• All proofs must be term-only; the token `by` must not appe...

Prompt

• All proofs must be term-only; the token `by` must not appear in proof code. • Prohibit: `sorry`, `admit`, custom `axiom`, `Classical.*`, `open Classical`, `Decidable.em`, `native_decide`, `decide` as automated proof, `Lean.Elab.Tactic.*`, `implemented_by`, and any tactic usage. • Do not use `opaque` for any definition in the execution model. • The main theorem must be nontrivial: it must connect the modeled result to the independent mathematical specification, not restate definitions or assume the conclusion. • Model `?T` as `Option T`, `!T` as `Except Error T` or equivalent. • For arrays/slices, represent indexing bounds explicitly. • For loops, use structural recursion or explicit fuel, with a formally proved invariant or recursive correctness lemma. Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY! module public import FormalConjecturesUtil @[expose] public section open Filter open scoped Topology Real namespace Erdos1 abbrev IsSumDistinctSet (A : Finset ℕ) (N : ℕ) : Prop := A ⊆ Finset.Icc 1 N ∧ (fun (⟨S, _⟩ : A.powerset) => S.sum id).Injective set_option linter.style.formal_proof_palomar false in @[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/tadamcz/erdos1/blob/0e395153306f34b3829d118b85bdd704136f2843/Erdos1/Resolutions/Erdos1_219usd_38h.lean#L2419"] theorem erdos_1 : ¬ ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card < N := by sorry @[category textbook, AMS 5 11] theorem erdos_1.variants.weaker : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card / A.card < N := by refine ⟨1/3, by norm_num, fun N A ⟨hA1, hA2⟩ hN => ?_⟩ have key : 2 ^ A.card ≤ A.card * N + 1 := by rw [← Finset.card_powerset] exact (Finset.card_le_card_of_injOn (Finset.sum · id) (fun S hS => Finset.mem_range.mpr <| Nat.lt_add_one_of_le <| (Finset.sum_le_card_nsmul S id N fun i hi => (Finset.mem_Icc.mp (hA1 (Finset.mem_powerset.mp hS hi))).2).trans (Nat.mul_le_mul_right N (Finset.card_le_card (Finset.mem_powerset.mp hS)))) (fun a ha b hb hab => by have := @hA2 ⟨a, ha⟩ ⟨b, hb⟩ hab; simp at this; exact this)).trans_eq (Finset.card_range _) rcases eq_or_ne A.card 0 with hc | hc · simp [hc]; positivity · rw [div_lt_iff₀ (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hc))] nlinarith [show (2 : ℝ) ^ A.card ≤ ↑A.card * ↑N + 1 from by exact_mod_cast key, show (1 : ℝ) ≤ ↑A.card from by exact_mod_cast Nat.pos_of_ne_zero hc, show (1 : ℝ) ≤ (N : ℝ) from by exact_mod_cast Nat.pos_of_ne_zero hN] @[category research solved, AMS 5 11] theorem erdos_1.variants.lb : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (1 / 4 - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry set_option linter.style.formal_proof_palomar false in @[category research solved, AMS 5 11, formal_proof using formal_conjectures at "https://github.com/MyTH-zyxeon/formal-conjectures/blob/362eed1a8864d142ae65f51af7981a7c7530956e/FormalConjectures/Scratch/HarperCompression.lean#L2174"] theorem erdos_1.variants.lb_strong : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (√(2 / π) - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry abbrev IsSumDistinctRealSet (A : Finset ℝ) (N : ℕ) : Prop := ↑A ⊆ Set.Ioc (0 : ℝ) N ∧ (A.powerset : Set (Finset ℝ)).Pairwise fun S₁ S₂ => 1 ≤ dist (S₁.sum id) (S₂.sum id) @[category research solved, AMS 5 11] theorem erdos_1.variants.real : ¬ ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℝ) (_ : IsSumDistinctRealSet A N), N ≠ 0 → C * 2 ^ A.card < N := by intro hreal apply erdos_1 rcases hreal with ⟨C, hC, hreal⟩ refine ⟨C, hC, ?_⟩ intro N A hA hN let e : ℕ ↪ ℝ := ⟨fun n => (n : ℝ), Nat.cast_injective⟩ let B : Finset ℝ := A.map e have hB : IsSumDistinctRealSet B N := by constructor · intro x hx change x ∈ B at hx simp only [B, Finset.mem_map] at hx rcases hx with ⟨n, hn, rfl⟩ have hnIcc := hA.1 hn simp only [Finset.mem_Icc] at hnIcc change (0 : ℝ) < (n : ℝ) ∧ (n : ℝ) ≤ (N : ℝ) exact ⟨by exact_mod_cast hnIcc.1, by exact_mod_cast hnIcc.2⟩ · intro S₁ hS₁ S₂ hS₂ hne change S₁ ∈ B.powerset at hS₁ change S₂ ∈ B.powerset at hS₂ have hS₁' : S₁ ⊆ B := Finset.mem_powerset.mp hS₁ have hS₂' : S₂ ⊆ B := Finset.mem_powerset.mp hS₂ let T₁ := S₁.preimage e (Set.injOn_of_injective e.injective) let T₂ := S₂.preimage e (Set.injOn_of_injective e.injective) have hm₁ : T₁.map e = S₁ := by ext x simp only [Finset.mem_map] constructor · rintro ⟨n, hn, rfl⟩ exact Finset.mem_preimage.mp hn · intro hx have hxB := hS₁' hx simp only [B, Finset.mem_map] at hxB rcases hxB with ⟨n, hn, rfl⟩ exact ⟨n, Finset.mem_preimage.mpr hx, rfl⟩ have hm₂ : T₂.map e = S₂ := by ext x simp only [Finset.mem_map] constructor · rintro ⟨n, hn, rfl⟩ exact Finset.mem_preimage.mp hn · intro hx have hxB := hS₂' hx simp only [B, Finset.mem_map] at hxB rcases hxB with ⟨n, hn, rfl⟩ exact ⟨n, Finset.mem_preimage.mpr hx, rfl⟩ have hT₁A : T₁ ⊆ A := by intro n hn have : e n ∈ S₁ := by rw [← hm₁] exact Finset.mem_map.mpr ⟨n, hn, rfl⟩ exact hS₁' this |> fun h => by simpa [B] using h have hT₂A : T₂ ⊆ A := by intro n hn have : e n ∈ S₂ := by rw [← hm₂] exact Finset.mem_map.mpr ⟨n, hn, rfl⟩ exact hS₂' this |> fun h => by simpa [B] using h have hTne : T₁ ≠ T₂ := by intro heq apply hne rw [← hm₁, ← hm₂, heq] have hsumne : T₁.sum id ≠ T₂.sum id := by intro hsum apply hTne have hsubeq : (⟨T₁, Finset.mem_powerset.mpr hT₁A⟩ : A.powerset) = ⟨T₂, Finset.mem_powerset.mpr hT₂A⟩ := hA.2 hsum exact congrArg Subtype.val hsubeq rw [← hm₁, ← hm₂, Finset.sum_map, Finset.sum_map] simp only [e, Function.Embedding.coeFn_mk, id_eq] have hs₁ : (∑ n ∈ T₁, (n : ℝ)) = ((T₁.sum id : ℕ) : ℝ) := by rw [Nat.cast_sum] rfl have hs₂ : (∑ n ∈ T₂, (n : ℝ)) = ((T₂.sum id : ℕ) : ℝ) := by rw [Nat.cast_sum] rfl rw [hs₁, hs₂, Real.dist_eq] rcases lt_or_gt_of_ne hsumne with hlt | hgt · rw [abs_of_nonpos (sub_nonpos.mpr (by exact_mod_cast Nat.le_of_lt hlt))] have hgap : T₁.sum id + 1 ≤ T₂.sum id := by omega have hgapR : (((T₁.sum id + 1 : ℕ) : ℝ)) ≤ T₂.sum id := by exact_mod_cast hgap norm_num at hgapR ⊢ linarith · rw [abs_of_nonneg (sub_nonneg.mpr (by exact_mod_cast Nat.le_of_lt hgt))] have hgap : T₂.sum id + 1 ≤ T₁.sum id := by omega have hgapR : (((T₂.sum id + 1 : ℕ) : ℝ)) ≤ T₁.sum id := by exact_mod_cast hgap norm_num at hgapR ⊢ linarith have := hreal N B hB hN simpa [B] using this @[category textbook, AMS 5 11] theorem erdos_1.variants.least_N_3 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 3 } 4 := by refine ⟨⟨{1, 2, 4}, ?_⟩, ?_⟩ · simp refine ⟨by decide, ?_⟩ let P := Finset.powerset {1, 2, 4} have : Finset.univ.image (fun p : P ↦ ∑ x ∈ p, x) = {0, 1, 2, 4, 3, 5, 6, 7} := by refine Finset.ext_iff.mpr (fun n => ?_) simp [show P = {{}, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}} by decide] omega rw [← Set.injOn_univ, ← Finset.coe_univ] have : (Finset.univ.image (fun p : P ↦ ∑ x ∈ p.1, x)).card = (Finset.univ (α := P)).card := by rw [this]; aesop exact Finset.injOn_of_card_image_eq this · simp [mem_lowerBounds] intro n S h h_inj hcard3 by_contra hn interval_cases n; aesop; aesop · have := Finset.card_le_card h aesop · absurd h_inj rw [(Finset.subset_iff_eq_of_card_le (Nat.le_of_eq (by rw [hcard3]; decide))).mp h] decide @[category textbook, AMS 5 11] theorem erdos_1.variants.least_N_4 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 4 } 7 := by refine ⟨⟨{3, 5, 6, 7}, ⟨by decide, by decide⟩, by decide⟩, ?_⟩ rintro N ⟨A, ⟨hA, hinj⟩, hcard⟩ by_contra! hN have hA6 : A ∈ (Finset.Icc 1 6).powersetCard 4 := Finset.mem_powersetCard.mpr ⟨fun x hx => by have := Finset.mem_Icc.mp (hA hx) exact Finset.mem_Icc.mpr ⟨this.1, by omega⟩, hcard⟩ have key : ∀ B ∈ (Finset.Icc 1 6).powersetCard 4, ¬ (fun (⟨S, _⟩ : B.powerset) => S.sum id).Injective := by decide exact key A hA6 hinj @[category research solved, AMS 5 11] theorem erdos_1.variants.least_N_5 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 5 } 13 := by refine ⟨⟨{6, 9, 11, 12, 13}, ⟨by decide, by decide⟩, by decide⟩, ?_⟩ rintro N ⟨A, ⟨hA, hinj⟩, hcard⟩ by_contra! hN have hA12 : A ∈ (Finset.Icc 1 12).powersetCard 5 := Finset.mem_powersetCard.mpr ⟨fun x hx => by have := Finset.mem_Icc.mp (hA hx) exact Finset.mem_Icc.mpr ⟨this.1, by omega⟩, hcard⟩ have key : ∀ B ∈ (Finset.Icc 1 12).powersetCard 5, ¬ (fun (⟨S, _⟩ : B.powerset) => S.sum id).Injective := by decide +kernel exact key A hA12 hinj @[category research solved, AMS 5 11] theorem erdos_1.variants.least_N_9 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 9 } 161 := by sorry end Erdos1