import Erdos1135.NumberTheory.Rhin.LiteralRemainder /-! # Exact cells for the relaxed Rhin remainder bound This file is generated by `scripts/generate_rhin_remainder_cells.py`. The 38 rational cells form a finite exact certificate for a continuous interval bound; they are not a historical low-frequency phase table. The cover, raw base `239/1000`, and loss cap `20` are derived exact G8 relaxations for Rhin's printed `Q₆` coefficient `-108`; Rhin does not print these certificate parameters, and they do not cover Wu's later `-104` variant. Python emits only the cell data and repetitive proof declarations. Each cell inequality and loss bound is proved by Lean's kernel-checked `norm_num`. No native compiler decision mechanism is used. -/ namespace Erdos1135 namespace NumberTheory namespace Rhin noncomputable section open scoped BigOperators set_option exponentiation.threshold 1000 set_option maxRecDepth 10000 set_option maxHeartbeats 1000000 structure RhinCell where left : ℚ right : ℚ deriving DecidableEq, Repr def rhinQ5Rat (x : ℚ) : ℚ := 17 * x ^ 2 - 102 * x + 144 def rhinQ6Rat (x : ℚ) : ℚ := 19 * x ^ 2 - 108 * x + 144 /-- Exact endpoint/vertex upper bounds for the six literal factors on a cell. -/ def rhinCellMax (c : RhinCell) : Fin 6 → ℚ := ![max |c.left - 3| |c.right - 3|, max |c.left - 2| |c.right - 2|, max |c.left - 4| |c.right - 4|, max |5 * c.left - 12| |5 * c.right - 12|, max (max |rhinQ5Rat c.left| |rhinQ5Rat c.right|) (if c.left ≤ 3 ∧ 3 ≤ c.right then 9 else 0), max (max |rhinQ6Rat c.left| |rhinQ6Rat c.right|) (if c.left ≤ 54 / 19 ∧ 54 / 19 ≤ c.right then 180 / 19 else 0)] def rhinCellExponent (c : RhinCell) (i : Fin 6) : ℕ := if 1 ≤ rhinCellMax c i then literalRhinUpperExponent i else literalRhinLowerExponent i def rhinCellCost (c : RhinCell) : ℚ := ∏ i : Fin 6, if rhinCellMax c i < 1 then (rhinCellMax c i)⁻¹ else 1 def rhinCellValid (c : RhinCell) : Prop := c.left < c.right ∧ 2 ≤ c.left ∧ c.right ≤ 4 ∧ (∀ i : Fin 6, 0 < rhinCellMax c i) ∧ (∏ i : Fin 6, rhinCellMax c i ^ rhinCellExponent c i) < (((239 : ℚ) / 1000) * c.left) ^ 1000 theorem rhinCellMax_pos_of_valid {c : RhinCell} (hc : rhinCellValid c) (i : Fin 6) : 0 < rhinCellMax c i := by exact hc.2.2.2.1 i def rhinCell23_00 : RhinCell := ⟨2, 33/16⟩ def rhinCell23_01 : RhinCell := ⟨33/16, 17/8⟩ def rhinCell23_02 : RhinCell := ⟨17/8, 69/32⟩ def rhinCell23_03 : RhinCell := ⟨69/32, 35/16⟩ def rhinCell23_04 : RhinCell := ⟨35/16, 141/64⟩ def rhinCell23_05 : RhinCell := ⟨141/64, 71/32⟩ def rhinCell23_06 : RhinCell := ⟨71/32, 143/64⟩ def rhinCell23_07 : RhinCell := ⟨143/64, 9/4⟩ def rhinCell23_08 : RhinCell := ⟨9/4, 73/32⟩ def rhinCell23_09 : RhinCell := ⟨73/32, 37/16⟩ def rhinCell23_10 : RhinCell := ⟨37/16, 75/32⟩ def rhinCell23_11 : RhinCell := ⟨75/32, 19/8⟩ def rhinCell23_12 : RhinCell := ⟨19/8, 39/16⟩ def rhinCell23_13 : RhinCell := ⟨39/16, 79/32⟩ def rhinCell23_14 : RhinCell := ⟨79/32, 5/2⟩ def rhinCell23_15 : RhinCell := ⟨5/2, 81/32⟩ def rhinCell23_16 : RhinCell := ⟨81/32, 41/16⟩ def rhinCell23_17 : RhinCell := ⟨41/16, 83/32⟩ def rhinCell23_18 : RhinCell := ⟨83/32, 21/8⟩ def rhinCell23_19 : RhinCell := ⟨21/8, 43/16⟩ def rhinCell23_20 : RhinCell := ⟨43/16, 11/4⟩ def rhinCell23_21 : RhinCell := ⟨11/4, 3⟩ def rhinCells23 : List RhinCell := [rhinCell23_00, rhinCell23_01, rhinCell23_02, rhinCell23_03, rhinCell23_04, rhinCell23_05, rhinCell23_06, rhinCell23_07, rhinCell23_08, rhinCell23_09, rhinCell23_10, rhinCell23_11, rhinCell23_12, rhinCell23_13, rhinCell23_14, rhinCell23_15, rhinCell23_16, rhinCell23_17, rhinCell23_18, rhinCell23_19, rhinCell23_20, rhinCell23_21] private theorem rhinCell23_00_valid : rhinCellValid rhinCell23_00 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_00] · norm_num [rhinCell23_00] · norm_num [rhinCell23_00] · intro i fin_cases i <;> norm_num [rhinCell23_00, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_00, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_00_cost_le : rhinCellCost rhinCell23_00 ≤ 20 := by norm_num [rhinCell23_00, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_01_valid : rhinCellValid rhinCell23_01 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_01] · norm_num [rhinCell23_01] · norm_num [rhinCell23_01] · intro i fin_cases i <;> norm_num [rhinCell23_01, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_01, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_01_cost_le : rhinCellCost rhinCell23_01 ≤ 20 := by norm_num [rhinCell23_01, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_02_valid : rhinCellValid rhinCell23_02 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_02] · norm_num [rhinCell23_02] · norm_num [rhinCell23_02] · intro i fin_cases i <;> norm_num [rhinCell23_02, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_02, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_02_cost_le : rhinCellCost rhinCell23_02 ≤ 20 := by norm_num [rhinCell23_02, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_03_valid : rhinCellValid rhinCell23_03 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_03] · norm_num [rhinCell23_03] · norm_num [rhinCell23_03] · intro i fin_cases i <;> norm_num [rhinCell23_03, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_03, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_03_cost_le : rhinCellCost rhinCell23_03 ≤ 20 := by norm_num [rhinCell23_03, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_04_valid : rhinCellValid rhinCell23_04 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_04] · norm_num [rhinCell23_04] · norm_num [rhinCell23_04] · intro i fin_cases i <;> norm_num [rhinCell23_04, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_04, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_04_cost_le : rhinCellCost rhinCell23_04 ≤ 20 := by norm_num [rhinCell23_04, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_05_valid : rhinCellValid rhinCell23_05 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_05] · norm_num [rhinCell23_05] · norm_num [rhinCell23_05] · intro i fin_cases i <;> norm_num [rhinCell23_05, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_05, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_05_cost_le : rhinCellCost rhinCell23_05 ≤ 20 := by norm_num [rhinCell23_05, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_06_valid : rhinCellValid rhinCell23_06 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_06] · norm_num [rhinCell23_06] · norm_num [rhinCell23_06] · intro i fin_cases i <;> norm_num [rhinCell23_06, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_06, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_06_cost_le : rhinCellCost rhinCell23_06 ≤ 20 := by norm_num [rhinCell23_06, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_07_valid : rhinCellValid rhinCell23_07 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_07] · norm_num [rhinCell23_07] · norm_num [rhinCell23_07] · intro i fin_cases i <;> norm_num [rhinCell23_07, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_07, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_07_cost_le : rhinCellCost rhinCell23_07 ≤ 20 := by norm_num [rhinCell23_07, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_08_valid : rhinCellValid rhinCell23_08 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_08] · norm_num [rhinCell23_08] · norm_num [rhinCell23_08] · intro i fin_cases i <;> norm_num [rhinCell23_08, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_08, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_08_cost_le : rhinCellCost rhinCell23_08 ≤ 20 := by norm_num [rhinCell23_08, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_09_valid : rhinCellValid rhinCell23_09 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_09] · norm_num [rhinCell23_09] · norm_num [rhinCell23_09] · intro i fin_cases i <;> norm_num [rhinCell23_09, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_09, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_09_cost_le : rhinCellCost rhinCell23_09 ≤ 20 := by norm_num [rhinCell23_09, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_10_valid : rhinCellValid rhinCell23_10 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_10] · norm_num [rhinCell23_10] · norm_num [rhinCell23_10] · intro i fin_cases i <;> norm_num [rhinCell23_10, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_10, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_10_cost_le : rhinCellCost rhinCell23_10 ≤ 20 := by norm_num [rhinCell23_10, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_11_valid : rhinCellValid rhinCell23_11 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_11] · norm_num [rhinCell23_11] · norm_num [rhinCell23_11] · intro i fin_cases i <;> norm_num [rhinCell23_11, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_11, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_11_cost_le : rhinCellCost rhinCell23_11 ≤ 20 := by norm_num [rhinCell23_11, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_12_valid : rhinCellValid rhinCell23_12 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_12] · norm_num [rhinCell23_12] · norm_num [rhinCell23_12] · intro i fin_cases i <;> norm_num [rhinCell23_12, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_12, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_12_cost_le : rhinCellCost rhinCell23_12 ≤ 20 := by norm_num [rhinCell23_12, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_13_valid : rhinCellValid rhinCell23_13 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_13] · norm_num [rhinCell23_13] · norm_num [rhinCell23_13] · intro i fin_cases i <;> norm_num [rhinCell23_13, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_13, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_13_cost_le : rhinCellCost rhinCell23_13 ≤ 20 := by norm_num [rhinCell23_13, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_14_valid : rhinCellValid rhinCell23_14 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_14] · norm_num [rhinCell23_14] · norm_num [rhinCell23_14] · intro i fin_cases i <;> norm_num [rhinCell23_14, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_14, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_14_cost_le : rhinCellCost rhinCell23_14 ≤ 20 := by norm_num [rhinCell23_14, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_15_valid : rhinCellValid rhinCell23_15 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_15] · norm_num [rhinCell23_15] · norm_num [rhinCell23_15] · intro i fin_cases i <;> norm_num [rhinCell23_15, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_15, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_15_cost_le : rhinCellCost rhinCell23_15 ≤ 20 := by norm_num [rhinCell23_15, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_16_valid : rhinCellValid rhinCell23_16 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_16] · norm_num [rhinCell23_16] · norm_num [rhinCell23_16] · intro i fin_cases i <;> norm_num [rhinCell23_16, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_16, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_16_cost_le : rhinCellCost rhinCell23_16 ≤ 20 := by norm_num [rhinCell23_16, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_17_valid : rhinCellValid rhinCell23_17 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_17] · norm_num [rhinCell23_17] · norm_num [rhinCell23_17] · intro i fin_cases i <;> norm_num [rhinCell23_17, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_17, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_17_cost_le : rhinCellCost rhinCell23_17 ≤ 20 := by norm_num [rhinCell23_17, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_18_valid : rhinCellValid rhinCell23_18 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_18] · norm_num [rhinCell23_18] · norm_num [rhinCell23_18] · intro i fin_cases i <;> norm_num [rhinCell23_18, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_18, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_18_cost_le : rhinCellCost rhinCell23_18 ≤ 20 := by norm_num [rhinCell23_18, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_19_valid : rhinCellValid rhinCell23_19 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_19] · norm_num [rhinCell23_19] · norm_num [rhinCell23_19] · intro i fin_cases i <;> norm_num [rhinCell23_19, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_19, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_19_cost_le : rhinCellCost rhinCell23_19 ≤ 20 := by norm_num [rhinCell23_19, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_20_valid : rhinCellValid rhinCell23_20 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_20] · norm_num [rhinCell23_20] · norm_num [rhinCell23_20] · intro i fin_cases i <;> norm_num [rhinCell23_20, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_20, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_20_cost_le : rhinCellCost rhinCell23_20 ≤ 20 := by norm_num [rhinCell23_20, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell23_21_valid : rhinCellValid rhinCell23_21 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell23_21] · norm_num [rhinCell23_21] · norm_num [rhinCell23_21] · intro i fin_cases i <;> norm_num [rhinCell23_21, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell23_21, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell23_21_cost_le : rhinCellCost rhinCell23_21 ≤ 20 := by norm_num [rhinCell23_21, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCells23_forall : List.Forall rhinCellValid rhinCells23 := by simpa only [rhinCells23, List.Forall] using And.intro rhinCell23_00_valid (And.intro rhinCell23_01_valid (And.intro rhinCell23_02_valid (And.intro rhinCell23_03_valid (And.intro rhinCell23_04_valid (And.intro rhinCell23_05_valid (And.intro rhinCell23_06_valid (And.intro rhinCell23_07_valid (And.intro rhinCell23_08_valid (And.intro rhinCell23_09_valid (And.intro rhinCell23_10_valid (And.intro rhinCell23_11_valid (And.intro rhinCell23_12_valid (And.intro rhinCell23_13_valid (And.intro rhinCell23_14_valid (And.intro rhinCell23_15_valid (And.intro rhinCell23_16_valid (And.intro rhinCell23_17_valid (And.intro rhinCell23_18_valid (And.intro rhinCell23_19_valid (And.intro rhinCell23_20_valid (rhinCell23_21_valid))))))))))))))))))))) private theorem rhinCells23_cost_forall : List.Forall (fun c ↦ rhinCellCost c ≤ 20) rhinCells23 := by simpa only [rhinCells23, List.Forall] using And.intro rhinCell23_00_cost_le (And.intro rhinCell23_01_cost_le (And.intro rhinCell23_02_cost_le (And.intro rhinCell23_03_cost_le (And.intro rhinCell23_04_cost_le (And.intro rhinCell23_05_cost_le (And.intro rhinCell23_06_cost_le (And.intro rhinCell23_07_cost_le (And.intro rhinCell23_08_cost_le (And.intro rhinCell23_09_cost_le (And.intro rhinCell23_10_cost_le (And.intro rhinCell23_11_cost_le (And.intro rhinCell23_12_cost_le (And.intro rhinCell23_13_cost_le (And.intro rhinCell23_14_cost_le (And.intro rhinCell23_15_cost_le (And.intro rhinCell23_16_cost_le (And.intro rhinCell23_17_cost_le (And.intro rhinCell23_18_cost_le (And.intro rhinCell23_19_cost_le (And.intro rhinCell23_20_cost_le (rhinCell23_21_cost_le))))))))))))))))))))) theorem rhinCells23_valid : ∀ c ∈ rhinCells23, rhinCellValid c := List.forall_iff_forall_mem.mp rhinCells23_forall theorem rhinCells23_cost_le : ∀ c ∈ rhinCells23, rhinCellCost c ≤ 20 := List.forall_iff_forall_mem.mp rhinCells23_cost_forall def rhinCell34_00 : RhinCell := ⟨3, 13/4⟩ def rhinCell34_01 : RhinCell := ⟨13/4, 27/8⟩ def rhinCell34_02 : RhinCell := ⟨27/8, 55/16⟩ def rhinCell34_03 : RhinCell := ⟨55/16, 7/2⟩ def rhinCell34_04 : RhinCell := ⟨7/2, 57/16⟩ def rhinCell34_05 : RhinCell := ⟨57/16, 115/32⟩ def rhinCell34_06 : RhinCell := ⟨115/32, 29/8⟩ def rhinCell34_07 : RhinCell := ⟨29/8, 117/32⟩ def rhinCell34_08 : RhinCell := ⟨117/32, 59/16⟩ def rhinCell34_09 : RhinCell := ⟨59/16, 119/32⟩ def rhinCell34_10 : RhinCell := ⟨119/32, 15/4⟩ def rhinCell34_11 : RhinCell := ⟨15/4, 121/32⟩ def rhinCell34_12 : RhinCell := ⟨121/32, 61/16⟩ def rhinCell34_13 : RhinCell := ⟨61/16, 123/32⟩ def rhinCell34_14 : RhinCell := ⟨123/32, 31/8⟩ def rhinCell34_15 : RhinCell := ⟨31/8, 4⟩ def rhinCells34 : List RhinCell := [rhinCell34_00, rhinCell34_01, rhinCell34_02, rhinCell34_03, rhinCell34_04, rhinCell34_05, rhinCell34_06, rhinCell34_07, rhinCell34_08, rhinCell34_09, rhinCell34_10, rhinCell34_11, rhinCell34_12, rhinCell34_13, rhinCell34_14, rhinCell34_15] private theorem rhinCell34_00_valid : rhinCellValid rhinCell34_00 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_00] · norm_num [rhinCell34_00] · norm_num [rhinCell34_00] · intro i fin_cases i <;> norm_num [rhinCell34_00, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_00, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_00_cost_le : rhinCellCost rhinCell34_00 ≤ 20 := by norm_num [rhinCell34_00, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_01_valid : rhinCellValid rhinCell34_01 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_01] · norm_num [rhinCell34_01] · norm_num [rhinCell34_01] · intro i fin_cases i <;> norm_num [rhinCell34_01, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_01, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_01_cost_le : rhinCellCost rhinCell34_01 ≤ 20 := by norm_num [rhinCell34_01, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_02_valid : rhinCellValid rhinCell34_02 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_02] · norm_num [rhinCell34_02] · norm_num [rhinCell34_02] · intro i fin_cases i <;> norm_num [rhinCell34_02, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_02, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_02_cost_le : rhinCellCost rhinCell34_02 ≤ 20 := by norm_num [rhinCell34_02, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_03_valid : rhinCellValid rhinCell34_03 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_03] · norm_num [rhinCell34_03] · norm_num [rhinCell34_03] · intro i fin_cases i <;> norm_num [rhinCell34_03, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_03, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_03_cost_le : rhinCellCost rhinCell34_03 ≤ 20 := by norm_num [rhinCell34_03, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_04_valid : rhinCellValid rhinCell34_04 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_04] · norm_num [rhinCell34_04] · norm_num [rhinCell34_04] · intro i fin_cases i <;> norm_num [rhinCell34_04, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_04, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_04_cost_le : rhinCellCost rhinCell34_04 ≤ 20 := by norm_num [rhinCell34_04, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_05_valid : rhinCellValid rhinCell34_05 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_05] · norm_num [rhinCell34_05] · norm_num [rhinCell34_05] · intro i fin_cases i <;> norm_num [rhinCell34_05, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_05, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_05_cost_le : rhinCellCost rhinCell34_05 ≤ 20 := by norm_num [rhinCell34_05, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_06_valid : rhinCellValid rhinCell34_06 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_06] · norm_num [rhinCell34_06] · norm_num [rhinCell34_06] · intro i fin_cases i <;> norm_num [rhinCell34_06, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_06, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_06_cost_le : rhinCellCost rhinCell34_06 ≤ 20 := by norm_num [rhinCell34_06, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_07_valid : rhinCellValid rhinCell34_07 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_07] · norm_num [rhinCell34_07] · norm_num [rhinCell34_07] · intro i fin_cases i <;> norm_num [rhinCell34_07, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_07, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_07_cost_le : rhinCellCost rhinCell34_07 ≤ 20 := by norm_num [rhinCell34_07, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_08_valid : rhinCellValid rhinCell34_08 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_08] · norm_num [rhinCell34_08] · norm_num [rhinCell34_08] · intro i fin_cases i <;> norm_num [rhinCell34_08, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_08, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_08_cost_le : rhinCellCost rhinCell34_08 ≤ 20 := by norm_num [rhinCell34_08, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_09_valid : rhinCellValid rhinCell34_09 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_09] · norm_num [rhinCell34_09] · norm_num [rhinCell34_09] · intro i fin_cases i <;> norm_num [rhinCell34_09, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_09, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_09_cost_le : rhinCellCost rhinCell34_09 ≤ 20 := by norm_num [rhinCell34_09, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_10_valid : rhinCellValid rhinCell34_10 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_10] · norm_num [rhinCell34_10] · norm_num [rhinCell34_10] · intro i fin_cases i <;> norm_num [rhinCell34_10, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_10, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_10_cost_le : rhinCellCost rhinCell34_10 ≤ 20 := by norm_num [rhinCell34_10, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_11_valid : rhinCellValid rhinCell34_11 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_11] · norm_num [rhinCell34_11] · norm_num [rhinCell34_11] · intro i fin_cases i <;> norm_num [rhinCell34_11, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_11, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_11_cost_le : rhinCellCost rhinCell34_11 ≤ 20 := by norm_num [rhinCell34_11, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_12_valid : rhinCellValid rhinCell34_12 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_12] · norm_num [rhinCell34_12] · norm_num [rhinCell34_12] · intro i fin_cases i <;> norm_num [rhinCell34_12, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_12, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_12_cost_le : rhinCellCost rhinCell34_12 ≤ 20 := by norm_num [rhinCell34_12, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_13_valid : rhinCellValid rhinCell34_13 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_13] · norm_num [rhinCell34_13] · norm_num [rhinCell34_13] · intro i fin_cases i <;> norm_num [rhinCell34_13, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_13, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_13_cost_le : rhinCellCost rhinCell34_13 ≤ 20 := by norm_num [rhinCell34_13, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_14_valid : rhinCellValid rhinCell34_14 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_14] · norm_num [rhinCell34_14] · norm_num [rhinCell34_14] · intro i fin_cases i <;> norm_num [rhinCell34_14, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_14, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_14_cost_le : rhinCellCost rhinCell34_14 ≤ 20 := by norm_num [rhinCell34_14, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCell34_15_valid : rhinCellValid rhinCell34_15 := by refine ⟨?_, ?_, ?_, ?_, ?_⟩ · norm_num [rhinCell34_15] · norm_num [rhinCell34_15] · norm_num [rhinCell34_15] · intro i fin_cases i <;> norm_num [rhinCell34_15, rhinCellMax, rhinQ5Rat, rhinQ6Rat] · norm_num [rhinCell34_15, rhinCellMax, rhinCellExponent, rhinQ5Rat, rhinQ6Rat, literalRhinUpperExponent, literalRhinLowerExponent, Fin.prod_univ_succ] private theorem rhinCell34_15_cost_le : rhinCellCost rhinCell34_15 ≤ 20 := by norm_num [rhinCell34_15, rhinCellCost, rhinCellMax, rhinQ5Rat, rhinQ6Rat, Fin.prod_univ_succ, Matrix.vecCons, Matrix.vecEmpty] private theorem rhinCells34_forall : List.Forall rhinCellValid rhinCells34 := by simpa only [rhinCells34, List.Forall] using And.intro rhinCell34_00_valid (And.intro rhinCell34_01_valid (And.intro rhinCell34_02_valid (And.intro rhinCell34_03_valid (And.intro rhinCell34_04_valid (And.intro rhinCell34_05_valid (And.intro rhinCell34_06_valid (And.intro rhinCell34_07_valid (And.intro rhinCell34_08_valid (And.intro rhinCell34_09_valid (And.intro rhinCell34_10_valid (And.intro rhinCell34_11_valid (And.intro rhinCell34_12_valid (And.intro rhinCell34_13_valid (And.intro rhinCell34_14_valid (rhinCell34_15_valid))))))))))))))) private theorem rhinCells34_cost_forall : List.Forall (fun c ↦ rhinCellCost c ≤ 20) rhinCells34 := by simpa only [rhinCells34, List.Forall] using And.intro rhinCell34_00_cost_le (And.intro rhinCell34_01_cost_le (And.intro rhinCell34_02_cost_le (And.intro rhinCell34_03_cost_le (And.intro rhinCell34_04_cost_le (And.intro rhinCell34_05_cost_le (And.intro rhinCell34_06_cost_le (And.intro rhinCell34_07_cost_le (And.intro rhinCell34_08_cost_le (And.intro rhinCell34_09_cost_le (And.intro rhinCell34_10_cost_le (And.intro rhinCell34_11_cost_le (And.intro rhinCell34_12_cost_le (And.intro rhinCell34_13_cost_le (And.intro rhinCell34_14_cost_le (rhinCell34_15_cost_le))))))))))))))) theorem rhinCells34_valid : ∀ c ∈ rhinCells34, rhinCellValid c := List.forall_iff_forall_mem.mp rhinCells34_forall theorem rhinCells34_cost_le : ∀ c ∈ rhinCells34, rhinCellCost c ≤ 20 := List.forall_iff_forall_mem.mp rhinCells34_cost_forall /-- A compact computable condition that a nonempty cell list is contiguous with the stated outer endpoints. -/ def rhinCellChain (a b : ℚ) : List RhinCell → Prop | [] => False | c :: cs => c.left = a ∧ c.left < c.right ∧ match cs with | [] => c.right = b | _ => rhinCellChain c.right b cs private theorem rhinCells23_chain : rhinCellChain 2 3 rhinCells23 := by norm_num [rhinCellChain, rhinCells23, rhinCell23_00, rhinCell23_01, rhinCell23_02, rhinCell23_03, rhinCell23_04, rhinCell23_05, rhinCell23_06, rhinCell23_07, rhinCell23_08, rhinCell23_09, rhinCell23_10, rhinCell23_11, rhinCell23_12, rhinCell23_13, rhinCell23_14, rhinCell23_15, rhinCell23_16, rhinCell23_17, rhinCell23_18, rhinCell23_19, rhinCell23_20, rhinCell23_21] private theorem rhinCells34_chain : rhinCellChain 3 4 rhinCells34 := by norm_num [rhinCellChain, rhinCells34, rhinCell34_00, rhinCell34_01, rhinCell34_02, rhinCell34_03, rhinCell34_04, rhinCell34_05, rhinCell34_06, rhinCell34_07, rhinCell34_08, rhinCell34_09, rhinCell34_10, rhinCell34_11, rhinCell34_12, rhinCell34_13, rhinCell34_14, rhinCell34_15] theorem exists_cell_of_chain {a b : ℚ} {cells : List RhinCell} (hchain : rhinCellChain a b cells) {x : ℝ} (hax : (a : ℝ) ≤ x) (hxb : x ≤ (b : ℝ)) : ∃ c ∈ cells, (c.left : ℝ) ≤ x ∧ x ≤ (c.right : ℝ) := by induction cells generalizing a with | nil => simp [rhinCellChain] at hchain | cons c cs ih => rcases hchain with ⟨hcLeft, hcLt, hrest⟩ cases cs with | nil => refine ⟨c, by simp, ?_, ?_⟩ · simpa only [hcLeft] using hax · simpa only [hrest] using hxb | cons d ds => by_cases hxc : x ≤ (c.right : ℝ) · refine ⟨c, by simp, ?_, hxc⟩ simpa only [hcLeft] using hax · obtain ⟨cell, hmem, hcell⟩ := ih hrest (le_of_not_ge hxc) exact ⟨cell, List.mem_cons_of_mem c hmem, hcell⟩ theorem exists_rhinCell23 {x : ℝ} (hx : x ∈ Set.Icc (2 : ℝ) 3) : ∃ c ∈ rhinCells23, (c.left : ℝ) ≤ x ∧ x ≤ (c.right : ℝ) := exists_cell_of_chain rhinCells23_chain hx.1 hx.2 theorem exists_rhinCell34 {x : ℝ} (hx : x ∈ Set.Icc (3 : ℝ) 4) : ∃ c ∈ rhinCells34, (c.left : ℝ) ≤ x ∧ x ≤ (c.right : ℝ) := exists_cell_of_chain rhinCells34_chain hx.1 hx.2 /-- The derived relaxed raw base gives the integer G8 contraction witness `13` after the later base-three LCM factor is included. This exact certificate is distinct from Rhin's printed exponent `13.3`. -/ theorem rhinRelaxedRateCertificate : (72 : ℚ) * ((3 : ℚ) * (239 / 1000)) ^ 13 < 1 ∧ (13 : ℚ) ≤ 133 / 10 := by norm_num end end Rhin end NumberTheory end Erdos1135