 248 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, HPow.hPow, OfNat.ofNat, HMul.hMul, HAdd.hAdd, 𝕔, HSub.hSub
Proof uses: Eq.mpr, HDiv.hDiv, id, Eq, congrArg, Real.rpow_sub, lt_of_not_ge, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, LT.lt, Neg.neg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos and 60 more
145 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma calculation_16 line 257 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LT.lt, HMul.hMul, OfNat.ofNat, HPow.hPow, defaultD, HAdd.hAdd
Proof uses: mul_lt_mul'', lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 52 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma calculation_7_7_4 line 264 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LE.le, OfNat.ofNat, HSub.hSub, HPow.hPow, HMul.hMul, defaultZ, HAdd.hAdd
Proof uses: Eq.mpr, id, Eq, congrArg, propext, le_sub_iff_add_le, Trans.simple, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit1, Bool, Bool.true, pow_right_mono₀, one_le_two and 10 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
lemma near_1_geometric_bound A bound on the sum of a geometric series whose ratio is close to 1 line 282 source
Source documentation
A bound on the sum of a geometric series whose ratio is close to 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Membership.mem, Set, Set.Icc, OfNat.ofNat, LE.le, ENNReal, Inv.inv, HSub.hSub, HPow.hPow, Nat, Neg.neg, HMul.hMul, ENNReal.ofReal
Proof uses: Eq.mpr, id, Eq, congrArg, propext, ENNReal.inv_le_iff_inv_le, ENNReal.mul_inv, Or.inl, Ne, Top.top, two_ne_zero, ENNReal.ofNat_ne_top, inv_inv, HDiv.hDiv, Eq.symm, ENNReal.div_eq_inv_mul, ENNReal.ofReal_ofNat, ENNReal.ofReal_one, ENNReal.ofReal_div_of_pos, Mathlib.Meta.Positivity.pos_of_isNat and 101 more
132 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma near_1_geometric_bound {t : ℝ} (ht : t ∈ Set.Icc 0 1) :
    (1 - 2 ^ (-t))⁻¹ ≤ 2 * (ENNReal.ofReal t)⁻¹ := by
…
```
lemma calculation_convexity_bound line 295 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, Membership.mem, Set.Icc, OfNat.ofNat, LE.le, ENNReal, Finset.sum, Finset.range, HPow.hPow, defaultD, Neg.neg, HMul.hMul, Inv.inv, ENNReal.ofReal
Proof uses: KernelProofData.four_le_a, le_trans, HSub.hSub, Trans.trans, Eq, tsum, SummationFilter.unconditional, Finset.sum_le_sum, Finset, pow_le_pow_left', Eq.mpr, id, congrArg, ENNReal.rpow_neg, propext, ENNReal.inv_le_inv, ENNReal.rpow_le_rpow, 𝕔, Nat.le_pow, LT.lt and 63 more
149 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma calculation_7_6_2 line 313 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LE.le, ENNReal, Finset.sum, Finset.range, HPow.hPow, defaultD, Neg.neg, HDiv.hDiv, defaultκ, OfNat.ofNat, HAdd.hAdd, HMul.hMul
Proof uses: Trans.trans, Eq, Inv.inv, ENNReal.ofReal, calculation_convexity_bound, κ_nonneg, κ_le_one, Eq.mpr, Membership.mem, Set.Icc, And, id, congrArg, propext, Set.mem_Icc, And.intro, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp and 99 more
144 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma calculation_150 line 329 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, Int, Set, PseudoMetricSpace, ProofData, LE.le, HMul.hMul, 𝕔, HDiv.hDiv, OfNat.ofNat, HPow.hPow, defaultκ
Proof uses: Eq.mpr, Neg.neg, id, Eq, congrArg, defaultκ.eq_1, neg_mul, Inv.inv, Real.rpow_neg, zero_le_two, mul_inv_le_one_of_le₀, congr, Eq.trans, Real.rpow_natCast, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Nat.rawCast, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat and 117 more
155 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma sq_le_two_pow_of_four_le line 344 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HPow.hPow
Proof uses: Nat.le_induction, Eq.mpr, HAdd.hAdd, HMul.hMul, id, Eq, congrArg, pow_succ, mul_two, add_sq, one_pow, mul_one, add_assoc, add_le_add, Trans.trans, mul_le_mul', le_refl, Eq.symm, sq
54 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma calculation_6_1_6 line 354 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HMul.hMul, HPow.hPow, HAdd.hAdd
Proof uses: Trans.trans, Eq, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Mathlib.Tactic.Ring.Common.pow_add, Mathlib.Tactic.Ring.Common.single_pow, Mathlib.Tactic.Ring.Common.mul_pow_mul, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_mul and 26 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.