30 lines
1.0 KiB
Lean4
30 lines
1.0 KiB
Lean4
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import Sunbeam.Model.Basic
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namespace Sunbeam
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/-- The sigmoid function σ(x) = 1 / (1 + exp(-x)). -/
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def sigmoid (x : Float) : Float :=
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1.0 / (1.0 + Float.exp (-x))
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/-! ## Trust boundary: sigmoid axioms
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These axioms capture IEEE-754 properties of sigmoid that hold for all finite
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float inputs. They cannot be proved inside Lean because `Float` operations are
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`@[extern]` (opaque to the kernel). The axioms form a documented trust boundary:
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we trust the C runtime's `exp` implementation.
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When TorchLean ships its verified Float32 kernel, these axioms can be replaced
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with proofs against that kernel.
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-/
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/-- Sigmoid output is always positive: exp(-x) ≥ 0 ⟹ 1+exp(-x) ≥ 1 ⟹ 1/(1+exp(-x)) > 0. -/
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axiom sigmoid_pos (x : Float) : sigmoid x > 0
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/-- Sigmoid output is always less than 1: 1+exp(-x) > 1 ⟹ 1/(1+exp(-x)) < 1. -/
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axiom sigmoid_lt_one (x : Float) : sigmoid x < 1
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/-- Sigmoid is monotonically increasing (derivative = σ(1-σ) > 0). -/
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axiom sigmoid_monotone {x y : Float} (h : x ≤ y) : sigmoid x ≤ sigmoid y
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end Sunbeam
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