theorem proved term proof

`pid_stability_one_statement`

No prose has been written for this declaration yet. The Lean source and graph data below render without it.

formal statement (Lean)

 172theorem pid_stability_one_statement :
 173    ∀ (f : ℝ → ℝ), RecognitionDescent f →
 174      f 1 = 1 ∧
 175      (∀ (r : ℝ), 0 < r → r ≠ 1 → Cost.Jcost (f r) < Cost.Jcost r) ∧
 176      (∀ (x : ℝ), 0 < x → (Cost.Jcost x = 0 ↔ x = 1)) := by

proof body

Term-mode proof.

 177  intro f hf
 178  refine ⟨hf.fixes_equilibrium, ?_, ?_⟩
 179  · intro r hr hne
 180    exact hf.strict_descent_off r hr hne
 181  · intro x hx
 182    exact cost_zero_iff_equilibrium hx
 183
 184end
 185
 186end PIDStabilityFromJCost
 187end Robotics
 188end IndisputableMonolith

depends on (3)

Lean names referenced from this declaration's body.

Cost in IndisputableMonolith.Measurement.RSNative.Core decl_use
cost_zero_iff_equilibrium in IndisputableMonolith.Robotics.PIDStabilityFromJCost decl_use
RecognitionDescent in IndisputableMonolith.Robotics.PIDStabilityFromJCost decl_use