openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /Conditional.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.OrderedLimits | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| namespace MatrixMultiplication.AllFieldConditional | |
| open AllFieldNativeCapacity | |
| structure NumericalConditions : Prop where | |
| denominator_gap : nativeB < nativeHigh | |
| stageA_positive : ∀ i : Fin 3, 0 < nativeLA i | |
| stageB_positive : ∀ i : Fin 3, 0 < nativeLB i | |
| stageC_positive : ∀ i : Fin 3, | |
| 0 < nativeCstar - (nativeLA i + nativeLB i) | |
| stageC_feasible : ∀ i : Fin 3, | |
| nativeCstar - (nativeLA i + nativeLB i) < nativeHigh | |
| volume_positive : 0 < AllFieldTerminalRates.S | |
| strict_rate_bound : | |
| 3 * (8 * Real.log 7 - | |
| (AllFieldInitialEntropy.nativeH0 + nativeCstar)) / AllFieldTerminalRates.S < | |
| (2371054887 : ℝ) / 10 ^ 9 | |
| theorem omega_lt_2371054887_div_1000000000_of_numerical_conditions | |
| (F : Type*) [Field F] (h : NumericalConditions) : | |
| Arithmetic.omega F < (2371054887 : ℝ) / 10 ^ 9 := by | |
| have hc (i : Fin 3) : 0 < nativeLC nativeLambda i := by | |
| rw [nativeLC_balanced h.denominator_gap i] | |
| exact h.stageC_positive i | |
| have hbound := AllFieldOrderedLimits.omega_le_native_ratio F | |
| h.denominator_gap h.stageC_feasible | |
| h.stageA_positive h.stageB_positive hc h.volume_positive | |
| exact lt_of_le_of_lt hbound h.strict_rate_bound | |
| end MatrixMultiplication.AllFieldConditional | |
| end OAI | |